The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education (International Studies in the History of Mathematics and its Teaching) 303104312X, 9783031043123

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International Studies in the History of Mathematics and its Teaching Series Editors: Alexander Karp · Gert Schubring

Fulvia Furinghetti Livia Giacardi   Editors

The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education

International Studies in the History of Mathematics and its Teaching Series Editors Alexander Karp, Teachers College, Columbia University, New York, NY, USA Gert Schubring, Universität Bielefeld, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil, Bielefeld, Germany

The International Studies in the History of Mathematics and its Teaching Series creates a platform for international collaboration in the exploration of the social history of mathematics education and its connections with the development of mathematics. The series offers broad perspectives on mathematics research and education, including contributions relating to the history of mathematics and mathematics education at all levels of study, school education, college education, mathematics teacher education, the development of research mathematics, the role of mathematicians in mathematics education, mathematics teachers' associations and periodicals. The series seeks to inform mathematics educators, mathematicians, and historians about the political, social, and cultural constraints and achievements that influenced the development of mathematics and mathematics education. In so doing, it aims to overcome disconnected national cultural and social histories and establish common cross-cultural themes within the development of mathematics and mathematics instruction. However, at the core of these various perspectives, the question of how to best improve mathematics teaching and learning always remains the focal issue informing the series.

Fulvia Furinghetti  •  Livia Giacardi Editors

The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education

Editors Fulvia Furinghetti Department of Mathematics University of Genoa Genoa, Italy

Livia Giacardi Department of Mathematics University of Turin Turin, Italy

ISSN 2524-8022     ISSN 2524-8030 (electronic) International Studies in the History of Mathematics and its Teaching ISBN 978-3-031-04312-3    ISBN 978-3-031-04313-0 (eBook) https://doi.org/10.1007/978-3-031-04313-0 © Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover caption: Cover image is based on the cuneiform tablet, known as Plimpton 322 (collection of Columbia University, New York). This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To our beloved mothers Teresa and Annetta

Preface

After all, [I believed that] the entire field of mathematical learning, from the modest beginnings in elementary school to the highest level of specialised scientific research, had to be recognised and organized as an organic whole. It became clearer and clearer to me that, if these further perspectives were to be neglected, even pure scientific research itself would have to suffer. [and] that, by closing off the varied, ongoing lively cultural developments, it would condemn itself to wilting, like a plant deprived of sunlight in a cellar. (Klein 1923)1 We sometimes wonder if the time we devote to questions concerning teaching would have been better used in scientific research. Well, our answer is that it is a social duty that forces us to deal with these problems ... Should we not facilitate human beings in acquiring knowledge, which is a source of both power and happiness? (Castelnuovo 1914)2 In no other living science is the part of mise en forme, transposition didactique, so important at a research level. In no other science, however, is the distance between the taught and the new so large. In no other science has teaching and learning such social importance. In no other science is there such an old tradition of scientists committed to educational questions. (Kahane 1990)3

These quotes by two presidents (Klein and Kahane) and a vice-president (Castelnuovo) of the Commission, which, born as the Commission Internationale

 Klein, Felix. 1923. Göttinger Professoren. Lebensbilder von eigener Hand. 4. Felix Klein. Mitteilungen des Universitätsbundes Göttingen 5, p. 24. The original text is: “Schließlich war das gesamte Gebiet mathematischen Lernens von den bescheidenen Anfängen in der Volksschule bis zur höchsten wissenschaftlichen Spezialforschung als ein organisches Ganzes zu erfassen und auszugestalten. Es wurde mir immer deutlicher, dass durch Vernachlässigung dieser weiteren Ausblicke auch die rein wissenschaftliche Forschung selbst leiden müsse, dass sie sich durch Abschluss von der vielseitigen, lebendig pulsierenden allgemein geistigen Entwicklung wie ein der Sonne entzogenen Kellerpflanze zur Verkümmerung verurteile.” 2  Castelnuovo, Guido. 1914. Discours de M. G. Castelnuovo. L’Enseignement Mathématique 16, p. 191. The original text is: “nous nous demandons parfois si le temps que nous consacrons aux questions d’enseignement n’aurait pas été mieux employé dans la recherche scientifique. Eh bien, nous répondons que c’est un devoir social qui nous force à traiter ces problèmes [...] Ne devons-­ nous pas faciliter à nos semblables l’acquisition du savoir, qui est à la fois une puissance et un bonheur?”. 3  Kahane, Jean-Pierre. 1990. A farewell message from the retiring president of ICMI. ICMI Bulletin 29, p. 6. 1

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de l’Enseignement Mathématique/ Internationale Mathematische Unterrichtskommission (CIEM/IMUK), became the present International Commission on Mathematical Instruction (ICMI), epitomize two aspects that we consider to be central in the life of ICMI: the importance of dealing with problems concerning the teaching of mathematics for scientific research too, and the social role of mathematics education. Mathematics is a universal discipline and therefore it is international par excellence. Mathematics and mathematics education, though naturally linked, are different as for areas of knowledge, methods, domains of practice, and academic role. ICMI is an excellent lens to grasp the evolution of mathematics education, changing relationships within the community of mathematicians, and the process of achieving international cooperation in the field of mathematical teaching and learning. At present, the ICMI’s objectives, as Hyman Bass and Bernard Hodgson write, could be globally described as offering researchers, practitioners, curriculum designers, decision makers, and others interested in mathematical education a forum for promoting reflection, collaboration, exchange and dissemination of ideas, and information on all aspects of the theory and practice of contemporary mathematical education, as seen from an international perspective4

These objectives developed and matured during the twentieth century – beginning with the initial aim of the Commission to carry out a comparative study of the syllabi and teaching methods of secondary schools in the various countries5 – thanks to the work of mathematicians, educators, and teachers and the stimuli of the international events of the time. The purpose of this volume is to understand and outline the evolution of the objectives and field of action of ICMI, from its creation in 1908  in Rome until 2008, through the description of the main events and protagonists of ICMI history. In the century of the Commission’s existence, it is possible to identify various phases that were the result of both external events, such as WWI and WWII, which influenced its activities, as well as changes in centers of interest and broadening of the field of action of the Commission itself. The period from the foundation up to the WWI was characterized by the creation of an important international network of national subcommissions to prepare reports both on the state of mathematics teaching in the various countries, and on specific topics. It was so strongly influenced by the first president Klein that it was called Klein’s Era. The effects of WWI undermined scientific internationalism and the shocking decision was made to ban researchers in the former Central Powers from most international activities. This led to the decision to dissolve the CIEM/IMUK in 1920–1921. International scientific cooperation, and therefore also the Commission,

 Bass, Hyman and Hodgson, Bernard. 2004. The International Commission on Mathematical Instruction What? Why? For Whom?. ICMI Bulletin 55, pp. 26-27. 5  Castelnuovo, Guido (ed.). 1909. Atti del IV Congresso Internazionale dei Matematici. Roma: Tipografia della R. Accademia dei Lincei, Vol. I, p. 51. 4

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was only restored in 1928 on the occasion of the ICM in Bologna. The reborn Commission was not able to produce new ideas and limited itself to concluding the projects started in the past. Its activities had a second forced arrest due to the outbreak of WWII. After the end of the war, in 1952, the Commission (now ICMI) was reconstituted as a permanent subcommission of the newly restored International Mathematical Union (IMU). In the following years, with some difficulties, ICMI defined its structure and relations with IMU and established both scientific and organizational collaborations with other associations thus leading to greater internationalization and to new approaches to mathematics education. At the end of the 1960s, the action of ICMI received a remarkable impetus from the initiatives carried out by its president Hans Freudenthal, who, with his organizational talent and above all his strong spirit of independence, founded a new international journal dedicated to mathematics education and organized the first International Congress on Mathematical Education (ICME). These important initiatives mark a turning point in the history of ICMI, projecting it towards the future. Recent decades have seen an important change in the relations between mathematicians and educators which led in 2006 to the evolution in the governance of ICMI for which the election of the ICMI Executive Committee was to be carried out directly by the General Assembly of ICMI itself and not by that of  IMU as in the past. The present volume consists of three parts, each exploring different facets of these 100 years of ICMI life. In Part I, four chapters retrace the various stages of the Commission’s life over its 100 years of history, through an analysis of published and unpublished sources. The periods we have singled out are the following: –– Foundation and early period up to WWI; –– Rebirth in 1952 as a permanent IMU subcommission  down to  Freudenthal’s innovations; –– “Renaissance” in the late 1960s and further development up to 2008. In Chap. 1, Gert Schubring describes the origins of the movement toward international cooperation and the establishment of an active network of mathematics educators, also presenting the key figures in this story. The work of CIEM/IMUK is described, showing the problems in the constitution of the Commission, then its “golden period” in the years before WWI, and the friction due to this war, leading to its dissolution in 1920. The last part of the chapter deals with the ephemeral re-­ constitution in 1928 until its “sleeping mode” from 1936 on. The whole chapter is based on a considerable amount of archival material. Chapter 2 by Fulvia Furinghetti and Livia Giacardi analyzes the evolution of ICMI after its establishment in 1952 as a subcommission of the International Mathematical Union (IMU). The changes in society and research, and movements such as New Math, made the need evident for new paradigms for approaching the problems of mathematical instruction. The collaboration with international bodies such as UNESCO and OEEC/OECD fostered new initiatives on every continent.

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The three main lines of investigation chosen to study the history of ICMI in the 20 years after WWII are the following: the relationships between IMU and ICMI, which often resulted in relationships between pure mathematicians and educators; the emergence of mathematics education as an autonomous field of research; and the change in ICMI’s objectives. These aspects occur in a transversal way in the work of the various Executive Committees that came one after another in this period. However, it was  only under the presidency of Freudenthal, who launched two important initiatives – a journal and a tradition of conferences specifically dedicated to mathematics education – that the concomitance of these three aspects ushered in a new season for ICMI. This chapter is largely based on the unpublished documents presented in Chap. 4. In Chap. 3, Marta Menghini outlines the life of ICMI after Freudenthal’s important and innovative initiatives. One of these was the creation of the International Congresses on Mathematics Education (ICMEs). Since 1969, these congresses have marked the life of ICMI: they have given voice to ICMI’s Executive Committee, its president, and its secretary, and also featured the principal topics and actors within mathematics education from an international perspective. Each congress has  become an important date in the life of researchers, teachers, and people involved in various ways in mathematics education. In this chapter, the evolution of ICMI is analyzed through a study based on the proceedings of successive ICMEs. Oral interviews released by prominent actors on the ICMI scene provide further information on this period of consolidation of the Commission and creation of new trends in mathematics education research. In Chap. 4, Livia Giacardi presents a wide selection of unpublished letters and documents belonging to the period 1952–1974, coming from different archives, especially from the IMU Archive. The purpose is to highlight—through the voice of the protagonists—unknown or lesser-known aspects of the ICMI history, such as, for example, the not always harmonious relations between ICMI and IMU as well as the internal dynamics of the Commission, and to discover the true motivations behind certain actions. Part II presents useful data about the life of ICMI: “The Timeline of ICMI 1908–2008” (F.  Furinghetti and L.  Giacardi); “The Central and Executive Committees of CIEM/IMUK and ICMI” (F. Furinghetti); “Terms of Reference for ICMI (1954–2007)” (L.  Giacardi); “Mathematics Education in the International Congresses of Mathematicians 1897–2006” (F. Furinghetti); “Maps on the Process of Internationalization of ICMI” (L. Giacardi); and “The Beginning of an Adventure: Glances at the First ICME” (Lyon 1969) (F. Furinghetti). Part III contains the biographical portraits of the 54 members of the Central/ Executive Committee of ICMI who passed away in the first 100 years of ICMI. Almost all of them were professional mathematicians; David E. Smith, who was an educator, is the most remarkable exception. Generally, these were scholars who were interested in problems concerning mathematics education and participated in various ways in the activities of ICMI. However, beginning in the 1950s, some mathematicians joined the ICMI Executive Committee as ex officio members

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for the sole reason that they occupied certain institutional positions – such as the IMU presidency – but whose involvement in ICMI activities was modest. Although for the famous figures various biographies are available, the portraits in Part III of the volume offer an opportunity to outline the actual involvement of these mathematicians in ICMI’s development and, more generally, in mathematics education. Each of these portraits consists in a section of general information and a section dedicated to contributions, if any, to education. In the same vein, the bibliographic references are divided into a section containing a succinct selection of works on the person in question and his mathematical works, and a section including selected publications linked in some way to mathematics education (if any). Part III also contains portraits of the scholars awarded the title of Honorary Member of the Commission during the International Congress of Mathematicians in Oslo (1936) and Charles-Ange Laisant, one of the founders of L’Enseignement Mathématique, the official organ of ICMI. The authors of the 63 portraits presented in Part III are 30 and belong to 18 countries. Genoa, Italy  Fulvia Furinghetti Turin, Italy  Livia Giacardi

Acknowledgments

At the end of this work—which has involved the collaboration of 31 authors, mathematicians, historians, and researchers in mathematics education with the purpose of giving the readers a comprehensive picture of the first century of ICMI—we wish to thank all those who have provided their collaboration. First of all, heartfelt thanks go to the authors and referees who made it possible to carry out this work, and to Alexander Karp and Gert Schubring, the editors of the Springer Series International Studies in the History of Mathematics and its Teaching, for their constant support. Our most wholehearted thanks also go to: Jill Adler, Ferdinando Arzarello, Jerry Becker, Guillermo Curbera, Renaud d’Enfert, Corinna Desole, Judith Goodstein, Bernard Hodgson, Geoffrey Howson, Scott Jung, Erika Luciano, Elena Anne Marchisotto, Vilma Mesa, Manuel Ojanguren, Peter Ransom, Antonio Salmeri, Marta Sanz-Solé, Elena Scalambro, Norbert Schappacher, Kim Williams, and Erich Wittman, who in various ways have provided their help. Special thanks go to the directors and personnel of the various archives we explored: Birgit Seeliger (IMU Archives, Berlin); Tara Craig (Rare Book & Manuscript Library, Columbia University, New  York); Sacha Auderset (UNIL, Université de Lausanne); Laura Garbolino, Antonella Taragna, Giulia Scarcia, and Giuseppe Semeraro (Biblioteca Matematica G. Peano, Università di Torino); Maria Barbieri (Biblioteca della Scuola di Scienze Matematiche, Fisiche e Naturali, Università di Genova); and Team dienstverlening Esther Graftdijk Noord-Hollands Archief. We are also very grateful to Augusto, Alessia and Andrea Astesiano.

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Contents

Part I ICMI 1908–2008  1 The  History of ICMI: The First Phase as IMUK and CIEM��������������    3 Gert Schubring  2 ICMI  in the 1950s and 1960s: Reconstruction, Settlement, and “Revisiting Mathematics Education” ��������������������������������������������   43 Fulvia Furinghetti and Livia Giacardi  3 The  New Life of ICMI: Pursuing Autonomy and Identifying New Areas of Action��������������������������������������������������������������������������������   95 Marta Menghini  4 The  Voice of the Protagonists: A Selection of Unpublished Letters ������������������������������������������������������������������������������������������������������  137 Livia Giacardi Part II Events and Data  5 Timeline  of ICMI 1908–2008������������������������������������������������������������������  241 Fulvia Furinghetti and Livia Giacardi  6 Central  and Executive Committees of CIEM/IMUK and ICMI ��������  279 Fulvia Furinghetti  7 The  Terms of Reference for ICMI (1954–2007)������������������������������������  287 Livia Giacardi  8 Mathematics  Education in the International Congresses of Mathematicians 1897–2006 ��������������������������������������������������������������������  297 Fulvia Furinghetti  9 The  Process of Internationalization of ICMI through Maps (1908–2008)����������������������������������������������������������������������������������������������  325 Livia Giacardi xv

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Contents

10 The  Beginning of an Adventure: Glances at the First ICME (Lyon 1969) ����������������������������������������������������������������������������������������������  335 Fulvia Furinghetti Part III The Portraits of the Central/Executive Committee Members and Other Eminent Figures 11 The  Central/Executive Committee Members����������������������������������������  349 Masami Isoda, Man Keung Siu, Henrik Kragh Sørensen, Livia Giacardi, Jeremy Kilpatrick, Fulvia Furinghetti, Gert Schubring, Giorgio T. Bagni, Sébastien Gauthier, Catherine Goldstein, Michela Malpangotto, Jaime Carvalho e Silva, Margherita Barile, Éric Barbazo, Harm Jan Smid, Sten Kaijser, Adrian Rice, Hélène Gispert, Milosav M. Marjanović, Stevo Todorčević, Osmo Pekonen, Reinhard Siegmund-Schultze, Michèle Artigue, Aline Robert, Ewa Lakoma, and László Surányi 12 Other Eminent Figures����������������������������������������������������������������������������  649 Gert Schubring, Sándorné Kántor Tünde Varga, Éric Barbazo, Gregg De Young, Ewa Lakoma, Livia Giacardi, Eduardo L. Ortiz, Fulvia Furinghetti, Snezana Lawrence, and Gert Schubring Author Index����������������������������������������������������������������������������������������������������  709 Subject Index����������������������������������������������������������������������������������������������������  729

About the Contributors

Michèle Artigue  entered the field of mathematics education after a doctorate in mathematical logic. In this field, beyond theoretical contributions on the relationships between epistemology and didactics, didactical engineering, the instrumental approach, and more recently the networking of theoretical frames, her main research areas have been the teaching and learning of mathematics at university level, and especially the didactics of calculus and analysis, and the integration of computer technologies into mathematics education. She is currently an emeritus professor at the University of Paris. She has been vice-president of ICMI from 1998 to 2006, then president from 2007 to 2009. She was awarded the ICMI Felix Klein Medal for her life-long research achievement in 2013, and the Luis Santaló Medal by IACME for her support to the development of mathematics education in Latin America in 2015. Giorgio  T.  Bagni  (1958–2009) obtained a degree in mathematics from the University of Padova (Italy). He taught at the University of Bologna, Rome, and Udine. His main interests in research were history of mathematics and didactics of mathematics. On these subjects, he published numerous and important works. Éric Barbazo  is Professeur agrégé of mathematics and docteur in history of sciences at the École des hautes études en sciences sociales (EHESS, School for advanced studies in the social sciences). Until 2012, he was member of IREM of Bordeaux. He was teacher at French lycée. He was professor at the University of Bordeaux (France), and afterwards at the University of Houston (Texas, USA). From 2009 to 2013, he was president of Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP, Association of Mathematics Teachers in Public Education). Margherita  Barile  received her master’s degree in mathematics from the University of Genoa, Italy, and her doctorate from the University of Osnabrück, Germany. She is a professor at the University of Bari, where she teaches courses in algebra and history of mathematics. xvii

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Jaime Carvalho e Silva  born in Coimbra (Portugal), holds a PhD in mathematics (Partial differential equations) and is an associate professor at the University of Coimbra, member-at-large of the Executive Committee of ICMI from 2006 to 2009, and secretary-general of ICMI from 2010 to 2012. He studied at the University of Paris 6, and was a visiting professor at the Arizona State University (USA). He is main advisor for master’s and PhD students in pure mathematics, mathematics education, and history of mathematics. From 1996 to 2003, he coordinated several committees that wrote the math syllabus used for 15 years in Portuguese Secondary Schools (10th to 12th grades), including new courses like “Applied Mathematics for the Social Sciences” and “Mathematics for Art Students,” and math modules for vocational schools. He coordinated a group that in 2020 produced a report with recommendations for the improvement of mathematics education in Portugal, commissioned by the Portuguese Government. Fulvia Furinghetti  is a retired professor of mathematics education at the University of Genoa (Italy). Her research concerns mathematics education (computer science in mathematics teaching, proof, beliefs, teacher education, history in mathematics teaching, and public image of mathematics) and the history of mathematics education. She was one of the organizers of the celebrations of the centenary of the journal L’Enseignement Mathématique in 2000 and the centenary of the International Commission on Mathematical Instruction (ICMI) in 2008. In both cases, she was one of the editors of the proceedings. With Livia Giacardi, she has developed a website on the history of the first 100 years of ICMI. In 2001–2004, she chaired the International Study Group on History and Pedagogy of Mathematics (HPM) affiliated to ICMI. Sébastien  Gauthier  is an associate professor at the University Claude Bernard Lyon 1 and a member of the Institut Camille Jordan (Lyon, France). His research activity in the history of mathematics focuses on the history of number theory and algebra in the nineteenth and twentieth centuries. He recently published “On the Youthful writings of Louis J.  Mordell on the Diophantine Equation y2  – k = x3”, Archive for History of Exact Sciences 73 (2019), 427–468 (with François Lê). Livia Giacardi  is a Full Professor of History of Mathematics at the University of Turin. Her research focuses on the history of mathematics – in particular on the history of geometry in Italy in the 19th and 20th centuries  – and on the history of mathematics education. She has been the Secretary of the Italian Society for the History of Mathematics (2000–2008), and a member of the Council (2000–2017), a member of the Italian Commission for Mathematics Teaching (2003–2012), a member of the IPC of the Symposium on the occasion of the 100th Anniversary of ICMI (2008) and was one of the editors of the proceedings. With Fulvia Furinghetti, she has developed a website on the history of the first 100 years of ICMI.

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Hélène Gispert  is Professor Emeritus of History of Science at the University of Paris Saclay. Her main topics of research are the history of the production and circulation of mathematical knowledge in France and Europe from the 1860s to the 1930s, the history of mathematical and scientific training in the nineteenth and twentieth centuries, and the history of the diffusion and popularization of science in the Third Republic France. Her latest publications focus on the circulation of mathematics via journals. Catherine Goldstein  is directrice de recherche at the Centre national de la recherche scientifique (CNRS) and is based at the Institut de mathématiques de Jussieu-­ Paris Gauche (SU, UP), in Paris (France). Her work focuses on the history of number theory, the social history of mathematical practices and knowledge, and the use of network analysis to study collective and long-term aspects of the development of mathematics. Her recent publications include an edition of Ernest Coumet’s Œuvres, vol. 2 (PUFC, 2019); “Long-term history and ephemeral configurations” in Proceedings of the International Congress of Mathematicians  – Rio de Janeiro, 2018 (World Scientific, 2019, vol. 1, p. 487–522), and the co-edition of Les travaux combinatoires en France (1870–1914) et leur actualité: un hommage à Henri Delannoy (PULIM, 2017). Masami  Isoda  Prof/PhD, is a professor in the Faculty of Human Sciences and director of the Center for Research on International Cooperation in Educational Development (CRICED) at the University of Tsukuba (Japan). He received a PhD in education from Waseda University. His contributions have earned him numerous awards: Honorary Professor by Universidad San Ignacio de Loyola (Peru), Honorary PhD by Khon Kaen University (Thailand), and others. He is an advisory board member of HPM and a council member of WALS. His recent publication is Teaching Multiplication with Lesson Study: Japanese and Ibero-American Theories for Mathematics Education from Springer (2021). Sten  Kaijser  is a retired professor of mathematics at Uppsala University. His research was mainly on Banach algebras and on interpolation of Banach spaces. He has had an interest in the history of mathematics, and one of his students wrote a thesis on the history of mathematics in Sweden in the eighteenth century. He was chairman of the Swedish Mathematical Society from 2003 to 2005, and he was main organizer of the ICME-10 Satellite meeting of the International Study Group on History and Pedagogy of Mathematics (HPM) at Uppsala University in 2004. Jeremy Kilpatrick  (1935-2022) was Regents Professor of Mathematics Education Emeritus at the University of Georgia, Athens, GA, USA. He has taught at European and Latin American universities, receiving four Fulbright awards. He holded  an honorary doctorate from the University of Gothenburg and was a Fellow of the American Educational Research Association, a National Associate of the National Academy of Sciences, and a member of the National Academy of Education. He received a Lifetime Achievement Award from the National Council of Teachers of

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Mathematics and the 2007 Felix Klein Medal from the International Commission on Mathematical Instruction. His research interests included proficiency in mathematics teaching, curriculum change and its history, assessment, and the history of research in mathematics education. He was especially interested in the history of research in mathematics education and the history of the school mathematics curriculum in various countries. Ewa Lakoma  is Rector’s Proxy for Quality of Education at the Military University of Technology in Warsaw. Her research concerns especially didactics of probability and statistics, history of mathematics, and epistemology of mathematics, with particular reference to the concepts of probability and statistics. She is the author of many publications on these subjects and co-author of a series of mathematics textbooks and teaching materials for 10–19-year-old students. She is a member (2020–2024) of the Executive Committee of the Study Group on the History and Pedagogy of Mathematics affiliated to ICMI (HPM), and a member of the scientific committees of the European Summer Universities on the History and Epistemology in Mathematics Education and ICME Satellite Conferences in the field of history and pedagogy of mathematics. She collaborated to the ICMI Study 9 and ICMI Study 14. Snezana Lawrence  is currently a senior lecturer in the Department of Engineering Design and Mathematics at Middlesex University, London. She is on the editorial board of Mathematics Today and writes for it a regular column “Historical Notes.” Snezana’s book, A New Year’s Present from a Mathematician, was published by Chapman and Hall (2019); she also co-edited a book with Mark McCartney, Mathematicians and Their Gods, published by OUP (2015). Snezana’s main interests are in the history of mathematics and mathematics education for architects, engineers, and, more recently, pilots. Snezana chairs the History and Pedagogy of Mathematics Study Group (affiliate of ICMI) for 2020–2024, and is the education officer of the British Society for the History of Mathematics, and the associate editor of the British Journal for the History of Mathematics. Snezana is a corresponding editor for the Nexus Network Journal, a global network of architects, engineers, and mathematicians researching links between these disciplines. Michela Malpangotto  directrice de recherche at CNRS, has directed the History of Astronomy Department at Paris Observatory. She is a member of the International Academy for the History of Science and editor-in-chief of the International Archives for the History of Science. Her work is at the crossroads of mathematics, astronomy, and humanism with studies published on Theodosius, G. Peurbach, A. Brudzewo, Regiomontanus, F. Maurolico, C. Clavius, and R. Baranzano. Milosav M. Marjanović  was born on August 24, 1931, in Nikšić (Montenegro). He received a diploma in mathematics from the Faculty of Natural Sciences, University of Belgrade in 1955, where his whole university carrier ran until he retired in 1998. He was elected to the Serbian Academy of Sciences and Arts as a corresponding member in 1976 and as a full member in 1991.

About the Contributors

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His main scientific result concerns complete classification of hyperspaces of 0-dim. compact, metric, spaces (see Chap. 4 of S. Todorčević, Topics in Topology, Springer 1997). As a university professor, M. Marjanović substantially contributed to the mathematics education at all levels, from the elementary school to the university, by producing highly influential textbooks and research papers and by having an important role in the innovation of school and university curricula. Marta Menghini  is an associate professor in the Department of Mathematics at the University of Rome Sapienza. She is the author of numerous published works in the fields of mathematics education, history of mathematics, and the history of mathematics education. She was the chief organizer of the international symposium on the occasion of the centenary of ICMI held in Rome in March 2008 and edited the proceedings. She held a regular lecture on the historical development of practical geometry at ICME-12 in Seoul; she was co-author of the chapter “History of teaching geometry” in the Handbook on the history of mathematics education, and the chapter “From mathematics and education to mathematics education” in the Third international handbook of mathematics education. She was involved in the translation and edition of Felix Klein’s third volume of Elementary Mathematics from a Higher Standpoint (2016). Eduardo  L.  Ortiz  is Emeritus Professor of Mathematics and of the History of Mathematics at Imperial College London. He is fellow of the Institute of Mathematics, Great Britain; the Royal Academy of Science, Spain; and the National Academy of Science, Argentina. He is Guggenheim Research Fellow, Department of History, Harvard University, USA. He is currently studying matters related to the communication of mathematics and mathematical physics in countries away from the main centers of research. Osmo  Pekonen  PhD, DSocSci, was born in Mikkeli, Finland (1960–2022). He studied mathematics in Paris, and was based at the University of Jyväskylä, Finland. He has published on differential geometry, string theory, k-theory, and history of mathematics. He was the book reviews section editor of The Mathematical Intelligencer. Adrian  Rice  is the Dorothy and Muscoe Garnett Professor of Mathematics at Randolph-Macon College in Ashland, Virginia, USA, where his research focuses on nineteenth- and early twentieth-century British mathematics. In addition to papers on various aspects of the history of mathematics, his books include Mathematics Unbound: The Evolution of an International Mathematical Research Community, 1800–1945 (with Karen Hunger Parshall); Mathematics in Victorian Britain (with Raymond Flood and Robin Wilson); and most recently Ada Lovelace: The Making of a Computer Scientist (with Christopher Hollings and Ursula Martin). He is a four-time recipient of awards for outstanding expository writing from the Mathematical Association of America.

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About the Contributors

Aline  Robert  has been a researcher in mathematics didactics since 1978, after several years of research in mathematics. She is a retired professor of Cergy Pontoire University and works in the LDAR, a didactical lab of the University of Paris named André Revuz. She first worked on students’ acquisitions in the first year of university. This led to the development of specific tools to study the proposed tasks in view of the targeted contents, assessed by a mathematical, curricular, and cognitive study. Her research then focused on the practices of secondary school teachers, according to the students’ activities they can provoke. All of this work fed into the development of a training program for secondary school teacher educators, based on a specific use of the previous tools and on hypotheses about the development of practices inspired by the results of her research. Gert Schubring  is a retired member of the Institut für Didaktik der Mathematik, a research institute at Bielefeld University, and at present a visiting professor of the history of mathematics at the Universidade Federal do Rio de Janeiro (Brazil). His research interests focus on the history of mathematics and the sciences in the eighteenth and nineteenth centuries and their systemic interrelation with social-cultural systems. One of his specializations is history of mathematics education. He has published several books, among which is Conflicts between Generalization, Rigor and Intuition: Number Concepts Underlying the Development of Analysis in 17th–19th century France and Germany (New York, 2005). He was chief editor of the International Journal for the History of Mathematics Education from 2005 to 2015 and is co-editor of the series International Studies in the History of Mathematics and its Teaching, published by Springer. Reinhard Siegmund-Schultze  is a German historian of mathematics working in Kristiansand (Norway). He has published, among other things, on the history of functional analysis, probability theory, and applied mathematics; on Rockefeller and mathematics, and emigration of mathematicians from NS-Germany; and on the work of Richard von Mises. Man Keung Siu  obtained a BSc in mathematics and physics from the University of Hong Kong and went on to earn a PhD in mathematics from Columbia University. Like the Oxford cleric in Chaucer’s The Canterbury Tales, “gladly would he learn, and gladly teach” for three decades at the University of Hong Kong before retirement in 2005, and is still doing the same. He has published research papers in mathematics and computer science, some more papers of a general nature in history of mathematics and mathematics education, and several books in popularizing mathematics. In particular, he is interested in integrating history of mathematics with the learning and teaching of mathematics, actively participating in an international community of history and pedagogy of mathematics since the mid-1980s.

About the Contributors

xxiii

Harm Jan Smid  (1945) studied mathematics at the Leyden University, He was a math teacher from 1966 until 1975 and a lecturer at a teacher training college from 1975 until 1981. From 1981 until his retirement in 2007, he was assistant professor, later associate professor, of mathematics and mathematics education at the Delft University of Technology. He earned his PhD in 1997 with a thesis on the history of Dutch math teaching in the nineteenth century. He has published several articles and contributed to conference proceedings on the history of mathematics education. Henrik Kragh Sørensen  is professor of the history and philosophy of the mathematical and computational sciences at the University of Copenhagen. His research has focused on early-nineteenth-century analysis and algebra, on mathematics and science in Denmark, and on philosophy of mathematical practice. Currently he is working on developing and deploying tools from big data, digital humanities, and machine learning to ask and approach questions in the history and philosophy of mathematics. László Surányi  was born in 1949 in Budapest. He won third prize and first prize at the International Mathematical Olympiad (IMO) in the years 1966 and 1967, respectively. He studied mathematics at the Eötvös Loránd University (ELTE) Budapest during 1967–1972, where he also earned his doctorate in mathematics in 1977. During 1972–1982, he worked as a researcher at the Mathematical Institute of the Hungarian Academy of Sciences, Department of Graph Theory, Set Theory and Logic. Between 1982 and 2013, he taught mathematics to secondary school classes with a specialization in advanced mathematics. His main research areas are “Sprachdenken” (the “dialogical thinkers” Ferdinand Ebner, Martin Buber, Franz Rosenzweig, and the Hungarian Lajos Szabó and Béla Tábor, see http://lajosszabo.com/BPDISKANG.html) and Metaaxiomatics (L.  Surányi, Metaaxiomatische Probleme, 1992/99, see http://lajosszabo.com/SL/ maxnem.html). He is also an author of several published essays and a book on music (in Hungarian see http://lajosszabo.com/SL/publist.htm#zene ) and literary criticism. Stevo Todorčević   is a Canada Research Chair at the University of Toronto and is a Directeur de Recherche at CNRS, Paris. He is a member of the Serbian Academy of Sciences and Arts and is a Fellow of the Royal Society of Canada. He has published so far more than 170 research papers in mathematics including 5 research monographs. Sándorné Kántor Tünde Varga  was born in Debrecen (Hungary) in 1935. She graduated from the University of Debrecen in 1957 as a mathematics and physics teacher and received her PhD in 1977. She was a high school teacher (1957–1971) and university lecturer (1971–2018). Between 1971 and 1976, she was an assistant professor in the Department of Geometry at the Institute of Mathematics at the University of Debrecen (Hungary). She retired in 1995, but continued to hold classes till 2018.

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Her areas of research: differential geometry, teaching mathematics and descriptive geometry, teacher training, in-service teacher training, mathematics competitions, talent management, history of mathematics and its education, and history of the University of Debrecen. In addition to professional articles (more than 200, in Hungarian, German, English), she wrote several books. Gregg  De Young  has taught in the Core Curriculum Program at the American University in Cairo since 1990. Among his research interests are history of the transmission of Euclidean geometry in the medieval Mediterranean, history and editing of mathematical diagrams, and history of the printing of geometry textbooks in the Middle East.

Abbreviations and Acronyms

ANL Accademia Nazionale dei Lincei APMEP/APM Association des Professeurs de Mathématiques de l’Enseignement Public CIEAEM Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques CIEM Commission Internationale de l’Enseignement Mathématique CTS Committee on the Teaching of Science Fondo Delessert Fondo André Delessert, Biblioteca Speciale di Matematica “Giuseppe Peano,” Torino Fonds de Rham Université de Lausanne EARCOME East Asian Regional Conference on Mathematics Education EM L’Enseignement Mathématique ESM Educational Studies in Mathematics HPM History and Pedagogy of Mathematics IA Archives of the International Mathematical Union (section of ICMI), Berlin IACME/CIAEM Inter-American Committee on Mathematical Education (in Spanish, Comité Interamericano de Educación Matemática) ICM International Congress of Mathematicians ICME International Congress on Mathematical Education ICMI Bulletin ICMI Bulletin of the International Commission on Mathematical Instruction ICSU International Council of Scientific Unions IMN Internationale Mathematische Nachrichten IMU International Mathematical Union IMU Bulletin Bulletin of the International Mathematical Union IMUK Internationale Mathematische Unterrichts-Kommission IOWO Instituut voor de Ontwikkeling van het Wiskunde Onderwijs IUCST (CIES) Inter-Union Commission on the Teaching of Science (in French, Commission Interunions de l’Enseignement des Sciences) NCTM National Council of Teachers of Mathematics xxv

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Abbreviations and Acronyms

NUG Niedersächsische Staats- und Universitätsbibliothek Göttingen OEEC (OECD) Organisation for European Economic Co-operation, later Organization for Economic Co-operation and Development PME Psychology of Mathematics Education RBML Rare Book & Manuscript Library, Columbia University, New York SCOTS Special Committee on the Teaching of Science SEACME Southeast Asian Conference on Mathematics Education SEAMS Southeast Asian Mathematical Society SMSG School Mathematics Study Group UNESCO United Nations Educational Scientific and Cultural Organisation ZDM The International Journal on Mathematics Education (?) word with uncertain decipherment (…) missing or damaged text [ ] addition by the editor // end of page of the manuscript a. u. archival unity cit. cited reference del. deleted n. d. no date n. n. p. not numbered page n. p. no place transl. translation

Part I

ICMI 1908–2008

1.1  Introductory Note In this part of the volume, four chapters retrace the various stages of the Commission’s life over its hundred years of history, through the analysis of published and unpublished sources. In this century of life, we have singled out three main periods: –– Foundation and the early period up to WWI; –– Rebirth in 1952 as a permanent subcommission of the International Mathematical Union (IMU) down to Freudenthal’s innovations; –– “Renaissance” in the late 1960s and further development up to 2008. In Chap. 1, Gert Schubring describes the origins of the movement toward international cooperation, the establishment of an active network of mathematics educators, and the key figures in this period. The work of CIEM/IMUK is described, showing the problems in its constitution, its “golden period” in the years before WWI, and the friction due to this war, leading to its dissolution in 1920. The last part of the chapter deals with the ephemeral reconstitution of the Commission in 1928 until its “sleeping mode” from 1936 on. The whole chapter is based on a considerable number of archival sources. Chapter 2 by Fulvia Furinghetti and Livia Giacardi analyzes the evolution of ICMI after its establishment in 1952 as a subcommission of IMU. The changes in society and research, and movements such as New Math, made the need evident for new paradigms for approaching the problems of mathematical instruction. The collaboration with international bodies such as UNESCO and OEEC/OECD fostered new initiatives all around the world. The three main lines of investigation chosen to study the history of ICMI in the 20 years after WWII concern the following aspects: relationships between IMU and ICMI, which often resulted in relationships between professional mathematicians and educators; the emergence of mathematics education as an autonomous field of research; and the change in the objectives of ICMI. These aspects affected in a transversal way the work of the various Executive Committees that came one after another in this period. However, it was only under

2

ICMI 1908–2008

the presidency of Freudenthal, who launched two important initiatives – a journal and a tradition of conferences specifically dedicated to mathematics education  – that the concomitance of these three aspects ushered in a new season for ICMI. This chapter is largely based on the unpublished documents presented in Chap. 4. In Chap. 3, Marta Menghini outlines the life of ICMI after Freudenthal’s important and innovative initiatives. One of these was the creation of the International Congresses on Mathematical Education (ICMEs). Since 1969 these congresses have marked the life of ICMI: they have given voice to ICMI’s Executive Committee, its president, and its secretary and also featured the principal topics and actors within mathematics education from an international perspective. Each congress has  become an important date in the life of researchers, teachers, and people involved in various ways in mathematics education. In this chapter, the evolution of ICMI is analyzed through a study based on the proceedings of successive ICMEs. Oral interviews released by prominent actors on the ICMI scene provide further information on this period of consolidation of the Commission and creation of new trends in mathematics education research. In Chap. 4, Livia Giacardi presents a wide selection of unpublished letters and documents (69) belonging to the period 1952–1974, from different archives, especially from the IMU Archive. The purpose is—through the voice of the protagonists—to highlight unknown or lesser-known aspects of the ICMI history, for example, the not always harmonious relations between ICMI and IMU and the internal dynamics of the Commission, and also to discover the motivations behind certain actions. .

Chapter 1

The History of ICMI: The First Phase as IMUK and CIEM Gert Schubring

1.1 Introduction1 One does not need to give a lengthy explanation of the importance of ICMI— International Commission on Mathematical Instruction2—for the development of mathematics education worldwide. Having survived several profound crises is already a significative sign for the need of such an organisation. Moreover, the results achieved confirm its role for the internationalisation of the national communities hitherto without an effective communication and interaction. Yet, in proposing and founding this international organisation, nothing foreshadowed any long-standing activity – or even a well-structured body. The task given was a rather usual one, namely to prepare reports for the next International Congress of Mathematicians, four years later. The first aim of this chapter will be, hence, to show how the change came about. And the other one will be to show the development and the successes, as well as the problems encountered – until World War II.  This chapter is based on two earlier, but largely revised papers (Schubring 2003; Schubring 2008).  Actually, this name in English dates only from the 1950s. Originally, there appeared only one official name, in French, due to the language of the first published document about the work of the Commission (Klein et al. 1908): Commission Internationale de L’Enseignement Mathématique, soon abbreviated as CIEM. Since Klein as president used always the German language in his correspondence, his naming and abbreviation became commonly used, too: Internationale Mathematische Unterrichts-Kommission, IMUK. Although this first document defined four official languages for the work of the Commission—English, French, German and Italian (ibid., p. 450)—names in English and in Italian were used only by the subcommissions of the United States and Italy. 1 2

G. Schubring (*) Institut für Didaktik der Mathematik, University of Bielefeld, Bielefeld, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_1

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1.2 Preparing and Founding One uses to recall that the first idea to create the Committee had been launched by David Eugene Smith (1860–1944),3 in the journal L’Enseignement Mathématique (hereafter EM), founded in 1899 by the French mathematician Charles-Ange Laisant (1841–1920) and the Swiss mathematician Henri Fehr (1870–1954). Actually, this was not only the first international activity for mathematics teaching, the journal had also a decisive role in instigating Smith’s proposal. Looking with more care into this primary impulse and in Smith’s reaction, one remarks that the journal’s initiative had a quite specific objective – not identical with the objectives later on decided for the Commission. There was a note published by the two editors in 1905, which was due to a concern about the relation between pure and applied sciences, and this basically in the context of higher education, as is shown in particular by the first and the last of the three questions put by the editors to the readers of the journal: 1.  What progress needs to be achieved in the organisation of the teaching of pure mathematics? 3. How should teaching be organised in such a way as to meet the requirements of pure and applied sciences better than in the past? (quoted apud Furinghetti 2003, p. 38).

One remarks, thus, that Smith’s proposal was an answer to the first question, considering not mathematics teaching in general, but regarding improving the teaching of pure mathematics: The best way to reinforce the organisation of the teaching of pure mathematics would be the establishment of a committee appointed by an international Congress and which would study the problem in its entirety (quoted in the translation by Schubring 2003, p. 54).

In the pedagogical section of the International Congress of Mathematicians (ICM) at Rome, in 1908, Smith presented his contribution about the teaching of mathematics at secondary schools in the United States,4 and after that he added an extended and revised version of his proposal of 1905, listing nine issues. Now, it was a general proposal, no longer focussed upon the relation between pure and applied aspects. This proposal is barely known and will hence be documented here, since it makes to understand Smith’s original concrete ideas and how this relates to the later practice:

 See the biography of Smith in this volume.  Smith had presented a second contribution in the same section on philosophical, historical, didactical questions (questioni filosofiche, storiche, didattiche) in its history subsection, on The Ganita-­ Sâra-­Sangraha of Mahâvïrâcârya. 3 4

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5

Questions for the Considerations of Such Congresses as This And now, in closing, I should like to express the hope that these International Congresses may add to their already great value as clearing houses of thought, by sometime investigating, through committees, a few questions relating to secondary education. Countries cannot be uniform in their curricula, their school systems, nor their methods of teaching, but the influence of a Congress like this might greatly help many who are earnestly seeking to improve the teaching of mathematics. Some of the questions that might profitably be considered are the following: 1. What have been the results of attempting to remove the barrier between such topics as algebra and geometry, or to teach the two simultaneously, and are we prepared as yet to make any recommendation in this matter? 2. What have been the results of attempting to teach demonstrative geometry before algebra? If they have been favourable, what is the nature of the geometry best adapted to this apparently psychological sequence? 3. What is the opinion of impartial observers of the work of the Méray geometry in France and of works like that of DE PAOLIS in Italy, as to the union of plane and solid geometry? 4. What is done in the various countries as to the union of plane geometry and trigonometry? 5. What is being done to advantage in the introduction of the elementary ideas of the calculus into the work in secondary algebra? 6. What is the safe minimum of Euclidean geometry, the calculus, and mechanics? 7. What is the safe relation to be established between secondary mathematics and physics? 8. What position should the secondary schools take with respect to the nature of applications and the relations of applied to pure mathematics? 9. What should be the relative nature of the courses in the secondary schools for those who do not intend to proceed to the universities, and for those who do intend to do so? In other words, of finishing and preparatory courses? These questions, and others like them, are occupying the serious thought of American teachers. As we have always turned to Europe for conservative but helpful suggestions, so some of us would be glad if this Congress might deem it advisable to appoint international committees, corresponding in any of the four languages admitted to these deliberations to consider matters of this kind. An agreement is not essential, but the interchange of views and suggestions could not fail to be very helpful (Atti 1909b, p. 476–477). All the issues concerned the teaching of mathematics in secondary schools, and mainly curricular issues. The relation between pure and applied aspects constituted one among other issues, without particular emphasis. Some of the later reform issues were addressed: the issue of fusion between algebra and geometry, and between plane and solid geometry – even the introduction of the elements of the calculus. The issue of rigour, later one of the IMUK studies, was addressed by issue

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no. 2. The issue of relations between secondary and higher education was contained, as no. 9. On the other hand, the then much debated questions of reforming teaching methods such as the Perry movement and the laboratory method were not included. One remarks the modest style of Smith’s proposals: he suggested, even quite timidly, to establish various committees to discuss the different issues – emphasising: just for discussions and exchanges, but no joint actions were implied, and not one well-organised international body. Furthermore, one remarks a discrepancy: the final vote was to establish the Committee during the same Congress  – whereas Smith had left it open, proposing the establishment of committees in his contribution, written before the ICM, at “congresses as this [one]”; and Lietzmann had explicitly told that Smith, in his oral summary of the printed contribution, proposed to establish the Committee—now already as just one—at the next congress (“auf dem nächsten Kongreß”; Lietzmann 1908, p. 83). One has to infer that it was the dynamic initiated by Smith’s oral talk which led to Archenhold’s formulation, to create one committee, and right now. It is not well-known, however, that the founding almost failed, due to the machinations of a German participant, himself committed to mathematics education. At first, all seemed to work well. The imminent failure of Smith’s proposal, however, is clear from Eileen Donoghue’s PhD thesis of 1987 devoted to Smith’s actions in promoting mathematics education as an academic discipline. In the fourth section of the Congress, on 9 April, while the German astronomer Friedrich Simon Archenhold (1861–1939) was presiding, Smith presented his paper and proposed establishing the international committee – yet, Archenhold’s summarising shows a decisive difference from the later decision that the committee should be “permanent”: Prof. Archenhold, supporting an idea already expressed by Prof. Smith in his communication, proposes that a permanent committee be established for studying questions concerning the teaching of mathematics in the secondary schools. (Atti 1909a, p. 45).5

In fact, Archenhold favoured the proposal and took the vote which passed.6 Nevertheless, Smith thought it necessary to have the proposal reconfirmed at the last meeting of the fourth section, on 11 April. Then, however, the presider was the German mathematics teacher and educator Max Simon (1844–1918). Simon, as Donoghue reports, “was one of the few who voiced an objection to Smith’s proposal.” She observes, “While Smith and his supporters were in the lobby discussing manoeuvres, Simon adjourned the meeting” (Donoghue 1987, p. 267). Because it was the last meeting of the section, Smith’s proposal would have been postponed until the next Congress, four years later! Donoghue reports how Smith prevented the premature death of his idea by tricky dealing with standing orders:

 The original text is: “Il Prof. ARCHENHOLD, in appoggio a un’idea già espressa dal Prof. SMITH nella sua Comunicazione, propone che sia istituito un Comitato permanente per lo studio delle questioni riguardanti l’insegnamento della Matematica nelle scuole secondarie.” 6  “La proposta è accolta favorevolmente dalla Sezione” (The proposal is approved by the Section) (Atti 1909a, p. 45). 5

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Since Italy was the official host country for the Congress, Smith urged Professors Castelnuovo and Enriques to exercise their procedural right and reconvene the meeting (Donoghue 1987, p. 268).

In fact, the Proceedings confirm this extraordinary procedure.7 Only then the decisive motion was voted on and transmitted to the General Assembly for its approval. Section IV, having recognized the importance of a comparative examination of the programmes and methods of the teaching of mathematics in the secondary schools of the various nations, entrusts to the Professors KLEIN, GREENHILL and FEHR the task of setting up an international committee to study the matter and report on it to the next Congress. Regarding this proposal, the Section requests the support of the General Assembly on. (Atti 1909a, p. 51).8

It was adopted in the closing session of the Congress that same afternoon, in the formulation presented by Guido Castelnuovo (1865–1952)9, elaborated that morning of 11 April (see below). A second source for the near failure of Smith’s proposal is a report given by Walther Lietzmann, who shortly thereafter became Klein’s assistant in handling IMUK matters. Lietzmann, who had participated in Rome for entirely private motives and without being tied to national or international reform work, reported on an incident at the 9 April session where Smith gave his talk. August Gutzmer, one of the key German promoters, together with Klein, of the reform of mathematics teaching, reported on the movement. In the discussion following Gutzmer’s talk, Max Simon criticised those reforms. Presiding at the last session, Simon used his position to defeat Smith’s proposal. Lietzmann quotes Simon’s comment upon Gutzmer’s report of the reform proposals: “The Good is not new; and the New is not good!” (“Das Gute ist nicht neu, und das Neue ist nicht gut”) Lietzmann characterised this as “malicious saying” (Lietzmann 1960, 44; my transl., G.S.).

 Having thus completed the Agenda, and since no one else asking to speak, the President [i.e. Simon] declares the works of the Section closed. The session is reopened by Prof. Enriques to ask the Assembly if it intends to discuss the setting up of the International Committee for the study of reforms concerning the teaching of mathematics in secondary schools. The Assembly agrees and so an animated discussion takes place, in which Professors Bonola, Smith, Conti, Archenhold, Castelnuovo, Stéphanos, Fehr participate. (Atti 1909a, p. 50– 51). The original text is: “Esaurito così l’Ordine del giorno, e nessun altro domandando la parola, il Presidente [i.e. Simon] dichiara chiusi i lavori della Sezione. Viene riaperta la seduta dal Prof. Enriques, per domandare all’Assemblea se intenda discutere sulla costituzione del Comitato internazionale per lo studio delle riforme riguardanti l’insegnamento della matematica nelle Scuole secondarie, L’Assemblea consentendo, ha luogo un’animata discussione, a cui prendono parte i Professori Bonola, Smith, Conti, Archenhold, Castelnuovo, Stéphanos, Fehr.” 8  The original text is: “La Sezione IV, avendo riconosciuto l’importanza di un esame comparato dei programmi e dei metodi dell’insegnamento delle matematiche nelle Scuole secondarie delle varie Nazioni, confida ai Prof.i KLEIN, GREENHILL e FEHR l’incarico di costituire un Comitato internazionale che studii la questione e ne riferisca al prossimo Congresso. La Sezione su tale proposta domanda l’appoggio dell’Assemblea generale” 9  See his biography in this volume. 7

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The conflict between Klein and the modernisers on the one hand and the defenders, like Simon, of traditional approaches to treat the calculus via infinitesimals and neglectable quantities, on the other hand, has been seen as a rivalry between two competing scientific schools in German mathematics education, with Max Simon as a prominent representative (Burscheid 1984). Max Simon was a mathematics teacher at a gymnasium who had grown up and been educated in a classically minded culture. As a mathematics teacher, his main interest was in geometry, where he advocated classical Euclidean methods. After 1903, he also lectured at Strasbourg University on the history and didactics of mathematics. Lietzmann, in his later publications, never missed an opportunity to include some critical or polemical point about Simon. Given the positive vote after the first discussion, on 9 April, why did Smith present the proposal a second time and thus nearly fail to reach his goal? Donoghue offers an explanation in her report on the first session (1987, p. 267): Smith’s proposal was so attractive that almost everyone present expressed interest in participating in the Commission. He realized “that to ensure the level of success he wanted, the Commission members needed to be strategically chosen.” This means that Smith had not really envisaged the concrete establishment of that committee when he presented it to the pedagogical section. He was surprised by the sudden success, due in particular to the German session president, and became now aware of the consequences of his proposal. In such general terms as stated by him, the endeavour might end in an uncoordinated chaos. It was only after the apparent success that Smith began to reflect on detailed terms for the committee. Between 9 and 11 April, the concept was elaborated to first form a three-man committee, the later Comité Central, to be complemented afterwards by national delegates. During this process, as Smith contacted numerous influential participants, the names of Klein, Fehr, and Greenhill emerged and were eventually accepted. Donoghue summarises this second stage: With this accomplished, Smith pre-empted the haphazard nomination process by proposing a three-man committee on organisation whose members would serve as officers of the Commission (Donoghue 1987, p. 268).

In fact, comparing the two votes, one immediately remarks the differences between a general proposition and a measure for concrete activity:

 For the original text see the footnote 7.  The original text is: “Il Congresso, avendo riconosciuto la importanza di un esame accurato dei programmi e dei metodi d’insegnamento delle matematiche nelle scuole secondarie delle varie nazioni, confida ai Professori KLEIN, GREENHILL e FEHR l’incarico di costituire un Comitato internazionale che studii la questione e ne riferisca al prossimo Congresso.” 10 11

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9 April

11 April

Prof. Archenhold, supporting an idea already expressed by Prof. Smith in his communication, proposes that a permanent committee be established for studying questions concerning the teaching of mathematics in the secondary schools. The proposal is approved by the Section. (Atti 1909a, p. 45; my transl.)10

The Congress, having recognised the importance of a careful investigation of the syllabi and of the methods of teaching mathematics in the secondary schools of the various nations, entrusts the professors Klein, Greenhill and Fehr with the task to constitute an international Committee, which should investigate the question and report about it at the next Congress” (Atti 1909a, p. 33; my transl.)11.

Comparing the two versions, one will remark moreover a telling difference: In the first proposal, the committee was to be constituted “permanently.” Lietzmann, in his report, mentions, too, that Smith had proposed a “permanent” (dauernd) constitution (Lietzmann 1908, p. 83). This proposal is not contained in Smith’s printed version of his proposal; he must have added it in his oral presentation on 9 April. In fact, as Lietzmann reports, Congress participants did not read the entire text as printed in the Proceedings but gave a shortened oral version; he told: “Smith (New York) gave a short summary of his report: The teaching of secondary Mathematics in the United States” (ibid.). In the definite vote, the task of the body to be created was drastically limited: only until the next congress, four years later. Furthermore, the second version neither contained a list of countries invited to send delegates, nor the quota for the number of delegates. Those details were elaborated later, as we will see. Finally, Smith had conceived of the committee’s work to be restricted to secondary schools. It was upon Klein’s initiative that all levels of education became included (Schubring 2003, p. 57).

1.3 The Steps for Constituting the Comité Central The next question one might ask upon reading the decision taken by the ICM is, “Why these three persons?” Klein was an entirely understandable choice given his reputation as a mathematician and his active involvement in the German reform movement.12 The Swiss Henri Fehr was likewise obvious: As editor of L’Enseignement Mathématique, he was well informed about national developments in mathematics teaching and about persons active in this field.13 But why George Greenhill? He was an applied mathematician at the Royal Artillery Institution in Woolwich, but was retired and had not been known to be involved in questions of school teaching.14 He had been knighted, however, and thus had the aura of “Sir George Greenhill.” He seems to have been nominated for purely political reasons:  Actually, Smith had originally planned to propose only persons present in Rome and had hence asked Gutzmer to be the president. Gutzmer had, however, proposed Klein as a better choice (Gutzmer 1917, 332). See the biography of Klein in this volume. 13  See the biography of Fehr in this volume. 14  See the biography of Greenhill in this volume. 12

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Smith gave as a reason that Britain would host the next ICM (Donoghue 1987, p. 268). And the British government was to play a special role in the formation of ICMI (see below). Lietzmann was quite puzzled when Greenhill had remained almost entirely silent during the first meeting of the Comité Central in September 1908 in Cologne (Lietzmann 1960, p. 45). Given that they were to collaborate to create the broader international committee, the choice of Klein, Fehr, and Greenhill posed an unprecedented challenge. How would these three succeed in establishing a common understanding and vision and arrive at working terms? None of them had previously been in cooperation or real contact with the other two – except for Klein and Greenhill who had corresponded, although not extensively, since 1886.15 Although the three did not live very far apart—Göttingen, Geneva and London—it was a person outside the group and living much farther away who got the three together, essentially serving as the midwife for IMUK: D.E. Smith again! Smith, who loved to travel and whose participation in Rome was part of a one-year-sabbatical leave, devoted much time during that summer 1908 to the birth of the Commission. The week after the Congress, Smith managed to see Fehr and Greenhill, who had been at Rome, meeting both, according to Donoghue, “to explain his ideas on the aim of the commission and how it should function” (Donoghue 1987, p. 269). She credits Smith to have done all the next steps, but the documents show that Fehr and Smith jointly urged that the work begin. Real activity began in June. A letter of June 22 from Fehr to Smith shows Fehr’s concerns about Klein’s alleged passivity. Klein had proposed that the committee meet first during the next Naturforscherversammlung, at the end of September in Cologne. Klein’s intention had apparently been to have an extensive discussion and thus to forge a common understanding. Fehr, however, wanted immediate action – but only through correspondence. He continued in showing that he and Smith had primarily discussed how to constitute the Commission at large and in particular the quota.16 He asked Smith to visit Klein to discuss matters with him. Klein, in contrast, fully recognised the importance of the new task, but realised that it implied not merely the presidency of the International Commission but also an active role in the German subcommittee which would have to be constituted, as he told Smith in a letter of 6 October (RBML, box 29, Klein to Smith). To assume this double obligation, he needed a reliable and capable assistant. Actually, Klein had begun activities early in June. He met Gutzmer for extended deliberations, on

 As evidenced by their letters, preserved in the Nachlass of Felix Klein in the Göttingen University Library, NUG, no. 9, fol.s 489–499. 16  “Mr. Klein […] despite my very detailed letter, does not yet seem to realize the useful role that this Commission can play. He would like to postpone any decision until September on the occasion of a Committee’s meeting that he proposes to be held in Cologne.” (RBML box 16). The original text is: “M. Klein […] malgré ma lettre très détaillée, ne semble pas encore se rendre compte du rôle utile que peut jouer cette Commission. Il voudrait remettre toute décision au mois de septembre à une réunion de Comité qu’il propose à Cologne.” 17  See the Lietzmann biography by Gert Schubring in this volume. 15

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11 and 12 June and was thus first-hand informed about the events in Rome. Gutzmer, however, did not accept the additional task of assistant for the IMUK work (Gutzmer 1917, p. 333). Klein had then another candidate in mind, but received no answer from him: One day in the summer 1908, Walther Lietzmann received a card from Klein grudgingly asking why he had not responded. Weeks before, Klein had written a long letter inviting him to participate in the IMUK work, but it had not reached Lietzmann, because of vacations in Italy.17 Lietzmann had shown himself capable of this function thanks to an excellent comparative international analysis of present trends in geometry teaching (Lietzmann 1908a). Lietzmann accepted, and together they prepared intensely for the meeting in Cologne. Moreover, Klein seems to have been the first to think about financial matters. He was sceptical that three delegates generously allotted by Fehr to the major countries could be financed in Germany. Smith had offered to visit Klein in Göttingen, and Klein had invited him to come for two days, 6 and 7 July. They discussed extensively Smith’s draft of a report and Fehr’s proposals for the tasks of the international committee and that of the national subcommissions. The text elaborated at this occasion contained already the general programme for the work of IMUK. After a US student, Clifford Upton, had typed their text, copies were sent to Fehr and to Greenhill. Fehr responded to Smith already on 18 July, very happy about the outcome. By then, Fehr, Klein and Smith had reached a common understanding.18 In particular, Fehr appreciated the active role now given to the Comité Central in choosing the delegates who would become members of the Commission – whereas his and Smith’s earlier draft had reserved that right to national associations or subcommissions. Missing in the constitution of the Comité Central was the cooperation of Greenhill. Smith, still acting as midwife, went to England in August to integrate Greenhill into the projected work. Letters from Greenhill to Smith in the second half of August and early September show that they had met several times in London, to acquaint Greenhill with the discussions among the other three. The letters show, moreover, that Smith – a devoted bibliophile – used the stay in London for working with historical material, but that he had also tried to meet John Perry, the famous propagator of reforms in mathematics teaching (RBML, box 20, Greenhill to Smith). Yet, Perry never became a British delegate! After these various bilateral meetings, the three members of the original committee, now naming itself Comité Central (CC), got together in Cologne, September 23–24, without Smith but with Klein’s assistant Lietzmann, during the Naturforscherversammkung and the yearly meeting of the Deutsche Mathematiker-­ Vereinigung. The meeting achieved, on the one hand, the definite consolidation, the  “You had very gainful sessions with Mr. Klein. In fact, the first survey you give me of the results shows that the Commission will be established on solid foundations with an excellent general plan. […] I completely agree with the changes you have made to my draft of the project.” (RBML, box 16, Fehr to Smith, letter of 18 July 1908) The original text is: “Vous avez eu des séances très fructueuses avec M.  Klein. Le premier aperçu que vous me donnez des résultats montre en effet que la Commission va être établie sur des bases solides avec un excellent plan général. […] Entièrement d’accord avec les changements que vous avez apportés à mon avant-projet.” 18

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self-discovery, of the leading group. On the other hand, it also achieved an elaboration of the plan for the entire work and a definition of procedures for constituting the network. I will comment on this in the next section. The participants were highly satisfied. Suffice it here to quote how much Greenhill, who until then had stayed a bit outside of the communication, had become integrated into the group. After the meeting, he wrote to Smith (12.10.08): “Klein set to work at once in fine style and drew up an excellent plan of our procedure for the next years. My misgiving vanished that the work was too vast, and we should not know how and where to begin” (RBML, box 20, Greenhill to Smith).

1.4 Building the Network of National Subcommittees As Lietzmann reported, the major work of the Comité Central at Cologne was to find out who should be chosen and approached as a delegate to the international committee. He credits Klein to knowing whom to ask: “One has always to admire Klein’s stunning capacity to select in all the countries the adapted person” (Hier war nun immer wieder die fabelhafte Fähigkeit Kleins zu bewundern, in all den Ländern den geeigneten Mann herauszusuchen.) (Lietzmann 1960, p. 45; my transl., G.S.). Actually, as the letters of the committee members show, it was rather a collective work, one still supported by D.E. Smith. A substantial part was, in fact, already elaborated at this first meeting. One has to be aware that it was an unprecedentedly large and differentially structured network that the committee had undertaken to be built up. Comité Central • President • Vice-President • Secretary-General Voting members: • Three Delegates (Germany, Austria, France, Great Britain, Hungary, Italy, Russia, Switzerland, the United States) • One Delegate (Belgium, Denmark, Spain, Greece, Netherlands, Norway, Portugal, Rumania, Sweden) Associated Countries • Australia, Brazil, Bulgaria, Canada, Cape Colony, Japan, Mexico, Serbia. Since the national subcommissions were to constitute the infrastructure necessary for producing successful inquiries and reports from the national delegates, there were to be 18 fully participating countries and 15 associated countries. Clearly, such an international network had never before been proposed or established for any discipline. To form the network, the organisational situation of the mathematical communities and the communities for teaching mathematics in all of the countries had to be explored.

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An additional complication emerged, however, with respect to finances. Klein had been aware from the beginning that the IMUK would need a budget to achieve its goals. At first, he had envisaged asking the national mathematical associations for financial contributions. As those contributions would clearly be limited, a new idea arose during the first meeting in Cologne: The work of IMUK should be considered not a private initiative, but a public enterprise. Hence, the formal way for constituting the national subcommissions and nominating the delegates for the international body should be to approach the respective governments that they form a subcommission, make nominations and provide a budget for these activities. To make such a procedure viable, the Comité Central would conduct its own search for capable candidates in the respective countries and then make sure, by convenient means, those candidates were made known to the governments and formally confirmed. Actually, the group in Cologne had envisaged a more complicated procedure. As Klein explained in his letter of 6 October to Smith: As soon as the situation in the main countries will be sufficiently clear, the English government shall request officially the other states to appoint delegates and to constitute national bodies. By this procedure we hope to provide the necessary stability to the endeavour, from the outset, also from the financial point of view (RBML, box 29, Klein to Smith; my transl., G.S.).

Klein even outlined what the letter of the British government should contain, in particular a national budget of about 500 pounds. I have never encountered an explanation why the British government was chosen to act as a super-power in favour of mathematics teaching, giving advice to the other governments. It is probable that this particular idea was proposed by Sir George Greenhill, but I am not aware that Britain had ever before acted internationally in favour of learning! At least, Greenhill was active for achieving such a tutoring activity of the British government, as a letter from Smith to William Osgood (1864–1943) from 14 October shows - in the context of Smith’s efforts to constitute the US subcommission (see Karp 2019b) -, after having received Klein’s letter quoted above: I enclose a letter just received from Professor Klein. Untangling his chirography as much as possible I judge that Greenhill will get the education Department of England to send a request to the various governments for the appointments. In that case such a request will probably reach our Board of Education. I am therefore rather inclined to write to Commissioner Brown and state the situation as it has thus far developed. Unless this strikes you as poor policy I will proceed accordingly. Professor Young is now in Europe, I understand, so that we cannot easily communicate with him (RBML, box 37).

Hence, in the first part of October, the CC had been confident that this unique approach will function. Yet, in the document released later in October, disseminating the call for national activities, such a patronage function was not mentioned. After the internal preparations, the starting signal for initiating the international activities was given by the publication of a document signed by Klein, Greenhill and Fehr, as of October 1908, in L’Enseignement Mathématique, in the last issue of 1908: Commission Internationale de L’Enseignement Mathématique. Rapport sur l’Organisation de la Commission et le Plan Général de ses Travaux (Klein et  al. 1908, pp. 445–458).

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This extensive document detailed: • The origin, due to the decision at ICM IV in Rome • The organisation of the Commission: the Comité Central and the national delegates  – listing the participant and the associated countries and the respective number of delegates; and financial dispositions • The official four languages (German, English, French and Italian), the organisation of the studies to be undertaken and the general objectives • The objectives of the national studies – differentiated into reports on the present state and on modern tendencies in mathematics teaching Issues to be studied in both parts were the exams, the teaching methods and the formation of teachers. Regarding financial dispositions, the document told that the ICM had not provided means for the works, so that an appeal was made to the governments of the participating countries to finance the expenses of the national delegates, of the national subcommission and to contribute to the expenses of the Commission. As a matter of fact, as Lietzmann noted assessing the nomination and constitution procedures as planned and envisaged in Cologne: The future has shown that all went well in all those countries, with just one exception, the request by the English government. (Lietzmann 1960, p. 45)19

1.5 The Works of the National Subcommissions Lietzmann was right. Despite the enormity of the work envisaged, and although no models existed for such a type of an international network, all went remarkably smoothly. Already in the autumn of 1908, the mode proposed in the October 1908 document was realised for some key countries: the delegates were proposed to the German and Italian governments and the subcommissions were constituted, France followed soon, etc. There were some problems with Belgium, where the first person asked (Paul Mansion) thought the task was unrealisable. Eventually, Joseph Neuberg (1840–1926) agreed. The main problem was presented by Britain, which occupied the minds of the CC for a long time. In fact, in Britain, the process of nominating delegates proved to be the most complicated. For over three years, Greenhill remained the only delegate, which is all the more astonishing because, at that time, the movement to reform geometry teaching, and to extend reform to mathematics teaching in general, was strong and successful. John Perry, who was asked, refused  The original text is: “Die Zukunft hat gelehrt, daß es überall geklappt hat, nur eins nicht, die Aufforderung der Regierungen durch die englische Regierung.” 20  Only a secretary for the national subcommittee has been appointed: Jackson (Woolwich). In the preface of the British reports, the Board of Education explained that it had appointed a special commission to elaborate the required reports and added “which acted at the same time as the British subcommission of the International Commission on the Teaching of Mathematics” (Reports 1912, vol. I). 19

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to become a delegate, and even Andrew Russell Forsyth of Cambridge University, though a moderate reformer, was strongly opposed to the entire endeavour (RBML, box 20, Greenhill to Smith, 24.6.1909). One aspect of the problem was that Scotland needed to be represented by a delegate of its own. Not until 1912, the year of the Congress in Cambridge, were the three delegates appointed – all were from England and none were from Scotland! Moreover, Britain was the only participating country that had not succeeded in forming a proper national subcommission. Greenhill addressed the Board of Education, hence the government, to find persons who would work on the needed national reports.20 Strangely, this first International Commission and the related British activities are not mentioned in any of the publications on the reform of mathematics teaching in England. The national subcommissions of the most active countries have been studied recently in a special volume, analysing their origin, context, functioning and publications (Karp 2019). The countries for which detailed studies of their subcommissions are thus available are: France, Germany, Great Britain, Italy, Russia and the United States. In fact, all the countries which had been assigned three delegates had shown considerable activity. The countries with just one delegate, however, used not to constitute a proper commission – actually, in general, it was there only one person who was promoting the IMUK activities in his country. Another point of concern was to activate the associated countries. Although Greenhill soon succeeded in getting representatives from Canada and the Cape Colony, and although Australia and Japan were very active, Latin America was not. Before the First World War, only Mexico and Brazil had nominated representatives. And only by the end of the war, there were perspectives that Argentina and Cuba would join. The sensitive issue of financing the IMUK/CIEM activities was apparently best resolved in the case of Germany, thanks to the steps taken by Felix Klein: Klein was able to report in this second meeting [1909] the success he experienced in asking for grants for financing the German IMUK activities: the Reich had allotted 5.000 Marks and Prussia 5.000 Marks, too. The other German states had not yet made such grants, but authors of special reports for a state would be granted aids. (Schubring 2019, p. 71).

Various other states of the Reich, being a confederation where the federal states enjoyed autonomy in particular for education, reimbursed authors for reports analysing the issues in their state (ibid., p. 73). The contrary case was presented by Great Britain. Smith communicated, on 6 February 1909, to Osgood the complaints by Greenhill not to succeed in obtaining funds there: I have just received a letter from Sir George Greenhill in which he says that they will not be able to get government support in England, but that the work will go on. He says that Professor Klein expects to get it in Germany, and that if he succeeds there they will make an attempt in England next year. I have replied telling him about the status of the matter here. (RBML, box 37).

Greenhill repeatedly complained about his dreadful and fruitless initiatives to obtain money from the government:

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G. Schubring All this is easy in Germany where education is organized by the Government; and the work there may be said to be complete already. But here in England there is no centralisation and our work will be very difficult. I am warned that it is a hopeless task for any money of our Government. (RBML, box 20, 26.1.09). we are now trying to secure Fletcher, of the Government Education Department. He will direct us how best to obtain a Grant from the Treasury; but the outlook is not favourable. (ibid., 24.6.09). We are relying on securing the patronage of the Education Department, and our Secretary C.  S. Jackson has laid the matter before his friend W.  C. Fletcher, who is an important official. But the date 1912 has been allowed to be seen, and so the business is not considered urgent, and so far we have had no definite response. It is only through the Government Channel that we can hope to receive assistance in money. Every other application has turned out a failure. (ibid., 18.7. 09). It is impossible to get an answer, Yes or No, out of our Board of Education. […] With the general election pending, our chance of a decision is more remote still. (ibid., 22.12. 09).

Due to the complex structure of the United States with local and regional competencies for education issues, Smith met great problems to find funding for the work; the various commissions created needed to have their expenses for correspondence, travels and printing be financed. The Bureau of Education had no funds at disposal and even the Carnegie Institution for Science was not able to contribute. Eventually, the American subcommission succeeded in obtaining donations by a number of universities and by commission members themselves (Karp 2019b, p. 208–212). The study on Russia does not mention explicitly a budget, but since the ministry became involved, some allowances can be supposed (Karp 2019a). For France and Italy, the respective studies do not mention financial issues (Giacardi 2019; d’Enfert & Ehrhardt 2019).

1.6 The “Golden Period” of IMUK/CIEM: 1909 to 1914 Besides these problems, the proper work of the Comité Central and of the Commission as a whole progressed remarkably well. The CC met regularly every year, and from 1910 on, there were general meetings of the entire Commission. These meetings can be considered, in effect, as having been the first international congresses on the teaching of mathematics, hence as precursors of the ICMEs (International Congress on Mathematical Education); the most pertinent such congress was the meeting in Paris, in April 1914, a few months before World War I (Fig. 1.1). As it became characteristic for the ICMEs, since ICME-3  in Karlsruhe 1976, thematic reports were prepared, presented and discussed: in Brussels 1910, Milan 1911, Cambridge 1912, and Paris 1914, the last congress having been prepared at a meeting of the CC 1913 in Heidelberg. As can be seen from the list in Appendix 1 showing the members (delegates) in 1914, the great majority of members were university professors of mathematics. Their relation to mathematics teaching was constituted by the fact that at least a considerable number of their students were intending to become secondary school

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Fig. 1.1  Announcement of “ICME I,” April 1914

teachers. Probably the only professional among them was David Eugene Smith who, from the beginning of his career, had worked in normal schools, that is, at teacher training institutions in the United States. His productivity was connected with his professorship at the Teachers College, a part of Columbia University in New York. His exclusive task there was to train mathematics teachers. He succeeded in firmly establishing mathematics education in the United States, by his own reflections on teaching methodology, by publishing school textbooks, and by extensive historical research. The fact that the overwhelming majority of IMUK members came from higher education constituted at the same time a strength and a weakness of IMUK’s work. It was strong in that these members were firmly rooted within the mathematical community. The first IMUK thus worked on educational problems as an integral part of mathematical issues – there were no conflicts, no divergences between mathematicians and educators. On the other hand, this “enracinement” within the mathematical community was a weakness of the first IMUK, since work was not undertaken from a proper perspective of school mathematics, that is, that of teaching mathematics in schools. This potential weakness did not become an obstacle or a problem, however: firstly, since proper communities of mathematics education had not yet emerged, and secondly since Felix Klein successfully avoided the one-sidedness which seemed to be an inevitable outcome of the domination of IMUK by higher education and by mathematicians’ concerns. It is therefore characteristic that, at ICM IV in Rome, IMUK had been commissioned to deal with mathematics instruction at secondary schools only. This bias in favour of secondary schools was a result of the fact that university mathematicians considered this sector the only field where they were possibly educationally competent.

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Felix Klein, and the Comité Central with him, modified this conception and undertook to extend their task to all school levels: from primary schools up to universities and technical colleges, and even including vocational schools! According to the wording of the resolution at the Rome Congress,, only the teaching in secondary schools is envisaged. But, given that the aim of these schools and the duration of their studies vary greatly from one State to another, the work of the Committee will cover the whole field of mathematical instruction, from the first initiation up to higher education. It will not be limited to institutions for general education schools leading to the University, but will also study the teaching of mathematics in technical or vocational schools. Therefore, it will be undertaken a study of the entirety of mathematics teaching at the different types of schools and at its various degrees.21 (Klein et al. 1908, 451–453).

The work of IMUK and of its national subcommissions on mathematics instruction at all these school levels was really impressive. The official list of publications established in 1920 upon the dissolution of IMUK documents 294 contributions published in 17 countries. Evidently, the quality of these papers varies significantly. In some countries, the reports were the result of truly collective and intensive work while, for other countries, the reports were prepared by individuals. The German reports were generally noted for being the best organised. Communication and cooperation had not been an aim in itself for the Comité; they were understood rather as a process with a direction, namely initiating reforms. The pivotal point for the launching of the first international reform movement in mathematics education was, in fact, that the decision of the Comité Central to complement the (more or less descriptive) national reports submitted by the national subcommittees with international comparative reports on a few key topics representing the major reform concerns. These topics, perhaps unsurprisingly, reflected the fact that the viewpoints of higher education were indeed dominant in the first IMUK. Of the eight topics for written reports, only three (1., 2., 5.) corresponded to the level officially designated as IMUK responsibility, that is, the level of secondary schools, whereas the majority of five topics concerned either the transition from secondary to higher education or related exclusively to higher education. The list of the thematic reports and the related four “congresses” evidence that the tasks had developed far beyond the objectives as defined at the ICM in Rome:

 The original text is: “Dans le texte même de la résolution du Congrès de Rome il n’est question que de l’enseignement dans les écoles secondaires. Mais, étant donné que le but de ces écoles et la durée de leurs études est très variable d’un État à l’autre, le Comité fera porter son travail sur l’ensemble du champ de l’instruction mathématique, depuis la première initiation jusqu’à l’enseignement supérieur. Il ne se bornera pas aux établissements d’instruction générale conduisant à l’Université, mais il étudiera aussi l’enseignement mathématique dans les écoles techniques ou professionnelles. Il s’agit donc d’une étude d’ensemble de l’enseignement mathématique dans les différents types d’écoles et à ses divers degrés.” 21

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1. The fusion of different branches of mathematics in the teaching in the secondary schools. (Milan 1911) 2. Rigour in the teaching of mathematics in the secondary schools. (Milan 1911) 3. The theoretical and practical teaching of mathematics for university students of physics et natural sciences. (Milan 1911) 4. The mathematical formation of physicists at the university. (Cambridge 1912) 5. Intuition and experience in the teaching of mathematics in the secondary schools. (Cambridge 1912) 6. The results obtained by introducing the elements of differential and integral calculus in the upper grades of secondary schools. (Paris 1914) 7. The mathematical formation of engineers in the different countries. (Paris 1914) 8. The education mathematics teachers for the secondary schools (planned for Munich 1915).

An overview of major aspects of these thematically comparative reports will be given here:

Fusion (1911) The report on this topic concentrated on the controversy whether it was preferable or not for geometry teaching to integrate planimetry and stereometry. Other possible concrete realisations of “fusion,” say, of integrating algebra and geometry, were not discussed. Although the rapporteur was a Frenchman, Bioche, the discussion was dominated by the Italians. The controversy whether planimetry and stereometry should be taught in an integrated manner, or not, had been dividing the Italian mathematical community for decades. The pivotal point of the controversy was the concept of the purity of mathematical method (EM vol. 13, 1911: pp. 468–471). Since the Italian syllabus of 1900, which admitted a plurality of teaching methods, and in particular to show the relations between plane and solid figures, had been replaced around 1910 by an anti-fusionist one, the traditional purists in Italy had prevailed (see Scarpis 1911, pp. 31–32).

Rigour (1911) The rapporteur for this topic was Guido Castelnuovo from Italy. He confined himself to geometry. While efforts in the other countries were towards reducing rigour in teaching geometry, and to introduce empirical methods to encourage the role of intuition, such reforms were refuted in principle in Italy. The majority of Italian mathematicians even opted, and acted, for reinforcing rigour in geometry. A presentation of geometry in a more and more axiomatic form over the school years is characteristic of this Italian approach (EM vol. 13, 1911, pp. 461–468; espec. p. 463).

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 ure and Applied Mathematics for Students P of the Sciences (1911) The rapporteur was this time a German, Heinrich Timerding. He presented his report along the lines of the German problematic: the hotly debated point being whether the mathematical training necessary for studying the natural sciences should be specially adapted to the future profession of the students, or whether a general education in mathematics as a common foundation for the study of the sciences was preferable. Timerding criticised the too narrow specialisation prevailing in Germany, emphasising the more general character of mathematical education in France and Italy (EM vol. 13, 1911, pp. 481–495). Since the two topical reports to the Cambridge Congress of 1912, where the mandate of IMUK was extended for another four years, discuss essentially variations of the first three topics, I will now concentrate on the last two, presented in 1914 at Paris.

Paris 1914: The Calculus and Training of Engineers It is quite remarkable that the culminating point of IMUK work was reached in 1914 at its Paris session where the two topics presented corresponded exactly to the two cornerstones of Klein’s German reform programme. In Paris, the subject that attracted the most attention and participation was “the evaluation of the introduction of calculus to secondary schools.” The topic was hotly debated, and the report on it was the most voluminous of all the international IMUK reports.22 This was also the topic that Klein had prepared more carefully than any other. He not only helped design the international questionnaire that dealt with this matter, but he also chose its coordinator and reporter, Emanuel Beke,23 a Hungarian scholar and former student of Klein, and one of the most fervent adherents of Klein’s programme. Beke was responsible for transmitting the reform ideas directly to Hungary and he was highly successful in initiating an analogous movement there. For the second topic of the Paris session, the mathematical preparation of engineers, Klein also chose a trustworthy person as chairman, namely Paul Stäckel, a close co-worker of Klein’s in matters of history and of applied mathematics. Klein, who did not participate in the Paris session himself, was immediately informed about the course of the debates by letters from his assistants and co-workers. Concerning the mathematical training of the engineers, Lietzmann was able to report, much pleased, that “the engineers want—this was the general opinion—to get their mathematics from the mathematician, not from the engineer.”24 Stäckel also reported that he was quite satisfied with the international response, particularly from the French engineers, the overwhelming majority of whom had stressed “the necessity of a culture générale for engineers.”  See L’Enseignement Mathématique, 16 (1914), 245–306.  See the biography of Beke in this volume. 24  Letter of Lietzmann to Klein, 4 April 1914, from Paris. NUG, Nachlass Klein, No. 51. 22 23

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Although Klein and his co-workers had expected, with regard to the calculus, more “palpable results,” the impact of his main reform agenda, namely to introduce the concept of “functional thinking” as basic notion pervading the entire mathematical curriculum, proved to be enormously successful on the international level. Sooner or later, the syllabus became modernised almost everywhere, supplanting the traditional restriction to the static ideas of elementary geometry which excluded all knowledge of variables and modelling of physical processes. I should complement this by some information on the administrational side of this work. It is instructive to see how the members of the CC presented themselves in their official letters; at first, the letterhead used by Klein, produced by a typewriter (Fig. 1.2). Then the French printed letterhead for CIEM as used by the Secretary-General Fehr (Fig. 1.3): And then the printed letterhead for ICTM as used by David Eugene Smith (Fig. 1.4). Greenhill used no letterhead at all. It is important to see how the next ICM, at Cambridge in 1912, had dealt with IMUK/CIEM. The final vote of the General Assembly documents the high esteem and the recognition for the work of this first international network (Fig. 1.5). Smith was not elected right away Vice-President at Cambridge. Actually, by decision of the Congress, he joined the CC. It was by a later letter of Fehr that this was interpreted in the way that Smith should act not as a simple member of the CC, but as an additional Vice-President (15.10. 1912). In fact, the CC felt legitimated the

Fig. 1.2  The IMUK letterhead as used by Klein

Fig. 1.3  The letterhead as used by Henri Fehr

Fig. 1.4  The letterhead as used by D. E. Smith

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Fig. 1.5  Vote of ICM V about IMUK (Proceedings 1912, p. 41)

next year, in September 1913, to co-opt three new members and to increase thus the number of its members from four to seven – all from countries hitherto not represented in the CC: Paul Appell (1855–1930) from France, Guido Castelnuovo (1865–1952) from Italy and Emanuel Czuber (1851–1925) from Austria. This increase should prove to be important in the later crisis. Since Appell did not accept, he was replaced by Jacques Hadamard (1865–1963). The other remark concerns the relation of the network to the national governments and to a super power. The vote underlines (see Fig. 1.5) the necessary cooperation with the national governments. Fehr commented upon the underlying intentions when he sent Smith this printed version, on 15 October, which should serve for addressing the governments. One understands by his explanation, on the one hand, that it were the delegates who were in need of a superior instance, which would urge their respective government to secure the budget for their work and, on the other hand, that it was by now no longer necessarily the British government, which had to act as such a superior instance, but that it might be substituted by the governments of two key representatives of the CC: Germany and Switzerland.25  We have already considered the question of transmitting the resolution adopted by the Cambridge Congress to the governments; for this purpose, I have had the attached text printed. The Congress having taken place in England I had asked the Board of Education to transmit the resolution to the various governments via the Foreign Office, indicating that it is sending it at the invitation of the Congress Committee and the Central Committee of the Commission. I’m waiting for the answer; if it is negative we will have recourse either to the German government or to the Swiss government. Since the Commission is already established, there is no problem this time taking the longer route, which is the diplomatic route. Many delegates have indeed asked that the renewal be done in this way so that the budget funds are more easily granted. (RBML, box 16, Fehr to Smith, 15.10. 1912). The original text is: “Nous avons déjà examiné la question de transmettre aux gouvernements la résolution adoptée par le Congrès de Cambridge; à cet effet j’ai fait imprimer le texte ci-joint. Le Congrès ayant eu lieu en Angleterre j’avais demandé au Board of Education de transmettre la résolution aux différents gouvernements par l’intermédiaire du Foreign Office, en indiquant qu’il fait cet envoi sur invitation du Comité du Congrès et du Comité central de la Commission. J’attends la réponse ; si elle est négative nous aurons recours soit au gouvernement allemand, soit au gouvernement suisse. Étant donné que la Commission est déjà constituée, il n’y a pas inconvénient à prendre cette fois la voie la plus longue qui est la voie diplomatique. Beaucoup de délégués ont en effet demandé que le renouvellement se fasse de cette manière afin que les crédits leur soient plus facilement accordés.” 25

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Fig. 1.6  Rendering of accounts of IMUK for ICM Cambridge

It is highly telling to remark that IMUK Secretary Fehr felt obliged to render the account of the expenses of the CC for the four years 1908 to 1912 to the ICM in Cambridge, and even audited by two controllers (Fig. 1.6).

1.7 World War I and the Crisis The Paris Congress, the climax of IMUK, had taken place in early April of 1914. On 28 April, it was decided that the next meeting of the Commission should take place in Munich, from 2 to 5 August 1914. To prepare for that meeting and for the next

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ICM in Stockholm in 1916, where the last work of the Commission was to be presented and thus the task and mandate given by the Congress of 1908 would be achieved, the CC met in Göttingen, from 23 to 26 July. A few days later, on 1 August, World War I began! The former unity and cooperation within the CC ended almost immediately. Henri Fehr, citizen of a neutral state, but by his socialisation sympathetic to French culture and scientists, acted on the side of the Western allies. During and after the war, he was an active promoter to aligning the Commission’s policy with that of the later victorious powers. Surely, one has to consider that the German army invaded Belgium, a neutral state, right from the start and effected terrible destructions. The first documented reaction to the war was an 14 August letter from Greenhill to Smith, written with almost the first emotion: This terrible catastrophe will throw Europe back a century in civilisation. It will take that time to recover. There cannot be any meeting at Munich next year, and I fear it is all [one illegible word] with the Congress at Stockholm. So it will be  – actum est de I.M.U.K (RBML, box 20, Greenhill to Smith).

The next document is a 3 September letter from Fehr to Smith, evidently written after the destruction of the Belgian town Louvain by the Germans at the end of August, condemning “the Austrian-German military caste” and wondering “what the opinion, in particular of the German academics in general, must in the heart of their hearts think [about it].”26 Fehr thought it would still be possible to hold the international conference planned for Munich, by transferring it to a neutral country. Moreover, he reported the common opinion from the Göttingen meeting that IMUK’s mandate should definitely be declared finished in Stockholm. In fact, the first letter dated 29 September that Fehr sent to Klein after the outbreak of the war expresses optimism and a willingness to continue the work as far as possible (Fig. 1.7). Fehr’s next letter to Smith, on 30 September, confirmed this position. He lamented that the unfortunate European war was delaying all international endeavours. He reported to have proposed Klein to continue the IMUK work and to publish an issue of L’Enseignement Mathématique dedicated to the reports planned for 1915, but without that Conference. He announced that he would write to the CC as soon as he received Klein’s answer. He continued, however, with a severe critique of the German efforts trying to convince foreigners that news about cruelties committed by German armies were mere propaganda.27  The original sentence is “La caste militaire austro-allemande a jeté l’Europe dans un triste état et je me demande parfois quelle doit être l’opinion, au fond de leur conscience, de nos collègues et en général des savants allemands.” RBML, box 16, Fehr to Smith 27  “Je reçois journellement des brochures allemandes pour la défense de la “vérité allemande.” Ces messieurs ignorant qu’en pays neutre on connait mieux la vérité complète que chez eux où la censure est d’une rigueur inouï” (RBML, box 16, Fehr to Smith). (I receive German brochures every day in defense of “German truth.” These gentlemen, ignoring that in a neutral country we know the whole truth better than in their homes where censorship is incredibly severe). 26

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Fig. 1.7  First letter of Fehr to Klein after the beginning of the war

Even on 8 November, Fehr reassured Smith that the work of the Commission would continue, but the 1915 meeting should be cancelled (RBML, box 16). Klein, in his response to Fehr of 7 October, had supported the proposal to continue the work as far as possible (NUG, draft, fol. 130). But then he received a letter of 2 December from Fehr that Klein annotated to be “the critical letter.” Fehr repeated what he had said to Smith on 10 December: All the work had to be stopped until better times arrived. Fehr even refused to publish the questionnaire for the study on teacher training. His reason, he explained that several colleagues had expressed doubts about completing the ongoing projects. He told that numerous professors had been mobilised, not just in countries at war, but also in neutral countries, so that important collaborators were missing. But he also referred to the escalation of violence in the war since the destruction of Louvain (NUG, fol. 132f.). Because Klein regarded Fehr’s propositions as very critical, he reflected intensely on his response and, uncharacteristically for him, drafted his answer, of 11 December. He presented two alternative ways for proceeding. In general, Klein underlined the need to concentrate all energy on finishing the ongoing projects and noted that what was not finished now would never be finished. He wanted that to be

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the CC position, too, and urged Fehr—representing a neutral country—to support this view within the CC. He then outlined the two possible procedures for the IMUK: • Either to continue the work as planned, that is, to collect and publish all available reports in EM, in particular the questionnaire, while desisting to have a discussion; a closing issue of EM in 1916 should be envisaged. • Or, to declare that under the given conditions the work of ICMI has to be given up, so that IMUK as an international association be formally dissolved – hoping for re-establishment under new conditions. Even in this second case, EM should publish the questionnaires. Klein expressed the expectation that IMUK might be constituted again one day and that younger colleagues should take over then.28 Despite Klein’s clear instructions to submit these proposals to the CC, Fehr did not consult the CC, but announced in the November issue of L’Enseignement Mathématique (actually published at the end of December) the postponement of the work of IMUK during the war (Fig. 1.8). Crowning his one-sided actions, Fehr not only sent an offprint of that notice to Klein, but also dealt a heavy and radical blow in the accompanying 28 December letter: He proposed that Klein renounce his position as President of IMUK and step back to the position of a Vice- President, ceding the presidency to Smith. Justifying  Either: we will continue the work of the Central Committee in the manner previously envisaged. So: You collect all the reports from different countries, which can be collected, and we publish the related issue, preceded by the questionnaire. Discussion then naturally cannot take place. We plan to publish the corresponding final issue in 1916. Or else: We declare that, due to the current circumstances, we desist completely to continue the general work for the IMUK, so that it can arise again if the circumstances permit it. Thus, we formally dissolve the IMUK as an international association. [ inserted.] with the new IMUK that may one day come into being I myself will then later, due to a different age or to health conditions, no longer participate, but will be very happy when younger people take my place. I recommend: In this second case, we would publish the questionnaires after all in L’Enseignement. [....] Would you please submit these alternatives, which I am making in an objective reaction to the actual conditions, on my behalf to the members of the Central Committee? German original: Entweder: wir setzen die Arbeit des Zentralkomitees in der bislang in Aussicht genommenen Weise fort. Also: Sie sammeln an Referaten, aus den verschiedenen Ländern, was sich sammeln lässt, und wir veröffentlichen unter Vordruck des Fragebogens bez. [ügliches] Heft. Diskussion fällt dann freilich von selbst fort. Wir nehmen in Aussicht, 1916 entsprech.[endes] Schlussheft zu veröffentlichen. Oder aber: Wir erklären, dass wir unter den gegenwärtigen Umständen die allg.[emeine] Arbeit an der IMUK überhaupt aufgeben, damit diese neu entstehen kann, wenn es die Umstände gestatten. Wir lösen also die IMUK als internationale Vereinigung formell auf. [inserted:] bei der neuen IMUK, die eines Tages entstehen mag Ich selbst werde dann später bei anderem Alter resp. Gesundheitszuständen freilich nicht mehr mitmachen, aber mich sehr freuen, wenn jüngere Kräfte an meine Stelle treten. Ich empfehle: In diesem 2. Falle würden wir im Enseignement doch noch die Fragebögen veröffentlichen. [....] Würden Sie bitte diese Alternativen, die ich in objektiver Erwiderung der tatsächlichen Zustände stelle, in meinem Namen den Mitgliedern des Zentralkomitees unterbreiten? (NUG, fol. 134). 28

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Fig. 1.8  Fehr’s one-sided announcement

Fig. 1.9  Excerpt from the Aufruf and from the signatures

this step, Fehr referred to Klein’s signature on the scandalous “Aufruf an die Kulturwelt” of October 1914 (Fig. 1.9). This appeal was signed by 93 German intellectuals and scientists, among them Felix Klein. It was a deeply deplorable document of nationalism; it argued that war crimes supposedly committed by German troops were just false propaganda and even that German militarism was necessary to safeguard German culture. The appeal

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was one of the factors that later led to the exclusion of German scientists from international congresses and organisations. And, for Klein, it led to his expulsion from the Paris Academy of Sciences. Only much later did it become known that many of the signatories, including all the non-Berliners had never seen the text they were asked to support. They had reacted positively to a telegraphic inquiry to sign a patriotic appeal against war propaganda (Ungern-Sternberg 1996, p. 23). Klein told later his former doctoral student Grace Chisholm Young (1868–1944) that he had not known the text – he had been asked only telegraphically whether he would sign a declaration which he assumed would contribute to calm the situation (Tobies 2019, p. 452). Fehr argued in his letter to Klein for adjournment on the one side for material reasons, the non-availability of numerous colleagues, and on the other side due to national irritabilities (“susceptibilités”) due to the war: As far as our committee is concerned, the issue is aggravated by the fact, that its President has signed the manifesto of German scientists. From the point of view of international relations among scientists, one has to recognize that this manifesto is deplorable, since it has dragged milieus into the fights, which had hoped to continue to work jointly on the fields of humanities, sciences and arts. (NUG, Fehr to Klein, fol. 26–28; my transl., G.S.).29

Klein was deeply dismayed at this blow. Many private notes document his concerns. A first such note stated how “ungehörig” (improper) it was by Fehr to publish the stop before the ongoing reflections had succeeded in a consensus, at least between the President and his Secretary-General (NUG, fol. 29). Having received Fehr’s letter just three days after it had been sent, Klein immediately wrote to Smith, on 9 January, 1915, to seek his advice. He communicated the proposal that Smith, as a member coming from a neutral country, should take over the presidency, while he would act as one of the Vice-Presidents (RBML, box 29). At the same time, Klein answered Fehr briefly, saying he had contacted “several sides” concerning the proposal and asked Fehr to wait until he had an answer. He added that if he ever would renounce the presidency, it would be because of the present situation and not for reasons of health (NUG, draft, fol. 33; 17. 1. 1915). It is remarkable to observe the differences of the viewpoints of the members of the CC. Smith, responding to Klein on 5 February, did not accept the offered presidency and instead urged Klein to continue. It is a moving letter, convincingly arguing for separating politics and science: Next, as to the Internationale Mathematische Unterrichtskommission. Of course we cannot meet at Munich, and I fear that the Stockholm meeting must be postponed a year at least. […] As to your presidency, and as to the organization of the Central Committee in general, I hope that we may make no change at present. This is not a conflict of scholars, and in the academic world there is no war. I was in England until November, and I heard only the kindest words in academic circles for all German friends. It would be a calamity for you not  The original text is “Pour ce qui concerne notre Commission cette question se trouve aggravée par le fait que son Président a signé le manifeste des savants allemands. En se plaçant au point de vue des relations internationales entre savants, il faut reconnaître que ce manifeste est regrettable, car il a entraîné dans la lutte, des milieux qui espéraient pouvoir continuer à travailler en commun sur le terrain des lettres, des sciences et des arts ». 29

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to direct this great international investigation until it is completed. Of course I shall be only too glad to act as your assistant, to send out any letters that you might suggest and to cooperate with Fehr in every possible way. I am at your disposal. I will do what your wisdom directs. I shall be only too glad to write to any scholar whom you might suggest, and in your behalf, and to carry out any plans you might have. Only let it be under your direction; let it continue to be known that you are at the head of the work. Thus shall we be sure of high international standing, and the world will be sure that the work will be worthy of the support.

On receipt of this plea, Klein felt determined to stay in his position and not to resign, at least for the time being, as he told Vice-President Castelnuovo in a letter of 4 March, 1915 (NUG, fol. 36). Castelnuovo’s answer of 10 March is likewise a moving document: He assured that all members of IMUK are fully aware that nobody else could replace him, and, moreover, in this complicated situation, no change in the organisation of international institutions should be enacted: On the contrary, one should make all efforts to make them survive so that they might facilitate to retake normal relations among the nations (NUG, Castelnuovo to Klein, fol. 186v.; my transl.)30

Meanwhile, Fehr had informed Smith of his proposal to Klein that Smith should become the President (17.2. 15). Answering on 4 March 1915, Smith first repeated the argument he had made to Klein, that it seemed to me that it would be a great misfortune if the Commission were deprived of his leadership. I stated that this was not a war of scholars, and that I did not believe there would be any personal feeling against a man like himself. I based this judgment upon my conversation with a number of English scholars before I returned from Europe, all of whom spoke in the kindest terms of him and of other German scholars (RBML, box 16, Smith to Fehr).

 Mr. Fehr never told me about your intentions. I understand and appreciate the reasons that inspired you. But, in agreement with Mr. Smith, and in the interests of our Commission, I would ask you to refrain from putting your proposals into effect for the moment. All my colleagues in the Commission, I am sure, recognise the admirable organisational work you have done and the impetus you have given to our work; they know very well that no one, in this respect, can replace you. Moreover, the serious moment that all international institutions are going through advises not to introduce any change in their organisation for fear that these weak organisms might succumb. On the contrary, efforts should be made to make them survive until peace is reached, so that they can facilitate the resumption of normal relations between peoples. (NUG, Castelnuovo to Klein, fol. 186r-186v.) The whole original text is: “M. Fehr ne m’a jamais parlé de vos intentions. Je comprends et j’apprécie les motifs qui vous ont inspiré. Mais, d’accord avec M. Smith, et dans l’intérêt de notre Commission, je vous prie de vouloir bien renoncer pour le moment à mettre à effet votre propos. Tous mes collègues de la Commission, j’en suis certain, reconnaissent l’œuvre admirable d’organisation que vous avez accomplie et l’impulsion que vous avez donnée à nos travaux; ils savent bien que personne, sous ce rapport, ne saurait vous remplacer. D’ailleurs le moment grave que traversent toutes les institutions internationales conseille de n’introduire aucun changement dans leur organisation de crainte que ces faibles organismes ne doivent succomber. Il faut au contraire s’efforcer de les faire survivre jusqu’à la conclusion de la paix, à fin qu’elles puissent faciliter la reprise des relations normales entre les peuples.” 30

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This report on English scientists and on himself is in stark contrast to the French and to Fehr himself. In fact, Smith had been in England throughout the autumn and had often communicated with the colleagues there. Greenhill’s position is likewise remarkable. In November 1914, he reacted to an apparent criticism of Klein by Smith, probably his signing the Aufruf: I think you are hard on Klein. As a patriot German he must stand up for his own country. The world looks up to German science and learning as our leaders; but it is a great defeat of civilisation that they are controlled by a military despotism. (RBML, box 20, Greenhill to Smith).

Unlike Fehr, Greenhill distinguished between scientists and the military complex. We see that Fehr held a minority position in the CC, backed only by the French member. In that 4 March letter to Fehr, Smith said that he had just offered to assist Klein with correspondence, but then he told Fehr that he would accept the presidency for the duration of the war: that we should do everything to recognize our indebtedness to Professor Klein, and should make no change excepting with a view to having him again our titular as well as actual leader as soon as the war closes. If, by taking nominal charge of the work, and acting under the direction of Professor Klein I can relieve him and assist in the progress of work, I shall be glad to do so. (RBML, box 16, Smith to Fehr).

While Klein had understood Smith’s answer as that he should continue as President, Greenhill had understood the contrary – probably due to a message by Fehr – and wrote to Smith on 27 March, 1915: I am glad you have accepted the Presidency of the Commission, as Fehr is still working hard, as best he can. (RBML, box 20, Greenhill to Smith).

Fehr had wavered several times between publishing and not publishing the questionnaire for the ongoing study on teacher training. Eventually, he followed Klein’s pleas and published it in the first half of 1915.31 In fact, despite his claim that many collaborators were mobilised, he told Smith to stay in active contact with most of the subcommittees regarding their work on the study (RBML, box 16, 7.12.1915 to Smith). Fehr informed Smith on 31 July 1915 that Klein had agreed to transfer him the presidency if they would judge it necessary for the continuation of the work.32 He added that this had no urgency and that he would submit this proposal to the CC in the autumn. Actually, he did not address the CC and thus Klein formally remained President all the time. From the end of 1915 on, the correspondence became extremely rare and almost all work stagnated. Even between Smith and Fehr, the correspondence stopped – only that between Smith and Greenhill continued, but it was on literary and  L’Enseignement Mathématique, 1915, vol. 17, pp. 61-65: Enquête sur la préparation théorique et pratique des professeurs de mathématiques de l’enseignement secondaire dans les divers pays. Questionnaire. 32  There is no proof for this claim in Klein’s correspondence. 31

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scientific subjects. There was no correspondence at all between Klein and Fehr between the end of 1915 and 1920! The last letter by Fehr to Smith was of 5 February, 1917. While other correspondents addressed concrete events of the War, Fehr again condemned German acts, in this case the unlimited submarine war: Expressing the hope that a parcel with offprints would reach him, he welcomed strong reactions by the US government.33 In fact, the United States entered the war later in 1917, too, and thus Smith no longer represented a neutral country.

1.8 The Dissolution Europe had experienced many wars, but never before had a war decisively interfered with cooperation between scientists of the states concerned. World War I was the first in which a ban against scientists of defeated countries was prepared and enacted. As Olli Lehto has impressively shown in his book on the history of IMU, it was French mathematicians who propagated the exclusion of German scientists from international cooperation and organisations. Gaston Darboux (1842–1917), permanent secretary of the French Academy of Sciences, inviting representatives of all Allied nations in 1916 to a meeting in Paris on international relations after the war, wrote: “Do you want, yes or no, to retake personal relations with your enemies? (Lehto 1998, p. 16).34 His letter expressed the clear conviction to regard fellow scientists from the Central Powers as enemies and to exclude them hence from cooperation. After Darboux’s death, Émile Picard (1856–1941) continued the same policy. It became official policy after two meetings: in London in October and in Paris in November 1918, where one prepared the constitution of the IRC (International Research Council), which would henceforth organise and coordinate international cooperation and associations. Institutions such as Academies were also to be excluded. The constitutive assembly of the IRC was held in Brussels from July 18–28, 1919. There, the ideological basis and practical implementation of the post-war international science policy were ratified. The formation of various scientific unions was prepared then, among them the IMU (Lehto 1998, p. 19). Fehr followed all these developments closely and with sympathy. He continued his one-sided, partial policy. He was the first after the war to contact members of the CC, but exclusively those from the victorious Allied countries. He first contacted Smith in a letter of 31 March, 1919. He told Smith that the Commission was bound by the decisions of London and Paris, that the French and Italian “delegations” had  This is not certain now when the Germans are extending their heinous barbaric acts to neutral ships. Your Government has responded appropriately. (RBML, box 16, Fehr to Smith) The original text is “Cela n’est pas certain, maintenant que les allemands étendent aux navires neutres leurs actes d’ignoble barbarie. Votre Gouvernement a répondu comme il convenait.” 34  The original text is: “Veut on, oui ou non, reprendre des relations personnelles avec nos ennemis?” 33

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asked him to proceed in this manner, that the Commission could not continue to function as originally constituted, and that the representatives of the Central Powers had to be excluded. Thus, Smith should now accept the presidency, and they should continue the last project, on teacher education, because of its key importance. Hence, they should not liquidate the Commission (RBML, box 16, Fehr to Smith). In his answer of 7 May, Smith said that—although he had not accepted the 1915 offer—“a change must now be made.” Still, he pleaded for a president having an “international reputation as a research scholar in mathematics” and being “at the same time specially interested in the teaching of the sciences.” Although nobody at the time showed those combined qualities, he would accept “the presidency for a time.” He asked that all members of the CC be consulted about this move, but was not explicit as to whether he accepted the exclusions (RBML, box 16, Smith to Fehr). Fehr did not reply until September 2, reporting that he had consulted the members of the CC, except Klein and Czuber, about the question of presidency and that he was missing only Greenhill’s answer. He mentioned that he had published in his journal the new conditions determining international scientific cooperation according to the Allied decisions. Thus, he expressly accepted the reduction of internationalism (EM, 21/1920, 59–60). On the other hand, he said that additional decisions had been made at a “recent” interallied conference in Brussels, which—although held in July—he claimed not yet to know. Actually, these decisions implied the dissolution of all organizations that had been constituted with the membership of the now-besieged countries. Adhering closely to all these decisions, Fehr saw no way to continue with IMUK and making Smith President (RBML, box 16, Fehr to Smith). It was therefore only in March 1920, in the wake of the preparations for the Strasbourg Congress of mathematicians, that Fehr again contacted Smith. Given that the Strasbourg Congress would represent only a part of those countries that had given the mandate, IMUK would have no relation to this Congress, and therefore no report had to be given there. Foreseeing a continuation of the exclusion policy, Fehr now urged liquidation of the Commission (RBML, box 16, Fehr to Smith, 23.3.20). It is not clear how strongly the Italian members supported the French and Fehr’s view, but Fehr’s next letter, dated 13 April, before receiving an answer from Smith, proves that the French mathematicians were the driving force and that Fehr wholeheartedly supported them. He informed Smith that he had just met with influential French mathematicians and that he and they had agreed that IMUK should be liquidated, which he proposed should be carried out by a committee consisting of Smith, Hadamard and himself. Only then did Fehr intend to submit the question to the other Commission members (RBML, box 16, Fehr to Smith). Smith saw no hurry in answering. He did so only on July 20, after having received word from Klein whom he had contacted in the interim. Smith expressed clear disagreement with the exclusion policy, in particular with regard to the Strasbourg Congress: I think the general feeling among scholars in Great Britain and America, so far as I have talked with them, is that we should as soon as possible make these Congresses genuinely international, whatever our present feelings may be.

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He also reported Klein’s response: Klein would no longer consider himself “the President of the I.M.U.K., but [...] Sir George Greenhill and I should act as Vice-­ Presidents and take what steps we think best to close up the work at Strasbourg” (RBML, box 16, Smith to Fehr). Smith had written to Klein on April 28, 1920 (his first letter in four years), to tell him that he saw “no chance of general cooperation in the work of our Commission for some time to come” so that its work should be stopped “at least for the present” with regard to its international dimension (NUG, fol. 60–61). In his answer of May 26, Klein lamented the destruction of the general association of scientific academies that he himself had strongly promoted. But he lamented even more that the joint work would thus remain unfinished. He explained that Smith’s proposal of a closing report in Strasbourg was not feasible, and asked Smith to contact Fehr in this regard, since he was sceptical of a direct communication with Fehr.35 Fehr himself had waited, actually, until 1 June, 1920, to contact Klein— more than a year and a half after the end of the war—to inform him of the inevitable liquidation of the Commission.36 He outlined four steps: • The secretary-general informs the members of the forthcoming liquidation and invites them to do the final work • Publishing the complete list of the reports given • Clearing all financial issues • Sending a closing circular to the members, including in particular the financial report37 Klein answered immediately on June 8, agreeing to the necessity of dissolving IMUK and to proceed according to the proposed four steps. He was concerned, however, by a remark in Smith’s April 28 letter to him—repeated again by Smith in the quoted letter to Fehr of July 20 —namely to give a report at the Strasbourg Congress (RBML, box 29, Klein to Smith). Klein pointed out that the Strasbourg

 “Ich wende mich nicht an Fehr direkt, weil er nach seinen elsässischen Beziehungen und als Westschweizer von vorneherein an die Entente gebunden erscheint, eine Korrespondenz meinerseits mit ihm über die vorliegende Frage also keinen Erfolg verspricht” (RBML, box 29, Klein to Smith). English translation: “I am not addressing Fehr directly because, due to his Alsatian relations and as a citizen of Western Switzerland, he appears to be bound to the Entente from the outset, so that a correspondence with him on the present question does not promise any success.” 36  The time seems to have come for the liquidation of the International Commission on Mathematical Education. This is the conclusion which one inevitably arrives at because of the conditions drawn up by the Interallied Conference of Academies […] for international scientific collaboration. Several delegations request it. (NUG, Fehr to Klein, 1.6.1920, fol. 102f.) The original text is: “Le moment semble venu de procéder à la liquidation de la Commission internationale de l’enseignement mathématique. Telle est la conclusion à laquelle on arrive fatalement en raison des conditions élaborées par la Conférence interalliée des Académies […] en vue de la collaboration scientifique internationale. Plusieurs délégations le demandent.” 37  This final report had in fact already been prepared by the German committee, which had published in 1917, a complete list of the German and of the foreign IMUK publications. 35

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Congress could, in no way, be a successor in interest (“Rechtsnachfolger”) of the former ICMs. Thus, no report “ex officio” was possible. In an analogous manner, Klein wrote to Greenhill (NUG, draft, fol. 92; 8.6. 1920). In this regard, there was no disagreement between the ex-President and his Secretary-General. Fehr affirmed in a June 29 letter that “our” committee had nothing to do with the Strasbourg Congress.38 Having received Klein’s agreement of 10 July, Fehr formulated a letter, predated July 5, to the entire Commission, informing them about the liquidation process (Fig. 1.10). Greenhill’s response to Klein on June 26 shows that he, too, disagreed with the French policy, not wanting to identify scientists with the policy of their states. His letter reveals, however, new dimensions and problems in international policy.

Fig. 1.10  Fehr informing the IMUK delegates of the liquidation of IMUK (excerpt; RBML, box 16, Fehr to Smith)

 I see that we are in complete agreement. Our Commission has nothing to do at the next Strasbourg Congress. (NUG, Fehr to Klein, fol. 104). The original text is “Je constate que nous sommes tout-à-fait d’accord. Notre Commission n’a rien à faire au prochain Congrès de Strasbourg.” 38

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Greenhill did not blame militarism or uncivilized behaviour of the Central Powers, but rather Jews: Fehr has prepared me for the dissolution of IMUK after his visit to Paris. I am sure D. E. Smith agrees with me in thinking we have been thrust aside in a very rude unceremonious manner, without a word of explanation or regret. A number of busy bodies are at work (Jews for the most part) trying to stir up strife, instead of helping the world to settle down peacefully. […] But considering how profitable the war has been to the Jews, and the profit they derive from the misfortunes of their fellow creatures, it is not surprising that the Jews are striving to keep the world at war. […] The object of all sensible men should be to resume harmonious relations, and carry on the comradeship of scientific work, with as little delay as possible.” (NUG, Greenhill to Klein, fol. 19–20).

Smith originally was not inclined to participate in Strasbourg, but eventually not only participated at a meeting in Brussels, preparing the foundation of IMU, but also participated in Strasbourg (22 to 30 September 1920) to try, as he explained to Klein, “in some slight way contribute to the more international point of view” (NUG, Smith to Klein, fol. 62–64; 20.7. 1920). In the end, three of the four members of the CC of Cambridge were present at Strasbourg: Fehr, Greenhill and Smith. They had agreed not to speak for the Commission.39 Greenhill was the only one to give a paper, contrasting the Fourier and the Bessel functions. It was classified as a history paper and therefore grouped in section IV, uniting philosophy, history and pedagogy. Fehr and Smith gave no papers; Greenhill and Smith were asked to preside one of the sessions of this section IV. Of its 11 contributions, only two concerned issues of teaching (Comptes-Rendus 1920). As the Proceedings of Strasbourg show, the Commission was not mentioned explicitly. IMUK no longer existed. There was no longer a mandate for the work. Except for a letter of protest from Emanuel Beke for the Hungarian Committee, there had been no resistance to the liquidation (NUG, Beke to Klein, fol. 23–24; 25.9.1920). Both Smith and Greenhill were deeply disappointed by the outcomes of the Strasbourg Congress. Greenhill was particularly harsh: We were obliged to come to Strasbourg, to attend the obsèques of IMUK. I fear our work passed unappreciated for the most part. And the new Commissions have made no sort of acknowledgement of the pioneering part we had taken. Reviewing the memory of the Congress, it strikes me as hardly worth the trouble of going so far. And the attendance was very small. Only three or four from England, one from Italy (Volterra), none from Spain, Sweden” (RBML, box 20, 3.11.20 to Smith).

 We do not have to speak officially on behalf of the Commission. (RBML box 16, Fehr to Smith, 23.8. 1920). The original text is: “nous n’avons pas à prendre la parole officiellement au nom de la Commission.” 39

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The only positive decision taken by the old CC had been to let the national network—that is, the national subcommission—survive. Klein pointed this out explicitly to Federigo Enriques, in a letter of 13 August, 1920.40 The liquidation process itself protracted a long time. First, the final list of publications had to wait until the report on mathematics teaching in Argentina, one of the new associated countries, was translated into French. That took until late 1921. Second, a number of national contributions to the budget of the International Commission were still pending and had to be admonished. Eventually, on 6 September, 1921, Fehr sent all members a last circular, the final report as published in the journal L’Enseignement Mathématique (vol. 21, fasc. no. V, 305 pages), and reminded them to pay any missing contributions. This was the last action of the first IMUK.

1.9 Re-establishment and Stagnation: The Interwar Years After the Strasbourg Congress, Greenhill had remarked sarcastically: “I fear Fehr is saying, like Othello, his occupation is gone” (RBML, box 20, Greenhill to Smith; 3.11. 20). In fact, although Fehr had been the active promoter of liquidation, the letterheads, which he used from 1922 on show a quite different behaviour: these were letterheads of CIEM with him as secretary-general and the only officer (Fig. 1.11)! At the next Congress, in September 1924, in Toronto (Canada), again banning mathematicians from the defeated countries, there were 16 communications in the now sixth section, uniting again philosophy, history and “didactics” – four of them related to teaching. One of them was a paper by Fehr: “L’Université et la preparation des professeurs de mathématiques.”

Fig. 1.11  Fehr’s letterhead after the dissolution

 Fortunately, I was able to reach a complete agreement with Fehr on the position of the IMUK. A final report will be made, although it will not be presented in Strasbourg. Our official work has thus come to an end, but in spite of that it will be carried on unofficially). (NUG, Klein to Enriques, fol. 67). The original text is: “Glücklicherweise habe ich mich mit Fehr über die Haltung der IMUK völlig verständigen können. Es wird ein abschliessender Bericht gemacht werden, der aber nicht in Strassburg vorgelegt wird. Unsere offizielle Arbeit ist damit zu Ende, wird aber trotzdem inoffiziell fortgeführt werden.” 40

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In 1928, the organisers of the Bologna ICM rejected the exclusion and boycott policy of the IMU (Lehto, 1998, 44), making it the first truly international Congress after the war. In the section VI, on teaching, Fehr presented himself as “Segretario generale” of the International Commission and gave, in its name, the final report of its activities since 1908. He did not mention the liquidation; rather, his lengthy report gave the impression of an uninterrupted development from 1908 to 1928. For instance, he spoke of “recent publications and report in preparation” (Atti 1928, I, p. 111). He declared: “If the Congress wishes it, we are ready” to continue the study of teacher education. In the ensuing discussion, however, one demanded “the reconstitution” of the Commission. The section prepared a vote, with three points for the Congress: a) thanking the various governments and institutions, which had supported IMUK, b) extend (“proroger”) the CC, now with Smith as President, Castelnuovo and Hadamard41 as Vice-Presidents, and Fehr as Secretary-General, and to co-opt a fifth member, and c) the CC should seek the support of the national governments.42 On October 2, 1928, Fehr informed Smith, who had not been able to attend, of the results and proposed that Lietzmann be co-opted as the fifth member. Smith agreed, as did Lietzmann. Smith wrote Fehr several times expressing concern that he heard nothing concrete about the work to be done. In mid-February, Fehr wrote back, saying that things were difficult. Old members had to be replaced, and delegates from newly invited countries had to be found. But his only concrete action had been to ask Loria who was still alive to redo his studies on teacher education, begun in 1914, and to join with two other experts, Lietzmann and Raymond Archibald (1875–1955) (RBML; box 16, Fehr to Smith and Smith to Fehr).  See his biography in this volume.  (1) The International Congress of Mathematicians expresses its thanks to the Governments, institutions and persons who have agreed to support the International Commission on Mathematics Teaching, and pays homage to the memory of deceased members 2) It decides to extend the powers of the Central Committee currently composed of MM. Dav. Eug. Smith (New York), president; Castelnuovo (Roma) and J. Hadamard (Paris), vice-presidents; H. Fehr (Geneva), Secretary-General, and which should be completed by the adjunction of a fifth member, appointed by co-option 3) It requests the Central Committee to complete the Commission so that all the nations participating in the Congress should be represented and to ensure the cooperation of their government. (Atti 1928, I, p. 113) The original text is: “1) Le Congrès international des mathématiciens adresse ses remerciements aux Gouvernements, aux institutions et aux personnes qui ont accordé leur aide à la Commission Internationale de l’enseignement mathématique, et rend hommage à la mémoire des membres décédés 2) il décide de proroger les pouvoirs du Comité central composé actuellement de MM. Dav. Eug. Smith (New York), président; Castelnuovo (Roma) et J. Hadamard (Paris), vice-présidents; H.  Fehr (Genève), Secrétaire-général, et qui devra être complété par l’adjonction d’un cinquième membre, désigné par cooptation 3) il prie le Comité central de compléter la Commission de manière que toutes les nations participant au Congrès y soient représentées et de s’assurer de la coopération de leur gouvernement” Actually, the six volumes of Proceedings of this ICM do not record a general session discussing proposals from the sections, but one can assume this proposal as having been accepted. 41 42

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At the end of May 1929, Fehr and Smith finally met to discuss how to reconstitute the Commission. The structure they adopted followed largely that of the original, with some simplifications; for example, to have just one delegate per member country.43 Financial contributions by each country to pay the expenses of the CC were also stipulated, without, however, charging the respective governments with those payments. Even the progress of the international study on teacher education was quite slow. Apparently, the problem was that the old members and activists were leaving the scene and that very few new persons were developing interest and initiative. It is characteristic that almost no new ideas and projects were proposed, and that the only project pursued was the last one of the old agenda. Even between Smith and Lietzmann, for instance, there were only three letters during the four years of common membership in the CC from 1928 to 1932, however, not really substantial ones. Likewise, the national subcommissions functioned at a quite feeble level. At the next ICM, in Zürich, in September 1932, the session of section VIII on September 7 was dedicated entirely to IMUK, entitled Sitzung der Internationalen Mathematischen Unterrichtskommission and presided by Smith (Verhandlungen 1932, I, 51). Fehr gave there a “rapport sommaire” about the last four years of IMUK activities. The main topic was the report on teacher training. Gino Loria, who had prepared the international report on teacher training since 1914, was now finally able to deliver this report, based on the additional new reports. Additionally to Loria’s report, Georg Hamel (1877–1954), President of the Mathematische Reichsverband, a union of various mathematical associations, took the initiative to report about the German situation of teacher training, without using the material collected earlier under Klein’s presidency of the German subcommission and without coordinating with the other members of the subcommission (Verhandlungen 1932, II, pp. 363–367).44 In this session, Smith declared not being able to continue as President, due to his age and health conditions. The Zürich ICM adopted a motion proposed by its 8th section to prolong the activities of CIEM, and approved officers of the CC: Hadamard as President, and Poul Heegaard (1871–1948) from Oslo, Lietzmann and Gaetano Scorza (1876–1939) from Naples as Vice-Presidents, and Fehr as Secretary-General. It even defined a new task: to elaborate a report on the current tendencies of the development of mathematics teaching in the different countries (ibid., 364). This corresponded to Smith’s intentions: He had even proposed, in a letter to the CC preparing the Zürich Congress, “that the work of presenting a world view of the status of mathematics [education] should continue.” Smith was also aware of the need to have younger members and had argued for a young person as president (RBML, box 16). Despite his obvious efforts to renew programmes and personnel, the decisions of the Congress were rather conservative.

 In the Smith papers, there is a handwritten draft by Smith developing a much more elaborated program of work. (RBML, box 16). 44  Loria’s general report and the various national reports were published separately by Fehr. 43

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At the next ICM, at Oslo in 1936, the IMU (International Union of Mathematicians), founded in 1920 with the same exclusion policy, was dissolved. Within the section VIII (Pedagogy), there was again a session for IMUK-CIEM. The agenda of IMUK for this Congress had been that the national delegates should present reports on the state of mathematics education in their respective countries. Eleven such reports were presented  – the only non-European one being from Japan. For Germany, Lietzmann had delivered that report (Lietzmann 1937), which was well received (Hollings et al. 2020, pp. 232–234). In response to a proposition by Fehr “to renew the mandate” of the Commission— hence, still regarded as established only temporarily—the General Assembly agreed to continue the Commission’s activities, without defining any concrete tasks or calling other persons into the CC and leaving it to the CC to define the activities: The Congress requests the International Commission on the Teaching of Mathematics to continue its work, prosecuting such investigations as shall be determined by the Central Committee (Comptes-Rendus 1936, p. 48).

The IMUK/CIEM officers were to continue in their functions. The closing meeting of the IMUK/CIEM session was called “séance administrative.” There, besides distinguishing various long-standing delegates as “honorary Commission members,”45 there were only exchanges of opinions about possible tasks to be prepared for the next Congress (Comptes Rendus 1936, Vol. II, p. 289).

1.10 Concluding Remarks Due apparently to the lack of concrete tasks and of younger devoted activists, there were no real activities of the CC or the Commission as a whole until 1939, and, thereafter, evidently no common activities were possible. Even in the journal L’Enseignement Mathématique, there were no activities reported after 1936. The activities of the national subcommissions during the interwar period was likewise without the former engagement and initiatives.46 A new epoch confronting new challenges was necessary to revive the idea of an effective international network. Klein died in 1925, Greenhill in 1927 and Smith in 1944. Fehr was the last of the founding activists: In 1952, he was the one to pass the torch of IMUK/CIEM/ICTM to the then new ICMI.

 The title of honorary member of the Commission was conferred to Emanuel Beke (Budapest), Charles Bioche (Paris), Guido Castelnuovo (Rome), Samuel Dickstein (Warsaw), Federigo Enriques (Rome), Farid Boulad Bey (Cairo), Gino Loria (Genoa), Mihailo Petrovitch (Belgrade) and Wilhelm Wirtinger (Vienna), see (Comptes rendus du Congrès International des Mathématiciens, Oslo 1936, Vol. II, p. 289). 46  See the chapters on the subcommissions for France, Germany, Great Britain, Italy, Russia, and the United States in: Karp 2019. 45

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Appendix Members of the IMUK/CIEM (L’Enseignement Mathématique, 1914, 16, p. 166) Allemagne: Felix Klein (Univ.); Paul Staeckel (Univ.); Albrecht Thaer (Sec. School) Australie. Horatio Scott Carslaw (Univ.) Autriche: Emanuel Czuber (Univ.); Wilhelm Wirtinger (Univ.); Rudolf Suppantschitsch (Univ.) Belgique: Joseph Neuberg (Univ.) Brésil: Raja Gabaglia (Sec. School) Bulgarie: Antonín Václav Šourek (Univ.) Colonie du Cap: Sydney Samuel Hough (Observatory) Danemark: Poul Heegaard (Univ.) Egypte: Farid Boulad, engineer Espagne: Luis-Octavio de Toledo (Univ.) Etats-Unis: David Eugene Smith (Teachers College); William Osgood (Univ.); Jacob William Albert Young (Univ.) France: Jacques Hadamard (Polytechn., Collège de France); Maurice d’Ocagne (Polytechn.); Charles Bioche (Sec. School) Grèce: Cyparissos Stéphanos (Univ.) Hollande: Jabob Cardinaal (Polytechn.) Îles Britanniques: Sir George Greenhill (Artillery college); Ernest William Hobson (Sec. School); Charles Godfrey (Naval College) Italie: Guido Castelnuovo (Univ.); Federigo Enriques (Univ.); Gaetano Scorza (Univ.) Japon: Rikitaro Fujisawa (Univ.) Mexique: Valentin Gama (Observatory) Norvège: Olaf Alfsen (Sec. School) Portugal: Gomes Teixeira (Polytechn.) Roumanie: Gheorghe Tzitzeica (Univ.) Russie: Nikolay Sonin (Univ.); Boris Kojalovic (Polytechn.); Constantin Possé (Univ.) Serbie: Michel Petrovitch (Univ.) Suède: Edvard Göransson (Sec. School) Suisse: Henri Fehr (Univ.); Carl Friedrich Geiser (Polytechn.); Johann Heinrich Graf (Univ.)

References Archival sources Rare Books and Manuscript Library, Butler Library New York: D.E. Smith Professional Collection. Correspondence with F. Klein, H. Fehr, W. Lietzmann, G. Greenhill. [RBML] Niedersächsische Staats- und Universitätsbibliothek Göttingen, Handschriftenabteilung: Cod. Ms. Felix Klein, Nr. 51. [NUG]

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Printed sources Atti del IV Congresso internazionale dei matematici. 1909a. Vol. I.  Roma: Tip. Accademia dei Lincei. ———. 1909b. Vol. III. Roma: Tip. Accademia dei Lincei. Proceedings of the fifth International Congress of Mathematicians. 1913. Vol. 1. Cambridge: University Press. Comptes Rendus du Congrès International des Mathématiciens. 1921. Toulouse: E. Privat. Atti del Congresso Internazionale dei Matematici. 1929. Vol. I. Bologna: Zanichelli. Verhandlungen des Internationalen Mathematiker-Kongresses. 1932a. Vol. I. Zürich: Orell-Füssli. ———. 1932b. Vol. II. Zürich: Orell-Füssli. Comptes Rendus du Congrès International des Mathématiciens. 1936. Vol. II. Oslo: Broeggers. Board of Education, ed. 1912. The teaching of mathematics in the United Kingdom: being a series of papers prepared for the International Commission on the Teaching of Mathematics. London: HMS Office. Burscheid, Hans-Joachim. 1984. Eine Schulenbildung unter den Gymnasialdidaktikern des ausgehenden 19. Jahrhunderts, Zentralblatt für Didaktik der Mathematik 16: 191–195. Donoghue, Eileen Frances. 1987. The origins of a professional mathematics education program at Teachers College, PhD Diss. New York: Columbia University, Teachers College. D’Enfert, Renaud, and Caroline Ehrhardt. 2019. The French subcommission of the International Commission on the Teaching of Mathematics (1908-1914): Mathematicians Committed to the Renewal of School Mathematics. In National Subcommissions of ICMI and their Role in the Reform of Mathematics Education, ed. Alexander Karp, 35–64. Cham: Springer. Furinghetti, Fulvia. 2003. In Mathematical instruction in an international perspective: The contribution of the journal L’Enseignement Mathématique. In One Hundred Years of L’Enseignement Mathématique. Moments of Mathematics Education in the Twentieth Century, ed. Daniel Coray, Fulvia Furinghetti, Hélène Gispert, Bernard R. Hodgson, and Gert Schubring, 19–46. Geneva: L’Enseignement Mathématique. Giacardi, Livia. 2019. The Italian Subcommission of the International Commission on the Teaching of Mathematics (1908-1920): Organizational and Scientific Contributions. In National Subcommissions of ICMI and their Role in the Reform of Mathematics Education, ed. Alexander Karp, 119–147. Cham: Springer. Gutzmer, August. 1917. Die Tätigkeit des Deutschen Unterausschusses der Internationalen Mathematischen Unterrichtskommission 1908-1916. Bericht anlässlich der Fertigstellung der ‚Abhandlungen’ (1916). In Lietzmann (1917), Band I, pp. 329-357. Hollings, Christopher, Reinhard Siegmund-Schultze, and Henrik Kragh Sørensen. 2020. Meeting under the integral sign?: the Oslo Congress of Mathematicians on the eve of the Second World War. Providence (Rhode Island): American Mathematical Society. Karp, Alexander. 2019. National Subcommissions of ICMI and their Role in the Reform of Mathematics Education. Cham: Springer. ———. 2019a. The Russian National subcommission of ICMI and the Mathematics Education Reform Movement. In National Subcommissions of ICMI and their Role in the Reform of Mathematics Education, ed. Alexander Karp, 149–191. Cham: Springer. ———. 2019b. The American National subcommission of ICMI. In National Subcommissions of ICMI and their Role in the Reform of Mathematics Education, ed. Alexander Karp, 193–234. Cham: Springer. Klein, Felix, George Greenhill, and Henri Fehr. 1908. Commission Internationale de L’Enseignement Mathématique. Rapport sur l’Organisation de la Commission et le Plan Général de ses Travaux. L’Enseignement Mathématique 10: 445–458. Lehto, Olli. 1998. Mathematics without borders: a history of the International Mathematical Union. New York: Springer. Lietzmann, Walther. 1908. Der IV. Internationale Mathematikerkongreß in Rom, Mathematisch-­ Naturwissenschaftliche. Blätter 5 (6): 81–84.

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———. 1908a. Die Grundlagen der Geometrie im Unterricht (mit besonderer Berücksichtigung der Schulen Italiens). Zeitschrift für Mathematischen und Naturwissenschaftlichen Unterricht 39: 177–191. ———, ed. 1917. Berichte und Mitteilungen, veranlasst durch die Internationale Mathematische Unterrichtskommission. In zwei Folgen. Leipzig/Berlin: Teubner. ———. 1937. Die gegenwärtigen Bestrebungen im mathematischen Unterricht der höheren Schulen Deutschlands. Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 68: 19–22. ———. 1960. Aus meinen Lebenserinnerungen. Im Auftrag von Walter u. Käthe Lietzmann hrsg. von Kuno Fladt. Göttingen: Vandenhoeck & Ruprecht. Scarpis, Umberto. 1911. L’insegnamento della matematica nelle scuole classiche. I. I successivi programmi dal 1867 al 1910. Bollettino della Mathesis. Supplemento. Atti della Sottocommissione italiana per l’insegnamento matematico 3: 25–33. Schubring, Gert. 2003. In L’Enseignement Mathématique and the First International Commission (IMUK): The emergence of international communication and cooperation. In One Hundred Years of L’Enseignement Mathématique. Moments of Mathematics Education in the Twentieth Century, ed. Daniel Coray, Fulvia Furinghetti, Hélène Gispert, Bernard R. Hodgson, and Gert Schubring, 47–65. Geneva: L’Enseignement Mathématique. ———. 2008. The origins and the early history of ICMI. International Journal for the History of Mathematics Education 3 (2): 3–33. ———. 2019. The German IMUK subcommission. In National Subcommissions of ICMI and their Role in the Reform of Mathematics Education, ed. Alexander Karp, 65–91. Cham: Springer. Tobies, Renate. 2019. Felix Klein - Visionen für Mathematik, Anwendungen und Unterricht. Berlin: Springer-Spektrum. von Ungern-Sternberg, Jürgen, and Wolfgang von Ungern-Sternberg. 1996. Der Aufruf “An die Kulturwelt!”: das Manifest der 93 und die Anfänge der Kriegspropaganda im Ersten Weltkrieg. Stuttgart: Steiner.

Chapter 2

ICMI in the 1950s and 1960s: Reconstruction, Settlement, and “Revisiting Mathematics Education” Fulvia Furinghetti and Livia Giacardi

2.1 Introduction This chapter deals with the life of the International Commission on Mathematical Instruction (ICMI) in the period from the International Congress of Mathematicians (ICM) in Cambridge (USA) in 1950, the first after World War II, to the end of the mandate of Hans Freudenthal as President of ICMI in 1970. These decades could rightly be called “the roaring years” in mathematics education due to concomitant events in politics, technology, research and society. The Cold War and the consequent atmosphere of tension between countries stimulated interest in science, including mathematics, and fostered the birth of specific projects aimed at research and teaching. The plans to help nations recover from the disasters of war supported national and international initiatives also in the field of education. Mathematical research developed according to new paradigms and computers opened new research perspectives. In society, the school became “for all,” that is, an institution that involved all citizens. The very way of looking at the world changed. Phenomena such as the progressive decolonization and the independence of some countries led to consider the problems of education—obviously also the mathematical one— in those countries. This is the context in which the Commission Internationale de l’Enseignement Mathématique (CIEM), founded in 1908, was revived. The primary objectives of the new Commission, now known as International Commission on Mathematical F. Furinghetti (*) University of Genoa, Genoa, Italy e-mail: [email protected] L. Giacardi (*) University of Turin, Turin, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_2

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Instruction (ICMI), were: to acquire its own identity as an agency of ideas and projects on mathematical education and to adapt its mission to the context changed from that of the early years after its foundation as CIEM. This chapter takes these challenges as a track to illustrate what happened in the period 1950–1970 and how the transition to a new standing—both administrative and cultural—was carried out. The old Commission was born and developed within the community of mathematicians and depended on the four-year International Congresses of Mathematicians (ICMs) for its mandate, but otherwise it was free to choose its own members and decide which projects to carry out. The new Commission ICMI was a permanent subcommission of the International Mathematical Union (IMU) and so depended on it both for the choice of the members of the Executive Committee (hereafter EC) and the funding. The lack of precise Terms of Reference to regulate the relationships between the two bodies IMU and ICMI caused friction that arose mainly from the ICMI’s desire to gain greater independence from IMU. Moreover, the old agenda of CIEM, which placed the emphasis above all on questions concerning curricula and organizational aspects of mathematics education in the various countries, was now outdated. The new needs of society, the new trends in mathematical research and in technology required a renewal in the subjects to be studied, in tools and in methods. The presidents of ICMI who succeeded each other in this phase—Heinrich Behnke, Marshall Stone, André Lichnerowicz and Hans Freudenthal—worked to different extents on these fronts, that is, on clarifying and modifying the relationship between ICMI and the community of mathematicians; on giving ICMI a cultural identity; on widening the range of ICMI action by involving the peripheries, including South East Asia and South America, and developing countries. In this chapter, we analyse the life of ICMI in the years 1950–1970 through the action of these presidents according to the following lines of investigation: relationships between IMU and ICMI; the emergence of mathematics education as an academic discipline; and the change in the ICMI’s objectives. In this respect, the rich unpublished correspondence1 kept in various archives adds new and illuminating details to the existing literature, see (Furinghetti and Giacardi 2010; Furinghetti et al. 2008, 2020; Howson 1984; Lehto 1998).

2.2 The Rebirth of IMU and the International Commission on the Teaching of Mathematics At the end of World War II, the community of mathematicians had to face again—as had happened at the end of World War I—the dramatic experience of emerging from a global conflict and look to the future. The first International Congresses of Mathematicians organized in the aftermath of the two world wars mark the starting point in this process. The addresses delivered during these two Congresses  A selection of these letters can be found in Chap. 4 by Livia Giacardi in this volume.

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highlighted how different was the approach to the problem in the two cases. In 1920, at ICM in Strasbourg, Émile Picard,2 President of the Congress, claimed3: As for certain relations, which have been broken by the tragedy of recent years, our successors will see if a sufficiently long time and a sincere repentance will allow to resume them one day, and if those who have excluded themselves from the concert of civilized nations are worthy to enter again. As for us, too close to events, we still share the beautiful claim made during the war by Cardinal Mercier, that to forgive certain crimes is to be their accomplice. (Proceedings ICM-1920 1921, p. XXXIII)

This passage shows that the horrors of the war were not forgotten. As a consequence of these feelings, the former Central Powers had been excluded from membership of the newly established IMU. Instead, in 1950, at the ICM in Cambridge (USA), the mathematicians tried to overcome the past and adopted a constructive approach. We read in the “Secretary report” (Proceedings ICM-1950 1952, Vol. 1, p. 145) that: The Congress was undoubtedly the largest gathering of persons ever assembled in the history of the world for the discussion of mathematical research. However, the real measure of its success lies not in the large number of persons present, but in the excellence of its scientific program and in the contributions which it made to the cause of closer cooperation among scientists and to the cause of international good will [our italics].

As Garrett Birkhoff noted, “The organizing of a successful International Congress at such a time of political tensions, and after a gap of fourteen years, has had its anxious moments” (Proceedings ICM-1950 1952, Vol. 1, p.  123). In this regard Oswald Veblen, President of the Congress, recalled: We are holding the Congress in the shadow of another crisis, perhaps even more menacing than that of 1940, but one which at least does allow the attendance of representatives from a large part of the mathematical world. It is true that many of our most valued colleagues have been kept away by political obstacles and that it has taken valiant efforts by the Organizing Committee to make it possible for others to come. Nevertheless, we who are gathered here do represent a very large part of the mathematical world. (Proceedings ICM-1950 1952, Vol. 1, p. 124)

These words allude to the difficult scenario of this post-war period due the Cold War, the new geographical and political order in the new polarized world. Other phenomena were affecting the world relationship, such as the end of colonialism and the emerging needs of the developing countries. Indeed, the mathematicians of  Émile Picard, the president of the International Research Council, absolutely disagreed with the reintegration of the former Central Powers, in particularly, Germany, in international scientific organizations and strongly boycotted the efforts by Salvatore Pincherle, the President of IMU and of the Unione Matematica Italiana to organize a truly international ICM in Bologna in 1928, see (Giacardi and Tazzioli 2021). 3  The original text is: “Quant à certaines relations, qui ont été rompues par la tragédie de ces dernières années, nos successeurs verront si un temps suffisamment long et un repentir sincère pourront permettre de les reprendre un jour, et si ceux qui se sont exclus du concert des nations civilisées sont dignes d’y rentrer. Pour nous, trop proches des évènements, nous laissons encore nôtre la belle parole prononcée pendant la guerre par le cardinal Mercier, que, pardonner à certains crimes, c’est s’en faire le complice”. In this chapter all the translations are by the authors. 2

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the Soviet Union and its satellite states could not participate, as emerges from the secretary’s report (Proceedings-1950 1952, Vol. 1, p. 122): Mathematicians from behind the Iron Curtain were uniformly prevented from attending the Congress by their own governments which generally refused to issue passports to them for the trip to the Congress. Their non-attendance was not due to any action of the United States Government.

The politicians at the head of the different nations have tried not to make the same mistakes of the past, and brought back to the fore (albeit not always realizing them) important values for true internationalization such as cooperation and collaboration. It is no coincidence that bodies inspired by these values, for example UNESCO (United Nations Educational Scientific and Cultural Organisation) and OEEC (Organisation for European Economic Co-operation, later on OECD, Organisation for Economic Co-operation and Development), were set up immediately at the end of the war, respectively in 1945 and in 1948. The political world organization United Nations (UN) was founded in 1945. Though its primary mandate was peacekeeping, the division between the USA and USSR often paralyzed its action. In this spirit of internationalization, the mathematicians tried to reconstitute the International Mathematical Union (IMU), which has been founded in 1920 and dissolved in 1932. After the failure of the attempts to re-establish the Union during the 1930s the new IMU was officially in existence as of 10 September 1951 (Lehto 1998, p. 86), but the activities of the new Union were only inaugurated by the General Assembly held in Rome (6–8 March 1952). The gestation was long and complex. Lehto (1998, pp. 74–88) carefully describes the steps leading to the foundation of the new IMU, in which Marshall Stone played a major role. Stone was not only a firstrate mathematician, but he was one of the more active promoters of the transformation of the American mathematical research community from a national to an international community, creating a new scientific centre alongside the European one. As President of the American Mathematical Society (AMS, 1943–1944), chairman of the Department of Mathematics at the University of Chicago from 1946, a member of the Emergency Committee of the AMS, whose purpose was to organize the ICM in Cambridge in 1950, he worked to enhance the scientific level of the mathematical community, to encourage the contacts with the Latin American mathematicians and in general to foster a broader mathematical cooperation. It is therefore understandable why he was very active in the foundation of the new IMU on a truly international basis without exclusions due to political reasons. To this end he made numerous trips around the world with the dual purpose of acquainting himself about what was being done in mathematics internationally, but also of communicating information and exchanging ideas on the reconstitution of IMU; see (Browder 1989; Parshall 2009).4 In 1952 he was elected President of IMU. Stone showed that other lessons, beside the need for cooperation and collaboration, had come from the war, for example, the relevance of science to “the high art of guiding human affairs at the level of complexity represented by the elaborately  See the biographical portraits of the members of the Executive Committee mentioned in this chapter in Part III of this volume. 4

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organized modern state,” as well as to “the contributions made by scientists in the various warring nations to the development of military material of extraordinary variety and effectiveness” (Stone 1947, p. 507). He also felt the importance of mathematics education, as shown by the following passage of his report on the first General Assembly of IMU held in Rome from 6 to 8 March 1952: The problem of determining the place of mathematics [in society] cannot be divorced from technical considerations concerning teaching methods. If we judge by the results, we must find it difficult to escape from the conclusion that our attempts to teach mathematics as part of a program of mass education have so far been, to put it bluntly, a colossal failure, traceable to our ignorance and complacency in respect to the art of teaching. (Stone 1952, p. 1974)

In this report Stone also claimed that one of the most important tasks of IMU was to revive the International Commission on the Teaching of Mathematics founded in 1908. In fact, during the ICM in Oslo (1936) the General Assembly had agreed that the Commission continued its activities “as shall be determined by the Central Committee” (Proceedings ICM-1936 1937, p. 48), but it did not define any concrete tasks.5 This mandate had not been renewed in the successive ICM of 1950, but the reconstitution of the Commission was requested on several occasions. During the ICM of 1950 William Betz had presented a communication (Betz 1952) where he expressed the hope for the recommencement of the work begun in 1908 by the International Commission on the Teaching of Mathematics. Later Henri Fehr, Secretary General of the Commission and Erich Kamke,6 President of the German Mathematical Society, independently asked for its reconstruction under the auspices of IMU. On 20 February 1952 Fehr sent the Interim Committee of IMU a letter, in which, in accordance with the decisions that emerged at a meeting in July 1951 in Paris attended by Jacques Hadamard, President of the Commission, he presented the following proposals to be submitted to IMU General Assembly, which was to meet in Rome: 1. The Assembly of Delegates of the Union declares itself in favour of the attachment of the International Commission for Mathematical Education to the International Mathematical Union. 2. It takes note of the collective resignation of the members of the said Commission. 3. It appoints a committee of 3 members in charge of reconstituting the Commission by calling on new forces. 4. The delegates will be appointed by the National Committees of the Union (Two delegates per country: one for secondary education, the other for higher education.) It goes without saying that I remain at the disposal of the new Committee during the transitional period. Its first meeting would be held in Geneva, where all documents concern-

 On the difficult period for international cooperation following World War I, see (Schubring 2008), Chap. 1 by Gert Schubring in this volume and (Gispert 2021). 6  See B. Jessen to H. Fehr, Copenhagen 25.2.1952, ICMI Archives (hereafter IA) 14A 1952–1954. For the letters and archival documents cited in this chapter, see Chap. 4 by Giacardi in this volume. 5

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F. Furinghetti and L. Giacardi ing the Commission are centralized.7 (H. Fehr to the Interim Committee of the Mathematical Union, Geneva, 20 February 1952, IA 14A 1952–1954).

Actually, a few days later Fehr wrote to Børge Jessen, secretary of the Interim Executive Committee of IMU, that after almost 44 years of service in the ICMI he would gladly leave his office to a successor. (H. Fehr to B. Jessen, Geneva, 1 March 1952, IA 14A 1952–1954). During the General Assembly of IMU in Rome the proposals were accepted. Enrico Bompiani, Secretary of the new IMU wrote: The interim Committee presented a letter from Professor Henri Fehr, General Secretary of the Commission Internationale de l’Enseignement Mathématique, suggesting that the Commission should be continued by the Union and offering the resignation of the present Commission. The Assembly agreed that the Commission should be attached to the Union and accepted the resignation of its present members, expressing hearty thanks for the important work that the commission has accomplished. Professors Behnke, A.  Châtelet, R.  L. Jeffery and Kurepa were appointed members of the Commission. The Assembly accepted with thanks an offer from Professor Fehr to place himself at the disposal of the new commission.8 (E. Bompiani to H. Fehr, Rome, 24 March 1952, IA 14A 1952–1954)

In continuation with the rules established at the time of its foundation the Commission kept English, French, German, and Italian as official languages and L’Enseignement Mathématique as the official organ of the Commission (Bompiani 1953, pp.  6–7). The name “International Mathematical Instruction Commission” (IMIC)9 and other denominations such as “International Committee on Mathematical Instruction” were introduced, even if for some time the old names were used as well. The use of the word “instruction” instead of “teaching” was proposed by Heinrich Behnke.10 Starting in 1954, International Commission on Mathematical Instruction (ICMI) gradually became the denomination internationally used.11 The installation of the new Commission experienced some initial delays. By the end of September, the officers had not yet been appointed. For this reason, Bompiani, IMU Secretary, wrote to Behnke and Kurepa that he was “very anxious to have the

 The original text is: “1. L’Assemblée des délégués de l’Union se déclare favorable au rattachement de la Commission internationale de l’enseignement mathématique à l’Union mathématique internationale. 2. Elle prend acte de la démission collective des membres de ladite Commission. 3. Elle nomme un comité de 3 membres chargé de reconstituer la Commission en faisant appel à des forces nouvelles. 4. Les délégués seront désignés par les comités nationaux de l’Union. (Deux délégués par pays: un pour l’enseignement secondaire, l’autre pour l’enseignement supérieur.) Il va sans dire que je reste à la disposition du nouveau Comité pendant la période transitoire. Sa première réunion se tiendrait à Genève, où se trouvent centralisés tous les documents concernant la Commission”. 8  See also L’Enseignement Mathématique (hereafter EM) 39, 1942–1950, p. 162. 9  See EM 40, 1951–1954, p. 80. 10  See H. Behnke to E. Bompiani, Münster (Westf.), 18 November 1952: “Ich bitte, auf den Namen des Committees zu achten nicht “teaching” sondern “instruction” (IA 14A 1952–1954). 11  See Internationale Mathematische Nachrichten (hereafter IMN, 35–36, 1954: 9). 7

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work of the Commissions started” and urged them for this purpose to appoint secretary and president as soon as possible (E.  Bompiani to H.  Behnke and D. Kurepa, [Rome], 22 September 1952, IA 14B 1952–1954). The EC of the new Commission was constituted during the meeting in Geneva on 20–21 October 1952 as follows: President Albert Châtelet,12 Vice-President Đuro Kurepa, Secretary Heinrich Behnke, Honorary President Henri Fehr.13 In his autobiography, Behnke explains why Châtelet was chosen: “So I proposed Professor Chatelet as President. He was the oldest and most dignified among us, and under no circumstances did I want national prestige to enter into the elections.”14 (Behnke 1978, p.  266) The French mathematician was seriously interested in educational issues. From July 1945 to August 1946, he was director of the Mouvements de jeunesse et de l’éducation populaire at the Ministère de l’éducation nationale and became vice-president of the Commission de la recherche scientifique et technique, responsible for designing a modernization plan for theoretical and applied research. Moreover, he was already involved in the activities of the International Commission on the Teaching of Mathematics: in fact, in 1929 he had been the rapporteur on the changes in the teaching of mathematics after 1910 in France (EM 28, 1929: 6–13). During the meeting in Geneva, Kurepa asked to elect another vice-president and proposed Saunders Mac Lane, whose name had been suggested by Stone (Séance de la Commission Internationale de l’Enseignement Mathématique, Genève, le 20 octobre 1952, IA 14B 1952–1954). In order to form the Commission, it was decided to ask the committee of each country adhering to IMU to appoint a representative to ICMI who would be responsible for setting up in his country a national mathematics subcommission comprising representatives of the various degrees of teaching to work with the Commission (ibidem). This decision gave rise to the first friction with IMU. Concerning this decision, Stone, the President of IMU, wrote to Châtelet: It is my understanding that the Commission has proposed an arrangement whereby it will seek the adherence of several nations and set up special national committees in the adhering nations to work with the Commission. I believe that activity of this kind is inappropriate for a Commission of the Union and that it would lead to intolerable confusion as to the relations between the Union, the Commission, and the nations adhering to one or the other. My own immediate suggestion as to the proposed way of handing the relations between the Commission and the national bodies interested in supporting it would be to urge all interested nations to adhere to the Union and to arrange for the appointment of suitable persons to the National Committees for Mathematics which have to be set up as part of the procedure of adhering to the Union. The Commission could then arrange for direct contacts with these National Committees by coopting as members or as liaison agents, appropriate mem-

 See the biographical portrait of Châtelet by Catherine Goldstein and Sébastien Gauthier in this volume. 13  See EM 39, 1942–1950, p. 162. 14  The original text is: “So schlug ich nun Professor Châtelet als Präsidenten vor. Er war unter uns der älteste und würdigste. Und unter keinen Umständen wollte ich in die Wahlen nationales Prestige einbringen”. 12

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F. Furinghetti and L. Giacardi bers of the National Committee. (M. Stone to A. Châtelet, Chicago, 3 November 1952, IA 14A, 1952–1954) [our italics]

Stone also participated in the meeting of ICMI, which was held in Paris on 21 February 1953, and Châtelet on that occasion complied with IMU’s requests regarding the national subcommissions so that the roles of IMU and ICMI were clearly defined.15 As Lehto (1998, p. 110) writes, “Thus the Commission returned to the mode of action that had been applied and proved successful in 1908–1920 and 1928–1939,” but now IMU wanted to have some sort of control over ICMI, including the appointments of the national subcommissions, whose role had been diminishing over time.16 Eventually, in the meeting in Paris (15 January 1954) the EC of ICMI was completed with the co-optation of other members: Saunders Mac Lane (USA, Vice-­ President), Aksel Frederik Andersen (Denmark, Member), Guido Ascoli (Italy, Treasurer), Evert W.  Beth (Netherlands, Member), Ralph L.  Jeffery (Canada, Member), Edwin A. Maxwell (UK, Member), Marshall H. Stone (USA, Ex officio Member as President of IMU); see (EM 40, 1951–1954: 81–82).

2.3 Challenges of the New Commission When the new Commission was established,17 the only officers of the old Central Committee who stayed on were Fehr, Hadamard, Walther Lietzmann and Eric Harold Neville. Fehr, Secretary-General in the old Central Committee, was appointed lifelong Honorary President of the new EC (EM 40 1951–1954, p.  93); he passed away in 1954. The other past officers were in their late eighties (Hadamard and  See the report of the meeting by Châtelet to Stone, Paris, 7 May 1953 (IA 14A 1952–1954): “To organize this work, we think it is necessary that in each member country of the International Mathematical Union, the National Commission designates a teaching sub-commission and that it let us know as soon as possible the name and the address of its president and secretary. For questions which concern it directly, this sub-commission should be in contact with our International Commission. On the other hand, questions which fall within the remit of the International Mathematical Union should be the subject of communications sent to the Union by the National Commissions”. The original text is: “Pour organiser ce travail, nous pensons qu’il est nécessaire que dans chaque pays membre de l’Union Mathématiques Internationale, la Commission Nationale désigne une sous-commission d’enseignement et qu’elle nous fasse connaître le plus tôt possible le nom et l’adresse de son président et de son secrétaire. Pour les questions qui la concernent directement, cette sous-commission devrait être en contact avec notre Commission Internationale. Par contre, les questions qui sont du ressort de l’Union Mathématique Internationale devront faire l’objet de communications transmises à l’Union par les Commissions Nationales”. 16  See Chap. 3 by Marta Menghini in this volume. 17  The final composition of the commission is as follows: Honorary President: H. Fehr; President: A. Châtelet; Vice-Presidents: Đ. Kurepa, S. Mac Lane; Secretary: H. Behnke; Treasurer: G. Ascoli; Members: A.  F. Andersen, E. W.  Beth, R.  L. Jeffery, E.  A. Maxwell, Ex officio: M.  H. Stone, President of IMU; (EM 40, 1951–1954, pp. 81–82). 15

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Castelnuovo, who died in April 1952) or for different reasons were less involved in academic issues (Lietzmann18 and Neville). Heinrich Behnke, who worked to prepare the reestablishment of ICMI, can ideally be considered an element of continuity with the old Commission for his explicit reference to Klein’s work. As evidence of this fact, we like to recall the talk entitled “Felix Klein und die heutige Mathematik” that he delivered at the Symposium on “The teaching of Geometry in Secondary School,” sponsored by ICMI and the Mathematics Institute of Aarhus University (30 May – 2 June 1960, Aarhus, Denmark). In fact, in this talk Behnke underlines firstly what distinguishes the contemporary mathematicians from Klein and among other things he says that Klein probably would have appreciated the Bourbakist approach due to his idea of fusionism (p. 15). This is not surprising because Behnke was a fervent advocate of the importance of pure mathematics. At the same time Behnke highlights his legacy, also considering mathematics education, the topic that especially interests us. He says that it was Klein who brought this branch of knowledge to the fore fifty years ago (p.  19). In particular he recalls the fundamental aspects of Klein’s thought: the fusionist principle understood in a broad sense, not only between the various parts of mathematics, but also between mathematics and culture (p.  13); the genetic approach to presenting a theory (p. 14); the importance of the introduction of infinitesimal calculus in secondary schools (p. 17). He illustrates the meaning and the value of intuition according to Klein (p. 9) but he also underlines the need of logical developments (p. 12). He concludes his speech with the following statement: Let us try, however, to modernize the teaching in a completely undogmatic manner in accordance with our modified conception of what mathematics is, and let us keep in touch with the neighboring sciences of physics and astronomy. Then we will still be Klein’s heirs in taking care of school mathematics.19 (Behnke 1960, p. 19)

As detailed below, during his terms as secretary and then as President of ICMI, Behnke sought to restore the Commission to the glory of its early years. At the moment of the rebirth of both IMU and ICMI the situation was new and complex. People had changed, as had the social, economic political, technological and scientific contexts. Also, with the appearance of the first computers the role of mathematics in the world was changing. As Lehto remarks: The correspondence reflects the difficulties under which the IMU began its work. There was no tradition on which to build. Administrative routines had to be developed from scratch, at a time when the world had not yet fully recovered from the war. Financial resources were very limited; bureaucracy hampered activities; the telephone was not of much use for international connections. Even many years later, meetings of the Executive Committee could

 For Lietzmann, one of the reasons might be his alignment with Nazi ideology; see his biographical portrait by Schubring in Part III of this volume. 19  The original text is: “Versuchen wir aber den Unterricht entsprechend unserer modifizierten Auffassung von dem, was Mathematik ist, ganz undogmatisch zu modernisieren und behalten wir Tuchfühlung mit den benachbarten Wissenschaften Physik und Astronomie. Dann bleiben wir auch in der Pflege der Schulmathematik noch Kleins Erben.” 18

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These factors affected the life of the scientific bodies. Moreover, ICMI was confronted with two elements of novelty that pushed towards a process of reflection on its nature in terms of statutory rules and way of operating in the field of mathematics education. Firstly, the old Commission operated with some autonomy inside the community of mathematicians, both in choosing the officers and deciding activities. The states were made responsible for the financial issues for their national subcommissions, see (Klein et al. 1908, p. 449). When the new Commission was created as an adhering commission of IMU, it depended on the Union and problems arising from this situation soon made it clear that precise terms of reference were necessary to avoid friction. Secondly, ICMI was not the only body discussing the questions of mathematics education. In the early 1950s local projects flourished, such as that of the University of Illinois Committee in School Mathematics (UICSM, in 1951), and international commissions, such as the Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques (CIEAEM, International Commission for the Study and Improvement of Mathematics Teaching) in 1952, which shifted attention to different approaches to the problems of mathematics education. In these new contexts, psychologists, philosophers, and schoolteachers had a say in the discussion on mathematics teaching.

Terms of Reference to Regulate ICMI-IMU Relationships The lack of precise rules concerning the relationships between ICMI and IMU were, from the beginnings, a source of friction, as we have seen above. The problems that arose principally concerned the constitution of the national subcommissions of ICMI and funding issues. Moreover, the lack of collaboration of President Châtelet with both the EC of IMU and the Secretary of ICMI, Behnke, made the situation even more difficult. Châtelet neither responded to Stone’s letters nor sent him reports, and replied only months later to Behnke’s requests and reminders.20 Various letters kept in the ICMI Archives testify to this. In July 1953 Stone wrote to Bompiani: There seems to be a great deal of confusion in connection with ICMI. I hope we can get it cleared up… The difficulties which have been encountered in getting our various commissions organized and launched into activity make me particularly aware of the fact that we  See, for example, H. Behnke to E. Bompiani, Münster 7 July 1953 (“Since four months I never heard a word from Châtelet and I am now nervous”); Münster 31 July 1953 (“Since Châtelet does not answer to my 3 last letters, I cannot do nothing in this moment”); Münster 11 September 1953 (“Six weeks are passed since I have written this letter, but Châtelet did not answer till today. And with my preceding letters it was the same”), IA 14A, 1952–1954. 20

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need to clarify our procedure for appointing the members of Commission. (M.  Stone to E. Bompiani, Chicago, 10 July 1953, IA 14B, 1952–1954).

A year later he reminded Châtelet to respect the indications of the EC of IMU concerning the establishment of all national subcommissions and in particular that of Greece, which due to internal conflicts had caused problems for IMU (Lehto 1998, p. 105): In connection with the Constitution of the National Sub-Commissions, I recall our agreement that each such Sub-Commission is to be in the first place a Sub-Committee of the National Committee for Mathematics in the country which it represents. (M.  Stone to A. Châtelet, Chicago, 29 July 1954, IA 14B, 1952–1954)

William Hodge, member of the EC of IMU, also expressed his concerns about the conduct of ICMI: About ICMI, I agree very strongly that something must be done to curb its activities. At a recent meeting of our national committee very grave concern was expressed at the fact that so many of the Commission’s activities were carried on behind our backs and that we were being let in for responsibilities we know nothing about. They are demanding all sorts of things for individuals who have merely been asked to help in minor capacities, and their behaviour is quite unfair to these individuals and to the National Committee. I learn, too, that they are assuming quite unjustifiable rights in regard to their membership; e.g., they claim the sole right to replace any individual member who resigns. I think it will be necessary to lay down very precise terms of reference for the Commission, and to define its powers very rigidly. It will also be necessary to select a president very carefully. I agree that we should get rid of Châtelet. (W.  Hodge to M.  Stone, 31 May 1954, quoted by Lehto 1998, p. 111)

The Terms of Reference governing the relationships between IMU and ICMI were introduced during the General Assembly of IMU, which took place in The Hague (31 August–1 September 1954). According to the new Terms, ICMI consisted of ten members-at-large and two national delegates named by each National Adhering Organisation of IMU. Precise rules concerning the officers and the EC were given. ICMI could have a relatively free hand in its internal organisation, but IMU retained control on this important point: the president and the ten members-at-large of ICMI would be elected by the General Assembly of IMU on the nomination of the Union’s president. The EC of ICMI would be composed of a president, a secretary, two vicepresidents, and three additional members elected from among the members of the Commission. The president of IMU was to be an ex officio member of all Commissions of the Union. Moreover, any National Adhering Organization wishing to support or encourage the work of the Commission might create, in agreement with its National Committee, a subcommission to maintain relations with the Commission in all matters pertinent to mathematics teaching.21 In keeping with these decisions, the General Assembly of IMU nominated the ten members-at-large who would be part of ICMI, whose EC had to be renovated starting in January 1955: Yasuo Akizuki (Japan), Guido Ascoli (Italy), Heinrich Behnke (Germany), Ram Behari (India), Paul J.  Dubreil (France), Johan 21

 See IMN 35/36, 1954: 12–13 and Chap. 7 by Giacardi in this volume.

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C. H. Gerretsen (Netherlands), Ralph L. Jeffrey (Canada), Ðuro Kurepa (Yugoslavia), Edwin A. Maxwell (England), Marshall H. Stone (USA); Behnke was designated as President of ICMI (EM 40, 1951–1954, p. 91). With the new rules, the way seemed paved to ensure a fruitful collaboration between ICMI and IMU, but new difficulties would soon arise.

 onfronting with New Arenas for Discussing C Mathematics Education The second aspect ICMI had to deal with was, as we have said, the emergence of new arenas where the questions related to teaching and learning mathematics were discussed. Among these arenas CIEAEM played a significant role. The short presentation of the group of scholars—mathematicians, psychologists, philosophers, educators and teachers22—gathered under the name of CIEAEM, makes evident the difference between its approach to the math education issues and that of ICMI: [CIEAEM] brings together true competences and results from the realization that the most powerful team that can be assembled today to tackle these problems [mathematics educational problems] must be made up of those who have shown in their work a concern covering several fields simultaneously: mathematics and psychology; history of mathematics as the history of the mental realizations of certain relationships; pedagogy as an activity encompassing the world of mathematical relationships mixed with transmission techniques and obstacles in the act of learning, etc. (Piaget et al. 1955, Préface, p. 5)23

A growing attention to the student and to the process of teaching and learning was the main concern of CIEAEM. The two books published in the 1950s by CIEAEM epitomize the hallmarks of the Commission. The first (Piaget et al. 1955) includes the article by Jean Piaget on mathematical structures and mental structures and reflections on mathematical contents and their teaching with particular attention to Modern Mathematics. The second book (Gattegno et  al. 1958) is oriented to

 According to Bernet and Jaquet (1998), when the CIEAEM was officially founded in 1952, the president was the mathematician Gustave Choquet, Vice-President was the psychologist Jean Piaget, and the Secretary was the mathematician and an expert in pedagogy Caleb Gattegno. Among the founding members there were the mathematician and philosopher Ferdinand Gonseth, the mathematicians Evert Willem Beth, Jean Dieudonné, André Lichnerowicz, Hans Freudenthal, Willy Servais, formerly mathematics teacher and later Préfet des Etudes, and some secondaryschool teachers, including Emma Castelnuovo and Lucienne Félix. 23  The original text is: “Elle réunit les vraies compétences et elle résulte de la prise de conscience que l’équipe la plus puissante qui se puisse constituer aujourd’hui pour aborder ces problèmes doit être formée de ceux qui ont montré dans leurs travaux une préoccupation couvrant en même temps plusieurs domaines: mathématiques et psychologie; histoire des mathématiques comme histoire des réalisations mentales de certaines relations; pédagogie comme activité englobant le monde des relations mathématiques mêlé à des techniques de transmission et des obstacles dans l’acte d’apprendre, etc.”. 22

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classroom practice, with a focus on the importance of experimentation, the use of concrete materials, and the role of the teachers. International inquiries and comparison of the curricula of the various countries, which were the core of the agenda of the old Commission, were not suitable to face the new situation. Lucienne Félix a secondary school teacher and member of the CIEAEM, explicitly stated that CIEAEM was created to counteract the insignificance (1986, p. 64) of the old Commission on the teaching of mathematics. We also add that the editorial line of the journal L’Enseignement Mathématique, which continued to be the official organ of ICMI, was shifting towards publishing mathematical research articles, see (Furinghetti 2009). The first important event that ICMI had to organize was the participation in the ICM to be held in Amsterdam in 1954, which can be considered a mirror of the efforts to both bring ICMI back to the international role of the Klein era, and to renovate the agenda. Due to the inertia of Châtelet, it was Behnke who organised the intervention of ICMI in Amsterdam, as attested by his correspondence with IMU President and Secretary, even in the face of resistance on the part of the organising committee of the congress. In a confidential letter he wrote to Stone: They [the members of the organizing committee] all expressed the ideas that lectures on mathematical instruction might not be worthy enough for the Congress. Thus, I showed them the reports of previous congresses and pointed out that after 1912  in Cambridge (England) Section VII (history and instruction) was as strongly accentuated as Section II (analysis). Later, from 1920–1950, Section VII had no part at all at the Congresses.24 After a report stating this fact, I recommended to rebuild Section VII. Finally, I succeeded ... I was given a highly unfavourable time for the report on the work of our commission, which would have never happened in the case of my scientific lectures. (H. Behnke to M. Stone, Oberwolfach, 11 August 1954, IA 14A, 1952–1954)

Behnke’s aspiration was to recreate the climate of fervour and international collaboration that had existed during Klein’s presidency, and he meant “to extend the influence of Section VII (Instruction) so that it will equal the importance it had at the Congress in 1914,” the congress organised by ICMI in Paris.25 He was aware of the organisation and policy problems ICMI had to face to revitalize the Commission: the difficulty of finding mathematicians active in research who were interested in teaching26; the difficulty of being visible in the world of mathematics, and thus the importance that the activities of the Commission be visible at the international congresses; the difficulty of obtaining funding; the importance of involving secondary teachers. Moreover, in Section VII, dedicated to Philosophy, History and Education, of the XII International Congress of Mathematicians (Amsterdam, 2–9 September

 This is not really true, see Chap. 8 by Fulvia Furinghetti in this volume.  See H. Behnke to E. Bompiani, Münster, 8 July 1954 (IA 14A, 1952–1954). 26  “It is very difficult matter to engage mathematicians, well-known for their research work, into problems of instruction. Most of our colleagues refuse to be active for our commission because they regard this kind of work of little value, and they even neglect to forward circulars”. (H. Behnke to M. Stone, Oberwolfach, 11 August 1954, IA 14A, 1952–1954) 24 25

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1954),27 awareness of the new needs of mathematics education seems to be gaining ground. The section included two invited lectures—one by Kay Piene on “School mathematics for Universities and for life” and the other by C. T. Daltry on “Self-­ education by children in mathematics using Gestalt methods, i.e. learning-through-­ insight”—and the report by Đuro Kurepa on “The role of mathematics and mathematician at present time” (Proceedings ICM-1954 1954–1957, Vol. 3, pp.  297–324), and many communications and reports expressly dedicated to the teaching of mathematics.28 The issues addressed are various, but those concerning the two inquiries proposed by ICMI prevail: the mathematical instruction for students between 16 and 21 years of age (programs, methods, etc.), and the role of mathematics and the mathematician in contemporary life. As Howson observed, the choice of this theme “was in marked contrast to much that had gone before: attention was now directed not so much at current practice, but at the changing social and scientific contexts within which mathematics curricula had to be designed” (1984, p. 81). Kurepa was charged with presenting a general report concerning this second issue. Among the points he dealt with, Kurepa emphasized in particular the need for a “interscientific fecundation,” that is, the collaboration between mathematicians and scientists in other sectors of research, and cited as examples the IBM; the Cowles Commission which relied on collaboration between mathematicians and economists; cybernetics, fruit of the collaboration between engineers and mathematicians. He concluded his report by reaffirming the fundamental role of mathematics in human activities and underlining as well its function as a mutual means of comprehension between individuals. From the new roles of mathematics in the entire gamut of human activities follows the importance of proceeding “to a considerable revision of methods of work, and instruction and to an appropriate selection of matters to be teached (sic)” (Proceedings ICM-1954 1954–1957, Vol. 3, p. 316). Similar considerations were made by Piene, who maintained that “the mathematics program in schools must in our days be determined according to what is useful later on – in studies or in life,” bearing in mind that the school had become a school for the masses (Proceedings ICM-1954 1954–1957, Vol. 3, p. 320). He proposed a minimum program that should be taught in all kinds of schools and invited teachers to recall the unitary character of mathematics and bear in mind the research work within psychology. Daltry instead focused on questions of method. He referred to Max Wertheimer, one of the founders of the Gestalt school of psychology, and to his book, Productive Thinking, and proposed a method of teaching mathematics that begins with problem-solving, bearing in mind Wertheimer’s suggestions: “first contemplation of the problem as a whole, second the following of ‘hunches’ or tensions, third the flash of insight … Children would then learn by doing, their  See EM 40, 1951–1954: 100–104, EM s. 2, 1, 1955: 93–191; Proceedings ICM-1954 1954–1957. The Congress recorded a large participation: 1553 regular members from 55 nations from all over the world. 28  Abstracts of 25 short lectures on mathematics education (12 in volume I and 13 in volume 2) are published in (Proceedings ICM-1954 1954–1957). 27

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education would be self-education – the most enduring… Self-education in mathematics would become a dynamic experience, it would touch the heart as well as the head” (Proceedings ICM-1954 1954–1957, Vol. 3, pp. 299–300, 304). In keeping with the decisions of the General Assembly of IMU, held in Amsterdam just before the starting of the ICM, Behnke was beginning his term as President of ICMI.29

2.4 The Years 1954–1958: New Education Approaches and New Policy Issues in Shaping ICMI In the period from the General Assembly of Rome (1952) to the end of the 1950s, the plan of the activities of ICMI was marked mainly by the fixed dates of the ICMs – 1954 in Amsterdam and 1958 in Edinburgh.30 These events provided the occasions for discussing both issues of mathematics teaching and the nature of ICMI from the organizational point of view. This fact marks a difference with the golden age of the Commission from 1908 to World War I: in 1912, the ICM had been just one of the appointments, whereas important meetings were organized independently from ICMs, for example, the conferences in Milan (1911) and in Paris (1914).31 To have no independent activities was one of the weaknesses of ICMI, which would be made clear in a short time. A few days after the conclusion of the ICM in Amsterdam, the outgoing IMU President Stone wrote a letter to the outgoing ICMI President Châtelet and the new President Behnke reiterating that ICMI was “an agency of IMU,” that it was necessary to seek other funds besides those made available by UNESCO, and that “the moving impulse must come from ICMI itself.” Consequently, he stressed that it was of primary importance to formulate a “long-range” work program and added: Such a program should take cognizance of the scientific advances made possible in the techniques of instruction as a result of psychological investigations, as well as of the needs for curricular reform presented by changing social conditions. It should appeal to the countries where new educational systems are being introduced as well as to the countries where established systems are undergoing the social influences characteristics of our times. Its aim should be better teaching of more mathematics for more students at each successive level in the hierarchy of mathematical instruction. (M. Stone to A. Châtelet and H. Behnke, Rome, 21 September 1954, IA 14A 1952–1954)

 It is worth noting that in this ICM the Dutch hosts were fully responsible for the official mathematical program of the Congress and the role of IMU was not as important as it would become in the late 1950s and permanently thereafter. 30  See (Giacardi 2008), in particular the periods 1937–1954, 1955–1959. 31  See Chap. 1 by Schubring in this volume. 29

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 ehnke’s Efforts to Improve ICMI Organization and Acquire B More Independence from IMU As we have seen, Châtelet’s footprint as President of ICMI in the two-year period 1952–1954 was not very significant and much of the work was done by Behnke. This fact may surprise because Châtelet was seriously interested in educational issues: he had contributed to the study of topics such as the laboratory of mathematics and the use of concrete materials; he had collaborated with Jean Piaget on a well-known book on introducing children to arithmetic; he worked for the modernization of the educational system in France. Moreover, he was very active at a high administrative and political level and during the Cold War period, he promoted scientific and cultural links with the Soviet Union, East Germany, and China as a means to further international peace.32 Probably these too many commitments together with the organizational and administrative difficulties of the new Commission had led him to entrust much of the work to Behnke. Behnke too was very committed to mathematics education. In particular, in 1951, he had succeeded in promoting the establishment in Münster of a groundbreaking institution, the “Seminar für Didaktik der Mathematik” – the first institutionalisation of m ­ athematics education at a German university. Moreover, he was an esteemed mathematician and was one of the editors of the Mathematische Annalen.33 Beginning his term as President, Behnke soon understood the problems arising from the application of the Terms of Reference and wrote a long report to the new IMU President Heinz Hopf in which various criticisms emerge. The decisions of IMU General Assembly in The Hague had been taken without consulting either the EC in charge of ICMI or the future president (Behnke himself). Following these decisions, the commission consisted of “about 40 members dispersed all over the world,” so not all the delegates to ICMI from IMU-adhering countries had the opportunity to participate in international congresses and “the big Commission was the weakest part of the structure of ICMI.” Behnke also emphasized that: [the sub-commissions] suffer from the interference of the national adhering organizations. … sub-commissions suffer from being ruled by university professors, for their influence is predominant through the national adhering organizations … although the number of the university professors in their countries (at least in Europe) represents but a very small part of the teachers of mathematics … the presidency of the national adhering organizations does not appreciate questions of mathematical instruction and inconsiderately uses its national power… The presidency of IMU has… to look upon the national sub-­commissions – as was the case already before 1914 – as sub-commissions of the I.C.M.I. and not of the national adhering organizations. Otherwise, the work of ICMI is made impossible. (Behnke 1955)

Behnke also believed that collaboration between mathematics teachers of all levels was fundamental:  See the biographical portraits by Sébastien Gauthier and Catherine Goldstein in in Part III of this volume. 33  See the biographical portrait by Schubring in Part III of this volume. 32

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I regard it a special, honorary mission of the I.C.M.I. to establish a contact among the teachers of all levels. The teachers have to get interested in the research work, and those active in the field of research have to get interested in the work of the teachers. (Behnke 1955)

In this regard, he proposed the realization of “an international encyclopedia of school mathematics” made in collaboration with mathematics teachers of all levels in order to maintain contacts between schools and universities. (Program of Work of the International Commission for Mathematical Instruction for the period of 1955/58, in IA, 14A 1955–1957). In fact, in 1958, the German subcommission of ICMI produced the first of the five-volume Grundzüge der Mathematik für Lehrer an Gymnasien sowie für Mathematiker in Industrie und Wirtschaft. In the preface to the first volume, the editors Behnke and Kuno Fladt stress that the work was above all aimed at teachers; see (Furinghetti and Giacardi 2010, p. 32). Two years later, the criticisms by Behnke prompted IMU President Hopf to set up a small ICMI Commission with the task of reviewing the regulations, consisting of Hopf himself, Jurjen Ferdinand Koksma (Amsterdam), Behnke, Stone, and an ICMI-elected Member, Julien Desforges, Commission secretary.34 In 1958, Behnke proposed a new draft of the by-laws, the main points of which were published in L’Enseignement Mathématique (EM s. 2, 4, 1958: 216–217). In the ICMI Archives (IA 14A 1958–1960) is conserved a draft of his proposal. The main points highlighted by Behnke are the following: the reduction to a single representative of each of the national subcommissions; “ICMI is also authorized to accept appropriate organizations as National Sub-Commission even from countries which are not members of IMU”; the right of the national subcommissions to co-opt additional members; “Each National Sub-Commission shall elect a Chairman. Generally, the Chairman shall be the representative to ICMI from his Sub-Commission …, but he also is entitled to delegate a substitute who will have full voting power”; the establishment of Regional Groups, see (Draft  – New Terms of Reference, IA 14A 1958–1960). The new Terms of Reference were adopted in 1960, but not all Behnke’s suggestions were accepted, as we shall see.35

Behnke’s Presidency: Emerging of New Educational Objectives On 2 July 1955, Behnke convoked the new EC of ICMI in Geneva. As a result of voting—as required by the Terms of Reference—the EC was formed as follows: Vice-Presidents: Đuro Kurepa, Stone; Secretary: Desforge; Members: Ram Behari, Edwin Arthur Maxwell, Kay Waldemar K.  Piene, Ex officio: Hopf (President of IMU) (EM s. 2, 1, 1955: 197). During the meeting, a plan of work was established

 Procès-verbal (Extrait) de la Réunion du Comité Exécutif de la Commission Internationale de l’Enseignement mathématique, tenue à Bruxelles le 3 juillet 1957, IA 14A 1955–1957. 35  See Chap. 4 by Giacardi in this volume. 34

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for the period 1955–1958. The following three themes were proposed for study: the first two concerned the investigations undertaken for the Amsterdam ICM on the role of mathematics and mathematicians at the present time, and the teaching of mathematics to students up to the age of 15 years. The third theme was the scientific basis of mathematics teaching in secondary schools and teachers training; see (EM s. 2, 1, 1955: 198). Hans Freudenthal, a renowned mathematician for his contribution to algebra and topology, but with a profound interest in education and history of mathematics and a founding member of the CIEAEM, was present. He observed that the first two subjects were too general and could lead to reports focused more on organization and administration issues than on strictly scientific questions. Instead ICMI should rather stimulate research and studies of scientific character. He, therefore, suggested these more limited and precise themes: the need for an intuitive approach at the first lessons in geometry; the aid that psychology can provide in the first phases of mathematics teaching; the importance of the teaching of geometry; logic and the teaching of mathematics. After an animated discussion, Freudenthal’s considerations led to replacing the first theme initially proposed by the following: a comparative study of the methods used in the initiation to geometry; see (EM s. 2, 1, 1955: 198–201, 268). This intervention by Freudenthal can be considered the starting point for changing the cultural objectives of ICMI. We see, for example, that just in 1955 the second series of the journal L’Enseignement Mathématique—confirmed as the official organ of ICMI (EM s. 2, 1, 1955: 201)36—was started with new objectives: This second series will be devoted to the reform and development of mathematical instruction; it will publish articles focusing on and explaining modern theories in a manner comprehensible to non-specialised mathematicians; deal with the arrangement and organisation of teaching; study the psychological formation of mathematical ideas; and publish accounts of the work done and surveys carried out by the I.C.M.I.37 (EM s. 2, 1, 1955, pp. 270–271)

Fehr, one of the two founders of the journal and one of the protagonists of ICMI’s life starting in 1908, had died in 1954 and was commemorated in a symposium on the occasion of the Geneva meeting. His death symbolically closed an era and the seeds of renewal were sown. In the following meetings of the EC of ICMI in Münster-Westfalen (27 May 1956), in Brussels (3 July 1957), and again in Münster-Westfalen (28 May 1958),38 the participation in the upcoming International Congress of Mathematicians in Edinburgh was organized and the speakers for the three themes proposed in Geneva were chosen. Howard F. Fehr, professor of mathematics education at the Columbia University, was to discuss about the mathematical instruction up to the age of 15   The inscription on the title page Organe officiel de La Commission Internationale de l’Enseignement Mathematique had appeared until 1939. In the title page of the volumes 39 (1942–1950) and 40 (1951–1954) this sentence is missing, but it reappears in the second series. 37  The path was really at the beginning and the necessary supports were lacking. For example, as we mentioned before, the intentions set out at the beginning of the second series of L’Enseignement Mathématique were not realized. 38  See EM s. 2, 2, 1956: 317–323; 3, 1957: 300–306; 4, 1958: 213–219. 36

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years; Behnke was to examine the scientific basis of mathematics in secondary school teaching; and Freudenthal was to talk about the methods used in the initiation to geometry by making a comparative analysis. Moreover, there was a direct intervention by the new IMU President Hopf in the agenda of ICMI; in fact, he invited the Commission to undertake a study on “the difficulties that arise in recruiting professors39 of mathematics and professors of natural sciences  – difficulties which are mostly due to the industrial development which necessitates a greater number of engineers”; see (EM s. 2, 3, 1957, p. 79): David van Dantzig, Howard Fehr, Kurepa and Willy Servais were in charge of preparing a questionnaire concerning the problem highlighted by Hopf; see (EM s. 2, 4, 1958: 220–223). From 14 to 21 August 1958, the International Congress of Mathematicians was held in Edinburgh. Section VIII was devoted to History and Education, and in his presidential address William Hodge commented on the ICMI’s work on mathematics education and added that: It is part of our duty to see that our pupils who go on to walks of life outside the academic field understand that mathematics is an integral part of world culture; not only a pillar of the technological civilization of today, but an essential item in the intellectual equipment of the good citizen. (Proceedings ICM-1958 1960, p. lii)

Kurepa was invited by the Organizing Committee to give one of the half-hour addresses on “Some principles of mathematical education” (Proceedings ICM-1958 1960, pp. 567–572); the three ICMI reports by Howard Fehr, Behnke, Freudenthal40 were presented and 14 talks on mathematics education are listed in (Proceedings ICM-1958 1960).41 In his talk, Kurepa underlined various issues at the base of mathematical education: the teacher-learner relationship, the environment factors in various senses—physical (e.g. small or big classrooms), biological (e.g. age of pupils), social (e.g. village, town), etc. In particular, we would like to stress his emphasis on “action and perception” in the teaching-learning process: In the teaching process, the hands are to be active (writing, showing), the tongue, ears, brain, i.e., all organs are more or less in active interdependence and co-operation. Let us remember that for a long time, even in instruction of geometry, and still more in arithmetic, the factors action and perception were either eliminated or at least neglected. (Proceedings ICM-1960, p. 570)

It deserves to be mentioned that Freudenthal, in the final part of the report on the “Comparative study of methods of initiation into geometry,” addressed the theme of the impact that psychological and pedagogical research may have on geometrical instruction in the initiating phase. He also cited Piaget’s research, which although highly interesting, had two flaws: “firstly because Piaget’s mathematical background has been rather weak, but mainly because Piaget’s approach hardly reflects the teaching situation in the classroom, but the rather unusual laboratory situation of  By the word “professor” the authors mean secondary school teachers.  The text of the reports of Howard Fehr and Freudenthal are published in EM s. 2, 5, 1959, pp. 61–78 and pp. 119–145, respectively. 41  For further details, see Chap. 8 by Furinghetti in this volume. 39 40

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the psychologist”; he added that “Mathematical teaching theory can be furthered by mathematical teachers who are able mathematicians and able educators” (EM II s., V, 1959, p. 139). In this statement, it is interesting to underline two aspects highlighted by Freudenthal—the “mathematical teaching theory” and the teacher mathematician-­educator—which are symptomatic of the evolution that was taking place in mathematics education. During the ICMI meetings of the period 1955–1959, two other important aspects emerged, both highlighted by Behnke, one more linked to fostering international collaboration and greater cohesion, namely the creation of Regional Groups, about which more will be said below, and the other, to some extent anticipating what will happen later under Freudenthal’s presidency, on the importance of having congresses organized by ICMI between one ICM and another: At the International Congresses of Mathematicians, ICMI plays a relatively small role, since these Congresses are dominated by reports and discussions on matters of research. It might therefore, be more appropriate for ICMI to hold smaller symposia in the years between Congresses. This has, in fact, been the case in the past. (H. Behnke, Memorandum, IA 14A 1958–1960)

ICMI tried to gradually reduce its Eurocentric nature by attempting to extend the Commission’s activities beyond Europe. In 1955, Ram Behari, as said above, was nominated a member of the EC of ICMI. He came from India, which in 1947 had gained independence from British domination. Other colonies of France and United Kingdom were becoming independent and were consequently aware of the importance of a good educational system. In 1956, ICMI was officially represented by its Vice-President Stone at the Conference on Mathematical Instruction in South Asia in Bombay, which was “the first of this kind” and saw the participation of the following countries from South Asia: Burma, Ceylon, India, Indonesia, Malaya, Pakistan, Singapore and Thailand (IMN 47/48, 1956: 9; Report of a Conference on Mathematical Education in South Asia, The Mathematics Student 24, 1956: 1–183; Stone 1956).

2.5 Acquiring Greater International Dimensions and Widening Education Horizons: The Role of Marshall Stone The General Assembly of IMU held in St. Andrews (Scotland) on 11–13 August 1958, on the occasion of the ICM in Edinburgh, welcomed as IMU members the socialist countries of Europe, including the USSR.  Aleksandr Danilovich Aleksandrov was elected member-at-large of ICMI and afterward member of the EC. Marshall Stone became the President of the Commission.42 During the two-year  Executive Committee of ICMI and members-at-large for the period 1 January 1959  – 31 December 1962 are as follows: President: M. H. Stone; Vice-Presidents: H. Behnke, D. Kurepa; 42

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mandate as president of IMU (1952–1954) and later as vice-president of ICMI (1955–1958), he was very active on the international side, especially for his contacts with UNESCO testified by his correspondence. In the decades of the 1950s and the 1960s, the role of UNESCO was crucial. This was the period of decolonization, which led to an increase of the number of the membership to UN. The newly independent countries lacked many structures for their development, systems of instruction among them. UNESCO, the UN agency responsible for education, had the task of establishing schools, especially primary schools; mathematics education was part of its projects. UNESCO financed IMU through the ICSU (International Council of Scientific Unions) and IMU in turn provided funding to the ICMI, but these funds could only be used to refund travel expenses for the meeting of the Commission; see (EM s. 2, 2, 1956: 318). Funds for education greatly helped ICMI. As illustrated in (Christiansen 1978; Jacobsen 1993, 1996), in the following years important projects flourished thanks to the cooperation on common purposes of ICMI and UNESCO, though, as Christiansen (1978, p. 7) points out, in some occasions the funds from UNESCO had to be supplemented with personal funds of the collaborators or funds from the governments of UNESCO member states, due to the number of people involved. Stone had the merit of involving UNESCO in the ICMI activities beyond primary level.

The Contacts with UNESCO Stone had called for collaboration between UNESCO and ICMI since the summer of 1952 but, as he wrote to Châtelet, that organization had “a rather limited interest in the work of the Commission since his educational program is devoted mainly to the primary level.” He added: “I hope that eventually we shall be able to persuade UNESCO to be interested in secondary education and even in university education so far as it affects the under-developed areas” 43 (M. Stone to A. Châtelet, Chicago, 3 November 1952, IA 14A 1952–1954). To foster the collaboration, he invited the UNESCO general director to send a delegate to participate in the section organized by ICMI at the ICM in Amsterdam, also suggesting the person of the UNESCO Secretary: G.  Walusinski; Members: Y.  Akizuki, A.  D. Aleksandrov, O.  Frostman; Ex officio: R. Nevanlinna (President of IMU); Members-at-large: Y. Akizuki, A. D. Aleksandrov, H. Behnke, P. Buzano, G. Choquet, Howard Fehr, H. Freudenthal, D. Kurepa, E. A. Maxwell. (EM s. 2, 5, 1959, pp. 151–152) Subsequently J. Karamata would be co-opted, as the editor of L’Enseignement Mathématique (EM s. 2, 5, 1959: 290–292). 43  See for example the letter by Gerald Wendt, responsible of the Natural science department of UNESCO to M. Stone: “For the present, at least, it will probably be wiser for us to concentrate on the methods and devices for the introduction of mathematical training into the primary schools, especially in the less developed countries. This, in turn, must begin with the improvement of mathematical instruction in the teachers’ training colleges. This is our first objective… One of our major problems is the effort to organize science teachers’ associations in the large number of countries where they do not exist”. (G. Wendt to M. Stone, 6 August 1952, IA 14A 1952–1954)

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Staff who seemed most suitable to him, namely the French mathematician and writer François Le Lionnais (M. Stone to P. Auger, Roma, 19 April 1952, IA 14A 1952–1954). In 1956, Stone, as said above, had represented ICMI at the Conference on Mathematical Instruction in South Asia, organized in Bombay at the Tata Institute of Fundamental Research—with the support of IMU and UNESCO—with the purpose “to discuss, with special reference to South Asia, the problems of mathematical education at all levels, and to formulate plans for its sound development” (IMN 47/48, 1956, p.  9). The address by Stone dealt with “some crucial problems of mathematical instruction” (Stone 1957) focusing in particular on university teaching in the USA, but considering mathematics education problems from the top-­ down, and giving suggestions for curricula. Stone, once President of ICMI, was Chairman of the Organizing Committee of the Inter-American Conference on Mathematical Education (IACME), which was held in Bogotá from 4 to 9 December 196144 and organized with the cooperation of UNESCO. In the summer of 1961, UNESCO drew up a draft contract with the IMU to study the relationships between university teaching of mathematics and physics. On this occasion, there were disagreements between Stone and IMU because he had not been consulted in advance and because plans had been made to appoint an IMU subcommittee to discuss collaboration with UNESCO. Stone wrote: I thought that I.C.M.I. was designated as I.M.U.’s organ for handling all matters in the field of mathematical education … Furthermore, the setting up of two bodies under I.M.U. designed to deal, even in cooperation, as independent agents respectively restricted to the secondary field and the university field is more likely than not to destroy I.C.M.I.” (M. Stone to B. Eckmann, Chicago, 17 February 1961, IA 14A 1961–1966); I have expressed myself quite forcefully to several members of the Executive Committee about the dangers of having two separate bodies under IMU dealing with educational problems, and some are certainly fairly sympathetic with my position”. (M. Stone to B. Eckmann, Chicago, 22 July 1961, IA 14A 1961–1966)

Among other things, Stone thought that studying the question of the connections between the teaching of mathematics and the teaching of physics or even science in general, was “a little premature, at least from the point of view of a mathematician” (M. Stone to B. Eckmann, Chicago, 23 June 1961, IA 14A 1961–1966). However, he agreed that a special Committee designated by IMU’s EC should study the details of the contract, but it “ought to go out of existence once the UNESCO contract had been successfully negotiated” (M. Stone to B. Eckmann, Chicago, 22 July 1961, IA 14A 1961–1966).45

 For the action of ICMI in South America, see (D’Ambrosio 2008).  The EC of IMU appointed Nevanlinna, Chandrasekharan, Kuratowski, de Rham and Stone himself as members of the Special Committee, saying that it “may very well be considered later on as part of ICMI” (B. Eckmann to M. Stone, Zurich, 10 August 1961, IA 14A 1961–1966). See also IMN 71, 1962: 4. 44 45

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Eventually, IMU created a Special Committee on the Teaching of Science (SCOTS), of which Stone was appointed chairman, in order to collaborate with UNESCO in the broader field of science education. During the ICMI meeting of 10 August 1962 in Saltsjöbaden, near Stockholm, Stone expressed the hope of a fruitful collaboration between SCOTS and ICMI in the future, but some members of the Commission lamented the fact that the SCOTS was not a subcommittee of ICMI and pointed out the risk that the SCOTS would take over the study of the teaching of mathematics at the university level, leaving only that of the secondary level to the ICMI (EM s. 2, IX, 1963: 110, 113). Other conferences and seminars were organized thanks to the collaboration with the OEEC or with local mathematical bodies. Among these, particularly important was the one in Royaumont (1959), chaired by Stone, but it is worth mentioning also those held in 1960 in Aarhus (sponsored by ICMI and the Mathematics Institute of Aarhus University), Zagreb-Dubrovnik (OEEC), Belgrade (ICMI and the Yugoslav Association of Mathematicians and Physicists), Lausanne (ICMI and the Swiss Mathematical Society), and the seminar held in 1961  in Bologna (ICMI and the Italian Commission for Mathematics Teaching). The importance of these conferences, as we will see, lies above all in the fact that they represent ICMI’s efforts to take into account the stimuli of Modern Mathematics to develop new curricula and new teaching methods.

Stimuli by “Modern Mathematics” In the decades from 1950 to 1970, the school world was affected by reform movements known as “New Math” or, in Europe, “Modern Mathematics.” In reality, as Vanpaemel and de Bock (2019) claim, these labels refer to very different phenomena in terms of history, content and context, but they have some commonalities: the desire to bring school mathematics closer to that of research, to update language and symbols, to fill the mathematical gap between secondary school and university. The main characteristic of these reform movements was the emphasis on the axiomatic, logical structure of mathematics, even at the earliest levels. It is commonly believed that the development of New Math was triggered by the launch of the Soviet Sputnik and the consequent concern of the USA for a technological breakthrough that would have political and military consequences. It is true that this event was a shock, and in the USA it favoured the disbursement of substantial funds for projects concerning science, but, in reality, the roots of these reform movements date back at least to the early 1950s. In the USA, the University of Illinois Committee in School Mathematics project (UICSM), headed by Max Beberman, was established in 1951. In the years that followed, there were further initiatives in mathematics education and, eventually, the launch of the Soviet Sputnik in October 1957 persuaded the US Congress to designate an unprecedented amount of dollars for science education. This event also led the OEEC—created to administer the funds allocated by the USA (Marshall Plan) to rebuild Western

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Europe after the Second World War—to deal with problems relating to the teaching of science and mathematics. In 1958, the OEEC opened in Paris the Office for Scientific and Technical Personnel to make science and mathematics education more efficient (Gispert and Schubring 2011, p. 89). In this context, as we will see below, the collaboration with ICMI developed. In 1958, the American mathematician Edward Begle was appointed director of the School Mathematics Study Group (SMSG), the largest and most influential of the so-called New Math curriculum projects in the USA. SMSG published and distributed extensive collections of books and films for teachers as well as a series of monographs for students, the New Mathematical Library.46 American educators feared that the Soviet Union was surpassing the USA in educational emphasis on science and mathematics, so in September 1959 a conference was held at Woods Hole on Cape Cod in the USA with the aim of improving science education in primary and secondary schools, bringing together scientists, mathematicians, educators, biologists, psychologists and other professionals. In his report, Jerome S. Bruner, an American psychologist who made significant contributions to human cognitive psychology, placed the accent on learning through structures and learning through discovery (Bruner 1960, pp. 17–32). In Europe, the Bourbaki group, which beginning in the 1930s had attempted to generalize, formalize, and unify all of pure mathematics, stimulated the emerging of the movement usually known as Modern Mathematics. Among the promoters were Dieudonné, Choquet, and Lichnerowicz, founding members of CIEAEM. In fact, Bourbaki’s views and Piaget’s ideas—which were considered at the time as mutually complementary through the links established by the notion of “structure,” which arises in both mathematics and developmental psychology—gave rise to a new approach to the teaching of mathematics. Among the various initiatives linked to Modern Mathematics, it is worth mentioning that in the years 1956 and 1957 in Paris the Société Mathématique de France (SMF) and the Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP) organized some lectures on Modern Mathematics at the initiative of Choquet and Gilbert Walusinski, President of the APMEP.47 These talks were addressed at mathematics teachers of all secondary schools and contributed to a change in mentality and to improving teachers’ knowledge of mathematical content; see (Félix 1986). On 28 May 1958, during the meeting of the Executive Committee of ICMI in Münster-Westfalen, Kay Piene, Member of ICMI, proposed, among the topics to be discussed by the Commission in the period 1959–1962, the study of which themes and applications of modern mathematics might find a place in the teaching programs of secondary schools (EM, s. 2, 4 1958: 218). It is in this climate that OEEC organized, from 23 November to 4 December 1959, the seminar at the Centre Culturel de Royaumont, Asnière-sur-Oise (France)  See https://www.maa.org/archives-of-american-mathematics-spotlight-the-new-mathematicallibrary-records (Retrieved June 2021). 47  See Structures algébriques et structures topologiques, 1958 (Monographie 7 de L’Enseignement Mathématique). 46

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on the new thinking in mathematics and mathematics education, and the implementation of reform. The participating countries were invited to send three delegates: an outstanding mathematician; a mathematics educator or person in charge of mathematics in the ministry of education; an outstanding secondary school teacher of mathematics. The basis for discussion should have been the answers to a questionnaire on the status of mathematics education, but it was sent in December, so the answers did not arrive in time. The questionnaire was published in Appendix B in the proceedings of the seminar.48 This meeting is best remembered for Dieudonné’s famous attack on Euclidean geometry, which overshadowed the other contributions. He said: If we had a curriculum at last freed from the dead-weight of “pure geometry”, what would we put in its place?... I would list the following ones: a. Matrices and determinants of order 2 and 3. b. Elementary calculus (functions of one variable). c. Construction of the graph of a function and of a curve given in parametric form (using derivatives) d. Elementary properties of complex numbers. e. Polar co-ordinates. (Dieudonné 1961, pp. 35, 38)

A different approach was expressed by William Douglas Wall, the director of the National Foundation for Educational Research in England and in Wales (NFER), and John B. Biggs, Assistant Research Officer there. In the proceedings of the seminar, their contribution is not included: it is summarized in a few pages (OEEC 1961, pp. 101–103 and 120), but the archival research by Gert Schubring made it possible to find the original text, “Psychological and Educational Researches into the Teaching of Arithmetic and Mathematics.” Their paper, as stressed by Schubring, “revealed itself to be a genuine and innovative research report on the state of empirical research into the teaching of mathematics” (Schubring 2014a, p.  164) at the international level up to the 1950s. Without going into details, we simply want to underline that the authors emphasize the importance of educational research into mathematics education and suggest carrying out experimental studies on the curricular changes that were proposed. Due to the predominance of secondary school teachers and university mathematicians, the work in Royaumont focused on the reform of curricular contents, although in his introductory address Stone, chair of the seminar and President of ICMI, outlined a new policy in mathematics education. He formulated a veritable “program of research in the teaching of mathematics” (study and experimentation), expressing his hopes for creation of ad hoc institutes for research, and promotion in the universities of research projects regarding the teaching of mathematics (Stone 1961, pp.  28–29). He also made a fervent appeal for attention to be paid also to primary schools (Stone 1961, p. 18) in the belief that mathematics in school should be for all. Howard Fehr, head of the Department of Teaching of Mathematics at the Teachers College of Columbia University and member-at-large of ICMI, drew up 48

 See OEEC 1961, pp. 221–237.

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the report of the meeting pointing out that its purpose was not “to formalize all mathematics instruction through an axiomatic and logical treatment,” but rather “to focus attention on the need to introduce modern concepts, clear definitions and such methods of teaching as will lead eventually to formal mathematical structure”49 (Fehr 1961, p. 209). In subsequent congresses, sponsored by ICMI, held in 1960 in Aarhus, Belgrade and Lausanne, dedicated respectively to the teaching of geometry, the coordination of the teaching of mathematics and physics and to the teaching of analysis, various topics discussed at Royaumont were taken up again, and during the seminar organized in Zagreb-Dubrovnik by OEEC new mathematics programs for secondary education were prepared following the auspices of the Royaumont seminar.50 Also in the reports presented by ICMI at the ICM in Stockholm in 1962, the introduction of Modern Mathematics in secondary instruction was amply discussed.51 The attitude towards Modern Mathematics in ICMI seems to be generally favourable as appearing from these reports; in particular four areas of Modern Mathematics were recommended by the majority: elementary set theory, introduction to logic, some topics from modern algebra, and introduction to probability and statistics. However, there was no agreement on how far the axiomatic approach should be applied to the teaching of mathematics and in particular of geometry, and there were also criticisms. For example, Freudenthal, member-at-large of ICMI, in his talk “Enseignement des mathématiques modernes ou Enseignement moderne des mathématiques?” presented in 1961 at the Seminar sponsored by ICMI in Bologna insisted on the importance of “frankly didactic research” (recherches franchement didactiques) and observed that one of the causes of the failure of mathematics teaching is the “didactic inversion of levels” (l’inversion didactique des niveaux): “we go down from higher to lower levels instead of going up from bottom to top,” and further,

 In fact, Dieudonné’s talk aroused both strong approval on the part of some participants and equally strong disagreement on the part of others, but ultimately the two groups reached a general agreement on not removing Euclid entirely from the secondary-school curriculum. (New Thinking in School Mathematics, p. 47). The original text of the conclusions of the Royaumont Seminar was published by Schubring (2014b), who also highlights the differences with Fehr’s report. 50  See Un programme moderne de mathématique pour l’enseignement secondaire (Paris: OEEC 1961). The developments of the Royaumont seminar are discussed in (Schubring 2014a) and (De Bock and Vanpaemel 2015). 51  The topics were the following: Which subjects in modern mathematics and which applications of modern mathematics can find a place in programs of secondary school instruction? (Reported by J. G. Kemeny); Connections between arithmetic and algebra in the mathematical instruction of children up to the age of 15 (Reported by S. Straszewicz); Education of the teachers for the various levels of mathematical instruction (Reported by K. Piene) (see Proceedings ICM-1962 1963, p. XXXVI). The reports were published respectively in EM, s. 2, 10, 1964: 152–176 and 271–293; EM, s. 2, 9, 1963: 116–127. 49

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“this anti-didactic inversion is taken to the extreme in recent axiomatic geometry programs” 52 (Freudenthal 1963, pp. 29, 34, 41). Freudenthal would be the driving force of ICMI in the following decade.

 Small Step Towards Greater Independence: The New Terms A of Reference (1960) The year 1960 was important for ICMI also from the organizational point of view. As we have seen (§ 2.4), Stone was a member with Behnke of the small commission appointed by IMU President Hopf with the task of reviewing the ICMI regulations. Behnke had proposed a draft for the new Terms of Reference which was discussed during the general Assembly of IMU held in St. Andrews (Scotland) on 11–13 August 1958. The General Assembly agreed on the necessity to make certain changes in the regulations of ICMI and invited the EC of IMU to collaborate with that of ICMI on this aim, and to present a joint proposal. Stone, the new ICMI president, did not share all of Behnke’s proposals; in fact, he wrote to the IMU secretary Eckmann: It seems to me that… the “New Terms of Reference” are extremely likely to make a great deal of trouble between groups in the adhering countries and between ICMI and IMU. I therefore consider them ill-advised and objectionable… By making the National Sub-­ commissions for Mathematical Instruction quite independent of the National Commissions for Mathematics… and letting the representatives of these Sub-commissions have effective voting control of ICMI, the way is cleared for the elimination of any real influence in ICMI from the side of the mathematicians who are acquainted with the higher levels of their subject and who are interested in research as well as in teaching and preparation for research.” (M. Stone to B. Eckmann, Chicago, 5 January 1959, IA 14A 1958–1960)

In a following letter, he further emphasized his contrast (“thorough disagreement”) with Behnke and added that “it would be much better to forget about adopting any reforms of a constitutional kind at this time and to let me try to make the scheme worked out very carefully in 1954 function properly” (M. Stone to B. Eckmann, Hong Kong, 28 February 1959, IA 14A 1958–1960). A year later, he wrote to IMU President Nevanlinna and presented the new Terms of Reference for ICMI, commenting on the changes that only partially accepted proposals by Behnke, who, in fact, was opposed to this new version.53 (M. Stone to R. Nevanlinna, n. p., 5 April 1960, IA 14A 1958–1960).

 The original text is: “on descend des niveaux supérieurs aux inférieurs au lieu de monter de bas en haut” … “cette inversion anti-didactique est poussé à l’extrême dans les programmes récents de géométrie axiomatique”. On Freudenthal’s educational thinking see (Howson 1985; Streefland 1993). 53  Akizuki, Frostman, Kurepa, Stone, Walusinski were in favour, Behnke was dissenting and Aleksandrov did not send his opinion on the matter. 52

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The new Terms of Reference were adopted in 1960.54 The differences from those approved at The Hague are the following: The number of national delegates was reduced from two to one; the clause “or recognize” is added in the point (e): any National Adhering Organization may create, or recognize, in agreement with its National Committee, a National Subcommission for ICMI; in the point (f) this sentence is added concerning the regional groups: “In the pursuit of this objective, the Commission shall cooperate, to the extent it considers desirable, with effective regional groups which may be formed spontaneously, within, or outside, its own structure”; a point (g) is added: “The Commission may, with the approval of the Executive Committee of IMU, coopt, as members of ICMI, suitably chosen representatives of non-IMU countries, on an individual basis.” Behnke’s proposals were downsized, and the IMU retained control over the Commission, but a small step towards greater independence had been taken. The next Terms of Reference would be introduced in 1982. In accordance with the point (f) of the new Terms of Reference, CIAEM/IACME (Comité Interamericano de Educación Matemática/Inter-American Committee on Mathematics Education) would be affiliated to ICMI as an effective regional group in 196555; and in line with the point (g), in 1964 Luxemburg and then Senegal would constitute their own national subcommissions (EM 12, 1966: 134), Tunisia in 1969 (IMN 93, 1969: 5; see also André Delessert to O. Frostman, Riex, 22 March 1969, IA, 14B 1967–1974). In 1994, there were 12 countries that were members of ICMI, but not of IMU (Lehto 1998, p. 257). One of the central tasks of the IMU, as Lehto states was “to maintain good mathematical cooperation across borders, irrespective of the political climate” (Lehto 1998, p. 122), but this aspiration was often disregarded. It is useful to recall that socialist countries were admitted to membership a few years after IMU’s rebirth: Poland in July 1956, USSR in March 1957, Bulgaria, Czechoslovakia, and Hungary in May 1957 and Romania in March 1958. In particular, the membership of the USSR was important for both scientific and political reasons because the Cold War divided the world in two opposing camps. In 1958, the Russian language was added as an official language alongside English and French (Lehto 1998, p. 109). More problems arose for the membership of East Germany and the People’s Republic of China (Lehto 1998, pp. 124–130). In any case, the world political situation caused trouble which also affected ICMI’s life. For example, Russians boycotted the Symposium in Belgrade in September 1960 and the desire of Kurepa, one of the organizers, to invite some socialists from Red China was vetoed by Jugoslav authorities (M. Stone to Eckmann, Paris, 22 November 1960, IA 14A 1958–1960). Another example: Aleksandrov, member of ICMI EC, proposed to ICMI President Stone to organize a European meeting in the USSR in 1961 (M. Stone to Eckmann, Chicago, 11 April 1960, IA 14A 1958–1960) and Stone discussed this with Sergei Sobolev, but nothing came of

54 55

 See IMN 68/69, 1961: 29; ICMI Bulletin 5 (April 1975): 5–6.  ICMI Bulletin 5 (April 1975): 6.

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it. Ultimately, the meeting was held in Bologna in Italy. A key moment in the development of the relationship between the international and the Soviet mathematical community was the ICM in Moscow in 1966.

2.6 Lichnerowicz’s Presidency: Collaboration with UNESCO Intensified and the ICMI’s Field of Action Expanded In the final report of his mandate “Report for the period 1959–1962” Stone had, among other things, underlined the inadequacy of the funding received by ICMI: in order to supplement subventions arriving from IMU, often subject to limitations of use, the Commission had decided to solicit voluntary contributions56 from the countries represented in ICMI, but the financial means were insufficient for “a satisfactory execution of its program of scientific meetings and publication” (EM s. 2, 9, 1963, p. 109). Global interest in the field of mathematical instruction was becoming intense, and ICMI found it difficult to play its role in relation to the movements for experimentation and reform. Beginning in the 1960s, numerous editorial projects and initiatives flourished. Suffice it to mention that in 1961 the “School Mathematics Project” was founded in Great Britain; it was aimed at secondary-school students 11 years old and up; in 1963, the first volume of Mathématique Moderne by George and Fréderique Papy appeared; and in 1964 the English “Nuffield Project” for mathematics started under the direction of Geoffrey Matthews: it was aimed at primary and lower secondary schools and was based on the so-called active and discovery methods, with influences from Piaget. Among the guidelines presented by Stone for the future EC, he stressed the importance of responding to the growing “demand for an international bibliographical informational service in the field of education” and of extending ICMI’s activity to new areas, such as Africa (EM s. 2, 9, 1963, p. 111). The new President of ICMI, André Lichnerowicz,57 accepted Stone’s suggestions by trying to obtain funding in order to be able to carry out more ambitious projects, and he succeeded in establishing collaborations with the USSR and Africa. He was a mathematician well known for his contributions in differential geometry, but also interested in the renovation of mathematics education: he was one the founding members of the CIEAEM and since 1955 had participated in the work of training teachers in Modern Mathematics supported by the “Association des Professeurs de Mathématiques de l’Enseignement Publique” (APMEP).  The following contributions were received by ICMI: in 1959, $100 from Denmark and $50 from Italy; in 1960, $25 from Luxemburg, $25 from the Netherlands and $25 from Sweden; in 1961, $50 from Italy and $25 from Luxemburg; in 1962, $50 from Italy and $1,250 from the USA; in 1963, $25 from Sweden, $50 from the Netherland and Italy; in 1964, $25 from the Netherlands, from Luxemburg and Sweden, $50 from Italy; in 1965, $25 from the Netherlands and Sweden, $50 from Italy (EM s. 2, 9, 1963: 108–109; 12, 1966: 136). 57  See the biographical portrait of Lichnerowicz by Hélène Gispert on Part III of this volume. 56

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Significantly, upon being elected President, Lichnerowicz asked to have the Russian mathematician Andrei Kolmogoroff and the American Edwin Moise as Vice-Presidents, and André Delessert, a high school teacher from French-speaking Switzerland, as secretary.58 Kolmogoroff refused59 and the choice fell on the Polish mathematician Stefan Straszewicz. The appointment of the secretary instead was successful, even if Delessert initially could not be appointed because he was not either an elected or co-opted member of ICMI. He was able to accept that position thanks to Marcel Rueff, the delegate in ICMI of the Swiss subcommission, who resigned from this post in favour of Delessert.60 Forming the new EC was problematic, so much so that Delessert in January 1964 wrote to Rueff: I inform you that the activities of CIEM during 1963 constitute an excellent example of an empty set. However, Mr Lichnerowicz has many plans. (A. Delessert to M. Rueff, Riex 25 January 1964, Fondo Delessert, Serie VI, u. a. 391)61

In fact, the ICMI EC met for the first time in Paris on 14 and 15 February 1964 (EM, s. 2, 10, 1964: 294–296), but on that occasion it made important decisions. In fact, it was decided to adhere to two important initiatives of UNESCO: the creation of a centre for documentation and information about mathematics teaching, and the preparation of a source book on this field (a list of manuals, periodicals, anthologies of articles, etc.). Moreover, the EC resolved to propose to the Organising Committee of the next ICM in Moscow a plenary lecture held by a Russian mathematician on the teaching of numerical analysis at the university, and to present reports on the following topics: a programme for the university formation of future physicists; the use of axiomatic system in the teaching at the secondary level; the development of student mathematical activities; the role of problems in this development. On 21 January 1964, Lichnerowicz managed to enter into special contracts directly between UNESCO and ICMI for $7,000 for the preparation of a report on mathematics teaching at the university level in eight countries (IA 14A, 1961–1966); EM s. 2, 10, 1964: 296; 12, 1966: 136), and the publication of a series of volumes entitled New Trends in Mathematics Teaching was planned. These would include new reports, reproduction of old articles and bibliographical information, and Anna Zofia Krygowska was entrusted with the task of overseeing the publication (IMN 83, 1966: 3). The first volume of the series appeared in 1967 (UNESCO 1967) with articles in French and in English with summaries, respectively, in English and in French. It gathered the papers on problems related to the teaching of mathematics presented at various conferences as well as original articles. Lichnerowicz wrote the  See G. de Rham to K. Chandrasekharan, Lausanne, 12 February 1963, Fonds de Rham, 5113-403.  See G. de Rham to K. Chandrasekharan, Lausanne, 18 May 1963, Fonds de Rham, 5113-403. 60  See K. Chandrasekharan to G. de Rham, Bombay, 18 February 1963, Fonds de Rham, 5113-403; and M. Rueff to G. de Rham, Zurich, 13 March 1963, Fondo Delessert. Serie 1, u. a. 7. 61  The original text is: “je vous informe que les activités de la CIEM au cours de 1963 constituent un excellent exemple d’ensemble vide. Toutefois M. Lichnerowicz a de nombreux projets.” 58 59

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preface—both in French and in English—where he emphasized the importance of Modern Mathematics: What we might call “the traditional mathematical disciplines” have been transformed into a unitary discipline of mathematics, our contemporary mathematics which might be characterized in the following terms: that instead of conforming to the mathematical structures and identifying them more or less by chance, modern mathematics has made an effort to dominate them. … All too often still mathematics as taught is dissected like a dead specimen or exhibited like a work of architecture, in finished and final form, and good minds not infrequently reach the point of wondering if it is still possible to do any creative work in mathematics… Modern mathematics teaching calls for all the teaching talent and skill of the teachers and it is encouraging to see that since 1945 there have been individual pioneers or centres all over the world “on the job”. The aim of “New Trends” is to provide a link between all those who are, or who would wish to be, concerned in a task as essential as it is absorbing.” (p. 17)

It is no coincidence that, after his term as ICMI President, Lichnerowicz would chair the French Ministerial Commission on the teaching of mathematics, known as the Commission Lichnerowicz, responsible for the renovation of mathematics education at primary and secondary levels according the principles of the Modern Mathematics. The volume also included a list of symposia on mathematics teaching, a first list of journals related to this subject and a list of centres where the problems concerning mathematics education were studied. It can rightly be considered the expression of a community, that of researchers in education, which was gradually assuming an identity and transforming mathematics education into a new scientific discipline. These were the years in which the need for a new field of research, which was not typical of either mathematicians or general educators, was emerging and the creation of university chairs in mathematical education was advocated by important figures such as Wittenberg, Freudenthal, Krygowska; see for example (Kilpatrick 2008; Gispert and Schubring 2011; Furinghetti et al. 2013; Schubring 2016).

I CMI Is Involved in International Congresses on Mathematics Education Thanks to the collaboration with UNESCO and other bodies, such as the IUCST (Inter-Union Commission on Science Teaching),62 chaired by Stone, the previous president of ICMI, a period began that was full of important international conferences on specific problems of mathematics education. Among those in which ICMI was involved, the main ones were held in Frascati (Italy, 8–10 October 1964) on “Mathematics at the coming to university. Real situation and desirable situation”; in

 IUCST (in French CIES, Commission Interunions de l’Enseignement des Sciences) was established in September 1961 by ICSU. CIES was a mechanism that served to coordinate the educational activities of the various scientific unions (Baker 1986). 62

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Utrecht (Holland, 19–22 December1964) on “Modern trends in the teaching of mathematics in secondary schools”; in Dakar (Senegal, 14–22 January 1965) on “Science teaching and economic progress”; in Echternach (Luxemburg, 30 May–4 June 1965) on “The repercussions of mathematics research and teaching”.63 The organizing committee of the colloquium in Utrecht was chaired by Freudenthal and thus reflected to some extent his vision of mathematics education. It was attended by 41 experts on mathematics education (IMN 80, 1965: 3, 8), among whom it is worth mentioning the two secondary school teachers Lucienne Félix and Emma Castelnuovo (EM s. 2, 12 1966: 195–199), members of the CIEAEM.  Most of the lecturers dealt with the modern trends in the teaching of mathematics in secondary schools not by merely summarizing programmes and test results, but rather in an outspoken pedagogical context. Particularly significant was the lecture on “Priorities and Responsabilities (sic) in the Reform of Mathematical Education: An essay in Educational Meta-theory” by Alexander Wittenberg from Canada (EM s. 2, 11, 1965: 287–308). Here he underlined the importance of addressing the meta-questions in education “so as to establish at least a framework of coherent and responsible discussion of the many issues involved” (p. 288); the necessity of creating university chairs for mathematics education; the urgency of founding an international journal specifically dedicated to mathematics teaching so that diverse approaches could be compared. He also dealt with the theme of Modern Mathematics, distinguishing between “pseudo-­ modernization” and “genuine modernization.” The first “is the most external trappings of mathematics-only terminology, some uncalled-for concepts that perform no useful function within actual teaching, some isolated and disconnected semblances of ‘rigorous’ proofs of theorems like this one: A line segment has only one middle point. Probably the most popular example of this pseudo-modernization is the introduction, for its own sake, of the ‘language of sets’ from kindergarten onwards” (p. 305). Instead, “genuine modernization” is that which aims at carrying into the schools some genuine mathematical theories, for instance, some genuine axiomatic geometry “at an appropriate level of care and sophistication” or “some genuine group theory comprising not only the definition of group and some disconnected examples, but a fair amount of substantial theory with applications” (ibidem). The conference in Dakar, Senegal, which was organized by the IUCST, deserves to be mentioned because it saw the participation of nine African and five Asian countries in addition to four American and nine European countries. Moreover, Lichnerowicz managed to organize with the Senegalese national commission a small symposium (13–16 January 1965) on “L’Enseignement des mathématiques dans ses rapports avec celui des autres sciences,”64 in connection with the main conference.

 For details see (Giacardi 2008), in particular Timeline 1960–1966.  EM s. 2, 12, 1966: 131–132; A.  Lichnerowicz to A.  Delessert, Paris, 27 May 1964, Fondo Delessert, Serie VI, u. a. 412. 63

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There are other reasons to mention the conference held in Echternach, because on that occasion some decisions were taken by the members of the Executive Committee of ICMI—unofficially convened by Lichnerowicz65—that aroused some friction within the Commission. It was decided that the reports on the three topics previously chosen for the ICM in Moscow would be presented respectively by Charles Pisot (the university education of future physicists), Hans Georg Steiner (the axiomatic method in secondary teaching), and Krygowska (development of student mathematical activities and the role of problems). When this proposal was brought to the attention of the Commission, it was approved by all but Otto Frostman, a member-at-large of the Commission, as shown by his correspondence. In fact, he wrote to Delessert You informed me of an informal meeting of certain members (who?) of the ICMI Executive Committee in Echternach, when some proposals were made as to the subjects of the reports and the rapporteurs to the Congress in Moscow 1966, and you submit them to me for approval. For such an important decision, I would have preferred a formal meeting of the Commission, or at least of the Executive Committee, so that the members could have the opportunity to discuss the proposals presented in Paris in February 1964. Can we say something non-trivial about the third subject? As for the first subject, I would like to point out that in Sweden we will soon have a much bigger problem: The need or not for special university courses for prospective secondary school teachers. I am not sure that the subjects proposed are the best possible, but, given the advanced time, they should perhaps be accepted without further discussion.66 (O.  Frostman to A.  Delessert, Djursholm 14 June 1965, Fondo Delessert, Serie III u. a. 177)

In his reply, Delessert specified that the Paris meeting (February 1964) in which those proposals were presented was an official meeting and explained why the choices were made. He also informed him of his idea to create a permanent institution, a kind of office where all the ICMI archives could be kept. In fact, due to the lack of a permanent secretariat, the contacts and actions implemented by the president and secretary of a certain period are no longer available for the following officers.

 Only the participants in the conference were present.  The original text is: “Vous m’avez communiqué d’une réunion officieuse de certains membres (qui?) du Comité Exécutif de la CIEM à Echternach, à laquelle on a fait quelques propositions quant aux sujets des rapports et aux rapporteurs au congrès à Moscou 1966, et vous me les soumettez pour approbation. Pour une décision d’une telle importance, j’aurais préféré une réunion officielle de la Commission, ou au moins du Comité Exécutif, pour que les membres auraient l’occasion de discuter les propositions présentées à Paris en février 1964. Est-ce qu’on peut dire quelque chose non-banal du troisième sujet? quant au premier sujet je voudrais remarquer que dans la Suède nous aurons bientôt un problème beaucoup plus important: La nécessité ou non de cours universitaires particuliers pour les futurs professeurs de l’enseignement secondaire. Moi, je ne suis pas sûr que les sujets proposés sont les meilleurs possibles, mais, vu le temps avancé, il faut peutêtre les accepter sans discussion ultérieure.” 65 66

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2.7 The Freudenthal Era: ICMI Renaissance In 1966, Lichnerowicz’s mandate ended, but the IMU Executive Committee had already begun to think about the future president of ICMI in May 1965. As appears from a letter of the IMU secretary Chandrasekharan dated 25 May 1965, the office was initially proposed to Børge Jessen, the former Secretary of the Interim Executive Committee of the IMU 1950–1952, because of his “high standing as a research mathematician, coupled with a keen interest in mathematical education and a wide experience of administrative affairs.”67 The choice fell ultimately on Freudenthal,68 a charismatic personality whose broad mathematical knowledge was joined by a profound interest in education, and who had a talent for organisation and the independent spirit necessary to mark a turning point in ICMI activities. His interest in mathematics education had its roots far back in his early days in the Netherlands (Smid 2009, 2016). Freudenthal’s official presence in ICMI started as a delegate appointed by the Dutch committee, and as mentioned above (§ 2.4) starting in 1955, he participated in the activities of ICMI. At the meeting of the ICMI EC held in Geneva on 2 July 1955, he outlined a new way of carrying out the inquiries of ICMI: He [Freudenthal] thinks that the subjects of inquiry proposed are too general in nature, and that they risk giving rise, on the part of the National Subcommissions, to reports which would involve questions of organization and administration rather than strictly scientific questions; however, ICMI must rather encourage research and studies of scientific kind. Freudenthal thinks that the best way to avoid the danger he points out is to propose limited and specific topics for investigation.69 (EM s. 2, 1, 1955: 200)

Moreover, he advocated involving secondary teachers because of their experience in the field and having experimental data available for carrying out studies on mathematics teaching; he maintained the importance of psychology in dealing with educational problems, and of research in didactics (§ 2.5). These ideas inspired his mandate as president in ICMI and marked the ICMI renaissance in the 1960s. In his actions he enjoyed the support of the most important scholars in mathematics education, including Maurice Glaymann, Krygowska, André Revuz, Willy Servais among others, some of them active in the CIEAEM milieu. The time was ripe for new initiatives in mathematics education to flourish. In fact, in 1967, the Nordic Committee for the Modernization of School Mathematics (Denmark, Finland, Norway, and Sweden) presented a new syllabus inspired by New Math. One of the best-known members of this Committee was Bent Christiansen (Denmark) who would become Vice-President of ICMI in 1975. In 1968, the  K. Chandrasekharan to B. Jessen, n. p., 25 May 1965, Fonds de Rham, 5113-403.  See the portrait by Schubring in Part III of this volume. 69  The original text is: “Il estime que les sujets d’enquête proposés ont un caractère trop général, et qu’ils risquent de donner lieu, de la part des Sous-commissions nationales, à des rapports qui feront intervenir des questions d’organisation et d’administration plus que des questions proprement scientifiques; or, la CIEM doit plutôt susciter des recherches et des études dans l’ordre scientifique. M. Freudenthal pense que le meilleur moyen, pour éviter le danger qu’il signale, est de proposer des sujets d’enquête limités et précis”. 67 68

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Zentrum für Didaktik der Mathematik was founded in Karlsruhe by Steiner and Heinz Kunle. This was followed in 1973 by the IDM (Institut für Didaktik der Mathematik), founded in Bielefeld by Heinrich Bauersfeld, Michael Otte and Steiner whose aims combined practice in school and theoretical research. Starting from 1969, the first IREMs (Instituts de Recherche sur l’Enseignement des Mathématiques) were established in Paris, Lyon, and Strasbourg. In 1971, Hans Freudenthal himself founded the Instituut Ontwikkeling Wiskunde Onderwijs (IOWO, Institute for the Development of Mathematics Teaching).

The Creation of a New Journal and the Starting of ICMEs At the beginning of 1967, the new EC of ICMI began its mandate. Freudenthal was flanked by the two Vice-Presidents, Edwin Moise and Sobolev, and Delessert was confirmed as secretary.70 From the very beginning, he intended to renovate ICMI activities. The inspiring principles of his actions emerged clearly at the UNESCO colloquium in Lausanne on “Coordination of Instruction of Mathematics and Physics” (18–20 January 1967), in which he participated together with other important protagonists of mathematics education of those years, such as Krygowska, Revuz and Servais. These principles, “the mile-stone in the philosophy of mathematical education” (Freudenthal 1968a, p. 3) arose from the results of an inquiry promoted by the journal Dialectica71 and were published as “Propositions on the teaching of Mathematics” in the first issue of the journal Educational Studies in Mathematics.72 The main points are the following: (1) Mathematics constitutes a unique and characteristic activity of human mind. All children have a right to be educated through mathematics. (2) … [Mathematics education] must provoke and develop in the first place the capacity of intellectual action instead of merely piling up knowledge. (3) Mathematics develops more and more towards a general science of structures. These structures charge it with a remarkable power of application, information and unifica-

 The Executive Committee of ICMI and the Members for the period 1967–1970 were the following: President: H.  Freudenthal, Vice-Presidents: E.  Moise, S.  Sobolev, Secretary: A.  Delessert, Members: H. Behnke, A. Revuz, B. Thwaites, Ex officio: H. Cartan (President of IMU). 71  See the detailed report by Delessert together with some comments by Freudenthal, Gonseth, Eric Emery, and Krygowska: Enquête sur l’enseignement de la mathématique, Dialectica 21, 1967: 204–223. 72  A document from the ICMI Archives shows the signatories: M. W. Servais (Belgium), M. R. Guy (Canada), M. J. Lichtenberg (Denmark), M. C. Pisot (France), Mme P. Gadon [?] and M. A. Rényi (Hungary), M. C. Cattaneo (Italy), M. H. Freudenthal and M. L. N. H. Bunt (the Netherlands), M. Z. Krygowska and M. S. Straszewicz (Poland), M. E. Blanc, M. A. Delessert, M. E. Emery, M. F. Gonseth, M. K. Grimm, M. J. de Siebenthal (Switzerland) and M. I. Smolec (Yougoslavia). (IA 14B, 1967–1974, Propositions sur L’Enseignement Mathématique, Lausanne le 18 Janvier 1967). 70

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F. Furinghetti and L. Giacardi tion. The knowledge and the mastership of these structures, its utilization in the grasp of reality are the real objectives of mathematics teaching. … (7) The reformation of mathematics teaching has to be considered a permanent process. This implies a continuous retraining of the teachers which is based on regular pedagogical research. (8) An effective global collaboration in this field will become indispensable. It is an urgent requirement to establish an international organism for information on the teaching of mathematics. (Educational Studies in Mathematics 1968a, 1, p. 244)

On 15 February, Freudenthal sent his first letter to the national subcommissions, in which he communicated the composition of the new EC and the list of “national nominees” and above all stressed the importance of having a plenary meeting of ICMI.73 He intended to involve all the components of ICMI more closely, and at the same time make relations between the members of the EC less formal, as he wrote to Delessert: In the future I will seek your help to communicate with CIEM. On the other hand, with regard to the internal communication of the office, it seems to me better that its members consider each other as equals and that each communicates with all the others without any formality.74

The ICMI Colloquium was held in Utrecht from 21 to 25 August 1967 and it was attended by 34 active and 34 passive participants.75 The theme chosen was “How to teach mathematics so as to be useful.” In his introductory address, Freudenthal justified this choice, explaining his “educational philosophy”: Systematization is a great virtue of mathematics, and if possible, the student has to learn this virtue, too. But then I mean the activity of systematizing, not its result. Its result is a system, a beautiful closed system, closed with no entrance and no exit. In its highest perfection it can be even be handled by a machine. But for what can be performed by machines, we need no humans. What humans have to learn is not mathematics as a closed system, but rather as an activity, the process of mathematizing reality and if possible, even that of mathematizing mathematics. (Freudenthal 1968a, p. 7)

Among the lectures there were those by Delessert, Arthur Engel, Trevor J. Fletcher, Glaymann, Krygowska, Henry O. Pollak, Revuz, Servais, and Steiner. It is worth mentioning that Delessert concluded his talk hoping for the creation of a new journal aimed at disseminating the results of research in mathematics education

 He wrote: “Since non plenar [sic] meeting of ICMI has been held for many years, I think it would be helpful to discuss ICMI’s activity anew. I propose you to have an ICMI-colloquium, which will be held in Utrecht (presumably from August 21 to 25).” (H.  Freudenthal to the National Subcommissions, 15 February 1967, IA, 14B 1967–1974). 74  The original text is: “A l’avenir je me servirai bien de votre aide pour communiquer avec CIEM.  D’autre part, quant à la communication interne du bureau, il me semble mieux que ses membres se considèrent l’un l’autre comme des égaux et que chacun communique avec l’ensemble des autres sans aucune formalité.” H. Freudenthal to A. Delessert, Utrecht 22 March 1967, Fondo Delessert, Serie V, u. a. 347. 75  IMN 91, 1969: 3. 73

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(Delessert 1968, p.  88).76 On 26 August 1967, at the end of the colloquium, an important meeting of the Executive Committee of ICMI took place; in fact, mainly under the stimulus of Freudenthal, important decisions were delivered both from an organizational point of view and for the issues to be discussed in the future congresses (EM s. 2, 13, 1967: 245–246). First of all, Freudenthal claimed that the practice of quadrennial reports of ICMI at the ICMs was not a good one because the national reports were generally useless; he launched the idea of an ICMI congress to take place the year before the ICM, with invited lectures and communications. The Assembly approved the proposal and Glaymann offered to organize such a congress in 1969 in France. Moreover, the need for a journal completely dedicated to the teaching of mathematics was again expressed. In particular, Revuz observed that the level of L’Enseignement Mathématique was too high and solicited the founding of a new journal expressly addressed to teachers at the secondary level. A commission that included Behnke, Châtelet, Peter Hilton, Josef Novak, Angelo Pescarini, Steiner, Bryan Thwaites and Freudenthal was created to study this question (EM 13, 1967: 245). A year later, Freudenthal launched the new journal Educational Studies in Mathematics (Hanna and Sidoli 2002). The first issue included the proceedings of the Utrecht Colloquium and the recommendations on the coordination of the teaching of mathematics and physics issued from the meeting in Lausanne mentioned above. Finally, the ICMI EC, in order to improve the activities of the national subcommissions, proposed increasing the number of the members-at-large of the Commission and to think about the creation of a permanent secretariat of ICMI. Actually, this second proposal, already put forward by Delessert (see above), had been rejected by the EC of IMU about two months earlier, because it was not “convinced of the urgent necessity for such a secretariat.” However, the EC also asked “what its exact functions would be, and what financial implications of this project would be” (O. Frostman to the International Commission on Mathematical Instruction, Djursholm, 29 June 1967, IA 14B, 1967–1974). With regard to the issues to be discussed in the future meetings, Freudenthal suggested not to include topics concerning programs and school organization, and for the next congress in Bucharest to be held in 1968 he proposed the following topics: mathematization; motivation; how to teach mathematics without a schoolteacher; comparative evaluation of the contents of mathematics courses; criteria of success; evaluation of the results of research in mathematics education; research methodology. These were largely new topics and the conference that took place in Bucharest from 23 September to 2 October 1968 did not accept all of these suggestions and dealt with “Modernization of Mathematics Teaching in European Countries”; perhaps one of the reasons was that it was organized by UNESCO, which was more interested in mathematics as an outpost of the contemporary scientific-technical

 The text of the talks given in Utrecht will be published in the new journal Educational Studies in Mathematics 1, 1–2, 1968: 1–243. 76

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revolution and in its role in modern society. Freudenthal, who represented ICMI together with Delessert, discussed a theme dear to him with a wealth of examples: the integration of mathematics into students’ general activity and with other disciplines (Freudenthal 1968b, 1969). Among the final recommendations, the role of ICMI was stressed and the hope was expressed that in the future the national subcommissions would be appointed by the national association of mathematics teaching.77 The collaboration between ICMI and UNESCO was consolidated by Freudenthal with the conclusion on 6 March 1967 of a contract between the two bodies for the publication of the second volume of the New Trends in the Teaching of Mathematics series, without any reference to IMU – as there should have been. (G. Laclavère to O. Frostman, Paris, 23 August 1968, with a copy of the contract, IA 14B, 1967–1974). The collaboration between ICMI and UNESCO became more and more intense in the following decades, helping in organizing and financing the International Congresses on Mathematical Education (ICMEs) and especially ICMI regional groups in Africa, Latin America, Southeast Asia (Jacobsen 1993, p. 11). A significant role was played by Edward Jacobsen who began to work for UNESCO in 1969 and was responsible for its Mathematics Education section from 1976 to 199278 (see Hodgson 2009 and Chap. 3 by Marta Menghini in this volume).

New Friction with IMU As shown by the correspondence held in the ICMI Archives, Freudenthal carried out the two initiatives—the new journal and the ICMI Congress—and signed a contract between ICMI and UNESCO, without either consulting or informing IMU.  The IMU President was an ex officio member of ICMI and so he had to be informed on its activities.79 Furthermore, the new directors of L’Enseignement Mathématique, François Châtelet, Georges  de Rham and Raghavan Narasimhan, had not been informed either. On 2 December 1967, IMU secretary Frostman wrote to Freudenthal in an attempt to dissuade him from both initiatives: I must admit that I am not too happy about the new pedagogical journal. Do you really think that there is a market for two international journals of that kind (I do not)? If you are not satisfied with L’Enseignement, ICMI’s official journal, perhaps it would be better to try to reform it. And I am afraid too that in a new journal the “modernizers” of the extreme sort would try to be very busy. At least I ask you to be cautious. I can agree with very much of your criticism of the meetings of ICMI at the International Congresses, but I am not sure that ICMI should isolate itself from those who have, primarily, a scientific interest but who have, nevertheless, very often taken part in the discussions  Colloque International UNESCO Modernization of Mathematics Teaching in European countries. Bucharest: Éditions didactiques et pédagogiques, 1968, p. 555. 78  See https://www.mathunion.org/icmi/news-and-events/2011-04-14/ed-jacobsen-receives-luissantalo-medal (Retrieved June 2021). 79  See (Furinghetti and Giacardi 2010) and (Furinghetti et al. 2020). 77

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of ICMI.  And a special ICMI congress in France in 1969 will cost a lot of money. (O. Frostman to H. Freudenthal, n. p., 2 December 1967, IA, 14B 1967–1974)

On 20 December, Freudenthal replied: I would like to reassure you about the new pedagogical journal. The provisional list of editors does not include any “radical”. In spite of its name, Enseignement has never been a pedagogical journal. Its contributions on education were not pedagogical but organisatory and administrative. I do not believe it is possible to reorganize a journal so fundamentally. (Freudenthal to Frostman, Utrecht, 20 December 1967, IA, 14B 1967–1974)

A letter from Delessert dated December of the same year seems to testify to his attempt to mediate between IMU and Freudenthal: The journal you want to create and l’Enseignement Mathématique do not have the same goals. However, I would very much regret if it was decided, tacitly, not to longer consider l’Enseignement Mathématique as the official organ of ICMI. Does the contract to be concluded with Mr. Reidel contain a clause on this subject? Let me address another point. I recently met Mr. Henri Cartan, President of UMI. He told me he had no information about our commission. I sent him the report of the Utrecht meeting and I think it would be good to keep him informed on the decisions affecting the future of ICMI. (A.  Delessert to H. Freudenthal, Riex, 24 December 1967, Fondo Delessert, Serie V, u. a. 349)80

De Rham also complained to Delessert about not being informed about the creation of the new journal (G. de Rham to A. Delessert, Lausanne, 24 December 1967, Fondo Delessert, Serie VI, u. a. 503) Concerning the new journal, Frostman wrote to Freudenthal once again on 2 January 1968: I am still a bit afraid that the market will be hard for two publications, even if the new journal will mainly stress other points than L’Enseignement. (O. Frostman to H. Freudenthal, 2 January 1968, in IA, 14B 1967–1974)

Delessert’s advice and the IMU protests did not help much. In fact, in March 1969, Frostman complained about not having received any report on ICMI activities (O.  Frostman to A.  Delessert, n. p., 16 March 1969, in IA, 14B 1967–1974). Delessert answered him, announcing the title of the new journal, Educational Studies in Mathematics, with Freudenthal as editor; the date and place of the first ICME (Lyon, 24–30 August 1969); the preparation of the journal Zentralblatt für Didaktik der Mathematik, published as a collaboration between ICMI and the Zentrum für Didaktik der Mathematik of the University of Karlsruhe. He excused the delay in providing information by saying that the greater part of the work of secretary was performed by Freudenthal’s secretary’s office and he was not always well informed about the planned activities (A. Delessert to O. Frostman, Riex, 22  The original text is: “La revue que vous désirez créer et l’Enseignement Mathématique ne visent pas les mêmes buts. Toutefois, je regretterais beaucoup que l’on décide, tacitement, de ne plus considérer l’Enseignement Mathématique comme l’organe officiel de l’ICMI. Le contrat à conclure avec M. Reidel contient-il une clause à ce sujet? Permettez-moi d’aborder encore un autre point. J’ai rencontré récemment M. Henri Cartan, président de l’UMI. Il m’a dit n’avoir aucune information sur notre commission. Je lui ai envoyé le compte-rendu de la séance d’Utrecht et je pense qu’il serait bon de le tenir au courant des décisions liant l’avenir de l’ICMI”. 80

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March 1969, in IA, 14B 1967–1974). Thus, IMU was faced with decisions already made. From 24 to 30 August 1969, the first International Congress on Mathematical Education (ICME-1) was held in Lyon. The French Government and UNESCO gave financial subventions. The congress, attended by 655 active participants from 42 countries, was distinguished from the previous congresses in both the structure and the setting, which encouraged discussion and active participation. It was a success for the high level of the lectures and marked a turning point in the history of the ICMI.81 The main resolutions concerned the modernization of the teaching of mathematics, both in content and method; the collaboration between teachers of mathematics and those of other disciplines; international cooperation; the permanent training of the teachers; the place of “the theory of mathematical education” in universities or research institutes (ICMI Bulletin, 1975, 5: 20–24). Among the many interesting interventions, we mention that of Edward G. Begle, which focuses on what the characteristics of mathematics education as a scientific discipline must be: I see little hope for any further substantial improvements in mathematics education until we turn mathematics education into an experimental science… We need to start with extensive, careful, empirical observations of mathematics teaching and mathematics learning. Any regularities noted in these observations will lead to the formulation of hypotheses. These hypotheses can then be checked against further observations, and refined and sharpened, and so on. To slight either the empirical observations or the theory building would be folly. They must be intertwined at all times. (Proceedings ICME-1 1969, p. 110, also in EM 2, 2/3, 1969–70: 242)

In the course of the ICMI meeting that took place the day before ICME-1, Freudenthal explained the reasons for the changes made: although the small congresses dedicated to well-defined topics could be useful, it was important to go beyond the circle of specialists and reach the teachers, thus large congresses were necessary. He also stressed that it was necessary that all the national subcommissions work and collaborate, and for this it was indispensable to choose people who were genuinely interested in mathematics education. IMU President Henri Cartan replied that the section of the ICMs dedicated to the teaching of mathematics had to be retained anyway and that the members of the national subcommissions of ICMI had to be designated by the IMU subcommittee, also by consulting the competent organizations.82 In the ICMI meeting report, Delessert also mentioned the collaboration between ICMI and IUCST. In fact, at the same time as the presidency of ICMI, Freudenthal held the position of president of the IUCST and as such he had reorganized it,  IMN 97, 1971: 4–5. For details, see Chap. 3 by Menghini and Chap. 10 by Furinghetti in this volume. 82  Compte-rendu de la séance de la CIEM tenue à Lyon, le 23 août 1969, à 14 heures, à l’occasion du premier Congrès International de l’Enseignement Mathématique, in IA, 14B, 1967–1974. See also Circulaire ICMI, 13 October 1969, Fondo Delessert, Serie III, u. a. 228. 81

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carrying out with the help of the secretary Pierre Fleury an intense activity also in collaboration with the national subcommissions of ICMI.  Frostman, learning of this, wrote to Delessert: I am also not very keen on mentioning that the President “took an active part in the work of IUCST,” because this Commission is dissolved in its present form (see report from General Assembly of ICSU, Paris, 28 Sept. 1968, p. 170). (O. Frostman to A. Delessert, Djursholm 19 October 1969, Fondo Delessert, Serie VI, u. a. 525)

Freudenthal, advised by Delessert, replied to Frostman with a harsh letter which once again shows the tension between ICMI and IMU in this period: The IUCST has existed till a few days ago, when it was actually dissolved. … I remark that the idea of reorganization of IUCST has been mine. IUCST did not work properly that is, besides the secretary and me, there were almost no active people in it. From IUCST I got just the day before yesterday a kind letter by which they thank me for the work I did as a President of IUCST. You wrote to Mr. Delessert that you would prefer not to mention this activity of mine. It does not regard me. I enjoyed the work I did in IUCST and I am sure that people who saw it, did appreciate it. This is the only thing that matters. My work in IUCST has come to an end, since, as it has come clear from the correspondence with ICSU, the Union has not nominated me as a member of the new Commission. It does not regard me either. If I have a few ambitions, they are of an entirely different kind. (H. Freudenthal to O. Frostman, n. p. 3 November 1969, Fondo Delessert, Serie VI u. a. 525)

Frostman responded and justified himself by saying that he knew nothing about the reorganization of the IUCST, which shows that certainly there was a lack of communication between the two bodies.

The End of the Mandate As the end of Freudenthal’s mandate approached, the IMU Executive Committee had to choose the future ICMI president. Cartan, President of IMU, consulted the outgoing president on this. The answer by Freudenthal83 clearly shows his policies and the qualities he considered essential for an incisive position in ICMI. He begins with the reflection that “the problem of CIEM is that it does not have either its own office or any other permanent centre. It depends solely on how active the president is and on the administrative assistance available to him” and thus if the president is one “less active or less inclined to innovation, the CIEM is lost.”84 The qualities that he believed a president had to possess are:

 H. Freudenthal to H. Cartan, Utrecht, 29.6.1970, IA 14B 1967–1970. This letter is transcribed in (Furinghetti and Giacardi 2010, pp. 46–47). 84  The original text is: “Le problème de la CIEM est ce qu’elle n’a ni de bureau ni d’autre noyau permanent. Elle dépend de l’activité́ du président et des facilités administratives dont il dispose. Jusqu’alors presque chaque président a commencé de nouveau; si c’était un homme moins actif ou moins inventif, la CIEM serait perdue”. 83

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–– Familiarity with questions relative to secondary or primary teaching –– A thorough awareness of the international situation and of the international sphere of those who are concerned with teaching –– A spirit of initiative He also notes that the situation could be complicated by the fact that up to that time the presidents had been chosen from among the university professors, taking also into account their nationality. The person that Freudenthal considered by far the worthiest and most capable from all points of view was Revuz, but he also mentioned Servais, a Belgian high school teacher and later study prefect, Krygovska, director of the Normal School in Kraków, and Bent Christiansen, director of an institute for teacher training in Copenhagen. But he feared that they would not be considered to have the “mathematical weight necessary for a president of the CIEM” (le poids mathématique nécessaire au président de la CIEM). Freudenthal named others as well: Pollak (Bell Telephone Laboratories, USA) and the Indian Jagat Narain Kapur of the Indian Institute of Technology in Kanpur, India, noting that all of them deserved to become members-at-large. Only at the end of his letter did Freudenthal note that since the second ICME would take place in Exeter, it was natural to seek a president from Great Britain: he cited Maxwell, Robert Cranston Lyness, and Bryan Thwaites, but not James Lighthill, who would actually become the president. These reflections are indicative of the importance he placed on a thorough understanding of questions relative to education, as well as his open-mindedness towards secondary school teachers, women, and emerging countries. Freudenthal’s suggestions were accepted by IMU EC only to a small extent. However, in a letter shortly before the IMU General Assembly held in Menton from 28 to 30 August 1970, Cartan wrote to Lighthill, future President of ICMI, suggesting that Freudenthal be kept in the Commission as past President of ICMI.  He also added: You will have to study the problem of choosing a Secretary; he must necessarily be part of the Commission, but not necessarily as a “member at large”. I believe that the current secretary, Delessert (Switzerland) could give way to a new man; in my opinion, the relationship between the President and the Secretary was not satisfactory, the Secretary having become a mere “mailbox” of the President. Perhaps would it be desirable that the Chairman and the Secretary belong to the same country, which would facilitate the exchange of views between them? this is not a suggestion on my part; it’s just a question I ask, and which I would like to have an answer to it. (H. Cartan to J. Lighthill, Die, 20 August 1970, in IA, 14B 1967–1974).85

 The original text is: “Vous aurez à étudier le problème du choix d’un Secrétaire; il doit nécessairement faire partie de la Commission, mais pas nécessairement comme “member at large”. Je crois que l’actuel secrétaire, Delessert (Suisse) pourrait céder la place à un homme nouveau; à mon avis, les relations entre le Président et le Secrétaire n’étaient pas satisfaisantes, le Secrétaire étant devenu une simple “boîte aux lettres” du Président. Peut- être serait-il souhaitable que le Président et le Secrétaire appartiennent au même pays, ce qui faciliterait les échanges de vues entre eux? ceci 85

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During the General Assembly of IMU in Menton, Lighthill was elected ICMI President for the coming four-year term. Lighthill had a great interest in mathematical education at all levels focusing on the teaching of applied mathematics, which was his field of research, and in the 1960s he was involved in an advisory capacity in the creation of the School Mathematics Project.86 The Assembly also decided that the past President of ICMI, the secretary of IMU and the representative of the International Council of Scientific Unions (ICSU) Committee on the Teaching of Science should be ex officio members of ICMI (IMN 99, 1971: 16).87 In the new EC,88 Freudenthal appeared only as an ex officio member. During the ICMI meeting, which took place on 5 September 1970 on the occasion of the ICM in Nice (EM s. 2, 16, 1970: 197–199), the outgoing ICMI President presented the IMU’s decisions regarding the composition of the ICMI Executive Committee for the period 1971–1974. From the discussion that followed, two objections emerged: –– The members of the Commission are elected by people who are not particularly competent in elementary mathematical education (statement by Papy, Freudenthal, Behnke, Kurepa); –– The elected members do not represent the various current trends in elementary mathematics education (Papy).89

Consequently, the section formulated the recommendation that the regulations establishing the ways that ICMI members were designated had to be modified; and that ICMI members were always to be chosen from among those who were effectively involved with mathematics teaching. Responding to Frostman as to the first objection, Freudenthal wrote: I admit it looks strong. This, however, reflects the actual discussion in which much stronger terms have been used. The disapproval of the way in which the new members at large were n’est pas une suggestion de ma part; c’est simplement une question que je pose, et à laquelle j’aimerais que l’on puisse répondre”. 86  See the biographical portrait by Adrian Rice in Part III of this volume. 87  It is worth mentioning that only the Terms of Reference of 1982 included explicitly the pastpresident among the Officers of the EC: “The Executive Committee consists of four officers, namely, President, two Vice Presidents, and Secretary, and of three further members. Furthermore, the outgoing President of ICMI, the President and the Secretary of IMU, and the representative of IMU at CTS (ICSU) are members ex-officio of the E.C.” ICMI Bulletin, 13 (February 1983): 5. 88  The new EC was formed as follows: President: M.  J. Lighthill; Vice-Presidents: S.  Iyanaga, J. Surányi; Secretary: E. A. Maxwell; Members: H. O. Pollak, S. L. Sobolev; Members-at-large of ICMI: M. Barner, F. Châtelet, A. Gleason, L. Lombardo Radice, Y. Mimura, J. Novak. The ex officio members of ICMI were: H. Freudenthal (past president of ICMI), H. Cartan (delegated by the president of IMU K. Chandrasekharan), O. Frostman (secretary of IMU), A. Lichnerowicz (CTS/ ICSU). (See ICMI Bulletin 1, 1972: 8–9; K. Chandrasekharan to J. Lighthill, Zurich, 18 January 1971, IA, 14B 1967–1974) 89  The original text is: “Les membres de la Commission sont élus par des personnes qui ne sont pas particulièrement compétentes en matière d’enseignement mathématique élémentaire (déclaration de MM. Papy, Freudenthal, Behnke, Kurepa); Les membres élus ne représentant pas les diverses tendances qui se manifestent actuellement dans l’enseignement mathématique élémentaire (M. Papy)”. (EM s. 2, 1970,16, p. 198)

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F. Furinghetti and L. Giacardi appointed was unanimous. … I would suggest that this is taken up as a serious problem by the new Executives of IMU and ICMI. (H. Freudenthal to O. Frostman, Utrecht, 23 October 1970, in IA, 14B 1967–1974)

Another point discussed during the meeting concerned the organisation of ICME-2: the Congress would take place in Exeter (UK) in 1972 and would be structured differently from the preceding one; a limited number of plenary lectures on themes of general interest was planned, and working groups would be constituted for addressing more specialised topics. Strictly speaking, the discussion on ICME-2 should have involved the new EC and its President Lighthill, who was not present at the meeting; moreover, on 11 October 1970 Freudenthal sent a letter to ICMI EC with a proposal for the program committee for ICME-2, which did not include Lighthill. He also asked the members of outgoing EC to approve his list, and said the Program Committee should be formed before the end of November.90 Upon receiving the letter, Cartan complained to Frostman: “Freudenthal once again worries me … F. overdoes it somewhat by putting the new commission in front of decisions already made.”91 At the same time, he wrote to Freudenthal specifying that it would be the new ICMI that would decide about the organization of ICME-2. He also requested the rectification of the sentence in the minutes of the session on 5 September 1970, concerning the constitution of the new executive committee of ICMI: “The constitution of the new Executive Committee of C.I.E.M. will be done by consultation among the members already designated to new Commission,”92 that is, Cartan underlined that first each national subcommission had to designate its representative. He also added that this task mainly concerned the new President Lighthill. Freudenthal replied that everything was done in agreement with Lighthill and as for the minutes of the meeting in Nice, he remarked that they were full of errors and that he had already invited Delessert to correct them (H.  Freudenthal to H.  Cartan, n. p., 19 October 1970, IA, 14B 1967–1974). In the past, Freudenthal had already criticized the fact that at each change of the ICMI’s EC there were few if any contacts between the old and the new commission; for this reason, as he wrote to Frostman: …it was for me the most natural thing that I took no step as a president without close contact with the new officers, in particular with Mr. Lighthill. It is a pity that I did not mention explicitly what I considered as the most obvious thing of the world, and by this way caused this strange consternation. (Freudenthal to Frostman, Utrecht, 23 October 1970, in IA, 14B 1967–1974)

With a practical and constructive spirit, Freudenthal stressed the importance of the continuity in the action of the ECs. Lighthill supported him in this attitude and  H.  Freudenthal to the Executive Committee of ICMI, Utrecht, 11 October 1970, IA, 14B 1967–1974. H. Cartan to O. Frostman, Paris, 15 October 1970, Ibidem. 91  The original text is: “Freudenthal me donne encore du souci… F. exagère un peu en mettant la nouvelle Commission internationale devant un fait accompli”. (H. Cartan to O. Frostman, Paris, 15 October 1970, IA, 14B 1967–1974) 92  “La constitution du nouveau Comité Exécutif de CIEM se fera par concertation entre les membres déjà désignés de la nouvelle Commission.”, Ibidem. 90

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showed appreciation for his predecessor when he proposed him to Frostman as Vice-President, but this proposal was not accepted: Actually, Maxwell and I do regard with very great urgency the importance of taking action during the meeting of the General Assembly of ICMI at Exeter to appoint Vice-Presidents with correct composition of our Executive Committee… Perhaps I might mention to you in confidence that the persons that we are intending to become Vice-Presidents are Professor Freudenthal and Professor Revuz. (J. Lighthill to O. Frostman, Cambridge 24 July 1972, IA, 14B 1967–1974)

During his term as President, Freudenthal was completely independent regarding financial matters as well, as shown by a letter from Frostman to Cartan: I have not paid anything to the ICMI secretariat during the last years... I don’t have exact information about ICMI’s affairs. (O. Frostman to H. Cartan, n. p., 15 November 1970, in IA, 14B 1967–1974)

This is confirmed by a letter to Lighthill by Cartan, who wrote that IMU had provided no funding for ICME in Lyon for the simple reason that nothing was ever requested, and that he had not asked for any funding from UNESCO because Freudenthal had gone to UNESCO directly. He also underlined once again that the decision to hold ICMI congresses independent of the ICMs was made by Freudenthal without his having ever consulted IMU, and he hoped that Lighthill would establish closer and more confidential relations with the Union (H.  Cartan to J.  Lighthill, Paris, 30 November 1970, IA, 14B 1967–1974). It is worth mentioning that the EC of IMU in 1972 proposed that ICMI officers other than the president be elected by the IMU General Assembly itself, but Lighthill wanted it to remain in the hands of ICMI (E. Maxwell to O. Frostman, Paris, 18 February 1973, IA, 14B 1967–1974).

2.8 Conclusions The old Commission was born and developed with the goals illustrated in Chap. 1 under the energetic guidance of Felix Klein. After World War II, those goals were outdated and the context required the definition of new suitable goals and an organization aimed at achieving them. In the first phase of the history of ICMI, the figure of Klein was fundamental in directing the activities of the Commission, but these appeared to be the result of an international network made up of the various national subcommissions. Instead in the period we have just analysed (1950–1970), the national subcommissions no longer played a productive role; indeed, for many countries, they barely existed. The history of ICMI in these two decades appears to be above all a history of the presidents who by turns directed the activities of the Commission by impressing the mark of their personality, their interests and the vision of the role that this Commission should, according to them, play at the international level. If we exclude Châtelet, who despite being a very politically committed mathematician in his own country, France, and active in the field of mathematics education, all the other

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Presidents—Behnke, Stone, Lichnerowicz and Freudenthal—pursued very specific objectives for ICMI by influencing its activities. It is surprising, for the reasons mentioned above, that Châtelet (1952–1954) took such little interest in ICMI. In the private correspondence between the officers of both ICMI and IMU, there are frequent references to his lack of collaboration both with the EC of IMU and with the then secretary of ICMI Behnke, who, for example, wrote rather anxiously to IMU secretary Bompiani: “Six weeks have passed since I have written this letter, but Châtelet did not answer until today. And with my preceding letters it was the same.”93 As we have underlined (§ 2.3) various factors may have caused Châtelet’s inertia: the difficulty of reconstructing ICMI after World War II, the lack of precise Terms of Reference to regulate the relations with IMU, too many organizational and institutional commitments in his own country, and also the fact that Behnke made up for his shortcomings. After the transition period under the presidency of Châtelet, there were initiatives in various directions carried out by the four successive presidents. With regard to policy issues, we may identify two lines of interest by the presidents. Behnke and Freudenthal worked to clarify the relationship with IMU in order to obtain a reasonable autonomy of ICMI from it, and to introduce new educational issues. For Behnke, the time were not yet ripe for radical action, which was later implemented by Freudenthal. The action of Stone and Lichnerowicz was more externally projected, establishing contacts with official institutions and expanding the scope of ICMI. Their period was the period of Modern Mathematics and they organized conferences and publications. Behnke, first as secretary (1952–1954) and then as President of ICMI (1955–1958) intended to recreate the climate of fervour and international collaboration that had characterized Klein’s era, but the historical situation of the 1950s and the needs of society were very different. He himself, to a certain extent, was conscious of this; in his autobiography he wrote about the ICM in Amsterdam in 1954: “At that time, essentially only a survey could be made – but not to the same extent, as Klein has carried out” (Behnke 1978, pp. 266–267).94 Moreover, he did not have the omnipresent role as organizer in both the national and international arenas. Nevertheless, he was aware of the organization and policy problems to be faced to revitalize the Commission: relationships with IMU, funding, visibility, and above all the difficulty of finding mathematicians active in research who were interested in teaching, as he wrote to Stone: “It is very difficult matter to engage mathematicians, well-­ known for their research work, into problems of instruction… because they regard this kind of work of little value.”95 Therefore, during his mandate he tried on the one hand to make ICMI less dependent on IMU, succeeding only to a modest extent and

 See footnote 20.  The original text is: “Damals konnte im wesentlichen nur eine Bestandsaufnahme geleistet werden - allerdings nicht in den Dimensionen, wie Klein sie durchgeführt hat.” 95  See footnote 26. 93 94

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on the other hand to make room for new educational issues (§ 2.4) which would be reprised in the following years. Stone, who after the presidency of IMU, held the position of Vice-President of ICMI (1955–1958) and then that of President (1959–1962), was very active on the international side, especially thanks to his contacts with UNESCO and OEEC and opened ICMI to the stimuli of the Modern Mathematics. He chaired important meetings, among which the “Inter-American Conference on Mathematical Education” held in Bogotá in 1961, and the famous Seminar at the Centre Culturel de Royaumont in 1959, where the introduction of Modern Mathematics in secondary instruction was amply discussed. Moreover, during his mandate a small step was taken towards a greater independence of ICMI with the possibility of co-opting representative of non-IMU countries and of forming regional groups (§ 2.5). Stone’s policies were continued by Lichnerowicz, President in 1963–1966, who managed to enter into special contracts directly between UNESCO and ICMI for the preparation of a report on mathematics teaching at the university level in eight countries and the publication of the series of volumes New Trends in Mathematics Teaching. Moreover, he succeeded in establishing collaboration with the USSR and Africa (§ 2.6). But already in this period the influence of the decidedly stronger and more proactive personality of Freudenthal—who had entered as a member of the EC of ICMI—was being felt. It is indeed Freudenthal, member of the EC in 1963–1966 and president in 1967–1970, who impressed a real turning point in the Commission’s activities— from both an organizational point of view and that of objectives—with the decision to hold ICMI congresses independent of the ICMs, and to create a new journal truly devoted to mathematics education. He also tried to establish continuity with the next EC in order to consolidate the results obtained, all of this at the cost of strong friction with the EC of IMU (§ 2.7). The three main lines of investigation that we have chosen to study the history of ICMI in the 20 years after World War II—the relationships between IMU and ICMI which often resulted in the relationships between pure mathematicians and educators; the emergence of didactics as an autonomous field of research; and the change in the ICMI’s objectives—occur in a transversal way in the work of the various ECs that have followed one another in this period. As we have shown, the friction that marked the relationships IMU-ICMI was due to different reasons: the lack of clear Terms of Reference to regulate relations between the two institutions; the lack of communication between the two ECs; ICMI’s desire for autonomy from too strong control on the part of mathematicians; and the prominent role given by the EC of IMU to the mathematical weight of the ICMI officers with respect to the commitment to mathematics education. Another reason for disagreement was, as we have seen, the impossibility of directly accessing the funding allocated by UNESCO, and instead having to go through the entire hierarchy of international bodies: UNESCO financed IMU through the ICSU and IMU in turn provided funding to ICMI, but these funds were subject to particular limitation. Because of this situation, Stone, as we have seen, insisted that ICMI be consulted on the occasion of the stipulation of a contract between UNESCO and

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IMU on matters relating to the teaching of mathematics (§ 2.5); Lichnerowicz and Freudenthal managed to enter into special contracts directly between UNESCO and ICMI (§ 2.6 and § 2.7). The tensions with IMU are also connected with the gradual emergence of mathematics education as an autonomous research field with its own features and methods, favoured by the interactions with other bodies interested in mathematics education, in particular CIEAEM, some members of which also belonged to the EC of ICMI. This fruitful relationship brought out new themes of investigation in the ICMI agenda (§ 2.3). The first volume of the series New Trends in Mathematics Teaching (1967)—the result of the collaboration between ICMI and UNESCO— could rightly be considered the expression of a new community, that of researchers in math education, which was gradually taking on an identity and creating a new field of research. This fact is epitomized by the Point 5 of the resolutions expressed at ICME-1 (Proceedings ICME-1 1969, p. 284), which reads: The theory of mathematical education is becoming a science in its own right, with its own problems both of mathematical and pedagogical content. The new science should be given a place in the mathematical departments of Universities or Research Institutes, with appropriate academic qualifications available.

The new status of mathematics education as a field of both research and practice, also emerges from the fact that, thanks to Freudenthal, this field could count on its own journal, to which others were added in a few years, and conferences expressly dedicated to it.96 Consequently, the objectives of ICMI also changed: from the comparative study of curricula and teaching methods at the various school levels through inquiries and reports by national subcommissions, distinctive of the first phase of ICMI’s history, they moved on to “inaugurating appropriate programs designed to further the sound development of mathematical education at all levels and to secure public appreciation of its importance.”97 With the presidency of Freudenthal, new topics became central such as: mathematization; motivation; comparative evaluation of the contents of mathematics courses; criteria of success; the connection between mathematics and other disciplines; the development of mathematical activity in pupils; research methodology; and evaluation of the results of research in mathematics education. As (Hodgson et  al. 2013) and (Hodgson and Niss 2018) show, an impressive richness of initiatives and international organizations devoted to mathematics education arose after the Freudenthal era. Hodgson (2009, p. 84) rightly uses the word “turbulence” when referring to the Freudenthal years, but we must recognize that these years were also an incubator of ideas that promoted the ICMI renaissance.

 The election in 2006 of Michèle Artigue, researcher in math education, as president of ICMI, can be considered as the crowning achievement of this parallel process of evolution of the Commission and the community of researchers in math education. In the first hundred years of ICMI among the presidents only David Eugene Smith had been a mathematics educator. 97  See the Terms of Reference of 1954 and 1960. 96

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Chapter 3

The New Life of ICMI: Pursuing Autonomy and Identifying New Areas of Action Marta Menghini

3.1 Introduction As seen in Chap. 2 (Furinghetti and Giacardi, this volume), Hans Freudenthal’s presidency marks a turning point in the life of ICMI, both from the point of view of the organizational structure and from the point of view of action. This gradually resulted on the one hand in a greater autonomy from the community of mathematicians, and on the other hand in the Commission’s definition of its cultural identity. To analyse this evolution, we take the International Congresses on Mathematical Education (ICMEs) as a timepiece: after the first one in 1969 and the second in 1972 they became four-year appointments and marked the life of ICMI.  They gave voice to ICMI’s Executive Committee—starting from its President and its Secretary—and also featured the principal topics and actors within mathematics education from an international perspective. The congresses became an important date in the life of researchers, teachers and people involved in various ways in mathematics education. It is unavoidable to intertwine the life of ICMI with the organization of the ICMEs. Indeed, The ICMI’s primary responsibility is to plan for the ICME’s, which entails choosing from among host country bids, appointing an international program committee to form the scientific program and select presenters, and overseeing progress of the congress preparations (Bass and Hodgson 2004, p. 642).

In the interview released to Schubring (2008), Heinz Kunle, co-founder (as he himself reminds us) with Hans Georg Steiner of the Zentrum für Didaktik der Mathematik (Centre of Didactics of Mathematics) in Karlsruhe and of the Zentralblatt für Didaktik der Mathematik, chair of the German national M. Menghini (*) Sapienza University of Rome, Rome, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_3

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subcommission of ICMI since 1970, and national representative after Heinrich Behnke, stated that ICMI relied heavily on the national commissions for the organization of the ICMEs, in order to know the mainstreams of research and projects in each country and decide about the topics to be faced in the plenaries and in the working groups. The present chapter covers the period from the first ICME in 1969 to ICME-11 in 2008, the year of ICMI’s centennial. Its purpose is to outline the main characteristics of the four-year interval identified by each ICME, with reference also to ICMI activities, to the ICMI studies, to research trends, and to journals. The website on the history of ICMI (Furinghetti and Giacardi 2008) presents not only the portraits of the officers and a rich timeline, but also a brief description of the ICMEs, interviews, documents (including the Terms of Reference), see (Furinghetti and Giacardi 2010; Bass 2008). We will often refer to the interviews as well as to the ICME proceedings, reminding that the latter do not have a uniform structure; for instance, results of individual working groups within an ICME were often published separately. The contents of the website are at the base of our reconstruction of the period under consideration, together with various papers that cover the same topic such as the history of the first 75 years of ICMI outlined by Geoffrey Howson (1984) and the brief note on the ICMI sketched by Bass and Hodgson (2004). The relationship between ICMI and the community of mathematicians is illustrated in the book on the history of the International Mathematical Union (IMU) by Olli Lehto (1998). The events are framed by the emerging of mathematics education as an academic discipline, as outlined by different authors (Kilpatrick 2008; Hodgson 2009; Furinghetti et al. 2013). Hodgson and Niss (2018) give an “insiders’ view” of the latest half century of the Commission. Their work provides important information and serves as a guide. Indeed, the two authors occupied leading positions within ICMI for a long period and their knowledge and experience are irreplaceable. The column “Once upon a time... Historical vignettes from the ICMI Archive,” launched in the ICMI Newsletter of March 2019 (p. 6–7) provides firsthand information and documents from the ICMI Archives gathered by their curator Bernard Hodgson. For information about the ICMI officers who passed away in the first century of the life of ICMI, we refer to the portraits published in this book. We will provide some information for the other officers and members at large. We start with some general observations: • ICMI’s Executive Committee (EC) is elected 1 year before the ICME (2 years in the case of ICME-1), while the organization of an ICME necessarily requires some years. So an ICME cannot be seen as an expression of the Executive Committee that is in force during its holding, but rather as a bridge between two ECs (with exception of ICME-1, which completely represented ICMI’s President Freudenthal). • ICMI is an official commission of the International Mathematical Union. This still defines the official position of ICMI. Furthermore, the majority of the funding of ICMI comes from IMU. Up to 2006 the Terms of Reference of ICMI have been established by the General Assembly of IMU, which was also responsible for the election of the Executive Committee, the administrative leadership of ICMI. Once

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these elections and budget matters were settled, ICMI worked with a large degree of autonomy (Bass and Hodgson 2004). According to new Terms of Reference in 2007, the Executive Committee of ICMI is elected by the General Assembly of ICMI itself, see (https://www.mathunion.org/icmi/terms-­reference-­icmi). • Appointments of members of the Executive Committee have often been influenced by political issues (Hodgson and Niss 2018). Moreover, giving a look to the officers of ICMI within a committee, we see the attempt to gradually (and slowly) approach geographic balance, with a particular attention to developing countries, and gender equity. We also observe that mathematicians usually managed the appointment of the ECs, while that of the IPCs of the various ICMEs was more frequently the prerogative of mathematics educators.

3.2 ICME-1, Lyon 1969. Freudenthal’s Mark on Mathematics Education In the period 1967–1970, ICMI was chaired by Freudenthal (Netherlands). The Vice-Presidents were Edwin Moise (USA) and Sergei L. Sobolev (USSR); André Delessert (Switzerland) had been elected Secretary and the other members were Behnke (Germany), André Revuz (France) and Bryan Thwaites (UK). Henri Cartan (France), President of IMU, was Member ex officio. It was Maurice Glaymann, deeply involved in the reforms for the teaching of mathematics in France and first director of the IREM (Institut de Recherche sur L’enseignement des Mathématiques) in Lyon, to propose to have the venue of the ICME-1 in Lyon (interview to Glaymann, Artigue 2008a, part 1) Chapter 2 (Furinghetti and Giacardi, this volume) analyses the reasons that led to this first ICME. The volume of the proceedings was edited by the editorial board of Educational Studies in Mathematics (Editorial Board of ESM 1969; see Fig. 3.1). It shows no date of publication, nor a list of participants. The contents of this volume are also published in the same year in Educational Studies in Mathematics (2: 135–418). Glaymann tells that the participants were over thousand, and included a large number of teachers (Artigue 2008a, part 1). According to the ICMI Bulletin 5, 20–24, which analyses the geographical distribution of the participants, these were 655 from 42 countries (https://www.math.uni-­bielefeld.de/icmi/bulletin/ ICMI_05_04_1975.pdf; also see Becker 1970). All the papers of the congress are also published in Educational Studies in Mathematics (1969–1970, Vol. II, 134–418). Presenting the contents of the book, the editors regret that the address of Zoltan P. Diènes, “La mathématique a l’école primaire,” could not be inserted. Concluding his allocution during the congress, Freudenthal invites the participants To use this week of scientific and social events as a great opportunity to exchange experiences and ideas, to meet people from nearby and far away, and to enjoy all good things that this country and this city can offer you (Editorial Board of ESM, 1969, p. 6)

So, internationalization was growing.

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Fig. 3.1  Frontispiece of the proceedings of ICME-1

The contents of the proceedings show continuity with topics born within the CIEAEM (Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques). In the contributions to ICME-1, we find mention of games, worksheets, films, manipulatives, and the “modern” overhead projector, which allowed lessons to be prepared in advance, to perform movements and overlapping. The use of the above-mentioned tools and of concrete materials is linked with a new methodology that includes working groups and classroom discussion. Frédérique Papy presented the Minicomputer, proposed for primary school. The presence of Dienes is further evidence of the interest of the ICMI community in concrete materials, such as Dienes’s logic blocks, and the beginning of a real interest in primary school. A clear reference to the role of computers in school mathematics is in the contribution by Thwaites (Furinghetti et al. 2008). About this new

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technology we note that a panel was dedicated to CAI (Computer-Aided Instruction), see (L’Enseignement Mathématique 1970, s. 2, 16, p. 116). Thwaites was an applied mathematician from Southampton University (UK), and a member of the ICMI Executive Committee. He was involved with the curriculum reforms of the 1960s in his country, and was one of the founders of the School Mathematics Project (SMP). Thwaites observes in the interview he gave to Hodgson that as the end of the 60s came, and the 70s came, there was a greater emphasis on [...] what one might almost call ‘individual scholarship’, in the field of mathematical education. Mathematical education itself became a kind of academic subject [...] in a way that it wasn’t certainly in the 50s or really in the early 60s. (Hodgson 2008b, part 1)

This observation by Thwaites can be found in the fifth point of the resolutions of the congress, which state that 1. The modernization of the teaching of mathematics should be pursued in all countries, both in content and method. “Content and method are inseparable and should be kept continually under scrutiny”.1 2. Collaboration between teachers of mathematics and those of other disciplines should be encouraged. 3. Each country should be more fully informed of activities in the other countries. “In particular, the ‘advanced’ countries should continue to collaborate with the developing countries in the search for solutions appropriate to them”. 4. It is necessary for the teacher of mathematics to pursue further professional study during his employment. 5. The theory of mathematical education is becoming a science in its own right. The new science should receive places in the Universities or Research Institutes. The first International Congress on Mathematical Education makes the following recommendations to I.C.M.I.: 1. To study the problems of national information, in particular that of the creation of an information bulletin; 2. To pay more attention, in the next congress, to pre-school education, elementary education, mathematical education for young people of all ages, and adult education (Editorial Board of ESM, 1969, p. 284). In that period, besides the founding of the new journal Educational Studies in Mathematics in 1968, other journals appeared explicitly devoted to mathematics education: in 1969, the German Zentralblatt für Didaktik der Mathematik (now ZDM –The International Journal on Mathematics Education); in 1970, the USA Journal for Research in Mathematics Education, and the British International Journal of Mathematical Education in Science and Technology (IJMEST). The first two journals cited had links with ICMI: Educational Studies signed a contract with  The text presented constitutes a summary of the Resolutions. Only the parts in quotation marks are original. 1

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UNESCO (Lehto 1998, p.  259) clearly with ICMI’s assistance (Hodgson 2019); Zentralblatt had ICMI as co-editor (Redaktionskomitee. 1969). Important international and national initiatives make evident the ferment of those years: after the SMP in the 1960s, in 1968 the above-mentioned Zentrum für Didaktik der Mathematik (Center for the Didactics of Mathematics) was founded in Karlsruhe by Steiner and Kunle, followed in 1973 by the IDM (Institut für Didaktik der Mathematik) founded at Bielefeld University by Heinrich Bauersfeld, Michael Otte and Steiner (see Schubring 2018). In 1969, the first IREMs were established in Lyon, Paris and Strasbourg. Many countries were presenting new syllabi inspired by New Math or by new methodologies. In 1971, Freudenthal himself founded the Institut Ontwikkeling Wiskunde Onderwijs (IOWO, Institute for the Development of Mathematics Teaching) (Furinghetti et al. 2013). Even though Freudenthal’s presidency lasted only one term, as was usual in those days, he introduced many novelties into ICMI and in mathematics education, so that his influence lasted long time after his presidency. For instance, Thwaites underlines Freudenthal’s predominance over the other people’s view (see Hodgson 2008b, part 1) and Revuz considers him the best President of ICMI at all (interview to Revuz, Artigue 2008c, part 1). “So, it is fair to use the term the “Freudenthal Era”—in the spirit of Bass (2008)—for the years 1967–1980” (Hodgson and Niss 2018, p. 232). Undoubtedly, such a dominant character had to cause some problems in the relations with IMU. In a letter written on October 15, 1970, to IMU Secretary Frostman, Cartan wrote that Freudenthal caused him worries (Cartan 1970). The reason for Cartan’s worries was that Freudenthal wanted the outgoing ICMI Executive Committee to appoint the International Programme Committee for ICME-2 with only 2  months left of his presidency (Furinghetti and Giacardi, this volume; Furinghetti et al. 2020). This inaugurated some tension between the ICMI President and the IMU leadership, arising again from time to time in the years to come (Hodgson and Niss 2018). In 1970 and in 1972, the second and third volume of UNESCO’s New Trends in Mathematics Teaching were published (volume I had been published in 1966) (UNESCO 1970, 1972). These books were the result of a cooperation between ICMI and UNESCO and appeared in the UNESCO series devoted to the teaching of basic sciences; the aim was to foster the improvement of mathematics education at all levels and in all regions of the world. The two first volumes consisted of a blend of reprinted materials (as summary or critical reports) and original papers. As stated in the preface of the third volume, a general overview was lacking; moreover the papers were printed either in English or in French, often without even a summary in the other language (UNESCO, 1972, p. 6). It was therefore decided that the third volume be composed of scholarly analyses of the trends in various aspects of an overall educational area (e.g. mathematics education) and published in at least English, French and Spanish. Volume 3 was prepared and discussed during a two-­ week meeting in Royaumont in September 1971. The contributions referred to all educational levels, from kindergarten to university and to the main mathematical areas (algebra, geometry, analysis etc.) and were assigned to leading experts from different countries (Alan Bishop, Howard Fehr, Freudenthal, Claude Gaulin,

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Glaymann, Brian Griffiths, M. Hastad, Jeremy Kilpatrick, Anna Zofia Krygowska, A. D. Nijdam, Zolzistaw Opial, Georges Papy, Revuz, Steiner, Willy Servais, János Surányi). We will see in the following that UNESCO was involved in different ICMI initiatives. As clarified by Jacobsen (1993), UNESCO is not a funding organization, having a reduced budget, so cooperation with ICMI depended upon common purposes. The main areas of cooperation were, besides the editions of the series “New trends in mathematics teaching,” the international meetings cosponsored by the two organizations all over the world and particularly in developing countries (Christiansen 1978). The interest of UNESCO in the diffusion of scientific culture is evidenced, among other things, by the creation of a mathematics education post, which Bent Christiansen occupied from 1972 to 1976, and Edward C.  Jacobsen from 1976 to 1992.

3.3 ICME-2, Exeter 1972. The Season of the Projects The period 1971–1974 was chaired by James Lighthill (UK), with Vice-Presidents Shōkichi Iyanaga (Japan) and Surányi (Hungary). The Secretary was Edwin A. Maxwell (UK) and members were Henry O. Pollak (USA) and Sobolev (USSR). As decided by the General Assembly of IMU in 1970, ex officio members were not only the President of IMU (Komaravolu Chandrasekharan (Switzerland)), but also Freudenthal (Netherlands) as Past President of ICMI, Otto Frostman (Sweden) as Secretary of IMU, and André Lichnerowicz (France) as Representative of IMU in the International Council of Scientific Unions (ICSU) Committee on the Teaching of Science (CTS).2 ICME-2 was held in Exeter, 29 August - 2 September 1972.3 The proceedings were edited by Howson. As written on the back cover, the book surveys the work of the conference, and presents a picture of developing trends in mathematical education. Moreover, we read that there were around 1400 participants from 73 countries (1384 participants and 76 countries, according to ICMI Bulletin 5, 1975: 20–24). The frontispiece shows the photos of George Pólya and Jean Piaget (Fig. 3.2). The

 Describing the Commission in ICMI Bulletin 1 (1972), Lighthill also mentions the members at large of ICMI, namely Sir James Lighthill, M.  Barner, F.  Chatelet, A.  Gleason, L.  Lombardo Radice, Y. Mimura, J. Novak, H. Pollack, S. L. Sobolev, J. Surányi and includes also Freudenthal among the members of the EC. Even if not explicit in the Terms of Reference of 1960, the EC is formed by the officers (namely, the President, the Secretary and the two Vice-Presidents) and by three members chosen among the members at large, one of which is the former ICMI President (see Hodgson 2020). 3  The information given in the proceedings, at the beginning of the survey, only indicates a generic end of August for the beginning, and in the (ICMI Bulletin 5, 1975: 20–24) the final day is September 3. 2

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Fig. 3.2  Frontispiece of the proceedings of ICME-2

proceedings open with a message of Prince Philip, Duke of Edinburgh, who stresses the important work of mathematics educators, in that “it requires real genius to light a flicker of understanding in the minds of those to whom mathematics is a clouded mystery”(Howson 1973, p. V). Not all the congress papers appear in the proceedings. As Howson tells, the committee made a selection, also with the idea to give guidelines for further publications in the field (interview to Howson, Hodgson 2008a, part 1). In his presidential address, Lighthill underlines the importance of the ICMEs:4 before their creation the study of mathematical teaching methods and curricula was supplemented and strengthened, among others, by international discussion every four years in the educational section of the International Congress of Mathematicians. Those useful discussions were, nevertheless, rather limited in scope and in the number of interested persons involved (Lighthill 1973, p. 88).  In several letters from Lighthill to Cartan, when Cartan was still President of IMU in 1970, we can see the efforts of Lighthill on the one hand to obtain funds for the Congress, on the other hand to keep good relations with IMU (ICMI Archives 14b, 1967–1974). 4

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If ICME-1 was the expression of ICMI’s President Freudenthal, ICME-2 can be seen as the expression of the English component of the program and the organizing committee, which included Lighthill himself, Thwaites, Trevor Fletcher and above all Howson. Several English projects had a major part in this congress, SMP in particular (Hodgson 2008b, part 1). In the introduction to the proceedings of ICME-2, Howson discusses the role of laboratories and concrete materials, and the need to reflect on the use of the related teaching methodologies. In Howson’s opinion, educators look at the 1960s as the period of New Math, but the main long-lasting idea of that period was the emergence of new styles of teaching and a more systemic transfer of teaching materials and ideas in the various countries (Howson 1973). One of the working groups—a new format introduced in ICME 2—deals with the use of television and films in the teaching of mathematics. The tradition already existed, but it was revived thanks to new technologies seen as necessary in mathematics education. At ICME-2, a WG explicitly addressed technology, and we also find Seymour Papert’s “turtle geometry,” aimed at primary school (Furinghetti et al. 2008). From that time on, the programs of ICME always encompassed official activities linked to computers. Another important feature, already present in ICME-1, were Piaget’s theories. At ICME-2, the written contribution of Piaget, who did not attend the conference, still outlines the analogy of Bourbaki’s three mother structures with the structures of thinking; he ascribes the failure of Modern Mathematics in school to the use of traditional teaching methods based on oral transmission. Both WGs on “Pre-school and primary mathematics” and on “Structure and activity” are linked to Piaget’s experiences. According to Mary Sime, the value of Piaget’s tests is educational rather than diagnostic, because the formation of concepts happens through the use of suitable materials. In his turn, Efraim Fischbein (present in the first two ICMEs) considers that Piaget’s theory is the most important reference in the psychogenetic field, proposing a compromise between the theories of Piaget and Bruner (Howson 1973). In the introduction to the proceedings, Howson stresses the importance that Piagetian psychology had for elementary school. He also observes that the WG for “the psychology of learning mathematics” was the most attended. According to Howson, the topic dealt with “underpins the whole of mathematics education” (Howson 1973, p. 15). Here we see the premises for the creation in 1976 at ICME-3 of the Study Group PME affiliated to ICMI. The invited lecture by Pólya comments about a list of quotations about the learning of mathematics, which underline the role of conjectures and intuition before proving. The two photographs on the cover of the proceedings of ICME-2 summarize the two main issues that emerged in the ICMI community: new methods of doing mathematics in the classroom (Pólya’s photo) and the interlacement with psychology (Piaget’s photo). The discussion about the identity of mathematics education, or didactics of mathematics—the preferred nomenclature in some countries—continued.

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Krygowska, for example, in her contribution to the Working Group on teacher training for prospective secondary teachers, which was chaired by Steiner, identified four aspects of didactics of mathematics: a synthesis of the appropriate mathematical, educational, cultural and environmental ideas; an introduction to research; the nature and situation of the child; and practical experience (see Howson 1973). A call for more in-depth research in mathematics education was also present in the plenary of the mathematician Hassler Whitney, who had in mind the failure of New Math. At ICME-2, New/Modern Math is still a matter of discussion, with positions against and in favour of it. The important contribution by René Thom (1973) stresses the contradiction of a teaching that is heuristic in principle, but is based on abstract mathematics. Thom thinks Piaget too much confident in the potentialities of mathematical formalism: Modern Mathematics has not produced new theorems and, as far as education is concerned, does not produce new knowledge. It has eliminated Euclidean geometry in favour of algebra, but it is exactly Euclidean geometry that connects natural language and abstraction. Because of Thom’s contribution, many authors date the end of Modern Mathematics back to ICME-2 (Furinghetti et al. 2008). The Executive Committee decided to endorse formally the following resolutions proposed by working groups: From the working group on “Mathematics in Developing Countries”: “That all possible encouragement and assistance should be given to developing countries to make changes in their mathematical syllabuses and curricula”. The cultural background of the pupils and the needs of national development are to be taken into account. From the working group on “Links with other Subjects at Secondary level”: The congress recommends that action be taken in providing support (including financial): to enable teachers of different areas to work together; to publish what has been done in the direction of interdisciplinarity; to encourage individuals and institutions to develop new teaching materials which cross disciplinary boundaries; to produce source materials suitable for use in secondary schools on topics linking mathematics with other subjects. Moreover the Congress agrees that steps be taken by ICMI to establish a center for the interchange and dissemination of information on all matters of interest in Mathematics Education; and to encourage cooperation between journals in different languages (Howson 1973).

During Lighthill’s term of office, in 1972, the ICMI Bulletin was established as a rather informal means of communication within the “ICMI family” (Hodgson and Niss 2018), appearing in average two times a year. The Bulletin lists the many conferences and symposia organized by ICMI, and thus the many interests that were developing in the field: for instance, the interest toward primary education, testified by the ICMI Symposium about the teaching of mathematics in primary education, held in 1973 in Eger, Hungary (ICMI Bulletin, 2, 1973, p. 5). Moreover a new policy of holding Regional Symposia “to facilitate wider discussion of mathematical education outside those areas of Europe and America where

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international meetings on the subject have mainly been held hitherto” (L’Enseignement Mathématique 1975, s. 2, 21, p. 330) was adopted, and numerous symposia were held with the co-sponsorship of the ICMI. Examples are the symposia held jointly with UNESCO, in Nairobi, Kenya (September 1–11, 1974) on “Interactions between linguistics and mathematical education,” and the regional conference about the development of a mathematics integrated curriculum for developing countries, held in New Delhi (India) in December 1975. Still in 1975, the fourth Interamerican conference on Mathematical Education was organized by IACME (Interamerican Committee on Mathematical Education, affiliated to ICMI in 1974) in Caracas Venezuela. At the IMU General assembly in Canada in 1974, the President Chandrasekharan stated that IMU, as a member of ICSU, is committed to the principle of the free circulation of scientists and that the object of the world mathematical community should be of encouraging the growth of mathematics in the disadvantaged areas of the world. In relation to this, the adopted resolutions expressed the Assembly’s “great appreciation of the activities of ICMI in every aspect of mathematical education, particularly in developing countries, and its hope that this work will grow, and that the mathematical needs of other disciplines will be taken into consideration” (quoted in Lehto 1998, p. 182). IMU also became less dependent on subventions from ICSU and UNESCO due to a different politics regarding the unit contributions, fixed at 600 Swiss Francs (Lehto 1998, p. 182).

3.4 

I CME-3, Karlsruhe 1976. The Birth of Affiliated Study Groups

The term 1975–1978 was chaired by Iyanaga (Japan), the Vice-Presidents were Christiansen (Denmark) and Steiner (Germany). The Secretary was Yukiyoshi Kawada (Japan), the members of the EC5 were Edward G. Begle (USA) and Lev D. Kudrjavcev (USSR). There also were four ex officio members: Lighthill (UK) as Past President of ICMI, Deane Montgomery (USA) as President of IMU, Jacques-­ Louis Lions (France) as Secretary of IMU, and Freudenthal (Netherlands) as Representative of IMU in CTS/ICS. Iyanaga summarizes as follows the period 1975–1978: In the wake of the success of the Bourbaki movement, the reform movement of mathematical instruction began in the early 1950s particularly in the U.S. by the SMSG (School Mathematics Study Group) directed by Begle. But the excess of this tendency was warned by mathematicians like André Weil or Morris Kline. In the international politics, on the other hand, the Soviet Union took a quite separate position from the “West” at the time. I believe that the general background of the period 1975–78 was largely like this.

 According to ICMI Bulletin (1975, 5, p.  7–8), members at large of ICMI were Begle, P. L. Bhatnagar, E. Castelnuovo, B. Christiansen, S. Iyanaga, L. D. Kudrjavcev, J. Lelong-Ferrand, B. H. Neumann, Z. Semadeni, J. Surányi. 5

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Fortunately, the tradition of the ICMI is rather apolitical (Iyanaga 2001).

The ICME-3 took place in Karlsruhe, from August 16 to 21. The International program committee was chaired by Steiner. The proceedings were edited by Hermann Athen and Kunle  (1977) Howson tells—in the mentioned interview by Hodgson—that the congress was very well organized, also thanks to a financial support from UNESCO, being Christiansen the UNESCO representative for high school mathematics (Hodgson 2008a, b, part 4). In the already mentioned interview by Schubring, Kunle recalls the organization of ICME-3. The preparation lasted 3 years, and much was due to Steiner’s international relations. Germany was well represented in the congress. An interesting aspect concerns the choice of the official languages, which was decided to be four: German, French, English and Russian.6 This required to find experts in at least two languages for the chairing of the different sessions in order to help with some translations. Indeed, as also Lehto underlines, in the ICMEs language was a much bigger problem than at the ICMs: “Mathematics education imposed greater linguistic demands on the speaker and hearer than did the presentation of mathematics. Mathematics educators lacked largely the international terminology and vocabulary of the mathematician and could not resort with equal facility to a universal set of symbols” (Lehto 1998, p. 260). Moreover, we can add that teachers have generally fewer opportunities for international contacts. Kunle tells that there were about 200 participants from USA, 80 from Japan, 70–80 from Africa. There were also many teachers, and also meetings among teachers. A novelty was the creation of a logo (Fig. 3.3) for the congress, inspired to the hosting city, to mathematics and to ICMI. The proceedings reflected the structure of the whole program, with the main lectures, the sections and poster sessions, the panel discussion, the working groups (still called Exeter working groups), study groups, workshops, the projects, films and exhibitions. The Survey reports and the list of participants were in separate booklets. The main work of the congress was focused on 13 sections, each opened by a survey report. The proceedings contain only a summary of the results, which were later collected in Vol. IV of the series published by UNESCO, New trends in mathematics teaching (1979), prepared by Fig. 3.3  The logo for ICME-3

 The proceedings are only in English. We note that the proceedings of the ICM in Vancouver the year before presented papers in English, French and Russian. 6

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the ICMI. As was the case for volume 3 of the same series, it was decided to link the papers of the fourth volume to the results of a conference (in this case the sections of ICME-3), with further meetings before and after the congress (UNESCO 1979, p. 2). The intense work of collaboration between ICMI and UNESCO that brought to the edition of what can be considered the true proceedings of ICME-3 is well described by Christiansen (1978). The planning of the volume started at the end of 1974 with an Advisory Meeting in Paris. It was decided that the volume should deal with broad topics of interest for mathematics teachers and educators: thirteen themes were identified, some of which were divided according to school levels. For each theme, a section should be established at the Karlsruhe Congress, the report being prepared by an international specialist, and later developed for Vol. IV of the series published by UNESCO. A particular role was also played by the projects: in the proceedings of ICME-3, the presentation of 15 projects is mentioned, among them SMP, IOWO, the Open university, IREMs. Many of them were already present in Exeter. The Resolutions of the congress can be summarized as follows: (a) ICMI should continue and increment its cooperation with other associations devoted to furthering mathematical and scientific education. (b) National Subcommittees if ICMI should be reactivated. In fact many countries are not active in both matters of national concern and in international cooperation and contacts. (c) Internationally composed committees, working groups and study groups have been established in order to study and further particular areas of research in mathematical education, as HPM or PME. These are invited to continue their work. Moreover it is recommended that the theme “Women and Mathematics” be an explicit theme of ICME 1980. An important moment of the congress was the establishing of the first Study Groups affiliated to ICMI: HPM (the International Study Group on the relations between the History and Pedagogy of Mathematics) and PME (the International Group for the Psychology of Mathematics Education). As to the first one, its history started with a working group at ICME-2 in 1972, where “there occurred a confluence between growing interest within the mathematics education community (seen notably in the NCTM’s celebrated 31st Yearbook of 1969, Historical Topics for the Mathematics Classroom) and an increased readiness of international bodies to take such interests and concerns on board” (Fasanelli and Fauvel 2008). Also concerning PME “the impetus to develop an organization with a psychological focus on mathematics education began much earlier when, in 1969 at the first International Congress on Mathematics Education […], Fischbein was invited by then ICME President Freudenthal to chair and organize a round table on the psychological problems of mathematics education. Fischbein, a cognitive psychologist and, at that time head of the department of Educational Psychology at the University of Bucharest (later he served as head of the Science Education Department at Tel Aviv University), was keen to take up Freudenthal’s call to improve mathematics education in schools by going beyond philosophical discussions of mathematics teaching

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and learning to advocating empirical scientific research in the field” (Nicol and Lerman 2008). In 2008, ICMI’s organization included five permanent Affiliated Study Groups, each focusing on a specific field of interest and study in mathematics education consistent with the aims of the commission. The Affiliated Study Groups are neither appointed by ICMI nor operate on behalf or under the control of ICMI. They are thus independent of ICMI for their work, also in terms of finances, but they collaborate with ICMI on specific activities, such as the ICMI Studies or components of the program of the ICMEs. They present reports on their activities to the General Assembly of the ICMI (Bass and Hodgson 2004). In 1977, the first conference of the Affiliated Group PME took place in Utrecht, with the opening address of Freudenthal (Freudenthal 1978). In the same issue of Volume 9 (1978), the lectures of the conference were published. During those same years, the ICMI Secretary was the Japanese Kawada, who initiated the Southeast Asian Conference on Mathematical Education (SEACME) series in 1978 with the inaugural conference in Manila. This conference, sponsored by ICMI and SEAMS (Southeast Asian Mathematical Society), was very important for the involvement of the Eastern countries in the international movement of math education (see Lim-Teo 2008). Whilst secondary education received most of the attention in the first 50 years of ICMI, primary and tertiary education now entered the field of interest as well. The already mentioned volume of UNESCO (1979), reflecting the Sections of ICME-3, contains chapters devoted to all school levels. Moreover, it contains chapters on the goals of mathematics teaching (by Ubiratan D’ Ambrosio), on applications (by Pollak) and on algorithms (by Arthur Engel), which went beyond the teaching of established mathematical areas and topics. In 1978, the BACOMET (Basic Components of Mathematics Education) group was initiated by Christiansen, Howson and Otte, elaborating research for improving teacher education.

3.5 ICME-4, Berkeley 1980. ICME Crosses the Ocean Whitney (USA) was the President of ICMI in the period 1979–1982, Vice-Presidents were Christiansen (Denmark) and Ubiratan D’Ambrosio (Brazil). The Secretary was Peter Hilton (USA), and members were Stanley H.  Erlwanger (Canada), Bernhard H. Neumann (Australia) and Zbigniew Semadeni (Poland). The ex officio members were Iyanaga (Japan) as Past President of ICMI, Lennart Carleson (Sweden) as President of IMU, Lions (France) as Secretary of IMU. The representative of IMU in CTS/ICSU was Christiansen, who was already in the EC as Vice-President.7  The ten members at large were Whitney, M. Barner, E. Castelnuovo, H. Halberstam, P. Hilton, Y. Kawada, L. D. Kudrjavcev, B. Malgrange, B. H. Neumann, Z. Semadeni. 7

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After the Indian Ram Behari (appointed as an officer in 1955) and the Japanese Yasuo Akizuki (appointed in 1959), Iyanaga and Kawada we see again a step forward toward internationalization with the appointment of the Brazilian D’Ambrosio from South America and Neumann from Australia as members of the ICMI EC. The congress ICME-4 was held in Berkeley, California, USA, August 10–16. The proceedings were edited by Marilyn J.  Zweng, Thomas Green, Kilpatrick, Pollak, and Marilyn Suydam (1983). According to the introduction to the proceedings, attendance was about 1800 full and 500 associate members from about 90 countries. In the acknowledgments, Zweng states that it is the first time in the history of ICMEs that all submitted papers were published. To stay within page limits, the editors often had to delete several pages or paragraphs of the submitted papers, without the approval of the authors. The chair of the International Program committee was Pollak. Pollak was an applied mathematician, director of the mathematics group at Bell Labs. He was, throughout his career, interested in mathematics education, with a major focus on mathematical modelling, a field he did much to implant in the school curriculum (Bass 2008, p. 14). He was also an active participant in the School Mathematics Study Group (SMSG) directed by Begle, one of the main American expressions of the “New Math” movement. He was active in ICMI affairs, serving on its Executive Committee in 1971–74 and in 1983–86 (Bass 2008). The volume of the proceedings consists of thematic chapters, each with short contributions; it is not possible to distinguish within working and study groups, mini-conferences etc. One of the chapters is devoted to critical variables in mathematical educations, in memory of Begle. A version of the plenary by Freudenthal appeared in Educational Studies in Mathematics, 12, 1981, 133–150. Pólya was not able to attend the conference, and his short address was delivered by Gerald L. Alexanderson. For the first time, no resolutions appear, there is no Presidential Address (at least not in the proceedings), and ICMI is not even mentioned. The logo of the congress is shown in Fig. 3.4. We still find echoes of the debate about New Math, in particular in the discussion about the movement Back to Basics, contained in Chap. 2 “Universal Basic Education” (Sobel 1983). Pólya’s brief abstract in the proceedings does not refer to

Fig. 3.4  The logo for ICME-4

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problem-solving. But Pólya’s problem-solving—which arises from the tradition of mathematicians reflecting on their own work—was the argument of one of the 38 chapters and would become a dominant theme in the successive ICMEs and in mathematics education up to the present day. According to Lehto, the period 1979–1982 suffered from a lack of adequate administration. Lehto considers the professional competence of Whitney and Hilton “in striking contrast to the Commission’s inefficient administration” (Lehto 1998, p. 258). The ICMI Bulletin even ceased to appear. Undoubtedly, the term of Whitney as President and Hilton as Secretary was a problematic one with reference to the relations between IMU and ICMI, and also to ICMIs internal relationships. In the minutes of a meeting of the IMU Executive Committee held in 1980, we can read: “The EC expresses concern about the lack of communications between IMU and ICMI.” And again in 1981: “Much concern concerning the difficulties that arose in the [ICMI] EC.” (IMU EC Minutes 1980, p. 14, and 1981, p. 25, quoted in (Hodgson and Niss 2018, p. 234)). Indeed during Whitney’s term the EC met only rarely; moreover Hilton felt that his role as Secretary be confused with that of a clerk rather than that of an organizer and decision-making executive officer. He expressed this in a letter to IMU Secretary Lions: “It is clear to me that I was expected by some of my colleagues on the EC to act purely in a ‘secretarial’ capacity”(Hilton 1980). That perception had led Hilton to present his resignation from his office. However, this resignation did not materialize and Hilton finally remained as the ICMI Secretary till the end of his term. Another controversial theme was due to the fact that members of the ICMI EC put forward as their candidate for the next President the Danish mathematics educator Christiansen, ICMI Vice-President for two terms since 1975. Instead, the IMU President Carleson, in a letter to the ICMI EC at the end of 1981, expressed the desire to have as next President a well-known mathematician with established interests in education. Reporting the letter, Hodgson and Niss comment—paraphrasing Clemenceau8—that for the IMU officers “mathematics education was far too important to be left to the mathematics educators” (Hodgson and Niss 2018, p. 235). Important events in that period were the creation of the journals For the Learning of Mathematics (FLM) and Recherches en Didactique des Mathématiques (RDM), both in 1980. In 1980, the ICMI General Assembly set up an IMO (International Mathematical Olympiad) Site Committee. Its task was to ensure that annual IMOs were held to assist the host country. It was customary that the appointments of the members to the Site Committee followed nominations by the IMO jury. “Although ICMI has no responsibility for financing and organizing the Olympiads, a link was thus created between it and the IMOs” (Lehto 1998, p. 263).

 Georges Clemenceau, 1886: “La guerre! C’est une chose trop grave pour la confier à des militaires.” 8

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In 1982, new terms were adopted by the General Assembly of IMU, there is no longer mention to members at large; moreover, the ex officio members are said to belong to the EC. The rest is quite unchanged: (a) The Commission shall consist of 1. the members of an Executive Committee as specified in (b) below, elected by IMU, and 2. one national delegate from each member nation as specified in (d) below. (b) The Executive Committee consists of four officers, namely, President, two Vice-Presidents, and Secretary, and of three further members. Furthermore, the outgoing President of ICMI, the President and the Secretary of IMU, and the representative of IMU at CTS (ICSU) are members ex-officio of the E.C. (c) In all other respects the Commission shall make its own decisions as to its internal organization and rules of procedure. (d) Any National Adhering Organization wishing to support or encourage the work of the Commission may create, or recognize, in agreement with its National Committee, a National Sub-Commission for ICMI to maintain liaison with the Commission in all matters pertinent to its affairs. The National Adhering Organization in question shall designate one member of the said Sub-­ Commission, if created, to serve as a delegated member of ICMI as mentioned in (a). The new terms were to be implemented by the next President (ICMI Bulletin, 13, February 1983, 5).

3.6 ICME-5, Adelaide 1984. The Birth of Ethnomathematics The next President, for the period 1983–1986, was Jean-Pierre Kahane (France), Vice-Presidents were Christiansen (Denmark) and Semadeni (Poland). The Secretary was Howson (UK). Other members were Bienvenido F.  Nebres (Philippines), Michael F. Newman (Australia), Pollak (USA). The ex officio members were Whitney (USA) as Past President of ICMI, Jürgen Moser (Switzerland) as President of IMU, Lehto (Finland) as Secretary of IMU and Henri Hogbe-Nlend (Cameroon) as the representative of IMU in CTS/ICSU. The 1984 ICME congress was held in Adelaide, Australia, August 24–30 (logo in Fig. 3.5). The proceedings were edited by Marjorie Carss (1986). In the interview by Artigue, Kahane mentions with great interest the Australian Mathematical Competitions (Artigue 2008b, part 2). Indeed, they were directed to a large number of students and were based on the principle that every student can find a challenge somewhere (Kahane, ICMI Bulletin 47). May be that this can be one of the reasons for the attribution to Adelaide of the organization of ICME-5. But no

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Fig. 3.5  From the frontispiece of the proceedings of ICME-5

particular reference to the competitions appears in the proceedings, with the exception of a dedicated Topic Area, which anyway existed also in previous ICMEs. Only, on that occasion, a first proposal for a Federation of National Competitions was made (in 1994, the Federation would become affiliated to ICMI). According to the Foreword, attendance numbered more than 1800 participants from over 70 countries. In the proceedings, there is the list of participants. In addition to the working sessions, there were three major plenary addresses, several specially invited presentations, and over 420 individual papers in the form of short communications, either as posters or brief talks. In addition, there were a variety of exhibits, film and video presentations, and workshops. In all, more than half of those attending the congress made a direct contribution to the scientific program. Most of the presentations and papers prepared for ICME 5 are not included in the proceedings. The Presidential Address by Kahane is in fact a plenary about “Mesures et dimensions.” In the Public Forum, as well as in the Specially Invited Presentations, some local situations about the teaching of mathematics are presented. The other two main invited addresses are held respectively by Pollak on the effects of technology on the curriculum and Phillip Davis about the nature of proof. Plenary lectures were held by Kilpatrick (“Reflection and recursion”), Renfrey Potts (“Discrete

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mathematics”), and d’Ambrosio (“Sociocultural bases of mathematical education”). This latter lecture represents a milestone, namely, the beginning of ethnomathematics. Some days before, the first Satellite meeting of HPM had taken place in Adelaide and d’Ambrosio had underlined the need to develop three separate histories of mathematics: “history as taught in schools, history as developed through the creation of mathematics, and the history of that mathematics which is used in the street and the workplace. In the plenary at ICME 5 he introduced the concept of ‘ethno mathematics’ as compared to ‘learned mathematics’ to deal with these differences (see Fasanelli and Fauvel 2008).” According to Hodgson and Niss (2018) the—perhaps implicit—mandate of Kahane and Howson was to put ICMI back on track, or at least to revitalize ICMI (also see the interview to Kahane, Artigue 2008b). Kahane (1926–2017) was a French mathematician of the first rank. During his long career at the University of Orsay, he extensively contributed to the development of mathematics knowledge in many fields such as analysis and probability. He also played a role in mathematics education: the French Minister of Education asked him to chair what came to be called the “Commission Kahane,” charged to bring recommendations for revision of the mathematics programs in the French schools (see https://smf.emath.fr/publications/jean-­pierre-­kahane-­numero-­special-­ gazette). Bur, already before, as ICMI President he had shown interest and deep understanding in topics that concerned the role of computers in mathematics education (Kahane 1987), in mathematics as a service subject (Kahane 1986a, b) and in general in the developments of mathematics education (Kahane 1988a). These topics provided the subject for the first ICMI studies. More than that, Kahane was “a thinker and communicator of philosophical depth, elegance, and eloquence” (Bass 2008, p. 20), qualities that made him an exemplary President of ICMI, from 1983 through 1990. One of the first decisions, took in Adelaide, was that the program committee for the next ICME be elected by the Executive Committee of ICMI (interview to Howson, Hodgson 2008 a, part 4), thus becoming international rather than local.9 As a consequence, ICME becomes even more emanation of ICMI. One of the problems of such a decision was a financial one, as it required travelling—other kinds of communication were not easy at that time—and ICMI did not have big incomes. The travel expenses also concerned the meeting of the EC, which became more frequent than before, namely about twice a year. Even more frequent were the meetings between Howson and Kahane (Hodgson 2008a, part 4). Kahane tried to involve more people in ICMI and he had the merit to establish good relationships with IMU and obtain a wider support. As Howson tells, the fact of taking on responsibilities like a major involvement of the EC helped in gaining a stable recognition from IMU. Christiansen, who had been Kahane’s competitor for

 The programme committee was quite international also before that decision, but may be that the local organizing committee had a major role in choosing the speakers. 9

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the presidency of ICMI, admitted that he would never have reached what Kahane managed to do (Interview to Howson, Hodgson 2008a, part 4). Another novelty of the Kahane-Howson term was the series of the ICMI studies, starting in 1984. The ICMI studies replaced the former tradition of the international inquiries (Furinghetti and Giacardi 2010). In general, an ICMI study focuses on an issue of current interest in mathematics education. Here too, an international program committee is established, which launches the Study through a Discussion Document published in L’Enseignement Mathématique and in other journals; researchers submit their contributions on the theme of the Study; on the basis of the contributions received, the Committee delivers the invitations to the ICMI Study meeting to about 80 participants; at the end, a book (the ICMI Study volume) is published to disseminate the results. The whole process lasts about 3 years. Kahane tells that the idea had already been discussed with Whitney, in 1982 (Kahane 2001). The first ICMI Studies dealt with the following topics: 1. The Influence of computers and informatics on mathematics and its teaching (Strasbourg, France, 1985). 2. School mathematics in the 1990s (Kuwait, 1986). 3. Mathematics as a service subject (Udine, Italy, 1987). 4. Mathematics and cognition (book prepared by PME, an Affiliated Study Group of ICMI), 1990. The first one was held in Strasbourg, France, in March 1985. The resulting book was edited by Robert F. Churchhouse, Bernard Cornu, Howson, Kahane, van Lint, François Pluvinage, Antony Ralston, and Masaya Yamaguti in 1986 as first volume of the ICMI Study Series, published by Cambridge University Press. A second edition was published by UNESCO in 1992, edited by Bernard Cornu and Anthony Ralston (Science and Technology Education No. 44).10 The second ICMI Study was not launched by a discussion document, the Study Volume was prepared by the editors Howson and Bryan Wilson following a closed international seminar held in Kuwait in February 1986. It was again published by Cambridge University Press as volume 2 of the ICMI Study Series. The third Study Conference was held in Udine, Italy, in April 1987, see (Kahane 1986a, b). The Study Volume was published in 1988, edited by Howson, Kahane, Pierre Lauginie and Elisabeth de Turckheim. Moreover, a volume of Selected Papers on the Teaching of Mathematics as a Service Subject was published by Springer Verlag in 1988, edited by Richard R. Clements, Lauginie and de Turckheim (CISM Courses and Lectures No. 305). Also the fourth study was not the result of a general study conference, the Study Volume was prepared under the responsibility of PME and published by Cambridge University Press in 1990, edited by Pearla Nesher and Jeremy Kilpatrick.  We will not list in the references the volumes of the ICMI studies. The complete list can be found on the ICMI website, https://www.mathunion.org/icmi/publicationsicmi-studies/icmi-study-volumes-niss. For further information and comments, see (Goos 2020; Hodgson 1991; Hodgson 1999; Holton 2008; Howson 2007). 10

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The themes 1, 2, 4 of the ICMI studies exactly reflected the wishes respectively of Kahane, Howson and Christiansen (Hodgson 2008a, part 4) And so, this first study served me as a model for what followed. The principle of the studies was this, as stated in English in the prefaces of the first studies. It is a question of identifying subjects that are, if not ‘on fire,’ at least ripe for a serious international study. And that we preside over such a study, that we provide the state of the art, but that we never make recommendations. The principle is that the study should provide a survey of the problems, and possibly the elements of solutions, but never a solution with an ‘ICMI’ label (from the interview to Kahane by Artigue, translated in Bass 2008, p.19).

Many other activities such as conferences and working sessions were organized, edited and published by the Affiliated Study Groups of ICMI (see ICMI website: https://www.mathunion.org/icmi/organization/affiliated-­organizations). Under Kahane’s presidency, old links were renewed and strengthened: in 1983, Kahane and Howson joined the editorial board of L’Enseignement Mathématique (hereafter EM), which then started publishing discussion documents for the newly instituted ICMI Studies. In 1987, a new study group was affiliated to ICMI, namely the IOWME – The International Organization of Women and Mathematics Education. According to Nancy Shelley, the history of the group started as early as in 1976 at ICME-3 in Karlsruhe, at a meeting arranged during the course of that congress to discuss the question of “Women and Mathematics.” The calling of that meeting was initiated by two Australian women, Jan Kennedy and Nancy Shelley, who were struck by the lack of representation of women as speakers, panel members or presiders, despite the fact that nearly 50% of those attending the congress were women. As Shelley (2008) put it Eight years later, a somewhat more enlightened view is taken about women and the study of mathematics, and it is now acknowledged that much human potential is being lost by the fact that so few women consider mathematics to be a subject for them to study. It may, therefore, be a surprise to some to learn what the reaction was to the calling of that first meeting, to holding it, and to its outcomes. For the record, however, it needs to be told.

As a subcommission of IMU, ICMI has always been apolitical and succeeded in staying out, for instance, of the long debate over China’s entry into the IMU. But surely the organization since 1969 of international congresses on mathematics education increased ICMI’s sovereignty. Lehto (1998) observed that this became manifest in 1986, when a movement began to bar the Republic of South Africa from the activities mounted under ICMI auspices. This would have been in violation of the non-discrimination policy of the ICSU, to which IMU and thus ICMI belong. The decision finally taken by the Executive Committee was that ICMI, as well as the affiliated study groups, should abide by ICSU’s rules. Nevertheless, the EC condemned the apartheid policies of the South African regime and, in the ICMI Bulletin of June 1986, members of the EC expressed their different opinions about changing or keeping ICSU’s rules. Christiansen was in favour of changing the rules and considered South African apartheid not as a political issue, but as a question of human rights. Howson, though understanding

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Christiansen’s opinion, believed however that, on balance, ICMI had more to lose than to gain from banning South Africans (Lehto 1998, p. 261).

3.7 ICME-6, Budapest 1988. The Legacy of Tamás Varga For the first time ICMI President and Secretary had the opportunity for a long-term planning. Indeed, in the period 1987–1990, both Kahane and Howson were reconfirmed. The remaining members of the EC changed: the new Vice-Presidents were Lee Peng-Yee (Singapore) and Emilio Lluis (Mexico). Members were Hiroshi Fujita (Japan), Kilpatrick (USA), Mogens Niss (Denmark) and the ex officio members were Ludwig Faddeev (USSR) as President of IMU, Lehto (Finland) as Secretary of IMU, Jacobus H. van Lint (Netherlands) as representative of IMU in CTS/ICSU. ICME-6 was held in Budapest, July 27 to August 30. The chair of the Hungarian Organizing Committee was Janos Szendrei, the chair of the IPC and of the Janos Bolyai Mathematical Society was Akos Csaszar. The proceedings were edited by Ann and Keyth Hirst (1988) (see the frontispiece in Fig. 3.6). No date of printing is indicated. According to the Foreword, there were 2414 registered participants from

Fig. 3.6  Frontispiece of the proceedings of ICME-6

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74 countries. The general pattern used in Adelaide was followed (in particular, the action groups are the same as the fifth congress). The reports of the Action and Theme Groups and of the topic Areas form the major part of the volume. An additional aspect of the groups’ work was the provision, in a number of cases, of a related Survey Lecture. The congress included also national presentations, exhibitions, poster presentations and short oral communications. Two books of abstracts were prepared for congress participants. The editors also write that the range of topics in these presentations was wide, the use of video materials was represented, but little emphasis was given to computers. An invisible architect of this conference was the well-known mathematics educator Tamas Varga. Varga had played a particular role at ICME-2 in Exeter in that he was one of the reviewers who decided about the papers to be accepted in the proceedings (Howson contribution delivered by video at the conference Varga 100, Budapest 2019). Moreover, Howson met him often when travelling to Budapest during the organization of ICME-6. But he died in November 1987. The Topic Area 5 about “Comparative Education,” in which Varga appears as Hungarian coordinator together with Douglas Quadling, became in fact a meeting in the congress dedicated to him, in recognition of his outstanding contributions to international cooperation in mathematics education (Quadling 1988). Another eminent Hungarian was remembered with more emphasis in the President’s address, namely Pólya, died in 1985 (Kahane 1988b). Of particular importance was the Fifth Day Special about “Mathematics, Education and Society,” the question of the interrelation between mathematics education and educational policies became the issue of a whole day. The contributions to this point were published in a separate book (Keitel et al. 1989) with the support of UNESCO. The ICMI studies launched in this period were the numbers 5. The Popularization of Mathematics (Leeds, UK, 1989), edited by Howson and Kahane and published in 1990 by Cambridge University Press; 6. Assessment in Mathematics Education (Calonge, Spain, 1991). Two volumes were edited by Mogens Niss and published by Kluwer in 1993, appearing as volume 1 and 2 of the New ICMI Study Series (NISS): Cases of Assessment in Mathematics Education, and Investigations into Assessment in Mathematics Education. The duo Kahane–Howson was very appreciated (Lehto 1998) and, in particular, Kahane praised Howson: “We had a very good and active Executive Committee, full of ideas and easy to chair, because everything was thought, planned and prepared by Geoffrey Howson” (Kahane 2001, p.1; also see Kahane 1988a). The year 1990 was the year of the retirement of Kahane as ICMI President. Bass (2008) lists the particularly insightful observations on mathematics and mathematics education that he offered in that occasion:

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But Kahane’s farewell message also pointed to open problems, as the fact that contacts between the ICMI Executive Committee and the National Representatives were far from satisfactory. According to him, it was not always clear who the National Representatives are, how they react to the ICMI Bulletin, if they do pass on the information (Kahane 1990).

3.8 ICME-7, Quebec 1992. The Solidarity Program The period 1991–1994 was chaired by Miguel de Guzmán (Spain) with Mogens Niss (Denmark) as Secretary, Vice-Presidents were Kilpatrick (USA) and Peng-Yee (Singapore). The three members were Yuri L.  Ershov (Russia), Eduardo Luna (USA) and Anna Sierpinska (Canada). Ex officio members were Kahane (France) as Past President of ICMI, Lions (France) as President of IMU, Jacob Palis (Brazil) as Secretary of IMU and van Lint (Netherlands) as Representative of IMU in CTS/ ICSU.  For the first time, the Executive committee includes a woman: Anna Sierpinska. ICME-7 was held in Québec-city from August 17 to 23 (see the logo in Fig. 3.7). Attendance was about 3500 participants from 94 countries. The proceedings were edited by Gaulin, Hodgson, David Wheeler, John Egsgard  (1994), while David Robitaille, Wheeler and Carolyn Kieran (1994) edited the selected lectures. The first Fig. 3.7  The logo of ICME-7

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and the second volume (containing the lectures) have a preface describing the structure (plenaries, topic groups, working groups, etc.) and the organization that is becoming the norm for the ICME congresses. In the presidential address, de Guzmán stresses the need for a Solidarity Program and a Solidarity Fund in Mathematics Education. In his concluding address, the Secretary-General Niss reminds the audience that the ICMEs are held on behalf of and under the auspices of ICMI. He gives a survey on the activities of ICMI and underlines the complexity of the field of mathematics education. Moreover, we observe the complex organization of such an event. There is an International Program committee (IPC) appointed by ICMI and a National Organizing Committee run by the host country. David Wheeler chaired the IPC, Hodgson chaired the National Organizing Committee, while Gaulin chaired the Local Organizing Committee. In addition, Robitaille chaired the Executive Committee. A relevant role was played by Gaulin. ICME-7 can be seen in many ways as “his” congress. “Even the fact that ICMI accepted the invitation of the Canadian community to host the congress in Québec is undoubtedly connected to Claude himself and to his deep and practical knowledge of the ICMI culture, tradition and expectations. He knew exactly what it means to organize an ICME and all the details that need to be attended. He was the mainspring behind the setting-up, in Canada and more particularly in Québec, of an infrastructure ensuring the success of the enterprise” (Dionne and Hodgson 2020, p. 15/ section 11). The Solidarity Program in Mathematics Education, strongly desired by de Guzmán, was based on two points: The first objective of the Solidarity Fund was to increase, in a variety of ways, the commitment and involvement of mathematics educators around the world in order to help the progress of mathematics education in those parts of the world where there is a need for it that justifies international assistance and where the economic and socio-political contexts do not permit adequate and autonomous development. This initiative thus aims to foster solidarity in mathematics education between well-defined quarters in developed and less-­ developed countries. Particular emphasis is placed on projects that enable the activation of a self-sustainable infrastructure within mathematics education in the region, country, or province at issue. Central to this program of international assistance was the establishment of a fund to provide financial support for the approved projects. The Solidarity Fund is based on voluntary donations from individuals and organizations and is kept separate from the ICMI’s general funds. The second component of the ICMI Solidarity Program aims at having a balanced representation from all over the world among the presenters and the general participants in activities such as the ICMI Studies or the ICME’s (Bass and Hodgson 2004).

In 1994, one more study group was affiliated to ICMI, namely the WFNMC – The World Federation of National Mathematics Competitions. As Peter Kenderov states, WFNMC was founded through the inspiration of Peter O’Halloran from Australia, who realized that there was a need “for an international organization to exchange ideas and information on mathematics competitions as well as to give encouragement to those mathematicians and teachers who are involved with the competitions.” The proposal had already been made  at ICME-5  in Adelaide,

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Australia, in 1984 (Kenderov 2008). Also see the paper by Kahane in (ICMI Bulletin 47, 1999). Three more ICMI studies were launched in that period: 7. Gender and Mathematics Education (Höör, Sweden, 1993); the Study Volume was published by Kluwer Academic Publishers in 1996, and edited by Gila Hanna with the title Towards Gender Equity in Mathematics Education, (New ICMI Study Series 3). A separate book with the Proceedings of the Study Conference was published by Lund University Press in 1995 (outside the ICMI studies series) edited by Barbro Grevholm and Hanna. 8. What is Research in Mathematics Education and what are its Results? (College Park, USA, 1994). The proceedings were edited in 1998 by Sierpinska and Kilpatrick with the title Mathematics Education as a Research Domain: A Search for Identity (2 voll., NISS 4) 9. Perspectives on the Teaching of Geometry for the 21st Century (Catania, Italy, 1995). The Study Volume was published in 1998 and edited by Carmelo Mammana and Vinicio Villani with the same title (NISS 5). With reference to the Study no. 7, Kahane remembers an episode (interview to Kahane, Artigue 2008b; also mentioned in Pollak, Niss, and Kahane 2004): in 1993, at Höör in Sweden, before the ICMI Study on Gender and Mathematics Education, there was a special session in the morning, prepared by the organizers in order to emphasize that there was a problem in the gender composition of the ICMI Executive Committee. All members of the EC who were present were invited to sit on the platform, all men, in front of an audience of women.11 The lesson was clear, and the ICMI EC was renewed later with a fair participation of women. In 1992, Jacobsen, having reached UNESCO’s retirement age, left the mathematics education post at UNESCO after 18 years. In this long period, UNESCO’s collaboration with ICMI had continued by helping in the organization and financing of ICMEs and especially the ICMI regional groups in Latin America, Southeast Asia, and Africa (IACME, SEAMS, and AMU) (Jacobsen 1993). It was Jacobsen who decided to replace in the 1980s the series “New Trends in Mathematics Teaching” with more frequent studies on mathematics education, merged into the series “Science and Technology Education,” which was published in English and Spanish (Jacobsen 1996). These latter volumes stemmed out from different occasions, for instance Mathematics for All came out from Theme Group 1 at ICME 5 (Damerow et al. 1986) and Mathematics, Education and Society summarized the special day at ICME 6 on the political dimension of mathematics education (Keitel et al. 1989, see above). The involvement of UNESCO through Jacobsen favoured the success of the regional conferences and brought to major cooperation with the local governments. A result was, for instance, the creation of a Faculty of Education at the University

 We remember that in that period the only woman in the EC was A. Sierpinska, not present on that occasion. 11

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of Botswana. Jacobsen was awarded in 2011 by IACME of the Luis Santaló Medal for his international commitment to mathematics education.

3.9 ICME-8, Sevilla 1996. Regular Lectures, for All Tastes The period 1995–1998 featured again de Guzmán (Spain) as President and Niss (Denmark) as Secretary. Vice-Presidents were Kilpatrick (USA) and Sierpinska (Canada). The members were Colette Laborde (France), Gilah Leder (Australia), Carlos E.  Vasco (Colombia), Zhang Dianzhou (China). The ex officio members were David Mumford (USA) as President of IMU and Palis (Brazil) as Secretary of IMU. Attendance to the ICME-8 Congress was about 3500 participants from 98 countries (see the logo in Fig. 3.8). The proceedings were edited by Claudi Alsina, José María Alvarez, Niss, Antonio Pérez, Luis Rico, and Anna Sfard (Alsina et al. 1998a, b). In the preface to the second volume (the Selected Lectures), Hodgson (for the Editing Committee: Alsina, Alvarez, C.  Laborde and Pérez) reports that the IPC decided to maintain the practice introduced at ICME-7 of having a significant number of lectures. Therefore, in addition to the plenary lectures, the working groups and topic groups, there were about sixty invited Regular Lectures, a selection of which (33) was published in Volume 2. This large number was necessary as they reflected the many research areas in which mathematics educators were split. Two more ICMI studies were launched in that period: 10. The Role of the History of Mathematics in the Teaching and Learning of Mathematics (Luminy, France, 1998). The Study Volume was published in the year 2000, edited by John Fauvel and Jan van Maanen with the title History in Mathematics Education: The ICMI Study  (NISS 6). 11. Teaching and Learning of Mathematics at University Level (Singapore, 1998). The proceedings were edited by Derek Holton and published in 2002 (NISS 7). Selected papers presented at the Study Conference were published as a special Fig. 3.8  The logo of ICME-8

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issue of the International Journal for Mathematical Education in Science and Technology (31: 1-160, 2000). Pursuing the goal of having a balanced representation at the ICMEs, ICMI implemented—starting with ICME-8—a general policy of forming for each ICME an ICME Solidarity Fund established by setting aside 10% of the registration fees in order to provide grants to congress delegates from non-affluent countries. Bass and Hodgson (2004) report that at each of the recent ICMEs, some 100 to 150 participants from economically challenged regions of the world have thus been given financial support to facilitate their presence at the congress. Another way to involve non-affluent countries is through the organization of the ICMI Regional Conferences, which continued despite the acquired international nature of ICMI’s position. These meetings are supported morally by the ICMI and sometimes with modest financial contributions as well. In 1999, the SEACME-8 – eighth South East Asian Conference on Mathematical Education was organized in Quezon City, Philippines. Kahane (Interview to Kahane, Artigue 2008b, part 3) praised very much the work of the duo de Guzmán–Niss, while Hyman Bass praised the dedication, skill, and efficacy in advancing the work of ICMI of the secretaries Howson, Niss, and later Hodgson. “Each of them, in his own right, has been a major international figure in mathematics education. One cannot overestimate the debt that the ICMI community owes them, as each of the ICMI Presidents will readily bear witness” (Bass 2008 p. 23).

3.10 ICME-9, Tokyo 2000. Grants to Support Mathematics Education In the period 1999–2002, the President was Hyman Bass (USA), with Bernard R.  Hodgson (Canada) as Secretary. Vice-Presidents were Néstor Aguilera (Argentina) and Artigue (France). Four more members composed the EC: Leder (Australia), Yukihiko Namikawa (Japan), Igor F. Sharygin (Russia), Jian-Pan Wang (China). Ex officio members were de Guzmán (Spain) as Past President of ICMI, Palis (Brazil) as President of IMU, Phillip Griffiths (USA) as Secretary of IMU. The participants to the ICME-9 conference were about 2300 from more than 70 countries (the logo is shown in Fig. 3.9). The proceedings were edited by Hiroshi Fujita, Yoshihiko Hashimoto, Hodgson, Peng Yee Lee, Stephen Lerman, and Toshio Sawada (Fujita et al. 2004). The International Program Committee was chaired by Fujita, the National Organizing Committee was chaired by Ken-Ichi Sugiyama. The congress was supported by the Science Council of Japan, led by Rector Yoshikawa. In the presidential address, Bass states: the great challenges now facing mathematics education around the world demand a much deeper and more sensitive involvement of disciplinary mathematicians than we now have, both in the work of educational improvement and in research on the nature of teaching and learning. There are many things that have impeded such boundary crossing and collaboration, such as the need to reconcile language, epistemology, norms of evidence, and, in

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Fig. 3.9  The logo of ICME-9

general, all of the intellectual and attitudinal challenges that face multidisciplinary research and development. ICME-9 brings together people who know and understand different things, to learn from each other, and hopefully to foster collaboration (Fujita et  al. 2004, p. XV)

In his closing remarks, the Secretary Hodgson observes that a great part of grant funds for non-affluent countries comes from individual domestic donations, the majority of these from persons not participating in ICME 9 but wishing to support mathematics education (Fujita et al. 2004, p. 391). In the meeting of April 2002 in Paris, new Terms of Reference for ICMI were approved by the Executive Committee of the International Mathematical Union. Among the modifications is a change in nomenclature regarding the position of “Secretary,” which was now designated by the term “Secretary-General,” as it was in the past (see ICMI Bulletin 51, 2002: 8–12). Moreover, the members of ICMI are now countries and not individuals. The ICMEs are explicitly mentioned as the place where the Commission shall meet every 4 years. So the General Assembly of the Commission consists of. (a) the members of the Executive Committee, as specified in (3) below, and (b) one Representative from each member country of ICMI, as specified in (5) below. The General Assembly of ICMI shall normally meet once in every 4 years, during the International Congress on Mathematical Education. … 3. The Executive Committee of the Commission consists of the following members, elected by IMU: Nine members, including the four officers, namely, the President, two VicePresidents, and the Secretary General. Ex-officio members: The outgoing President of ICMI, the President and the Secretary of IMU. Co-opted members: In order to provide for missing coverage or representation, the ICMI Executive Committee may co-opt up to two additional members.

There is no longer reference to the representative of the Committee on the Teaching of Science in ICSU.

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In 2003 a new study group was affiliated to ICMI, the International Study Group for Mathematical Modelling and Applications (ICTMA). According to Ken Houston, Peter Galbraith, Gabriele Kaiser, all began in 1973 with the McLone Report. This work surveyed what recent mathematics graduates were doing in their employment, how relevant their education was to their work, and how satisfied their employers were with their performance. Some years later, in the UK David Burghes, who could well be described as the Father of ICTMA, decided to try to enliven the school mathematics curriculum by working with teachers to produce interesting modelling investigations for pupils at secondary level. There were many influences from many directions, and things began to happen (Houston et al. 2008). Two new ICMI Studies, the numbers 12 and 13, were launched in these years, namely: 12. The Future of the Teaching and Learning of Algebra (Melbourne, Australia, 2001). The Study Volume was published by in 2004, edited by Kaye Stacey, Helen Chick and Margaret Kendal (NISS 8). 13. Mathematics Education in Different Cultural Traditions: A Comparative Study of East-Asia and the West (Hong Kong, 2002). The Study Volume was published by Springer in 2006, edited by Frederick K.S. Leung, Klaus-D. Graf and Francis J. Lopez-Real (NISS 9). The tradition of regional conferences continued, with the organization of several conferences: the All-Russian Conference on Mathematical Education (Dubna, Russia, 2000); the ICMI-EARCOME-2  – Second ICMI East Asia Regional Conference on Mathematics Education (Singapore, 2002); the XI-IACME – 11th Inter-American Conference on Mathematics Education (Blumenau, Brazil, 2003), and the Espace Mathématique Francophone (EMF) (Tozeur, Tunisia, 2003). In October 2000, in Geneva, the 100  years of the journal L’Einseignement Mathematique were celebrated. Founded in 1899 by Charles-Ange Laisant and Henri Fehr, this journal played an important role as official organ of ICMI within the international community of mathematics educators. This role diminished with time, but the meeting was an important occasion to rethink at the evolution of mathematics education, looking at the teaching of specific mathematical topics (Coray et al. 2003). On the occasion of the 50th issue of ICMI Bulletin, de Guzmán reflects on the meaning of ICMI, asking “what would be nowadays the main tasks in which ICMI could be involved in a natural way?” (De Guzmán 2001). Undoubtedly, the “star activity” of ICMI is the International Congress on Mathematical Education (ICME), notwithstanding the fact that someone questions the need for such an energy-­ consuming activity and asks if it still has the meaning and the impact it should have, considering the other activities as the “ever more influential” ICMI studies and the meeting organized by the affiliated groups. But, according to de Guzmán, there are two problems which appear much more important nowadays and which concern both ICMI and the International Mathematical Union (IMU). These two problems should be at the center of the regular activities of both bodies:

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The main problem with which ICMI should be concerned, as an organism responsible for the health of mathematics education at a global level, as well as IMU, as an organism which has to attend to the good state of the mathematical activity, is the huge gap in many places around the world between those members of the mathematical community whose main activities are related to education, and those whose main occupation is the furtherance of mathematical research, be it oriented towards its more theoretical or its more applied aspects. … For a number of years, a traditional standpoint adopted by the community of those involved in mathematical research (especially university faculty) towards theoretical and practical pedagogical issues, which are of deep concern to another important segment of the mathematical community - such as mathematicians interested by the processes of mathematical learning or those interested in their daily tasks in identifying ways to facilitate this learning at any level -, has been to look at those issues with contempt. Maybe they disregard pedagogical studies and occupations on the basis that they constitute a field of second or third category, where it is very easy to decide at any time what are the appropriate options and where one who has taught for a few years has as much authority as anyone to express a valid opinion (De Guzmán 2001, p. 1).

To solve this gap, it is necessary, according to de Guzmán, to fully recognize that ICMI is the education commission of IMU. The risk of a split into two different organizations must be avoided. This risk is not negligible, as ICMI had a relatively vigorous life before IMU existed.

3.11 ICME-10, Copenhagen 2004. The ICMI Awards The term 2003–2006 was again chaired by Bass (USA), with Vice-Presidents Jill Adler (South Africa) and Michèle Artigue (France). The Secretary-General was still Hodgson (Canada). The other members were Carmen Batanero (Spain), Nikolai Dolbilin (Russia), Maria Falk De Losada (Colombia), Peter L. Galbraith (Australia), Petar S. Kenderov (Bulgaria), Frederick Koon-Shing Leung (Hong Kong). Ex officio members were John Ball (UK), President of IMU, and Phillip Griffiths (USA), Secretary of IMU. We observe that Africa had its first officer, Jill Adler. ICME-10 was held in Copenhagen in 2004, from July 4 to 11 (the logo is shown in Fig. 3.10). Participants numbered about 2300, from nearly 100 different countries. The chair of the IPC was Niss, who also edited the proceedings. Morten Blomhøj chaired the Local Organizing Committee. The proceedings included 64 papers based on the regular lectures (out of 74). According to the editor’s foreword, it has not been possible to include reports on several other important congress activities such as the five national presentations by Korea, Mexico, Romania, and Russia, and the Nordic host countries (Denmark, Finland, Iceland, Norway, and Sweden), the 46 workshops, the 12 sharing experiences groups, the more than 220 posters, the five ICMI Affiliated Study Groups, and the several informal meetings. The closing

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Fig. 3.10  The logo of ICME-10

address was given, as usual, by the Secretary-General of ICMI, Bernard Hodgson. Among the various innovations of this congress, he particularly mentions the creation of five so-called Survey Teams, each having as a mandate to survey the state-­of-­the-art with respect to a certain theme or issue, paying particular attention to the identification and characterization of new knowledge, recent developments, new perspectives and emergent issues (Niss 2008). ICME-10 also featured the first awarding of two medals in mathematics education research, inaugurated by ICMI (officially assigned in 2003). The Felix Klein Medal for lifetime achievement was awarded to Guy Brousseau from France. The Hans Freudenthal Medal for a major program of research was awarded to Celia Hoyles of the UK (information about these awards and citations of the work of the laureates can be found on the ICMI website, http://www.mathunion.org/ICMI/).12 ICME-10 was also the occasion to launch a travelling exhibition titled “Why Mathematics?” on which ICMI was collaborating with UNESCO.  This international exhibition on mathematical objects and phenomena was aimed particularly at young people, their parents, and their teachers and would later travel to various places. The ICMI studies launched in that period were the numbers  In the next years, the recipients of the Felix Klein Award were Ubiratan d’Ambrosio (Brazil) in 2005 and Jeremy Kilpatrick (USA) in 2007. The Hans Freudenthal Award was assigned to Paul Cobb (UK/USA) in 2005 and to Anna Sfard (Israel) in 2007.  In 2016 also the Emma Castelnuovo Award for Excellence in the Practice of Mathematics Education was established. 12

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14. Applications and Modeling in Mathematics Education (Dortmund, Germany, February 2004). The Study Volume was published by Springer in 2007 with the title Modeling and Applications in Mathematics Education, and was edited by Werner Blum, Peter L. Galbraith, Hans-Wolfgang Henn and Mogens Niss (New ICMI Study Series 10). 15. Professional Education and Development of Teachers of Mathematics (Águas de Lindóia, Brazil, May 2005). The Study Volume published by Springer in 2009 had the same title and was edited by Ruhama Even and Deborah Loewenberg Ball (NISS 11). 16. Challenging Mathematics in and beyond the Classroom (Trondheim, Norway, June 2006). This is also the title of the Study Volume published by Springer in 2009, edited by Edward J. Barbeau and Peter J. Taylor (NISS 12). 17. Technology in Mathematics Education (Hanoi, Vietnam, December 2006). The Study Volume published by Springer, in 2010 was titled Mathematics Education and Technology - Rethinking the Terrain, and was edited by Celia Hoyles and Jean-Baptiste Lagrange (NISS 13). Regional conferences held in the same period were the EARCOME-3, third ICMI East Asia Regional Conference on Mathematics Education (Shanghai, China, 2005), and EMF 2006  – Espace Mathématique Francophone (Sherbrooke, Canada, 2006). At ICM-2006, the French Michèle Artigue became the first woman appointed as President of ICMI. Up to 2008, of the 107 officers, only nine were women. Former Vice-President Artigue was not only the first female ICMI President ever, but also the first ICMI President after Smith whose primary expertise is mathematics education rather than research mathematics. This was linked to the objective of changing the rules of ICMI, an objective that Bass had set for his presidency. As Bass tells (Bass 2020, section 10), one of his conditions to become President of ICMI was the change in the terms of reference, what indeed happened in 2007, when the election of ICMI’s EC became a task of the General Assembly of ICMI itself.

3.12 2008: The Centennial of ICMI in Rome and ICME-11 in Monterrey The term 2007–2009 was chaired by Artigue (France), with Vice-Presidents Adler (South Africa) and Bill Barton (New Zealand). The Secretary-General was again Hodgson (Canada). The other members were Mariolina Bartolini Bussi (Italy), Jaime Carvalho e Silva (Portugal), Hoyles (UK), Kumaresan S. aka Kumaresan Somaskandan (India), Koon-Shing Leung (Hong Kong SAR), Alexei L. Semenov. Other members were Bass (USA), Past President of ICMI, László Lovász (Hungary), President of IMU and Martin Grötschel (Germany), Secretary of IMU. Having in mind to pave the way for a smooth transition to the new governance structure, the new EC was established by the 2006 IMU General Assembly for a 3-year term, 2007–2009. To ensure continuity from the past to the future, Hodgson

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Fig. 3.11  The logo of the Centennial

was exceptionally asked to serve as the Secretary-General for a third term, also a complete novelty (besides the first female President) since the time when Henri Fehr (1870–1954) served for decades as Secretary-General of the “Old ICMI” (Hodgson and Niss 2018, p. 14). An important event preceded ICME-11. It was the celebration of the centennial of the foundation of ICMI. An international symposium entitled “The First Century of the International Commission on Mathematical Instruction. Reflecting and Shaping the World of Mathematics Education” was held in Rome from 5 to 8 March 2008. Once again, as it did in 1952, when ICMI was reconstituted as a subcommission of IMU, Palazzo Corsini, home of the Accademia Nazionale dei Lincei, provided the venue for the congress, along with Palazzo Mattei di Paganica, home of the Enciclopedia Italiana (the logo in Fig.  3.11) The congress was attended by about 180 participants representing 43 countries. The program included ten plenary lectures, eight parallel lectures, five working groups and a panel discussion. An afternoon was reserved for the Italian teachers and was broadcasted in Italian schools. The last day featured an excursion that recalled that of 1908, and took the participants to visit the Villa d’Este and Hadrian’s Villa in Tivoli (see Castelnuovo 1909). The conviction that history is a powerful means not only for giving an account of the past but also for building the future, inspired the activities of the symposium as well as the publication. The papers in the Proceedings touch on a wide variety of themes: the origins of the ICMI; its rebirth at the end of the 1960s and the emergence of the new field of research of mathematics education; the dialectic between rigor and intuition; the relationships between pure and applied mathematics and the emphasis to be given to each; the interactions between research and practice; the comparison between centres and peripheries of the world; the relationships between mathematics and mathematics teaching; the training of teachers; and the relationship of mathematics education to technology, society and other disciplines. It emerges that ICMI has mirrored the development of mathematics education as a field of

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Fig. 3.12  The logo of ICME-11

study and practice, and stimulated new directions of research, opening new horizons (Menghini et  al. 2008). On that occasion also the website mentioned before was established by Fulvia Furinghetti and Livia Giacardi (Furinghetti and Giacardi 2008). Some months later, ICME-11 was held in Monterrey, Mexico, from July 6 to July 13 (the logo is shown in Fig. 3.12). For the first time an ICME was held in a developing (or “non-affluent”) country. The bid was made by the Mexican Mathematical Society with the support of the Ministry of Education. The Chair of the International Program Committee was Marcela Santillán, while Carlos Signoret chaired the Local Organizing Committee. No proceedings have been edited, some contributions and summaries are available online in the ICMI’s website (ICME-11 2008). On the congress website we read that between 2000 and 2500 professionals from 100 countries were expected: www.mathunion.org/fileadmin/ICMI/Conferences/ICME/ICME11/www. icme11.org/index.html Retrieved 23 February 2021. In her opening address, Artigue comments on the role of the ICMEs: There is no doubt that, in the last decade, the number of conferences in mathematics education has exploded, leading to question what is the role, the specificity of a congress such as ICME. As are the International Congresses of Mathematicians for mathematics, ICMEs are unique events for mathematics education. Why? This is not only due to their size, to the international representation they gather, but also because reflecting ICMI values, they address to all those involved in mathematics education all around the world: educational researchers, teachers of mathematics and teacher educators, mathematicians, curriculum designers, educational policy makers and administrators. ICMI ambition is to provide them all with a unique forum for exchanging, discussing, disseminating ideas and realizations, a unique opportunity for accessing information about the most recent advances in the field of mathematics education, an information covering the multiplicity of its dimensions, and sensitive to the diversity of the voices that exist in it. This explains the diversity of proposed activities […]. (https://www.mathunion.org/fileadmin/ ICMI/files/Digital_Library/ICMEs/Opening_Michele_Artique_President_ICMI.pdf, Retrieved 1 November 2020)

The IPC proposed to launch the academic activities of ICME-11 through a dialogue on issues of crucial interest for mathematics education. Artigue and Kilpatrick initiated the dialogue posing the following questions: What do we know that we did not know 10 years ago in mathematics education, and how have we come to know it?

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• What kind of evidence is accessible, and what has to be looked for in mathematics education? • What are the societal expectations regarding our field, and how do we situate ourselves regarding them? • Up to what point can visions of teaching and learning mathematics and evidence in the field transcend the diversity of educational contexts and cultures? • What are the main challenges that mathematics education faces today? The paper that appears on the website collecting the materials of ICMI-1113 reflects the dialogue proposing the different positions of the two authors. In the same year, the ICMI study 18 on Statistics Education in School Mathematics: Challenges for Teaching and Teacher Education, was organized jointly by ICMI and the International Association for Statistical Education (IASE). The study Conference was held in Monterrey, México, July 2008. The Study Volume was published by Springer in 2011, with the title Teaching Statistics in School Mathematics-Challenges for Teaching and Teacher Education and was edited by Carmen Batanero, Gail Burrill, Chris Reading (New ICMI Study Series, 14). In 2008, a new IMU/ICMI project came to life: the Klein Project, whose aim is to produce mathematics resources for secondary teachers on contemporary mathematics. It materializes through “vignettes,” which are a short, readable piece on a topic of contemporary mathematics (http://blog.kleinproject.org/?page_id=363). The project was inspired by Felix Klein’s book Elementary Mathematics from a Higher Standpoint, first published 100  years earlier (in the year of the birth of ICMI). It aims at representing a stimulus for mathematics teachers, so to help them to make connections between the mathematics they teach, or could teach, and the field of mathematics, while taking into account the evolution of this field over the last century.

3.13 Conclusions In the year of the centenary of ICMI, there are 84 member countries of ICMI, 68 of which are also members of IMU and 2 are associate members (ICMI Bulletin 2008, 62, p.  3). Each country, whether an IMU member or not, is invited to appoint a Representative to ICMI, who acts as a liaison between ICMI and the mathematics education community in the country. Of course, 84 is less than the half of the total countries in the world, but the effort to expand the presence of the ICMI in the world is undeniable. An example is the more recent CANP (Capacity & Networking Project), started in 2011 and promoted by ICMI, IMU, UNESCO and ICIAM (International Congress of Industrial and Applied Mathematics).

 https://www.mathunion.org/fileadmin/ICMI/files/Digital_Library/ICMEs/Plenary_1_MA_JK_ final_01.pdf (Retrieved 1 November 2020). 13

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CANP aims to enhance mathematics education at all levels in developing countries so that their people are capable of meeting the challenges these countries face. It wants to develop the educational capacity of those responsible for mathematics teachers, and create sustained and effective regional networks of teachers, mathematics educators and mathematicians, also linking them to international support (https://www.mathunion.org/icmi/news-­and-­events/2011-­08-­09/canp-­project). The CANP project, as well as the Klein project mentioned above, help to face two of the “evergreen” questions within ICMI: the relationship with mathematicians and the engagement of non-affluent countries. The evolution of the organization of the successive ICMEs leads us to ask the question that Howson asked himself (Howson 2004, p. 1): “Since [ICME-1] there have been many changes, but have we arrived at a suitable form and balance, better than anything that has gone before?” And, in reference to the current context, another question arises: “What is the role of the ICMEs in this context?” De Guzmán wondered if such huge conferences still made sense, Artigue (indirectly) answered that they have, because they reflect ICMI values, and therefore address to all those involved in mathematics education all around the world. We can add that, being an ICME a large container for a wide variety of topics related to mathematics education, everybody finds their place in it. Probably this very diversity helps to keep the community of mathematics educators together.

References ICME Proceedings Alsina, Claudi, José María Alvarez, Mogens Niss, Antonio Pérez, Luis Rico, and Anna Sfard, eds. 1998a. Proceedings of the 8th international congress on mathematical education. Sevilla: S.A.E.M. Thales. Alsina, Claudi, José María Alvarez, Bernard Hodgson, Colette Laborde, and Antonio Pérez, eds. 1998b. 8th international congress on mathematical education. Selected lectures. Sevilla: S.A.E.M. Thales. Athen, Hermann, and Heinz Kunle, eds. 1977. Proceedings of the third international congress on mathematical education. Karlsruhe: Zentralblatt für Didaktik der Mathematik. Carss, Marjorie, ed. 1986. Proceedings of the fifth international congress on mathematical education. Basel: Birkhäuser. Fujita, Hiroshi, Yoshihiko Hashimoto, Bernard R. Hodgson, Peng Yee Lee, Stephen Lerman, and Toshio Sawada. 2004. Proceedings of the ninth international congress on mathematical education. Dordrecht: Kluwer Academic Publishers. The book is accompanied by a CD whose revised and completed version appeared in 2005. Gaulin, Claude, Bernard R. Hodgson, David H. Wheeler, and John Egsgard, eds. 1994. Proceedings of the seventh international congress on mathematical education. Québec: Les Presses de l’Université Laval. Hirst, Ann, and Keith Hirst, eds. 1988. Proceedings of the sixth international congress on mathematical education. Budapest: János Bolyai Mathematical Society.

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Howson, A.  Geoffrey, ed. 1973. Developments in mathematical education. Proceedings of the second international congress on mathematical education. Cambridge: Cambridge University Press. ICME-11. 2008. Materials that have been collected from ICME-11  in Monterrey (México). https://www.mathunion.org/icmi/publications/icme-­proceedings-­and-­publications. Retrieved 1 Nov 2020. Niss, Mogens, ed. 2008. Proceedings of the tenth international congress on mathematical education. Roskilde: IMFUFA, Roskilde University. Robitaille, David, David H.  Wheeler, and Carolyn Kieran, eds. 1994. Selected lectures from the seventh international congress on mathematical education. Québec: Les Presses de l’Université Laval. The Editorial Board of Educational Studies in Mathematics. 1969. Proceedings of the first international congress on mathematical education. Dordrecht: Reidel Publishing Company. Zweng, Marilyn, Thomas Green, Jeremy Kilpatrick, Henry Pollak, and Marilyn Suydam, eds. 1983. Proceedings of the fourth international congress on mathematical education. Basel: Birkhäuser.

Papers and Books Bass, Hyman. 2005. Mathematics, mathematicians, and mathematics education. Bulletin of the American Mathematical Society 42 (4): 417–430. ———. 2008. Moments in the life of ICMI. In The first century of the international commission on mathematical instruction (1908–2008): Reflecting and shaping the world of mathematics education, ed. Marta Menghini, Fulvia Furinghetti, Livia Giacardi, and Ferdinando Arzarello, 9–24. Roma: Istituto della Enciclopedia Italiana. ———. 2020. I accepted to be nominated as president of ICMI with the intention of assuring that the process of my nomination would not be repeated - a follow up to the historical vignette. ICMI Newsletter, section 10, July, 14. Bass, Hyman, and Bernard R.  Hodgson. 2004. The international commission on mathematical instruction. What? Why? For whom?, Notices of the AMS, June/July, 639–644. Becker, Jerry P. 1970. Some notes on the first international congress on mathematical education. American Mathematical Monthly 77 (3): 299–302. Cartan, Henri. 1970. Letter to Otto Frostman, October 15, 1967–1974. Paris: ICMI Archives 14b. Castelnuovo, Guido. 1909. Atti del 4. Congresso internazionale dei matematici: Roma 6–11 Aprile 1908, Vol. I. Tipografia della R. Accademia dei Lincei. Christiansen, Bent. 1978. The cooperation between ICMI and UNESCO. ICMI Bulletin 10: 4–10. Coray, Daniel, Fulvia Furinghetti, Helène Gispert, Bernard R. Hodgson, and Gert Schubring. 2003. One Hundred Years of L’Enseignement Mathématique, Monographie n. 39 de L’Enseignement Mathématique. Damerow, Peter, Mervyn Dunkley, Bienvenido Nebres, and Bevan Werry. 1986. Mathematics for all. In Science and technology education, document series. Paris: UNESCO. De Guzmán, Miguel. 2001 The meaning of ICMI Today, ICMI Bulletin n. 50. https://www. mathunion.org/fileadmin/IMU/Organization/ICMI/bulletin/50/meaning.html. Retrieved 1 Nov 2020. Dionne, Jean J., and Bernard R. Hodgson. 2020. Claude Gaulin (1938-2020). In Memoriam. ICMI Newsletter, section 11, July, 15–16. Freudenthal, Hans. 1978. Address to the first conference of I.G.P.M.E. (International Group for the Psychology of Mathematical Education), at Utrecht 29 August 1977. Educational Studies in Mathematics 9: 1–5.

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Furinghetti, Fulvia, and Livia Giacardi. 2010. People, events, and documents of ICMI’s first century. Actes d’història de la ciència i de la tècnica, nova època 3 (2): 11–50. Furinghetti, Fulvia, Marta Menghini, Ferdinando Arzarello, and Livia Giacardi. 2008. ICMI renaissance: The emergence of new issues in mathematics education. In The first century of the international commission on mathematical instruction (1908–2008): Reflecting and shaping the world of mathematics education, ed. Marta Menghini, Fulvia Furinghetti, Livia Giacardi, and Ferdinando Arzarello, 131–147. Roma: Istituto della Enciclopedia Italiana. Furinghetti, Fulvia, José Manuel Matos, and Marta Menghini. 2013. From mathematics and education, to mathematics education. In Third international handbook of mathematics education, ed. M.A.  Ken Clements, Alan Bishop, Christine Keitel, Jeremy Kilpatrick, and Frederick K.S. Leung, 273–302. New York etc.: Springer. Furinghetti, Fulvia, Livia Giacardi, and Marta Menghini. 2020. Actors in the changes of ICMI: Heinrich Behnke and Hans Freudenthal. In “dig where you stand” 6. Proceedings of the sixth International Conference on the History of Mathematics Education, ed. Evelyne Barbin, Kristin Bjarnadóttir, Fulvia Furinghetti, Alexander Karp, Guillaume Moussard, John Prytz, and Gert Schubring, 247–260. Münster: WTM-Verlag. Goos, Merrilyn. 2020. Review of ICMI studies: Some initial findings. ICMI Newsletter, section 4, July, 5–7. Hilton, Peter. 1980. Letter to Jacques-Louis lions, IMU secretary, 24 September 1980. ICMI Archives, Box 14B: 1967–1980. Hodgson, Bernard R. 1991. Regards sur les études de la CIEM. L’Enseignement Mathématique s. 2 (37): 89–107. ———. 1999. The ICMI studies: Background information and projects. ICMI Bulletin: 46. ———. 2009. ICMI in the post-Freudenthal era: Moments in the history of mathematics education from an international perspective. In Dig where you stand. proceedings of the conference on On-going research in the history of mathematics education, ed. Kristín Bjarnadóttir, Fulvia Furinghetti, and Gert Schubring, 79–96. Reykjavik: University of Iceland – School of Education. ———. 2019. Once upon a time… historical vignettes from the ICMI archive: Episodes from the Freudenthal era, ICMI Newsletter, section 5, July, 8–10. ———. 2020. Once upon a time… historical vignettes from the archives of ICMI: A dilemma related to the ICMI terms of reference, ICMI Newsletter, section 6, November, 6–8. Hodgson, Bernard R., and Mogens Niss. 2018. ICMI 1966-2016: A double insiders’ view of the latest half century of the international commission on mathematical instruction. In Invited lectures from the 13th international congress on mathematical education, ed. Gabriele Kaiser, Helen Forgasz, Mellony Graven, Alain Kuzniak, Elaine Simmt, and Xu Binyan, 229–247. Cham: Springer. Holton, Derek. 2008. The process of an ICMI study. In The first century of the international commission on mathematical instruction (1908–2008): Reflecting and shaping the world of mathematics education, ed. Marta Menghini, Fulvia Furinghetti, Livia Giacardi, and Ferdinando Arzarello, 217–223. Roma: Istituto della Enciclopedia Italiana. Howson, A. Geoffrey. 1984. Seventy-five years of the international commission on mathematical instruction. Educational Studies in Mathematics 15: 75–93. ———. 2004. Reflections on ICMEs. Papers from unpublished issues of the ICMI Bulletin. https://www.mathunion.org/icmi/publicationsicmi-­bulletin/papers-­unpublished-­issues-­icmi-­ bulletin. Retrieved 19 Dec 2020. ———. 2007. Some notes on the early ICMI studies. Papers from unpublished issues of the ICMI Bulletin. https://www.mathunion.org/icmi/publicationsicmi-­bulletin/papers-­unpublished-­ issues-­icmi-­bulletin. Retrieved 19 Dec 2020. Iyanaga, Shōkichi. 2001. My remembrances from the period 1975-1978. ICMI Bulletin 50. https:// www.mathunion.org/fileadmin/IMU/Organization/ICMI/bulletin/50/remem.html. Retrieved 1 Nov 2020. Jacobsen, Edward. 1993. The cooperation between ICMI and UNESCO. ICMI Bulletin 34: 11–12.

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Jacobsen, Edward C. 1996. International co-operation in mathematics education. In International handbook of mathematics education, ed. Alan J.  Bishop, Ken Clements, Christine Keitel, Jeremy Kilpatrick, and Colette Laborde, 1235–1256. Dordrecht/Boston/London: Kluwer. Kahane, Jean-Pierre. 1986a. Enseignement mathématique, ordinateurs et calculettes. In International congress of mathematicians, ed. Andrew M.  Gleason, vol. II, 1682–1696. Providence: American Mathematical Society. ———. 1986b. Mathématique comme discipline de service. Bullettin de l’APMEP 353: 161–184. ———. 1988a. ICMI and recent developments in mathematics education. In Wissenschaftliches Kolloquium Hans-Georg Steiner zu Ehren, ed. Bernard Winkelmann, 13–23. Occasional paper 116. ———. 1988b. La grande figure de Georges Pólya. In Proceedings of the sixth international congress on mathematical education, ed. Ann Hirst and Keith Hirst, 79–97. Budapest: János Bolyai Mathematical Society. ———. 1990. Farewell message. ICMI Bulletin 29: 3–8. ———. 2001. Some aspects of my terms as ICMI president. ICMI Bulletin 50. https://www. mathunion.org/fileadmin/IMU/Organization/ICMI/bulletin/50/asp_pres.html. Retrieved 1 Nov 2020. Keitel, Christine, Alan Bishop, Peter Damerow, and Paulus Gerdes, eds. 1989. Mathematics, education, and society, science and technology education, document series. Paris: UNESCO. Kilpatrick, Jeremy. 2008. The development of mathematics education as an academic field. In The first century of the international commission on mathematical instruction (1908–2008): Reflecting and shaping the world of mathematics education, ed. Marta Menghini, Fulvia Furinghetti, Livia Giacardi, and Ferdinando Arzarello, 25–39. Roma: Istituto della Enciclopedia Italiana. Lehto, Olli. 1998. Mathematics without Borders: A history of the International Mathematical Union. NewYork: Springer. Lighthill, James. 1973. The presidential address. In Developments in mathematical education. Proceedings of the second international congress on mathematical education, ed. A. Geoffrey Howson, 88–100. Cambridge University Press. Lim-Teo, Suat Khoh. 2008. ICMI activities in east and Southeast Asia: Thirty years of academic discourse and deliberations. In The first century of the international commission on mathematical instruction (1908–2008): Reflecting and shaping the world of mathematics education, ed. Marta Menghini, Fulvia Furinghetti, Livia Giacardi, and Ferdinando Arzarello, 247–252. Roma: Istituto della Enciclopedia Italiana. Menghini, Marta, Fulvia Furinghetti, Livia Giacardi, and Ferdinando Arzarello, eds. 2008. The first century of the international commission on mathematical instruction (1908–2008): Reflecting and shaping the world of mathematics education. Roma: Istituto della Enciclopedia Italiana. Pollak, Henry O., Mogens Niss, and Jean-Pierre Kahane. 2004. In memoriam – Jacobus H. Van lint (1932-2004). ICMI Bulletin 55: 80–84. Quadling, Douglas. 1988. Topic area 5: Comparative education. In Proceedings of the sixth international congress on mathematical education, ed. Ann Hirst and Keith Hirst, 342–345. Budapest: János Bolyai Mathematical Society. Redaktionskomitee. 1969. Zu den Aufgaben des Zentralblatts für Didaktik der Mathematik. ZDM 1 (1): 1. Schubring, Gert. 2018. Die Geschichte des IDM Bielefeld als Lehrstück. Düren: Shaker Verlag. Sobel, Max. 1983. Back-to-basics. In Proceedings of the fourth international congress on mathematical education, ed. Marilyn Zweng, Thomas Green, Jeremy Kilpatrick, Henry Pollak, and Marilyn Suydam, 29–32. Basel: Birkhäuser. UNESCO. (1966/1970/1972-73). New trends in mathematics teaching/ Tendances nouvelles de l’enseignement des mathématiques. In Prepared by the international commission on mathematical instruction. The teaching of basic sciences. Mathematics, Vol. 1, 2, 3. Paris: UNESCO. ———. 1979. New trends in mathematics teaching/ Tendances Nouvelles de l’enseignement des mathématiques. In Prepared by the international commission on mathematical instruction. The teaching of basic sciences. Mathematics, Vol. 4. Paris: UNESCO.

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Interviews Artigue, Michèle. 2008a. Interview to Maurice Glaymann. In (Furinghetti and Giacardi 2008a). https://www.icmihistory.unito.it/clips.php. Retrieved 1 Nov 2020. ———. 2008b. Interview to Jean-Pierre Kahane. In (Furinghetti and Giacardi 2008b). https:// www.icmihistory.unito.it/clips.php. Retrieved 1 Nov 2020. ———. 2008c. Interview to André Revuz. In (Furinghetti and Giacardi 2008c). https://www.icmihistory.unito.it/clips.php. Retrieved 1 Nov 2020. Hodgson, Bernard. 2008a. Interview to Geoffrey Howson. In (Furinghetti and Giacardi 2008a). https://www.icmihistory.unito.it/clips.php. Retrieved 1 Nov 2020. ———. 2008b. Interview to Bryan Thwaites. In (Furinghetti and Giacardi 2008b). https://www. icmihistory.unito.it/clips.php. Retrieved 1 Nov 2020. Schubring, Gert. 2008. Interview to Heinz Kunle. In (Furinghetti and Giacardi 2008). https://www. icmihistory.unito.it/clips.php. Retrieved 1 Nov 2020.

Websites Fasanelli, Florence, and John G. Fauvel. 2008. HPM. The first twenty-five years 1976–2000. Furinghetti, Fulvia, and Livia Giacardi. 2008. The first century of the international commission on mathematical instruction (1908–2008). The history of ICMI. http://www.icmihistory.unito.it/. Retrieved 1 Nov 2020. Houston, Ken, Peter Galbraith, and Gabriele Kaiser. 2008. ICTMA: The first twenty-five years.. https://www.icmihistory.unito.it/ictma.php#1. Retrieved 1 Nov 2020. Kenderov, Petar. 2008. The history of WFNMC. https://www.icmihistory.unito.it/wfnmc.php. Nicol, Cynthia, and Steve Lerman. 2008. A brief history of the. In With assistance from Joop van Dormolen, ed. Carolyn Kieran, Gerard Vergnaud, Kath Hart, and Heinrich Bauersfeld. PME: International Group for the Psychology of Mathematics Education. https://www.icmihistory. unito.it/pme.php. Retrieved 1 Nov 2020. Shelley, Nancy. 2008. A brief history of IOWME. https://www.icmihistory.unito.it/iowme.php#2. Retrieved 1 Nov 2020.

Chapter 4

The Voice of the Protagonists: A Selection of Unpublished Letters Livia Giacardi

4.1 Introduction Unpublished correspondences and archival documents allow those who study the past to reconstruct the spirit of an era in its various facets, and to revive “the articulate audible voice of the Past, when the body and material substance of it has altogether vanished like a dream” (Carlyle 1841, p. 258), as the Scottish mathematician and historian Thomas Carlyle wrote. The selection of letters and documents that I present in this chapter has the purpose to highlight, through the voice of the protagonists, unknown, or lesser-known aspects of the history of ICMI such as, for example, the not always harmonious relations between ICMI and IMU, the internal dynamics of the Commission, and also to discover the true motivations behind certain actions. The selection, guided by the above purpose, was also conditioned by the international COVID-19 pandemic, which made access to certain archives more difficult. Fortunately, the research that I had undertaken on the occasion of the ICMI centenary in 2008, photocopying, filing, and studying a good part of the documents kept at the IMU Archive, then located in Helsinki, greatly facilitated my work. These documents belong to the period from 1952 to 1974 and the authors are all deceased. That earlier research was made possible by Michèle Artigue, Bernard Hodgson, then president of ICMI and secretary-general of ICMI respectively, and by Guillermo Curbera, curator of the Archive of IMU.

L. Giacardi (*) University of Turin, Turin, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_4

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Other documents come mainly from the Fondo Delessert relating to ICMI, donated by Delessert’s daughter Edith Samba to the Biblioteca Matematica “Giuseppe Peano” in Turin, from the Fonds Georges de Rham, in Lausanne, and from the Smith Professional Papers at Columbia University. Unfortunately, the archival sources about ICMI preserved in the IMU Archive date from 1952, when ICMI became a permanent subcommission of IMU.  While the Klein era is well documented, mainly by the correspondence of David E. Smith, the period 1920–1952 is not; actually the life of the Commission after the International Congress of Mathematicians in 1920 in Strasbourg—during which its term was not renewed— had ups and downs due to the difficulty of restoring international scientific collaboration after World War I, and due to the lack of a charismatic and driving figure like Klein had been.1 Furthermore, the fact that the role of ICMI became marginal on the international scene2 makes exploration of the archives broader and more complex.3

Editorial Conventions Adopted The criteria adopted for transcription of letters and documents aim to respect the original text as much as possible, both in spelling and punctuation, bearing in mind that various authors write in a language other than their own. The words underlined in the original are also underlined in the transcription, while those deleted in the draft of the letters are inserted in a footnote preceded by the abbreviation Del. The handwritten additions in the margins are reported in footnotes as well, while those in the text are written in italics. In some cases, deciphering of the text was very difficult and the omitted illegible words were marked with an ellipsis (…); the words with uncertain transcription are followed by the symbol (?). The end of the page is marked with the symbol //. Additions by the editor are indicated by […]. For reasons of graphic uniformity, the placement of the date and signature in each letter has been normalized according to the traditional form of pagination. For the sake of simplicity, the acronym ICMI is used to indicate the various names assumed by the Commission over time: Commission Internationale de l’Enseignement Mathématique (CIEM), Internationale Mathematische Unterrichtskommission (IMUK), and International Commission on Mathematical Instruction, although that acronym was officially introduced only in 1954.

 See Schubring 2008, and Chap. 1 by Gert Schubring in this volume.  See Gispert 2021, who underlines the “narrowness” of vision of ICMI in this phase, while L’Enseignement Mathématique, its official journal, “continued to reflect a much broader and more diverse mathematical world.” (p. 85). 3  It should be noted, however, that recent research has brought to light interesting unpublished documents relating to this period: see the report by Walther Lietzmann on the German delegation at the ICM in Oslo 1936 (Hollings et al. 2020, pp. 271–277), in which ICMI’s contributions are also highlighted. 1 2

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For the people mentioned in the letters and documents that I present below, I have entered in a footnote the dates of birth and death and the position held in ICMI, referring to the detailed biographical profiles of the members of the Central/ Executive Committees of ICMI in Part III of this volume. Few biographical notes have been provided for the lesser-known figures, with particular attention to elements useful for understanding the text. The bibliographical data of the cited articles and books have been reported in full in the footnotes, for the convenience of the reader, while the general bibliography appears at the end of the chapter. Acknowledgements My most wholehearted thanks go to Judith Goodstein, Elena Anne Marchisotto, Norbert Schappacher, Gert Schubring, and to all those who in various ways have provided their help: Guillermo Curbera, Dirk de Bock, Corinna Desole, Renaud d’Enfert, Fulvia Furinghetti, Denis Gailor,  Geoffrey Howson, Scott Jung, Erika Luciano, Manuel Ojanguren, Kim Williams, and Erich Wittmann. Special thanks go to the directors and personnel of the various archives I explored: Birgit Seeliger (IMU Archive, Berlin), Laura Garbolino, Antonella Taragna, Giulia Scarcia, Giuseppe Semeraro (Biblioteca Matematica G.  Peano, Università di Torino), Tara Craig (Rare Book & Manuscript Library, Columbia University, New York), and Sacha Auderset (UNIL, Université de Lausanne).

4.2 Letters and Documents Henri Fehr4 to David E. Smith,5 Geneva, 18 July 1908 Rare Book & Manuscript Library, Columbia University, New  York, Smith Professional Papers, Series I: Cataloged Correspondence (RBML-SPP from now on), Box 16 Handwritten letter L’Enseignement Mathématique Revue internationale paraissant tous les deux mois Rédaction: C.-A. Laisant, 162, Avenue Victor-Hugo, Paris. H.  Fehr, 72, Route de Florissant, Genève.  – A.  Buhl, 6, Rue Villefranche, Montpellier. Administration: 72, Route de Florissant, Genève Genève, le 18 juillet 1908

 Henri Fehr (1870–1954), secretary-general of CIEM/IMUK, was a Swiss mathematician and cofounder with Charles-Ange Laisant of the journal L’Enseignement Mathématique. 5  David Eugene Smith (1860–1944) was an American mathematician, interested in mathematics education and history of mathematics. At that time, he taught at Teachers College in New York, and was an associate editor of the Bulletin of the American Mathematical Society. He was president of CIEM/IMUK from 1928 to 1932. 4

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Mon cher monsieur Smith, Vous avez eu des séances très fructueuses avec M. Klein. Le premier aperçu que vous me donnez du résultat montre en effet que la Commission va être établie sur des bases solides avec un excellent plan général. Je vous félicite et je vous remercie de ce travail qui facilitera le mien dans la suite. Entièrement d’accord avec les changements que vous avez apportés à mon avant-­ projet qui devait simplement servir de premier point de départ. Il est en effet préférable de prendre nous-même la responsabilité de la Commis // sion dans le choix des délégués. Nous pouvons ainsi nous assurer plus facilement des hommes compétents et actifs. J’espère que vous parviendrez à gagner entièrement M. Greenhill6 à notre cause. A la suite d’un entretien que nous avons eu à Rome avec M. Bryan7 nous avons convenu qu’on entrerait en relation avec M. Pendlebury8 40 Glazebury Road West Kensington London W. Dès que j’aurai examiné le texte détaillé que vous m’annoncez je vous écrirai plus longuement. Donc merci de votre précieuse appui et croyez-moi votre bien dévoué H. Fehr Felix Klein9 to David E. Smith, Göttingen, 6 October 1908 RBML-SPP, Box 29 Handwritten letter Göttingen, 6. Okt. 08 Verehrter Herr Kollege!10  Alfred George Greenhill (1847–1927), was a British mathematician well known for his research on applications of elliptic integrals in electromagnetic theory. 7  George Hartley Bryan (1864–1928) was a British applied mathematician who was well-known for his work in thermodynamics and aeronautics. He participated in the International Congress of Mathematicians (ICM) in Rome (1908). 8  Charles Pendlebury (1854–1941) was a British mathematician. In 1885 he joined the Association for the Improvement of Geometrical Teaching, became its secretary in 1886 and held this office till his resignation in 1936. 9  Felix Klein (1849–1925) was a German mathematician who was not only active in research (in particular, group theory, complex analysis, and non-Euclidean geometry), but was also deeply interested in the reform of secondary and university mathematics teaching. He was the first president of CIEM/IMUK. 10  The English translation is: “Esteemed Colleague! the consultations in Cologne with Fehr and Greenhill were quite successful. We have discussed your organizational and working draft in detail and Fehr will work out a new version of it in the near future, which will then be used as the basis for further procedures. Once the situation in the main countries will have been sufficiently clarified by the correspondence to be initiated in this way, the English government should officially request 6

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Die Cölner Beratungen mit Fehr und Greenhill verliefen durchaus erfolgreich. Wir haben Ihren Organisations- und Arbeitsentwurf genau durchgesprochen und Fehr wird danach in nächster Zeit Neuredaktion ausarbeiten, die dem weiteren Vorgehen zu Grunde gelegt werden soll. Ist durch die solcherweise einzuleitende Korrespondenz die Sachlage in den Hauptländern hinreichend geklärt, so soll die englische Regierung die anderen Staaten offiziell zur Ernennung von Delegierten und zur Bildung nationaler Beiräte auffordern. Wir hoffen dadurch dem Unter // nehmen von vornherein die erforderliche Festigkeit auch nach der finanziellen Seite zu geben. (Wir dachten, dass in dem Briefe der englischen Regierung, soweit es sich um Hauptländer handelt, die Aufforderung enthalten sein könnte, den nationalen Comités ein Pauschquantum von mindestens 500 L. zur Verfügung zu stellen). Dabei ist natürlich so zu verfahren, dass die mathematischen Gesellschaften der einzelnen Länder (oder die etwaigen sonst in Betracht kommenden Instanzen) ihren Regierungen vorweg die Namen der zu ernennenden Delegierten oder Beiräte in Vorschlag bringen. So ist denn in Cöln die Deutsche Mathematiker // Vereinigung bereits in die Personenfrage eingetreten und hat die Anregungen, die wir ihr erteilt hatten, sich angeeignet. Die Vorschläge (die ich inzwischen bereits der Deutschen Regierung zur Kenntnis brachte) sind:

the other states to appoint delegates and to form national advisory councils. We hope that this will give the undertaking the necessary financial stability from the beginning. (We thought that the letter from the English government, as far as the main countries are concerned, could contain the request to provide the national committees with a flat sum of at least 500 L.). In this process, it is of course necessary to proceed in such a way that the mathematical societies of the individual countries (or any other relevant bodies) suggest in advance the names of the delegates or advisory councils to be appointed to their governments in advance. Thus, in Cologne the German Association of Mathematicians has already entered into the question of persons to be named and has taken up the suggestions that we had given it. The suggestions (which I have already brought to the attention of the German government) are: 1 ) German delegates: Treutlein and Klein 2) German national advisory board: Gutzmer, Pietzker, Poske, Schotten, Stäckel and a representative of the engineering association. As far as the other countries are concerned, I just want to say that we are well informed of the meeting of the Italian mathematicians, and that in France we would like to try and get Appell and Tannery alongside Blutel as far as possible; Fehr is // in Paris and wants to try to settle the matter with personal consultation. The only concern that troubles me is the large amount of work I have to handle. If it was just our international commission, that would be alright, but I have so much to think about at the same time, such as how I rush from one meeting to another during this vacation. I will be all the more grateful for any support which I am asking for, especially from your side. With best regards and a request for friendly regards to everyone. Your devoted F. Klein”. (trans. by the author)

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1) Deutsche Delegierte: Treutlein11 und Klein 2) Deutscher nationaler Beirat: Gutzmer,12 Pietzker,13 Poske,14 Schotten,15 Stäckel16 und ein Vertreter des Ingenieurvereins. Was die anderen Länder angeht, so will ich nur bemerken, dass wir von dem Zusammentritt der italienischen Mathematiker genau unterrichtet sind, und dass wir in Frankreich doch versuchen wollen, neben Blutel17 möglichst Appell18 und Tannery19 zu gewinnen; Fehr ist // eben in Paris und will versuchen, die Sache in persönlicher Rücksprache zu ordnen. Die einzige Sorge, die mich drückt, ist der große Umfang der von mir selbst zu bewältigenden Arbeitslast. Wäre es nur unsere internationale Kommission, so möchte es hingehen, aber ich habe so sehr viel nebenbei zu bedenken, wie ich denn z.B. in diesen Ferien von einer Versammlung zur anderen eile. Um so dankbarer werde ich für alle Unterstützung sein, die ich insbesondere von Ihrer Seite erbitte. Mit den besten Grüssen und der Bitte um fr. [freundliche] Empfehlung allerseits Ihr ergebener F. Klein George Greenhill to David E. Smith, London, 12 October 1908 RBML-SPP, Box 20 Handwritten letter I Staple Inn W.C. London October 12, 1908  Peter Treutlein (1845–1912) was a reformist of geometry instruction, a schoolbook author, a committed teacher and school principal with many years of teaching experience. In 1911 he published the book Der Geometrische Anschauungsunterricht als Unterstufe eines zweistufigen geometrischen Unterrichts an unseren höheren Schulen (Leipzig und Berlin: B. G. Teubner). In the introduction Klein writes that he found everywhere in this book the same principles that he followed in his own teaching approach. Treutlein was a correspondent of Smith. 12  August Gutzmer (1860–1924) was a German mathematician who chaired some German commissions committed in mathematics education. He was a correspondent of Smith. 13  Friedrich Pietzker (1844–1916) was a German mathematics teacher, a textbook author, and a politician. 14  Friedrich Poske (1852–1925) was a German educator and scientist. 15  Heinrich G. Schotten (1856–1939) was a German mathematician and mathematical pedagogue, well known for his work on reforms in the teaching of geometry. He directed the journal Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht from 1908 to 1914. He was a correspondent of Smith. 16  Paul Stäckel (1862–1919) was a German mathematician interested in mathematics education and in history of mathematics. In 1905 he was president of the Deutsche Mathematiker-Vereinigung. 17  Emile Blutel (1862–1945) was a mathematics teacher at the Lycée Saint-Louis in Paris and inspector of Public Instruction. 18  Paul Émile Appell (1855–1930) was a French mathematician committed to the promotion of research and national and international solidarity. His mathematical work extends to projective geometry, algebraic functions, differential equations, and complex analysis. 19  Jules Tannery (1848–1910) was a French mathematician, committed to mathematics education. He was a correspondent of Smith. 11

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Dear Mr. Smith, we held our first meeting of the Committee at Cologne,20 during the Naturforscher Versammlung. Klein set to work at once in fine style and drew up an excellent plan of our procedure for the next year. My misgiving vanished that the work was too vast and we // should not know how and where to begin. The Congress on Electrical Units begins in London today: this will give us some useful hints of procedure. Klein was to be very busy travelling about for a month. When he has settled down in Göttingen in November, we shall start again on our work. The enclosed, the earliest representation of a cannon, 1326, may interest you. If spoils by folding, I can send another. Your sincerely G. Greenhill George Greenhill to David E. Smith, London, 24 June 1909 RBML-SPP, Box 20 Handwritten letter Staple Inn W. C. June 24, 1909 ____________ Dear dr. Smith, I expect you will find me here in London in August. We are moving with our Committee here very slowly. We are not organized bureaucratically, and the University resents government interference. I was not able to persuade either Forsyth21 or Perry22 for the English Committee. I am glad you approve of Godfrey,23 and we are now trying to secure Fletcher24 of the Government Education Department. He will direct us how best to obtain a grant from the Treasury; but the outlook is not favourable as we are promised seven lean years of

 The meeting in Cologne was held from 23 to 24 September 1908 to decide about the organization of the Commission. Klein, Greenhill and Fehr were present, Smith did not participate. 21  Andrew Russell Forsyth (1858–1942) was a British mathematician who played an influential part in mathematical reform in England. He was a plenary speaker at the International Congress of Mathematicians (ICM) in 1908 in Rome. 22  John Perry (1850–1920) was an Irish mathematician and engineer, professor of mechanics and mathematics at the Royal College of Science in London from 1896. He introduced the idea of laboratory for mathematics. His most famous book is Elementary practical mathematics, London: Macmillan and Co., 1913. 23  Charles Godfrey (1873–1924) was a British mathematician. In September 1899 he was appointed Senior Mathematical Master at Winchester College, where he modernized the teaching and started a laboratory for practical mathematics. He is important not only as a textbook writer, but also for his commitment to getting reforms considered and making them possible in his country. 24  Greenhill is referring to W.  C.  Fletcher (1865–1959), an English mathematician in 1904 was appointed to the newly created post of chief inspector of secondary schools at the Board od Education and there he stayed till his retirement in 1926. He was president of the Mathematical Association from 1939 to 1943. 20

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Retrenchment and skinflint economy. As for the colonies, Bovey25 will act for Canada, and Hough26 for S. Africa; but Australia and India are not get settled. Germany has been working for the last 5 years at least; I think I have most of their publications here bearing on the subject; in England there is nothing to show, except for Perry’s Addresses,27 and Godfrey’s paper at Rome.28 Yours sincerely George Greenhill Ans. Munich 2/4/09 George Greenhill to David E. Smith, n. p., 14 August 1914 RBML-SPP, Box 20 Handwritten letter Dear Eugene Smith, This terrible catastrophe29 will throw Europe back a century in civilization. It will take that time to recover. I first met you marooned in London one of the 10,000; and have let it go so far as a shipwrecked manner that you were reluctant to show yourself. But I am glad to hear you are in comfortable quartiers, and need not hurry back home. There cannot be any meeting at Munich next year, and I fear it is all up with the Congress at Stockholm.30 So, it will be – actum est de I.M.U.K. I would send this enclosure about General Leman31 to Fehr, if I was sure he was not a fervent pro-­ German. It is interesting as showing how Leman was a pioneer in our work and his Sur l’Enseignement de l’analyse infinitesimal 1901, should not have escaped Fehr’s attention. I have no recollection of seeing him at our little congress in the Brussels Exhibition 1910. We shall want a General Leman in our war office. I transcribe a Greek inscription (titulus antiquissimus) for you to put away till required. It is a gravestone of one of your ancestors from the island of Melos, or Milo, where the Venuses [^Mrs Smith] come from. So, you may claim descent from the sculptor’s model. There is a story current now about the immortal William and

 Henry Taylor Bovey (1852–1912) was a British engineer, fellow of the Royal Society of Canada.  Sydney Samuel Hough (1870–1923) was a British applied mathematician and astronomer. In 1908 he became the first president of the Royal Society of South Africa. 27  John Perry. 1902. Discussion on the teaching of Mathematics. London: Macmillan and Co. 28  Charles Godfrey. 1909. The Teaching of Mathematics in English Public Schools for Boys. In Proceedings ICM 1909, Vol. 3, pp. 449–464. 29  Greenhill is referring to World War I, which had just broken out. 30  The ICM scheduled for 1916 in Stockholm did not take place because of World War I. 31  Gérard Mathieu J.  G. Leman (1851–1920) was a Belgian general. In January 1914 he was appointed commander of the fortress of Liège and when in August 1914 this town was attacked by the Germans he refused to surrender. Ultimately the town was destroyed and Leman was captured and imprisoned in Germany. He returned to Belgium after the Armistice, and died at Brussels in 1920. In 1901 he had published the book Sur l’enseignement de l’analyse infinitésimale. 25 26

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the Kaiser which is an adaptation of a Greek epigram, supposed to have been spoken by the Lady when she was looking at the statue “When did I pose to the sculptor?” Yours sincerely G. Greenhill Aug. 14, 1914 You do not countenance the answer on the page of your books. So, I am leaving the inscription for you to work out. Buxton32 is an interesting centre: fine old timber houses near.33 Henri Fehr to David E. Smith, Geneva, 30 September 1914 RBML-SPP, Box 16 Typewritten letter with signature Commission Internationale de L’enseignement MathÉmatique ComitÉ Central Président: F. Klein, G.R.R., Wilhelm Weberstr, Goettingue (Allemagne) Vice-présidents: Sir G. Greenhill, 1, Staple Inn, Londres – D.E. Smith, Teachers College, Columbia University, New York Secrétaire-général: H. Fehr, 110, Florissant, Genève (Suisse) Genève, le 30 septembre 1914 Monsieur le Prof. D.E. Smith, LONDRES Mon cher collègue, Vous trouverez ci-inclus l’épreuve du questionnaire que nous avons élaboré à Goettingue. Veuillez me le retourner avec les observations que vous jugerez utiles. Dites-moi en même temps quel est la personne qui pourrait répondre à M. Loria34 pour les États Unis. Peut-être vous vous en chargeriez avec M. Young35 (Chicago). Cette malheureuse guerre européenne36 va retarder les œuvres internationales. La nôtre était en si bonne voie. J’ai proposé à notre président que l’on continue les travaux le plus possible, et que l’on consacre un fascicule aux rapports projetés pour 1915 sans faire de réunion, cela va sans dire. M. Loria est d’accord pour se mettre à

 Buxton is a spa town in Derbyshire, England.  Addition written in the margin of the letter. 34  Gino Loria (1862–1954) was an Italian mathematician and historian of mathematics interested in mathematics education. At the end of the international conference of ICMI held in Paris from 1 to 4 April 1914 the Central Committee entrusted Loria with the task of preparing the report on theoretical and practical training of mathematics teachers at the different school levels. Because of the war and the subsequent vicissitudes of ICMI, Loria could present this report only at the ICM in Zurich in September 1932 (Loria 1933). 35  Jacob William Albert Young (1865–1948) was an American mathematician. In 1909  he  was nominated delegate of the US national subcommission by Smith. He was the author of many books for primary and secondary schools. 36  World War I (1914–1918). 32 33

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l’œuvre si on peut lui procurer les documents nécessaires. Dès que j’aurai la réponse de M. Klein j’écrirai aux membres du Comité Central. Je regrette vivement que les circonstances actuelles me privent du plaisir du vous voir. Je reçois journellement des brochures allemandes pour la défense de la « vérité allemande ». Ces messieurs ignorent qu’en pays neutre on connait mieux // la vérité complète que chez eux où la censure est d’une rigueur inouïe. Nos collègues sont bien à plaindre ! La caractéristique des toutes ces brochures est qu’elles sont particulièrement violentes contre l’Angleterre. Les Allemands espéreraient que les Anglais se tiendraient hors de ce conflit. En Suisse au contraire on sympathise beaucoup avec les Anglais qui prennent si vaillamment la défense de la Belgique dont les Allemands ont si ignoblement violé la neutralité. Si vous avez l’occasion de voir notre ami Sir George37 veuillez lui transmettre mes cordiales salutations et le mettre au courant de notre entretien. Sans nouvelles de votre ami M. Carson,38 qui m’avait annoncé un article pour « l’Ens. Math ». Croyez, mon cher collègue, à mes sentiments les meilleurs. H. Fehr P.S. Je fais expédier 25 exemplaires du Compte rendu de Paris à votre adresse à New York. D’autres exemplaires sont à votre disposition. George Greenhill to David E. Smith, n. p., 19 November 1914 RBML-SPP, BOX 20 Handwritten letter Dear Eugene Smith, I was pleased to receive your letter, and to hear you feel yourself again. I dare say Mrs. Smith will be glad to be back again in safety at her own fireside. I think you are hard on Klein. As a patriot German he must stand up for his own country. The world looks up to German science and learning as our leaders: but it is a great defeat of civilization that they are controlled by a military despotism. There was much bad blood in Europe; but now the enemies have fought into a respect for each other, and are prepared like school boys after a fight to be better friends. Peace might be made if it was not for diplomatic39 interference, and so the war must go on till Europe is smashed to ruins. Fehr writes hopefully of the Stockholm Congress, but Munich is out of the question. I wish I could share his optimism. Jackson40 gave me a call. His work is converted into a branch of military science.

 Alfred George Greenhill.  George Edward St. Lawrence Carson (1873–1934), mathematician and educationist, had published in 1913 the book Essays on mathematical education with an introduction by Smith. He was a correspondent of Smith. 39  Del. difficulties. 40  Greenhill is probably referring to C. S. Jackson, who was secretary of the British subcommission; see Chap. 1 by Gert Schubring in this volume. 37 38

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Hime41 convened a meeting of the Roger Bacon Committee to wind up and dissolve; but we found suicide was more difficult than we thought. However, we have a little balance in hand after all liabilities are paid. And the Clarendon Press has not lost money by coming to our aid, and has the satisfaction of a noble action, and no loss incurred. (…) is in the thick of the fight. Yours sincerely Nov 19, 1914 G. Greenhill Felix Klein to Guido Castelnuovo, Göttingen, 4 March 191542 Accademia Nazionale dei Lincei, Archivi privati, Guido Castelnuovo Handwritten letter Göttingen 4.III.15 Sehr geehrter Herr Kollege! Ich weiß nicht, ob Fehr Ihnen geschrieben hat, dass ich mit Rücksicht auf die Zeitverhältnisse daran dachte, das Präsidium der „IMUK“ niederzulegen und

 Hime, Henry William Lovett of the Royal Artillery at the beginning of the twentieth century, published the theory that Roger Bacon’s Epistola de Secretis Operibus Artis et Naturae et de Nullitate Magiae contained a cryptogram giving a recipe for the gunpowder. He was a correspondent of Smith. 42  This letter can be accessed at Furinghetti and Giacardi 2008: https://www.icmihistory.unito.it/LetteraKleinLAST.pdf. It is also published in (Luciano and Roero 2012, pp. 135–136). The English translation is: “Most honourable Colleague, I don’t know if Fehr has written you that I had thought, out of consideration for the circumstances of the times, of resigning the presidency of the I.M.U.K and asking D.E. Smith to take over my place. Regarding this I had asked Smith himself and have now received from him an amicable letter which has made me decide to remain in my place, at least for the time being. Smith would like, if difficulties arise, to take over my place in the correspondence. In the meantime, it would be valuable to me to hear what you think about the present possible developments. Independent of these reflections, the work of the German subcommittee carries on just the same. You will have received the study by Stäckel regarding the technical colleges and you will be as happy as I am with the successful results. The 4 “studies” still in arrears are already at a pretty advanced stage, and should in any case be ready to be published within a year, the same can be said also for the 4 “reports” (3 of which are already printed but will only be issued after some time). We will then have finished our work, as far as Germany is concerned, without having sacrificed anything of our original plan. Above all we will try, as long as we are not directly caught up in the present circumstances, to keep our scientific undertakings alive as much as possible. To this belongs in particular the work on the Mathematical Enzyklopädie. I am very happy in this regard to hear from Mohrmann that the two reports written by yourself and by Enriques concerning algebraic surfaces, are well on their way to completion. Concerning the report on correspondences, Mohrmann first turned to Berzolari, whose earlier diligence had earned him the particularly favourable regard of the editorial group of the third volume. Loria’s report is almost ready for printing, the very detailed and extremely precise report by Segre is being translated by Mohrmann. I hope that you are well during this crucial period. All of us are caught up in immense events, if not personally, then through some family members, but this is not the place to comment on it. I send my best regards, Your most devoted Klein”. 41

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D.E. Smith zu bitten, an meine Stelle zu treten. Ich hatte mich in dieser Sache an Smith selbst gewandt und bekomme von diesem jetzt einen freundschaftlichen Brief, der mich bestimmt, vorläufig jedenfalls noch auf meinem Platz zu bleiben, Smith will, wenn Schwierigkeiten entstehen, gern statt meiner in die Korrespondenz eintreten. Inzwischen wäre mir lieb, zu hören, wie Sie über die hier vorliegenden Eventualitäten denken. Unabhängig von diesen Überlegungen gehen übrigens die Arbeiten des Deutschen Unteraus // schusses weiter. Die Abhandlung von Stäckel43 über die technischen Hochschulen werden Sie erhalten haben und sich mit mir der wohlgelungenen Arbeit freuen. Die 4 noch rückständigen „Abhandlungen“ sind schon ziemlich weit gefördert und sollen jedenfalls binnen Jahresfrist herauskommen, ebenso noch 4 „Berichte“ (von denen schon 3 gedruckt sind aber erst in einiger Zeit zur Ausgabe gelangen). Wir sind dann, was Deutschland angeht, mit unseren Arbeiten fertig, ohne etwas von unserem ursprünglichen Plan geopfert zu haben. Überhaupt suchen wir, soweit wir durch die Zeitverhältnisse nicht unmittelbar in Anspruch genommen sind, unsere wissenschaftlichen Betriebe nach Möglichkeit aufrecht zu erhalten. Es gehört dahin namentlich auch die Arbeit an der mathematischen Enzyklopädie.44 Es freut mich, in dieser Hinsicht von Mohrmann45 zu hören, dass die Fertigstellung der von Ihnen und Enriques46 bearbeiteten zwei Referate über algebraische Flächen dem guten Ende entgegengeht. Wegen Übernahme des Referates über Korrespondenzen hat sich Mohrmann zunächst an Berzolari47 gewandt, dessen sorgfältige frühere Arbeit ihm die besondere Sympathie der Redaktion von Bd. III gesichert hat. Das Referat Loria48ist so gut wie fertig gedruckt, das sehr eingehende und mit großer Genauigkeit gearbeitete Referat von Segre49 bei Mohrmann in Übersetzung. Hoffentlich geht es Ihnen in dieser inhaltsschweren Zeit gut. Jeder von uns ist, wenn nicht persönlich so doch durch seine Angehörigen in die ungeheuren  Klein is referring to the volume by Paul Stäckel, Die mathematische Ausbildung der Architekten, Chemiker und Ingenieure an den deutschen technischen Hochschulen. Leipzig-Berlin: Teubner, 1915, included in the series Abhandlungen über den mathematischen Unterricht in Deutschland. 44  Klein is referring to the Encyklopädie der mathematischen Wissenschaften published by Teubner in Leipzig from 1898 to 1935. 45  Hans Mohrmann (1881–1941) was a German mathematician who edited several volumes of the Encyklopädie der mathematischen Wissenschaften. 46  Guido Castelnuovo and Federigo Enriques. Grundeigenschaften der algebraischen Flächen. In Encyklopädie der mathematischen Wissenschaften, III.2.1, Leipzig: Teubner, 1903–1915, pp. 635–673; Die algebraischen Flächen vom Gesichtspunkte der birationalen Transformationen aus. Ibidem, pp. 674–768. 47  Luigi Berzolari. Allgemeine Theorie der höheren ebenen algebraischen Kurven. In Encyklopädie der mathematischen Wissenschaften, III.2.1, Leipzig: Teubner, 1903–1915, pp. 313–455. 48  Gino Loria. Spezielle ebene algebraischen Kurven von höherer als den vierten Ordnung. In Encyklopädie der mathematischen Wissenschaften, III.2.1, Leipzig: Teubner, 1903–1915, pp. 571–634. 49   Corrado Segre. Mehrdimensionale Raüme (1912). In Encyklopädie der mathematischen Wissenschaften, III.2.2A, Leipzig: Teubner, 1921–1928, pp. 769–972. 43

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Ereignisse verstrickt, aber es ist hier nicht der Ort, darüber Betrachtungen anzustellen. Mit besten Grüßen verbleibe ich Ihr ergebenster Klein Guido Castelnuovo to Felix Klein, Rome, 10 March 191550 Niedersächsische Staats  – und Universitätsbibliothek Handschriftenabteilung: Cod. Ms. Felix Klein, Nr. 51

Göttingen,

Handwritten letter Rome, 10 Mars 1915 (via Boncompagni 16) Monsieur et cher collègue M. Fehr ne m’a jamais parlé de vos intentions. Je comprends et j’apprécie les motifs qui vous ont inspiré. Mais, d’accord avec M. Smith, et dans l’intérêt de notre Commission, je vous prie de vouloir bien renoncer pour le moment à mettre à effet votre propos. Tous mes collègues de la Commission, // j’en suis certain, reconnaissent l’œuvre admirable d’organisation que vous avez accomplie et l’impulsion que vous avez donnée à nos travaux; ils savent bien que personne, sous ce rapport, ne saurait vous remplacer. D’ailleurs le moment grave que traversent toutes les institutions internationales conseille de n’introduire aucun changement dans leur organisation de crainte que ces faibles organismes ne doivent succomber. Il faut au contraire s’efforcer de les faire survivre jusqu’à la conclusion de // la paix, à fin qu’elles puissent faciliter la reprise des relations normales entre les peuples, dès que la guerre sera terminée. Alors, si vous voudrez, je pourrai me charger de consulter mes collègues de la Commission (non pas les italiens, dont l’opinion m’est bien connue) au sujet de la Présidence, et de vous renseigner exactement sur les réponses que me seront données. Par le moment il suffit que chaque pays, qui a la force de le faire, continue les travaux qu’il a projetés. Pour ma part je vous dirai que tout à l’heure // je me suis accordé avec M. Loria et M. Pincherle (relateur pour l’Italie) afin de rédiger la réponse au questionnaire sur la préparation des enseignants.51 À propos de l’enseignement, certain que vous agréerez la nouvelle, je vais vous communiquer que les programmes (modernes)52 de l’enseignement mathématique que j’ai fait adopter dans les lycées modernes, ont été si bien accueillis que le  This letter has already been published in (Luciano and Roero 2012, pp. 212–213).  See footnote 34.  The report on the Italian situation was not prepared by Salvatore Pincherle (1853–1936), professor at the University of Bologna, but by the ministerial inspector Alfredo Perna. 52  The liceo moderno was established in 1911 by the Minister Luigi Credaro. Castelnuovo was given the task of preparing the mathematics syllabi and the instructions regarding the teaching methods to be adopted for the new courses; in particular, following Klein’s reform programme, he introduced the concepts of function, derivative and integral, suggesting that they be illustrated by applications to the experimental sciences, and he attached great importance to numerical approximations (Giacardi and Scoth 2014, pp. 218–219). 50 51

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Ministère de l’Instr. P. pense maintenant de les introduire même dans les lycées classiques et // dans les instituts techniques, en développant davantage, dans ces dernières écoles, le programme de calcul infinitésimal. _______ J’apprends avec plaisir que les travaux de l’Encyclopédie avancent. Le choix de M.  Berzolari pour l’article sur les correspondances algébriques est très bon.53 M. Berzolari est en effet un rédacteur scrupuleux et diligent. Avec l’espoir que nos // relations personnelles et scientifiques ne seront pas entamées par les événements de cette année terrible, je vous prie, Monsieur et cher collègue, de vouloir bien agréer l’expression de ma considération la plus haute. G. Castelnuovo Henri Fehr to David E. Smith, Geneva, 31 March 1919 RBML-SPP, Box 16 Typewritten letter with signature with handwritten additions54 Commission Internationale de l’Enseignement Mathématique Secrétariat-général H. Fehr 110 Florissant, Genéve (Suisse) Genève, le 31 mars 1919 Monsieur le Professeur D.E. Smith, Columbia University, New York Mon cher collègue, Avec l’armistice et l’établissement très prochain des préliminaires de paix, le moment est venu d’examiner la situation de la Commission internationale de l’Enseignement mathématique et d’adopter une organisation conforme à l’orientation actuelle des associations internationales. Nous sommes d’ailleurs liés, dans une certaine mesure, par le programme de la collaboration scientifique internationale préparé par la Conférence interalliée réunie à Londres en octobre 1918 et adoptée à Paris fin novembre de la même année.55

 Luigi Berzolari. Algebraische Transformationen und Korrespondenzen. In Encyklopädie der mathematischen Wissenschaften, III.2.2b, Leipzig: Teubner, 1921–1934, pp. 1781–2218. 54  The handwritten additions appear in italics. 55  On 9–11 October 1918 the Inter-Allied Conference on International Scientific Organizations, held in London at the Royal Society, adopted resolutions prescribing, inter alia, that existing scientific institutions should be dissolved; new types of scientific institutions should be established; and access to these institutions will not be open to all scientists. In November of the same year another meeting was held in Paris during which the establishment of the International Research Council (IRC) was discussed. The constituent assembly of the IRC was held in Brussels on 18–28 July 1919: even if the declared objective of the meeting was promoting international cooperation in science and creating new scientific unions, it was decided to exclude the former Central Powers— Germany, Austria-Hungary, Bulgaria, and Turkey. The majority of the participants in the assembly came from France and Belgium (Lehto 1998, pp. 18–20). 53

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Les délégations françaises et italiennes viennent précisément de faire une démarche auprès de moi dans ce sens. Se basant sur les principes adoptés à Londres et à Paris, elles estiment que notre Commission ne peut plus continuer à fonctionner telle qu’elle était constituée et qu’il y a lieu d’exclure les représentants des puissances centrales. Elles me chargent de vous demander de bien vouloir accepter la présidence de la Commission pendant cette dernière période de nos travaux. Vous ayant déjà pressenti à ce sujet, il y a plus de quatre ans, je ne puis que me joindre à leurs vœux, certain qu’entre vos main la Commission ne tardera pas à reprendre son activité ancienne et pourra procéder dignement à l’accomplissement de son programme. Sur la demande des délégations ci-dessus je suis prêt à continuer mes fonctions de secrétaire-général, si vous êtes également d’accord. “L’Enseignement mathématique” continuerait à servir d’organe officiel de la Commission. Liquidation ou continuation des travaux ? – Si l’on renonce à la dernière question inscrite au programme, on pourrait se borner à une simple liquidation de la Commission, le rapport de M.  Archibald étant considéré comme rapport d’ensemble. – Toutefois la question de la préparation des professeurs est d’une telle importance qu’il serait préférable, dans l’intérêt général, d’en faire une étude approfondie en suivant le plan général adopté en 1914. Les délégations françaises et italiennes sont disposées à répondre aux questions arrêtées il y a cinq ans; d’autres pays suivront sans doute dès que la situation mondiale sera redevenue à peu près normale. Votre présidence permettra, je l’espère, d’obtenir l’adhésion de quelques États non encore représentés, ainsi que des États nouveaux. Dans l’attente de vous lire, je vous prie d’agréer, mon cher collègue, l’expression de mes sentiments cordiaux, H. Fehr P.  S.  – Avez-vous déjà fait quelque projet de voyage en Europe pour 1919 ou 1920? David E. Smith to Henri Fehr, n. p., n. d.56 RBML-SPP, Box 47 Handwritten draft of letter Orig+3 thins57 Commission paper My dear Professor Fehr, your letter of March 31 was delayed in the mails, and hence it reached me only a few days ago. As you recall, professor Klein wrote to me early in the war and asked me to take his place as President of the Commission. At that time, I told him and you that58 I

 Of this letter there are both the handwritten draft and the typewritten version. I have transcribed the first. On the second the date 7 May 2019 is indicated. 57  Smith is probably referring to “onion skins”, a thin paper used to make carbon copies. 58  Del. the matter. 56

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felt that the matter would better lie in abeyance for a while. It is evident, however, that a change59 must now be made, and hence I have been giving much thought to60 your proposition that I accept the presidency. I wish it were possible to find a man of international reputation as a research scholar in mathematics61 who is at the same time especially interested in the teaching of the science. We were fortunate in having such a man as professor Klein to inaugurate the work eleven years ago. At present I do not know of anyone who combines these // two important elements and who is willing to, or62 should be tempted to give the time necessary to do the work of President of the Commission.63 There is also another consideration, namely one which I mentioned to you when the question was first raised. After the war the world must necessarily seek to work64 in peace if this is possible. Whatever we may think of the action of our enemies, we need65 to know what they are doing in the teaching of mathematics, and66 the world requires that they should also know what we are doing. Of this account, it is perhaps as well that one who comes from a country67 remote from the scenes of the war should become the president.68 // After considering the above statement, if you and the other members of the Central Committee feel that I should assume the presidency for69 a time, I shall be glad to do so and to do my best to carry on the work with the assistance which I know you will so generously give. I suggest that each member of the Central Committee, including Sir George Greenhill, be asked to70 express his opinion upon the matter if this has not already been done. My American colleagues have expressed their approval of my taking the position under the circumstances which I have mentioned. In case the Central Committee decides that I should take the position, I would like to send a letter to the South American countries and Canada asking that they prepare brief reports on // the present status of mathematics in their territories. I hope that you will feel disposed to send a similar letter to the European countries and Japan.71 Either you or I might also write to China, Australia, and India. The  Del. has.  Del. the. 61  Del. and. 62  Del. who. 63  Del. If you know of anyone, I shall be pleased to vote for him. 64  Del. in as peacefully much in harmony. 65  Del. their. 66  Del. they need. 67  Del. that is. 68  Del. It is probable that he could contribute something to the post-war international spirit. Much as However bitter the feelings // of the world at present eventually we must get together. If you feel that. 69  Del. the present. 70  Del. vote. 71  Del. and possibly. 59 60

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world view of the aims and methods of the teaching of mathematics has undergone a great change in ten years, at any rate in England and America. Brief national reports of these changes72 would be very valuable. I can promise that America will be glad to prepare such a report. I wish you would kindly send me the names and addresses of the members of the Central Committee as it will henceforth stand. This Summer is not a favorable time for going to Europe, but I think I shall arrange to spend next summer there. It is probable that I shall be in Geneva for a little time early in July, 1920. // With best personal regards I remain, my dear Colleague and Friend, Yours sincerely Henri Fehr to David E. Smith, Geneva, 23 March 1920 RBML-SPP, Box 16 Typewritten letter with signature and handwritten additions73 Commission Internationale de l’Enseignement Mathématique Secrétariat-général H. Fehr 110 Florissant, Genève (Suisse) Genève, le 23 mars 1920 Monsieur le Professeur David Eugène Smith, Columbia University, New York Mon cher collègue, Permettez-moi tout d’abord de vous féliciter de votre élection à la présidence de la Mathematical Association of America.74 Vous aurez là un champ d’action très vaste qui vous mettra en relation non-seulement avec les professeurs des États-Unis, mais encore avec ceux de l’Amérique centrale et de l’Amérique du sud. A peu près en même temps que votre lettre, j’ai reçu la circulaire relative au Congrès de Strasbourg.75 Bien qu’elle ait déjà été lancée, je ne manquerai pas de faire part de vos observations à MM. Picard et Koenigs.76  The reports on the changes in the teaching of mathematics since 1910 would be published in L’Enseignement Mathématique in the years 1929, 1930, 1931, 1933. 73  The handwritten additions appear in italics. 74  Smith was elected president of the Mathematical Association of America for the term 1920 to 1921. 75  Fehr refers to the ICM in Strasbourg (22–30 September 1920). On 20 September 1920 the IMU was founded in Strasbourg and the former Central Powers were excluded. 76  Charles Émile Picard (1856–1941) was a French mathematician, who was president of IRC from 1919 to 1931 and honorary president of IMU from 1920 to 1932. He was one of the chief architects of postwar international science policy that excluded the former Central Powers from IRC and IMU. Gabriel Koenigs (1858–1931) was secretary-general of IMU and a supporter of this policy of exclusion. 72

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Il y a lieu de se demander quelle sera la situation de notre Commission vis-à-vis de ce congrès. Je vais consulter les membres du Comité central, et j’en parlerai à la fin de la semaine à nos collègues de la délégation française. Pour ce qui me concerne, après avoir examiné la question sous toutes ses faces, j’estime que nous devons procéder à la liquidation de la commission. Nous n’avons pas de rapport à présenter à Strasbourg, d’une manière officielle étant donné que la réunion ne comprendra qu’une partie des pays ayant pris part aux grands congrès internationaux et que plusieurs de nos collègues de la commission n’ont pas accès au Congrès. A la suite des décisions prises par les conférences des Académies scientifiques interalliées (Londres et Paris) et par le Conseil international de recherches (Bruxelles)77 la Commission ne peut plus subsister sous sa forme actuelle. Dans ces conditions, il y aurait lieu de procéder à la liquidation de la Commission créée en 1908. Toutefois il y aurait lieu, une fois que le Comité central serait fixé, de soumettre la question aux membres de la commission et de publier encore une liste complète de tous les travaux. Le détail de la liquidation serait confié à un comité restreint dont vous seriez le Président et qui comprendrait en outre M. Hadamard et le soussigné. Notre collègue de Paris étant actuellement aux États-Unis, vous pourriez prendre rendezvous et examiner la question de près. Une fois qu’il serait rentré à Paris, je m’entendrai avec lui. Toutefois il sera bon que vous m’écriviez vos propositions au lendemain de votre entrevue, afin de gagner du temps. // Des vœux en faveur de la reprise des travaux ayant été exprimés de divers côtés, le Congrès de Strasbourg pourra, s’il le juge utile, proposer la création d’une nouvelle commission poursuivant le même but et confier ce mandat éventuellement à l’ancien Bureau.78 C’est un point au sujet duquel il conviendra également de vous entendre avec M. Hadamard. J’en parlerai de mon côté avec MM. Picard et Koenigs. J’ai reçu récemment 80 exempl. Du Bulletin, 1918, N. 9 du Bureau of Education79 comprenant votre liste des périodiques mathématiques. Je l’ai fait distribuer aux membres de la Commission. Veuillez, je vous prie, remercier le Département de l’Intérieur à Washington au nom du Bureau central. Veuillez agréer, mon cher Collègue, l’expression de mes sentiments les meilleurs. Votre bien cordialement dévoué H. Fehr

 See footnote 55.  The mandate of ICMI was not renewed as it was done at the ICM in Cambridge in 1912. 79  Fehr is referring to the Bulletin of the Bureau of Education of the Department of the Interior, United States. 77 78

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Henri Fehr to David E. Smith, Geneva, 13 April 192080 RBML-SPP, Box 16 Typewritten letter with signature and handwritten additions81 Commission Internationale de l’Enseignement Mathématique Secrétariat-général H. Fehr 110 Florissant, Genéve (Suisse) Genève, le 13 avril 1920 Monsieur le Professeur David Eugène Smith, Columbia University, New York Mon cher Collègue, Comme suite à ma lettre du 23 mars, je tiens à vous mettre au courant de l’entretien que j’ai eu à Paris avec la délégation française. Étaient présents M.  P. Appell, président d’honneur de la délégation française et MM.  Bioche82 et d’Ocagne.83 Je ne puis que vous confirmer ce que je vous écrivais le 23 mars, nous avons été entièrement d’accord au sujet de la liquidation dont le détail serait confié à un comité restreint composé de MM. D.E. Smith, Hadamard et du Secrétaire-général. Il y aurait lieu de soumettre la question aux membres de la commission, afin de leur donner l’occasion de formuler leur avis et d’émettre des vœux dont on pourrait tenir compte dans le rapport final. Les travaux de clôture comprenaient: 1. La publication de la liste complète des travaux de la commission avec un rapport sommaire du secrétaire-général;84 2. Un rapport administratif destiné uniquement aux membres. Peut-être serez-vous disposé de rédiger une sorte de préface au dernier fascicule que publiera le comité central. J’espère que vous aurez l’occasion de voir M.  Hadamard. Je lui transmets un double de cette lettre (à l’Université de Yale). Veuillez agréer, mon cher Collègue, l’expression de mes sentiments distingués. Votre H. Fehr

 In L’Enseignement Mathématique Fehr includes an excerpt of a letter by Smith dated April 1920, in which it is affirmed: “… les conditions des temps presents ne sont pas favorables à ce genre d’investigation internationale, aussi il semble préférable de cloôturer l’oeuvre de la Commission” (21, 1920–1921, pp. 316–317). 81  The handwritten additions appear in italics. 82  Charles Bioche (1859–1949) was a French mathematician who devoted the main part of his career to teaching mathematics at secondary level. He was appointed honorary member of ICMI during the ICM in Oslo in 1936. 83  Maurice d’Ocagne (1862–1938) was a French engineer and mathematician. He founded the branch of mathematics called nomography. 84  See La Commission internationale de l’Enseignement Mathématique de 1908 à 1920. Compte rendu sommaire suivi de la Liste complete des travaux publiés par la Commission et les SousCommissions nationales. L’Enseignement Mathématique (from now on EM) 21, 1920–1921:305–342. 80

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Henri Fehr to David E. Smith, Geneva, 5 July 1920 RBML-SPP, Box 16 Typewritten letter Commission Internationale de L’enseignement Mathématique Comité Central Président: F. Klein, G.R.R., Wilhelm Weberstr, Goettingue (Allemagne) Vice-présidents: Sir G. Greenhill, 1, Staple Inn, Londres – D.E. Smith, Teachers College, Columbia University, NewYork Secrétaire-général: H. Fehr, 110, Florissant, Genève (Suisse) Genève, le 5 juillet 1920 Le Secrétaire-général À Messieurs les Membres de la Commission internationale de l’enseignement mathématique, Messieurs et très honorés Collègues, D’après la résolution adoptée par le 5ème Congrès international des mathématiciens (Cambridge, 21–28 août 1912),85 le mandat de la Commission internationale de l’Enseignement mathématique institué par le Congrès de Rome (1908), avait été prolongé de quatre ans, pour prendre fin au 6ème Congrès international qui devait avoir lieu à Stockholm. En raison de la guerre mondiale et des conditions nouvelles imposées aux relations scientifiques internationales, il ne peut être question, pendant longtemps, de réunir la Commission. On sait d’ailleurs qu’à la suite de la déclaration de principe et des résolutions relatives aux organisations scientifiques internationales votées par la Conférence interalliée des Académies scientifiques (Londres et Paris, octobre et novembre 1918; Bruxelles, juillet 1919),86 les Associations ou Commissions internationales créées avant la guerre procèdent tour à tour à leur dissolution. Consultés individuellement par lettres, les membres du Comité Central ont reconnu que, dans les conditions actuelles, la dissolution de la Commission internationale de L’Enseignement mathématique devient inévitable. La liquidation se fera de la manière suivante: 1. Circulaire du Secrétaire-général annonçant la dissolution prochaine de la Commission. Les délégations d’un caractère officiel voudront bien en informer leur Gouvernement. Messieurs les délégués sont priés d’adresser au Secrétaire-­ général, dans le plus bref délai possible: a) Les renseignements concernant la clôture des travaux de leur sous-­ commission  – b) la liste des travaux, revue et complétée, s’il y a lieu  – c) Leurs observations concernant la dissolution. // 2. Publication, par le Secrétaire-général, d’un rapport sommaire (1914–1920) et de la liste complète des rapports. Toutefois, cette publication ne pourra se faire que 85 86

 In Proceedings ICM 1913, the date of the congress is 22–28 August 1912.  See footnote 55.

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si les délégations qui n’ont pas encore payé leurs cotisations veulent bien les verser sans retard au Secrétaire-général. 3. Retour aux Sous-commissions du solde des rapports en dépôt au siège de la Commission. 4. Dernière circulaire aux membres, avec un rapport financier. Au cas où certaines Sous-commissions nationales désirent achever leurs travaux et poursuivre leur action en vue de reformes de l’enseignement mathématique de leur pays, elles auront la faculté de subsister sous leur forme actuelle. Elles voudront bien adresser leurs rapports aux membres de l’ancienne Commission, ou s’entendre, en vue de leur publication, avec la Rédaction de “L’Enseignement Mathématique” qui remplissait jusqu’à ce jour le rôle d’organe officiel de la Commission.87 Si nous devons renoncer à fournir encore une étude d’ensemble de la préparation théorique et pratique des professeurs de mathématiques,88 nous avons du moins la satisfaction de constater que la plus grande partie des travaux projetés il y a 12 ans ont pu être accomplis. Des documents fort précieux ont pu être réunis, et, dans chaque pays, nos travaux laissent des traces profondes et durables. Dans bon nombre de pays les travaux sont d’ailleurs terminés. Sans doute, les circonstances nouvelles créées par la guerre auront répercussions sur l’organisation de l’enseignement scientifique. De nouveaux problèmes se posent dans tous les pays, mais il serait prématuré de les examiner dans le domaine international avant de pouvoir présenter des résultats basés sur une expérience de quelques années. La Commission met fin à ses travaux après avoir produit plus de 320 rapports répartis sur plus de 190 fascicules ou volumes89 et embrassant tous les ordres de l’enseignement scientifique et professionnel. Les Sous-commissions et associations nationales s’efforceront à faire connaître ces documents si riches et si complets rédigés par les représentants les plus distingués de l’enseignement mathématique à tous les degrés. C’est au corps enseignant et aux autorités scolaires qu’incombe maintenant le devoir d’en tirer parti en vue de réaliser de nouveaux progrès dans l’enseignement scientifique. En vous priant de répondre le plus tôt possible aux questions soulevées dans cette circulaire, je vous présente, Messieurs et très honorés Collègues, l’expression de mes sentiments très distingués. Le Secrétaire-général, H. FEHR90

  The inscription on the title page, Organe officiel de la Commission Internationale de L’Enseignement Mathematique, had appeared until 1939–1940. In the title page of the volumes 39 (1942–1950) and 40 (1951–1954) this sentence is missing, but it reappears in the second series. 88  See footnote 51. 89  See EM 21, 1920–1921: 339. 90  Del. N.B. Cotisations arriérées. 87

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George Greenhill to David E. Smith, London, 3 November 1920 RBML-SPP, Box 20 Handwritten letter Dear Eugene Smith, I hope you had a fine passage home, and are comfortably settled at work again. I wish I could say the same. We were obliged to come to Strasbourg, to attend the obsequies of I.M.U.K.91 I fear our work has passed unappreciated for the most part. And the new Commission has made no sort of acknowledgement of the pioneering part we have taken. I fear Fehr is saying, like another Othello,92 his occupation is gone. Reviewing the memory of the Congress, it strikes me as hardly worth the trouble of going so far. And the attendance was very small. Only three or four from England, one for Italy (Volterra), none from Spain, Sweden. And in the French representation, the attendance was meagre of their chief men. Was Strasbourg too small to hold Appell and Picard93 at the same time? Old Camille Jordan was a marvel and worked very hard, to keep his mind occupied, I fear; he was tried so cruelly by the war in family losses.94 He came out to the excursion to St. Odille95 and walked up and down the mountains among the youngest. The absence of news makes me fear that the prospect of any call96 or lecture in your country has vanished. I suppose I must reconcile myself to the position of the superannuated man. I should have heard before this if there was any hope. Your sincerely Nov. 3 1920 G. Greenhill I Staple Inn W. C. P.S. I open this letter again to enclose the Liste des communications. This had not been issued before you left, although contributors were desired to send in their titles a month or two beforehand. There was a general air of careless absent mindedness about the Congress. And Strasbourg struck me as a place where there was no sitting accommodation never (?) expected we should like to sit down. I have just turned up your review of

 During the ICM in Strasbourg (22–30 September 1920) the mandate of ICMI was not renewed.  William Shakespeare, Othello, Act 3, Scene 3: “O you mortal engines, whose rude throats The immortal Jove’s dead clamors counterfeit, Farewell! Othello’s occupation’s gone”. 93  Both Appell and Picard appear in the list of members of the Congress (Proceedings, ICM 1921, pp. IX–XIV). Picard gave the introductory speech, Appell did not give a lecture. 94  Camille Jordan (1838–1922) actively participated in the ICM in Strasbourg and held the lecture “Sur la classification des constellations” (Proceedings, ICM 1921, pp. 410–436). Between 1914 and 1916 three of his six sons were killed in World War I. 95  Mont Sainte-Odile is a 764-metre-high peak in Alsace in France. 96  Del. to work. 91 92

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Euclid in Greek. As I see a red pencil, I suppose it means I have acknowledged it already and make a confession of my feelings at seeing you call it a baize-covered table, when the cloth came from Athens and represents the colours of the seven concentric walls of the Median capital, Ecbatane,97 as described by Herodotus98 and in the book of Judith.99 I must at the same time (from the note I see) have described Whitehead’s recitation when he reads Euclid’s own definition of point for the first time: “A point is nothing at all” – Σημεῖόν οὗ μέρος οὐθέν.100 I cannot follow a metaphysical discussion. And in Strasbourg we were bored with incessant Fourth Dimension. I turned up Liddle (sic)-Scott101 to make sense of the account and they say, the darlings, III In Arist. An. Post.,102 it seems to be taken for στιγμή, a point. This dictionary takes a classical pride in ignoring Greek Science. Some of your young lions should go through it to point out absurdities and ignorance. G Henri Fehr103 to the Interim Committee of the Mathematical Union, Geneva, 20 February 1952 Archive of the International Mathematical Union (section of ICMI), Berlin (IA from now on) 14A 1952–1954 Typewritten letter with signature Commission Internationale de l’Enseignement Mathématique Secretariat-général H. Fehr, professeur à l’Université 110, Florissant Genève (Suisse) Genève, le 20 février 1952 Au Comité intérimaire de l’Union Mathématique internationale Copenhague

 Ecbatane was an ancient city in Media in western Iran.  Herodotus (V sec. b. C.) was a Greek historian. 99  The Book of Judith is a text contained in the Catholic Christian Bible, but not accepted in the Hebrew Bible (Tanakh). 100  The definition by Euclid is “Σημεῖόν ἐστιν, οὗ μέρος οὐθέν” (A point is that of which there is no part) and is the first in the Book 1 of the Elements. 101  The Liddell-Scott is a Greek–English Lexicon compiled by Henry George Liddell and Robert Scott. 102  Greenhill is referring to the Posterior Analytics by Aristotle and in particular to 27. 87a 36 where the philosopher uses the word στιγμή for point. See William David Ross 2001. Aristotle’s Prior and Posterior Analytics, a revised Text with Introduction and Commentary. Oxford: The Clarendon Press. 103  Fehr has been secretary-general of ICMI from its foundation in 1908 to 1952, according to his statement in this letter. See also (Lehto 1998, p. 316). 97 98

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Messieurs et très honorés collègues, Nous avons l’honneur de vous proposer le rattachement de la Commission internationale de l’enseignement mathématique à l’Union mathématique internationale. Les buts que poursuit la Commission font partie de ceux que prévoient les statuts de l’Union. La Commission a été créée par le Congrès international des mathématiciens tenu à Rome en 1908. Son mandat a été confirmé successivement par le congrès de Cambridge (Angleterre), de Bologne, de Zurich et d’Oslo.104 Le compte rendu ci-joint vous donne un bref aperçu de l’activité du Comité central et des délégations nationales. Plus de trois cents rapports ont été publiés sous les auspices de la Commission. La guerre mondiale et les difficultés qui s’en suivirent ont interrompu nos travaux, difficultés d’ordre financier, impossibilité de maintenir le contact avec de nombreux délégués, etc. Mentionnons aussi le décès de plusieurs de nos collègues qui n’ont pas été remplacés: S.  Dickstein  – G.  Feigl  – S. (sic) Heegaard  – A. Mineur – Dav. Eug. Smith – G. Scorza. – G. Tzitzeica. – W. Wirtinger.105 Lors d’une séance, tenue à Paris, au début de juillet 1951, chez Monsieur J. Hadamard, président de la Commission106, et à laquelle assistait Monsieur Pierre Auger,107 de l’UNESCO, nous avons envisagé la solution suivante: Rattachement de la Commission Intern. de l’ens. math. à l’Union math. Intern. – Démission en bloc de la Commission.  – Son renouvellement par les soins de l’Union. La plupart des membres actuels étant en fonction depuis plus de quarante ans, il y lieu de procéder à un renouvellement complet.  – L’aide financière de l’UNESCO pourra être obtenue en passant par l’Union. Nous sommes ainsi amenés à vous présenter les propositions ci-après en vous priant de les soumettre à l’assemblée que se réunira à Rome le mois prochain:108 // 1. L’assemblée des délégués de l’Union se déclare favorable au rattachement de la Commission internationale de l’enseignement mathématique à l’Union mathématique internationale. 2. Elle prend acte de la démission collective des membres de ladite Commission. 3. Elle nomme un comité de 3 membres chargé de reconstituer la Commission en faisant appel à des forces nouvelles.  The ICMs in Bologna, Zurich and Oslo took place in 1928, 1932, and 1936 respectively.  Samuel Dickstein (1851–1939); Georg Feigl (1890–1945); Poul Heegaard (1871–1948); David Eugene Smith (1860–1944); Gaetano Scorza (1876–1939); Gheorghe Țițeica (or George Tzitzéica, 1873–1939); Wilhelm Wirtinger (1865–1945). Mineur could be the mathematician Paul Mineur who prepared the report on the Écoles de Commerce for ICMI, see EM 21, 1920–1921: 331. 106  The old Commission was composed as follows: president: Jacques Hadamard; vice-presidents: Poul Heegaard, Walter Lietzmann, Gaetano Scorza; secretary-general: Henri Fehr; member (coopted): Eric Harold Neville. 107  Pierre Auger (1899–1993) was the Director of the UNESCO Natural Sciences Section/ Department from April 1948 to December 1958. 108  Fehr is referring to the First General Assembly of the reconstituted IMU which took place on 6–8 March 1952 in Rome. 104 105

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4. Les délégués seront désignés par le comités nationaux de l’Union. (Deux délégués par pays: un pour l’enseignement secondaire, l’autre pour l’enseignement supérieur.) ______________ Il va sans dire que je reste à la disposition du nouveau Comité pendant la période transitoire. Sa première réunion se tiendrait à Genève, où se trouvent centralisés tous les documents concernant la Commission. Avec mes meilleurs vœux de succès de l’Union mathématique internationale reconstituée, je vous prie d’agréer, Messieurs et très honorés collègues, l’assurance de mes sentiments distingués. H. Fehr Secrétaire général Enrico Bompiani109 to Heinrich Behnke, Albert Châtelet, Ralph L.  Jeffery, Đuro Kurepa and Henri Fehr, Rome, 24 March 1952 IA 14A 1952–1954 Typewritten letter with signature To Professors: H.  Behnke, A.  Châtelet, R.L.  Jeffery, Ð.  Kurepa and Professor H. Fehr Rome, March 24, 1952 It is my privilege to send you article 14 of the proceedings of the General Assembly regarding the Commission Internationale de l’Enseignement Mathématique: “(14) The interim Committee presented a letter from Professor H.  Fehr, General Secretary of the Commission Internationale de l’Enseignement Mathématique, suggesting that the work of this Commission should be continued by the Union and offering the resignation of the present Commission. The Assembly agreed that the Commission should be attached to the Union and accepted the resignation of its present members expressing hearty thanks for the important work that the commission has accomplished, Professors Behnke, A. Châtelet, R. L. Jefferey and Kurepa were appointed members of the Commission. The Assembly accepted with thanks an offer from Professor Fehr to place himself at the disposal of the new commission”.110 I remain at your disposal to give you any help you may desire. With kindest regards, Sincerely yours Enrico Bompiani

 Enrico Bompiani (1889–1975) was an Italian mathematician well known for his contributions to differential geometry. He was secretary of IMU from 1952 to 1956. 110  Heinrich Behnke, Albert Châtelet, Ralph Lent Jeffery, and Ðuro Kurepa were designated by the International Mathematical Union (IMU) as members of ICMI during the First General Assembly of the reconstituted IMU in Rome. On that occasion Henri Fehr was requested to remain at the disposal of the new Commission during the transition period (EM 39, 1942–1950: 162). 109

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Gerald Wendt111 to Marshall Stone,112 n. p., 6 August 1952 IA 14A 1952–1954 Typewritten letter, carbon copy COPY 6 August, 1952 Dear Dr. Stone, Prof. Auger has asked me to acknowledge your letter of July 21 with a tear-sheet from Dept. of State Bulletin on the Report of the First Session of the International Mathematical Union. He, Dr. Ging-Hsi-Wang113 and I have all read it with great interest and are delighted at the successful establishment of the Union. It is your interest in the teaching of mathematics and the problem of mathematical illiteracy that gives me the privilege of replying to you, since I have the responsibility for the teaching and dissemination of science in this Department. I cannot well assure you that we are as deeply convinced of the need of better mathematical training as you are. But I can assure you that we are much impressed by the worldwide problem of incorporating mathematical thought into all education through the schools. It is of course impossible for us with a limited budget and limited staff to achieve a revolution in the teaching of all science subjects in all countries. We trust that the IMU, through its Committee on Mathematical Instruction, will interest itself in the problem of secondary mathematical teaching and, even more specifically, in the incorporation of competent mathematical instruction in the training of young scientists. For the present, at least, it will probably be wiser for us to concentrate on the methods and devices for the introduction of mathematical training into primary schools, especially in the less developed countries. This, in turn, must begin with the improvement of mathematical instruction in the teachers’ training colleges. This is our first objective. UNESCO can influence teaching in the various countries only at the request of the individual National Commissions or individual Governments. At the moment, the number of requests for such assistance exceeds the number of experts that are available to be sent as councillors to the various nations. This is, however, one of our more active fields. Your letter also asks what UNESCO is doing toward the educational problems in science as a whole. This would be a long story but the officers of the Mathematical Union should certainly be informed. Briefly the situation is this: We do select, brief, and supervise to a degree the experts that are sent to the various countries under financial arrangements with the Technical Assistance Board of the United Nations. Here in Paris, we deal directly with those experts and in addition have the following major projects under way.

 Gerald Wendt (1891–1973) headed UNESCO’s division of teaching and dissemination in the Department of Natural Science in Paris headquarters from 1950 to 1954. 112  Marshall Stone (1903–1989) was president of IMU from 1952 to 1954. 113  Ging-Hsi Wang (1897–1968) was a Chinese neurophysiologist. 111

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A series of ten Handbooks or Manuals for teachers of science. These are not textbooks but are practical teaching suggestions in the various sciences from primary through the secondary level which emphasize laboratory method and especially the use of simple, home-made laboratory equipment when none other is available. The manual on mathematics is being edited by Mr. Sawyer,114 of the United Kingdom. The entire series is being edited by Mr. Smithies115 who has 22 years of experience in the teaching of science in British West Africa. When completed the series will be published by the Oxford University Press. We also have in preparation an extend series of catalogues or lists of laboratory equipment for all sciences and at all levels, including the engineering and medical sciences. These suggest curricula and list specific items for each type of class, together with the commercial source and cost. This series is composed of the following volumes: // VOL. I – Primary schools VOL. II – Universities VOL. III, part 1 – Technical Schools – Veterinary sciences   part 2 – Technical Schools – Physics-Chemistry   part 3 – Technical Schools – Agricultural Sciences The first volume is published in English, French, Spanish and Chinese and the remaining volumes in English, French and Spanish. The first volume is at present out of print but has been revised and will be reprinted shortly. Parts 4 to 8 of Vol. III will be published during the course of 1953 and 1954. Finally, we have now under preparation by a scientific apparatus manufacturer in Denmark a series of 150 blueprints of simple laboratory equipment which will be published in book form early in 1953, together with detailed specifications and instructions for use, which are intended to stimulate the manufacture of scientific equipment by small industries in countries where there is no such manufacture now and where the problems of exchange make it very difficult to purchase such materials from Europe or America. In addition to these specific projects, we work through the Association of Science Teachers or of Science Masters in countries where they exist and use every method for promoting the exchange of journals, teaching aids and even of experts between the various countries. One of our major problems is the effort to organize science teachers associations in the large number of countries where they do not exist.

 Walter Warwick Sawyer (1911–2008) was an English mathematician and mathematics educator. In 1943 he published his first and most successful book Mathematician’s Delight (1943), whose aim was “to dispel the fear of mathematics”. 115  Frank Smithies (1912–2002) was a Scottish mathematician who gave contributions to integral equations, functional analysis, and the history of mathematics. See Frank Smithies (Ed.) 1953–1957. UNESCO Handbooks on the Teaching of Science in Tropical Countries. London: Oxford University Press. 114

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I must also make reference to the fact that the Department of Education makes comprehensive studies of the status of education in all fields and in all countries and has just published a Bulletin on the Teaching of Natural Science in Secondary Schools which is issued as Publication No. 139 of the International Bureau of Education in Geneva. I presume that you have already established cordial relations with the Secretary of the U.  S. National Commission for Unesco, Mr. Max McCullough116 of the Unesco office, U.S. Dept. of State. In that case, or in any case, I am sure that you can easily obtain a copy of this publication, if it interests you, from Mr. McCullough. I am happy to make this effort to answer your questions and to orient you on our work. It would be a privilege to have your comments in the future and specifically to have your leadership in developing our work in the directions that seem wise to you. Perhaps you should also have in mind the expression of your opinion to the Committee on Unesco of the U.  S. National Research Council, the Chairman of which is Prof. Maurice B. Visscher,117 of the University of Minnesota. Sincerely yours, Gerald Wendt, Natural Sciences Department Marshall Stone to Albert Châtelet,118 Chicago, 3 November 1952119 IA 14A 1952–1954 Typewritten letter, carbon copy 303 Eckart Hall University of Chicago Chicago 37, Illinois November 3, 1952 M. A. Châtelet Doyen de la Faculté des Sciences Université de Paris, France Dear M. Châtelet, I am very happy to learn that the International Commission for Mathematical Instruction120 recently held a meeting121 and organized itself under your chairmanship. I hope that the Executive Committee of the Union will soon receive from you a report of the meeting together with the budgetary proposals which were adopted there.

 Max McCullough was secretary of the U.  S. National Commission for UNESCO, see U. S. National Commission UNESCO. News, December 1950, IV.6: 7. 117  Maurice Bolks Visscher (1901–1983) was an American cardiovascular physiologist. 118  Albert Châtelet (1883–1960) was the president of ICMI from 1952 to 1954. 119  This letter has already been published in (Furinghetti and Giacardi 2010, pp. 36–37). 120   The name International Commission on Mathematical Instruction was not yet well-established. 121  Stone is referring to the Geneva meeting (20–21 October 1952), which is the first meeting of the new ICMI (EM 39, 1942–1950: 161–162). 116

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At the present moment I have seen only a somewhat informal communication from Professor Hille.122 From this I am led to believe that there will be need for bringing about a clear understanding of the position of the Commission as a Commission of the International Mathematical Union. I already called Professor Behnke’s attention to the bearing of the statutes and bylaws of the Union upon this question, but have received no reply from him. I feel sure that there will be no great difficulty in working out a suitable method of transacting the business operations of the Commission under the statutes and bylaws of the Union. However, there is one proposal made by Geneva meeting123 which must be very carefully discussed before any action is taken. It is my understanding that the Commission has proposed an arrangement whereby it will seek the adherence of the several nations and set up special national committees in the adhering nations to work with the Commission. I believe that activity of this kind is inappropriate for a Commission of the Union and that it would lead to intolerable confusion as to the relations between the Union, the Commission, and the nations adhering to one or the other.124 My own immediate suggestion as to the proposed way of handling the relations between the Commission and the national bodies interested in supporting it would be to urge all interested nations to adhere to the Union and to arrange for the appointment of suitable persons to the National Committees for Mathematics which have to be set up as part of the procedure of adhering to the Union. The Commission could then arrange for direct contacts with these National Committees by coopting as members or as liaison agents, appropriate members of the National Committees. I would welcome your own suggestions for reaching a suitable method of procedure. During the last six months I have been trying to make contacts in UNESCO which would be useful to your Commission. Some of the most important information I have gathered should already have been available to the Commission at the Geneva meeting as I had Professor Bompiani125 send copies of the correspondence containing it. I believe it is quite clear that UNESCO will have a rather limited interest in the work of the Commission since its educational program is devoted mainly to the primary level. I hope that eventually we shall be able to persuade UNESCO to be interested in secondary education and even in university education so far as it affects the under-developed areas. With best personal greetings, Sincerely yours, Marshall. H. Stone, President MHS:mre126

 Einar Hille (1894–1980) was an American mathematician. He participated in the First General Assembly of the reconstituted IMU (Rome, 6–8 March 1952) as a delegate of the USA. (Lehto 1998, p. 94). 123  See footnote 121. 124  About this question see Chap. 2, §. 2.2 by Fulvia Furinghetti and Livia Giacardi in this volume. 125  See footnote 109. 126  This is an acronym that first identifies the writer in capital letters (in this case MHS = Marshall H. Stone) and second in lower case the secretary who typed the letter. 122

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Marshall Stone to Enrico Bompiani, Chicago, 10 July 1953 IA 14A, 1952–1954 Handwritten letter The University of Chicago Chicago 37 – Illinois Department of Mathematics July 10, 1953 Dear Prof. Bompiani, there seems to be a great deal of confusion in connection with the ICMI. I hope we can get it cleared up. So, I recall the original agreement it was that M. Châtelet would prepare a letter (addressed to me) setting forth ICMI’s program and would send a large supply of copies of it to you for transmission to the National Committees with a covering letter from the officers of the Union. We prepared this letter of transmittal long ago and have been waiting for the communication from M. Châtelet. I suppose your office in Rome has copies of the letter of transmittal; I do not have anything but rough draft. I had been led to assume evidently erroneously that M. Châtelet had sent this to you. When I saw him in Paris during the latter half of May, he told me that he had been held up by delays in receiving Prof. Behnke’s // report as secretary on the February meeting of ICMI127 but that he would (or had already been able to?) send his letter on to you. When I returned to Chicago on June 15, I found on my desk a letter from M. Châtelet dated May 7, evidently the letter which he wishes to have transmitted to the National Committees. Of course, I assumed what a copy must have gone to you at the same time. Otherwise, there was little sense in sending the letter to me in America by surface mail, and making no mention of its despatch in the later conversation I had with M. Châtelet. In any case, here is the letter and the problem is to get it distributed as rapidly possible along with our covering letter. I agree with you that ICMI should provide the copies required, and suggest that if Behnke is unable to assume the responsibility you could offer to have the work done in your office at the expense of ICMI (though we might contribute the postage). It would certainly be // desirable to have enough copies sent each National Committee to take care of the Subcommittee members who are to be appointed in accordance with Prof. Châtelet’s request (sup. 2 of his letter). If you have someone who can type French there in Pittsburgh, you might prefer to handle the work there – or, at least, you might get copies of the letter of M. Châtelet for our personal use in case the letter should be lost or destroyed in its further travels. The long delay, which will only be increased by all this confusion is regrettable. It may be that Prof. Behnke will want to send out additional material at the same time with Prof. Châtelet’s letter. I think it is certain that he will want to send out  Stone is referring to the meeting of ICMI in Paris on 21 February 1953 (EM 40, 1951–1954: 81–82). 127

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material before the requested sub-committees are finally organized. Hence in any case, I think we should permit him to send out informational material (directly to the National Committees) intended for the attention and use of these subcommittees and marked accordingly – and this from the moment that Prof. Châtelet’s letter // has been circulated. Or perhaps you can suggest a happier solution of these various difficulties! The difficulties which have been encountered in getting our various commissions organized and launched into activity make me particularly aware of the fact that we need to clarify our procedures for appointing the member of Commissions. At the present time it is not clear what the term of membership is, though there is a basis for thinking that all commission memberships expire at the end of the calendar year following a General Assembly, so that the Assembly can decide who shall be a member from the beginning of the following calendar year. Most Commissions should probably be appointed by the President (with the advice – and perhaps only with the consent – of the Executive Committee). During his term of service though certain ones intended to have permanent functions (of these ICMI is a good example) might better be elected by the General Assembly, nominations being made as for other offices of the Union. Hastily Marshall Stone Marshall Stone to Albert Châtelet, Chicago, 29 July 1954 IA 14A, 1952–1954 Typewritten letter with handwritten notes, carbon copy International Mathematical Union128 Professor A. Châtelet, President International Commission for Mathematical Instruction Paris, France Chicago, July 29, 1954 Dear Mr. President, I acknowledge with thanks the agenda and minutes of the meeting of the Executive Committee of your Commission held in Paris on July 2. I trust that you will permit me to comment on several points suggested by these documents. In relation to the Financial Situation, I think I may assume that the Commission will indicate to Professor Bompiani, as Treasurer of this Union, its various needs and desires in drawing on appropriations from ICSU129 funds. It is important, if we are to avoid the  In the margin this handwritten note appears: “Dear Professor Bompiani: please have copies of this letter sent to prof. Behnke and to all members of the Executive Committee”. 129  UNESCO (United Nations Educational, Scientific and Cultural Organization) financed IMU through ICSU (International Council of Scientific Unions) and IMU in turn provided funding to ICMI. ICSU was founded in 1931 as a successor of the International Research Council. On the history of UNESCO see (Kulnazarova and Ydesen  2016); see also (International Council of Scientific Unions and Certain Associated Unions 1965, pp. 2–6). 128

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difficulties which arose last year when the Secretary of your Commission found himself for a long time unable to draw from Paris the funds, he needed to discharge his duties, that Professor Bompiani arranges to pay over these funds in the different currencies appropriate to the particular purposes envisaged, as far as that is possible. This he cannot do unless he is given detailed statements as to your planned operations. In relation to the publication “L’Enseignement Mathematique”130 I note that you expect to pose the question of the necessary credits to the Executive Committee of I.M.U. for consideration at its meeting on August 30. I would be very happy to receive your proposals in advance of the meeting. In this connection I believe I should suggest to you the importance of offering a well-thought-out plan for financing the publication, based in substantial measure upon contributions from the organizations expressing direct interest in the work of the Commission. It seems to me unlikely that the Executive Committee of the Union will be ready to recommend that the Union participate in the financing of the journal until adequate contributions from other sources can be guaranteed. In connection with the Constitution of the National Sub-Commissions, I recall our agreement that each such Sub-Commission is to be in the first place a Sub-­ Committee of the National Committee for Mathematics in the country which it represents. This agreement has particular importance in the case of Greece and I most urgently request you to be extremely scrupulous in applying the strict letter of the agreement. The only body in Greece which can request the formation of a Sub-­ Commission for Greece in your Commission is the Greek National Committee for Mathematics as constituted by the Academy of Athens, the National Adhering Organization for Greece. I caution you explicitly against heeding the requests or claims of any other organization. My reason for doing so is that the Executive Committee of the Union has had to deal with a most delicate situation in Greece arising out of conflicting claims, and would not wish to see a renewal of the conflict after the settlement which was reached through the good-will of the Academy of Athens.131 In this same connection, I wish to ask you to enjoin all your Sub-Commissions to maintain close contacts with their National Committees as is required by the letter and the spirits of our agreement as to the constitution of the Sub-Commissions. Will you please despatch a suitable letter dealing with this question of organization to all your Sub-Commissions and emphasize the importance of the matter in addressing the meeting of your Commission at Amsterdam? It is necessary for me to report that the National Committee for Mathematics in one of the leading countries, which is concerned about proper conduct of international affairs in science, has expressed dissatisfaction on this point. 130  As already mentioned in footnote 87, until 1939–1940 the journal L’Enseignement Mathématique had been the “Organ official de la Commission Internationale de l’Enseignement Mathématique”. This inscription disappeared from the front cover in the period 1924–1939/1940, even if it generally appeared in the frontispiece of each issue. The inscription reappeared starting from the first volume of the second series in 1955. 131  Greece was a member of IMU, but due to internal disagreements it was not clear who the national representatives were (Lehto 1998, p. 105).

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In general, it is very satisfactory to note that good beginnings have been made toward an efficient organization of the work of the // Commission, and that an interesting and useful program is in prospect for the Congress.132 As you know, there will be a discussion in the General Assembly of the work being done by the various Commissions of the Union, and the Assembly will designate the members and officers of the Commissions for the term of four years beginning January 1. I believe that the Secretary of the Union, Professor Bompiani, called the latter point to your attention in a circular letter sent to all Commissions. The importance of the work which can be done by the International Commission for Mathematical Instruction is so great that everything possible should be done to guarantee the best results from its activities over the next four years. It has been suggested in the Executive Committee that some of the organizational frictions noted above should be reduced by working out more explicit terms of reference for the Commission. It is evident in addition that new blood should be brought into the Commission by suitable changes in the membership and constitution of the Commission. Your comments and these of other members of the Commission relating to any of these matters would be greatly appreciated. With best personal regards, I am Sincerely yours, Marshall H. Stone President133 Heinrich Behnke to Marshall Stone, Oberwolfach (Schwarzwald), 11 August 1954 IA 14A, 1952–1954 Typewritten letter with signature Prof. H. Behnke As secretary of the Int. Math. Instr. Comm. Oberwolfach (Schwarzwald) August 11 1954 To prof. Marshall Stone President of the Int. Math. Union University of Chicago Chicago, Ill.

 Stone is referring to the ICM in Amsterdam (2–9 September 1954). The section VII would include two invited lectures – one by Kay Piene on School mathematics for universities and for life and the other by C.T. Daltry on Self-education by children in mathematics using Gestalt methods, i.e., learning-through-insight – the report by Đuro Kurepa on The role of mathematics and mathematician at present time, and many communications and reports expressly dedicated to the teaching of mathematics. See Chap. 2, §  2.3 by Fulvia Furinghetti and Livia Giacardi in this volume. 133  At the bottom of the letter there is this handwritten note: “cc: Professor Behnke; Members of the Executive Committee, IMU”. 132

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U.S.A. Confidential Dear Mr. President, Your letter of July 29th 1954 addressed to Prof. Châtelet, President of the International Mathematical Instruction Commission, induces me to give you the following information: 1) It is very difficult matter to engage mathematicians, well-known for their research work, into problems of instruction. Most of our colleagues refuse to be active for our commission because they regard this kind of work of little value, and they even neglect to forward circulars. Twice I sent your letter asking to set up sub-commissions and to designate delegates, to all adhering national organizations using a list with addresses which I had received from Prof. Bompiani. There were only a few answers. Most of the addresses were furnished to me by the personal intervention of interested colleagues. But even now only 15 countries sent word.134 2) The work of our commission reveals its purpose and meaning only when we give lectures and exhibitions at the international mathematical congress at Amsterdam. It was not easy to convince the organization committee at Amsterdam of the value of our efforts. In the courses of last winter and spring several Dutch colleagues successively paid me a visit at Münster to learn what plans we had in mind. They all expressed the idea that lectures on mathematical instruction might not be worthy enough for the Congress. Thus, I showed them the reports of previous congresses and pointed out that, after 1912, in Cambridge (England) Section VII (history and instruction) was as strongly accentuated as Section II (analysis).135 Later, from 1920–1950, Section VII had no part at all at the congresses. After a report stating this fact, I recommended to rebuild Section VII. Finally, I succeeded. // 3) When M.  Cardot136 of the Ministère de l’Education de la France was in Amsterdam, in order to prepare the exhibition of school book as representative of our commission, he did not have much influence. He reported that the treat-

 The 15 countries which had designated their two delegates to ICMI where the following: Germany: W.  Süss, E.  Kamke; England: R.  P. Gillespie, A.  P. Rollett; Canada: P.  A. Petrie, A. Gauthier; Denmark: S. Bundgaard, M. Pihl; Finland: K. Väisälä, I. Simola; France: A. Châtelet, J.  Desforge; Greece: K.  Papaioannou, Michalopoulos; Israel: A.  Fraenkel, N.  Elyoseph; Italy: L.  Brusotti, M.  Villa; Japan: S.  Iyanaga, M.  Hukuhara; The Netherlands: H.  Freudenthal, J.  H. Wansink; Sweden: O.  Frostman, L.  Sandgren; Switzerland: G. de Rham, E.  Trost; USA: E. H. C. Hildebrandt, Syer; Yugoslavia: I. Bandic, L. Gabrovsek (EM s. 2, 1 1955: 197–198). 135  Behnke is referring to the ICM held in Cambridge from 21 to 27 August 2012. See Proceedings ICM 1913. Concerning the Section of history and education in the ICMs, see in this volume Chap. 8 by Fulvia Furinghetti. 136  André Cardot was the head of the Département de l’édition scolaire et du matériel didactique et scientifique at the Centre national de documentation pédagogique (CNDP), which depended on the French Ministry of Education. 134

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ment had been almost past endurance. Later Prof. Beth137 had struggled for the necessary rooms for the exhibition. 4) In Germany, too, I have had bad experiences with the adhering organization, although the presidency of D.M.V. (deutsche Mathematiker-Vereinigung) is held by old acquaintances. At the last meeting of the D.M.V. in Mainz, September 1953, I was given a highly unfavourable time for the report of the work of our commission, which would have never happened in the case of my scientific lectures. For the congress at Amsterdam the German sub-commission publishes a volume (of about 320 pages) on the mathematical instruction in Germany. The authors are 30 teachers of every type of school. In order to manage the work several meetings had to be called. For these sessions and for the printing of the volume I needed much money. I wanted to apply for the money at the German ministries by the aid of the presidency of the D.M.V. However, the D.M.V.138 did not lend its interposing assistance. So, I had to negotiate with the German ministries myself as representative of a special organization. 5) In accordance with arrangements by the organization committee of the Congress at Amsterdam I have invited 10 German teachers of mathematics, known to me by publications, to give lectures in Section VII at Amsterdam. Again the D.M.V. was not able to grant the travelling expenses for these reporters and again it was pointed out to me to undertake my own steps at the ministries. As chairman of the German sub-commission of the International Mathematical Instruction Commission I succeeded in carrying my project through. The D.M.V. commits a grave mistake in representing practically only the professors and assistants of university level and not the teachers (Gymnasium) who are much more numerous. Thus, the membership of the D.M.V. had reduced to 300, whereas there are approximately 5.000 teachers in Germany who have taken their university degree (the end of the studies being reduced at about 24 years of age). By taking the side of these teachers I am successful at the ministries. It would be useless for our German sub-commission to be made dependent on the presidency of the D.M.V. // At the close of this letter, I would like to recommend that the seat of the Instruction Commission stays at Paris, since communication thereto is very convenient. I have always stated with vivid regret that the delegates of the USA and Canada were not present. Yours very truly, Behnke P.S. I hope that you can receive our new volume on Sept. 1st 1954. Confidential

 Evert Willem Beth (1908–1964), a Dutch logician, was a member of the Executive Committee of ICMI from 1952 to 1954. 138  D.M.V. is the acronym for Deutsche Mathematiker-Vereinigung. 137

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Marshall Stone to Albert Châtelet and Heinrich Behnke, Rome, 21 September 1954139 IA 14A 1952–1954 Typewritten letter, carbon copy International Mathematical Union Istituto Matematico – Città Universitaria – Roma – Italy Rome, September 21, 1954 Professor A. Châtelet, Chairman I.C.M.I., Paris, France Professor H, Behnke, Chairman Elect, I.C.M.I., Münster, Germany Dear Colleagues, Following the meetings of the IMU at the Hague140 and of I.C.M.I in Amsterdam, I hope you will permit me to make a few comments on the actions taken there – comments which I would like to place on record for the Executive Committee of I.M.U. as well as for your Commission, and which I am therefore transmitting also to Professors Hodge and Mac Lane because of their interest in both connections. The organization of I.C.M.I adopted by the General Assembly of I.M.U confirms the general scheme which has been worked out over the last two years or so, but it also emphasizes and clarifies the position of ICMI as an agency of IMU.141 The fact that the members of ICMI, the chairman included, are to be named (and indeed have been named for the period January 1, 1955 to December 31, 1958) by the General Assembly, and the fact that the Executive Committee of ICMI has to be chosen in part from this group of ten Members, establish the responsibility of ICMI to the General Assembly of IMU. On the other hand, the fact that the two delegates named by each interested adhering country, through its National Committee for Mathematics are members of ICMI on an equal footing with the members designated by IMU emphasizes the importance of direct national participation in the work of ICMI. It is clearly desirable that the interested nations should have an opportunity of changing their delegates for the period January 1, 1955 to December 31, 1958, if they so wish. In principle, I suppose, it would be appropriate for any participating nation to change its delegates, at any time it may desire to do so, provided the appropriate notifications are made to ICMI and to IMU. In practice it would clearly be desirable for such changes to place normally at the time when the membership of the ICMI is reconsidered by the General Assembly of IMU at its regular meetings. It is also essential that the national delegates of the participating nations should be able to take part in the election of the officers of ICMI (other than the Chairman who is

 This letter has already been published in (Furinghetti and Giacardi 2010, pp. 38–40).  Stone is referring to the Second General Assembly of IMU, which took place in The Hague on 31 August – 1 September 1954. See Internationale Mathematische Nachrichten, 1954, 35/36: 4–14. 141  See the Terms of Reference of 1954 in Chap. 7 by Livia Giacardi in this volume. 139 140

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named by the IMU) and of the additional members of the Executive Committee of ICMI, since it is in this way that the participating nations can give direct expression of their views as to the organization and aims of ICMI (less direct expression having already been given through the General Assembly // of IMU). It is my understanding that at Amsterdam there was some expression by Dutch speakers of dissatisfaction with the choice of Mr. Gerretsen142 as a member of ICMI for the period beginning January 1, 1955. Possibly this dissatisfaction arises out of a misunderstanding. It is evident that the Netherlands is free to determine its two national delegates in any way it may decide as convenient and appropriate. It would certainly be reasonable for the delegates to be chosen among experts in secondary, or even primary, instruction, if that were agreeable to the various groups influencing the decision. In principle, it seems to me, ICMI should have a strong element of such experts, and in consequence I doubt that there is any substantial conflict in the case of Mr. Gerretsen. It must be stated, however, that as a result of the elections by the General Assembly of IMU Mr. Gerretsen is to remain a member of ICMI as a representative of IMU, until the next elections or until the General Assembly decides to replace him for one reason or another (in which case the General Assembly would not be bound to replace him by a citizen of the Netherlands). The arrangements for the financial support of ICMI should be a matter for very serious consideration in the immediate future. There has not been sufficient awareness till recent of the very strong limitations placed upon the use of funds deriving from UNESCO via ICSU143. The realization that these funds are to be used exclusively for travel and subsistence expenses incidental to meetings of ICMI underlines the necessity for uncovering other sources of support for the work of ICMI. At the First General Assembly of IMU at Rome, the discussions brought out the hope that ICMI would be able to mobilize supports from the Ministries of Education in the participating nations and from the various national organizations directly interested in the problems of mathematical instruction. So far, we have virtually nothing to mobilize such support, and we cannot go on much longer without making a real effort along this line. I have no doubt that the newly elected Executive Committee of IMU will be prepared to cooperate closely with ICMI in this effort, but the moving impulse must come from ICMI itself. The support which will be given to ICMI will depend largely upon the attractiveness of the general program formulated and announced by ICMI. Consequently, it is of the first importance that a general program which has long-range, fundamental objectives // should be formulated over the next year or two. Such a program should take cognizance of the scientific advances made possible in the techniques of instruction as a result of psychological investigations, as well as of the needs for curricular reform presented by changing social conditions. It should appeal to the  Johan Cornelis Hendrik Gerretsen (1907–1983) was a professor at the University of Groningen from 1946 to 1977. He made contributions in the fields of geometry, function theory and topology, and was also interested in problems relating to mathematics education and scientific popularization. 143  See footnote 129. 142

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countries where new educational systems are being introduced as well as to the countries where established systems are undergoing the social influences characteristic of our times. Its aim should be better teaching of more mathematics for more students at each successive level in the hierarchy of mathematical instruction. The various contributions to the basis of such a program already made at Amsterdam, whether in the form of addresses or in the very impressive exposition of factual material,144 reminds us that the work of ICMI will depend upon access to a good working library of reference documents, whether its own or that of some cooperative but independent organization. The meeting of ICMI which is planned to be held in Paris in October145 should be able to do many useful things toward starting off the new Commission which will succeed it on January 1 next. The presentation to the new Commission of a slate of officers and members of the Executive Committee would be helpful as the new Commission could then proceed at once to elect its officers and organize itself as required by the Statutes of the IMU and by the recent action of the General Assembly at the Hague. More important a start can be made toward developing a program which will meet the requirements of the General Assembly as expressed by a resolution passed at the Hague146 and which will open a brilliant future for ICMI. With felicitation for what has already been accomplished under the leadership of Professor Châtelet, and the highest hopes for what will be accomplished following the lead of Professor Behnke, I remain Faithfully yours, (Marshall H. Stone) President I.M.U. Conformal copy: E Bompiani147 Report of the president [Heinrich Behnke] of the International Commission of Mathematical Instruction to the president of the International Mathematical Union, [April 1955] 148 IA 14A 1955–1957 Typewritten document with a handwritten annotation149

 On the occasion of the ICM in Amsterdam 1954, exhibitions were organized: in the Municipal Museum of Amsterdam, the graphical work of M. C. Escher, which shows many remarkable connections with the mathematical thought, was exhibited, see Proceedings ICM 1954. Vol. 1, pp. 157–158. 145  The ICMI meeting took place in Paris on 29 October 1954. See Proceedings (Procès-Verbal de la réunion du Comité Exécutif de la Commission Internationale de l’Enseignement Mathématique (C.I.E.M.) tenue à Paris, le 29 Octobre 1954, IA, 14A, 1952–1954). 146  See the Terms of Reference of 1954, in Chap. 7 by Livia Giacardi in this volume. 147  Bompiani’s signature is handwritten. 148  An excerpt of this Report has already been published in Furinghetti and Giacardi 2010, pp. 41–42. 149  In the upper margin of the first page is written: “From Pres. Hopf. Zurich April 20, 1955”. 144

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1. History and Task of the I.C.M.I. The International Commission of Mathematical Instruction (Commission Internationale de l’Enseignement Mathématique; Internationale Mathematische Unterrichtskommission) was founded in Rome in 1908 at the congress of mathematics and had been working until the Second World War (in Oslo, at the international Congress, Henri Fehr gave a detailed report on the work which the I.C.M.I. had achieved). In 1952, at the session of the presidency of the Mathematical Union the I.C.M.I. was reconstituted. The first session of the reconstituted I.C.M.I. was held in Geneva, Oct. 1952.150 The old tradition was consciously revived and Henri Fehr, who had been the secretary of the I.C.M.I. from its foundation to the Second World War, was elected honorary president. In accordance with the statutes of the I.C.M.I. adopted at The Hague on Sept. 1st 1954151 – issued without having consulted the Executive Committee then in office nor the new president elected on Sept. 1st the task of the I.C.M.I. is defined by paragraph g) of the statutes as follows: The Commission shall be charged with the conduct of the activities of IMU, bearing on mathematical and scientific education, and shall take the initiative in inaugurating appropriate programs designed to further the sound development of mathematical education at all levels and to secure public appreciation of its importance.

2. How the I.C.M.I. works The I.C.M.I. consists of 1 ) The Executive Committee (7 persons), 2) The actual Commission to which belong 10 members nominated by the I.C.M.I. and two representatives of each nation that has nominated delegates, 3) The national sub-commissions. The actual work has always been done by the national sub-commissions. The Executive Committee gathers stimulations and suggestions from the sub-­ commissions in order to fix working programs; it discusses these suggestions, takes resolutions and asks the sub-commissions to carry out the work of the programs they have established. The national sub-commissions give reports on their work at the international congresses. The program of the I.C.M.I. at the international congress at Amsterdam, Sept. 1954, included the following items: 1) An exhibition of schoolbooks. It consisted of approximately 1000 books of different nations and is now available to those nations that want the collections for purposes of exhibition. Luxembourg was the first to ask for the material. 2) Reports on mathematical instruction for students between 16 and 21 years of age.

150 151

 See footnote 121.  See footnote 141.

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3) Reports on the part of mathematics in contemporary life, which amounted to approximately 20 talks.152 In the session of the Section life was very active. There was a splendid atmosphere of understanding among the representatives of // the different nations. About 100 regular participants were present. Looking through the reports of congress it becomes evident that section VII (didactics and history) was even more intensely discussed in Cambridge (England) in 1912153 than it was in Amsterdam in 1954 (this section then offered as many reports as were given for the whole field of analysis). On the other hand, didactics were practically ignored at the congresses between 1912 and 1954.154 3. The actual International Commission of Mathematical Instruction. (the name appearing in the resolutions of Sept. 1st 1954). The actual International Commission of Mathematical Instruction consists of about 40 members dispersed all over the world. They function as voters for the Executive Committee and are – perhaps in a higher extent – correspondents with the Executive Committee. At the big congresses the commission does not come into appearance because the members of the national sub-commissions from countries not too far away usually assist in full number, whereas, on the other hand many delegates of distant countries that is to say members of the actual I.C.M.I. do not appear. Thus, the big Commission is the weakest part of the structure of I.C.M.I. I am endeavouring to impart intellectual life to this corporation (which is comparable to a kind of chief council). In this context I may mention my project of a symposia (sic) on mathematical instruction in Geneva in 1956 and the suggestion to create an international encyclopedia of elementary mathematics. 4. The Sub-commissions of the I.C.M.I. The national sub-commissions of the I.C.M.I. have been established by the national adhering organizations. Apart from this, however they have used their rights of cooption and have chosen experts to become new members of their commission. Thus, from the necessity of fulfilling their tasks, they have become bodies which really represent all teachers of mathematics. Yet they suffer from the interference of the national adhering organizations. In many countries these national sub-­ commissions of the I.C.M.I. are considered neutral bodies into which those persons are called as delegates that have to be distinguished and for whom presently are no better positions available.

 See footnote 132.  Behnke is referring to the ICM held in Cambridge in 1912. Concerning the sections devoted to education and to philosophy, history and education during the ICMs in Cambridge and in Amsterdam respectively, see Proceedings ICM 1913, Vol. 2, pp.  449–458, 545–641, and Proceedings ICM 1954, Vol. 1, pp. 543–557 and Vol. 2, pp. 413–429. 154  See Chap. 8 by Fulvia Furinghetti in this volume. 152 153

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Besides, the national delegates who should sensibly act as kind of chairman of the national sub-commissions are changed at random. This happens without in the least considering the fact that the work of the sub-­ commissions is thus disturbed and without any interest in the qualification of the new delegate. There were cases which made me wonder how far these new delegates have occupied themselves with problems of instruction in the public of their native countries to be regarded sufficiently apt for such a leading position. The national adhering organizations lack the awareness of responsibility towards the I.C.M.I.  This derives from the depreciation of the commission for instruction through the mathematicians doing research work. These people, however, overlook the fact that in the field of research work the existence of a lace-corporal is as frequent as it is in the field of instruction. Thus, it happens that the national sub-commissions suffer from being ruled by university professors, for their influence is predominant through the big (King’s) academies, although the number of the university professors in their countries (at least in Europe) represents but a very small part of the teachers of mathematics. // Quite a number of countries sent us contradictory nominations of their national delegates. In other countries the delegates were rapidly changed. As president of the International Commission of Mathematical Instruction it is my duty to see that the members of the Commission are not university professors only. It is true that no government by university professors is actually intended. Yet it is realized in fact because the presidency of the national adhering organizations does not appreciate questions of mathematical instruction and inconsiderately uses its national power. From the Netherlands already four different delegates have contacted me as chairman of their international sub-commission since January 1954. As for Germany, relations are completely absurd. The German national adhering organization is the German corporation of mathematicians. It has the status of a corporation of about 450 persons. Yet the total number of German mathematicians with academic degrees lies between five and ten thousand; among them far the biggest number comprises teachers of secondary level (Gymnasiallehrer). These do not recognize the DMV (Deutsche Mathematiker-Vereinigung) as their organization but a much more comprehensive corporation: the Verein zur Förderung des mathematisch-­naturwissenschaftlichen Unterrichts. The DMV is so little interested in problems of instruction that despite considerable efforts, I did not succeed to have reserved two hours for the section of instruction at a six days’ national Congress. Besides, the DMV holds by no means the function of a King’s academy. In Germany there is a second organization equal in scientific reputation to the DMV: the GAMM (Gesellschaft für angewandte Mathematik und Mechanik). The GAMM is not yet represented in the German sub-commissions of the I.C.M.I.  On that account there are presently difficult debates between the DMV and myself. Underlying all these discussions is the question whether the national sub-­ commissions of the I.C.M.I., have become, on account of the new statutes of Sept. 1st, 1954, sub-commissions of the national adhering organizations. If this has actually happened it means a catastrophe for the I.C.M.I. An international conduct of the

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I.C.M.I. then has lost any sense whatsoever. The work of the I.C.M.I. being done by the national sub-commissions this would involve that, of the whole I.C.M.I., only the Executive Committee, consisting of 7 persons, would be left over, deprived of any basis. The new statutes have never been discussed in the Executive Committee before they became effective. At the session in Paris, Oct. 2nd, 1954, the withdrawing president155gave the comment: the national sub-commissions of the I.C.M.I. are created by the national adhering organizations. According to this, however, they are totally independent of the national adhering organizations – apart from their rights of cooption. This conception guarantees the only possible basis for the work of the I.C.M.I. The president of the I.C.M.I. has to take into consideration the sensibility of the different nations and can never interfere with the constitution of the sub-­commissions. He has to make inciting suggestions for the subject of work to be done which implies the effect that the national sub-commissions, enlarged by experts, becomes a workable body, preventing that they are regarded as a dead track for the national adhering organizations. The presidency of the IMU has to give up (at least with a view to the work of the I.C.M.I.) the principle of national bundling – which is an obsolete principle also in Europe – and has to look upon the // national sub-commissions – as was the case already before 1914 – as sub-commissions of the I.C.M.I. and not of the national adhering organizations, Otherwise the work of the I.C.M.I. is made impossible. 5. The work of the ICMI during 1955/58 The program of work planned for the I.C.M.I. cannot be adopted before the session of the newly constituted Executive Committee has been held. The first session of this Executive Committee will probably take place in Geneva this coming July. But, according to a discussion on the work of the I.C.M.I. for the next years at the last session of the former Executive Committee in Paris, Oct. 1954, the following program was suggested: 1) the proposition is to be made to the national sub-commissions to work on the subject of “The Scientific Basis of School Mathematics” and to compose for their countries or groups of countries a book for the scientific consultation of the teachers. For this book it is of primary importance that teachers of mathematics of all levels cooperate. I regard it is a special, honorary mission of the I.C.M.I. to establish a contact among the teachers of all levels. The teachers have to get interested in the research work, and those active in the field of research have to get interested in the work of the teachers. I have already succeeded in being assured of the readiness of cooperation for the second volume of the German I.C.M.I. report (“Mathematical Instruction for the Early Youth in the Federal Republic of Germany”) from professors of the

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 The withdrawing president was Albert Châtelet.

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academies for education (Pädagogische Akademien) and through them from the teachers of primary level. At the interim meeting in 1956 (symposia for the scientific basis of school mathematics) the experiences shall be compared gathered by the different nations in projecting this book. In this context I may mention the suggestion to create an international encyclopedia of elementary mathematics. I do not yet see a way to realize this project because the school systems and therefore the material of instruction deviate too considerably from one another in the different countries. Yet this project will be submitted at the session of the Executive Committee in July. This way it may be possible to approach the suggestion made by M. Stone to create an international work of instruction. This plan might at first sound simply fantastic for everyone who knows the diversity of national conditions of instruction in different countries. It would, indeed involve an entirely new way of working for the I.C.M.I. since, for the first time, it would not simply have to coordinate national work, but would have to realize an important international work. As a matter of course, considerable financial means would be necessary for the realization of such a project, because the different collaborators would have to be in constant communication during the time of accomplishing this work. 2) The inquiry “The Part of Mathematics on Contemporary Life”156 has to be examined more fully and with much more gravity than has been done up to now. The investigation of this bulk of questions is closely connected with the technical development of the different countries. In America, f. i., there exists a supervision of production at the instant of production. This plays a particularly important part in iron industry of small quantities, thus preventing refuse. The establishment of such a supervision and such a controlling office is, to a high degree, dependent on exact mathematical calculations. Our colleague Ulrich Graf,157 who died last year, was about to introduce the same establishment in Germany. If this is done on a larger scale in the region of the Ruhr, for example, large numbers of mathematicians will be required. All this has to be planned before-hand. // Similar questions arise for the use of large-size calculating machines in the industrial field. It is thinkable that, in the coming years, the applications of these machines might expand enormously. This involves the new vocation of the industrial mathematician. The firm Siemens-Halske158 in Munich has now opened a large department for the development of calculating machines and has already called from Münster four of our young doctors of mathematics. Thus, questions are raised which have to be discussed on an international basis.  During the ICM in Amsterdam in 1954 Đuro Kurepa presented a report on The role of mathematics and mathematician at present time. See footnote 132. 157  Ulrich Graf (1908–1954) was a German mathematician and one of the pioneers of modern mathematical statistics in Germany after the Second World War. 158  Siemens-Halske was a German electrical engineering company which later became part of Siemens. 156

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All pains taken by the I.C.M.I. can be summed up by this formula: To contact people of different qualities and abilities, people of different nations and different teaching professions (as long as they are seriously interested in mathematics) in order to make them work together. There resides the great obligation and chance for the I.C.M.I. I personally try to be an example for this possible, rather comprehensive kind of work 1) with my meetings at Whitsuntide159 aiming at the maintenance of relation between universities and schools (Pfingsttagungen zur Pflege des Zusammenhangs von Universität and Gymnasien); regular attendance of approximately 250 persons; 2) with the international interim meetings of the I.C.M.I. which will be introduced (symposia for school mathematics) and the sessions of the Section at international congresses; 3) with our series of books on mathematical instruction in the different countries; 4) with the national encyclopedias of school mathematics; 4a) possibly with an international encyclopedia of school mathematics. Heinrich Behnke to Beno Eckmann160, Münster, 4 March 1957 IA, 14A 1955–1957 Typewritten letter with handwritten corrections and signature161 Mathematische Institute der Universität Münster Direktoren: Prof. Dr. Heinrich Behnke Prof. Dr. Hans Petersson Münster (Westf.), den 04.03.1957 Schloßplatz 2 Fernruf *40739/430 Be/We Herrn Prof. Dr. B. Eckmann Eidgen. Techn. Hochschule Zürich / Schweiz Lieber Herr Eckmann!162  Pentecost.  Beno Eckmann (1917–2008) was secretary of IMU from 1956 to 1961. 161  The handwritten parts appear in italics. 162  The English translation is: “I am extremely dismayed about your letter of 27.2. The Activity Report was translated by Mrs. Irwin, who is not only an American but moreover studies English. It is undeniable that she has a perfect command of the English language. I have therefore given her the ‘Rapport sur les Activités’ and charged her to translate and send it to you. She’s also very intel159 160

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Ich bin auf das äusserste über Ihren Brief vom 27.2. bestürzt. Der Activity-Report ist von Frau Irwin übersetzt worden, die nicht nur Amerikanerin ist sondern überdies Englisch studiert. Dass sie die englische Sprache einwandfrei beherrscht, ist unbestreitbar. Ich habe ihr deshalb den „Rapport sur les Activités“ gegeben und sie beauftragt, die Übersetzung herzustellen und sie Ihnen zuzusenden. Sie ist auch sehr intelligent – aber flüchtig – und indem sie damit rechnete, dass ich den Report nicht mehr durchläse, hat sie eine so nachlässige Arbeit geliefert. Ich kann aber nicht mehr leisten als in meiner Kraft steht, und ich habe micht deshalb auf M. Desforge und Frau Irwin verlassen. Das ist ja eine furchtbare Katastrophe! Ich kann Ihnen im Augenblick nur das Original von M. Desforge163 zusenden. Irgendwelche Abschriften besitze ich nicht. Und Frau Irwin ist schon in die Ferien gefahren. Ich wäre Ihnen also sehr dankbar, wenn Sie mir das Original wieder zusenden würden. Indem ich sehr um Entschuldigung bitte, dass ich Ihnen soviel Arbeit mache, verbleibe ich mit herzlichen Grüssen Ihr sehr ergebener Behnke Heinrich Behnke to Heinz Hopf,164 n. p., 27 November 1957 IA, 14A 1955–1957 Typewritten letter with handwritten notes and signature165 Abschrift Herrn Prof. Dr. Eckmann 27. November 1957 BE/sl Herrn Professor Dr. Heinz Hopf Zollikon bei Zürich alte Landstraße 37 Schweiz

ligent – but hasty – and in expecting me not to read the report again, she did such a sloppy job. But I cannot do more than I can, which is why I have relied on Mr. Desforge and Mrs. Irwin. It’s a terrible disaster! At the moment I can only send you the original by Mr. Desforge. I do not have any copies. And Mrs. Irwin has already gone on vacation. Thus, I would be very grateful if you would send me the original again. While I apologise for causing you so much work, I send you best regards. Your very devoted, Behnke”. (trans. by the author) 163  Julien Desforge (1891–1984) was secretary of ICMI from 1955 to 1958. 164  Heinz Hopf (1894–1971) was president of IMU from 1955 to 1958. 165  The handwritten parts appear in italics.

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Lieber Herr Hopf,166 Von diesem Brief schicke ich wegen seiner Wichtigkeit sogleich eine Kopie an Herrn Eckmann, um Ihnen keine Mühe mit der Übersendung zu machen. Am 6. oder 8. Februar wollen wir in Paris167 die zunächst für November angesetzte Sitzung der IMUK abhalten. Es wird voraussichtlich die einzige Sitzung sein, die wir zwischen Juli 1957 und dem Kongress 1958 abhalten. Die Reisekosten für die europäischen Teilnehmer hat Herr Eckmann mit freundlicherweise zugesagt. Aber auf dieser Sitzung soll auch der Fragebogen besprochen werden, durch den wir Aufschluß bekommen möchten über den Mangel an Mathematiklehrern in den verschiedenen Ländern. Herr Kurepa168 hat einen Fragebogen zusammengestellt, der völlig unmöglich ist, auch seine revidierte Form kommt nicht in Frage. Herr van Dantzig169 ist so ärgerlich darüber, daß er keine Lust hat zu kommen, wenn nicht Herr Fehr170 aus New York kommt, der etwas von solcher Statistik versteht. Fehr ist Mitglied dieser Kommission und ist bereit zu kommen. Aber von den 600 Dollar, die eine solche Reise ihn kostet, möchte er 450 Dollar ersetzt bekommen. Das ist

 The English translation is: “Dear Mr. Hopf, given its importance, I send right now a copy of this letter to Mr. Eckmann, so as not to bother you with sending it. On February 6th or 8th we intend to hold the IMUK meeting in Paris, initially scheduled for November. It is likely to be the only meeting we will realise between July 1957 and the 1958 Congress. Mr. Eckmann has kindly agreed to cover the travel expenses for the European participants. However, during this meeting, the questionnaire will also be discussed, by which we would like to get information about the shortage of mathematics teachers in the various countries. Mr. Kurepa has elaborated a questionnaire that is completely impossible, and its revised version is also out of the question. Mr van Dantzig is so angry about it that he does not want to participate unless Mr. Fehr, who knows something about such statistics, comes from New York. Fehr is a member of this commission and is ready to come. However, of the $ 600 that such a trip costs him, he would like that $450 will be refunded. From his point of view, this proposal is very conciliatory. But can you think of such large amounts? Here it will be decided whether IMUK will be able to realise international meetings, or whether has to be limited to Europe. You have expressed some criticisms of our meetings, saying that they are only for Europeans and organised only in the European spirit. I have not denied that. However, if we want to go significantly beyond Europe in our work, then it should at least be possible for Mr Fehr to be reimbursed for the greater part of his travel expenses. I am not in favour of arguments of principle. The preceding lines should only be an introduction to the question: Can you help us obtaining the $ 450 for Mr. Fehr? You would do it for a valuable person and for an important cause that you yourself had suggested. Best regards, Your Behnke”. (trans. by the author) 167  The planned meeting in Paris did not take place, but it was held in Munster-Westfalen on 28 May 1958 (EM, s. 2, 4 1958: 213–219). 168  David Van Dantzig, Howard F. Fehr, Đuro Kurepa and Willy Servais were assigned to the preparation of the questionnaire proposed by Hopf on the difficulties that arise in recruiting professors of mathematics and professors of natural sciences. See the text of the questionnaire in EM, s. 2, 4 1958: 220–223. 169  David van Dantzig (1900–1959) was a Dutch mathematician well known for his contributions to topology as well as probability and statistics. He was a member of the Significs Group interested in the science of meaning and communication with attention to philosophical, linguistic, logical and psychological aspects. 170  Howard F. Fehr (1901–1982) was an American mathematics educator, professor at the Columbia University Teachers College. 166

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von seiner Sicht aus gesehen sehr konziliant. Aber können Sie an solche großen Summen denken? Hier entscheidet es sich, ob die IMUK überhaupt internationale Sitzungen abhalten kann oder ob sie sich auf Europa beschränken muß. Sie haben eine gewisse Kritik an unseren Sitzungen geübt, indem sie sagten, sie seien nur für Europäer und nur im europäischen Geist abgehalten. Das habe ich nicht bestritten. Aber wenn wir in unserer Arbeit wesentlich über Europa hinausgreifen sollen, dann muß es mindestens möglich sein, daß Herr Fehr seine Reisekosten zu einem größeren Teil ersetzt bekommt. Ich bin nicht für prinzipielle Auseinandersetzungen. Die vorstehenden Zeilen sollten nur eine Einleitung zu der Frage sein: Können Sie uns helfen, die 450 Dollar für Herrn Fehr zu beschaffen? Sie würden es für einen wertvollen Menschen und für eine wichtige Sache tun, die Sie selbst angeregt haben. Herzliche Grüße, Ihr Behnke Heinz Hopf to Heinrich Behnke, Zollikon, 6 December 1957 IA, 14A 1955–1957 Typewritten letter with handwritten corrections and initials of name and surname 171 Kopie Zollikon, den 6. Dezember 1957 Herrn Professor Dr. H. Behnke Münster Lieber Herr Behnke!172  The handwritten parts appear in italics.  English translation: “Dear Mr. Behnke! Your request of 27.11 concerning Mr. Fehr’s travel, crossed Eckmann’s letter of 27.11 and his letter, attached to this my letter, to the president of the Commission dated 18. 11 and you will have seen that these two letters contain, in principle, the answer to your question: the payment of a large amount for travel expenses to Mr. Fehr in February would run counter to the plan of our Executive Committee, to allocate the amounts that are available in the year 1958 or which can be raised for travel expenses, primarily in connection with the Edinburgh Congress. In the case of Mr. Fehr, this general point of view is supplemented by the special fact that Mr Fehr will be one of the main speakers at the IMUK in Edinburgh and therefore it would be only natural if a considerable part of his expenses for travelling to Edinburgh were covered either from IMUK-IMU or by the organisers of the Congress leadership; but to pay the same person twice in a year for trips from America to Europe and back would be quite uneconomically, from whatever source the money may flow. However, I think that these considerations also suggest that it would be practical to include the item on the agenda for which the presence of Mr. Fehr is so strongly desired, not at the February meeting but at the IMUK meeting during the Congress in Edinburgh. Such a meeting in Edinburgh would have a much more international character than the small meeting in Paris in February and could demonstrate in a particularly convincing way the fact that the IMUK activity is not limited to Europe. I would like to add here that we in the IMU always try to reconcile the dates of the Executive Committee’s meetings with the already existing travel plans of our two non-European members (Chandrasekharan and Mac Lane); so far, we have always been able to arrange that at least one of the two gentlemen attended the 171 172

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Ihre Anfrage vom 27.11. wegen der Reise von Herrn Fehr hat sich mit Eckmanns Brief vom 27.11. und seinem, diesem Brief beigelegten Schreiben an die Kommissions-Vorsitzenden vom 18.11. gekreuzt, und Sie werden sehen gesehen haben, daß diese beiden Schreiben im Prinzip die Antwort auf Ihre Anfrage enthalten: die Ausrichtung einen hohen Reisekostenbeitrages an Herrn Fehr im Februar würde im Widerspruch stehen zu dem Plan unseres Exekutiv-Komitees, die Beträge, die im Jahre 1958 für Reisekosten zur Verfügung stehen oder beschafft werden können, in erster Linie im Zusammenhang mit dem Kongress in Edinburgh173 zu verwenden. Im Falle von Herrn Fehr kommt zu diesem allgemeinen Gesichtspunkt noch der spezielle hinzu, daß Herr Fehr ja einer der Hauptreferenten der IMUK in Edinburgh sein174 wird und daß es daher nur natürlich175 ist, wenn ein beträchtlicher Teil seiner Reisekosten nach Edinburgh entweder von IMUK-IMU oder von der Kongressleitung übernommen176 wird; aber derselben Person zweimal während eines Jahres die Reisen von Amerika nach Europa und zurück zu bezahlen, das wäre doch zu unökonomisch, aus welchen Quellen die Gelder auch immer fließen mögen. Diese Überlegungen legen es aber, wie ich finde, auch nahe, daß es praktisch wäre, den Punkt der Tagesordnung, für dessen Diskussion die Anwesenheit von Herrn Fehr so dringend erwünscht ist, nicht in der Februar-Sitzung, sondern in einer IMUK-Sitzung während des Kongresses in Edinburgh zu behandeln. Eine solche Sitzung in Edinburgh würde einen viel stärker internationalen Charakter haben als die kleine Sitzung im Februar in Paris und könnte in besonders über // zeugender Weise die Tatsache demonstrieren, daß die Aktivität der IMUK nicht auf Europa beschränkt ist. Ich möchte hier einfügen, daß wir in der IMU überhaupt immer versuchen, die Termine der Sitzungen des Exekutivkomitees mit den ohnehin bestehenden Reiseplänen unserer beiden außereuropäischen Mitglieder (Chandrasekharan und meeting without the travel expenses we have to pay having increased significantly. Furthermore, we have had excellent experiences with the cooperation of members who were unable to attend a meeting but who made written comments on individual items on the agenda; and it is precisely the geographically distant members who take part in the activities of the IMU continuously and with great interest. Incidentally, regarding the critical remark I made earlier and which is mentioned in your last letter that the IMUK meetings are too focused on Europe, I ask you not to misunderstand me: I definitely think that the European group, which performs a positive work, is an utmost rewarding and important institution – certainly more important than a supra-continental organisation that only exists in theory; however, I have considered and also consider the problem of ‘How can groups, distributed in many parts of the world, which promote or wish to promote mathematics teaching fruitfully, be brought together?’ This problem should therefore be dealt with more intensively than has been done so far. The problem is difficult, and for the moment I have raised it only because we – you and I – are obliged to submit proposals for new statutes of ICMI as soon as possible; but the basis for such statutes must surely consist in a certain position on this problem. So, we will have to talk about it soon, in writing or orally. Kind regards! Your H. Hopf”. (trans. by the author). 173  Hopf is referring to the ICM in Edinburgh (14–21 August 1958). 174  Del. soll. 175  Del. wäre. 176  Del. würde.

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Mac Lane)177 in Einklang zu bringen; bisher konnten wir es immer einrichten, daß wenigstens einer der beiden Herren an der Sitzung178 teilnahm, ohne daß sich die von uns zu zahlenden Reisespesen wesentlich erhöht hätten. Ferner haben wir auch mit der brieflichen Mitarbeit der Mitglieder, die an der Teilnahme an einer Sitzung verhindert waren, sich aber schriftlich zu einzelnen Punkten der Tagesordnung äußerten, sehr gute Erfahrungen gemacht; und gerade die geographisch entfernten Mitglieder nehmen an den Geschäften der IMU dauernd und mit großem Interesse teil. Was übrigens meine früher geäußerte und in Ihrem letzten Brief erwähnte kritische Bemerkung betrifft, die IMUK-Sitzungen seien zu sehr auf Europa konzentriert, so bitte ich Sie, mich nicht mißzuverstehen: ich bin durchaus der Ansicht, daß die positiv arbeitende europäische Gruppe eine höchst erfreuliche und wichtige Institution ist – wichtiger sicher als eine nur theoretisch existierende überkontinentale Organisation; aber ich meinte und meine auch, daß man das Problem „Wie kann man die in vielen Teilen der Welt existierenden Gruppen, die den mathematischen Unterricht fördern oder fördern wollen, miteinander in fruchtbare Verbindung bringen?“  – daß man also dieses Problem intensiver in Angriff nehmen sollte, als es bisher geschehen ist. Das Problem ist schwierig, und ich habe es im Augenblick eigentlich nur deswegen zur Sprache gebracht, weil wir –Sie und ich- ja verpflichtet sind, nächstens Vorschläge für neue Statuten der ICMI zu machen; die Grundlage für solche Statuten muß aber wohl doch in einer gewissen Stellungnahme zu diesem Problem bestehen. Wir werden uns also bald, schriftlich oder mündlich darüber aussprechen müssen. Herzlichen Grüssen! Ihr H. H. Heinrich Behnke, Suggestions on the subject of forming “Regional Groups” within I.C.M.I. n.d. [1958]179 IA 14A 1958–1960 Typewritten mimeographed document180 Professor Behnke: Suggestions on the subject of forming “Regional Groups” within I.C.M.I. A) Since its foundation in 1908 in Rome, ICMI’s aim has been to compare experiences in the teaching of mathematics in all types of educational establishments, and to discuss possible reforms in the teaching of mathematics. This program includes the following points: ( 1) Reports on mathematical instruction. (2) Discussions on eliminating obsolete parts of material hitherto used.  Komaravolu Chandrasekharan (1920–2017) and Saunders Mac Lane (1909–2005) were members of the Executive Committee of IMU from 1955 to 1962 and from 1955 to 1958 respectively. 178  Del. teilnehmen konnte. 179  This document has already been published in (Furinghetti and Giacardi 2010, pp. 43–44). 180  In the upper margin there is the following handwritten annotation: “Memorandum von Herrn Behnke über die Bildung von Gruppen”. 177

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(3) Suggestions and discussions on introducing new mathematical points of view into curriculum (for instance to introduce the concept of “structure” already at Secondary School level). (4) To establish and cultivate contacts between various types of schools – as far as mathematical teaching is concerned – particularly where pupils progress from one school to the other. B) In dealing with any particular problem included in this general program, we must be quite clear about the particular age group and the particular educational level of those pupils to whom this problem applies. But this is not easy because conditions vary considerably from country to country, as the national school systems are very often based on different principles. Therefore, the work of the National Sub-Commissions must be the basis of all life in ICMI. It is then one of the main tasks of ICMI to create and cultivate the exchange of ideas and experiences between the sub-commissions of different countries. This exchange is obviously easiest between those countries where the school systems are similar. Therefore, it is rather natural that the sub-commissions of the countries sharing the old European traditions in educational matters (namely France, Germany, Great Britain, Italy, Scandinavia, etc.) – which I shall briefly call the WNE-countries (West and North European)  – have up to now found closer contact with one another than with national sub-commissions from other parts of the world. Consequently, the activity of ICMI during the 50 years of its existence was mainly concerned with these WNE-countries. If ICMI now makes the attempt to extend its activities to all parts of the world, then it would be appropriate to form “groups” of national sub-commissions, so that countries whose educational systems are similar, belong to the same group. This is in accordance with the resolution passed by the General Assembly of the IMU at St. Andrews, August 1958. C) At the International Congresses of Mathematicians, ICMI plays a relatively small role, since these congresses are dominated by reports and discussions on matters of research. It might, therefore, be more appropriate for ICMI to hold smaller symposia in the years between Congresses. This has, in fact, been the case in the past; especially during the periods 1909–1914181 and 1953–1958,182 such symposia have taken place annually. For financial reasons, it is necessary to restrict each of these meetings to some countries not too far from each other. Thus, the geographical aspect must also be considered in the formation of the  Behnke is referring to the meetings in Brussels (10–16 August 1910) and in Milan (18–21 September 1911) held between the two ICMs in Rome (1908) and in Cambridge (1912) during the presidency of Felix Klein (see Chap. 1 by Gert Schubring in this volume). 182  Behnke is probably referring to the meeting in Geneva (2 July 1955) when a symposium was organized to commemorate the important work undertaken by Henri Fehr for ICMI (1908–1954) and in directing L’Enseignement Mathématique, and to the meeting in Brussels (1–3 July 1957) when the Journées de la C.I.E.M. were organized by the Belgian subcommission (EM s. 2, 3, 1957: 155). These meetings were held between the ICMs in Amsterdam (1954) and in Edinburgh (1958). 181

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proposed groups. Fortunately, these two aspects, the similarity of educational systems and the geographical one, coincide in the most cases. // Led by these considerations, I make the tentative suggestions that to begin with, the following “Regional Groups” of ICMI might be formed: 1. The WNE-countries 2. The East European Countries 3. South East Asia 4. Central and South America 5. Australia and New Zealand. One might imagine that very large countries, like USA and USSR, would have little interest in joining a regional group. The organization of the symposia mentioned above would be such that each year a certain group arranges a meeting in which the reports and discussions deal in first place with the particular interests of the members of that group. However, representatives of other groups should be invited to take part. With regard to financial arrangements, it remains to be seen whether enough money would be available from IMU or whether the group itself would have to find other resources. D) The climax of the whole work would be a meeting of all national sub-­commissions (at present numbering 23). It would be appropriate for this to take place in connection with each International Congress of Mathematicians. Marshall Stone183 to Beno Eckmann, Chicago, 5 January 1959 IA 14A 1958–1960 Handwritten letter The University of Chicago Chicago 37 – Illinois Department of Mathematics January 5, 1959 Professor B. Eckmann, Secretary, IMU Zurich, Switzerland Dear Professor Eckmann, Thank you for your letter of December 19 last, giving me the financial information which I sought in relation to ICMI. When ICMI has chosen a Secretary, I shall discuss with him what we should do about making use of the appropriation for secretarial help to the President, and I will let you know what we decide. I have

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 Marshall Stone was the new president of ICMI. His mandate lasted from 1959 to 1962.

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lately received the draft of the “New Terms of Reference” for ICMI.184 Would you kindly lay before the President of IMU and the members of the Executive Committee my comments as detailed below. It seems to me that (a2), (c), (d), (e) of the “New Terms of Reference” are extremely likely to make a great deal of trouble between groups in the adhering countries, and between ICMI and IMU. I therefore consider them ill-advised and objectionable. The tendency of these constitutional arrangements would be to make ICMI a body of individuals primarily interested in the educational aspects of mathematics and independent of IMU to the extent that it could raise funds from sources interested // in mathematical education. By making the National Subcommission for Mathematical Instruction quite independent of the National Commission for Mathematics (instead of being subordinate organizations as at present) as would obviously be the case under (c), (d), and (e), and letting the representative of these Subcommissions have effective voting control of ICMI, the way is cleared for the elimination of any real influence in ICMI from the side of the mathematicians who are acquainted with the higher levels of their subject and who are interested in research as well as in teaching and preparation for research. When each Subcommission was entitled to two representatives a proper compromise between the research mathematicians and the mathematical educators was possible. Now under the New Terms, these two groups will have to fight it out in each country to determine which one will be able to name the sole representative under (a2). I would like to see the last sentence of (c) given much more careful analysis and elaboration than is offered in this document. I doubt that ICMI should have the power to accept such adherence without individual approval by IMU.  There are surely many pitfalls here! In short, I do not look forward to having to operate under the New Terms while I am President of ICMI! Faithfully yours Marshall H. Stone Heinrich Behnke to Beno Eckmann, Münster, 28 December 1959 IA 14A 1958–1960 Typewritten letter with signature I. Mathematisches Institut der Universität Münster Direktor: Prof. Dr. Heinrich Behnke Münster (Westf.) 28.12.1959

 Behnke had proposed a draft for the new Terms of Reference which was discussed during the General Assembly of IMU held in St. Andrews (Scotland) on 11–13 August 1958. The General Assembly agreed on the necessity to make certain changes in the regulations of ICMI and invited EC of IMU to collaborate with that of ICMI at this aim, and to present a joint proposal (EM s. 2, 4, 1958: 226–228). 184

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Schloßplatz 2 Be/Ro Fernruf *40739/430 Herrn Prof. Dr. B. Eckmann Zürich / Schweiz Lieber Herr Eckmann!185 Anbei die Quittung für die 39 Dollars, die zur teilweisen Bezahlung meiner IMUK-Reise nach Paris Marshall Stone mir gnädigst zurückerstattet hat. Meinen Bericht über die recht bedrückende Sitzung in Paris habe ich an Herrn Hopf gesandt. Ich darf annehmen, daß Sie davon Kenntnis genommen haben. Es gibt nur einen Trost: Marshall Stone sagte zum Abschluß: er werde so leicht keine solche Vorstandssitzung der IMUK wieder einberufen. Ich übersetze diesen Ausspruch so: daß die nächste Sitzung im Winter 1961/62 sein wird, d.h. in zwei Jahren. Inzwischen macht Herr Bundgaard186 in Aarhus187 eine Tagung der Europäischen Unterkommissionen. Wenn die gut verläuft, werde ich vorschlagen, etwas Ähnliches im Jahre 1961 in einem anderen Teil von Europa zu wiederholen. Aber es fehlt einfach das Leben, das es 6 Jahre lang (1952–1958) in der IMUK gegeben hat. Ich bleibe um unseren Freund Hopf besorgt. Anfang März möchte ich gern für 2 Tage nach Zürich kommen und dabei vor allem mich nach Hopf umsehen. Herzliche Grüße Ihr sehr ergebener Behnke Marshall Stone to Rolf Nevanlinna, n. p. 5 April 1960 IA 14A 1958–1960 Typewritten letter with signature and handwritten additions188 5 April 1960  English translation: “Dear Mr. Eckmann! Attached is the receipt for the 39 dollars that Marshall Stone kindly reimbursed me to partially pay for my IMUK travel to Paris. I sent my report on the rather depressing meeting in Paris to Mr. Hopf. I can assume that you have taken note of it. There is only one consolation: Marshall Stone concluded by saying that he would not easily invite such a board meeting of IMUK again. I translate this saying this way: that the next meeting will be in the winter of 1961/62, i.e., in two years. Meanwhile, Mr Bundgaard is organising a meeting of the European subcommissions in Aarhus. If that goes well, I will propose that something similar be repeated in another part of Europe in 1961. But the life that existed for 6 years (1952–1958) in IMUK is simply missing. I remain worried about our friend Hopf. Early in March, I would like to come to Zürich for 2 days and above all to have a look at Hopf. Best regards your very devoted, Behnke”. (trans. by the author) 186  The meeting on The Teaching of Geometry in Secondary School took place in Aarhus (Denmark) from 30 May to 2 June 1960 (Internationale Mathematische Nachrichten, 66, 1961: 1). 187  Svend Bundgaard headed the Institute of Mathematics of the University of Aarhus from 1954 until he was appointed rector of that university in 1971. 188  At the bottom of the first page of the letter there is this handwritten note by Stone: “No note has been received from Professor A.D. Alexandroff”. The other handwritten additions appear in italics. 185

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International mathematical Union Dear president Nevanlinna:189 As requested by the 1958 General Assembly of IMU, the Executive Committee of ICMI has studied the proposed “new terms of reference” for ICMI and wishes to express the following majority views concerning them (Professors Akizuki, Frostman, Kurepa, Stone, Walusinski concurring, and professor Behnke dissenting).190 The three principal points raised by the proposed “new terms” are: (1) the formation of regional groups; (2) the modification of the relations among IMU, ICMI and the National Sub-Commissions for Mathematical Instruction (NSCMI); (3) the participation of non-IMU countries in the work of ICMI. While appreciating the goals envisaged under the “new terms”, the majority of the Executive Committee feels that the means proposed for attaining them have inherent defects and would tend to weaken the relations between IMU and ICMI or to weaken ICMI itself. The majority therefore favors retaining the existing constitution of ICMI, as established by actions of the 1954 and 1958 General Assemblies of IMU, and using its provisions in such a way as to move toward the desired goals. First, as to regional groups, it is felt that the formation of permanent groups of this character inside ICMI would seriously divide and weaken ICMI’s activities. Furthermore, studies by a special committee chaired by Professor Hopf191 have led to the conclusion, reported informally by the Chairman, that the formation of permanent regional groups representing Europe would be difficult to achieve. On the other hand, ICMI is prepared to cooperate fully with effective regional groups which may be formed spontaneously outside structure of ICMI. The majority therefore recommends that the constitution of ICMI should contain no provision for regional groups. Secondly, as to the relations among IMU, ICMI and the NSCMI, it is felt that the proper subordination of ICMI and of the NSCMI to IMU requires the NSCMI to continue to be subordinate to their respective National Committees for Mathematics, as provided by the 1954 General Assembly of IMU,192 and not to become independent of both the National Committees and IMU as proposed in the “new terms”. It is clear that independence for the NSCMI would weaken in a serious way the bonds between ICMI and IMU. On the other hand, it is agreed that the National Committees for Mathematics should be under an obligation to make the NSCMI // broadly representative of the national interests in mathematical instruction and to assist them in every way to find adequate financial support. The majority therefore recommends that the NSCMI continue as subcommittees of their respective National Committees for Mathematics, and that the Presidents of IMU and of ICMI exert their influence  Rolf Nevanlinna (1895–1980) was president of IMU from 1959 to 1962.  Y. Akizuki, A.D. Aleksandrov (sometimes Alexandrov or Alexandroff) , H. Behnke, O. Frostman, Ð. Kurepa, M. Stone, and G. Walusinski were the members of the Executive Committee of ICMI from 1959 to 1962. 191  Heinz Hopf was a member of the Executive Committee of IMU as past-president. 192  See the Terms of Reference of 1954, in Chap. 7 by Livia Giacardi in this volume. 189 190

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with the latter to make the NSCMI active and effective organs for the improvement of mathematical instruction. Finally, as to the participation of non-IMU countries in the work of ICMI, this is most cordially desired but needs to be arranged in such a way that IMU may retain ultimate control of relations with such countries. The “new terms” would permit any non-IMU country to associate itself with ICMI without any control whatsoever by IMU, and would thus operate to weaken the ties between ICMI and IMU and perhaps also to create embarrassing situations in the case where such a country might eventually wish to adhere to IMU. A recently suggested solution would be to permit ICMI, with the approval of IMU in each individual case, to coöpt as members of ICMI suitably chosen representatives of such non-IMU countries. This solution seems to be both simple and effective, requiring no new legislation, and is therefore favored by the majority. Hence, the majority recommends that the participation of non-IMU countries be individually arranged through coöptation of suitably chosen representatives as members of ICMI in accordance with the existing Statutes and By-Laws of IMU. For the Executive Committee of ICMI: Marshall H. Stone MHS/lg Copies to Professors Eckmann, Hopf, Morse, Choquet193 Marshall Stone to Beno Eckmann, Chicago, 11 April 1960 IA 14A Archives 1958–1960 Handwritten air letter 194 April 11, 1960 Dear Professor Eckmann The ICMI program for 1960 comprises a Seminar in Aarhus, May 30–June 2, and a Colloquium in Belgrade, September 19–24,195 jointly, supported by ICMI and the Jugoslav – Association of Mathematicians and Physics. ICMI has allocated from its 1960 budget the sum of $1000 to each of these activities. Thus, ICMI has on hand only the balance remaining from the item in the budget for secretarial expenses, and a small contribution from the Danish National Sub-Commission. According to the chairman of the organizing committee for the Aarhus Seminar, Prof. S.  Bundgaard, three of the four days will be devoted to the discussion of school geometry, which is certainly one of the most difficult and controversial topics at the secondary level. It is proposed to invite five speakers at $ 100 apiece, to  B. Eckmann, H. Hopf, M. Morse, G. Choquet were members of the Executive Committee of IMU in the period 1959–1962. 194  In addition to the address of the recipient, the sender’s address also appears on the envelope: Prof. B.  Eckmann, Secretary International Mathematical Union, Eid. Technische Hochschule, Zurich, SWITZERLAND; M.  H. Stone, 303 Eckhart Hall, University of Chicago, Chicago 37, Ill., USA. 195  The symposium in Belgrade (19–24 September 1960) dealt with The Co-ordination of the Teaching of Mathematics and Physics. 193

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subsidize four of the members of the organizing committee (namely, Behnke, Piene, Servais, Straszewicz, the others being Bundgaard, Kemeny [U.S.A.] and Rindung)196 to the extent of $100 apiece, and to hold $100 at the disposal of the organizing committee. It seems to me that this break-down of the seminar budget does not give the organizing committee very much freedom. If the IMU could add a little to the budget the meeting might be considerably improved at small additional expense. I hope the door can be kept open for such a grant, in case Prof. Bundgaard’s final budget shows a clear need for it. He will be in touch with you about the payments to be made out of ICMI’s poor appropriation for the seminar, in any case I see no reason why the Belgrade seminar should receive additional support from ICMI, beyond // that already provided. Only travel expenses to and from Yugoslavia need to be provided and the ICMI grant will take care of most of the European invitees not otherwise provided for. There seems to be a good chance that OEEC will give some support to this meeting because of initiatives I have taken, but the modalities have still to be settled. For 1961 we contemplate two possible meetings. Professor A.D. Alexandrov197 has suggested the possibility of a meeting in the USSR with interested Russian organizations in the spring (June?). This of course is strictly confidential at this time and should not go outside the Ex. Comms. of IMU and ICMI – and here we are exploring a North American – South American meeting in the latter part of the year. It is too early to be able to propose budgets, but both meetings should call forth quite an effort. Sincerely yours Marshall Stone Pres. ICMI Marshall Stone to Beno Eckmann, Paris, 22 November 1960 IA 14A 1958–1960 Handwritten letter Morgan Guaranty Trust Company 14, Place Vendôme, Paris November 22, 1960 Dear professor Eckmann, your letter of 25 – X – 60 just missed me in Athens, apparently, and did not reach me till a few days ago. I am glad to have this information about next year’s budget,

 Kay W.K. Piene (1904–1968) was a member of the Executive Committee of ICMI in 1955–1958; Willy Servais (1913–1979) was secretary of the Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques (CIEAEM); Stefan Straszewicz (1889–1983) was vice-president of ICMI in 1963–1966; János György Kemény (1926–1992), American mathematician of Hungarian origin, was a computer scientist and an educator; Ole Rindung (1921–1984), lector at Virum in Denmark and mathematical specialist at the Ministry of Education, was committed in the modernization of mathematics teaching in Danish schools. 197  Alexandr Danilovich Aleksandrov (1912–1999) was a member of the Executive Committee of ICMI in 1959–1962. 196

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about the reactivation of the Swiss Committee, and about the Swiss interest in a meeting of ICMI next year. The question of meetings of ICMI next year is something that ICMI itself will have to decide. Up till now the plan has been to have only two meetings, one in America, one in Europe. As you know a strong (but not widely advertised) effort is being made to have the European meeting in USSR. If there is a meeting there, we shall have to spend all the funds now in sight in order to secure an adequate representation there. If there is no Russian meeting in 1961 the Italians have already invited ICMI to meet with UMI in Firenze – Bologna next September.198 A separated (or a combined) meeting in Switzerland would certainly be a possibility, if funds are adequate. However, I submit that $2000 to cover meetings of ICMI as a whole and of the Executive Committee of ICMI separately is barely enough to allow for one of each. Bundgaard’s experience with the Aarhus meeting shows what the difficulty is. He spent about $1200; I believe. There was no quorum of the EC (ICMI) there, and if it had been desired to secure one under the most favorable conditions it would have been necessary to bring two or three additional people from European points. The balance which would have remained in that case would have been barely sufficient to arrange a single independent meeting of the EC (ICMI). Furthermore, the Aarhus meeting while very successful, was small and not as representative as it might have been (though far better in this respect than used to be the case). Of course, it is often possible to take advantage of special circumstances so as to reduce the cost of a meeting (e.g. in Belgrade we had a good deal of indirect help from OEEC,199 which would not necessarily be available under other circumstances). Also, Professors Morse200 and Fehr, in raising money for ICMI’s American meeting, will try to interest some of the foundations in support for other ICMI activities; and we can hope for continued UNESCO support as well. At the moment, no decision seems to be possible, since the possibility of a Russian meeting is still open. At the ICMI meeting in Belgrade, our business session consisted solely // in a thorough review of our program through 1962. There were no decisions to be taken, and there was no formal meeting of the EC though we had present a quorum consisting of Frostman, Kurepa, Walusinski,201 and myself. Professor Behnke was invited to the meeting but failed to attend. So, also in the case

 The meeting was held in Bologna (4–8 October 1961) and dealt with A discussion of the Aarhus and Dubrovnik reports on the teaching of geometry at the secondary level (EM s. 2, 9, 1963: 1–104). 199  The acronym OEEC stands for Organisation for European Economic Co-operation. In September 1961 the OEEC was superseded by OECD (Organisation for Economic Co-operation and Development). 200  Marston Morse (1892–1977) was an American mathematician, well known as the father of the so-called Morse theory. He was one of the two vice-presidents of IMU in 1959–1962. 201  Ð. Kurepa, G.  Walusinski and O.  Frostman, were respectively vice-president, secretary and member of the Executive Committee of ICMI in 1959–1962. 198

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of professor Alexandrov and Akizuki.202 It was clear that the Russian boycotted the Belgrade meetings as a whole, since no one came from USSR at all. On the other hand, Kurepa’s desire to invite to the Symposium some scientist from Red China was vetoed by the Jugoslav authority. At the moment therefore, I do not see any reason to call a meeting of the EC (ICMI) before I return to America on December 23 (I leave Paris for London already on December 12). In any case I expect to visit both Behnke and Frostman before I leave Paris. Once we have clear indications from UNESCO and from Russia about 1961, some decisions will have to be taken, either in a meeting or by mail. There is some prospect that I shall go to India on a 2-month educational mission in the spring, in which case I could call a meeting on my way eastward (unless my time is too short because of the scheduling of my trip) With best regards, Sincerely yours, Marshall Stone Marshall Stone to Beno Eckmann, Chicago, 17 February 1961 IA 14A 1961–1966 Handwritten air letter203 February 17, 1961 Dear Professor Eckmann, I have just received a letter from Dr. Roderick204 in which he says, inter alia: “During the meeting in Zurich, we have discussed some more questions about our future coöperation and I was very impressed by the Preliminary Plan of the IMU to undertake the study of the Mathematics Curricula. We intend to support this work financially and a contract will probably be concluded in the near future concerning the preparation of the comparative survey on mathematics teaching at university level with a group of the Union which is devoted to mathematics education. It is hoped that in the future, the group would become UNESCO’s adviser in the field of mathematics teaching”. I thought that I.C.M.I was designated as I.M.U.’s organ for handling all matters in the field of mathematical education. In any case this seems to me to be a matter on which the President of I.C.M. I should have been consulted. No doubt if these plans referred to by Dr. Roderick are still under discussion only inside the Executive Committee of I.M.U. there is still time to consider what their proper relation to the work of I.C.M.I may be.

 Yasuo Akizuki (1902–1984) was member of the Executive Committee  of ICMI from 1959 to 1966. 203  In addition to the address of the recipient, the sender’s address also appears on the envelope: Prof. B. Eckmann, Eid. Technische Hochschule, Zurich, Switzerland; M. H. Stone, 303 Eckhart Hall, University of Chicago, Chicago 37, Illinois, USA. 204  Hilliard Roderick was the deputy-director of UNESCO Natural Sciences Department. 202

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I should point out, however, that an attempt to separate the problems of secondary instruction and university instruction is completely impossible. Furthermore // the setting up of two bodies under I.M.U. designed to deal, even in cooperation, as independent agents respectively restricted to the secondary field and the university field is more likely than not to destroy I.C.M.I. Sincerely yours Marshall H. Stone President, I.C.M.I. Marshall Stone to Beno Eckmann, Chicago, 23 June 1961 IA 14A 1961–1966 Handwritten air letter205 June 23, 1961 Dear Professor Eckmann, Since my most recent remarks to you on the Cuban situation I have obtained further information, indirectly, from Professor Mario Gonzalez.206 The latter has left Cuba as a “voluntary” exile, as I previously informed you, but he reports that Dr. Gutierrez retains his post (as of May 1961) and can be addressed as follows: Dr. Lino Gutierrez Novos Facultad de Ciencias Unversidad de la Habana Habana, Cuba. This means that except possibly for a replacement for Prof. Gonzalez on the Cuban National Committee for Mathematics, the relations between Cuba and IMU are in order. In particular the OC of IACME207 will presumably invite Dr. Gutierrez to participate in the Bogotá meeting.208 In addition, there will be an invitation to the Cuban Government as to all American governments, to send an (…) at its own expense. How do you think the proposals of Prof. Kovda209 in his letter to you of June 15 (he sent me a copy) should be handled? I have been thinking for some time about how ICMI should approach the question of the connections between the teaching of mathematics and the teaching of physics (or even of science in general), but I feel  In addition to the address of the recipient, the sender’s address also appears on the envelope: Prof. B. Eckmann, Secretary IMU, Eid. Technische Hochschule, Zurich, Switzerland; M. H. Stone, 303 Eckhart Hall, University of Chicago, Chicago 37, Illinois, USA. 206  Mario Octavio González (1913–1999) was a Cuban mathematician and mathematics educator. 207  The acronym IACME stands for Inter-American Committee on Mathematical Education. 208  The meeting was held in Bogotá (Colombia) from 4 to 9 December 1961, see Internationale Mathematische Nachrichten, 71, 1962: 2. 209  Victor A. Kovda (1904–1991) was the director of the UNESCO Department of Sciences from January 1959 to December 1964. 205

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very strongly (as I said at Royaumont210 and in another way at Belgrade211) that the mathematicians must first put own house in order. Only after we see quite clearly what mathematics should be taught and how it should be arranged // can we wisely begin to discuss the coördination of our teaching with that of physicists and other scientists. Consequently, the activities of UNESCO as described by Prof. Kovda strike me as being a little premature, at least from the point of view of a mathematician. It is not good to be drawn prematurely into a situation which might easily involve us in professional commitments of one kind or another. Hence, I regret that we were not consulted earlier as to our views and I now feel that the most we should do212 at this stage is to try to provide a consultant so chosen that he can collaborate effectively with the representatives from the five (unnamed) states involved in the UNESCO project. Would you kindly inform me of the accounting status of the money provided in 1959 by OEEC for ICMI, as asked in my earlier correspondence? Sincerely yours Marshall Stone. Pres. ICMI Marshall Stone to Beno Eckmann, Chicago, 22 July 1961 IA 14A 1961–1966 Handwritten letter 303 Eckhart Hall The University of Chicago Chicago 37 – Illinois Department of Mathematics July 22, 1961 Dear Professor Eckmann. Thank you very much for your letter of July 11, with the answers to my various questions. I quite agree that we should be as helpful as possible to UNESCO, though not to the extent of sacrificing our own best interests. In the matter of the relation between mathematics and physics we have to be cautious. In particular I think that if we feel UNESCO’s actions in the matter to be premature, we should not hesitate to express our opinion. It would also be well for us to make sure that whatever UNESCO does, is not at cross-purposes with what OEEC has already done and is continuing to do. I am glad to note that the thinking in the Executive Committee about UNESCO contract tends to the conclusion that “most of the tasks contained in the ‘contract’ be given to ICMI”. I have expressed myself quite forcefully to several members of  Stone is referring to his introductory address during the seminar held at the Centre Culturel de Royaumont, Asnière-sur-Oise (France) (23 November–4 December 1959), on new thinking in mathematics, new thinking in mathematical education, and implementation of reform. See in this volume Chap. 2.§  2.5 by Fulvia Furinghetti and Livia Giacardi. 211  See footnote 195. 212  Del. now. 210

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the Executive Committee about the dangers of having two separate bodies under IMU dealing with educational problems, and some are certainly fairly sympathetic with my position. It is quite appropriate at this stage that the details of the contract arrangements be studied by a special sub-committee of the EC, and I would be glad to serve on such a sub-committee. (I would attend a meeting in Princeton on October 4 next. I assume my travel expenses would be covered). I would think, however, that the sub-committee ought to go out of existence once the UNESCO contract had been successfully negotiated. I have two financial questions to raise. First, // I was surprised to learn that the amount of money made available from the UNESCO grant for IACME was less (by $700) than the sum of $ 4000 which had been promised by Dr. Roderick and by IMU (after it was decided that part of the grant would go to another purpose). May I please have an explanation? Secondly, can you inform me whether the amount of $700 appropriated by the EC of ICMI for the Lausanne Seminar213 was used up as intended? I hear that the Lausanne Seminar went off very well, but I am still in the dark as to whether any arrangements have been made to publish the proceeding, or part of them at least. Professor Rueff214 has been an exceedingly poor correspondent, as you know (I am not referring to the loss of one letter, which was not his fault, nor mine) and has in particular never responded to my inquiries with regard to publication plans. The plans for the Bologna Seminar215 now seem to be developing fast. I shall ask the EC of ICMI to authorize expenditure of the balance of our funds for 1961 for this Seminar (we already appropriated $ 1000). What we shall actually spend will depend on how our arrangements with speakers work out – but even if some of our speakers are supported financially by other organizations, we could use our resources to bring some other mathematicians as active but “non-speaking” participants. I do not think I shall call a meeting of the EC at Bologna, as all our plans for 1961–1962 are quite well fixed and any plans for later on might to be left to the new ICMI instead of being imposed by the present one. With best regards, Sincerely yours Marshall Stone, President, ICMI N.B. Address Aug.10–Oct.4 – c/o Morgan-Guaranty Trust Co., 14 Place Vendôme, Paris I, France

 Stone is referring to the Seminar on the teaching of analysis and relative manuals, sponsored by ICMI and the Swiss Mathematical Society held in Lausanne from 26 to 29 June 1961 (EM s. 2, 6, 1960: 311). 214  Marcel Rueff (1910–1997) was the delegate in the ICMI of the Swiss subcommission. 215  See footnote 198. The conference papers were published in EM s. 2, 9, 1963: 1–104. 213

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Beno Eckmann to Marshall Stone, Zurich, 10 August 1961 IA 14A 1961–1966 Typewritten letter, copy Professor Marshall H. Stone c/o Morgan-Guaranty Trust Co. 14 Place Vendôme Paris 1e, France Zurich, August 10, 1961 Dear Professor Stone, thank you very much for your letter of July 22. May I first answer your two financial questions: a) IACME. In the contract between UNESCO and IMU, of which you received a photostat, it was explicitly stated that the sum of $4000 – would be paid in two parts, the first being $3800 – to be paid immediately after signing the contract and the second being $200 –, to be paid after the final report has been submitted to UNESCO. b) ICMI Seminar Lausanne. The amount of $700 – has been used by the Seminar as intended. There was a contribution of the Swiss Mathematical Society in the amount of Swiss Francs 4000 –; furthermore, the Swiss National Sub-­Commission of ICMI added a small sum. You will get a copy of the financial report. The Plans for the Bologna Seminar seem to develop very well. Hopf told me that he plans to participate in the meeting and that you got first rate speakers for that occasion. I am very glad that you agree to participate in the Sub-Committee of our EC which will prepare the cooperation between UNESCO and IMU in the field of mathematics teaching at University level. In connection with the Princeton meeting on October 14, your travel expenses will be covered. So far, the following people have been nominated as members of the Committee: President Nevanlinna ex officio, Chandrasekharan, Kuratowski216 and de Rham217 (who will replace Hopf and Choquet in the Princeton EC Meeting). The EC of IMU will meet on October 12 and 13 and take final decision on the Status of the Sub-Committee. The Committee may very well be considered later on as part of ICMI; I agree with your viewpoint that it would be wrong to separate its task from those of ICMI. However, this new activity was not foreseen by the last General Assembly of IMU.  In view of the importance of the new task and of the difficulties that you have already mentioned in your letter, it seems appropriate that the whole matter be handled very carefully by the Special Committee. It will then be up to the next General Assembly of IMU

 At that time Komaravolu Chandrasekharan and Kazimierz Kuratowski (1896–1980) were members of the Executive Committee of IMU. 217  Georges de Rham (1903–1990) will become president of IMU in 1963. 216

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to decide whether this Committee shall continue its work and what its composition for the following four years’ term should be. With best regards, Yours sincerely Georges de Rham to Komaravolu Chandrasekharan, Lausanne, 12 February 1963 Fonds Georges de Rham, Université de Lausanne (Fonds de Rham from now on), 5113–403 Typewritten draft of letter Lausanne, 7 av. Bergières le 12 février 1963 Mon cher Chandra,218 Je reçois votre lettre du 6 février. J’espère que vous avec reçu maintenant ma lettre du 10 contenant les reçus signés, avec d’autres documents. Pardonnez-moi le retard! Pour notre prochaine réunion, je n’ai pas reçu d’autre réponse que celle de Hirzebruch.219 A ce propos, auriez-vous l’obligeance de me faire envoyer la liste des membres de notre comité avec les adresses où il faut leur écrire. Je puis trouver la plupart dans le “World Directory”, mais celle de Lavrentieff220 par exemple n’y est pas. J’aimerais leur écrire un mot personnel, pour les encourager à venir, du moins ceux qui le peuvent. Et pour Lavrentieff, il faut que je prévienne notre service diplomatique afin qu’on lui accorde rapidement le visa. Je reçois un mot de Lichnerowicz221 qui a pensé pour son Comité exécutif, à Kolmogoroff et Moise222 comme vice-présidents, Akizuki, Freudenthal et Behnke223 comme membres, et Karamata comme observateur. Pour secrétaire, il aimerait un Suisse français professeur de Lycée et il nomme M. Delessert,224 que je connais fort bien. J’en parlerai encore à Karamata. Pour les vice-présidents et les trois membres, son choix me semble bon.

 George de Rham and Komaravolu Chandrasekharan were respectively president and secretary of IMU in the period 1963–1966. 219  Friedrich Hirzebruch (1927–2012) was a member of the Executive Committee of IMU in the period 1963–1966. 220  Michail Alekseevič Lavrent’ev (1900–1980) was a member of the Executive Committee of IMU in the period 1963–1966. 221  André Lichnerowicz (1915–1998) was president of ICMI in the period 1963–1966. 222  Only Edwin Moise (1918–1998) would be vice-president of ICMI together with Stefan Straszewicz (1889–1983) in the years 1963–1966. 223  Y. Akizuki, H. Freudenthal, and H. Behnke would be members of the Executive Committee of ICMI in the years 1963–1966. 224  At that time Jovan Karamata (1902–1967) was one of the directors of the journal L’Enseignement Mathématique together with Jean Favard and François Châtelet. 218

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Le Secrétariat de ICSU m’envoie son Bulletin d’information no. 4, de janvier 1963. Dites-moi si vous désirez que je vous l’envoie. Il ne contient pas le texte de la résolution mentionné dans la lettre que je vous ai envoyée. Avec mes meilleures salutations, G. de Rham Komaravolu Chandrasekharan to Georges de Rham, Bombay, 18 February 1963 Fonds de Rham, 5113–403 Typewritten letter with signature International Mathematical Union Tata Institute of Fundamental Research Colaba, Bombay 5, India Telegrams: ZETESIS Telephone: 213181

REF: IMU-GR: 33/63 18 February 1963

My dear de Rham, thank you for your letter of 12 February 1963. I send herewith a list of addresses of all members of the Executive Committee. I am sorry, I did not do so earlier. Copies are also being sent to all the other members. Yes, I also received word from Lichnerowicz about the proposed new Executive Committee of ICMI. His suggestion seems to me fine, except for one small difficulty. M. Delessert225 is not a member of ICMI, either elected or co-opted. I, therefore, suggested that Karamata be made the Secretary, and Delessert associated unofficially with the work of the Secretary. Lichnerowicz himself had mentioned this as a possibility. I hope it will work out all right. I have received a reply from South Africa to the letter I wrote to them at the instance of the EC (Princeton meeting). I shall, of course, include it in the agenda papers for the April meeting, which I hope to mail out in the next couple of days. Please do not bother to send me the ICSU Bulletin. I have a copy already. With warmest personal regards, Cordially yours, Chandra Professor G. de Rham 7 avenue Bergière Lausanne Switzerland

 André Delessert (1923–2010), Swiss mathematician and renowned sculptor, is the future secretary of ICMI. 225

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Otto Frostman226 to André Delessert, Djursholm, 14 June 1965 Fondo André Delessert, Biblioteca Speciale di Matematica “Giuseppe Peano”, Torino (Fondo Delessert from now on), 1965, Serie III, u. a. 177 Typewritten letter with signature Institut Mittag-Leffler Djursholm Suède 14.6.1965 Monsieur le Professeur A. Delessert Riex, Suisse Cher Monsieur Delessert, Vous m’avez communiqué d’une réunion officieuse de certains membres (qui ?) du Comité Exécutif de la CIEM à Echternach,227 à laquelle on a fait quelques propositions quant aux sujets des rapports et aux rapporteurs au congrès à Moscou 1966,228 et vous me les soumettez pour approbation. Pour une décision d’une telle importance, j’aurais préféré une réunion officielle de la Commission, ou au moins du Comité Exécutif, pour que les membres auraient l’occasion de discuter des propositions présentées à Paris en février 1964. Est-ce qu’on peut dire quelque chose non-­ banal du troisième sujet ? Quant au premier sujet je voudrais remarquer que dans la Suède nous aurons bientôt un problème beaucoup plus important: la nécessité ou non de cours universitaires particuliers pour les futurs professeurs de l’enseignement secondaire. Moi, je ne suis pas sûr que les sujets proposés sont les meilleurs possibles, mais, vu le temps avancé, il faut peut-être les accepter sans discussion ultérieure. Je vous présente, cher Monsieur Delessert, mes salutations empressées. Otto Frostman Komaravolu Chandrasekharan to Børge Jessen, n. p., 25 May 1965 Fonds de Rham, 5113–403 Typewritten letter with handwritten additions229 COPY Ref: IMU-3/2/65  Otto Frostman (1907–1977), member of the EC of ICMI from 1959 to 1962 and future secretary of IMU in the years 1967–1970. 227  The ICMI meeting was held in Echternach on the occasion of the Symposium on The repercussions of mathematics research and teaching (30 May–4 June 1965). 228  Frostman is referring to the International Congress of Mathematicians which would be held in Moscow (16–26 August 1966). The three topics proposed by ICMI for this ICM were: the university education of future physicists, the axiomatic method in secondary teaching, and the development of students’ mathematical activities and the role of problems. The rapporteurs were respectively Charles Pisot, Hans Georg Steiner, and Anna Zofia Krygowska. 229  The handwritten additions appear in italics. 226

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25 May 1965 Dear Professor Jessen, The Executive Committee of IMU is thinking of possible nominees for the position of President of the International Commission on Mathematical Instruction for the four-year period beginning January 1, 1967. The actual nomination will have to be made by the present President of the Union, Professor G. de Rham, and approved by the General Assembly of the Union due to be held in Moscow in August 1966. Several of us would like you to be President because of your high standing as a research mathematician, coupled with a keen interest in mathematical education and a wide experience of administrative affairs. I should be grateful if you would be prepared to accept the nomination, and to serve as President of ICMI, if elected. I need hardly add that this [is] a personal and confidential letter, and that in my opinion you would be doing a great // service to the Union, of which you were Foundation-­ Secretary, if you agreed to serve it in a new capacity. With kind regards, Sincerely yours, 230

professor K. Chandrasekharan Secretary International Mathematical Union Professor Børge Jessen H.C. Andersens Boulevard 35 Copenhagen V. Denmark Copy to: Prof. G. de Rham Lausanne KCrr 25/5/65 Hans Freudenthal to the National Subcommissions, Utrecht, 15 February 1967 IA, 14B 1967–1974 Typewritten letter with signature Commission Internationale de l’Enseignement Mathématique International Commission on Mathematical Instruction. _______________________________________________________________ February 15, 1967 To the National Subcommissions of the International Commission on Mathematical Instruction.

 Børge Jessen (1907–1993) was a Danish mathematician, who had been secretary of the Interim Executive Committee of IMU from 1950 to 1952. 230

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1. At the Fifth General Assembly of the Int. Math. Union at Dubna, Aug. 13–16, 1966, ten members at large of the ICMI have been appointed for the period 1967–1970: H. Freudenthal (President) (see below) Y. Akizuki, 418 Higashi-Koiso, Ōiso, Kanagawa, Japan A. Lichnerowicz, 6 ave Paul Appell, Paris 14, France E.E. Moise, Longfellow Hall, Harvard Univ. Cambridge (Mass) 02138 USA J. Novak,231 Math. Inst. Acad. Zitná 25, Praha 1, Czechoslovakia G. Papy,232 57 av. de l’Université, Bruxelles 5, Belgium A. Revuz, 16 Rue de Rome, Les-Essarts-de Roi (S&O) France S. Sobolev,233 Inst. Math. Sibirskogo Otdelenia AN, SSSR, Novosibirsk 90, USSR H.G. Steiner,234 Math. Inst. Schlossplatz 2, 4400 Münster i.W. Germany B. Thwaites, Dept. of Math. University of Southampton, England 2. The names of the national nominees (as far as received till now) are: Allemagne - H. Behnke, 44 Münster i.W. Schlossplatz 2 Argentine -… Australie -… Autriche  - E.  Hlawka, Math. Inst. Universität, Wien A-1090 Strudlhofgasse 4 Belgique -… Brésil - L. Nachbin, University, Rio de Janeiro Bulgarie -… Canada  - A.L.  Dulmage, University of Manitoba, Winnipeg 19 Manitoba Danmark -… Espagne - P. Abellanas, Secc. de Mat. Université de Madrid. Finlande - Y. Juve, Korppaantie 3 E 28, Helsinki 30 France - M. Glaymann, 14 rue de Chavril, 69 Sainte-Foy Grande-Bretagne - E.A. Maxwell, Queen’s Coll. Cambridge Grèce -…

 Josef Novák (1905–1999) was a Czech mathematician, internationally recognized for his chairmanship of the first five Prague Topological Symposia (1961–1981). 232  Georges Papy (1920–2011) was a Belgian mathematician well known for his contributions to the renewal of mathematics teaching. Together with his wife Frédérique, he contributed to the foundation of the Institute for Mathematics and Computer Science which played an important role in the modernization of school curricula of mathematics. 233  Sergej L’vovič Sobolev (1908–1989) was a Soviet mathematician, well known for the so-called Sobolev spaces. He was vice-president of ICMI from 1967 to 1970, and member from 1971 to 1974. 234  Hans-Georg Steiner (1928–2004), was one of the pioneers of the modern development of mathematics education as a scientific discipline. In 1973, together Heinrich Bauersfeld and Michael Otte, he was one of the founders in Bielefeld of the Institut für Didaktik der Mathematik. He was vice-president of ICMI from 1975 to 1978. 231

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Hollande - P.G.J. Vredenduin, Kneppelhoutweg 12, Oosterbeek Hongrie - J. Surányi, Math. Inst. de l’Université, Budapest 8 Indes  - K.  Balangangadharan, Tata Inst. of Fund. Res. Colaba, Bombay 5 Irlande - M.D. McCarthy, Royal Irish Academy, Dublin Israël - S.A. Amitsur, Dept. of Math. Hebrew University, Jerusalem Italie - L. Campedelli, Via Crimea 6, Firenze // Japon - S. Iyanaga, Université de Tokyo, Dept. Math. Bunkyo ku, Tokyo Luxembourg - L. Kieffer, 1 rue Jean Jaurès, Luxembourg Norvège - K. Piene, Wergelandsv 15, Oslo 1 Pakistan - M. Raziuddin Siddiqi, University of Islamabad, Rawalpindi Pologne - S. Straszewicz, Inst. Mat. P.A.N. ul. Sniadeckich 8, Varsovie Portugal - J. Sebastiao e Silvia, Rua Fernam Gomes 17 (Restelo) Lisboa 3 Sénégal - S. Niang, Université de Dakar, Fac. d. Sciences, Dakar Suède - A. Pleijel235 (Univ. de Lund): Dept. of Math. Haile Selassie I University, Addis Abbeba, Ethiopia Suisse - M. Rueff, École polytechnique fédérale, Zürich URSS -… USA - L. Gillman, University of Rochester, N.Y. Yougoslavie - … 3. Acting according to the regulations of the ICMI sub b and c,236 I propose you the following composition of the Executive Committee Vice-presidents: E. Moise S. Sobolev Secretary: A. Delessert Members: H. Behnke A. Revuz B. Thwaites If no objection will be received before March 20, I suppose you approve this proposal. 4. Since no plenary meeting of ICMI has been held for many years, I think it would be helpful to discuss ICMI’s activity anew. I propose you to have a ICMI meeting this summer, at the end (presumably from August 21 to 25). 5. Meanwhile I would appreciate to learn about your personal ideas on the future activity of ICMI.

 Åke Pleijel (1913–1989) was a Swedish mathematician who became a professor at the Royal Technical University in Stockholm in 1949, then from 1953 he taught at the Lund University and finally from 1967 at the Uppsala University. In the last years in Lund, he was on leave to contribute to the development of higher education in mathematics in Ethiopia. He returned to Addis Ababa for two years after his retirement from Uppsala. 236  See the Terms of Reference for ICMI of 1960: “(b) The officers of the Commission shall consist of a President, a Secretary, and two Vice-Presidents. The President shall be elected by the General Assembly on the nomination of the President of IMU from the members-at-large of the Commission; (c) The Executive Committee of the Commission shall consist of the officers of the Commission together with three additional members elected by the membership of the Commission”. See Chap. 7 by Livia Giacardi in this volume. 235

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Sincerely yours, H. Freudenthal (Prof. dr. H. Freudenthal) Matematisch Instituut der Rijksuniversiteit, Boothstraat 1c, Utrecht The Netherlands Hans Freudenthal to André Delessert,237 Utrecht, 22 March 1967 Fondo Delessert, 1967, Serie V, u. a. 347 Typewritten letter with signature Mathematisch Instituut Der rijksuniversiteit te utrecht Utrecht, le 22 mars 1967 Boothstraat 1c, 6 en 17 Tel. (030) 1 1965 M. le Professeur A. Delessert 1097 Riex (Vaud) Zwitserland Cher Delessert, Je ne suis pas sûr si les règlements admettent vraiment que vous êtes coopté au Bureau de la CIEM. En tous cas, il n’y avait pas d’objections. A l’avenir je me servirai bien de votre aide pour communiquer avec CIEM. D’autre part, quant à la communication interne du bureau, il me semble mieux que ses membres se considèrent l’un l’autre comme des égaux et que chacun communique avec l’ensemble des autres sans aucune formalité. Recevez, cher Collègue, mes salutations les meilleures. H. Freudenthal Otto Frostman to the International Commission on Mathematical Instruction, Djursholm, 29 June 1967 IA 14B, 1967–1974 Typewritten letter, copy To the International Commission on Mathematical Instruction, At the meeting of the Executive Committee of IMU in Oxford, June 12–13, 1967, the question of a Permanent Secretariat for ICMI was discussed, and the following resolution was unanimously adopted:

 Hans Freudenthal (1905–1990) was president of ICMI in the years 1967–1970 and André Delessert was the secretary. 237

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Pursuant to Resolution No. 4 of the Record of the Fifth General Assembly of IMU,238 the Executive Committee of IMU at its meeting in Oxford on June 12–13, 1967 was of the opinion that with the financial resources at the disposal of the Union at present, which could be utilized for purely administrative purposes, it would not be able to give any substantial financial support for such a project. Nor was the Committee convinced of the urgent necessity for such a secretariat. It would be glad, however, to know from the Executive Committee of ICMI whether it considers this question urgent, whether it has any plans ready for setting up such a secretariat, what its exact functions would be, and what the financial implications of this project would be. As to the due to IUCTS,239 $ 200 per annum, the Executive Committee expressed the opinion that these should be paid through ICMI. Referring to the By-Laws of the Union, IV, 10,240 the Committee also requested ICMI to give to IMU an audited financial statement, showing totals of income and expenditure, before the dues to IUCTS for 1966 and 1967 were refunded by IMU. Djursholm, June 29, 1967 /Otto Frostman/ [Otto Frostman]241 to Hans Freudenthal, n. p., 2 December 1967 Typewritten draft IA, 14B 1967–1974 December 2, 1967 Professor H. Freudentahl Mathematisch Instituut Utrecht Dear Freudenthal, According to the Terms of Reference of ICMI (Paris, April 1960)242, (h) and (i), I should have a report for 1967 and the budget for 1968 to be presented to the

 The Fifth General Assembly of IMU was held in Dubna, a small town on the river Volga, a hundred kilometers to the north of Moscow on 13–16 August 1966, see (Lehto 1998, § 8.1). 239  Frostman writes IUCTS instead of IUCST. IUCST, that is Inter-Union Commission on Science Teaching (in French CIES, Commission Interunions de l’Enseignement des Sciences), was established in September 1961 by ICSU. In 1968 ICSU decided to dissolve the IUCST and create a Committee on the Teaching of Science (CTS). (Baker 1986, p. 16). 240  The article IV of the Statutes of IMU concerns “Finance”. 241  See the answer by Freudenthal in the following letter. 242  The points (h) and (i) state that: 238

“(h) The budget of the Commission shall be submitted to the Executive Committee of IMU and the General Assembly, for approval, at such times as may be determined by agreement between the Commission and the Executive Committee of IMU. (i) The Commission shall file annual report of its activities with the Executive Committee of IMU, and shall file a quadrennial report at each regular meeting of the General Assembly”. See Chap. 7 by Livia Giacardi in this volume.

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Executive Committee of IMU. Would it be convenient for you to send these things to me before March 1, 1968? I think I should also have a specification of the use of the $ 2.000 that you received for the Symposium in Utrecht (I don’t think I have got such a report and I might need it early in January 1968). I must admit that I am not too happy about the new pedagogical journal.243 Do you really think that there is a market for two international journals of that kind (I do not)? If you are not satisfied with L’Enseignement, ICMI’s official journal, perhaps it would be better to try to reform it. And I am afraid too that in a new journal the “modernizers” of the extreme sort would try to be very busy. At least, I ask you to be cautious. I can agree with very much of your criticism of the meetings of ICMI at the International Congresses, but I am not sure that ICMI should isolate itself from those who have, primarily, a scientific interest but who have, nevertheless, very often taken part in the discussions of ICMI. And a special ICMI Congress in France in 1969244 will cost a lot of money. In 1969 IMU has already engaged itself rather heavily in an International Conference on Analysis or Functional Analysis in Tokyo. Yours sincerely, Copy to Cartan. Hans Freudenthal to Otto Frostman, Utrecht, 20 December 1967 IA, 14B 1967–1974 Typewritten letter with signature Utrecht, December 20, 1967 Professor O. Frostman International Mathematical Union Djursholm – Zweden Dear Frostman, Owing to a long illness first and a number of travels afterwards, I did not write earlier. As you see, I have used only a small part of the subvention. I got a number of other subventions. Do you wish the remainder returned or may it be cleared with the subvention ICMI receives for administrative use? I would like to reassure you about the new pedagogical journal. The provisional list of editors does not include any “radical”. In spite of its name, Enseignement245

 Frostman is referring to Educational Studies in Mathematics.  Frostman is referring to the First International Congress on Mathematical Education (ICME-1) which would be held in Lyon from 24 to 30 August 1969. 245  Freudenthal is referring to L’Enseignement Mathématique.

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has never been a pedagogical journal. Its contributions on education were not pedagogical but organisatory and administrative. I do not believe it is possible to reorganize a journal so fundamentally. I would also like to reassure you about the 1969 congress. It would be easier to deal with these questions orally than by letter. Sincerely yours, H. Freudenthal André Delessert to Hans Freudenthal, Riex, 24 December 1967 Fondo Delessert, 1967, Serie V, u. a. 349 Typewritten draft Riex, le 24 décembre 1967 Monsieur le Professeur H. Freudenthal Président de l’ICMI 1c, Boothstraat Utrecht, Pays-Bas Cher Monsieur, Votre lettre du 4 décembre m’est bien parvenue. Je vous en remercie très vivement. Je me permets de vous faire part d’une remarque touchant la création de la nouvelle revue dont vous parlez. La revue que vous désirez créer et l’Enseignement Mathématique ne visent pas les mêmes buts. Toutefois, je regretterais beaucoup que l’on décide, tacitement, de ne plus considérer l’Enseignement Mathématique comme l’organe officiel de l’ICMI. Le contrat à conclure avec M. Reidel246 contient-il une clause à ce sujet ? Permettez-moi d’aborder encore un autre point. J’ai rencontré récemment M. Henri Cartan,247 président de l’UMI. Il m’a dit n’avoir aucune information sur notre commission. Je lui ai envoyé le compte-rendu de la séance d’Utrecht et je pense qu’il serait bon de le tenir au courant des décisions liant l’avenir de l’ICMI. Je vous présente tous mes souhaits pour les fêtes de l’an et pour 1968 et je vous prie de recevoir, cher Monsieur, mes salutations très dévoués. A. Delessert 1097 Riex (Vaud) Suisse Georges de Rham to André Delessert, Lausanne, 24 December 1967 Fondo Delessert, 1967, Serie VI, u. a. 503 Handwritten card

 Delessert is referring to the contract for the publication of the new journal Educational Studies in Mathematics with D. Reidel Publishing Company in Dordrecht. See Chap. 2, § 2.7 by Fulvia Furinghetti and Livia Giacardi, in this volume. 247  Henri Cartan (1904–2008) was the president of IMU from 1967 to 1970 and ex-officio member of ICMI in the same period. 246

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Lausanne, le 24-12-67 Cher Monsieur, Je vous envoie une copie de cette lettre, pensant qu’elle peut vous intéresser. Je n’ai bien entendu, comme vous savez, nulle intention de faire obstruction à un nouveau périodique orienté comme vous m’avez dit. Mais je suis surpris que M. Châtelet248 n’ait pas été informé. Bien cordialement G. de Rham249 // 250 t.s.v.p. Naturellement, si vous avez des précisions sur le contrat envisagé, cela me serait agréable d’en avoir connaissance. [Otto Frostman] to Hans Freudenthal, n.p., 2 January 1968 IA, 14B 1967–1974 Typewritten draft Professor Dr. H. Freudentahl Mathematisch Instituut Utrecht January 2, 1968 Dear Freudenthal, Many thanks for your letter of December 20, 1967; I sincerely hope that you have recovered from your illness. From your preliminary report on the Colloquium at Utrecht I see that you have used only about $700 of the $2.000 given by IMU. Very strictly, the rest should be reimbursed, but it is not practical and would only mean a loss of money at the exchange. So, I agree that you keep the rest on ICMI’s account for future use and for the payments to CIES ($400 for 1967 and 1968). Moreover, as I think I have told you in a letter, in the budget of IMU there is a reservation of $300 a year for secretarial help. If you have had any expenses to that purpose, I should welcome a statement. As to the plans for a new pedagogical journal, I am still a bit afraid that the market will be hard for two publications, even if the new journal will mainly stress other points than L’Enseignement. In any case, I suppose you will discuss the matter with the publishers of L’Enseignement, just to avoid duplications. With my best wishes for the new year, Yours sincerely /Otto Frostman/  François Châtelet (1912–1987) was one of the directors of L’Enseignement Mathématique.  Georges de Rham (1903–1990) was the outgoing president of IMU (1963–1966) and, after the death of Jovan Karamata, he became one of the directors of L’Enseignement Mathématique together with François Châtelet and Raghavan Narasimhan (1937–2015). 250  The acronym means: “tournez s’il vous plait” that means “please turn the page”. 248 249

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Copies to Cartan and Delessert Georges Laclavère251 to Otto Frostman, Paris, 23 August 1968 IA 14B, 1967–1974 Typewritten letter with handwritten notes and signature252 Conseil International des Unions Scientifiques  – International Council of Scientific Unions Secrétariat: 7 via Cornelio Celso 00161 Rome (Italie) Tél.: 862555 et 863445

Président: Dr. J. M. Harrison (Canada) Secrétaire Général: Prof K. Chandrasekharan (India) Trésorier: Ing. Gén. G.R. Laclavère (France) 136 bis, Rue de Grenelle – Paris 7e

Paris, 23 August 1968 Prof. O. Frostman Auravagen 21 Djursholm Suède Dear Professor Frostman, I am sorry for the delay in answering your letter of 20 June 1968 but I have been travelling in France and in Africa since early July and the high pressure of work which followed is the cause of this delay. I send you herewith a copy of the contract between Unesco and the International Commission on Mathematical Instruction which was signed on 6 March 1967.253 I wish to let you know that a number of Commissions and Committees of Scientific Unions Members of ICSU negotiate contracts directly with Unesco without reference to the mother-Union. If this procedure is not accepted by the International Mathematical Union, I feel that it is to the Union to give appropriate instructions to its Commissions and Committees. Yours sincerely, G. Laclavère André Delessert to Otto Frostman, Riex, 22 March 1969 IA, 14B 1967–1974 Typewritten letter with signature Commission Internationale de l’Enseignement Mathématique  Georges Laclavère (1906–1994) was a French engineer. He became director of the Institut Géographique National in 1963 and member of the Bureau of ICSU in 1959, and he was the treasurer from 1961 to 1968. See (Louis 2019). 252  At the bottom of the letter there is this handwritten note: “Copy of this letter and attached copies sent to Cartan, Chandrasekharan and de Rham on Aug. 26, 1968 O. F.”. 253  The contract is signed by Freudenthal, ICMI President. 251

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ICMI Riex, le 22 mars 1969 Monsieur le Professeur O. Frostman Secrétaire de l’I.M.U. Institut Mittag-Leffler Auravagen Djursholm Suède Cher Monsieur, J’ai bien reçu votre lettre du 16 mars et je vous en remercie. Je ne vous ai pas envoyé de rapport sur l’activité de la CIEM cette année. C’est une faute de ma part. Je vous en demande pardon. Voici les réponses à vos diverses questions. i) La République Démocratique d’Allemagne a été admise comme sous-­ commission nationale dès le ler janvier 1969. La Tunisie a été admise comme sous-commission spéciale de la CIEM dès le janvier 1969.254 A ce sujet, je précise que la procédure suivie a été la même que celle de l’admission du Luxembourg et du Sénégal. ii) M.  Freudenthal a bien participé à un congrès sur “Integration of Science Teaching”, à Varna (Bulgaria) du 11 au 19 septembre 1968. Ce congrès était patronné par l’“Inter-Union Commission on Science Teaching of I.C.S.U.” avec l’assistance de l’UNESCO. iii) M.  Freudenthal a demandé aux sous-commissions nationales des renseignements sur les concours mathématiques en vue d’élaborer un rapport. iv) La nouvelle revue de pédagogie mathématique est: “Educational Studies in Mathematics” Editor: H. Freudenthal Address: D. Reidel Pub. Co. Box 17, Dordrecht-Holland La parution est de 4 numéros (environ 500 pages) formant un volume annuel. Le prix du volume est $ 22.50. Le volume 1, no. 1/2 a paru en mai 1968; le no 3 a paru en janvier 1969. v) Le premier congrès international de l’enseignement mathématique se tiendra à Lyon (France) du 24 au 30 août 1969. Le Bureau est présidé par M. Freudenthal. Le secrétaire en est M.  Glaymann, 43 Boulevard du 11 novembre 1918, 69-Villeurbanne (France).255 // Je me permets d’ajouter quelques points. Une réunion de la CIEM aura lieu à Lyon, dans le cadre du congrès, le 23 août 1969, après-midi.  See Internationale Mathematische Nachrichten, 1969, 93: 5.  IMU was faced with decisions already made. See Chap. 2, § 2.7 by Fulvia Furinghetti and Livia Giacardi in this volume. 254 255

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Le président et le secrétaire de la CIEM ont participé à un colloque consacre à “L’enseignement des mathématiques dans les pays européens”, à Bucarest, du 23 septembre au 2 octobre 1968.256 Ce colloque était organisé par la Société des Sciences mathématiques de la République socialiste de Roumanie, la commission nationale roumaine pour l’Unesco et sous l’égide de l’Unesco. Une revue: “Zentralblatt für Didaktik der Mathematik”257 est en préparation. Elle est éditée en liaison avec le “Zentrum für Didaktik der Mathematik an der Universität Karlsruhe” et la CIEM. L’administrateur en est M. H. Wäsche,258 West-Universität, Bau 35, Hertzstrasse l6 7500 Karlsruhe. Je ne sais rien de plus sur sa parution. Je signale enfin que, pour l’essentiel, le travail de secrétariat de la CIEM est effectué par le secrétariat du Prof. Freudenthal. Cela explique que je ne sois pas toujours très bien informe de ce qui se prépare. J’espère avoir répondu à votre lettre. Je vous prie d’excuser les difficultés que je vous ai causées. Et je vous présente, cher Monsieur, mes salutations empressées. A. Delessert 1097 Riex (Vaud Suisse) Otto Frostman to André Delessert, Djursholm, 19 October 1969 Fondo Delessert, 1969, Serie VI, a. u. 525 Typewritten letter with signature International Mathematical Union The Secretary – Professor Otto Frostman Djursholm – Sweden October 19, 1969 Professor A. Delessert Secretary of ICMI 1097 Riex (Vaud) Dear Professor Delessert, Many thanks for your letter of October 11 with Freudenthal’s report and for the report from the Lyon Congress.259 I have made some redactional alterations, and I  See Colloque International UNESCO Modernization of Mathematics Teaching in European Countries. Bucharest, Editions didactiques et pédagogiques, 1968. 257  See the presentation of the first issue appeared in 1969: Zu den Aufgaben des Zentralblatts für Didaktik der Mathematik. ZDM Zentralblatt für Didaktik der Mathematik, 1 1969: 1. 258  Hans Wäsche (1903–1980) studied mathematics, physics and philosophy first in Marburg and then in Berlin. After taking the exams for teaching, he taught in various German secondary schools. A man with vast knowledge not only on science and philosophy, but also on anthropology and psychology, he turned his interests to mathematics education and when he retired, he accepted the management of the journal Zentralblatt für Didaktik der Mathematik. Thanks to his various scientific contacts, he soon succeeded to induce numerous well-known authors to collaborate. 259  This Congress is ICME-1 and took place in Lyon from 24 to 30 August 1969. See Chap. 3 by Marta Menghini in this volume. 256

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ask you to approve them or to make the corrections you find desirable. Though I am not quite sure, I suppose that the resolutions and recommendations adopted by the Congress in Lyon should be printed.260 Then I am not at all sure about item (v), in particular about the Conference in Carbondale. In your letter some words must be missing. What is CEMREL?261 I am also not very keen on mentioning that the President “took an active part in the work of IUCST”,262 because this Commission is dissolved in its present form (see report from the General Assembly of ICSU, Paris, 28 Sept. 1968, p. 170). Would you please return the draft of my report after possible corrections and with your comments? Cordially Otto Frostman Hans Freudenthal to Otto Frostman, n. p., 3 November 1969 Fondo Delessert, 1969, Serie VI, a. u. 525 Typewritten letter November, 3 1969 Professor O. Frostman International Mathematical Union Djursholm Sweden Dear Frostman, Delessert sent me a few questions of yours. 1) CEMREL means Central Midwestern Regional Education Laboratory. They run the Comprehensive School Mathematics Project in Carbondale, Ill.263 2) The IUCST has existed till a few days ago, when it was actually dissolved. I have worked for IUCST first as an Acting President and then as a President, for together three years, since the original president answered no letters. I have written thousands of letters for IUCST and assisted to at least twenty meetings, I have with Mr. Fleury264 who as secretary of IUCST did the greater part of the

 See Proceedings ICME-1. 1969, and Educational Studies in Mathematics 2, 1969: 135–418.  CEMREL stands for Central Midwestern Regional Education Laboratory. See the letter that follows. 262  See footnote 239 and (Lehto 1998, pp. 252–257). 263  Carbondale is a city in Jackson County, Illinois, United States. 264  Pierre Fleury (1894–1976) is a French physicist. From 1936 he held the chair of Physique générale dans ses rapports avec l’industrie at the Conservatoire national des arts et métiers. He took an active part in the scientific organization in France and abroad, and was a member of many international bodies including ICSU. 260 261

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work, prepared the Varna Congress265 of IUCST and I have presided it. I am preparing a IUCST colloquium for 1970. I remark that the idea of reorganization of IUCST has been mine. IUCST did not work properly that is, besides the secretary and me, there were almost no active people in it. From IUCST I got just the day before yesterday a kind letter by which they thank me for the work I did as a President of IUCST. You wrote to Mr. Delessert that you would prefer not to mention this activity of mine. It does not regard me. I enjoyed the work I did in IUCST and I am sure that people who saw it, did appreciate it. This is the only thing that matters. My work in IUCST has come to an end since, as it has come clear from the correspondence with ICSU, the Union has not nominated me as a member of the new Commission. It does not regard me either. If I have a few ambitions, they are of an entirely different kind. There is, however, another thing that bothers me, and since I love frankness, I will tell you about it. My activity as a president of ICMI and of IUCST has been the aim of rumors that reached me time ago. If the decision of the Union not to designate me as a member of the new Commission was caused by such rumors, I have a right to know it. As far as I know those rumors I can tell you that they are untrue. Cordially yours cc. A. Delessert H. Cartan (Hans Freudenthal) Hans Freudenthal to Henri Cartan, Utrecht, 29 June 1970 Fonds de Rham, 5113-403, also in IMU Archive: SF1 / Ser 6.5 Typewritten letter with signature266 Mathematisch Instituut der Rijksuniversiteit te Utrecht Utrecht, le 29 juin 1970 Universiteitscentrum De Uithof Budapestlaan Tel. (030) 539111 Professeur Henri Cartan 95 Boulevard Jourdan 75 PARIS (14) Frankrijk

 Freudenthal is referring to the Congrès sur l’integration des enseignements scientifiques held in Droujba, near Varna (Bulgaria) from 11 to 19 September1968. See (Fleury 1968, pp. 5–7). 266  At the top of the letter there are these handwritten notes: “Copie pour le Professeur G. de Rham” and “reçue le 2 Juillet”. This letter has already been published in (Furinghetti and Giacardi 2010, pp. 46–47). 265

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Cher Ami, Je suis heureux que vous me posiez la question de ma succession à la présidence de la CIEM. Je désire que l’existence et l’activité de la CIEM soient bien assurées quand je sors de la chaire de président. Le problème de la CIEM est ce qu’elle n’a ni de bureau ni d’autre noyau permanent. Elle dépend de l’activité du président et des facilités administratives dont il dispose. Jusqu’alors presque chaque président a commencé de nouveau; si c’était un homme moins actif ou moins inventif, la CIEM serait perdue. C’est alors plus important qu’il n’a jamais été, parce que le président de la CIEM doit jouer un grand rôle dans la préparation du Congrès de 1972267 qui aura lieu à Exeter (Gr. Br.). Si le président de la CIEM n’est pas assez actif, l’organisation de tels congrès internationaux de l’enseignement glissera des mains de la CIEM. Je pense que le président de la CIEM doit remplir les conditions suivantes: –– état de préoccupations plus que superficielles et continues depuis des années avec les questions de l’enseignement secondaire ou même primaire, –– familiarité avec la situation internationale et le milieu international de ceux qui s’intéressent à l’enseignement mathématique, –– goût d’initiative et d’activité. Il y aurait assez de gens qui pourraient satisfaire à ces demandes, mais des conditions supplémentaires pourraient compliquer le choix. Jusqu’alors on a toujours choisi un professeur d’université de poids scientifique pas trop léger et on a tenu compte de la nationalité du candidat. Je n’ai aucun doute que parmi ceux que je connais et que j’ai considérés, de très loin le plus digne et le plus capable en quelque respect que ce soit et qui continuera et revivra le mieux les affaires, est A. Revuz,268 mais je crains qu’après deux présidents de nationalité française on n’ose pas donner à la CIEM un troisième. D’autres noms: W. Servais, professeur de lycée à Morlanwelz (Belgique), Mme A. Z. Krygovska,269 directeur d’école normale de Cracovie, M. B. Christiansen,270 directeur d’un institut de recyclage de maîtres à Copenhague, me semblent excellents: ils sont mieux connus dans les milieux de l’enseignement mathématique, mais je // ne suis pas sûr qu’on leur attribue le poids mathématique, nécessaire au président de la CIEM.  Un cas un peu différent serait M.  H.O.  Pollak271 (Bell Telephone Laboratories USA), excellent et même brillant et bon mathématicien,  This Congress is ICME-2 that took place in Exeter from 29 August to 2 September 1972 (EM s. 2, 19, 1973: 167–170). See Chap. 3 by Marta Menghini in this volume. 268  André Revuz (1914–2008) was a French mathematician who carried out an impressive activity for the improvement of mathematics education and the development of didactic research, He was a member of the Executive Committee of ICMI from 1967 to 1970. 269  Anna Zofia Krygowska (1904–1988) was a Polish mathematician and mathematics educator. 270  Bent Christiansen (1921–1996), a Danish mathematician and mathematics educator, was vicepresident of ICMI from 1975 to 1986. 271  Henri Otto Pollak (1927–) is an applied mathematician who worked at Bell Laboratories for many years, and a professor at Teachers College, Columbia University. He was President of the Mathematical Association of America in 1975–1976. 267

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mais encore une fois un américain. Un homme que je ne connais personnellement que depuis quelques mois et qui me semble excellent serait M.  J.N.  Kapur272 de «Indian Institute of Technology, Kanpur, India»; je regrette que je ne l’aie pas connu quand nous avons choisi les conférenciers de Lyon et cherché en vain quelque représentant de l’Asie. Si je devais nommer un président de la CIEM qui ne vient pas de l’Europe ou de l’Amérique, ce serait lui que je choisirais. Mais la CIEM devrait disposer de moyens plus amples qu’alors, pour se permettre un président loin de l’Europe et l’Amérique. M. Kapur a publié beaucoup par rapport à l’enseignement, c’est un homme avec des idées aussi fraiches que solides, il est actif et bon mathématicien. C’est dommage qu’il n’a jamais appartenu à la CIEM. En tout cas, il mériterait de devenir Member at Large.273 Le même vaut pour les autres que j’ai mentionnés. Vu que le second Congrès de l’Enseignement Mathématique aura lieu en Grande Bretagne, il serait assez naturel de chercher un président britannique. En Grande Bretagne il y a un grand nombre d’hommes bien capables d’être président de la CIEM. E.A. Maxwell274 a longuement et excellemment servi la CIEM, M.R.C. Lyness275 serait un candidat aussi digne. Il me semble que, chez lui et à l’étranger M.B. Thwaites276 jouit de la plus grande autorité. Il est toujours enthousiaste (parfois un peu trop, ce que j’aime), il n’aime pas trop le formalités (ce qui peut être un avantage et un désavantage). Voilà mes considérations et mes candidats. Peut-être j’ai oublié l’un ou l’autre. Le choix sera difficile, mais j’espère qu’il sera un bon choix. Cordialement, H. Freudenthal Henri Cartan to James Lighthill,277 Die, 20 August 1970 IA, 14B 1967–1974 Typewritten letter with signature Copie pour O. Frostman Die, le 20 août 1970  Jagat Narain Kapur (1923–2002) was an outstanding Indian mathematician with a keen interest in mathematics education and, at the end of his life, in the history of mathematics. He was the founder of the Mathematical Association of India. 273  On the meaning of “member at large”, see Chap. 6 by Fulvia Furinghetti in this volume. 274  Edwin Arthur Maxwell (1907–1897) was a member of the Executive Committee of ICMI in 1952–1958 and secretary in 1971–1974. 275  Robert C. Lyness (1909–1997), an English mathematician, was “Head of Mathematics” at the Bristol Grammar School. He participated in the Colloquium How to teach mathematics so as to be useful held in Utrecht from 21 to 25 August 1967. 276  Bryan Thwaites (1923–) is an English applied mathematician interested in mathematics education. He co-founded, in 1964, the Institute of Mathematics and its Applications (IMA), alongside James Lighthill. 277  Michael James Lighthill (1924–1998) was president of ICMI from 1971 to 1974. 272

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Professor M. J. Lighthill, University of Cambridge Department of Applied Mathematics and Theoretical Physics Silver Street Cambridge CB3 9 EW (U. K.) Cher Professeur Lighthill, Le Professeur Otto Frostman, Secrétaire de l’Union Mathématique Internationale, m’a communiqué le texte de votre lettre du 10 août. Permettez-moi de vous remercier chaleureusement d’avoir bien voulu accepter la charge de Président de la Commission Internationale de l’Enseignement Mathématique (International Commission of (sic)  Mathematical Instruction, I.C.M.I.) au cas où l’Assemblée Générale de l’Union Mathématique Internationale ratifierait ce choix par son vote.278 Je regrette qu’il ne vous soit pas possible de participer à cette Assemblée Générale, ce qui m’aurait donné la possibilité de discuter avec vous des problèmes de l’I.C.M.I. Je désire néanmoins entrer en contact avec vous avant cette Assemblée, bien que le temps soit maintenant très court ! Je ne sais si O. Frostman vous a donné beaucoup de détails sur le fonctionnement de l’I.C.M.I. C’est l’une des Commissions de l’Union Mathématique Internationale, mais elle a un fonctionnement un peu particulier, qui lui est propre. Le règlement de l’I.C.M.I., dont le texte vous sera bien entendu communiqué, dit notamment ce qui suit: “(a) The Commission shall consist of ten members-at-large elected by the General Assembly of IMU on nomination of the President of IMU, and of one national delegate from each member nation, as specified below. (b) The officers of the Commission shall consist of a President, a Secretary, and two Vice-Presidents. The President shall be elected by the General Assembly on the nomination of the President of IMU from the membership-at-large of the Commission. (c) The Executive Committee of the Commission shall consist of the officers of the Commission together with three additional members elected by the membership of the Commission. (d) In all other respects the Commission shall make its own decisions as to its internal organization and rules of procedure.” //279 C’est le 27 août que le Comité exécutif de l’Union se mettra d’accord sur les propositions de nominations que le Président de l’Union soumettra au vote de l’Assemblée Générale (28–30 août).

 The General Assembly of IMU was held in Menton (France) on 28–30 August 1970 (IMU Bulletin, 1, 1971: 5 ff.) 279  See the Terms of Reference of 1960 in Chap. 7 by Livia Giacardi in this volume. 278

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S’il était possible que d’ici là, malgré le délai très court, vous me fassiez connaître votre opinion éventuelle sur le choix des 9 autres “members at large” de la Commission, je vous en serais très reconnaissant. Voici, à titre d’information, la composition de la Commission sortante, en ce qui concerne les 10 “members at large”: H. Freudenthal, Président; Y. Akizuki, A. Lichnerowicz, E. Moise, J. Novak, G. Papy, A. Revuz, S. Sobolev, Steiner, Thwaites. Rien n’empêche a priori qu’un member at large soit réélu. D’ailleurs je crois qu’il est souhaitable que le Président sortant appartienne à la Commission suivante, car il peut ainsi faire profiter le nouveau Président de son expérience. C’est ainsi que Lichnerowicz, qui a présidé la Commission 1963–66 (élue en 1962), a fait partie de la Commission 1967–70 (élue en 1966). Je pense que, pour cette raison, Freudenthal devrait être choisi comme “member at large” de la nouvelle Commission. Vous aurez à étudier le problème du choix d’un Secrétaire; il doit nécessairement faire partie de la Commission, mais pas nécessairement comme “member at large”. Je crois que l’actuel secrétaire, Delessert (Suisse) pourrait céder la place à un homme nouveau; à mon avis, les relations entre le Président et la Secrétaire n’étaient pas satisfaisantes, le Secrétaire étant devenu une simple “boîte aux lettres” du Président. Peut- être serait-il souhaitable que le Président et le Secrétaire appartiennent au même pays, ce qui faciliterait les échanges de vues entre eux ? ceci n’est pas une suggestion de ma part; c’est simplement une question que je pose, et à laquelle j’aimerais que l’on puisse répondre. Je pense que vous pourriez discuter de ce problème avec le Professeur Burkill280 (qui est votre voisin); il pourrait vous dire notamment ce qu’il penserait du choix éventuel de Thwaites comme Secrétaire. Il faudra évidemment tenir compte du fait que l’I.C.M.I. a décidé d’organiser à Exeter281, en 1972, son deuxième congrès international. Il y aura donc une tâche d’organisation qui reviendra plus particulièrement aux collègues anglais. Veuillez m’excuser de vous avoir écrit si longuement en français. Naturellement, vous pouvez me répondre en anglais: je lis la langue anglaise plus facilement que je ne l’écris. En vous remerciant à nouveau, je vous prie de croire, cher Professeur Lighthill, à mes sentiments les plus dévoués. H. Cartan Hôtel du Parc, 06-Menton (France) (à partir du 26 août)

 John Charles Burkill (1900–1993), an English mathematician, was a member of the Executive Committee of IMU in 1963–1966. 281  ICME-2 was held in Exeter (UK) from 29 August to 2 September 1972. 280

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Hans Freudenthal to the Executive Committee of ICMI, Utrecht, 11 October 1970 IA, 14B 1967–1974 Typewritten letter with signature Commission Internationale de l’Enseignement Mathématique International Commission on Mathematical Instruction To the Executive Committee of I.C.M.I.

October 11, 1970

Dear Colleagues, According to a decision taken by I.C.M.I. in Nice282 with regard to the Second International Congress on Mathematical Instruction at Exeter in 1972 the Executive Committee of I.C.M.I. will appoint a Programme Committee to prepare that Congress. I propose you that this Committee283 consists of the British members Mr. D. G. Crawforth Dr. T. J. Fletcher Mr. M. Goldsmith Dr. A. G. Howson Prof. G. Matthews Mrs. J. Stephens Prof. Dr. B. Thwaites Mrs. E. M. Williams (chairwoman)284 and the International members Prof. dr. H. Freudenthal, Math. Dept. Univ. of Utrecht, Netherlands Prof. J. Novak, Math. Inst., Acad. Zitná 25, Praha 1, Czechoslovakia

 Freudenthal is referring to the ICMI meeting held in Nice on the occasion of the ICM (1–10 September 1970). 283  All the persons listed by Freudenthal appear in the Congress Committees, except D. G. Crawforth and M.  Goldsmith, see Proceedings ICME-2. 1973, p.  299. Denis G.  Crawforth from the Department of Education, University of Exeter, was the honorary secretary of ICMI-2 Congress. 284  Trevor James Fletcher (1922–2018), a pioneer maker of mathematical films, wrote various successful books and papers on mathematics education, with particular attention to modern mathematics and the use of computers in schools at both elementary and secondary levels. Geoffrey Howson had begun his scientific career as a researcher in algebra, then devoted himself to mathematics education, making also significant contributions to the history of mathematics education. He was secretary of ICMI for two terms from 1983 to 1990. Geoffrey Matthews (1917–2002), a British mathematician and teacher, was the chairman of the teaching committee of the Mathematical Association, and president of the Association in 1977–1978. In 1964, when the Nuffield Foundation decided to extend its work in secondary science teaching into primary mathematics education, Matthews was its organizer. Elizabeth M. Williams (1895–1986) was a British mathematician and educationist who became president of the Mathematical Association in 1965. 282

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Prof A. Pescarini, Via Montenero 6, Ravenna, Italy285 Dr. H. O. Pollak, Bell Telephone Laboratories, Math. Res. Center, Murray Hill, USA Prof. A. Revuz, 16 rue de Rome, Les Essarts-le-Roi, France Prof. dr. S.L. Sobolev, Morskoi prospekt, Novosibirsk 90, USSR Prof. dr. H.G. Steiner, Bayreuth, Germany Prof. I. Wirszup, Dept. of Math. Chicago Univ. USA286 I would like the constitution of the Committee to be settled not later than the end of November. May I ask you to let me know whether you agree or not with the proposed list, before November 15, 1970. Sincerely yours, H. Freudenthal Mathematisch Instituut R. U. Budapestlaan, Utrecht, Netherlands Henri Cartan to Otto Frostman, Paris, 15 October 1970 IA, 14B 1967–1974 Typewritten letter with signature Université de Paris Faculté des Sciences 91 Orsay Mathématique Bâtiment 425 Téléphone: 920 88-21 Lignes groupées Paris, le 15 octobre 1970 Cher Frostman, Freudenthal me donne encore du souci. Hier j’ai reçu (et vous aussi, sans doute ?) le “compte-rendu de la séance de la C.I.E.M.” du 5 septembre 1970, à Nice, envoyéé́ par Delessert, et rédigé́ en français. Il y est dit (page 3): “La constitution du nouveau Comitéeʺ Exécutif de la C.I.E.M. se fera par concertation entre les membres déjȁ désignés de la nouvelle Commission”.287  Angelo Pescarini (1919–2000) was an Italian mathematician who dedicated himself to improving mathematics teaching in Italy, holding important positions both nationally and internationally. He was the first, starting from the 1950s, to introduce in Italy the ideas of Caleb Gattegno, one of the founding members of the Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématique (CIEAEM). 286  Izaak Wirszup (1915–2008) was a Polish mathematician and a Holocaust survivor. He emigrated to the USA after World War II.  In 1983, Wirszup cofounded the University of Chicago School Mathematics Project (UCSMP) to bring mathematics education in the United States up to the standards of Japan, Western Europe, and the Soviet Union. 287  See Compte-rendu de la séance de la C.I.E.M. tenue à Nice, lors du Congrès International des Mathématiciens, le 5 septembre 1970, à 14 heures 30, dans les auditoires 2.7 et 2.8 du Bâtiment des salles de cours de la Faculté des Sciences, p. 3, IA, 14B 1967–1974. 285

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Comme cette phrase suit l’annonce de la liste des 10 “members at large” et des 3 membres ex-officio élus par l’Assemblée Générale de Menton, elle signifie que seuls ces 13 personnes participeront au choix du nouveau C.E de la C.I.E.M. Or ceci n’est pas correct puisque la nouvelle commission internationale doit d’abord être complétée par la désignation d’un membre par chaque commission nationale. Aussi ai-je écrit dès hier à Delessert pour lui demander de rectifier le compte-rendu sur ce point. Je pense que vous devriez faire quelque chose de votre côté, par exemple envoyer d’urgence une lettre circulaire aux différentes commissions nationales pour leur demander d’envoyer le plus tôt possible à Lighthill le nom de leur représentant à la C.I.E.M.  De plus, il serait bon, si c’est possible, de leur envoyer la partie des “records” de l’Assemblée Générale concernant la C.I.E.M. Ce matin, c’est une lettre de Freudenthal au Comitéeʺ exécutif de l’I.C.M.I. (= C.I.E.M.), c’est-à-dire à l’ancien C.E., qui me trouble. Avez-vous reçu cette lettre datée du 11 octobre ? Il s’agit de la désignation d’un “Programme Committee” pour préparer le Congrès d’Exeter; Fr. propose une liste de 8 noms de membres britanniques et de 8 noms de membres internationaux; le nom de Lighthill n’y figure pas. Freudenthal demande aux membres de l’actuel C.E. d’approuver sa liste, et dit que le Comité du programme doit être constitué avant la fin de novembre. Je trouve que c’est bien de ne pas se laisser mettre en retard pour la préparation du Congrès d’Exeter, mais que tout de même F. exagère un peu en mettant la nouvelle Commission internationale devant un fait accompli. En principe, F. a le pouvoir jusqu’au 31 décembre, mais il ne devrait rien faire d’aussi important sans être d’accord au moins avec le nouveau Président de l’I.C.M.I. Je vais lui écrire dans ce sens, avec copie de ma lettre pour Lighthill. Je vous enverrai une copie par le prochain courrier, mais je ne veux pas attendre davantage pour vous alerter; vous verrez vous-même si vous devez agir d’une manière ou d’une autre. Bien cordialement, H. Cartan Hans Freudenthal to Henri Cartan, n. p., 19 October 1970 IA 14B 1967–1974 Typewritten letter, carbon copy Le 19 octobre 1970 Professeur Henri Cartan, Président Union Mathématique Internationale 95 Boulevard Jourdan 75 – Paris (14) – Frankrijk Cher Ami, J’ai participéeʺ à une réunion du Comité britannique pour le Congrès de 1972288 sous la présidence de M.  Lighthill. Avant cette réunion j’ai discutéeʺ tous les 288

 Freudenthal is referring to ICME-2 to be held in Exeter in 1972, see footnote 281.

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problèmes de la CIEM avec M. Lighthill. Les lettres que j’ai adressées aux membres de la CIEM ont été approuvées par écrit par M. Lighthill, la lettre adressée au Comité Exécutif a été approuvé en principe par le Comité britannique sous la présidence de M. Lighthill. Il n’était pas recommendable d’attendre avec la nomination du Comité de programme du Congrès de 1972 jusqu’en 1971. Avant la fin de cette année le Comité britannique aura eu sa quatrième réunion. J’en aurai assisté à deux. A la réunion de Nice à laquelle assistait le secrétaire britannique, M. Crawforth, il y avait une discussion très utile sur les projets britanniques. Elle a montré que pour bien préparer la partie scientifique du Congrès il faut une coopération internationale très intensive. Pour réaliser les idées britanniques sur le congrès, il faut former très tôt des petits groupes internationaux préparatoires, et ce n’est que possible, si le Comité de programme international peut fonctionner très tôt. Je souligne que M. Lighthill était complètement d’accord avec la formation du Comité suivant la résolution de Nice, en attendant le fonctionnement de la nouvelle CIEM à partir du 1 janvier 1971. Les Comptes Rendus de la séance du 6 septembre 1970,289 que je n’ai pas approuvés, contiennent un grand nombre de fautes que je viens d’indiquer à M. Delessert. En outre, c’est clair de ma lettre adressée aux membres de la CIEM que je ne partage pas l’avis de M. Delessert. Mes salutations, cordiales (H. Freudenthal) Copies à M. J. Lighthill O. Frostman Otto Frostman to Hans Freudenthal, Djursholm, 20 October 1970 IA, 14B 1967–1974 Typewritten letter with signature International Mathematical Union The Secretary – Professor otto Frostman Djursholm – Sweden October 20, 1970 Professor H. Freudenthal Mathematisch Instituut R. U. Budapestlaan Utrecht / Netherlands Dear Freudenthal, The letter from Cartan to you reached me yesterday. I was myself a bit astonished when I received your circular letter of October 11 and, at the same time, the minutes of the ICMI meeting in Nice from Delessert. On p. 4 in these minutes it is said: “Un comité d’organisation, formé de cinq anglais et cinq étrangers, préparera 289

 Freudenthal writes 6 September instead of 5 September.

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le programme.” (underlinings made by me). Is the “Programme Committee” proposed by you a new committee and, if so, what has happened to the old one? Referring to my letter to Lighthill of September 17, I wonder if you or Lighthill have taken any steps to get the national delegates nominated? As I very well know, such a procedure takes some time, and it may be good to start fairly soon. In my opinion, the Secretary of ICMI must be charged with the work; he has all the names and addresses of the national members of ICMI which, if I am right, are not quite the same as those of IMU. I suppose the national members should send the names of their representatives to Lighthill, but of course, it is for you, Lighthill and Delessert to agree. I should like to make another comment to the minutes from Nice. On p. 3 it is stated: “Les membres de la Commission sont élus par des personnes qui ne sont pas particulièrement compétentes en matière …”.290 Isn’t this a bit too strong? Cordially Otto Frostman Copies to H. Cartan M. J. Lighthill A. Delessert Hans Freudenthal to Otto Frostman, Utrecht, 23 October 1970 IA, 14B 1967–1974 Typewritten letter with signature Mathematisch Instituut der Rijksunversiteit te Utrecth Utrecht, October 23, 1970 Universiteitscentrum De Uithof Budapestlaan Tel. (030) 539111 Professor O. Frostman International Mathematical Union Djursholm – Zweden Dear Frostman, I got your letter of October 20, 1970. Meanwhile my letter  – to Cartan  – of October 19,291 which explains things, will have reached you. The Committee mentioned in the Minutes will be appointed by the Executive Committee of I.C.M.I. (The numbers 5292 were meant approximately.) My circular letter of October 11 made a  See footnote 287.  See the two letters which precede this one. 292  Freudenthal is referring to the number of the members of the Programme Committee of ICME-2 in Exeter; see p. 4 of the Compte-rendu de la séance de la C.I.E.M. tenue à Nice,… cited in footnote 287. 290 291

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proposal about it. This proposal has been formulated in complete agreement with Mr. Lighthill. As to the second part of your letter, I suppose that you did not receive or did not notice the circular letter of the secretary of I.C.M.I., in which the I.C.M.I. members were asked to appoint their national representatives not later than November 15. This letter has been entirely approved by Mr. Lighthill. I will check with Mr. Delessert whether this circular letter has been mailed. It is true that at previous periodical changes of I.C.M.I. there was little if any contact between the old and the new I.C.M.I. Maybe this is the reason why you are alarmed. I have criticized this at any earlier instance so it was for me the most natural thing that I took no step as a president without close contact with the new officers, in particular with Mr. Lighthill. It is a pity that I did not mention explicitly what I considered as the most obvious thing of the world, and by this way caused this strange consternation. As to this third paragraph of your letter, I admit it looks strong. This, however, reflects the actual discussion in which much stronger terms have been used. The disapproval of the way in which the new members at large were appointed was unanimous. As an attendant to this election, I could only say that the procedure was in complete agreement with the formal regulations. I would suggest that this is taken up as a serious problem by the new Executive of I.M.U. and I.C.M.I. Cordially yours, H. Freudenthal Copies to: H. Cartan M. J. Lighthill Delessert [Otto Frostman] to Henri Cartan, n. p., 15 November 1970 IA, 14B 1967–1974 Typewritten letter, carbon copy November 15, 1970 Dear Cartan, As to the Congress in Exeter, I think that IMU should contribute financially, if necessary. Maybe ICMI, through IMU, could get some contribution from UNESCO to the Congress, because it is educational questions that will be discussed there, but I am not sure. I cannot recall that IMU made an application to UNESCO for the Lyon Congress – in fact, I am fairly sure we did not because such a thing would show in my correspondence. As you will remember, IMU paid $ 2.000 to the first ICMI Colloquium in Utrecht in 1967, but only about $ 700 was used and the rest has been used for other activities. See the reports from Freudenthal of January 3, 1969 and February 3, 1970. There still remains about $ 250. I should add that I have not paid anything to the ICMI secretariat during the last years.

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Moreover, ICMI has some money of its own – f.i. some countries pay an annual contribution to ICMI. Maybe I should but, in fact, I don’t have exact information about ICMI’s affairs. Cordially yours. Henri Cartan to James Lighthill, Paris, 20 November 1970 IA, 14B 1967–1974 Typewritten letter, carbon copy293 Paris, le 20 novembre 1970 Professor M.J. Lighthill University of Cambridge Department of Applied Mathematics Silver Street Cambridge CB3 EW (U.K.) Cher Professor Lighthill, Je vous prie d’excuser le retard avec lequel je réponds à votre lettre du 6 novembre 1970 concernant le financement du Congrès d’Exeter 1972. Vous avez soulevé un problème un peu délicat, au sujet duquel j’ai cru devoir consulter quelques personnes; or ce mois-ci la poste marche mal en France (par suite de grèves répétées), et cet échange de lettres a pris plus de temps que je n’aurais voulu. Il a dû se produire un malentendu au sujet des déclarations que le Professeur Freudenthal a pu vous faire lors de la réunion du 2 octobre dernier. En effet: 1) l’Union Mathématique International (IMU) n’a fourni aucune contribution financière pour le Congrès de Lyon, pour la simple raison que l’ICMI ne lui a jamais adressé de demande de subvention pour ce Congrès; 2) l’IMU n’a pas non plus fait de démarche auprès de l’UNESCO en faveur du congrès de Lyon, car le Professeur Freudenthal ne nous l’a jamais demandé, mais s’est adressé lui-même directement à l’UNESCO sans d’ailleurs en informer l’IMU. Pendant les quatre années où j’ai exercé la présidence de l’IMU, j’ai regretté à plusieurs occasions cette manque d’informations réciproques entre l’UMI294 et l’ICMI. En particulier, la décision de tenir des congrès internationaux spéciaux sur l’enseignement des mathématiques, indépendamment des congrès internationaux réguliers des mathématiciens, a été prise par l’ICMI sans consultation de l’IMU. Je serai très heureux que sous votre Présidence des relations plus étroites et confiantes soient établies, et je suis certain que mon successeur, le Professeur Chandrasekharan, pense comme moi.

 At the top of the letter there are these handwritten notes: “Cher Frostman, merci de votre lettre du 15 novembre. Cordialement H.M.” and “COPIES aux Professeurs Chandrasekharan et Frostman”. 294  Here Cartan writes UMI (Union Mathématique Internationale) according to the French denomination of IMU. 293

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Durant les dernières années, l’ICMI n’a utilisé qu’une petite fraction des sommes que l’IMU avait mises à sa disposition. Je pense que le nouveau Comité exécutif de l’IMU qui entrera en fonctions le ler Janvier prochain sera disposé à aider financièrement l’ICMI pour certaines de ses activités. Il devra discuter pour savoir s’il décide de donner son appui direct au Congrès d’Exeter. Mais sans attendre le ler Janvier, je ferai volontiers une démarche comme Président en exercice de l’Union, auprès de l’UNESCO en faveur du Congrès d’Exeter, si vous en êtes d’accord. Je vous prie de croire, cher Professeur Lighthill, à mes sentiments les plus dévoués. Henri Cartan, 95 Boulevard Jourdan, 75 – Paris James Lighthill to Henri Cartan, Cambridge, 30 November 1970 IA, 14B 1967–1974 Typewritten letter with signature University of Cambridge Department of Applied Mathematics and Theoretical Physics Silver Street, Cambridge CB3 9EW Telephone: Cambridge (0223) 51645 30th November 1970 Professor H. Cartan, 95 Boulevard Jourdan, 75 – Paris (14), France Dear Professor Cartan, I am most grateful for your letter of 20th November, as well as for having sent me copies of your earlier correspondence with Professor Freudenthal. I do greatly appreciate your concern about the importance of close liaison between ICMI and its parent body, the IMU. I do believe that some of this liaison can be achieved very effectively through Professor Frostman, to whom I am sending a copy of this letter, and in particular through his membership of ICMI. I propose, however, to make a special effort to keep both yourself and Professor Frostman informed of the activities of ICMI, and in particular those related to the forthcoming Congress. I believe that you will appreciate that because the Congress is planned for 1972, it was quite essential for us, already in the summer of this year, to take on much of the detailed work of planning the meeting, particularly as far as the local arrangements were concerned. It was, for example, necessary for us to book the required accommodation a full two years in advance. There is, furthermore, a lot of work to do on the planning of the programme for such a large meeting. I am happy to say that I have initiated all this in the closest possible collaboration with Professor Freudenthal, and that it has been with his approval that a Local Organizing

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Committee has been formed and has started to do all the required preliminary work. He has been present at two of our // meetings. There is a strong belief that the Congress can be made more effective by the careful preparation of activities by Working Groups on specific major areas of concern to those in mathematical education. The work on all this is progressing well, so that when the new Executive Committee of ICMI is formed we shall have some useful and rather concrete proposals to put before them for discussion. In the meantime, you will see from the enclosed copy of my letter to Professor Freudenthal, that good progress has been made on the nomination of the international delegates of ICMI, and that the nomination of those who are to be members of the Executive Committee should soon become possible. Naturally I should be delighted to have the benefit of your advice on how to proceed in this last most important matter. With best wishes, Yours sincerely M. J. Lighthill Henri Cartan to M. James Lighthill, Paris, 2 December 1970 IA, 14B 1967–1974 Typewritten letter with signature Paris, 2 décembre 1970 Professor M.J. Lighthill University of Cambridge Department of Applied Mathematics and Theoretical Physics Silver Street, Cambridge CB3 9EW (U.K.) Cher Professeur Lighthill, Je vous suis très reconnaissant de votre aimable lettre du 30 novembre 1970, et vous en remercie vivement. Je me réjouis de constater que vous partagez mon désir de maintenir un contact étroit entre l’IMU et l’ICMI, en particulier par l’intermédiaire du Professeur Frostman, qui comme Secrétaire de l’Union est maintenant membre ex-officio de l’ICMI. Moi-même je vais quitter la présidence de l’IMU ce 31 décembre; je suppose que vous entrerez bientôt en contact avec le nouveau Président, le Professeur K. Chandrasekharan. Je suis parfaitement conscient de la nécessité dans laquelle le Professeur Freudenthal et vous-même vous êtes trouvés à travailler à la préparation du Congrès d’Exeter dans l’attente que le nouveau Comité de l’ICMI soit définitivement constitué et soit entré en fonctions. Et je me réjouis de la collaboration étroite que vous avez établie avec le Professeur Freudenthal. Il est certainement urgent de demander une aide financière de l’UNESCO à l’ICMI pour le Congrès d’Exeter. Après avoir consulté le Professeur Chandrasekharan qui connaît mieux que moi le mécanisme de l’UNESCO, je pense que la meilleure méthode consisterait en ce que Freudenthal ou vous-même (ou les deux ensemble) adressiez une requête a l’UNESCO, analogue à celle que l’ICMI avait faite pour le

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Congrès de Lyon; dès que Frostman et moi aurons reçu copie de votre demande à l’UNESCO, je pourrai intervenir pour l’appuyer au nom de l’IMU (si les choses se font avant la fin de ce mois de décembre). Je vous prie de croire, cher Professeur Lighthill, à mes sentiments les plus dévoués. Henri Cartan 95 Boulevard Jourdan 75 – Paris (14), France Copies à:  O. Frostman K. Chandrasekharan James Lighthill to Otto Frostman, Cambridge, 24 July1972 IA 14B 1967–1974 Typewritten letter with signature University of Cambridge Department of Applied Mathematics and Theoretical Physics Silver Street, Cambridge CB3 9EW Telephone: Cambridge (0223) 51645 24 July 1972 Professor Otto Frostman, Auravagen 21, 18262 Djursholm, Sweden. Dear Professor Frostman, I am most grateful to you for a number of suggestions in your recent correspondence with Dr. Maxwell295 about the affairs of ICMI. Actually, Maxwell and I do regard with very great urgency the importance of taking action during the meeting of the General Assembly of ICMI at Exeter to appoint Vice-Presidents according to the terms of reference. I do fully intend that we shall have the correct complement of Vice-Presidents with correct composition of our Executive Committee by the time the General Assembly is over. The properly constituted Executive Committee is then due to meet on Saturday 2 September. Perhaps I might mention to you in confidence that the persons that we are intending to become Vice-Presidents are Professor Freudenthal and Professor Revuz.296 Any comments that you may have on our plan will be of the greatest interest to us. May I express my warm gratitude to the IMU for its support in the organization of the Exeter Congress. Preparations for this are going extremely well. There has been an enormous response to the announcement of the Congress and there will be

 Edwin Arthur Maxwell (1907–1987) was secretary of ICMI for the period 1971–1974.  Actually, the vice-presidents for the period 1971–1974 were Shōkichi Iyanaga and János Surányi (ICMI Bulletin 1, 1972: 3) 295 296

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representatives present from over 60 countries.297 Those of us involved in the organization are confident that it will make a contribution of outstanding value to the discussion of almost all the most important aspects of mathematical education today. With best wishes, Yours sincerely, James Lighthill Professor Sir James Lighthill James Lighthill to Otto Frostman, Cambridge, 13 September 1972 IA 14B 1967–1974 Typewritten letter with signature University of Cambridge Department of Applied Mathematics and Theoretical Physics Silver Street, Cambridge CB3 9EW Telephone: Cambridge (0223) 51645 13 September 1972 Professor Otto Frostman, Auravagen 21, 18262 Djursholm, Sweden. Dear Professor Frostman, I am most happy to report to you now about the Second International Congress on Mathematical Education and about the General Assembly of ICMI that took place during the Congress. All of us who have been involved during the past 2 years with preparations for the Second Congress felt extremely gratified by the response to the meeting involving over 1400 full members from about 70 countries, and by the general success of the organization and the very great amount of important work for mathematical education that was carried out during the Congress. 17 of the countries attending provided national presentations that gave exceedingly valuable accounts of the general state of development of mathematical education in the countries concerned. The primary work of the Congress was however centered into the 38 working groups on different specialized aspects of the subject centered either on the treatment of particular parts of mathematics at particular levels of education or on particular aspects of method or technology and on the fundamental studies underlying the choice of method. I have had excellent reports about the very successful working of practically all these working groups. Splitting up the Congress in this way into units of manageable numbers whose members could really get to know one another and conduct a continuous discussion on the matters of special interest to them proved to

 ICME-2  in Exeter was attended by 1384 participants from 73 countries, see Proceedings ICME-2, p. 13. 297

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have been an excellent decision. In the published Proceedings of the Congress298 there will be published the lectures given to Plenary Sessions and a selection of the most important papers presented to working groups together with a very long extended summary of the activities of the working groups that is being put together by a special editorial committee from reports of about 1000 words each prepared by the member of each working group designated by its Chairman as the person to make such a report. The Proceedings will be sent free of charge to each member of the Congress and will also be on sale to the general public and we anticipate that this publication will have very great sales indeed. // I was very satisfied with the general sense of coherence of ICMI that emerged as a result of the multifarious activities at the Congress. This was confirmed at the very harmonious meeting of the General Assembly. A large number of proposals were brought forward at this meeting, many of them involving ideas for specialized and regional symposia in the period between now and the third Congress. We have been corresponding recently about the importance of bringing the arrangements for the Executive Committee into line with the regulations governing the work of ICMI. I am delighted to be able to report that in the light of your advice and discussions with all the people principally involved I was able to bring this about most satisfactorily at Exeter and that the first meeting of the new Executive Committee proved the wisdom of having restricted its members in the way which you pointed out as desirable and necessary. After very careful consideration I finally nominated Professor Iyanaga of Japan and Professor Surányi of Hungary as the Vice-Presidents of ICMI.299 The General Assembly then elected the following persons to serve as members of the Executive Committee in addition to the President, the Secretary and two Vice-Presidents: Professor Freudenthal of the Netherlands, Dr. Pollak of the USA, Academician Sobolev of the USSR. The whole Executive Committee met the following day and took rather a large number of decisions. You will be receiving a full account of the work of the General Assembly and the Executive Committee quite shortly, but I might remark that we determined to hold eight specialized regional symposia during the next three years.300 Of these symposia five would be held in Europe on specialized subjects.  See Proceedings ICME-2.  János Surányi (1918–2006) was one of the vice-presidents of ICMI in the period 1971–1974, together with Shōkichi Iyanaga (1906–2006) who was president of ICMI from 1975 to 1978. 300  The Symposia were the following: Seminar on New aspects of mathematical applications at school level in Echternach from 4 to 9 June 1973) (ICMI Bulletin, 2, 1973: 3–4); Symposium on The theoretical problems of teaching mathematics in primary schools in Eger (Hungary) from 18 to 22 June 1973 (ICMI Bulletin, 2, 1973: 5); Symposium on Interactions between linguistics and mathematical education in Nairobi from 1 to 11 September 1974 (ICMI Bulletin, 3, 1974: 3–7, ICMI Bulletin, 4, 1975: 6–8); Symposium on The teaching of Geometry in Bielefeld from 16 to 20 September 1974 (ICMI Bulletin, 4, 1975: 8–9); Symposium on Curriculum and teacher training for mathematical education in Tokyo from 5 to 9 November 1974 (ICMI Bulletin, 4, 1975: 9); Colloquium on The evolving a mathematical attitude in the secondary education (age range 14–18 years) in Nyiregyháza (Hungary) from 18 to 22 August 1975 (ICMI Bulletin, 6, 1975: 7–9); Symposium on Combinatorics and probability in primary schools in Warsaw from 25 to 28 August 1975 (ICMI Bulletin, 6, 1975: 10–11); Second world conference on computers in education in 298 299

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The choice of location of the other three was made in a definite attempt to ensure that ICMI operates in a fully international way and looks after the interests of countries in all areas of the globe. These three symposia are of a regional character to be held in Africa, India and Japan. There was also discussion of the importance of regional activities in Latin America but activity in this region was postponed especially in the light of the forthcoming Third Congress of the Inter-American Committee on Mathematical Education at Bahia Blanca. ICMI has also agreed to undertake important new activities in the exchange of information in the general field of mathematical Education in relation to all the decisions taken the financial aspects were very carefully considered. A minimum scale of ICMI financial support to every supported symposium was agreed at $500 but it was recognized that a number of symposia would have to apply for additional support, and it was agreed that decisions // on such applications would be taken in the light of the current financial position of ICMI and making the maximum use of applications to every possible outside body that might be well disposed to the work of ICMI. I need hardly say that we shall be most grateful for any additional help that IMU is able to give to the important and extensive programme of ICMI that is planned for the next three years. I do assure you at the same time that we shall make every feasible effort to obtain supporting funds in addition from a wide range of other bodies. Naturally this is an interim report on the present very vigorous activity within ICMI. I shall be most interested however to receive any comment from you at this stage. All of your many friends at the Second Congress were very sorry that you were not able to be present in person, but I hope that we will have a chance to meet some time in the near future and talk about how all these matters are progressing. With best wishes, Yours sincerely, James Lighthill Professor Sir James Lighthill Edwin Maxwell to Otto Frostman, Cambridge, 18 February 1973 IA 14B 1967–1974 Handwritten letter 15, Gilbert Road Cambridge Tel. 53187 18/2/73

Marseille from 1 to 5 September 1975 (ICMI Bulletin, 7, 1976: 4–7); The fourth Inter-American conference on mathematical education in Caracas from 1 to 6 December 1975 (ICMI Bulletin, 7, 1976: 19–21); Regional conference on development of integrated curriculum in mathematics for developing countries in Asia in Bharwari (India) from 15 to 20 December 1975 (ICMI Bulletin, 7, 1976: 8–18). See also (Giacardi 2008. Timeline 1972–1976).

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Dear Professor Frostman, Many thanks for the IMU Bulletin – an admirable example of a lot of material in a short space! Professor Sir James Lighthill and I would both prefer, if possible, that the election of Officers (other than the President) should remain in the hands of I.C.M.I as it present.301 There seems to me to be great advantage in allowing the President, subject to I.C.M.I. of course, to nominate the committee with whom he will have to work. I hope you will find this agreeable. With best wishes to yourself and to Mrs. Frostman, Yours sincerely, E. Maxwell Otto Frostman to James Lighthill, n. p., 7 December 1974 IA 14B 1967–1974 Typewritten letter, with handwritten initials of name and surname Professor Sir James Lighthill Department of Applied Mathematics and Theor. Physics University of Cambridge, England 7 December 1974 Dear Sir James, Please try to forgive me for not having answered immediately to your letter of 14 November 1974. As an excuse, I could say that I have only just, a short time ago, received the final version of the Record of the General Assembly at Harrison Hot Springs,302 and the minutes of the meeting of the Executive Committee, from Chandrasekharan, and I have been preparing the printing of this material in No. 8 of the IMU Bulletin.303 It is quite as you say: Already in January 1974 Lichnerowicz wrote and told me that he wanted to be free from the CTS.304 I wanted to take up the problem of his replacement at the meeting of the Executive Committee in Zürich in March, but Chandrasekharan said it could wait until the ICSU General Assembly in Istanbul in September 1974. So, in fact, I was rather surprised at Harrison Hot Springs to hear that Chandrasekharan had asked Freudenthal to replace Lichnerowicz – and maybe even more surprised that Freudenthal had accepted!305 Because I remember quite  The Executive Committee of IMU in 1972 had proposed that ICMI officers other than the president be elected by the IMU General Assembly itself. 302  The General Assembly of IMU was held in Harrison Hot Springs (Canada) from 17 to 19 August 1974 (IMU Bulletin, Seventh General Assembly 1974). 303  See IMU Bulletin 8 December 1974. 304  CTS is the Committee on the Teaching of Science of ICSU. See footnote 239. In the 1970s, the IMU representative was Hans Freudenthal. When he resigned the CTS, he wrote to the IMU President Deane Montgomery: “I never experienced such frustration as I did in ICSU CTS”. (H. Freudenthal to D. Montgomery, 16 June 1978, quoted by (Lehto 1998, p. 256). 305  See IMU Bulletin 8 December 1974: 6. 301

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well that there was earlier some sort of disagreement between Freudenthal and some of the IMU members (maybe in particular Cartan). I quite agree with you: Freudenthal is a much more active man in mathematical education than Lichnerowicz, not afraid of taking initiatives. I know him rather well since many years as I was the national representative of Sweden on ICMI for many years until I became Secretary of the Union. So, I think it will be a good change. I am a bit more dubious about Iyanaga, but I admit I don’t know him very well. Now it is time for the Japanese to show that they are active and generous and prepared to pay a lot of money at least for their own travel! You can’t have all the meetings in Japan. OF James Lighthill to Otto Frostman, Cambridge, 19 December 1974 IA 14B 1967–1974 Typewritten letter with signature University of Cambridge Department of Applied Mathematics and Theoretical Physics Silver Street, Cambridge CB3 9EW Telephone: Cambridge (0223) 51645 19 December 1974 Professor Otto Frostman, Auravagen 21, 18260 Djursholm Sweden. My dear Frostman, Warmest thanks for taking the trouble to clear up the slight mystery regarding the IMU representative on the Committee on Science Teaching. You have now made the situation completely clear, and I have now made a beginning in entering into negotiations with the Chairman and Secretary of the Committee with the object of setting up some kind of joint activity between that Committee and ICMI. Yes, I am delighted at the vigorous activity which Professor Iyanaga has displayed since he became President-elect of ICMI. He has made two important visits to Europe during this period, one to the ICMI Symposium in Bielefeld306 which was held in September when he took a lot of trouble to discuss our future plans for the Third International Congress to be held in Karlsruhe in 1976,307 and a second visit in November when a large number of us who will be concerned with the Karlsruhe Congress met together at UNESCO Headquarters308 (at the expense of UNESCO) to

 The ICMI Symposium in Bielefeld was held from 16 to 20 September, see ICMI Bulletin 4, 1975: 8–9. 307  The congress is ICME-3 held in Karlsruhe from 16 to 21 August 1976. 308  The headquarters of UNESCO were in Paris. See (60 ans d’histoire de l’UNESCO 2007). 306

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take the detailed planning of that Congress a stage further. Iyanaga’s contribution during both these visits was first rate. In particular, he showed admirable tact in dealing with all the personalities from all the different countries and this is a quality of great value in the President of an international organization. I believe that you need have no anxiety and that he will prove to have been an excellent choice. With all best wishes for Christmas and the New Year, Yours sincerely, James Lighthill Professor Sir James Lighthill

References 60 ans d’histoire de l’UNESCO. 2007. 60 ans d’histoire de l’UNESCO: Actes du colloque international, Paris, 16–18 novembre 2005. Paris: UNESCO. Baker, F. W.G. (Mike). 1986. ICSU-UNESCO. Forty years of cooperation. Paris: ICSU Secretariat. Carlyle, Thomas. 1841. The Hero as Man of Letters. In On heroes, hero-worship, and the heroic in history. London: James Fraser. Fleury, Pierre. 1968. Préparation et déroulement du Congrès. In Congrès sur l’integration des enseignements scientifiques [Congress on the integration of science teaching, Droujba (Bulgarie)], 11–19 Septembre 1968, 5–7. Paris: CIES. Furinghetti, Fulvia, and Livia Giacardi. 2010. People, events, and documents of ICMI’s first century. Actes d’història de la ciència I de la tècnica, nova època 3 (2): 11–50, at pp. 36–37. ———. this volume. ICMI in the 1950 and 1960s: Reconstruction, settlement, and “revisiting mathematics education”. In The International Commission on Mathematical Instruction, 1908–2008: people, events, and challenges in mathematics education, ed. Fulvia Furinghetti and Livia Giacardi. Cham: Springer. Giacardi, Livia. 2008. Timeline 1908–1976. In The first century of the International Commission on Mathematical Instruction (1908–2008). History of ICMI, eds. Fulvia Furinghetti and Livia Giacardi. https://www.icmihistory.unito.it. Giacardi, Livia, and Roberto Scoth. 2014. Secondary mathematics teaching from the early nineteenth century to the mid-twentieth century in Italy. In Handbook on history of mathematics education, ed. Alexander Karp and Gert Schubring, 201–228. New York: Springer. Gispert, Hélène. 2021. L’Enseignement Mathématique and its internationalist ambitions during the Turmoil of WWI and the1920s. In Mathematical communities in the reconstruction after the Great War 1918–1928. Trajectories and institutions, ed. Laurent Mazliak and Rossana Tazzioli, 63–88. Basel: Birkhäuser. Hollings, Christopher, Reinhard Siegmund-Schultze, and Kragh Sørensen. 2020. Meeting under the integral sign? The Oslo congress of mathematicians on the eve of the Second World War. Providence: American Mathematical Society. International Council of Scientific Unions and Certain Associated Unions. 1965. Hearing before the Subcommittee on International Organizations and Movements of the Committee on Foreign Affairs House of Representatives. Eighty-ninth congress, first session. Washington, DC: U.S. Government Printing Office. Kulnazarova, Aigul, and Christian Ydesen, eds. 2016. UNESCO without borders: Educational campaigns for international understanding. London: Routledge. Lehto, Olli. 1998. Mathematics without borders: A history of the International Mathematical Union. New York: Springer.

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Loria, Gino. 1933. La préparation théorique et pratique des professeurs de mathématiques de l’enseignement secondaire dans les divers pays. L’Enseignement Mathématique 32: 5–20. Louis, Michel, ed. 2019. Georges Laclavère (1906–1994). Son oeuvre au service des sciences géographiques. Colléction du Bureau des longitudes, vol. 2. https://site.bdlg.fr/wp-­content/ uploads/2019/12/BDL2_Laclavere.pdf. Retrieved June 2021. Luciano, Erika, and C. Silvia Roero. 2012. From Turin to Göttingen: Dialogues and Correspondence (1879–1923). Bollettino di Storia delle Scienze Matematiche 32 (1): 7–232. Menghini, Marta. this volume. The new life of ICMI: Pursuing autonomy and identifying new areas of action. In The International Commission on Mathematical Instruction, 1908–2008: people, events, and challenges in mathematics education, ed. Fulvia Furinghetti and Livia Giacardi. Cham: Springer. Proceedings ICM 1909. Castelnuovo, Guido, ed. 1909. Atti del IV Congresso Internazionale dei Matematici (Roma, 6–11 aprile 1908) 3 Vols. Roma: Accademia R. dei Lincei. ——— 1913. Hobson, Ernest W., and Augustus Love, eds. 1913. Proceedings of the fifth international congress of mathematicians (Cambridge, UK, 22–28 August 1912), 2 Vols. Cambridge: Cambridge University Press. ——— 1921. Villat, Henri, ed. 1921. Comptes Rendus du Congrès International des Mathématiciens (Strasbourg, 22–30 September 1920). Toulouse: É. Privat (Librairie de l’Université). ——— 1954. Gerretsen, Johan C. H., and Johannes de Groot, eds. 1954–1957. Proceedings of the international congress of mathematicians (Amsterdam, Holland, 2–9 September 1954), 3 Vols. Groningen: Noordhoff – Amsterdam: North-Holland Publishing Co. Proceedings ICME-1. 1969. Actes du premier congrès international de l’enseignement mathématique / Proceedings of the first international congress on mathematical education. Dordrecht: D. Reidel Publishing Company. Proceedings ICME-2. Howson, A. Geoffrey, ed. 1973. Developments in mathematical education. Proceedings of the second international congress on mathematical education. Cambridge: Cambridge University Press. Schubring, Gert. 2008. The origins and the early history of ICMI. International Journal for the History of Mathematics Education 3 (2): 3–33. ———. this volume. The history of ICMI: The first phase as IMUK and CIEM. In The International Commission on Mathematical Instruction, 1908–2008: people, events, and challenges in mathematics education, ed. Fulvia Furinghetti and Livia Giacardi. Cham: Springer.

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George Greenhill to David E. Smith, 14 August 1914 (Courtesy of the Rare Book & Manuscript Library, Columbia University, New York)

Felix Klein to David E.  Smith, Göttingen, 6 October 1908 (Courtesy of the Rare Book & Manuscript Library, Columbia University, New York)

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Marshall Stone to Beno Eckmann, Chicago, 23 June 1961 (Courtesy of the IMU Archive, Berlin)

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Hans Freudenthal to Henri Cartan, Utrecht, 29 June 1970 (Courtesy of the IMU Archive, Berlin)

Part II

Events and Data

1.1  Introductory Note This part of the volume presents data about the life of ICMI that complement the information given in Part I. The timeline of ICMI 1908–2008 (Chap. 5 by Fulvia Furinghetti and Livia Giacardi) points out the most significant moments in the history of ICMI – former CIEM/IMUK – and shows how the activities of the Commission broadened and differentiated during the second half of the twentieth century, in order to provide a useful chronological framework for the chapters and the biographical profiles that make up this volume. Chapter 8 by Fulvia Furinghetti gives an overview of the activities related to mathematics education carried out during the International Congresses of Mathematicians (ICMs) from 1897 to 2006. After 1908 these activities often included contributions by CIEM/IMUK/ICMI. Chapter 9 by Livia Giacardi illustrates, using the maps created with Palladio software, the process of internationalization of ICMI over the first century of its life, taking into consideration the geographical origin of the delegates of the member countries until 2002 and subsequently the member countries of the Commission until 2008. Information about the Central and Executive Committees of CIEM/IMUK/ICMI (by Fulvia Furinghetti); the Terms of Reference for ICMI (1954–2007) (by Livia Giacardi); and participants at the First ICME in 1969 (by Fulvia Furinghetti) are provided in Chaps. 6, 7, and 10.

Chapter 5

Timeline of ICMI 1908–2008 Fulvia Furinghetti and Livia Giacardi

This Timeline is intended to point out the most significant moments in the history of ICMI – former CIEM/IMUK – and to show how the activities of the Commission broadened and differentiated during the second half of the twentieth century, in order to provide a useful chronological framework for the chapters and the biographical profiles that make up this volume. Each fact is documented with references to L’Enseignement Mathématique (EM), to the ICMI Bulletin, and to other documentation that was deemed of interest. In the first decades, the events were limited in number and geographical location. Later, when the new social and political contexts made evident the need to change the pattern of communication in most fields of human activities and to direct attention also to “peripheries”, ICMI broadened its field of action, so the events became more numerous and involved various parts of the world. This also led to the creation of specific documentation of its activities (bulletins, dedicated sites, conference proceedings, book series) which testifies to the most recent evolution. As for the last decades we have prevalently reported on the initiatives that have had a regular development after the first edition. The authors of this Timeline are Livia Giacardi (for the years 1908–1989) and Fulvia Furinghetti (for the years 1990–2008). Further details on the first part of this Timeline can be found in the Timeline (Giacardi 2008a, 2008b) on the website dedicated to the history of ICMI.1  The authors are very grateful to Geoffrey Howson for his comments and useful suggestions.

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F. Furinghetti (*) University of Genoa, Genoa, Italy e-mail: [email protected] L. Giacardi (*) University of Turin, Turin, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_5

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5.1 1908–1911 1908  – Rome (Italy) The Commission Internationale de l’Enseignement Mathématique (CIEM), Internationale Mathematische Unterrichtskommission (IMUK) is founded during the Fourth International Congress of Mathematicians (ICM) (Rome 6–11 April 1908): Section IV [devoted to philosophical, historical, and didactic questions], having recognized the importance of a comparative examination of the programs and methods of the teaching of mathematics in the secondary schools of the various nations, entrusts to the Professors KLEIN, GREENHILL and FEHR the task of setting up an international committee to study the matter and report on it to the next Congress.2 (Proceedings ICM 1909, Vol. I, p. 51).

The three mathematicians designated by the Congress constitute the First Central Committee3: President: F. Klein. Vice-President: G. Greenhill. Secretary-General: H. Fehr. 23–24 September 1908  – Meeting in Cologne (Germany) (EM 10, 1908: 445–458). During the meeting the organization of CIEM/IMUK is decided. Commission defines its priority: “Make an inquiry and publish a general report on the current trends in the teaching of mathematics in various countries” bearing in mind not only secondary schools but all kinds and levels of schooling.4 (EM 10 1908, p. 450).

L’Enseignement Mathématique is the official publication of CIEM/IMUK. The official languages are those admitted at the ICMs: German, English, French, and Italian. The Commission is made up of delegates from countries which have participated in at least 2 ICMs with an average of at least two members5:

 The original text is: “La Sezione IV, avendo riconosciuto l’importanza di un esame comparato dei programmi e dei metodi dell’insegnamento delle matematiche nelle Scuole secondarie delle varie Nazioni, confida ai Prof.i KLEIN, GREENHILL e FEHR l’incarico di costituire un Comitato internazionale che studii la questione e ne riferisca al prossimo Congresso”. 3  For further details on the Central/Executive Committees of ICMI see Chap. 6 by Fulvia Furinghetti in this volume. 4  The original text is: “Faire une enquête et publier un rapport général sur les tendances actuelles de l’enseignement mathématique dans les divers pays.” 5  Table 5.1 and Table 5.2 at the end of the timeline show the countries involved in the Commission at the beginning and after 100  years. With regard to the process of internationalization of the Commission, see Chap. 9 by Livia Giacardi in this volume. 2

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5  Timeline of ICMI 1908–2008 Table 5.1  Countries represented on the Commission (EM 10, 1908: 447–448) Countries which have participated in at least 2 ICMs with an average of at least two members Austria Belgium Denmark Spain USA France Germany Greece Holland Hungary the British Isles Italy Norway Portugal Romania Russia Sweden Switzerland

Associated countries whose delegates are permitted to follow the activities of the Commission, without having the right to vote Argentina Australia Brazil Bulgaria Canada Chile China Cape Colony Egypt Indian Raj Japan Mexico Peru Serbia Turkey

5–6 April 1909 – Meeting in Karlsruhe (Germany) (EM 11, 1909: 193–204). The delegates of 16 of the 18 countries to be represented on CIEM/IMUK are appointed. The National subcommissions are to constitute the infrastructure necessary for producing successful inquiries and reports. 10–16 August 1910 – Meeting in Brussels (Belgium), Congrès international de l’enseignement moyen (EM 12 1910: 353–415). During this meeting, the activities of the various countries represented on CIEM/ IMUK are presented. The work being carried out by the German subcommission is particularly striking. 18–21 September 1911 – Congress in Milan (Italy) of CIEM/IMUK (EM 13, 1911: 437–511). During the meeting the following issues are addressed: –– Rigor in middle school teaching and the fusion of the various branches of mathematics. –– The teaching of mathematics to students of physical and natural sciences.

5.2 1912–1921 21–27 August 1912, Cambridge (United Kingdom) – Congress of CIEM/IMUK (EM 14, 1912: 441–537) held on the occasion of the Fifth ICM (Proceedings ICM 1913).

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Results of the inquiries about the following subjects are presented: 1. The mathematical training of the physicist in the University (EM 14, 6, 1912: 495–507). 2. Intuition and experiment in mathematical teaching in the secondary schools (EM 14, 6, 1912: 507–534).6 The mandate of CIEM/IMUK is renewed (Proceedings ICM 1913, p. 41). Central Committee for the years 1912–1920: President: F. Klein. Vice-President: G. Greenhill, D. E. Smith. Secretary-General: H. Fehr. Members coopted in 1913: G. Castelnuovo, E. Czuber, J. Hadamard. 1–4 April 1914 – Paris Congress of CIEM/IMUK (EM 16, 1914: 165–226; EM 16, 1914: 245–356). The congress is attended by more than 160 participants from 17 countries. G. Castelnuovo in his opening speech (in place of Klein) among other things, reports on the broadening of the field of investigation of the Commission, which is no longer to be limited to secondary schools, but which now covers schools of every type and level. In addition, he underlines the Commission’s aim: It is … becoming increasingly necessary to know, even in the field of education, what our neighbours are doing and to benefit from their experience. Knowledge, moreover, does not impose action, but action would be blind without knowledge.7 (EM 16, 1914: 189).

During the congress the following issues are addressed: –– The results obtained in secondary schools by the introduction of differential and integral calculus in different countries. –– The mathematical training of engineers in different countries. This congress could be considered the first International Congress on Mathematical Education. 1914–1918 – First World War 1916  – The ICM scheduled for this year in Stockholm does not take place because of World War I. 1916–1920 – The CIEM/IMUK does not meet, and brief reports of the work of some national subcommissions and several papers on the technical and practical training of mathematics teachers in secondary schools are published in L’Enseignement Mathématique (EM 18, 1916: 335–361, 429–439; EM 21, 1920–1921: 281–304; EM 22, 1921–1922: 286–290).

 About the didactic sections in the ICMs, see Chap. 8 by Fulvia Furinghetti in this volume.  The original text is: “Il devient … toujours plus nécessaire de connaître, même en matière d’instruction, ce que font nos voisins et de profiter de leur expérience. La connaissance d’ailleurs n’impose pas l’action, mais l’action serait aveugle sans la connaissance.” 6 7

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18–28 July 1919, Brussels (Belgium) – The Constitutive Assembly of the International Research Council (IRC), is held. The initial membership of IRC is restricted to the Allied Powers; the Central Powers are excluded. The first steps toward the establishment of the International Mathematical Union (IMU) are taken. 22–30 September 1920, Strasbourg (France) – ICM under the chairmanship of E. Picard. (Proceedings ICM 1921). During the congress IMU is established. The Central Powers are excluded. Given that the Strasbourg ICM represents only a part of those countries that had given the mandate to CIEM/IMUK, the Commission does not present any report.8 The mandate of the Commission is not renewed. April 1921 – H. Fehr announces the dissolution of the CIEM/IMUK: The new conditions imposed on official international scientific relations oblige the international associations or commissions created before the war to proceed to their dissolution or to their reorganization… The members of the Central Committee recognized that dissolution has become inevitable in the current situation.9 (EM 21, 1920–1921: 317).

National sub-commissions may, however, continue with their work and may still send them to L’Enseignement Mathématique for publication. Fehr presents a report on the activities of the Commission from 1908 to 1920 complete with a list of the publications of the Central Committee and of the national sub-commissions. (EM 21, 1920–1921: 305–337).

5.3 1922–1945 11–16 August 1924, Toronto (Canada) – ICM under the chairmanship of J. C. Fields. (Proceedings ICM 1928). The IMU policy of exclusion is confirmed even if many mathematicians oppose the boycott against colleagues from the former Central Powers. 1925 Felix Klein dies. (EM 24, 1924–1925: 287–290). 3–10 September 1928, Bologna (Italy) – The ICM is held without restrictions under the chairmanship of S. Pincherle, who is president both of the Unione Matematica Italiana and of IMU. The participation of the German mathematicians, non-members of the IRC, strongly desired by Pincherle in the name of internationality of science, violates the rules of IMU.  For this reason, during the General

 See Chap. 1 by Gert Schubring in this volume.  The original text is: “Les conditions nouvelles qui se trouvent ainsi imposées aux relations scientifiques internationales d’un caractère officiel, obligent les associations ou commissions internationales créées avant la guerre à procéder tour à tour à leur dissolution ou à leur réorganisation… Les membres du Comité central ont reconnu que, dans la situation actuelle, la dissolution de la Commission est devenue inévitable.” 8 9

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Assembly of IMU, he submits his “absolutely irrevocable” resignation from his position as president (Proceedings ICM 1929, Vol. 1, pp. 5–10). During the congress H. Fehr, Secretary-General of CIEM/IMUK, presents a detailed report on the commission’s activities since 1908 (Proceedings ICM 1929, Vol. 1, pp. 106–113). At the end of his speech, the participants in Section VI (Elementary mathematics) agree on the following two points: that the commission be reconstituted in such a way that all the countries participating in the Congress are represented; that the existing Committee be reconfirmed, and that it elects the other members of the Commission (Proceedings ICM 1929, Vol. 1, p. 113). Central Committee for the years 1928–1932: President: D. E. Smith. Vice-Presidents: G. Castelnuovo, J. Hadamard. Secretary-General: H. Fehr. Member: W. Lietzmann. 5–12 September 1932, Zurich (Switzerland) – ICM under the chairmanship of R. Fueter (Proceedings ICM 1932). G. Loria presents the general report on “La préparation théorique et pratique des professeurs de mathématiques de l’enseignement secondaire dans les divers pays”, a summary of the reports submitted by the various national delegations of CIEM/ IMUK. It will be published in full in EM, 32, 1933: 5–20. The CIEM/IMUK is invited to continue its activities for a further four years. Central Committee for the years 1932–1936: President: J. Hadamard. Vice-Presidents: P. Heegaard, W. Lietzmann, G. Scorza. Secretary-General: H. Fehr. Member: (co-opted) E. H. Neville. A new subject for study is proposed: “Present trends in the teaching of mathematics in the various countries” (EM 31, 1932: 266). 15 September 1935 – Promulgation of the Nuremberg Laws. A.  Hitler’s national socialist regime institutionalizes its discrimination against Jews. Hundreds of Jewish university lecturers are removed from their teaching posts and some of the most important German scientists are forced to leave Germany. 13–18 July 1936, Oslo (Norway) – ICM under the presidency of C. Størmer (Proceedings ICM 1937). Various national delegations of CIEM/IMUK present their reports on the “Present trends in the teaching of mathematics in the various countries”, which will be published in full in EM 36, 1937: 236–262; 357–388, and in EM 37, 1938: 205–211. Lars Ahlfors (University of Helsinki) and Jesse Douglas (Institute of Technology of Cambridge, Massachusetts) are awarded the first Fields Medals (Proceedings ICM 1937, Vol. 1, p. 45).

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15 July 1936  – The Commission confers the title of Honorary Member on: E.  Beke (Budapest), Ch. Bioche (Paris), G.  Castelnuovo (Rome), S.  Dickstein (Warsaw), F.  Enriques (Rome), Farid Boulad Bey (Cairo), G.  Loria (Genoa), M. Petrovitch (Belgrade), and W. Wirtinger (Vienna) (Proceedings ICM 1937, Vol. 2, p. 289). During the congress CIEM/IMUK is requested to continue its work, and to prosecute such investigations as decided by the Central Committee (Proceedings ICM 1937, Vol. 2, p. 289, and EM 35, 1936: 388), but because of World World II the Commission will remain inactive until 1952. 1939–1945 – World War II The International Congress of Mathematicians which was to have taken place in Cambridge (USA) is postponed until a more favorable moment (EM 38, 1939–1940: 165).

5.4 1946–1954 1946 – UNESCO (United Nations Educational Scientific and Cultural Organisation) is founded with the aim of promoting international collaboration in the areas of education. 1948 – OEEC (Organization for European Economic Co-operation), an international organization for economic cooperation and development, is created. In 1961 it will be substituted by OECD (Organization for Economic Co-operation and Development). 30 August-6 September 1950, Cambridge (USA)  – ICM under the chairmanship of O. Veblen (Proceedings ICM 1952). CIEM/IMUK does not have a place of its own in the program of the congress. W.  Betz in his communication “Mathematics for the million, or for the few?” expresses the hope for the recommencement of the work begun in 1908 by the Commission (Proceedings ICM 1952, Vol. 1, p. 752). 6–8 March 1952, Rome (Italy) – The First General Assembly of the reconstituted International Mathematical Union (IMU) takes place in the Villa Farnesina. During the Assembly, the International Commission on the Teaching of Mathematics is transformed into a permanent subcommission of the IMU. The official languages remain English, French, German, and Italian, and the official organ is L’Enseignement Mathématique. (Internationale Mathematische Nachrichten (IMN) 27–28, 1953: 6–7). IMU designates H. Behnke, A. Châtelet, R.L. Jeffery, Đ. Kurepa as members of the new commission, and requests that H. Fehr remains at the disposal during the transition period (EM 39, 1942–1950: 162–163).

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April 1952  – CIEAEM, Commission Internationale pour l’Étude e l’Amélioration de l’Enseignement des Mathématiques is officially established. The president is the mathematician Gustave Choquet (Paris), the vice-president is the psychologist Jean Piaget (Geneva), and the secretary is the mathematician and an expert in pedagogy Caleb Gattegno (London). February 1953 and January 1954, Paris  – The Executive Committee of ICMI for the years 1952–1954 is completed as follows: Honorary President: H. Fehr. President: A. Châtelet. Vice-Presidents: Đ. Kurepa, S. Mac Lane. Secretary: H. Behnke. Treasurer: G. Ascoli. Members: A. F. Andersen, E. W. Beth, R. L. Jeffery, E. A. Maxwell. Ex-officio: M. H. Stone, president of IMU (EM 40, 1951–1954: 81–82). 31 August-1 September 1954, The Hague – The Second General Assembly of IMU determines the Terms of Reference of the Commission on the Teaching of Mathematics and the name International Commission on Mathematical Instruction (ICMI) is adopted.10 With regard to the aims of the ICMI the point g) says: The Commission shall be charged with the conduct of the activities of IMU, bearing on mathematical and scientific education, and shall take the initiative in inaugurating appropriate programs designed to further the sound development of mathematical education at all levels and to secure public appreciation of its importance (IMN 35-36, 1954: 12–13).

2–9 September 1954, Amsterdam (Netherlands)  – ICM under the presidency of J. A. Schouten (Proceedings ICM 1954–1957). The presence of ICMI is significant. Đ. Kurepa presents the report on “The role of mathematics and mathematician at present time”, and various communications are devoted to mathematical instruction for students between 16 and 21 years of age (Proceedings ICM 1954–1957, Vol. 3, pp. 297–324). 2 November 1954 – Henri Fehr, Secretary-General of ICMI since its founding and honorary president since 1952, dies (EM s. 2, 1, 1955: 4–17).

5.5 1955–1958 2 July 1955, Geneva (Switzerland)  – H.  Behnke convokes the new Executive Committee of ICMI for the period 1955–1958, which is constituted as follows: President: H. Behnke. Vice-Presidents: Đ. Kurepa, M.H. Stone. 10

 About these and subsequent Terms of reference, see Chap. 7 by Livia Giacardi in this volume.

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Secretary: J. Desforge. Members: Ram Behari, E.A. Maxwell, K. Piene. Ex officio: H. Hopf (President of IMU). The Members at large are: Y. Akizuki, G. Ascoli, H. Behnke, Ram Behari, P. J. Dubreil, J. C. H. Gerretsen, R. L. Jeffery, Đ. Kurepa, E. A. Maxwell, M. H. Stone (EM s. 2, 1, 1955: 195–198). Within a short time 21 countries nominate their two delegates. 1955 – The second series of L’Enseignement Mathématique begins. 22–28 February 1956, Bombay (India) – The First “South Asian Conference on Mathematical Education” is held at the Tata Institute of Fundamental Research. ICMI is represented by its vice-president, M. Stone, who gives one of the invited talks (Report of a Conference on Mathematical Education in South Asia, The Mathematics Student 24, 1956: 1–183). 4 October 1957 – The Soviet launch of Sputnik stimulates debate in the USA and in Europe about the need for major reforms in scientific and mathematical education. 11–13 August 1958, St. Andrews (Scotland) – The General Assembly of IMU agrees on the necessity to make certain changes in the regulations of ICMI in such a way that the Commission is made up of 10 members at large elected by the General Assembly of the IMU upon nomination by its President, and of 1 representative of each national subcommittee (EM s. 2, 4, 1958: 227–228). The General Assembly elects the following ten members-at-large of ICMI: Y. Akizuki, A. D. Aleksandrov, H. Behnke, P. Buzano, G. Choquet. Howard Fehr, H. Freudenthal, Đ. Kurepa, E. A. Maxwell, M. H, Stone. Marshall Stone is elected President of ICMI. (IMN 68–69, 1961: 26 and 29). 14–21 August 1958, Edinburgh (Scotland) – ICM under the presidency of W.V.D. Hodge (Proceedings ICM 1960). ICMI’s participation in the congress has been carefully prepared in two meetings of the ICMI Executive Committee in Münster-Westfalen (27 May 1956 and 28 May 1958: EM s. 2, 2, 1956: 317–323; 4, 1958: 213–219). Among other things three reports were presented on: “Mathematical instruction up to the age of fifteen years” (Howard Fehr); “The scientific bases of mathematics in secondary education” (H.  Behnke); “Comparative study of methods of initiation into geometry” (H. Freudenthal). Executive Committee of ICMI and Members at large of ICMI for the years 1959–1962: President: M. H. Stone. Vice-Presidents: H. Behnke, Đ. Kurepa. Secretary: G. Walusinski. Members: Y. Akizuki, A. D. Aleksandrov, O. Frostman. Ex officio: R. Nevanlinna (President of IMU). Members at large: Y. Akizuki, D. Aleksandrov, H. Behnke, P. Buzano, G. Choquet, H. Fehr, H. Freudenthal, Đ. Kurepa, E. A. Maxwell.

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Successively J.  Karamata, will be co-opted, as the editor of L’Enseignement Mathématique. (EM s. 2, 5, 1959: 290–292).

5.6 1959–1961 23 November–4 December 1959, Asnière-sur-Oise (France) – At the initiative of the OEEC, a seminar is held at the Centre Culturel de Royaumont on the new thinking in school mathematics. Chairman of the seminar is the president of ICMI, M. Stone, who gives the introductory address, formulating a veritable “program of research in the teaching of mathematics” (study and experimentation) (Appendix B in New Thinking in School Mathematics, OEEC, 1961, p. 28). One of the most influential talks is that of Jean Dieudonné, whose proposals for the reform of the teaching of mathematics are explicitly inspired by Bourbaki’s ideas. He pronounces the famous sentence: “If the whole program I have in mind had to be summarized in one slogan it would be: Euclid must go!” (New Thinking in School Mathematics, OEEC, 1961, p. 35). 1959 – The first International Mathematical Olympiad (IMO) is held in Romania. Paris, April 1960  – New Terms of Reference for ICMI are adopted (IMN 68–69, 1961: 29; ICMI Bulletin, 5, 1975: 5–6). 1960–1961 – Various seminars and symposia are organized by ICMI in collaboration with other associations and contacts made with OEEC, UNESCO, and other international organizations: 30 May – 2 June 1960, Aarhus (Denmark) – Symposium on “The teaching of geometry in secondary school” (IMN 66, 1961: 1). 21 August – 19 September 1960, Zagreb – Dubrovnik (Yugoslavia) – A seminar is held in order to prepare a program for secondary schools that can be used as the basis for the preparation of textbooks and experimental courses (Un programme moderne de mathématique pour l’enseignement secondaire, OEEC, 1961; Bollettino della Unione Matematica Italiana s. 3, 17, 1962: 204). 19–24 September 1960, Belgrade (Yugoslavia) – International Symposium on “The Coordination of the Instruction of Mathematics and Physics” (IMN 66, 1961: 4–5; EM s. 2, 6 1960: 140–141). 26–29 June 1961, Lausanne (Switzerland) – Seminar on the teaching of analysis and relative textbooks. (EM s. 2, 6, 1960, p. 311; EM s. 2, 8, 1962: 93–178). 4–8 October 1961, Bologna (Italy) – Seminar on “A discussion of the Aarhus and Dubrovnik reports on the teaching of geometry at the secondary level” (Bollettino della Unione Matematica Italiana s. 3. 17, 1962: 199–212; EM s. 2, 9, 1963: 1–104). 4–9 December 1961, Bogotá (Colombia)  – The First Inter-American Conference on Mathematical Education is organized by ICMI with the co-­ operation of UNESCO and other local organizations (IMN 70 1962: 9; 71, 1962: 2).

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5.7 1962–1965 10 August 1962, Saltsjöbaden (Sweden) – During the meeting of ICMI, president M. Stone, in his “Report for the period 1959–1962”, points out the Commission’s financial problems, and urges “that ICMI study methods and means for satisfying the growing demand for an international bibliographical informational service in the field of education and extend its activity to new areas, such as Africa” (EM s. 2, 9, 1963: 111).  11-13 August 1962, Saltsjöbaden – The General Assembly of IMU takes place  15–22 August 1962 Stockholm (Sweden)  – ICM under the presidency of R. Nevanlinna (Proceedings ICM 1963). ICMI organizes three meetings on the modern mathematics at the secondary school and on training of mathematics teachers: 1. Which subjects in modern mathematics and which applications of modern mathematics can find a place in programs of secondary school instruction? 2. Connections between arithmetic and algebra in the mathematical instruction of children up to the age of 15. 3. Education of the teachers for the various levels of mathematical instruction (Proceedings ICM 1963, p. XXXVI; EM s. 2, 10, 1964: 152–176). Executive Committee of ICMI and Members at large for the years 1963–1966: President: A. Lichnerowicz. Vice-Presidents: S. Straszewicz, E. Moise. Secretary: A. Delessert. Members: Y. Akizuki, H. Behnke, H. Freudenthal. Ex officio: G. de Rham (President of IMU). Members at large: S. Bundgaard, G. Choquet, O. Frostman, R.L. Jeffery, J. Karamata. (EM s. 2, 10, 1964: 297). 14–15 February 1964, Paris (France) – During the meeting, ICMI, in agreement with the President of IMU, decides to recognize the status of national subcommission to national commissions of countries that are not members of IMU (EM s. 2, 12, 1966: 134). This decision is immediately put into effect in the case of Luxemburg, and successively in that of Senegal. The Executive Committee decides to adhere to two important initiatives of UNESCO concerning the creation of a Centre for documentation and information about mathematics teaching, and the preparation of a source book including a list of textbooks, periodicals, anthologies and translations of articles, etc. (EM s. 2, 10, 1964: 295–296). February 1964 – The Technical Committee 3 (TC 3 –Education) of the International Federation for Information Processing (IFIP, established in 1960) holds its initial meeting in Paris.

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1964–65 – A very rich period of noteworthy activity begins for ICMI thanks to the collaboration with UNESCO and other organizations. Many international symposia are held: 8–10 October 1964, Frascati (Italy) – “Mathematics at the coming to university. Real situation and desirable situation” (Archimede 16, 1964: 314–320). 19–22 December 1964, Utrecht (Netherlands) – “Colloquium on modern curricula in secondary mathematical education” (IMN 80, 1965: 3, 8; EM s. 2, 12, 1966: 195–199). In his lecture “Priorities and responsibilities in the reform of mathematical education: an essay in educational meta-theory”, A. Wittenberg underlines the necessity of creating university chairs for mathematical education and the urgency of founding an international journal specifically dedicated to mathematics teaching so that different approaches can be compared (EM s. 2, 11, 1965: 287–308). 14–22 January 1965, Dakar (Senegal) – “Congress on science teaching and its role in economic progress”. A parallel meeting is organized by ICMI in collaboration with the National Senegalese Commission of the teaching of mathematics from 13 to 16 January (EM s. 2 12, 1966:131). 30 May  – 4 June 1965, Echternach (Luxemburg)  – “The repercussions of mathematics research and teaching” (IMN 83, 1966: 3; EM 12, 1966: 132). With the support of UNESCO, the publication of a series of volumes entitled New Trends in Mathematics Teaching is planned, and Anna Zofia Krygowska is entrusted with the task of overseeing the publication (IMN 83, 1966: 3).

5.8 1966–1970 20 May 1966 – The outgoing Executive Committee of ICMI, given the remarkable broadening of the Commission’s activities, requests the creation of a permanent Secretariat for ICMI (EM s. 2, 12, 1966: 138), which will be refused (O. Frostman to the International Commission on Mathematical Instruction, Djursholm, 29 June 1967).11 13–16 August 1966, Dubna (USSR)  – The General Assembly of IMU takes place  16–26 August 1966 Moscow (USSR)  – ICM under the chairmanship of I. G. Petrovsky (Proceedings ICM 1968). In the session dedicated to ICMI, the following reports are presented: 1. Ch. Pisot, “Rapport sur l’Enseignement des mathématiques pour les physiciens”. (EM s. 2, 12, 1966: 201–216); 2. Z.  Krygowska, “Développement de l’activité mathématique des élèves et role des problèmes dans ce développement” (EM s. 2, 12, 1966: 293–322).

11

 See this letter in Chap. 4 by Livia Giacardi in this volume.

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Executive Committee of ICMI for the years 1967–1970: President: H. Freudenthal. Vice-Presidents: E. Moise, S. Sobolev. Secretary: A. Delessert. Members: H. Behnke, A. Revuz, B. Thwaites. Ex officio: H. Cartan (President of IMU). 16–20 January 1967, Lausanne (Switzerland) – The UNESCO colloquium on “Coordination of Instruction of Mathematics and Physics” in held. H. Freudenthal and W. Servais participate. Important resolutions concerning the teaching of mathematics are adopted. These principles were published as “Propositions on the teaching of mathematics” in the first issue of the journal Educational Studies in Mathematics (1, 1968: 244). July 1967  – The first volume prepared by ICMI and published by UNESCO, New Trends in Mathematics Teaching, I, 1966 (Paris, 1967) is published. 21–25 August 1967, Utrecht (Netherlands) – The colloquium “How to teach mathematics so as to be useful” is held. A year later, the proceedings will be published in the new journal Educational Studies in Mathematics (1, 1–2, 1968: 1–243). 26 August 1967 – During the meeting of ICMI president H. Freudenthal affirms that the practice of quadrennial reports at the ICMs is not a good one because the national reports are generally useless, and proposes the idea of an ICMI congress to take place the year before the ICM. A. Revuz solicits the founding of a new journal expressly aimed at the secondary school teachers (EM s. 2, 13, 1967: 245–246). May 1968 – Freudenthal launches the new journal, Educational Studies in Mathematics (ESM), a truly international journal devoted to mathematics education. The initiative is funded by UNESCO, and IMU is not consulted beforehand. IMU President H. Cartan complains that he was not informed about the ICMI initiative at the Paris meeting in May 1968 (see Chap. 2 by F. Furinghetti and L. Giacardi in this volume; Lehto 1998, p. 259). 23 September – 2 October 1968, Bucharest (Romania) – The international colloquium “Modernization of Mathematics Teaching in European Countries” is organized by UNESCO.  H. Freudenthal, president of ICMI, A.  Delessert and A.  Revuz, members of the Executive Committee, participate. Among the recommendation of the colloquium, the cooperation with ICMI is solicited (Modernization of Mathematics Teaching in European Countries. Bucharest: Éditions didactiques et pédagogiques, 1968, pp. 549–556). 1969  – H.  G. Steiner founds—based on the Zentrum für Didaktik der Mathematik—the Zentralblatt für Didaktik der Mathematik (ZDM), the first truly international journal on the didactics of mathematics in Germany (IMN 97, 1971: 5). On the back cover of the issues, I to XI is written: “Zentralblatt für Didaktik der Mathematik herausgegeben in Verbindung mit ... und der Internationalen Mathematischen Unterrichtskommission (IMUK)”.

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24–30 August 1969, Lyon (France)  – The First International Congress on Mathematical Education (ICME-1) takes place.12 (Proceedings ICME-1 1969). The Congress is attended by 655 active participants from 42 countries (IMN 97, 1971: 4–5, ICMI Bulletin 5, 1975: 20–24). The main resolutions of the congress concerned: the modernization of the teaching of mathematics, both in content and method; the collaboration between teachers of mathematics and those of other disciplines; international cooperation; the permanent training of the teachers; the place of “the theory of mathematical education” in universities or research institutes. In particular, the Point 5 of the resolutions says: The theory of mathematical education is becoming a science in its own right, with its own problems both of mathematical and pedagogical content. The new science should be given a place in the mathematical departments of Universities or Research Institutes, with appropriate academic qualifications available (Proceedings ICME-1 1969, p. 416).

ICME-1, as G. Howson affirms, “was a landmark in ICMI’s history. Over six hundred mathematics educators from forty-two countries met in an unprecedented fashion” (ICMI Bulletin 16, 1984: 6). 28–30 August 1970, Menton (France)  – The General Assembly of IMU takes place. During the Assembly, M. J. Lighthill is elected ICMI president, along with the members at large, for the coming four-year term and it is decreed that the past president of ICMI, the secretary of IMU and the representative of the International Council of Scientific Unions (ICSU) Committee on the Teaching of Science should be members ex officio of ICMI (EM s. 2, 16, 1970: 198). 1–10 September 1970, Nice (France) – ICM under the presidency of J. Leray. The section Enseignement des mathématiques, includes four invited talks (Z.  Krygowska, H.  B. Griffiths, H.  O. Pollak, S.  L. Sobolev) (Proceedings ICM 1971, Vol. 3, pp. 335–367). 5 September 1970 – During the congress the ICMI meeting takes place. Two recommendations are formulated: –– The regulations that establish the ways that ICMI members are designated must be modified; –– It is mandatory to ensure that ICMI members are always chosen from among those who are effectively involved with mathematics teaching (EM s. 2, 16, 1970: 198). Executive Committee of ICMI and Members at large for the years 1971–1974: President: M. J. Lighthill. Vice-Presidents: S. Iyanaga, J. Surányi. Secretary: E.A. Maxwell. Members: H. O. Pollak, S. L. Sobolev.  For the ICMEs see Chap. 3 in this volume and https://www.icmihistory.unito.it/icme1.php (Retrieved June 2021) by Marta Menghini; see also Chap. 10 by Fulvia Furinghetti in this volume. 12

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Ex Officio: H. Freudenthal (Past President of ICMI), K. Chandrasekharan (President of IMU), O. Frostman (Secretary of IMU), A. Lichnerowicz (CTS/ICSU). Members at large: M.  Barner, F.  Châtelet, A.  Gleason, L.  Lombardo Radice, Y. Mimura, J. Novak. 38 countries nominate their delegate (ICMI Bulletin 1, 1972: 5–7). 1970 – The Journal for Research in Mathematics Education is founded.

5.9 

1971–1976

January 1971 – The first issue of the Bulletin of the International Mathematical Union (IMU-Bulletin) appears. 26 January 1971, Utrecht (Netherlands) – The Instituut voor de Ontwikkeling van het Wiskunde Onderwijs (IOWO, Institute for the Development of Mathematics Instruction) is established by H.  Freudenthal, who develops a new approach to mathematics education, the so-called Realistic Mathematics Teaching (ESM 7(3), 1976). 29 August – 2 September 1972, Exeter (United Kingdom) – ICME-2. The Congress, attended by 1384 participants from 73 countries, is structured differently than the preceding one: there are only 9 invited lectures on themes of general interest and 38 working groups on more specific questions (Proceedings ICME-2 1973, p. 13; ICMI Bulletin 5, 1975: 20–24).13 October 1972 – The first issue of the ICMI Bulletin appears. In his Foreword ICMI President James Lighthill writes: It is my sincere hope that these Bulletins will make really useful contributions to the promotion of international understanding among those involved in teaching that uniquely international mode of thought: mathematics (ICMI Bulletin 1, 1972: 1).

1973 – Heinrich Bauersfeld, Michael Otte and Hans Georg Steiner found the Institut für Didaktik der Mathematik (IDM) at the University of Bielefeld (ICMI Bulletin 3, 1974: 7–8). 1973–1974 – A new policy of holding Regional Symposia is adopted by ICMI in order “to facilitate wider discussion of mathematical education outside those areas of Europe and America where international meetings on the subject have mainly been held hitherto” (Seventh General Assembly, IMU Bulletin, 1974: 8). 1–11 September 1974, Nairobi (Kenya) – The first Regional Symposium is sponsored jointly by ICMI, UNESCO, and CEDO (Centre for Educational Development Overseas), on theme: “Interactions between linguistics and mathematical education” (ICMI Bulletin 3, 1974: 3–7; 4 1974: 6–8).

 In (ICMI Bulletin 5, 1975: 20–24) the dates of the congress are “29 August – 3 September 1972” and the number of countries is 76. 13

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5–9 November 1974, Tokyo (Japan) – The second Regional Symposium is sponsored jointly by ICMI and JSME (Japan Society of Mathematical Education), on the theme “Curriculum and teacher training for mathematical education” (ICMI Bulletin 4, 1974: 9). 17–19 August 1974, Harrison Hot Springs (Canada) – The General Assembly of IMU takes place 21–29 August 1974, Vancouver (Canada) – ICM is held under the presidency of H. S. M. Coxeter (Proceedings ICM 1975). 1974 – The ICMI Executive Committee resolves to consider the Inter-American Committee on Mathematical Education (IACME) as an effective regional group (ICMI Bulletin 4, 1974: 11). Executive Committee of ICMI and Members at large for the period 1975–1978: President: S. Iyanaga. Vice-Presidents: B. Christiansen, H. G. Steiner. Secretary: Y. Kawada. Members: E. Begle, L. D. Kudrjavcev. Ex officio: J.  Lighthill (Past President of ICMI), D.  Montgomery (President of IMU), J.-L. Lions (Secretary of IMU), H. Freudenthal (CTS/ICSU). Members at large: E. Castelnuovo, J. Lelong-Ferrand, B. H. Neumann, Z. Semadeni, J. Surányi, P. L. Bhatnagar (EM 21, 1975: 331–333). 1975  – Numerous congresses are held: 18–22 August 1975, Nyiregyháza (Hungary); 25–28 August 1975, Warsaw (Poland); 1–5 September 1975, Marseille (France); 1–6 December 1975, Caracas (Venezuela); 15–20 December 1975, Bharwari (India) (ICMI Bulletin 6, 1975: 7–11; 7, 1976: 4–21). 26–31 July 1976, Rabat (Morocco)  – The First Pan-African Congress of Mathematicians is held (ICMI Bulletin 8, 1976: 4–8). 16–21 August 1976, Karlsruhe (Federal Republic of Germany)  – ICME-3 (Proceedings ICME-3 1977). The congress is attended by 1831 full members from almost 80 countries; in addition, 237 associate members participate (EM s. 2, 22, 1976: 317–320). ICME-3 differs in many ways from its immediate predecessor: little emphasis is laid on full plenary sessions, and the congress is built around thirteen sections covering most aspects of education. Each of these sections would correspond to a chapter of a new volume of the series New Trends in Mathematics Teaching. This volume is foreseen “to have the character of a process, and not to consist in the author’s development of their own ideas in finished form” (Christiansen, Bent. 1978. The collaboration between ICMI and UNESCO. ICMI Bulletin 10: 6). UNESCO plays an important role in shaping the congress. The General Assembly of ICMI is held on that occasion (ICMI Bulletin 8, 1976: 14–19).

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1976 – The International Study Group on the Relations between the History and Pedagogy of Mathematics (HPM), an affiliated Study Group14 of ICMI, is established with its first president Ph. S. Jones (ICMI Bulletin 10, 1978: 26–27). The International Group for the Psychology of Mathematics Education (PME), an affiliated Study Group of ICMI, is established with E. Fischbein as its first president (ICMI Bulletin 16, 1984: 8; Proceedings ICME-3 1977, pp. 385–386).

5.10 1977–1979 1977, Utrecht (Netherlands) – The first PME Conference. 23–27 August 1977, Pécs (Hungary) – International Conference on the problems of training of teachers of mathematics. The conference is attended by 156 participants, 93 from Hungary. Among the speakers there are J. Surányi, H. Freudenthal, H.G. Steiner e A.Z. Krygowska (ICMI Bulletin 10, 1978: 11–13). 1978 – Two special issues on “Change in mathematics education since the late 1950s –- ideas and realization” are published in Educational Studies in Mathematics (Part I in 9.2; Part II in 9.3, 1978) (ICMI Bulletin 9, 1977: 19). In the introduction Freudenthal suggests a change with respect to the past: The issues would consist of critical reports on what has happened in various countries. In order to make clear how they may like to interpret this theme, I will first of all stress what the reports are not expected to present: No formal descriptions of existing or prescribed instruction, such as official teaching programmes, more or less official guidelines, examination regulations, lists of textbooks and other materials, formal lists of objectives, extensive statistical material (ESM 9.2, 1978: 143).

Australia, Bangladesh, France, Great Britain, Hungary, India, Iran, The Netherlands, Nigeria, Poland, Sierra Leone, Sri Lanka, Sudan, Thailand, USA, and West Indies presented a report. 29 May – 3 June 1978, Luxembourg City – ICMI Seminar on “Calculators in school teaching”. The meeting is attended by 135 participants from 12 countries. The seminar can be considered the continuation of those of Echternach started in 1965 (ICMI Bulletin 11, 1978: 11–12). 29 May  – 3 June 1978, Manila (Philippines)  – The First Southeast Asian Conference on Mathematical Education (SEACME). The conference is attended by 1085 local and 20 foreign mathematicians from Japan, France, Poland, Australia, Indonesia, Malaysia, Singapore, Thailand, and Hong Kong. Elementary, secondary, and tertiary-graduate mathematical education are taken into consideration. ICMI president S. Iyanaga gives the following plenary

 For information on the affiliated Study Groups of ICMI see: https://www.icmihistory.unito.it/ group.php (Retrieved June 2021) 14

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lecture: “The role of mathematical organizations in the improvement of mathematical education”15 (ICMI Bulletin 11, 1978: 5–10). 11–12 August 1978, Otaniemi (Finland) – General Assembly of IMU. During the Assembly 10 members at large of ICMI are elected and the president of ICMI is nominated (IMU Bulletin 14, 1978: 13). Executive Committee of ICMI for the years 1979–1982: President: H. Whitney. Vice-presidents: U. D’Ambrosio, B. Christiansen. Secretary: P. Hilton. Members: S.H. Erlwanger, B.H. Neumann, Z. Semadeni. Ex officio: S. Iyanaga (past president of ICMI), L. Carleson (president of IMU), J. L. Lions (IMU secretary), B. Christiansen (Representative of IMU in CTS/ICSU). (ICMI Bulletin 11, 1978: 3–4) 15–23 August 1978, Helsinki (Finland)  – ICM under the presidency of O. Lehto (Proceedings ICM 1980, pp. 21–43). 16–21 August 1978 – During the ICM, the ICMI-Symposium on “What knowledge, experience and understanding of mathematics should a mathematics teacher have?” is held with the cooperation of UNESCO and the Institut für Didaktik der Mathematik, Bielefeld (IDM). The symposium is attended by about 200 participants. The lectures are given by: H. G. Steiner, H. B. Griffiths, W. Kuijk, D. Wheeler, H. Mehrtens, F. Adams, Y. Kawada, H. Wussing, Mary G. Kantowski, T. J. Fletcher, Th. J. Cooney, Z. Semadeni, W. Dörfler, U. D’Ambrosio, S. Touré, J. A. Marasigan, A.Z. Krygowska, G.L. Lukankin, A. Revuz, M. Shumaker, and B. Leendert. They are followed by discussion panels (ICMI Bulletin 11, 1978: 13–15). 17–23 September 1978, Bielefeld (FRG)  – Joint ICMI/ICPE/CTS 16 / UNESCO/IDM Conference on cooperation between science teachers and mathematics teachers. The proceedings of the conference are published in 1979  in the series Schriftenreihe des IDM. A series of booklets, edited by Alan Rogerson, arises out of the conference on behalf of ICSU-CTS and ICMI. (ICMI Bulletin 11, 1978: 16–18 and ICMI Bulletin 13, 1983: 20–21). 1979 – The Volume IV of the series New Trends in Mathematics Tteaching, edited by H. G. Steiner and B. Christiansen, is published. This volume is largely based upon the preparation for and proceedings of the ICME-3 held in Karlsruhe in August 1976.

 The title of the paper by Iyanaga in the Proceedings of the First Southeast Asian Conference on Mathematical Education, Philippine International Convention Center, May 29–June3, 1978, Eds. Iluminada C. Coronel, Josefina C. Fonacier, Bienvenido F. Nebres, S.J. and Norman F. Quimpo (Mathematical Society of the Philippines, 1979) is: “The Role of Mathematical Societies in the Development of Mathematics”. 16   International Committee or Physics Education (ICPE); Committee on the Teaching of Science (CTS) of the International Council of Scientific Unions (ICSU). 15

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259

1980–1982

1980 – The Solidarity movement comes into being in Poland, asking for more freedom and respect for human rights. Its popularity, rapidly increased among the Poles, is seen as a threat by the European socialist countries and with increasing nervousness from the Polish government. 1980 – The first issues of the journals For the Learning of Mathematics and Recherches en Didactique des Mathématiques are published. 10–16 August 1980, Berkeley (USA) – ICME-4 (Proceedings ICME-4 1983). The congress is attended by about 1800 full members and 500 associate members from 80 countries (ICMI Bulletin 12, 1981: 8). The program includes lectures, panels, debates, posters, mini-conferences (in memory of E. Begle), and meeting of working and study groups. In addition, numerous major projects from around the world are presented. 14 August 1980, Berkeley – General Assembly of ICMI. The problem of the relationship between ICMI and IMU is discussed and future activities are planned. On invitation of B. Christiansen, E. Jacobsen, program specialist in mathematics education at UNESCO headquarters, gives examples of common tasks of ICMI and UNESCO and expresses the hope that the cooperation between the two bodies would continue in the future (ICMI Bulletin 12, 1981: 9). 8–16 September 1981, Oxford (England) – First Soviet-British Seminar on Mathematical Education. The seminar was born from a scientific and cultural collaboration between USSR and England: ten leading British mathematics educators meet with six Soviet participants for an intensive program of information sharing and discussion about the teaching and learning of mathematics at all school levels. Each theme is introduced by two speakers, one from each country. The seminar is a success, for this reason, a second edition is planned (ICMI Bulletin 13, 1983: 29). 13 December 1981 – Wojciech W. Jaruzelski imposes martial law in Poland in a bid to crush the Solidarity movement. Many mathematicians around the world express their moral protest against this provision which inevitably has a direct impact on the organization of the ICM in Warsaw in 1982. Ultimately the EC of IMU decides to postpone the Congress (Lehto 1998, pp. 219–224). In the last two years the EC of IMU had expressed several times concern about the lack of communications between ICMI and IMU. Moreover, members of the EC of ICMI had proposed as a candidate for the next President the Danish mathematics educator Bent Christiansen, ICMI vice-president for two terms since 1975, but the EC of IMU thought that the ICMI President had to be “a well-known mathematician with established interests in education” (Hodgson and Niss 2018, pp. 234–235; see also Hodgson 2009). 8–9 August 1982, Warsaw – General Assembly of IMU (IMU Bulletin, Special Number. Ninth General Assembly, 1982; IMU Bulletin 19, 1982) The issue of the Warsaw ICM in 1983 is discussed.

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New Terms of Reference for ICMI are introduced: A new idea is expressed in the Terms of Reference (ANNEX 4); it would be more reasonable that the General Assembly of IMU would choose the complete Executive of ICMI and not any part of the Commission itself, leaving to this Executive the task of reorganizing the work of ICMI in consideration of the Commission itself, that is that its General Assembly will be integrated in a suitable way with its work. (IMU Bulletin 19, 1982, p. 6; see also ANNEX 4, p. 39).

Executive Committee of ICMI for the years 1983–1986: President: J.-P. Kahane. Vice-Presidents: B. Christiansen, Z. Semadeni. Secretary: A. G. Howson. Members: B. F. Nebres (Philippines), M. F. Newman (Australia), H. O. Pollak (USA). Ex officio: H.  Whitney (past president of ICMI); J.  Moser (president of IMU); O. Lehto (secretary of IMU); H. Hogbe-Nlend (representative of IMU in CTS/ICSU). (ICMI Bulletin 13, 1983: 6–7). 9–13 August 1982, Sheffield (England) – First International Conference on Teaching Statistics (ICOTS).17 The conference is attended by 400 full participants from over 60 countries. The event, sponsored by the International Statistical Institute and UNESCO, aims to improve the quality of the teaching of statistics by promoting the exchange of ideas, materials, methods, and contents between teachers of all levels (ICMI Bulletin 12, 1981: 18): https://www.stat.auckland.ac.nz/~iase/cblumberg/OrderingICOTS.htm (Retrieved June 2021).

5.12 1983–1984 1983 – Seventy-fifth birthday of ICMI. On this occasion G. Howson, secretary of ICMI, publishes on the Bulletin the paper “Seventy-five years of ICMI” on the history of the Commission (ICMI Bulletin 14, 1983: 10–14; 15, 1984: 21–24; 16, 1984: 4–9). A fuller version (Howson 1984) of this article appears in ESM 15, 1984: 75–93. 16–24 August 1983, Warsaw (Poland)  – ICM under the presidency of Czesław Olech (Proceedings ICM 1984). Total attendance is about 2200 mathematicians from 65 countries, about a quarter less than what had been at the ICM in Helsinki; more than a third are Poles (830). The socialist countries are well represented, the USSR has 280 participants, more than five times the number in Helsinki. On the contrary, the participation from the United States has dropped from over 600 in Helsinki to 110, a thing to be expected  Since 1982, an ICOTS has been held every 4 years in a different part of the world. After ICOTS-3 it was clear that an international association for statistics education was needed, and this resulted in the creation of the International Association for Statistical Education (IASE) in 1991. 17

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if we take into account the responses to the questionnaire sent by members of the U.S. National Committee on Mathematics about attending an International Congress of Mathematicians in Warsaw: over 80% of mathematicians replied that they would not attend even if they received travel funds (Proceedings ICM 1984, pp. XI–XIV; Lehto 1998, pp. 231–233). 17–22 August, Warsaw 1983 – ICMI Symposium on “What should be the goals and the content of general mathematical education” under the chairmanship of A. Z. Krygowska and H. G. Steiner. The Leitmotiv of the symposium is the theme Mathematics for all and, in connection with it, the social dimension of mathematics and the situation of the African continent are considered (ICMI Bulletin 15, 1984: 5–10). 10–14 October 1983, Tokyo (Japan) – ICMI-JSME Regional Conference on Mathematical Education on “School mathematics in and for changing societies”. The Conference is organized by the Japan Society of Mathematical Education (JSME) and ICMI, in collaboration with UNESCO, the National Institute for Educational Research of Japan, and the Mathematical Society of Japan. It was attended by 380 participants, 73 of whom were not from Japan. Various aspects of the general theme “school mathematics in and for changing societies” are discussed, and in particular the impact of computers in school mathematics. Among the plenary lectures, that by E. Jacobsen, UNESCO’S Activities in the Field of Mathematics Education and a Critical Estimation on the Results of the Tokyo Conference, can be mentioned (ICMI Bulletin 15 1984: 11–12 and ICMI Bulletin 18, 1985: 8). 24–30 August 1984, Adelaide (Australia)  – ICME-5 (Proceedings ICME-5 1986). 1786 full members from 69 countries participate (ICMI Bulletin 17, 1984: 4). The congress focuses on the following thematic areas: evaluation, examination, and assessment; theory of mathematics education; research and teaching; language and mathematics; women and mathematics; competitions; teaching of geometry; teaching of probability and statistics. Before ICME-5, HPM group holds its first satellite meeting. From now on the HPM group will organize a satellite meeting on the occasion of each ICME (ICMI Bulletin 17, 1984: 17; https://www.icmihistory.unito.it/hpm.php Retrieved June 2021). 26 August 1984, Adelaide – General Assembly of ICMI. The Assembly is attended by 26 national representatives plus 3 observers from non-ICMI countries. In his report, A. G. Howson emphasizes, among other things, the improvement in the ICMI finances (especially thanks to the increasing financial support of UNESCO) and the approval of the “planning procedures” for the organization of future ICMEs (ICMI Bulletin 17, 1984: 6–10; see also Proposals concerning the planning of future ICME programs, ICMI Bulletin 15, 1983: 17).

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5.13 1985–1987 25–30 March 1985, Strasbourg – ICMI Study 1 Conference on “The influence of computers and informatics on mathematics and its teaching”. There are about 70 participants from Australia, Belgium, Canada, Federal Republic of Germany, Finland, France, Italy, Ivory Coast, Japan, Malaysia, Netherlands, United Kingdom, and USA (ICMI Bulletin 18, 1985: 7; ICMI Bulletin, 20, 1986: i–xii: here a report from the Zentralblatt für Didaktik der Mathematik is reprinted). Discussion document for the ICMI Study 1: R.  F. Churchhouse, B.  Cornu, A.  G. Howson, J.-P.  Kahane, J.  H. van Lint, F.  Pluvinage, A.  Ralston, and M. Yamaguti. 1984. The influence of computers and informatics on mathematics and its teaching. EM s. 2, 30, 1984: 161–172. The first volume of the series ICMI Studies is published in 1986 (ICMI Study-1 1986). Each volume in this series identifies the key problems within a specific area of mathematics education, gives an up-to-date account of relevant research and practice, and provides a framework to facilitate further study and development. 1986 – In the ICMI Bulletin 20 1986 there is a long section entitled ICMI and South Africa. Although the ICMI EC is unanimous “in condemning the ‘Apartheid’ policies of present South African Regime and in deploring the riots and killings there” (p. 9), decides that it “should abide by ICSU rules”. This body has a clear policy concerning the free circulation of scientists: “ICSU maintains that scientists from all parts of the world have the right to participate in its activities without regard to race, religion, political philosophy, ethnic origin, citizenship, language or sex”. (p. 9). In order “to carry forward the debate and to bring conflicting views into the open” ICMI members are given the opportunity to express their opinions in this issue of the ICMI Bulletin (pp. 9–17). 1–6 February, Kuwait 1986  – ICMI Study 2 Conference on “School Mathematics in the 1990s” under the patronage of the Kuwait Ministry of Education and the Foundation of the advancement of Sciences. The Discussion document for the ICMI Study 2 is published as a supplement to the ICMI Bulletin (ICMI Bulletin 18, 1985: 7). The resulting volume is published in 1986 (ICMI Study-2 1986). See the description of the ICMI Study by B. Wilson in ICMI Bulletin 22, 1987: i–iv. 3–7 February 1986, Monastir (Tunisia)  – International Symposium on “Informatics and the teaching of mathematics in developing countries”. The meeting is organized by the International Federation for Information Processing and is supported by ICMI (ICMI Bulletin 18, 1985: 9, ICMI Bulletin 19, 1985: 6). 31 July-1 August 1986, Oakland (USA) – General Assembly of IMU. During the Assembly the Executive Committee of ICMI is elected unanimously. Executive Committee of ICMI for the years 1987–1990:

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President: J.-P. Kahane. Vice-presidents: Lee Peng-Yee, E. Lluis Riera. Secretary: A. G. Howson. Members: H. Fujita, J. Kilpatrick, M. Niss. Members ex-officio: L.  D. Faddeev (president of IMU), O.  Lehto (Secretary of IMU), and J.H. van Lint (IMU representative at ICSU/CTS). (Report of the l0th General Assembly of IMU, IMU Bulletin 26, 1986: 6). 3–11 August 1986, Berkeley (USA)  – ICM under the chairmanship of Andrew Gleason (Proceedings ICM 1987; Report of the International Congress of Mathematicians 1986, IMU Bulletin 26, 1986: 7–8). Jürgen Moser, president of IMU, in his opening address says: At a time of increasing specialization and of rapid proliferation of mathematics into many subfields, these Congresses play a particularly important role in bringing together mathematicians of different interests and back- grounds. The danger of fragmentation of our science into many separate branches cannot be overemphasized. (Proceedings ICM 1987, Vol 1, p. XXI).

6–10 April 1987, Udine (Italy) – ICMI Study-3 Conference on “Mathematics as a service subject”. This meeting is attended by about 40 participants from 18 countries from all the continents. The meeting ends with this Final Statement: Mathematics is of increasing importance in all sciences and in everyday life. It is an essential part of the general culture needed by every citizen in order to understand our world and treat information and data with a critical mind. … Public opinion and governments should be made aware of the urgency of meeting these new needs. The status of service teaching and service teachers must be improved. New appointments, new means and increased resources are vital. (ICMI Bulletin 22, 1987: 12).

Discussion document: A.G. Howson, J.-P. Kahane, P.J. Kelly, P. Lauginie, T. Nemetz, F. H. Simons, C. A. Taylor and E. de Turckheim. 1986. Mathematics as a service subject. EM s. 2, 32, 1986: 159–172. The resulting volume is published in 1988 (ICMI Study-3 1988). 19–25 July 1987, Montreal (Canada) – PME meeting. The meeting is attended by 309 participants. The themes and the authors of the volume of the ICMI Study-4 are selected during this meeting. The resulting volume is published in 1990 (ICMI Study-4 1990). This ICMI Study 4 was not prepared by a discussion document. 1987 –The International Organization of Women and Mathematics Education (IOWME) becomes an Affiliated Study Group of ICMI. IOWME is an international network of individuals and groups who share a commitment to achieving equity in education and who are interested in the links between gender and the teaching and learning of mathematics: https://www.mathunion.org/icmi/ organisation/affiliated-­organisations/iowme (Retrieved June 2021).

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5.14 1988–1989 27 July  – 3 August 1988, Budapest (Hungary)  – ICME-6 (Proceedings ICME-6 1988). The congress is attended by 2414 participants from 74 countries (ICMI Bulletin 24, 1988: 11–23; 25, 1988: 1). A novelty with respect to the previous ICMEs is represented by the “Fifth Day Special: Mathematics, Education and Society,” a day focused on the relationships among mathematics education, educational policies and social context. The program is organized in four timeslots: mathematics education and culture; society and institutionalized mathematics education; educational institutions and the individual learner; mathematics education in the global village (ICMI Bulletin 24, 1988: 15–29). The contributions to the “Fifth Day Special” were published in C. Keitel, A. Bishop, P. Damerow and P. Gerdes (eds.). 1989. Mathematics, Education, and Society. Paris: UNESCO, Science and Technology Education, Document Series N. 35. 26 July 1988, Budapest (Hungary) – General Assembly of ICMI. A report on the activities of ICMI in the years 1984–1988 is presented (ICMI Bulletin 25, 1988: 2–5). 17–22 September 1989, Leeds (United Kingdom) – ICMI Study-5 Conference on “The popularization of mathematics”. The meeting is attended by more than 80 participants from 20 different countries. The program covered a wide range of themes concerning the popularization of mathematics (ICMI Bulletin 27, 1989: 7–8). Discussion Document: A.  G. Howson, J.-P.  Kahane and H.  Pollak. 1988. The popularization of mathematics. EM s. 2, 34, 1988: 205–213; ICMI Bulletin 24, 1988: 2–9). The resulting volume is published in 1990 (ICMI Study-5 1990). Concurrent with the seminar, the Royal Society/Joint Mathematical Council organized the ‘Pop Maths Roadshow’, “probably the largest mathematical exhibition ever held. This received considerable media coverage and drew further attention to the work of ICMI” (IMU Bulletin 31 Special Number, 1990: 11). 9 November 1989 – Fall of the Berlin Wall. The fall of the Wall is one of the events that marked the end of the Cold war. A year later, free elections are held in the reunified Germany, the first since 1933.

5.15 1990–1992 18–20 August 1990, Kobe (Japan) – General Assembly of IMU. 21–29 August 1990, Kyoto (Japan)  – ICM under the chairmanship of H. Komatsu. Five 45-minute lectures are invited by ICMI (Proceedings ICM 1991).

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24–29 September 1990, Sevilla (Spain) – First Ibero-American Conference on Mathematics Education (CIBEM), with Spanish, Portuguese, and Latin American participants. 13 October 1990, Hans Freudenthal dies. Executive Committee of ICMI for the years 1991–1994: President: M. de Guzmán. Vice-Presidents: J. Kilpatrick, P.-Y. Lee. Secretary: M. Niss. Members: Y. L. Ershov, E. Luna, A. Sierpinska. Ex officio: J.-P. Kahane (Past ICMI President), J.-L. Lions (IMU President), J. Palis Jr. (IMU Secretary), J. H. van Lint (Representative of IMU in CTS/ICSU). (ICMI Bulletin 29, 1990: 1–2) 11–16 April 1991, Calonge (Spain) – ICMI Study-6 Conference “Assessment in mathematics education and its effects”. Discussion Document for the ICMI Study 6: “Assessment in mathematics education and its effects” (ICMI Bulletin 28, 1990: 2–12; EM s. 2, 36, 1990: 197–206; ESM 21, 1990: 101–107). The resulting volumes are published in 1993 (ICMI Study-6 1993a; ICMI Study-6 1993b). With these volumes the New ICMI Study Series (NISS), published by Kluwer starts. 14 September 1991, Rijksuniversiteit Utrecht (Netherlands) – Inauguration of the Freudenthal Institute (ICMI Bulletin 31, 1991: 25). 5–8 August 1991, Beijing (China) – First ICMI-China Regional Conference on Mathematics Education (ICMI Bulletin 32, 1992: 15). 6 May 1992 – Jacques-Louis Lions, IMU President, declares the year 2000 the World Mathematics Year (ICMI Bulletin 33: 15–16, 1992). 17–23 August 1992, University of Laval in Quebec – ICME-7 (Proceedings ICME-7 1994). About 3500 participants from 94 countries attend the congress. In the Presidential Opening address Miguel de Guzmán proposes the project of an “ICMI Solidarity Program”; a “Solidarity Fund” based on private contributions by individuals, associations, etc. is established (ICMI Bulletin 33, 1992: 3–8). In conjunction with the conference, on 18 August 1992, the General Assembly of ICMI is held at the University of Laval (ICMI Bulletin 33, 1992: p. 9).

5.16 1993–1994 5–10 July 1993, Paris (France)  – Meeting for the Project 2000 PLUS_SPI Declaration. International forum on scientific and technological literacy for all (UNESCO) (ICMI Bulletin 35, 1993: 3–6). 23–26 August 1993, Perugia (Italy)  – First scientific meeting of the International Association for Statistical Education (IASE), the new section of

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the International Statistical Institute. The first regular Executive Committee for the period 1993–1995 is established (ICMI Bulletin 34, 1993: 18). 7–12 October 1993, Höör (Sweden) – ICMI Study-7 Conference on “Gender and mathematics education. Key issues and questions”. Discussion Document for the ICMI Study-7: “Gender and mathematics education. Key issues and questions” (ICMI Bulletin 32, 1992: 3–12; EM s. 2, 38, 1992: 189–198; ESM 23, 1992: 434–441). The resulting volume is published in 1996 (ICMI Study-7 1996).18 1 April 1994, The World Federation of National Mathematics Competitions (WFNMC) obtains its affiliation to ICMI. 8–11 May 1994, University of Maryland, College Park, Washington, DC – ICMI Study-8. Discussion Document of the ICMI Study 8: “What is research in mathematics education, and what are its results?” (ICMI Bulletin 33, 1992: 17–23; EM s. 2, 39, 1993: 179–186; ESM 23, 1992: 625–630). The resulting volume is published in 1998 (ICMI Study-8 1998). Report on the ICMI Study-8 by Jeremy Kilpatrick and Anna Sierpinska: EM s. 2, 47, 2001: 409–411). July 31–August 1 1994, Lucerne (Switzerland) – General Assembly of IMU. 3–11 August 1994, Zurich (Switzerland) – The ICM is held under the chairmanship of H. Carnal (Proceedings ICM 1995). There are four ICMI lectures.

5.17 1995–1996 Executive Committee of ICMI for the years 1995–1998: President: M. de Guzmán (Spain). Vice-Presidents: J. Kilpatrick (USA), A. Sierpinska. Secretary: M. Niss (Denmark). Members: C. Laborde, G. Leder, C. E. Vasco, D. Zhang. Ex officio: D. Mumford (IMU President), J. Palis Jr. (IMU Secretary). (IMU Bulletin 37, 1994). (ICMI Bulletin 38, 1995: 1–2) 19–23 April 1995, Monash University of Melbourne  – ICMI regional conference. September 27 – October 2, 1995 – The ICMI Study-9 Conference is held at the University of Catania (Italy).

 The proceedings were also published: Grevholm, Barbro and Hanna, Gila (eds.). 1995. Proceedings of the Study Conference: Lund: Lund University Press. 18

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Discussion Document for the ICMI Study-9: “Perspectives on the teaching on geometry for the 21st century” (ICMI Bulletin 37: 6–16, 1994; EM s. 2, 40: 345–357, 1994; ESM 28: 91–98, 1995). The resulting volume of the ICMI Study 9 is published in 1998 (ICMI Study-9 1998). Report on the ICMI Study 9 by Vinicio Villani: EM s. 2, 46, 2000: 411–415. 18–21 December 1995: National Institute of Education, Singapore  – First Asian Technology Conference in Mathematics (ATCM-1). December 1995, The ICMI Bulletin is availabe on the WWW in the pages of IMU-server. June 1996, ICMI Bulletin 40 (p.  7)  – Publication of the rules adopted by ICMI EC in 1995 about the minimum requirements for granting the status of ICMI Regional Meeting to a conference under planning. 14–21 July 1996, University of Sevilla (Spain)  – ICME-8 (Proceedings ICME-8 1998). The congress has an attendance of about 3500 delegates from almost a hundred countries. A novel feature is instigated: a 10% solidarity tax is imposed on all registration fees in order to provide (partial) financial support of the attendance of some 250 delegates from about 55 non-affluent countries. In conjunction with the conference, on 17 July 1996, the General Assembly of ICMI is held at the University of Sevilla (ICMI Bulletin 41, 1992: 3–9).

5.18 1997–1998 July 1997 – First issue of the Newsletter of the Commission on Mathematical Education of the African Mathematical Union (AMUCME). July 1997 – First issue of the new research journal in mathematics education Research in Mathematics Education published by the Korean Society of Mathematical Education. 1997– The ICMI WMY 2000 Committee is formed with the task of planning ICMI’s involvement in the World Mathematical Year 2000. 20–25 April 1998, Luminy Centre International de Rencontres Mathématiques (CIRM, France) – ICMI Study-10 Conference. Discussion Document for the ICMI Study-10: “The role of the history of mathematics in the teaching and learning of mathematics” (ICMI Bulletin 42, 1997: 6–16; EM s. 2, 43, 1997: 199–203; ESM 34, 1997: 255–259). The resulting volume is published in 2000 (ICMI Study-10 2000). The report on ICMI Study-10 by Fulvia Furinghetti is published (ICMI Bulletin 58, 2006: 38–44; EM 51, 2005: 365–372). 15–16 August 1998, Dresden (Germany) – the General Assembly of IMU. 17–21 August 1998, Korea National University of Education, Chungbuk – First East Asia Regional Conference on Mathematics Education (ICMI-­ EARCOME 1).

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18–27 August 1998, Berlin (Germany) – ICM (Proceedings ICM 1998). For the first time in many years, there were no specific ICMI lectures, symposium, or suchlike at the 1998 ICM. The ICMI-related activities were integrated into the program of Section 18 “Teaching and Popularization of Mathematics.” In the final program, outgoing and incoming ICMI officers (de Guzmán, M. Niss, B. R. Hodgson, and M. Artigue) were involved as invited speakers in talks or panels (Hodgson and Niss 2018, p. 239). 27–31 August 1998, Osnabrück – First Conference of the European Society for Research in Mathematics Education (CERME-1). The project for establishing this new Society was discussed in Osnabrück on 2–4 May 1997 by representatives of 16 European countries. 8–12 December 1998, Singapore National Institute of Education (Nanyang Technological University) – ICMI Study-11 Conference. Discussion Document for the ICMI Study-11 “The teaching and learning of mathematics at university level” (ICMI Bulletin 43, 1997: 3–13; EM s. 2, 43, 1997: 381–390; ESM 36, 1998: 91–103). Selected papers presented at the ICMI Study-11 Conference are published in a special issue of The International Journal of Mathematical Education in Science and Technology 31, 2000: 1–160. The resulting volume is published in 2001 (ICMI Study-11 2001). Report on the ICMI Study 11 (Personal thoughts on the ICMI Study: “The teaching and learning of mathematics at university level”) by Derek Holton (EM s. 2, 53, 2007: 429–436).

5.19 1999–2000 Executive Committee of ICMI for the years 1999–2002: President: H. Bass. Vice-Presidents: N. Aguilera, M. Artigue. Secretary: B. R. Hodgson. Members: G. Leder, Y. Namikawa, I. F. Sharygin, J.-P. Wang. Ex officio: M. de Guzmán (Past President of ICMI), J. Palis Jr. (President of IMU), Ph. Griffiths (Secretary of IMU). (ICMI Bulletin 45, 1998: 4) 15–17 July 2000, Grenoble – ICMI Regional Conference “Mathematics education in French-speaking countries in the XXth century and prospects for the beginning of the XXIst century” (launched by the French Sub-Commission of ICMI on the occasion of the World Mathematical Year 2000). The series of Espace Mathématique Francophone conferences is built on a notion of ‘region’ defined in linguistic rather than geographical terms, French being a common language among participants.

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31 July – 6 August 2000, Makuhari/Tokyo (Japan) – ICME-9 (Proceedings ICME-9 2004). The participants are about 2300 from more than 70 countries.  In conjunction with the conference, on 4 August 2000, the General Assembly of ICMI is held at Makuhari/Tokyo (ICMI Bulletin 48, 2000: n. n. p.). 18–23 September 2000, Dubna (Russia)  – The All-Russian Conference on Mathematical Education devoted to the theme “Mathematics and Society. Mathematical Education at the Frontier of Centuries”. (ICMI Bulletin 50, 2001: 33–34). 20–22 October 2000, Geneva – International Symposium organized jointly by the University of Geneva and ICMI, as a contribution to the celebration of the WMY 2000, on “One Hundred Years of L’Enseignement Mathématique”. (ICMI Bulletin 49, 2000: 45–47, 20; EM s. 2, 46, 2000: 219–222; ESM s. 2, 41, 2000: 311–312; EM s. 2, 47, 2001: 181–183. http://www.unige.ch/math/EnsMath/EM-­ICMI/welcome.html) Proceedings of the Geneva Symposium for the centenary of L’Enseignement Mathématique are published: –– Coray, Daniel, Fulvia Furinghetti, Hélène Gispert, Bernard R.  Hodgson, and Gert Schubring (eds.). 2003. One Hundred Years of L’Enseignement Mathématique, Monographie n. 39 de L’Enseignement Mathématique.

5.20 2001–2002 11 September 2001, New  York (USA)  – A terroristic attack destroys the Twin Towers. 20–23 February 2001, Goa (India) – International Conference on Science, Technology & Mathematics Education for Human Development. The ICMI Secretary B. Hodgson takes part. On this occasion, after an interruption of almost a decade, ICMI reinitiates contacts with UNESCO. (ICMI Bulletin 51, 2002: 79–82). 19–24 July 2001, ICMI co-sponsors an international workshop entitled “International Perspectives on Standards and Goals for K-12 Mathematics Education” organized in Utah, USA, in the context of the annual “Park City Mathematics Institute” hosted by the Institute for Advanced Study (Princeton, NJ, USA). (ICMI Bulletin 51, 2002: 48). 9–14 December 2001, University of Melbourne (Australia) – ICMI Study-12 Conference on “The future of the teaching and learning of algebra”. Discussion Document for the ICMI Study-12: “The future of the teaching and learning of algebra” (ICMI Bulletin 48, 2000: 6–13; EM s. 2, 46, 2000: 209–217; ESM 42, 2000: 215–224).

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The resulting volume is published in 2004 (ICMI Study-12 2004).19 Report on the ICMI Study 12 by Kaye Stacey: EM s. 2, 55, 2009: 397–402. 2001: ICMI reinvigorates its contact with L’Enseignement Mathématique and more reports on ICMI activities are published. (ICMI Bulletin 51, 2002: 49). 2 February 2001, Mariánské Lázně  – The self-organized group of junior researchers YERME (Young European Researchers in Mathematics Education) is established during CERME-2. (ICMI Bulletin 51, 2002: 75–76). April 12–13, 2002, Paris – In the IMU EC meeting, at which the president and secretary-general of ICMI are invited, a revised version of ICMI Terms of Reference is adopted by the EC of IMU (ICMI Bulletin 51, 2002: 9–12; EM s. 2, 50, 2004: 187–190). 17–18 August 2002, Shanghai (China) – General Assembly of IMU to which President and Secretary-General of ICMI are invited as ex officio observers; 20–28 August 2002, Beijing (China) – The ICM is held under the chairmanship of Wen-Tsun Wu (Proceedings ICM 2002). Section 18 entitled “Mathematics education and popularization of mathematics” contains a panel and three contributions. There is also an invited panel on “International comparisons in mathematics education: an overview”. 20–25 October 2002, University of Hong Kong – ICMI Study-13 Conference on “Mathematics education in different cultural traditions: A comparative study of East Asia and the West”. Discussion Document for the ICMI Study-13: “Mathematics Education in Different Cultural Traditions: A Comparative Study of East Asia and the West” (ICMI Bulletin 49, 2000: 16–33; EM s. 2, 47, 2001: 185–201; ESM 43, 2000: 95–116). The resulting volume is published in 2006 (ICMI Study-13 2006).

5.21 2003–2004 Executive Committee of ICMI for the years 2003–2006: President: H. Bass. Vice-Presidents: J. Adler, M. Artigue. Secretary-General: B. R. Hodgson (Canada). Members at large20: C. Batanero, M. Falk De Losada, N. Dolbilin, Peter L. Galbraith, P. S. Kenderov, F. K.-S. Leung. Ex officio: J. Ball (President of IMU), P. Griffiths (Secretary of IMU).

 The proceedings are also published: Chick, Helen, Stacey, Kaye, Vincent, Jill and Vincent, John (eds.). 2001. Proceedings of the ICMI Study Conference “The future of the teaching and learning of algebra”. Melbourne: The University of Melbourne. 20  The Terms of Reference of 2002 say: “The Executive Committee of the Commission consists of the following members. Elected by IMU: Nine members, including the four officers, namely, the President, two Vice- Presidents, and the Secretary-General. Ex-officio members: The outgoing President of ICMI, the President and the Secretary of IMU. Co-opted members: In order to provide for missing coverage or representation, the ICMI Executive Committee may co-opt up to two additional members.” Members at large are not mentioned. 19

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In accordance with the new Terms of Reference the position of “Secretary” is designated by the term “Secretary-General”. Since a tie in the voting, six members at large instead of four are coopted (IMU Bulletin 49, 2002: 9). 2–3 April 2003, Princeton (USA) – Meeting of the EC of IMU: an Ad Hoc Committee on “Supporting Mathematics in Developing Countries” is appointed. ICMI is represented by Vice-President Michèle Artigue (ICMI Bulletin 54: 32–48, 2004). July 2003, The International Study Group for Mathematical Modelling and Applications (ICTMA) becomes an Affiliated Study Group of ICMI. 11–12 February 2004, Dortmund – Presentation of the logo of ICMI during the meeting of the EC of ICMI. 13–17 February 2004, Universität Dortmund, Germany  – ICMI Study-14 Conference on Applications and modelling in mathematics education”. Discussion Document for the ICMI Study-14: “Applications and modelling in mathematics education”. ICMI Bulletin 51, 2002: 23–43; EM 49, 2003: 205–214; ESM 51, 2002: 149–171. The resulting volume is published in 2007 (ICMI Study-14 2007). 18–19 March 2004, Dublin – ICMI is invited by the Director of Education at OECD (Organisation for Economic Co-operation and Development) to participate in a Forum on education and social cohesion organized by OECD on the occasion of a meeting of Education Minister. This meeting is an opportunity for a direct link of ICMI with the OECD Directorate for Education (IMU Bulletin 53, 2006: 95–96). 2004, IMU appoints Victor A. Vassiliev to act as an IMU liaison with the ICMI EC (IMU Bulletin 53, 2006: 87). 2–4 July 2004, Copenhagen  – The “Pipeline Project” related to various aspects of the teaching and learning of mathematics at the higher level is initiated by ICMI in collaboration with IMU during the meeting of ICMI EC with the participation of IMU President John Ball. 4–11 July 2004, Technical University of Denmark (DTU) in Copenhagen (Denmark) – ICME-10 (Proceedings ICME-10 2008). This congress is organized through a collaboration of neighboring countries (the Nordic countries Denmark, Finland, Iceland, Norway, Sweden), a first in the life of ICMI. It is attended by 2324 participants from 93 countries. In conjunction with the conference, on 9 July 2004, the General Assembly of ICMI is held at the Technical University of Denmark (DTU) in Copenhagen (ICMI Bulletin 54, 2004: 24–25). The ICMI medals for 2003 are awarded to Guy Brousseau (Klein medal), Celia Hoyles (Freudenthal medal) (ICMI Bulletin 54, 2004: 8–11; EM s. 2, 50, 2004: 183–186). The Felix Klein Medal, named from the first president of ICMI (1908–1920), honors a lifetime achievement. The Hans Freudenthal Medal, named from the eighth president of ICMI (1967–1970), recognizes a major cumulative program of research. These awards are to be made in each odd numbered year, with presentation of the medals, and invited addresses by the medallists at the following

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International Congress on Mathematical Education (ICME) (ICMI Bulletin 51, 2002: 14–15; EM s. 2, 48, 2002: 377–378; ESM 51, 2002: 173–174). The mathematical exhibition entitled “Experiencing mathematics”, whose aim is to improve the image of mathematics among the general public, is launched at ICME-10 with the support of UNESCO. http://www.mathex.org/MathExpo/ (Retrieved June 2021) (IMU Bulletin 53, 2006: 96–97). 16–17 October 2004, Abdus Salam International Centre for Theoretical Physics (ICTP) in Trieste (Italy) – First meeting of the Developing Countries Strategy Group (DCSG), established by IMU, with the charge of increasing, guiding, and coordinating IMU’s activities in support of mathematics and mathematics education in the developing world. ICMI is represented in the DCSG by Michèle Artigue.

5.22 2005–2006 15–21 May 2005, Aguas de Lindóia (Brazil) – ICMI Study-15 Conference on “The professional education and development of teachers of mathematics”. Discussion Document for the ICMI Study-15: “The professional education and development of teachers of mathematics” (ICMI Bulletin 54, 2004: 12–22; EM s. 2, 50, 2004: 191–200; ESM 56, 2004: 359–372). The resulting volume is published in 2009 (ICMI Study-15 2009). 22–25 June 2005, University of the Witwatersrand in Johannesburg (South Africa) – First Africa Regional Congress of ICMI (AFRICME). 15–18 December 2005, Indira Gandhi National Open University in New Delhi (India) – National Conference on Mathematics Education at the National Council of Educational Research and Training (NCERT), and a meeting of the ICMI EC at the invitation of IMU EC member Madabusi S.  Raghunathan. The direct contacts between the Executive Committees of IMU and ICMI were more frequent during the period 2002–2005 (IMU Bulletin 53, 2006: 87–88). 27 June–3 July 2006, Norwegian University of Science and Technology, in Trondheim (Norway) – The ICMI Study-16 Conference on “Challenging mathematics in and beyond the classroom: the 16th ICMI study”. Discussion Document for the ICMI Study-16: “Challenging mathematics in and beyond the classroom” (ICMI Bulletin 55: 32–46, 2004; EM s. 2, 51: 165–176, 2005; ESM 60: 125–139, 2005). The resulting book is published in 2009 (ICMI Study-16 2009). 19–20 August 2006, Santiago de Compostela (Spain) – General Assembly of IMU. Following the agreement made in 2000 with the IMU EC, the President and Secretary-General of ICMI are invited as ex officio observers. Presentation of the new procedures for the election of the Executive Committee of ICMI agreed upon by the IMU and ICMI ECs in 2004, and fine-tuned in 2006, see (IMU Bulletin 2006, 54: 53–63).

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22–30 August 2006, Madrid (Spain) – The ICM is held under the chairmanship of Manuel de León (Proceedings ICM 2007). 3–8 December 2006, Hanoi University of Technology, Vietnam  – ICMI Study-17 Conference on “Digital technologies and mathematics teaching and learning: Rethinking the terrain”. Discussion Document for the ICMI Study-17: “Digital technologies and mathematics teaching and learning: Rethinking the terrain” (ICMI Bulletin 57, 200521; EM s. 2, 51, 2005: 351–363; ESM 60, 2005: 267–268). The resulting volume is published in 2010 (ICMI Study-17 2010).

5.23 2007–2008 Executive Committee of ICMI for the years 2007–2009: President: M. Artigue. Vice-Presidents: J. Adler, B. Barton. Secretary-General: B. R. Hodgson. Members: M.  Bartolini Bussi, J.  Carvalho e Silva, C.  Hoyles, S.  Kumaresan, F. K.-S. Leung, A. L. Semenov. Ex officio: H.  Bass (Past President of ICMI), L.  Lovász (President of IMU), M. Grötschel (Secretary of IMU). Artigue is the first woman elected as an ICMI President. She is the second educator after Smith in this position. The committee remains in office for three years due to the transition rules (IMU Bulletin 54, 2006: 22). December 2007: The first issue of the electronic Newsletter “ICMI News” is published. 5–8 March 2008, Palazzo Corsini, home of the Accademia Nazionale dei Lincei and Palazzo Mattei di Paganica, home of the Enciclopedia Italiana in Rome – The ICMI Centennial Symposium. In this occasion two permanent websites were created: –– http://www.unige.ch/math/EnsMath/Rome2008/ for the congress (Retrieved June 2021). –– http://www.icmihistory.unito.it for the history of the first 100  years of ICMI (Retrieved June 2021). The resulting book is (Menghini et al. 2008). 4–9 March 2008, Accademia dei Lincei in Rome – The second meeting of the current ICMI EC discusses the idea of the Klein Project, first mentioned during the June 2007 Executive Meeting of ICMI. The basic idea of this project is to revisit

 This Bulletin and the issue 59 of 2006 are reported in the ICMI website as plain text, thus the number of pages is not available. 21

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the vision of Felix Klein presented in his book Elementarmathematik vom höheren Standpunkte aus, whose aim was to provide prospective senior secondary teachers with a book that would lay out the connections between the curricula they would be teaching and research mathematics. The reflections of the ICMI EC on this project are pursued at its two meetings of 2008 and also in conjunction with the IMU EC. It is decided to form a Design Team of 8–9 people who would be responsible for the project. The composition of this team, chaired by Bill Barton, is decided by the end of 2008 (IMU Bulletin 56, 2008: 34–35). 30 June  – 4 July 2008, Instituto Tecnológico y de Estudios Superiores de Monterrey (ITESM) in Monterrey (Mexico) – ICMI Study-18 Conference held as a satellite conference to ICME-11. Discussion Document for the ICMI Study-18: “Statistics education in school mathematics: challenges for teaching and teacher education” organized jointly with the International Association for Statistical Education, posted in the website (Retrieved June 2021): https://iase-­w eb.org/documents/papers/rt2008/Discussion_Document.pdf? 1402524989 (ICMI Bulletin 59, 2006; EM 53, 2007: 179–191). The resulting volume is published in 2011 (ICMI Study-18 2011). 6–13 July 2008, Universidad Autonoma de Nuevo Leon (UANL) in Monterrey (Mexico) – ICME-11. There are no published proceedings. On the congress website, we read that between 2000 and 2500 professionals from 100 countries were expected. The ICMI medals are awarded. For the year 2005 Ubiratan d’Ambrosio is Klein medallist, and Paul Cobb is Freudenthal medallist (ICMI Bulletin 58, 2006: 6–10; EM s. 2, 52. 2006: 187–190). For the year 2007 Jeremy Kilpatrick is Klein medalist, and Anna Sfard is Freudenthal medalist (ICMI Bulletin 62, 2008: 6–11; EM s. 2, 54, 2008: 399–403). In conjunction with the conference, on 6 July 2008, the General Assembly of ICMI is held in Monterrey (ICMI Bulletin 62, 2008: p. 14).  In accordance with the new rules, see (IMU Bulletin 53, 2006: 37–40), for the first time, this General Assembly is responsible for the election of the ICMI Executive Committee. During the Congress ICMI obtains from Springer the permission to post the content of the New ICMI Study Series volumes freely accessible on ICMI website. 2008: Progress is made on the digitization of the ICMI material to be included in the ICMI Digital Library.

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Table 5.2  List of the 85 member countries of ICMI in 2008 Argentina Armenia Australia Austria Bangladesh (*) Belgium Bosnia and Herzegovina Botswana (*) Brazil Brunei Darussalam (*) Bulgaria Cameroon Canada Chile China Colombia Costa Rica (*) Croatia Cuba Czech Republic Denmark Ecuador (am)

Egypt Estonia Finland France Georgia Germany Ghana (*) Greece Hong Kong Hungary Iceland India Indonesia Iran Ireland Israel Italy Ivory Coast Japan Kazakhstan Kenya (am)

Republic of Korea Kuwait (*) Kyrgyzstan (am) Latvia Lithuania Luxembourg (*) Malawi (*) Malaysia (*) Mexico Mozambique (*) Netherlands New Zealand Nigeria Norway Pakistan Peru Philippines Poland Portugal Romania Russia

Saudi Arabia Senegal (*) Serbia Singapore Slovakia Slovenia South Africa Spain Swaziland (*) Sweden Switzerland Thailand (*) Tunisia Turkey Ukraine United Kingdom United States of America Uruguay Venezuela Vietnam Zambia (*)

(*) indicates the 13 members of ICMI that are not members of IMU; (am) indicates associated members of IMU

References and Sources 1. Essential Bibliography Giacardi, Livia. 2008a. Timeline 1908–1976. In The First Century of the International Commission on Mathematical Instruction (1908–2008). History of ICMI, eds. Fulvia Furinghetti and Livia Giacardi: https://www.icmihistory.unito.it. ———. 2008b. Cento anni della International Commission on Mathematical Instruction. In Associazione Subalpina Mathesis. Conferenze e Seminari 2007–2008, 247–314. Torino: Kim Williams Books. See also the Italian version. Hodgson, Bernard R. 2009. ICMI in the post-Freudenthal era: Moments in the history of mathematics education from an international perspective. In Dig where you stand. Proceedings of the Conference on On-going Research in the History of Mathematics Education, ed. Kristín Bjarnadóttir, Fulvia Furinghetti, and Gert Schubring, 79–96. Reykjavik: University of Iceland – School of Education. Hodgson, Bernard R., and Mogens Niss. 2018. ICMI 1966-2016: A double insiders’ view of the latest half century of the International Commission on Mathematical Instruction. In Invited Lectures from the 13th International Congress on Mathematical Education, ed. Gabriele

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Kaiser, Helen Forgasz, Mellony Graven, Alain Kuzniak, Elaine Simmt, and Xu Binyan, 229–247. Cham: Springer. Howson, A. Geoffrey. 1984. Seventy five years of the International Commission on Mathematical Instruction. Educational Studies in Mathematics 15: 75–93. See also ICMI Bulletin 14, 1983: 10–14; 15, 1984: 21–24; 16, 1984: 4–9. Lehto, Olli. 1998. Mathematics without borders: A history of the International Mathematical Union. New York: Springer. Menghini, Marta, Fulvia Furinghetti, Livia Giacardi, and Ferdinando Arzarello, eds. 2008. The first century of the International Commission on Mathematical Instruction (1908–2008). Reflecting and shaping the world of mathematics education. Rome: Istituto della Enciclopedia Italiana.

2. ICME Proceedings Proceedings ICME-1 1969. The Editorial Board of Educational Studies in Mathematics. 1969. Proceedings of the First International Congress on Mathematical Education. Dordrecht: Reidel Publishing Company. Note: The contents of this volume are also published in the same year in Educational Studies in Mathematics 2: 135–418. Proceedings ICME-2 1972. Howson, A.  Geoffrey (ed.). 1973. Developments in Mathematical Education. Proceedings of the Second International Congress on Mathematical Education. Cambridge: Cambridge University Press. Proceedings ICME-3 1976. Athen, Hermann, and Heinz Kunle (eds.). 1977. Proceedings of the Third International Congress on Mathematical Education. Karlsruhe: Zentralblatt für Didaktik der Mathematik. Proceedings ICME-4 1980. Zweng, Marilyn, Thomas Green, Jeremy Kilpatrick, Henry Pollak, and Marilyn Suydam (eds.). 1983. Proceedings of the Fourth International Congress on Mathematical Education. Basel: Birkhäuser. Proceedings ICME-5 1984. Carss, Marjorie (ed.). 1986. Proceedings of the Fifth International Congress on Mathematical Education. Basel: Birkhäuser. Proceedings ICME-6 1988. Hirst, Ann, and Keith Hirst (eds.). 1988. Proceedings of the Sixth International Congress on Mathematical Education. Budapest: János Bolyai Mathematical Society. Proceedings ICME-7 1992. Gaulin, Claude, Bernard R. Hodgson, David H. Wheeler, and John Egsgard (eds.). 1994. Proceedings of the Seventh International Congress on Mathematical Education. Québec: Les Presses de l'Université Laval. Proceedings ICME-8 1996. Alsina, Claudi, José Maria Alvarez, Mogens Niss, Antonio Perez, Luis Rico, and Anna Sfard (eds.). 1998. Proceedings of the 8th International Congress on Mathematical Education. S.A.E.M. Thales. Proceedings ICME-9 2000. Fujita, Hiroshi, Yoshihiko Hashimoto, Bernard R. Hodgson, Peng Yee Lee, Stephen Lerman, and Toshio Sawada (eds.). 2004. Proceedings of the Ninth International Congress on Mathematical Education. Dordrecht: Kluwer Academic Publishers. Proceedings ICME-10 2004. Niss, Mogens (ed.). 2008. Proceedings of the Tenth International Congress on Mathematical Education. Roskilde: IMFUFA, Roskilde University.

3. ICMI Studies ICMI Study-1 1986. Churchhouse, Robert F., Bernard Cornu, A. Geoffrey Howson, Jean-Pierre Kahane, Jacobus H. van Lint, François Pluvinage, Anthony Ralston, and Masaya Yamaguti (eds.). 1986. The influence of computers and informatics on mathematics and its teaching. Cambridge/etc.: Cambridge University Press, ICMI Study series. II edition: Cornu, B. & Ralston, A. 1992. Paris: UNESCO, Science and Technology Education 44. ICMI Study-2 1986. Howson, A. Geoffrey, and Bryan J. Wilson. 1986. School mathematics in the 1990s. Cambridge/etc.: Cambridge University Press, ICMI Study series. ICMI Study-3 1988. Howson, A.  Geoffrey (ed.). 1988. Mathematics as a service subject. Cambridge: Cambridge University Press, ICMI Study series.

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ICMI Study-4 1990. Nesher, Pearla, and Jeremy Kilpatrick (eds.). 1990. Mathematics and cognition. A research synthesis by the International Group for the Psychology of Mathematics Education. Cambridge: Cambridge University Press, ICMI Study series. ICMI Study-5 1990. Howson, A. Geoffrey, and Jean-Pierre Kahane (eds.). 1990. The popularization of mathematics. Cambridge/etc.: Cambridge University Press, ICMI Study series. ICMI Study-6 1993a. Niss, Mogens (ed.). 1993. Cases of assessment in mathematics education. Dordrecht/etc.: Kluwer Academic Publisher, New ICMI Studies series 1. ——— 1993b. Niss, Mogens (ed.). 1993. Investigations into assessment in mathematics education. Dordrecht/etc.: Kluwer Academic Publisher, New ICMI Studies series 2. ICMI Study-7 1996. Hanna, Gila (ed.). 1996. Towards gender equity in mathematics education. An ICMI study. Dordrecht/etc.: Kluwer Academic Publisher, New ICMI Studies series 3. ICMI Study-8 1998. Sierpinska, Anna, and Jeremy Kilpatrick (eds.). 1998. Mathematics education as a research domain: a search for identity: an ICMI study. Dordrecht/etc.: Kluwer Academic Publisher. New ICMI Studies series 4. ICMI Study-9 1998. Mammana, Carmelo, and Vinicio Villani (eds.). 1998. Perspectives on the teaching of geometry for the 21st century: An ICMI study. Dordrecht/etc.: Kluwer Academic Publishers, New ICMI Studies series 5. ICMI Study-10 2000. Fauvel, John, and Jan van Maanen (eds.). 2000. History in mathematics education. The ICMI Study. Dordrecht/etc.: Kluwer Academic Publishers, New ICMI Studies series 6. ICMI Study-11 2001. Holton, Derek (ed.). 2001. The teaching and learning of mathematics at university level: An ICMI study, Dordrecht/etc.: Kluwer Academic Publishers, New ICMI Studies series 7. ICMI Study-12 2004. Stacey, Kaye, Helen M.  Chick, and Margaret Kendal (eds.). 2004. The future of the teaching and learning of algebra. The 12th ICMI Study. Dordrecht/etc.: Kluwer Academic Publishers, New ICMI Studies series 8. ICMI Study-13 2006. Leung, Frederick Koon-Shing, Klaus-D. Graf, and Francis J. López-Real (eds.). 2006. Mathematics education in different cultural traditions. The 13th ICMI Study. New York/etc.: Springer, New ICMI Studies series 9. ICMI Study-14 2007. Blum, Werner, Peter L. Galbraith, Hans-Wolfgang Henn, and Mogens Niss (eds.). 2007. Modelling and applications in mathematics education. The 14th ICMI Study. New York/etc.: Springer, New ICMI Studies series 10. ICMI Study-15 2009. Even, Ruhama, and Deborah Loewenberg Ball (eds.). 2009. The professional education and development of teachers of mathematics. The 15th ICMI Study. New  York: Springer. New ICMI Study Series 11. ICMI Study-16 2009. Barbeau, Edward J., and Peter J.  Taylor (eds.). 2009. Challenging Mathematics In and Beyond the Classroom. The 16th ICMI Study. New York: Springer. New ICMI Study Series 12. ICMI Study-17 2010. Hoyles, Celia, and Jean-Baptiste Lagrange (eds.). 2010. Mathematics Education and Technology-Rethinking the Terrain. The 17th ICMI Study. Dordrecht etc.: Springer. New ICMI Study Series 13. ICMI Study-18 2011. Batanero, Carmen, Gail Burrill, and Chris Reading (eds.). 2011. Teaching Statistics in School Mathematics-Challenges for Teaching and Teacher Education. A Joint ICMI/IASE Study: The 18th ICMI Study. Dordrecht etc.: Springer. New ICMI Study Series 14.

4. ICM Proceedings Proceedings ICM 1908. Castelnuovo, Guido (ed.). 1909. Atti del IV Congresso Internazionale dei Matematici. Roma: Tipografia della R. Accademia dei Lincei. Proceedings ICM 1912. Hobson, Ernest W., and Augustus Love (eds.). 1913. Proceedings of the fifth International Congress of Mathematicians. Cambridge: Cambridge University Press. Proceedings ICM 1920. Villat, Henri (ed.). 1921. Comptes Rendus du Congrès International des Mathématiciens. Toulouse: É. Privat (Librairie de l’Université).

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Proceedings ICM 1924. Fields, John C. (ed.). 1928. Proceedings of the International Mathematical Congress. Toronto: The University of Toronto Press. Proceedings ICM 1928. 1929. Atti del Congresso Internazionale dei Matematici. Bologna: N. Zanichelli. Proceedings ICM 1932. Saxer, Walter (ed.). 1932. Verhandlungen des Internationalen Mathematiker-Kongresses Zürich 1932. Zürich – Leipzig: Orell Füssli. Proceedings ICM 1936. Lange-Nielsen, Fr., Edgar B. Schieldrop, and Nils Solberg. 1937. Comptes Rendus du Congrès International des Mathématiciens. Oslo: A. W. Brøggers Boktrykkeri A/S. Proceedings ICM 1950. Graves, Lawrence M., Paul A.  Smith, Einar Hille, and Oscar Zariski (eds.). 1952. Proceedings of the International Congress of Mathematicians. Providence, RI: American Mathematical Society. 2 Vols. Proceedings ICM 1954. Gerretsen, Johan C.  H., and Johannes de Groot (eds.). 1954–1957. Proceedings of the International Congress of Mathematicians. Groningen: E.  P. Noordhoff N. V.; Amsterdam: North-Holland. 3 Vols. Proceedings ICM 1958. Todd, John A. (ed.). 1960. Proceedings of the International Congress of Mathematicians. Cambridge: Cambridge University Press. Proceedings ICM 1962. Stenström, V. (ed.). 1963. Proceedings of the International Congress of Mathematicians. Djursholm, Sweden: Institut Mittag-Leffler. Proceedings ICM 1966. Petrovsky, Ivan G. (ed.). 1968. Proceedings of the International Congress of Mathematicians. Moscow: Printing House “MIR”. Proceedings ICM 1970. Berger, Marcel, Jean Dieudonné, Jean Leray, Jacques-Luis Lions, Paul Malliavin, and Jean-Pierre Serre (eds.). 1971. Actes du Congrès International des Mathématiciens. Paris: Gauthier-Villars, 3 Vols. Proceedings ICM 1974. James, Ralph D. (ed.). 1975. Proceedings of the International Congress of Mathematicians. Canadian Mathematical Congress. Proceedings ICM 1978. Lehto, Olli (ed.). 1980. Proceedings of the International Congress of Mathematicians. Helsinki: Academia Scientiarum Fennica. Proceedings ICM 1983. Ciesielski, Zbigniew, and Czesław Olech (eds.). 1984. Proceedings of the International Congress of Mathematicians. Warszawa: PWN  – Polish Scientific Publishers; Amsterdam – New York – Oxford: North Holland. Proceedings ICM 1986. Gleason, Andrew M. (ed.). 1987. Proceedings of the International Congress of Mathematicians. Washington, DC: MAA, 2 Vols. Proceedings ICM 1990. Satake, Ichiro (ed.). 1991. Proceedings of the International Congress of Mathematicians. Tokyo/Berlin/Heidelberg, etc.: The Mathematical society of Japan, Springer. Proceedings ICM 1994. Chatterji, Srishti D. (ed.). 1995. Proceedings of the International Congress of Mathematicians. Basel/Boston/Berlin: Springer. Proceedings ICM 1998. Fischer, Gerd, and Ulf Rehmann (eds.). 1998. Proceedings of the International Congress of Mathematicians. Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, Extra volume ICM 1998. Proceedings ICM 2002. LI, Tatsien (LI Daqian) (ed.). 2002. Proceedings of the International Congress of Mathematicians. Beijing: Higher Education Press. Proceedings ICM 2006. Sanz-Solé, Marta, Javier Soria, Juan L. Varona, and Joan Verdera (eds.). 2007. Proceedings of the International Congress of Mathematicians. Zurich: European Mathematical Society.

Chapter 6

Central and Executive Committees of CIEM/IMUK and ICMI Fulvia Furinghetti

6.1 Central Committees of CIEM (Commission Internationale de l’Enseignement Mathématique), IMUK (Internationale Mathematische Unterrichts-­Kommission), International Commission on the Teaching of Mathematics, Commissione Internazionale per l’Insegnamento Matematico 1908–1912 President: Felix Klein (Germany). Vice-President: Alfred George Greenhill (UK). Secretary-General: Henri Fehr (Switzerland). 1912–1920 President: Felix Klein (Germany). Vice-Presidents: Alfred George Greenhill (UK), David Eugene Smith (USA). Secretary-General: Henri Fehr (Switzerland). Members (co-opted in 1913): Guido Castelnuovo (Italy), Emanuel Czuber (Austria), Jacques Hadamard (France).

F. Furinghetti (*) University of Genoa, Genoa, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_6

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1928–1932 President: David Eugene Smith (USA). Vice-Presidents: Guido Castelnuovo (Italy), Jacques Hadamard (France). Secretary-General: Henri Fehr (Switzerland). Member: Walter Lietzmann (Germany). 1932–1936, 1936– President: Jacques Hadamard (France). Vice-Presidents: Poul Heegaard (Norway), Walter Lietzmann (Germany), Gaetano Scorza (Italy). Secretary-General: Henri Fehr (Switzerland). Member (co-opted in 1932): Eric Harold Neville (UK). In 1936, during the International Congress of Mathematicians in Oslo “The Congress requests the International Commission on the Teaching of Mathematics to continue its work, prosecuting such investigations as shall be determined by the Central Committee.” (L’Enseignement Mathématique 1936, 35, p. 388), but because of World War II, the Commission remained inactive until 1952 when it was transformed in a permanent subcommission of IMU.

6.2 

 xecutive Committee of IMIC (International E Mathematical Instruction Commission)

1952–1954 Honorary President: Henri Fehr (Switzerland). President: Albert Châtelet (France). Vice-Presidents: Ðuro Kurepa (Yugoslavia), Saunders Mac Lane (USA). Secretary: Heinrich Behnke (Germany). Members: Aksel Frederik Andersen (Denmark), Guido Ascoli (treasurer, Italy), Evert W. Beth (Netherlands), Ralph L. Jeffery (Canada), Edwin A. Maxwell (UK). Ex officio: Marshall H. Stone (USA) – President of IMU. Behnke, Châtelet, Fehr, Jeffery, and Kurepa were elected by the 1952 General Assembly of IMU without specifying their offices. Later in 1952 the Commission itself chose the officers and co-opted additional members. The name International Mathematical Instruction Commission (IMIC) was ephemeral, already in 1952–1954 the name International Commission on Mathematical Instruction (ICMI) was gradually introduced and became official in the 1954 IMU General Assembly, which also adopted the Terms of Reference for ICMI.  According to the By-Laws, the President of IMU is an ex officio member of all Commissions of the Union.

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 xecutive Committees of ICMI (International E Commission on Mathematical Instruction)

1955–1958 President: Heinrich Behnke (Germany). Vice-Presidents: Ðuro Kurepa (Yugoslavia), Marshall H. Stone (USA). Secretary: Julien Desforge (France). Members: Ram Behari (India), Edwin A. Maxwell (UK), Kay Piene (Norway). Ex officio: Heinz Hopf (Switzerland) – President of IMU. 1959–1962 President: Marshall H. Stone (USA). Vice-Presidents: Heinrich Behnke (Germany), Ðuro Kurepa (Yugoslavia). Secretary: Gilbert Walusinski (France). Members: Yasuo Akizuki (Japan), Aleksandr D.  Aleksandrov (USSR), Otto Frostman (Sweden). Ex officio: Rolf Nevanlinna (Finland) – President of IMU. 1963–1966 President: André Lichnerowicz (France). Vice-Presidents: Edwin Moise (USA), Stefan Straszewicz (Poland). Secretary: André Delessert (Switzerland). Members: Yasuo Akizuki (Japan), Heinrich Behnke (Germany), Hans Freudenthal (Netherlands). Ex officio: Georges de Rham (Switzerland) – President of IMU. 1967–1970 President: Hans Freudenthal (Netherlands). Vice-Presidents: Edwin Moise (USA), Sergei L. Sobolev (USSR). Secretary: André Delessert (Switzerland). Members: Heinrich Behnke (Germany), André Revuz (France), Bryan Thwaites (UK). Ex officio: Henri Cartan (France) – President of IMU. The 1970 IMU General Assembly decided that the Past President of ICMI, the Secretary of IMU, and the representative of the Union in the ICSU Committee on the Teaching of Science (CTS) shall be members ex officio of the Executive Committee of ICMI. 1971–1974 President: James Lighthill (UK). Vice-Presidents: Shōkichi Iyanaga (Japan), János Surányi (Hungary).

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Secretary: Edwin A. Maxwell (UK). Members: Henry O. Pollak (USA), Sergei L. Sobolev (USSR). Ex officio: Hans Freudenthal (Netherlands) – Past President of ICMI, Komaravolu Chandrasekharan (Switzerland)  – President of IMU, Otto Frostman (Sweden)  – Secretary of IMU, André Lichnerowicz (France) – Representative of IMU in CTS/ ICSU. 1975–1978 President: Shōkichi Iyanaga (Japan). Vice-Presidents: Bent Christiansen (Denmark), Hans-Georg Steiner (Germany). Secretary: Yukiyosi Kawada (Japan). Members: Edward G. Begle (USA), Lev D. Kudrjavcev (USSR). Ex officio: James Lighthill (UK)  – Past President of ICMI, Deane Montgomery (USA) – President of IMU, Jacques-Louis Lions (France) – Secretary of IMU, Hans Freudenthal (Netherlands) – Representative of IMU in CTS/ICSU. 1979–1982 President: Hassler Whitney (USA). Vice-Presidents: Bent Christiansen (Denmark), Ubiratan D’Ambrosio (Brazil). Secretary: Peter Hilton (USA). Members: Stanley H.  Erlwanger (Canada), Bernhard H.  Neumann (Australia), Zbigniew Semadeni (Poland). Ex officio: Shōkichi Iyanaga (Japan) – Past President of ICMI, Lennart Carleson (Sweden) – President of IMU, Jacques-Louis Lions (France) – Secretary of IMU, Bent Christiansen (Denmark) – Representative of IMU in CTS/ICSU. 1983–1986 President: Jean-Pierre Kahane (France). Vice-Presidents: Bent Christiansen (Denmark), Zbigniew Semadeni (Poland). Secretary: A. Geoffrey Howson (UK). Members: Bienvenido F.  Nebres (Philippines), Michael F.  Newman (Australia), Henry O. Pollak (USA). Ex officio: Hassler Whitney (USA)  – Past President of ICMI, Jürgen Moser (Switzerland) – President of IMU, Olli Lehto (Finland) – Secretary of IMU, Henri Hogbe-Nlend (Cameroon) – Representative of IMU in CTS/ICSU. 1987–1990 President: Jean-Pierre Kahane (France). Vice-Presidents: Peng-Yee Lee (Singapore), Emilio Lluis Riera (Mexico). Secretary: A. Geoffrey Howson (UK). Members: Hiroshi Fujita (Japan), Jeremy Kilpatrick (USA), Mogens Niss (Denmark).

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Ex officio: Ludwig Faddeev (USSR) – President of IMU, Olli Lehto (Finland) – Secretary of IMU, Jacobus H. van Lint (Netherlands) – Representative of IMU in CTS/ICSU. 1991–1994 President: Miguel de Guzmán (Spain). Vice-Presidents: Jeremy Kilpatrick (USA), Peng-Yee Lee (Singapore). Secretary: Mogens Niss (Denmark). Members: Yuri L.  Ershov (Russia), Eduardo Luna (USA), Anna Sierpinska (Canada). Ex officio: Jean-Pierre Kahane (France) – Past President of ICMI, Jacques-Louis Lions (France) – President of IMU, Jacob Jr. (Brazil) – Secretary of IMU, Jacobus H. van Lint (Netherlands) – Representative of IMU in CTS/ICSU. 1995–1998 President: Miguel de Guzmán (Spain). Vice-Presidents: Jeremy Kilpatrick (USA), Anna Sierpinska (Canada). Secretary: Mogens Niss (Denmark). Members: Colette Laborde (France), Gilah Leder (Australia), Carlos E.  Vasco (Colombia), Dianzhou Zhang (China). Ex officio: David Mumford (USA) – President of IMU, Jacob Palis Jr. (Brazil) – Secretary of IMU. 1999–2002 President: Hyman Bass (USA). Vice-Presidents: Néstor Aguilera (Argentina), Michèle Artigue (France). Secretary: Bernard R. Hodgson (Canada). Members: Gilah Leder (Australia), Yukihiko Namikawa (Japan), Igor F. Sharygin (Russia), Jian-Pan Wang (China). Ex officio: Miguel de Guzmán (Spain)  – Past President of ICMI, Jacob Palis Jr. (Brazil) – President of IMU, Phillip Griffiths (USA) – Secretary of IMU. 2003–2006 President: Hyman Bass (USA). Vice-Presidents: Jill Adler (South Africa), Michèle Artigue (France). Secretary-General: Bernard R. Hodgson (Canada). Members: Carmen Batanero (Spain), Maria Falk De Losada (Colombia), Nikolai Dolbilin (Russia), Peter L.  Galbraith (Australia), Petar S.  Kenderov (Bulgaria), Frederick Koon-Shing Leung (Hong Kong SAR). Ex officio: John Ball (UK) – President of IMU, Phillip Griffiths (USA) – Secretary of IMU.

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2007–2009 President: Michèle Artigue. Vice-Presidents: Jill Adler (South Africa), Bill Barton (New Zealand). Secretary-General: Bernard R. Hodgson (Canada). Members: Mariolina Bartolini Bussi (Italy), Jaime Carvalho e Silva (Portugal), Celia Hoyles (UK), Kumaresan S. aka Kumaresan Somaskandan (India), Frederick Koon-Shing Leung (Hong Kong SAR), Alexei L. Semenov. Ex officio: Hyman Bass (USA) – Past President of ICMI, László Lovász (Hungary) – President of IMU, Martin Grötschel (Germany) – Secretary of IMU. Starting from 2007 we find the expression “member-at-large” instead of “member” in some ICMI documents. The term “member-at-large” was used in the past in a different context, see (Hodgson 2020). To avoid ambiguity we simply use the term “member” as it is done on the ICMI website https://www.mathunion.org/icmi/organization/icmi-­executive-­committee/past-­icmi-­executive-­committees (retrieved 23 July 2021). The following acronyms are used: CTS Committee on the Teaching of Science ICSU International Council of Scientific Unions IMU International Mathematical Union

6.4 Names and Terms of Reference In 1908, when the old Commission was founded, the official languages were those used at the International Congresses of Mathematicians: English, French, German, and Italian (L’Enseignement Mathématique 1908, 10: 449–450). The usual denomination for the Commission was Commission Internationale de l’Enseignement Mathématique with its acronym (CIEM). The German denomination Internationale Mathematische Unterrichts-Kommission) (IMUK) was also used, especially under the presidency of the German Felix Klein. In English and Italian, the other two official languages of the Commission, there were no official acronyms; in fact, these languages were used locally in the countries speaking those languages and in a few official documents. In the British journal The Mathematical Gazette, the commission was termed “International Commission on Mathematical Teaching” or “International Commission on the Teaching of Mathematics”. This last denomination was used by David E. Smith. In the presentation of the questionnaire for the inquiry into training of secondary teachers of mathematics (L’Enseignement Mathématique 1915, 17: 129–145) we find “International Commission on Mathematical Education” in the English translation and “Commissione Internazionale dell’Insegnamento Matematico” in the Italian one. In 1952, after the stop of World War II, the Commission was resurrected with a new name and acronym  – International Mathematical Instruction Commission

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(IMIC) – that soon was changed into the present name “International Commission on Mathematical Instruction” (ICMI). Already in the proceedings of the International Congress of Mathematicians of 1954 the present name and acronym of the Commission are used. In its first 100 years, the life of ICMI has been regulated by successive Terms of Reference approved in 1954, 1960, 1982, 1986, 2002, and 20071. After the celebration of the centenary  – from 2009 onwards  – further Terms of Reference were approved. Denominations and rules changed. For example, the term “Secretary-­ General”, which was used in the first decades of the life of the Commission for Henri Fehr, became “Secretary” in 1952 and again “Secretary-General” within the Terms of Reference of 2002 (ICMI Bulletin 51, p. 13). Also the notion of “Member of ICMI” has a story, which dates back to the first version of the Terms of Reference of ICMI issued in 1954, two years after its rebirth in Rome. The Commission consisted of ten members-at-large elected by the General Assembly on the nomination of the President of the International Mathematical Union (IMU) and of two national delegates named by each of the National Committee for Mathematics of countries members of IMU.  The Officers were a President, a Secretary, and two Vice-­ Presidents. The President was elected by the General Assembly on the nomination of the President of IMU from the membership at large of the Commission. The Executive Committee consisted of the Officers together with three additional members elected by the membership of the Commission. In the revision of the Terms of Reference of 1982, the notion of member-at-large of ICMI was abolished. To align with the rules of IMU, since the revision of the Terms of Reference of 2002 ICMI members are countries and not individuals. Although for a long time a General Assembly was held regularly, in particular during the International Congresses on Mathematical Education (ICMEs), only in the 2002 Terms of Reference the notion of “ICMI General Assembly” was introduced. This General Assembly was composed of the members of the Executive Committee and a delegate for each member nation.

6.5 Procedures for Election of the Executive Committee of ICMI The IMU General Assembly in Santiago de Compostela held on 19–20 August 2006 adopted new procedures for the election of the Executive Committee of ICMI, whose most important feature was to consider transferring the election of the Executive Committee of ICMI from the IMU General Assembly to the ICMI General Assembly usually held during the ICMEs, see the preliminary report in (Bass and Hodgson 2004). According to this new procedure, the election happens in a context of a close collaboration between the IMU and ICMI communities, but the  See Giacardi 2008 and Chap. 7 in this volume.

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final decision rests with ICMI. Since ICMI Executive Committee was in office for four years (between ICMEs), and the ICME was two years out of phase with the ICM, it was necessary to design a transition strategy for moving from the current system to the new one. Then a new ICMI Executive Committee was elected by the IMU General Assembly in 2006, but the next such election was by the ICMI General Assembly at ICME-11 in Monterrey, in 2008. The terms of the two ICMI Executive Committee were for three years each (2007–2009 and 2010–2012), instead of four. In 2012 at ICME-12 in Seoul the new process, approved in Santiago de Compostela, was fully functioning for the election of the 2013–2016 ICMI Executive Committee.

References and Sources The main sources of information are the journals: L’Enseignement Mathématique, which is the official organ of ICMI, the ICMI Bulletin (first issued in 1972), the Bulletin of the International Mathematical Union (first issued in 1971). Until 1970, before the birth of the IMU Bulletin, the journal Internationale Mathematische Nachrichten published the news concerning the community of mathematicians. Bass, Hyman, and Bernard R.  Hodgson. 2004. Inaugurating a new election procedure for ICMI. ICMI Bulletin 55: 18–22. Hodgson, Bernard. 2020. Once upon a time... Historical vignettes from the Archives of ICMI: A dilemma related to the ICMI Terms of reference. ICMI Newsletter (November): 6–8. IMU rules for the election of the Executive Committee of IMU, updated in 1999: IMU Bulletin 48 (June 2002): 8. Lehto, Olli. 1998. Mathematics without Borders: A history of the International Mathematical Union. New York: Springer Verlag. Old Terms of Reference of ICMI: ICMI Bulletin 47 (December 1999): 35–36. Terms of Reference for ICMI adopted in the second General Assembly of IMU in 1954 (The Hague) in L’Enseignement Mathématique 1951–1954: 40, 92–93. Terms adopted by the Executive Committee of IMU in 1960: ICMI Bulletin 5 (April 1975): 5–6. Terms adopted by the General Assembly of IMU in 1982: ICMI Bulletin 13 (February 1983): 5. Terms adopted by the General Assembly of IMU in 1986: ICMI Bulletin 47 (December 1999): 35–36. Terms of Reference for ICMI adopted by the Executive Committee of IMU in 2002: ICMI Bulletin 51 (December 2002): 8–12.

The Texts of the Terms of Reference Can Be Found in: Giacardi, Livia. The Terms of Reference for ICMI (1954–2007) (this volume). ———. 2008 Terms of Reference. In Furinghetti, Fulvia, and Livia Giacardi. 2008. The first century of the International Commission on Mathematical Instruction (1908–2008). The history of ICMI. http://www.icmihistory.unito.it/ (Retrieved June 2021). Terms of Reference of ICMI in https://www.mathunion.org/icmi/terms-reference-icmi (Retrieved 6 January 2021).

Chapter 7

The Terms of Reference for ICMI (1954–2007) Livia Giacardi

The Terms of Reference for the International Commission on Mathematical Instruction (ICMI) were introduced in 1954 during the General Assembly of the International Mathematical Union (IMU) in The Hague, to regulate relationships between ICMI and IMU, following some friction due to the lack of precise rules. The previous year, IMU president Marshall Stone wrote: “There seems to be a great deal of confusion in connection with ICMI.  I hope we can get it cleared up” (M.  Stone to E.  Bompiani, Chicago 10 July 1953, Archives of the International Mathematical Union (section of ICMI), Berlin 14B, 1952–1954). During the first century of ICMI life, the Terms of Reference were modified several times (1954, 1960, 1982, 1986, 2002, 2007), mainly for two reasons: the desire of ICMI to have greater independence from IMU and to have a better international organization. In the 1954 Terms of Reference, the denomination International Commission on Mathematical Instruction is introduced, but the acronym ICMI does not appear yet, even if it had been in use since 1952. According to these Terms of Reference, the Commission consisted of ten members-at-large1, elected by the General Assembly on the nomination by the IMU president, and of two national delegates nominated by their National Committees for Mathematics. The officers of the Commission were the president, the secretary, and two vice-presidents. The president was elected by the General Assembly on the nomination of the IMU president from the members-­ at-­ large of the Commission. The Executive Committee (EC) consisted of the  The denomination “member-at-large of ICMI” was abolished in the Terms of Reference of 1982. About the use of this denomination (see Hodgson 2020). 1

L. Giacardi (*) University of Turin, Turin, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_7

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officers together with three additional members elected by the membership of the Commission. Any National Adhering Organization wishing to support or encourage the work of the Commission might create, in agreement with its National Committee, a Sub-commission, to maintain relations with the Commission in every issue pertinent to mathematics teaching. Although ICMI could have a relatively free hand in its internal organization, IMU retained a strong control over it. The amendments adopted in 1960 and 2002 are particularly noteworthy. In the 1960 Terms of Reference, the acronym ICMI is introduced and some points of the 1954 version are changed: national delegates decrease from two to one; the clause “or recognize” is added in the point (e): any National Adhering Organization may create, or recognize, in agreement with its National Committee, a National Sub-Commission for ICMI. Furthermore, regional groups are introduced at the point (f): “In the pursuit of this objective, the Commission shall cooperate, to the extent it considers desirable, with effective regional groups which may be formed spontaneously, within, or outside, its own structure”. Finally, a point (g) is added: “The Commission may, with the approval of the Executive Committee of IMU, coopt, as members of ICMI, suitably chosen representatives of non-IMU countries, on an individual basis”. The main modifications introduced in 2002 are mentioned here. The composition of the Commission is clarified thus: as for IMU, members of ICMI are countries, not individuals; the “General Assembly of ICMI” is officially introduced; the notion of “ICMI Representatives” is specified; the number of members of the ICMI Executive Committee is increased and the possibility to coopt additional EC members is introduced; the notion of ICMI Affiliated Study Groups is also formally introduced. The 2002 IMU General Assembly introduced a Nominating Committee to propose a slate of nominees for the IMU EC members. At first, IMU suggested that this same Nominating Committee would also propose a slate of candidates for the ICMI election (IMU Bulletin 48, 2002: [8]). This solution was immediately criticized by ICMI because it would not pay sufficient attention to the ICMI community: The ICMI EC concluded that these proposals of IMU did not pay sufficient attention to the specificity of ICMI and were de facto moving away from a context where the ICMI community could play a significant role in the selection of its governing body. (Hodgson 2004, p. [16])

In 2004 the following discussions resulted in introducing a specific ICMI Nominating Committee charged with proposing a slate of candidates for the ICMI EC members. The election of officers and other EC members of ICMI should be at and by the General Assembly of ICMI (not IMU) at the International Congress of Mathematical Education (ICME, not ICM) (ICMI Bulletin 55, 2004: 18). This substantial change was approved during the IMU General Assembly held on 19–20 August 2006 in Santiago de Compostela, Spain (IMU Bulletin 2006, 53: 37–40). The first election according to the new procedure only took place at ICME-11 in Monterrey in 2008. A phase of transition was necessary since the terms of office of ICMI EC members lasted four years (between the ICMEs) and the ICMEs were two years staggered with respect to the ICMs.

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To understand the reasons for the changes that gradually were introduced and the debates that preceded, see the articles (Hodgson 2002), (Bass, Hodgson 2004), and Chapter 2 by Fulvia Furinghetti and Livia Giacardi in this volume. For the reader’s convenience, the texts of the Terms of Reference issued in the first century of ICMI are transcribed below.

7.1 The Texts of the Terms of Reference The Hague, 1 September 1954 Texte des décisions prises par l’Assemblée Générale de l’Union Mathématique Internationale (U.M.I.), à La Haye, le 1erseptembre 1954, concernant la Commission Internationale de l’Enseignement Mathématique (C.I.E.M.). L’Enseignement Mathématique, 40, 1951–1954: 92–93 (French version). Internationale Mathematische Nachrichten, 35/36, 1954: 12–13 (English version). Resolution on the Commission for Mathematical Instruction. Resolved: That the terms of reference and the constitution of the International Commission on Mathematical Instruction be clarified and determined as follows: a) The Commission shall consist of ten members-at-large elected by the General Assembly on nomination of the President of IMU, and of two national delegates named by each of the National Committees for Mathematics as specified below; b) The officers of the Commission shall consist of a President, a Secretary, and two Vice-Presidents. The President shall be elected by the General Assembly on the nomination of the President of the IMU from the membership at large of the Commission; c) The Executive Committee of the Commission shall consist of the Officers of the Commission together with three additional members elected by the membership of the Commission; d) During his lifetime Professor Fehr (Geneva) shall remain Honorary President of the Commission in recognition of his longtime interest in the cause of Mathematical education and his devoted services to it; e) In all other respects, the Commission shall make its own decisions as to his internal organization and rules of procedure; f) Any National Adhering Organization wishing to support or encourage the work of the Commission may create, in agreement with its National Committee, a subcommittee to maintain liaison with the Commission in all matters pertinent to its affairs. The National Adhering Organization in question shall designate two members of the said subcommittee, if created, to serve as delegated members of the Commission; g) The Commission shall be charged with the conduct of the activities of IMU, bearing on mathematical and scientific education, and shall take the initiative in inaugurating appropriate programs designed to further the sound development of

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mathematical education at all levels and to secure public appreciation of its importance; h) The budget of the Commission shall be submitted to the Executive Committee of IMU and the General Assembly for approval at such times as may be determined by agreement between the Commission and the Executive Committee; i) The Commission shall file an annual report of its activities with the Executive Committee of IMU and shall file a quadrennial report at each regular meeting of the General Assembly. Paris, April 1960 Terms of Reference of the International Commission on Mathematical Instruction (ICMI). ICMI Bulletin, 5 (April 1975): 5–62 Terms of Reference of the International Commission on Mathematical Instruction (ICMI) (a) The Commission shall consist of ten members-at-large elected by the General Assembly of IMU on nomination of the President of IMU and of one national delegate from each member nation, as specified below. (b) The officers of the Commission shall consist of a President, a Secretary, and two Vice- Presidents. The President shall be elected by the General Assembly on the nomination of the President of IMU from the members-at-large of the Commission. (c) The Executive Committee of the Commission shall consist of the officers of the Commission together with three additional members elected by the membership of the Commission. (d) In all other respects, the Commission shall make its own decisions as to its internal organization and rules of procedure. (e) Any National Adhering Organization wishing to support or encourage the work of the Commission may create, or recognize, in agreement with its National Committee, a National Sub-Commission for ICMI to maintain liaison with the Commission in all matters pertinent to its affairs. The National Adhering Organization in question shall designate one member of the said Sub­ Commission, if created, to serve as a delegated member of ICMI as mentioned in (a). (f) The Commission shall be charged with the conduct of the activities of IMU, bearing on mathematical or scientific education and shall take the initiative in inaugurating appropriate programmes designed to further the sound development of mathematical education at all levels, and to secure public appreciation of its importance. In the pursuit of this objective, the Commission shall cooperate, to the extent it considers desirable, with effective regional groups which may be formed spontaneously, within, or outside, its own structure,

 See L’Enseignement Mathématique s. 2, 4, 1958: 227-228, Internationale Mathematische Nachrichten, 68/69, 1961: 29. 2

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(g) The Commission may, with the approval of the Executive Committee of IMU, coopt, as members of ICMI, suitably chosen representatives of non-IMU countries, on an individual basis. (h) The budget of the Commission shall be submitted to the Executive Committee of IMU and the General Assembly, for approval, at such times as may be determined by agreement between the Commission and the Executive Committee of IMU. (i) The Commission shall file annual report of its activities with the Executive Committee of IMU, and shall file a quadrennial report at each regular meeting of the General Assembly. Warsaw, 8–9 August 1982 Terms of Reference of the International Commission on Mathematical Instruction (Accepted in August 1982 by the General Assembly of IMU). IMU Bulletin 1982, Annex 4, p. 39; ICMI Bulletin, 13 (February 1983): 5. Terms of Reference of the International Commission on Mathematical Instruction (Accepted in August 1982 by the General Assembly of IMU) (a) The Commission shall consist of 1. the members of an Executive Committee as specified in (b) below, elected by IMU, and 2. one national delegate from each member nation as specified in (d) below. (b) The Executive Committee consists of four officers, namely, President, two Vice Presidents, and Secretary, and of three further members. Furthermore, the outgoing President of ICMI, the President and the Secretary of IMU, and the representative of IMU at CTS (ICSU) are members ex-officio of the E.C. (c) In all other respects, the Commission shall make its own decisions as to its internal organization and rules of procedure. (d) Any National Adhering Organization wishing to support or encourage the work of the Commission may create, or recognize, in agreement with its National Committee, a National Sub-Commission for ICMI to maintain liaison with the Commission in all matters pertinent to its affairs. The National Adhering Organization in question shall designate one member of the said SubCommission, if created, to serve as a delegated member of ICMI as mentioned in (a). (e) The Commission shall be charged with the conduct of the activities of IMU, bearing on mathematical or scientific education and shall take the initiative in inaugurating appropriate programmes designed to further the sound development of mathematical education at all levels, and to secure public appreciation of its importance. In the pursuit of this objective, the Commission shall cooperate, to the extent it considers desirable with effective regional groups which may be formed spontaneously, within, or outside, its own structure. (f) The Commission may, with the approval of the Executive Committee of IMU, coopt, as members of ICMI, suitably chosen representatives of non-IMU countries, on an individual basis.

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(g) The budget of the Commission shall be submitted to the Executive Committee of IMU and the General Assembly, for approval, at such times as may be determined by agreement between the Commission and the Executive Committee of IMU. (h) The Commission shall file an annual report of its activities with the Executive Committee of IMU, and shall file a quadrennial report at each regular meeting of the General Assembly. Oakland (California, USA), 31 July-1 August 1986 International Commission on Mathematical Instruction (ICMI). Terms of Reference (1986). ICMI Bulletin, 47 (December 1999): 35–36.3 International Commission on Mathematical Instruction (ICMI). Terms of Reference (1986) 1. The Commission shall consist of a. the members of an Executive Committee as specified in (2) below, elected by IMU, and b. one delegate from each member nation as specified in (4) below. 2. The Executive Committee consists of four officers, namely, President, two Vice-­ Presidents, and Secretary, and of three further members. Furthermore, the outgoing President of ICMI, the President and the Secretary of IMU, and the representative of IMU at CTS (ICSU) are members ex-officio of the E.C. 3. In all other respects, the Commission shall make its own decisions as to its internal organization and rules of procedure. 4. Any Adhering Organization4 wishing to support or encourage the work of the Commission may create, or recognize, in agreement with its Committee for Mathematics, a Sub-Commission for ICMI to maintain liaison with the Commission in all matters pertinent to its affairs. The Adhering Organization in question shall designate one member of the said Sub-Commission, if created, to serve as a delegated member of ICMI as mentioned in (1). 5. The Commission shall be charged with the conduct of the activities of IMU, bearing on mathematical or scientific education and shall take the initiative in inaugurating appropriate programmes designed to further the sound development of mathematical education at all levels, and to secure public appreciation of its importance. In the pursuit of this objective, the Commission shall cooperate, to the extent it considers desirable with effective regional groups which may be formed spontaneously, within, or outside, its own structure.

 See: https://www.mathunion.org/fileadmin/IMU/Organization/ICMI/bulletin/47/ICMI_terms_ of_reference.html (Retrieved June 2021) 4  The adjective “national” was deleted from the IMU Statutes and consequently also from the Terms of Reference of the IMU commissions. See IMU Bulletin 26 (December 1986): 4. 3

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6. The Commission may, with the approval of the Executive Committee of IMU, coopt, as members of ICMI, suitable chosen representatives of non-IMU countries, on an individual basis. 7. The budget of the Commission shall be submitted to the Executive Committee of IMU and the General Assembly, for approval, at such times as may be determined by agreement between the Commission and the Executive Committee of IMU. 8. The Commission shall file an annual report of its activities with the Executive Committee of IMU, and shall file a quadrennial report at each regular meeting of the General Assembly. Paris, 12–13 April 2002 International Commission on Mathematical Instruction (ICMI). Terms of Reference (2002). ICMI Bulletin, 51 (December 2002): 9–10.5 International Commission on Mathematical Instruction (ICMI) Terms of Reference (2002) (Adopted by the Executive Committee of the International Mathematical Union at its meeting held at Institut Henri-Poincaré in Paris on April 12–13, 2002) 1. The members of the International Commission on Mathematical Instruction (ICMI) consist of (a) those countries which are members of the International Mathematical Union (IMU), and (b) other countries which are co-opted, as specified in (7) below. The term “country” is to be understood as described in the Statutes of IMU. 2. The General Assembly of the Commission consists of ( a) the members of the Executive Committee, as specified in (3) below, and (b) one Representative from each member country of ICMI, as specified in (5) below.  he General Assembly of ICMI shall normally meet once in every 4 years, T during the International Congress on Mathematical Education. 3. The Executive Committee of the Commission consists of the following members. Elected by IMU: Nine members, including the four officers, namely, the President, two Vice-Presidents, and the Secretary-General. Ex-officio members: The outgoing President of ICMI, the President, and the Secretary of IMU. Co-­ opted members: In order to provide for missing coverage or representation, the ICMI Executive Committee may co-opt up to two additional members. 4. In all other respects, the Commission shall make its own decisions as to its internal organization and rules of procedure.

 See (Hodgson. 2002, pp. 8-12).

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5. Appointment of the Representative to ICMI is the responsibility of the Adhering Organization of IMU, for those countries which are members of IMU, and of the Adhering Organization of ICMI, for those countries co-opted under item (7) below. Any Adhering Organization wishing to support or encourage the work of the Commission may create, or recognize, in agreement with its Committee for Mathematics in the case of a member country of IMU, a Sub-­Commission for ICMI to maintain liaison with the Commission in all matters pertinent to its affairs. The Representative to ICMI, as mentioned in (2) above, should be a member of the said Sub-Commission, if created. 6. The Commission shall be charged with the conduct of the activities of IMU, bearing on mathematical or scientific education and shall take the initiative in inaugurating appropriate programmes designed to further the sound development of mathematical education at all levels, and to secure public appreciation of its importance. In the pursuit of this objective, the Commission shall cooperate, to the extent it considers desirable, with effective regional groups which may be formed spontaneously, within, or outside, its own structure. 7. The Commission may, with the approval of the Executive Committee of IMU, co-opt as members of ICMI countries that are not members of IMU, on an individual basis. 8. The Commission may approve the affiliation to ICMI of Study Groups, focusing on a specific field of interest and study in mathematics education consistent with the aims of the Commission. These Affiliated Study Groups are independent of ICMI, financially and otherwise, but they shall produce quadrennial reports to be presented at the General Assembly of ICMI. The Commission will cooperate, to the extent possible, with the work of the Study Groups, for example, by regularly publishing information on their activities in the ICMI Bulletin. 9. The budget of the Commission shall be submitted to the Executive Committee of IMU and the General Assembly of IMU, for approval, at such times as may be determined by agreement between the Commission and the Executive Committee of IMU. 10. The Commission shall file an annual report of its activities with the Executive Committee of IMU, and shall file a quadrennial report at each regular meeting of the General Assembly of IMU. 11. At each regular meeting of the General Assembly of ICMI, the Commission shall file a quadrennial report of its financial situation and of its activities. Procedures for the Election of the Executive Committee of ICMI The rules for the election of the Executive Committee of ICMI are similar to those for the election of the Executive Committee of IMU with the same Nominating Committee. The existing Executive Committee of ICMI shall request proposals for the membership of the EC of ICMI from the Representatives to ICMI. The EC of IMU shall request proposals for the membership of the EC of ICMI from the Committees for Mathematics, who shall consult the Representatives to ICMI for suggestions. The EC of IMU will conduct extensive consultations with the

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existing Executive Committee of ICMI before proposing slates to the Nominating Committee. No person can be a candidate for more than one office. 7 February 2007, Terms of Reference of ICMI (2007). This version of the Terms of Reference was adopted by the EC of IMU by an electronic vote: https://www.mathunion.org/icmi/terms-­reference-­icmi-­2007 (Retrieved June 2021) 1. The members of the International Commission on Mathematical Instruction (ICMI) consist of 1. those countries which are members of the International Mathematical Union (IMU), and 2. other countries which are co-opted, as specified in (7) below. The term “country” is to be understood as described in the Statutes of IMU. 2. The General Assembly of the Commission consists of 1 . the members of the Executive Committee, as specified in (3) below, and 2. one Representative from each member country of ICMI, as specified in (5) below. The General Assembly of ICMI shall normally meet once in every 4 years, during the International Congress on Mathematical Education. 3. The Executive Committee of the Commission consists of the following members: 1. Elected by the ICMI General Assembly: Nine members, including the four officers, namely, the President, the two Vice-Presidents, and the Secretary-­ General. The President shall serve for one, non-renewable, term. 2. Ex-officio members: The immediate Past President of ICMI, the President, and the Secretary of IMU. 3. Co-opted members: In order to provide for missing coverage or representation, the ICMI Executive Committee may co-opt up to two additional members. 4. In all other respects, the Commission shall make its own decisions as to its internal organization and rules of procedure. 5. Appointment of the Representative to ICMI is the responsibility of the Adhering Organization of IMU, for those countries which are members of IMU, and of the Adhering Organization of ICMI, for those countries co-opted under item (7) below. Any Adhering Organization wishing to support or encourage the work of the Commission may create, or recognize, in agreement with its Committee for Mathematics in the case of a member country of IMU, a Sub-­Commission for ICMI to maintain liaison with the Commission in all matters pertinent to its affairs. The Representative to ICMI, as mentioned in (2) above, should be a member of the said Sub-Commission, if created. 6. The Commission shall be charged with the conduct of the activities of IMU bearing on mathematical or scientific education and shall take the initiative in

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inaugurating appropriate programs designed to further the sound development of mathematical education at all levels, and to secure public appreciation of its importance. In the pursuit of this objective, the Commission shall cooperate, to the extent it considers desirable, with effective regional groups which may be formed spontaneously, within, or outside, its own structure. 7. The Commission may, with the approval of the Executive Committee of IMU, co-opt as members of ICMI countries that are not members of IMU, on an individual basis. 8. The Commission may approve the affiliation to ICMI of Study Groups, focusing on a specific field of interest and study in mathematics education consistent with the aims of the Commission. These Affiliated Study Groups are independent of ICMI, financially and otherwise, but they shall produce quadrennial reports to be presented at the General Assembly of ICMI. The Commission will cooperate, to the extent possible, with the work of the Study Groups, for example by regularly publishing information on their activities in the ICMI Bulletin. 9. The budget of the Commission shall be submitted to the Executive Committee of IMU and the General Assembly of IMU, for approval, at such times as may be determined by agreement between the Commission and the Executive Committee of IMU. 10. The Commission shall file an annual report of its activities with the Executive Committee of IMU, and shall file a quadrennial report at each regular meeting of the General Assembly of IMU. 11. At each regular meeting of the General Assembly of ICMI, the Commission shall file a quadrennial report of its financial situation and of its activities.

References Bass, Hyman, and Bernard Hodgson. 2004. Inaugurating a new election procedure for ICMI. ICMI Bulletin 55: 18–22. Hodgson, Bernard. 2002. New Terms of Reference for ICMI. ICMI Bulletin 51: 8–12. ———. 2004. Report on ICMI activities in 2004. IMU Bulletin 51: https://www.mathunion.org/fileadmin/IMU/Publications/Bulletins/2000_2006/51_December2004.pdf (Retrieved June 2021) ———. 2020. Once upon a time... Historical vignettes from the Archives of ICMI: A dilemma related to the ICMI Terms of reference. ICMI Newsletter (November): 6-8. Giacardi, Livia (ed.) Terms of Reference. In Furinghetti Fulvia and Livia Giacardi, The First Century of the International Commission on Mathematical Instruction (1908-2008). The History of ICMI: https://www.icmihistory.unito.it/terms.php (Retrieved June 2021) ICMI Bulletin. In https://www.mathunion.org/icmi/publications/icmi-­bulletin (Retrieved June 2021) IMU Bulletin. In https://www.mathunion.org/membership/imu-­bulletins (Retrieved June 2021) Terms of Reference: https://www.mathunion.org/icmi/organization/icmi-­organization (Retrieved June 2021)

Chapter 8

Mathematics Education in the International Congresses of Mathematicians 1897–2006 Fulvia Furinghetti

This chapter aims to give an overview of the activities related to mathematics education carried out during the International Congresses of Mathematicians (ICMs) from 1897 to 2006 listed in Table 8.1. Before mathematics education developed as a full-fledged academic discipline with its specific conferences and journals, the opportunities for international meetings in this field were limited, with few exceptions, to the special sections within the ICMs. After 1897 the program of the ICMs included a section dedicated to didactics of mathematics, often associated with other topics such as logic, history, or philosophy. The contributions concerning popularization of mathematics were included among those on education. In a few cases also special activities on didactics of mathematics were organized. In the didactic sections, important decisions were taken, for one the creation of CIEM/ IMUK (Commission Internationale de l’Enseignement Mathématique/Internationale Mathematische Unterrichts-Kommission) in 1908. In some cases, the ICMs, gathering many researchers from all over the world, were also an opportunity to organize collateral activities dedicated to mathematics education. The proceedings of the ICMs were edited according to different criteria, and then information about the contributions concerning mathematics education is not homogeneous: sometimes it is detailed and complete, and sometimes poor. Usually, these contributions were presented in the form of short communication, but there had been also half-hour addresses, 45-minute addresses, and addresses invited by the Program Committee or by the sections on didactics. Some special activities were organized by ICMI. As the number of participants increased, it was often decided not to publish the texts or the abstracts of short communications in the official proceedings. In these cases, during the congress, books were distributed with F. Furinghetti (*) University of Genoa, Genoa, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_8

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Table 8.1  List of the International Congresses of Mathematicians Zurich, Switzerland Paris, France Heidelberg, Germany Rome, Italy Cambridge, UK Strasbourg, France Toronto, Canada Bologna, Italy Zurich, Switzerland Oslo, Norway Cambridge, USA Amsterdam, Holland Edinburgh, UK Stockholm, Sweden Moscow, USSR Nice, France Vancouver, Canada Helsinki, Finland Warsaw, Poland Berkeley, USA Kyoto, Japan Zurich, Switzerland Berlin, Germany Beijing, China Madrid, Spain

9–11 August 1897 6–12 August 1900 8–13 August 1904 6–11 April 1908 22–28 August 1912 22–30 September 1920 11–16 August 1924 3–10 September 1928 5–12 September 1932 13–18 July 1936 30 August–6 September 1950 2–9 September 1954 14–21 August 1958 15–22 August 1962 16–26 August 1966 1–10 September 1970 21–29 August 1974 15–23 August 1978 16–24 August 1983 3–11 August 1986 21–29 August 1990 3–11 August 1994 18–27 August 1998 20–28 August 2002 22–30 August 2006

information on the short communications. In the IMU website that contains the texts of ICM proceedings from 1893 to 2018, these publications, which are not a part of the official proceedings, are not present, but they are kept in the IMU archives or in some university libraries, from which the information reported below comes. Over the years the official languages have been English, French, German, Italian, and Russian. For the reader’s convenience, the Russian contributions are presented only in English, when there was not the transliteration from Cyrillic of the texts.

8.1 Zurich (Switzerland), 9–11 August 1897 Rudio, Ferdinand (ed.). 1898. Verhandlungen des ersten Internationalen Mathematiker-Kongresses. Leipzig: B. G. Teubner. One volume: VIII +  306 pages.

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There were no sections on didactics. The following contribution in the section Geschichte und Bibliographie (History and Bibliography, pp.  274–306) concerns higher mathematical instruction: –– Klein, F., Zur Frage des höheren mathematischen Unterrichts, pp. 300–306.

8.2 Paris (France), 6–12 August 1900 Duporcq, Ernest (ed.). 1902. Compte Rendu du deuxième Congrès International des Mathématiciens. Gauthier-Villars: Paris. One volume: 455 pages. There were six sections. The fifth section, entitled Bibliographie et histoire (Bibliography and history, pp. 379–403), and the sixth section, entitled Enseignement et méthodes (Teaching and methods, pp. 405–450), were brought together, but the contributions appear separate. The full text of the talk given in the sixth section entitled “Sur les problèmes futurs des mathématiques” (On the future problems of mathematics, pp. 58–114) by David Hilbert was published among the plenary talks because of its great importance. In Section 6 there are: –– De Galdeano, Z. G., Note sur la critique mathématique, p. 405. –– Veronese, G., Les postulats de la Géométrie dans l’enseignement (Translated by R. Bricard and E. Duporcq), pp. 433–450.

8.3 Heidelberg (Germany), 8–13 August 1904 Krazer, Adolf (ed.). 1905. Verhandlungen des dritten Internationalen Mathematiker-­ Kongresses. Leipzig: B. G. Teubner. One volume: X + 756 pages. There were six sections. The sixth section, entitled Pädagogik (Pedagogy, pp. 582–713), contains: –– Greenhill, A. G., Teaching of mechanics by familiar applications on a large scale, pp. 582–585. –– Gutzmer, A., Über die auf die Anwendungen gerichteten Bestrebungen im mathematischen Unterricht der deutschen Universitäten, pp. 586–593. –– Loria, G., Sur l’enseignement des mathématiques en Italie, pp. 594–602. –– Fehr, H., L’enquête de “L’Enseignement Mathématique” sur la méthode de travail des mathématiciens, pp. 603–607. –– Stäckel, P., Über die Notwendigkeit regelmäßiger Vorlesungen über elementare Mathematik an den Universitäten, pp. 608–614. –– Fricke, R., Bemerkungen über den mathematischen Unterricht an den technischen Hochschulen in Deutschland, pp. 615–621. –– Andrade, J., L’enseignement scientifique aux écoles professionnelles et les “Mathématiques de l’ingénieur”, pp. 622–626.

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–– Schotten, H., Welche Aufgabe hat der mathematische Unterricht auf den deutschen Schulen und wie passen die Lehrpläne zu dieser Aufgabe?, pp. 627–638. –– Simon, M., Über komplexe Zahlen; über den Lehrgang in der sphärischen Trigonometrie; literarisch-historische Notizen, pp. 639–640. –– Thieme, H., Wirkung der wissenschaftlichen Ergebnisse auf den Unterricht in der elementaren Mathematik, pp. 641–650. –– Šourek, A., Über den mathematischen Unterricht in Bulgarien, pp. 651–666. –– Meyer, F., Über das Wesen mathematischer Beweise, pp. 667–686. –– Finsterbusch, J., Über eine neue einfache und vor allem einheitliche Methode, die Rauminhalte der Körper zu bestimmen, deren Querschnittsfunktion den dritten Grad der Höhe nicht Übersteigt, und ihre Verallgemeinerung, pp. 687–706. –– Brückner, M., Über die diskontinuierlichen und nicht-konvexen gleicheckig-­ gleichflächigen Polyeder, pp. 707–713.

8.4 Rome (Italy), 6–11 April 1908 Castelnuovo, Guido (ed.). 1909. Atti del IV Congresso Internazionale dei Matematici. Roma: Tipografia della R.  Accademia dei Lincei. Vol. 1: IV  +  217 pages. Vol. 2: 319 pages. Vol. 3: 588 pages. There were four sections. The fourth section, entitled Questioni filosofiche, storiche, didattiche (Philosophical, historical and didactical questions, Vol. 3, pp. 373–579), contains: –– Hessenberg, G., Zählen und Anschauung, pp. 377–379. –– Gutzmer, A., Ueber die Reformbestrebungen auf dem gebiete des mathematischen Unterrichts in Deutschland, pp. 441–448. –– Godfrey, C., The teaching of Mathematics in English public schools for boys, pp. 449–464. –– Smith, David Eugene. The teaching of mathematics in the secondary schools of the United States, pp. 465–477. –– Suppantschitsch, R., L’applications (sic) des idées modernes des mathématiques à l’enseignement secondaire en Autriche, pp. 478–481. –– Vailati, G., Sugli attuali programmi per l’insegnamento della matematica nelle scuole secondarie italiane, pp. 482–487. –– Fehr, H., Les mathématiques dans l’enseignement secondaire en Suisse, pp. 500–509. –– Archenhold, F.  S., Ueber die Bedeutung des mathematischen Unterrichtes im Freien in Verbindung mit Reformvorschlägen für den Lehrgang, pp. 510–513. –– Andrade, J., Quelques observations psychologiques recueillies dans les enseignements scientifiques d’initiation, pp. 514–518. –– Conti, A., Sull’iniziazione alle matematiche e sulla preparazione matematica dei maestri elementari in Italia, pp. 519–528.

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–– De Galdeano, Z.  G., Quelques mots sur l’enseignement mathématique en Espagne, p. 529. –– Beke, E., Ueber den jetzigen Stand des mathematischen Unterrichtes und die Reformbestrebungen in Ungarn, pp. 530–533. –– Emch, A., Der Rechenkünstler Winkler und seine Methoden, pp. 538–541. –– De Amicis, E., L’equivalenza in planimetria indipendentemente dalle proporzioni e dal circolo, pp. 563–568. –– Delitala, G., La tetragonometria piana nelle scuole secondarie, pp. 572–579. In section IIIB (Various applications of mathematics) there is:  - Castelli, G., Insegnamento della matematica finanziaria e attuariale nelle scuole professionali italiane, Vol. 3, pp. 327–333.

8.5 Cambridge (UK), 22–28 August 1912 Hobson, Ernest W. and Augustus Love (eds.). 1913. Proceedings of the fifth International Congress of Mathematicians. Cambridge: Cambridge University Press. Vol. 1: 500 pages. Vol. 2: 657 pages. There were four sections. The fourth section (Philosophy, history, didactics, Vol. 2, pp. 447–653) contains 29 contributions divided into the following subsections: a and b (Philosophy, history, didactics, Vol. 2, pp. 447–458), a (Philosophy, history, Vol. 2, pp. 459–541), and b (Didactics, Vol. 2, pp. 543–653): –– Gérardin, A., Sur quelques nouvelles machines algébriques, pp. 572–573. –– Hill, M. J. M., The teaching of the theory of proportion, pp. 545–571. –– Hatzidakis, N., Systematische Rekreationsmathematik in den mittleren Schulen, pp. 569. –– Le Chatelier, H., L’enseignement des mathématiques à l’usage des ingénieurs, pp. 574–577. –– Carson, G. St L., The place of deduction in elementary Mechanics, pp. 578–581. –– Nunn, T. P., The calculus as a subject of school instruction, pp. 582–590. –– Fehr, H., La Commission International de l’Enseignement Mathématique de 1908 à 1912, pp. 591–597. –– Runge, C., Report of Sub-Commission B of the International Commission on the Teaching of Mathematics: The mathematical training of the physicist in the University, pp. 598–607 including the discussion. –– Goldziher, K., Bericht über die Herausgabe einer Bibliographie des mathematischen Unterrichts 1900–1912, pp. 608–610. –– Smith, D. E., Report of Sub-Commission A of the International Commission on the Teaching of Mathematics: Intuition and experiment in mathematical teaching in secondary schools, pp. 611–632. –– Godfrey, C., Methods of intuition and experiment in English secondary schools, pp. 633–641. –– Liste des Publications du Comité Central et des sous-commissions nationales, pp. 642–653.

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F. Furinghetti

CIEM/IMUK met in Cambridge from 21 to 28 August 1912, at the same time as International Congress of Mathematicians. Following an agreement between the Congress Committee and the Central Committee of the Commission, the sessions were held jointly with those of Section IV b.

8.6 Strasbourg (France), 22–30 September 1920 Villat, Henri (ed.). 1921. Comptes Rendus du Congrès International des Mathématiciens. Toulouse: É. Privat (Librairie de l’Université). One volume: XLVIII + 672 pages. There were four sections. The fourth section, entitled Questions philosophiques, historiques, pédagogiques (Historical, philosophical and pedagogical questions, pp. 611–664), contains: –– Dubecq, Sur l’enseignement en République Argentine, p. 664, in absentia read by G. Königs. –– Zervos, P., Sur l’enseignement mathématique à Athènes (not printed in the proceedings). The following plenary talk concerns mathematics teaching at university level: –– Volterra, V., Sur l’enseignement de la physique mathématique et de quelques points de l’analyse, pp. 81–97 (plenary talk).

8.7 Toronto (Canada), 11–16 August 1924 Fields, John C. (ed.). 1928. Proceedings of the International Mathematical Congress. Toronto: The University of Toronto Press. Vol. 1: 935 pages. Vol. 2: 1006 pages. There were six sections. The sixth section (Vol. 2, pp. 929–989), entitled History, philosophy, didactics, contains: 1 . Crelier, L. J., Observations pratiques de méthodologie, pp. 973–974. 2. Fehr, H., L’université et la préparation des professeurs de mathématiques, p. 987.

8.8 Bologna (Italy), 3–10 September 1928 1929. Atti del Congresso Internazionale dei Matematici. Bologna: N.  Zanichelli. Vol. 1: 338 pages. Vol. 2: 365 pages. Vol. 3: 472 pages. Vol. 4: 429 pages. Vol. 5: 494 pages. Vol. 6: 554 pages.

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There were seven sections (some of them divided into subsections). The sixth section, entitled Matematiche elementari, Questioni didattiche, Logica matematica (Elementary mathematics, Didactical questions, Mathematical logic, Vol. 3, pp. 373–458), contains: –– De Vos van Steenwijk, J. E., Some remarks on the teaching of mathematics in secondary schools, pp. 393–396. –– Furlani, G., Nuovi indirizzi nell’insegnamento della matematica in Italia, pp. 397–402. –– Sittignani, M.  G., L’insegnamento della matematica negli Istituti di cultura generale, pp. 403–410. –– Artom, E., Intorno all’insegnamento dell’aritmetica e dell’algebra, pp. 411–414. –– Marcolongo, R., Il calcolo vettoriale nell’insegnamento secondario, pp. 415–419. –– Heegaard, P., La représentation des points imaginaires de Sophus Lie et sa valeur didactique, pp. 421–423. –– Clapier, C., Méthode de recherche didactique en géométrie élémentaire, pp. 425–427. –– Zervos, M., Sur une expression nouvelle de la définition des nombres premiers, pp. 429–431. –– Maroni, A., Osservazioni sui poliedri, pp. 433–437. –– Hatzidakis, N., Due proposte per l’insegnamento medio, pp. 439–441. –– Faragó, A., Közepiskolai Matematikai és Fizikai Lapok, pp. 453–456. –– Sakellariou, N., Projet pour la constitution d’une Commission Internationale pour l’Enseignement des Mathématiques, pp. 457–458.

8.9 Zurich (Switzerland), 5–12 September 1932 Saxer, Walter (ed.). 1932. Verhandlungen des Internationalen Mathematiker-­ Kongresses Zürich 1932. Zürich - Leipzig: Orell Füssli. Vol. 1: 335 pages. Vol. 2: XX + 367 pages. There were eight sections. Section 8, entitled Pädagogik und Verhandlungen der Internationalen Mathematischen Unterrichtskommission (Pedagogy and reports of the International Mathematical Commission on the Teaching of Mathematics), is divided into two subsections: Pedagogy (Vol. 2, pp. 353–359) and Reports of the International Mathematical Commission on the Teaching of Mathematics (Vol. 2, pp. 360–367). It contains: –– Carrus, S., Sur l’enseignement mathématique, pp. 355–356. –– Establier, A., Résumé du rapport de l’Institut International de Coopération intellectuelle concernant la coordination de l’économie scientifique internationale, pp. 356–357. –– Cowley, E. B., Technical vocabularies for plane and solid geometry, pp. 357–358. –– Zervos, M., Sur quelques définitions et théorèmes de l’arithmétique, pp. 358–359.

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Verhandlungen der Internationalen Mathematischen Unterrichtskommission (Reports of the International Mathematical Commission on the Teaching of Mathematics): –– Smith, D.  E., The International Commission on the teaching of mathematics, pp. 360–361. –– Fehr, H., La Commission Internationale de l’enseignement mathématique de 1928–1932. Rapport sommaire, pp. 361–362. –– Loria, G., La préparation théorétique et pratique des professeurs de mathématiques de l’enseignement secondaire, p. 363. –– Hamel, G., Der gegenwärtige Zustand der Frage der Ausbildung der Mathematik-­ Lehrer in Deutschland, pp. 363–364. –– Beschluss der Internationalem Mathematischen Unterrichtskommission, pp. 366–367.

8.10 Oslo (Norway), 13–18 July 1936 Lange-Nielsen, Fr., Edgar B. Schieldrop and Nils Solberg. 1937. Comptes Rendus du Congrès International des Mathématiciens. Oslo: A. W. Brøggers Boktrykkeri A/S, Vol. 1: 316 pages. Vol. 2: XVI + 289 pages. There were eight sections. The eighth section, entitled Pédagogie (Pedagogy, Vol. 2, pp. 281–287), contains: –– Fairthorne, R.  A., Farnborough: A method for demonstrating the qualitative properties of differential equations by means of cinematograph films, p. 283. –– Przibram, H., Beliebiges Wurzelziehen als Rechnungsart ohne Logarithmen, pp. 283–287. –– Commission Internationale de l’Enseignement Mathématique, Séance dans la section VIII Mercredi le 15 juillet, pp. 287–289.

8.11 Cambridge (USA), 30 August–6 September 1950 Graves, Lawrence M., Paul A. Smith, Einar Hille, and Oscar Zariski (eds.). 1952. Proceedings of the International Congress of Mathematicians. Providence RI: American Mathematical Society. Vol. 1: 769 pages. Vol. 2: 461 pages. There were seven sections. The seventh section is entitled History and education. The following abstracts are published in the proceedings, divided into a first part labeled “History” (Vol. 1, pp. 737–751) and a second part labeled “Education” (Vol. 1, pp. 752–760): –– Pólya, G., On plausible reasoning, pp. 739–747 (invited address in the section). –– Betz, W., Mathematics for the million, or for the few?, p. 752.

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–– Boyer, L. E., A note on the teaching of general mathematics, p. 753. –– Fawcett, H. P., Unifying concepts in mathematics, p. 754. –– Griffin, F.  L., Further experience with undergraduate mathematical research, p. 755. –– Gundlach, B. H., Gestalt theory in the teaching of mathematics, pp. 755–756. –– Kramer, M. S., The introduction of applied problems for the enrichment of classroom instruction in the schools and colleges, p. 756. –– May, K. and McVoy, K., Simplification of rigorous limit proofs, p. 757. –– Overn, O. E., Current trends in the teaching of plane trigonometry, p. 757–758. –– Richardson, M., Fundamentals in the teaching of undergraduate mathematics, p. 758. –– Rossing, E., The teaching of mathematics in Denmark, p. 759. –– Swain, R. L., Condensed graphs, p. 760.

8.12 Amsterdam (Holland), 2–9 September 1954 Gerretsen, Johan C. H. and Johannes de Groot (eds.). 1954–1957. Proceedings of the International Congress of Mathematicians. Groningen: E. P. Noordhoff N. V.; Amsterdam: North-Holland. Vol. 1: 582 pages. Vol. 2: 440 pages. Vol. 3: 560 pages. There were seven sections. In the seventh section (Philosophy, history and education), 36 short lectures were presented (19 reported in Vol. 1, pp. 541–558; 17 in Vol. 2, pp. 411–428) and 3 stated (half-hour) addresses (Vol. 3, pp. 296–324). Stated addresses: –– Daltry, C. T., Self-education by children in mathematics using Gestalt methods i.e. learning-through-insight, Vol. 3, pp. 297–304. –– Kurepa, G., International inquiry of the International Mathematical Instruction Commission (IMIC) on The role of mathematics and mathematician at present time, Vol. 3, pp. 305–317. –– Piene, K., School mathematics for universities and for life, Vol. 3, pp. 318–324. Short lectures: –– Amato, V., Sull’insegnamento matematico nelle scuole secondarie e sui testi scolastici, Vol. 1, p. 543. –– Ascoli, G., Le rôle de la mathématique et du mathématicien dans la vie contemporaine, Vol. 1, pp. 543–544. –– Bundgaard, S., Report on mathematics instruction in Denmark, no manuscript. –– Bunt, L. N. H., Report of the Dutch sub-commission on the teaching of mathematics in the Netherlands to students of 16–21 years of age, Vol. 1, pp. 545–546. –– Cartwright, M.  L., Report of the British sub-commission on mathematical instruction for students between 16 and 21 years of age, Vol. 1, pp. 546–547. –– Châtelet, A., Éducation in France, no manuscript. –– Créspo Pereira, R., The teaching of mathematics, Vol. 1, pp. 547–548.

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–– Darmois, G., Le rôle du mathématicien dans la vie contemporaine, Vol. 1, pp. 548–549. –– Dolinsky, R., Determinanten im Unterricht allgemeinbildender Schulen, Vol. 1, pp. 549–550. –– Fehr, H. F., The administration of mathematics education in the United States of America, Vol. 1, pp. 550–551. –– Frostman, O., Summary of a report on the mathematical instruction in Sweden for students between 16 and 21 years of age, Vol. 1, pp. 552–553. –– Hohenberg, F., Der mathematische Unterricht in Österreich, pp. Vol. 1, 553–554. –– Mac Lane, S., Intermediate mathematical instruction in the United States, Vol. 1, pp. 554–555. –– Palazzo, E., Pedagogia scientifica inquadrata in una teoria matematica, Vol. 1, pp. 555–556. –– Villa, M., L’insegnamento della matematica in Italia per i giovani dai 16 ai 21 anni, Vol. 1, p. 556. –– Athen, H., Die Vektorrechnung im deutschen Schulunterricht, Vol. 2, p. 413. –– Baur, A., Anschaulichkeit und strenge im mathematischen Unterricht der deutschen Oberschule, Vol. 2, pp. 414–415. –– Behnke, H. A. L., Der mathematische Unterricht der 16–21-jährigen Jugend in der Bundesrepublik Deutschland, Vol. 2, p. 415. –– Bridger, M., The mathematical laboratory in the grammar school and the technical college, Vol. 2, pp. 415–416. –– Bunt, L. N. H., Didactical research in the field of mathematics at the Institute of Education of the University of Utrecht, Vol. 2, pp. 416–417. –– Chevrier, J. L. P., Les notions de structure en mathématiques élémentaires, Vol. 2, pp. 417–418. –– Cramer, H., Mathematik an Gymnasien (höheren Schulen), Vol. 2, pp. 418–419. –– Drenckhahn, F., Strukturstufen der Schulmathematik in Anpassung an alterstypische Auffassungsweisen, Vol. 2, pp. 419–420. –– Guggenbuhl, L., Henri Brocard and the geometry of the triangle, Vol. 2, pp. 420–421. –– Hansen, H. K., Geometrie und Wirklichkeit, Vol. 2, pp. 421–422. –– Kamke, E., Werden und Sicherheit mathematischer Erkenntnis, Vol. 2, pp. 422–423. –– Rosskopf, M. F., Trends in the content of high school mathematics in the United States, Vol. 2, pp. 424–425. –– Rüping, H., Philosophische Vertiefung des mathematischen Unterrichts, Vol. 2, p. 425. –– Sengenhorst, P., Arbeitsunterrichtliche Methoden in der Mathematik der Höheren Schulen, Vol. 2, pp. 426–427. –– Wigand, K., Intuitive Methoden im mathematischen Unterricht, Vol. 2, pp. 427–428. –– Wolff, K.  G., Das projektive denken im mathematischen Unterricht, Vol. 2, pp. 428–429.

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There were demonstration of some electronic devices and exhibitions of mathematical books, didactical works, and graphical works by M. C. Escher, and tablecloths showing the Gaussian primes in the complex plan were sold.

8.13 Edinburgh (UK), 14–21 August 1958 Todd, John A. (ed.). 1960. Proceedings of the International Congress of Mathematicians. Cambridge: Cambridge University Press. One volume: LXIV + 573 pages. Summaries of the short communications were printed in the volume of abstracts issued to members during the Congress. In the eighth section (History and education), there were 2 half-hour addresses: 25 short communications (not printed in the proceedings) and special sessions were arranged by ICMI. Half-hour address in the eighth section: –– Kurepa, G., Some principles of mathematical education, pp. 567–572. The list of the short communications is given in the proceedings (pp. XLI-XLII). In the book Abstracts of Short Communications and Scientific Programme, only the abstracts of Bunt, Cassina, Drenckhahn, May, Ness, Piene, and Storer are reported, and the talks of Tucker, Vaughan, Duren, Baley Price, and Allendoerfer are grouped under the label “Reports by speakers nominated by the United States Committee on Mathematical Instruction” (with slightly different titles): –– Allendoerfer, C. B., Teaching mathematics on television, p. XLI. –– Baley Price, G. National Science Foundation Summer Schools, p. XLII. (Institutes for mathematics teachers, p. 173 in the book Abstracts of …). –– Bunt, L. N. H., An investigation into the possibility of teaching probability and statistics in Dutch secondary schools, p. XLI. –– Cassina, U., A study of the present state of teaching the elements of geometry in Italy, p. XLI. –– Drenckhahn, F., Elementargeometrie in Unterricht: vom Gestaltliehen aus und in didaktischen Experimenten, p. XLI. –– Duren, W.  L. Jr., The reform of college mathematics teaching in the United States, p. XLI. –– Geary, A., Diploma in Technology: a new departure in technical education in England, p. p. XLII. –– Ilić-Dajović, M., Sur la réalisation d’un enseignement moderne de la géométrie élémentaire, p. XLII. –– Jaeger, A. Contributions to the American undergraduate instruction for the prospective mathematician, p. XLII. –– May, K. O., Stimulating undergraduate research, p. XLII. –– Ness, W., Beispiel und Gegenbeispiel in der Mathematik, p. XLII. –– Piene, K., Statistics in secondary schools, p. XLII.

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–– Storer, W.  O., Symbolism and the rules of operations in school mathematics, p. XLII. –– Tanner, R. C. H., ‘Equal and unequal’, p. XLII. –– Tricomi, F. G., Quo vadimus?, p. XLII. –– Tucker, A. W., The work of the (American) Commission in school of Mathematics. (The work of the (American) Commission on Mathematics, p.  172  in book Abstracts of …). –– Vaughan, H.  E., The U.I.C.S.M.  Secondary School Programme. (The Illinois experiment in high school mathematics, p. 172 in book Abstracts of …). Special sessions arranged by ICMI (p. XLV): –– I.C.M.I., First Topic, Mathematical instruction up to the age of fifteen years, reported by H. F. Fehr. –– I.C.M.I., Second Topic, The scientific bases of mathematics in secondary education, reported by H. Behnke. –– I.C.M.I., Third Topic, Comparative study of methods of initiation into geometry, reported by H. Freudenthal.

8.14 Stockholm (Sweden), 15–22 August 1962 Stenström, V. (ed.). 1963. Proceedings of the International Congress of Mathematicians. Djursholm, Sweden: Institut Mittag-Leffler. One volume: L  + 595 pages. A booklet containing the program, a booklet contains the names of the participants, and a book entitled Abstracts of Short Communications were issued to members during the Congress. In the program and in the text of talks, the Russian language is also used. There were eight sections; in Section 8, entitled Education, there were no invited talks. The short communications are listed at p. XXXIII of the proceedings and some abstracts of them in the dedicated book: –– Acton, F.  S., The growing importance of mathematical models in medical research, p. 208. –– Bidger, M., The use of desk calculating machines in schools, p. 208. –– Bunt, L. N. H., Statistics in schools; basic notions for testing a hypothesis, p. 208. –– Faragó, L., Difficulties of the analytical operation of thinking in the learning of mathematics, p. XXXIII. –– Fehr, H. F., Instruction in geometry for the secondary school, p. 208–209. –– Harkin, D. C, Mathematics and language, p. XXXIII. –– Herriot, S. T., Impact of the new school mathematics study group curriculum on college-bound students, p. 209. –– Hood, R. T., Inverting the conics, p. 209. –– Ilić-Dajović, M., Sur la réalisation d’un enseignement moderne de géométrie, p. XXXIII.

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309

Maschler, M., Mathematics curriculum for humanistic studies, p. 209. May, K. O., Undergraduate research in the United States, 209. Ness, W., Über die harmonische Reihe, p. 210. Orloff, C., Concrétisation des notions algébriques dans l’enseignement secondaire, p. 210. Perkus, R., What mathematics for the twelfth year?, p. XXXIII Sawyer, W. W., Not modern; not traditional; but mathematics as a whole, p. 210. Shanks, M. E., The role of rigor in school mathematics, p. 210. Smith, E.  M. R., The challenge and opportunities of mathematics teaching in Africa, p. XXXIII. Suppes, P., The learning of mathematical concepts, p. 210. Thwaites, B., An experiment in new syllabuses, p. 211. Wigand, K., Modernisierung des Mathematikunterrichts durch mathematische Geräte und praktische Verfahren, p. 211.

Within Section 8 there were also three special meetings organized by ICMI with reports followed by discussion on the following topics: 1. Which subjects in modern mathematics and which applications of modern mathematics can find a place in programmes of secondary school instruction? Reported by J. G. Kemeny, see (1964. L’Enseignement Mathématique s. 2, 10: 152–176). 2. Connections between arithmetic and algebra in the mathematical instruction of children up to the age of 15. Reported by S.  Straszewicz, see (1964. L’Enseignement Mathématique s. 2, 10: 271–293). 3. Education of the teachers for the various levels of mathematical instruction. Reported by K.  Piene, see (1963. L’Enseignement Mathématique s. 2, 9: 116–127). There were exhibitions of books, textbooks organized by ICMI, and exhibitions of computers.

8.15 Moscow (USSR), 16–26 August 1966 Petrovsky, Ivan G. (ed.). 1968. Proceedings of the International Congress of Mathematicians. Moscow: Printing House “MIR”. Vol. 1: 727 pages; Vol. 3: 198 pages; Vol. of abstracts: 796 pages. There were 15 sections. Section 15 (History and pedagogical questions, pp. 664–680) contains one historical contribution. In Vol. 3, among the half-hour addresses, there is: –– Papy, G., La géométrie dans l’enseignement moderne de la mathématique, pp. 82–89. In the volume of abstracts, the 15 sections are numbered separately; Section 15 (28 pages) contains 73 abstracts, 34 of which concern mathematics education:

310

F. Furinghetti

–– Biggs, E.  E., Discovery mathematics. An experiment with teachers of young children, pp. 3–4. –– Brischle, T., Gruppen- und Körperisomorphismen in Unterricht, p. 4. –– Daltry, C. T., From problems to discovery: through invention to research, p. 5. –– Davis, H. M., Pre-University training for statistics, p. 5. –– Hayman, M. R., Mathematical competitions in England, p. 6. –– Hilton, P. J., The continuing activities of the Cambridge conference on school mathematics, p. 6. –– Laitone, E. V., An ordinary difference equation with useful pedagogical properties, pp. 6–7. –– Levi, H., A new mathematical base for the teaching of calculus, p. 7. –– Lyness, R. C., New mathematics in English secondary schools, p. 7. –– Matthews, G., The Nuffield mathematics teaching project, p. 7. –– Ra, A. N., A new aid in the teaching of arithmetic, p. 8. –– Roman, T., Sur l’activité mathématique supplémentaire, organisée pour les élèves, p. 8. –– Ronveaux, A., Équations aux différences finies pour le cours secondaires, pp. 8–9. –– Stergiou, G., L’importance de la rhétorique dans la méthode didactique des mathématiques, p. 9. –– Turner, N. D., National aspects of the MAA – SA contest in the development of talent, p. 10. –– Van Yzeren, J., The approach to linear algebra, p. 10. –– Andronov, I. K., Three stages in the development of international school mathematics education in 19–20 century, p. 11. –– Arzumanyan, G. S. On principles of the developing scientific methodology of mathematics teaching, p. 11. –– Glagoleva, E. G. and Gel’fand, I. M., Moscow University Mathematical School by correspondence, pp. 13–14. –– Gol’dberg, Y. I. and Cherkasov, R. S., Experience of conducting Diploma works on methodology of mathematics in the Pedagogical Institutes, p. 14. –– Dobrovol’sky, V.  A. Mathematics in higher technical and military institutions, p. 15. –– Zykov, A. A., On the experimental curriculum for an elementary school, p. 15. –– Kureno, D., Mathematical structures in teaching mathematics in secondary school, p. 18. –– Kureno, N. I., Role of vectors in the secondary school, p. 18. –– Margulis, A. Ya., Moskow mathematical society and secondary school, p. 19. –– Maslova, G. G., Neshkov, K. I., and Semushkin, A. D., An experiment on the modernization of the course of secondary school mathematics, pp. 19–20. –– Matskin, M. S., Towards the issue of studying analytic geometry, mathematical analysis and mathematical logic in the secondary school, p. 20. –– Pototsky, M. V., On the issue of mastering mathematics, p. 22. –– Prinits, O. I., The development of mathematics education in Estonian school, p. 23. –– Stolyar, A. A., Logico-mathematical language in teaching mathematics, p. 24.

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–– Shvartsburd, S. I. and Ashkinuze, V. G., On one direction of the mathematical specialization in secondary school, p. 24. –– Shikhanovich, Y. A., On teaching mathematics to the humanity students, p. 25. –– Yaglom, I. M. and Boltyansky, V. G., Moscow university and secondary school students, p. 26. –– Harley, R. M., The joint school project in Ghana, pp. 26–27. The following general reports of ICMI, presented at the International Congress of Mathematicians in Moscow, were published in L’Enseignement Mathématique: –– Pisot, Ch. 1966. Rapport sur l’Enseignement des mathématiques pour les physiciens. L’Enseignement Mathématique s. 2, 12: 201–216. –– Krygowska, A. Z. 1966. Développement de l’activité mathématique des élèves et rôle des problèmes dans ce développement. L’Enseignement Mathématique s. 2, 12: 293–322. The USA Conference Board of Mathematical Sciences published the booklet The Role of Axiomatics and Problem Solving in Mathematics, which was made available for distribution at the ICM in Moscow, 1966.

8.16 Nice (France), 1–10 September 1970 Berger, Marcel, Jean Dieudonné, Jean Leray, Jacques-Luis Lions, Paul Malliavin, and Jean-Pierre Serre (eds.). 1971. Actes du Congrès International des Mathématiciens. Paris: Gauthier-Villars. Vol. 1: XXXIII + 532 pages. Vol. 2: 959 pages. Vol. 3: 371 pages. Each participant received a book published by GauthierVillars (1970) entitled Les 265 communications individuelles. There were six sections. Volume 3 contains the contributions presented in Section F Histoire et enseignement (History and teaching, pp. 331–367). This section is split into F1 Histoire des mathématiques (History of mathematics, pp. 333–334) and F2 Enseignement des mathématiques (Mathematics teaching, pp.  335–367), which contains: –– Krygowska, Z., Problèmes de la formation moderne des professeurs de mathématiques, pp. 347–351). –– Griffiths, H. B., Mathematical insight and mathematical curricula, pp. 335–345. –– Pollak, H. O., On teaching application of mathematics, pp. 353–357. –– Sobolev, S.  L., Quelques aspects de l’enseignement des mathématiques en U.R.S.S., pp. 359–367. In the book of the individual communications (not presented orally), Section F2 contains the abstracts: –– Amir-Moéz, A. R., Residue classes and string figures, p. 287. –– Rimer, D., Une application de la théorie des graphes dans la didactique, p. 288. –– Turner, N. D., Sex differences in mathematics achievement, p. 289.

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8.17 Vancouver (Canada), 21–29 August 1974 James, Ralph D. (ed.). 1975. Proceedings of the International Congress of Mathematicians. Vancouver, B. C.: Canadian Mathematical Congress. Vol. 1: LV  + 552 pages. Vol. 2: VIII +  600 pages. There were 20 sections; Section 20, entitled History and education (Vol. 2, pp. 547–586), contains the address: –– Matthews, G., Science as handmaiden of mathematics, pp. 571–576. In the long list of short communications (Vol. 2, pp. 587–600, without abstracts), the following title refers to didactics: –– Deakin, M., Curriculum priorities in developing countries, p. 589.

8.18 Helsinki (Finland), 15–23 August 1978 Lehto, Olli (ed.). 1980. Proceedings of the International Congress of Mathematicians. Helsinki: Academia Scientiarum Fennica. Vol. 1: 506 pages. Vol. 2: 516 pages. A book with abstracts of short communications and poster sessions was distributed during the congress. In the introductory chapter of the proceedings, we read “Unofficial mathematical activities also included a three-day symposium organized by the International Commission on Mathematical Instruction” (p. 7, Vol. 1). There were 19 sections. Section 19 (Vol. 2, pp. 1005–1020), entitled History and education, contains: –– Banchoff, T.  F., Computer animation and the geometry of surfaces in 3- and 4-Space, pp. 1005–1013, pp. 1005–1013. In the book of abstracts, there are the following contributions concerning mathematics education: –– –– –– –– –– –– –– –– ––

Bajpaj, D. “Mathematical creation”. A preliminary report, p. 213. Owden, L., The heuristics of schoolboy approximation to n!, p. 214. Broman, A., A film on four-dimensional geometry, p. 214. Farkas, M., A dynamic theory of simultaneous learning, pp. 216–217. Chalvron, J. and Curien, N., Un enseignement en mathématiques applicables, une expérience pédagogique, p. 215. Delange, G., Reflections sur la formation des professeurs de mathématiques, p. 215. Eslinger, R., A program of undergraduate mathematical research, p. 216. Flegg, G., Presentation on open university mathematics courses, pp. 217. Glocke, T., Pädagogische Hochschule Erfurt/Mülhausen DDR “Uber die Einbeziehung künftiger Mathematiklehrer in die Forschungen auf mathematischem und pädagogischem Gebiet”, pp. 217–218.

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De Puya, I. and Delta, E., Der Unterricht der Topologie, p. 218. Malik, M. A., Pedagogical aspects of the definition of function, p. 218 (poster). Melter, R. A., A mathematics course for marine scientists, p. 218–219 (poster). Rogers, L. F., International Study Group on the Relations between History and Pedagogy of Mathematics: aims and Activities, p. 219. Rowley, C., Learning mathematics from textbooks, p. 219 (poster). Smith, E.  M.  R., Factors affecting attitudes of High School students towards Mathematics, pp. 219–220. Soares, R., “Qualification” de paramètres – Domain cognitive, p. 220. Turner, N. D., Careers of individuals who ranked high in secondary school mathematics contexts, pp. 220–221. Wheeler, D., Understanding mathematization, pp. 221–222. Wussing, H., Teaching history of mathematics, p. 222. Miles, E.  P. Jr., Computer related modules for mathematics instruction, pp. 224–225.

An ICMI Symposium was organized during this ICM with the cooperation of UNESCO and IDM (Institut für Didaktik der Mathematik) on “What knowledge, experience and understanding of mathematics should a mathematics teacher have?” (1978. ICMI Bulletin 11: 13–15).

8.19 Warsaw (Poland), 16–24 August 1983 Ciesielski, Zbigniew and Olech, Czesław (eds.). 1984. Proceedings of the International Congress of Mathematicians. Warszawa: PWN  - Polish Scientific Publishers; Amsterdam – New York – Oxford: North Holland. Vol. 1: LXII + 848 pages. Vol. 2: X + 882 pages. In the first volume, ICMI is mentioned (p. XIII, Vol. 1): “A Symposium of the International Commission on Mathematical Instruction was held in four afternoons, accompanied by a seminar of the International Group on the Relation between History and Pedagogy of Mathematics affiliated to ICMI.” There were 19 sections. Section 19, entitled History and education (Vol. 2, pp. 1695–1728), contains: –– Freudenthal, H., The implicit philosophy of mathematics history and education, pp. 1695–1709. –– Pogorelov, A. V., O prepodavanii geometrii v shkole [On the teaching of geometry in school, in Russian], pp. 1711–1716. In the book of the abstracts of short communications offered to the Congress members upon registration, there are the following contributions in Section 19: –– Ilić, J. L., Expérience mentale, p. 13. –– Feluk, B. and Zawdowski, W., Teaching mathematics by games, p. 14. –– Frank, B., Zur Funktion und Gestaltung des Geometrieunterrichts an der allgemeinbildenden Schule, p. 15.

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–– Jahnke, T., Eine Bemerkungen zum Analysisunterricht / Some remarks on teaching calculus, p. 16. –– Kantor, J. M., Concrétisation de la mathématique, p. 17. –– Kantowski, M. G., The microcomputer and problem solving, p. 18. –– Lubański, M., Einiger Bemerkungen zur wahrheitsfrage in der Mathematik, p. 19. –– Ray, A. K., Meaning of research, p. 20. –– Sierpinska, A., Obstacles in the learning of limits and continuity, p. 21. –– Slomska, A., On some results of investigations of the communicative function of the language of teaching mathematics, p. 22. –– Turner, N.  D., A follow-up study of winners of the U.S.A. mathematical Olympiad, p. 23. –– Vetulani, Z., “Live computer” – A didactically useful model of computers, p. 24. –– Yaseen, A. A. K., Space of knowledge: a conceptual structure for a development in mathematics education, p. 25. During the ICM (17–22 August), the ICMI Symposium entitled “What should be the goals and the content of general mathematical education” is organized under the chairmanship of A.  Z. Krygowska and H.  G. Steiner (ICMI Bulletin 15, 1984, pp. 5–10).

8.20 Berkeley (USA), 3–11 August 1986 Gleason, Andrew M. (ed.). 1987. Proceedings of the International Congress of Mathematicians. Washington DC: MAA. Vol. 1: CII  + 870  + A2 pages. Vol. 2: III  +  838 pages. A book with abstracts was distributed to the members of the congress. There were 19 sections. In Section 19 (Teaching of mathematics), there were short communications, whose titles and authors are listed at the pages LXXV-­ LXXVI of Vol. 1. Some of them have the abstract in the book of abstracts (their pages are indicated with Arabic numerals in the following list): –– Adda, J., Apprentissage de la résolution de problèmes de didactique des mathématiques, p. LXXV. –– Ale, S.  O., Poor performance in school mathematics  - Causes and remedies, p. LXXVI. –– Bebbe-Njoh, E., Considerations on the dialectic construction of mathematics: Its didactic implications, p. LXXVI. –– Burgues, C., New materials for teaching fractions at the elementary level, p. LXXVI, p. 345. –– Cooney, M. P., A seminar on women and mathematics, p. LXXVI, p. 345. –– Demana, F. D., Establishing fundamental concepts of algebra through numerical problem solving, p. LXXVI, p. 345. –– Durapau, V. J. and Fontova, E. M., An undergraduate colloquium in mathematics and computer science, p. LXXVI, p. 346.

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–– Feldman, L., Introductory algebra: A participatory approach for college students, p. LXXVI, p. 346. –– Fjelstad, P., Encouraging student creativity: One result - trigonometries galore, p. LXXVI, p. 346. –– Fusaro, B.  A., The mathematical competition in modeling (MCM), p. LXXVI, p. 347. –– Galdon, J. M., The systematic use of audiovisual teaching methods in the introduction to calculus, p. LXXVI, p. 347. –– Hammer, J., Teaching and learning in teams, p. 349. –– Hartnett, W.  E., History of modern mathematics (1850-) courses, p. LXXVI, p. 349. –– Kajikawa, Y. (Presented by Hitosumatsu Shin), Teaching by the level of achievement, p. LXXVI, p. 349. –– Katsifli, D. and Fyfe, D. J., Computer aided learning for numerical analysis problem solving (GALNAPS), p. LXXVI, p. 350. –– Kishore, M., Division in decimal and binary number systems, p. LXXVI, p. 350. –– Linfoot, J. J., Some considerations on the difficulty of questions in arithmetic, p. LXXVI, p. 350. –– Loase, J., An alternative to computer assisted mathematical instruction  Individualized learning at Westchester Community College, p. LXXVI, p. 350–351. –– Michael, M., Using a microcomputer to simulate certain 2-manifolds, p. LXXVI, p. 351. –– Miles, E. P., Printing applications for nested self-dual model subsets of the color continuum, p. LXXVI, p. 351. –– Pustilnik, S.  W., Action research on reasoning and practice in mathematics achievement, p. LXXVI, p. 351. –– Rakover, B. D., The use of algorithmic procedures in mathematical instruction, p. LXXVI, pp. 352. –– Reis, M.  R., On the integrated teaching of the different geometries, p. LXXVI, p. 352. –– Riley, P. E., Teaching styles and motivation in upper division and graduate mathematical education, p. LXXVI, p. 352, p. 352. –– Sanatani, S., Can mathematics be taught?, p. LXXVI, p. 352–353. –– Thompson, T. M. and Wiggins, K. L., Limits and continuity, p. LXXVI, p. 354. –– Towers, D.  A., The role of mathematics education courses in undergraduate mathematics teaching, p. LXXVI, p. 354. –– Waits, B.  K., Using microcomputers to enhance the learning of pre-calculus mathematics, p. LXXVI, p. 354. –– Wilson, J. H., Mathematical instruction-Focus on the development of reasoning skills, p. LXXVI, p. 354. –– Yanosko, B. J., How long does it take to forget mathematics?, p. LXXVI, p. 354. Invited addresses at the section meeting on teaching mathematics (Vol. 2):

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–– Grabiner, J. V., The centrality of mathematics in the history of Western thought, pp. 1668–1681. –– Kahane, J.-P., Enseignement mathématique, ordinateurs et calculettes, pp. 1682–1696. –– Semadeni, Z., Verbal problems in arithmetic teaching, pp. 1697–1706. This ICM provided an opportunity for sessions sponsored by ICMI, the International Study Group on Mathematics Education in the Far East, the Commission on Development and Exchange of the IMU, and the International Study Group on the Relations Between History and Pedagogy of Mathematics (IMU Bulletin 26, 1986, pp. 7–8).

8.21 Kyoto (Japan), 21–29 August 1990 Satake, Ichiro (ed.). 1991. Proceedings of the International Congress of Mathematicians. Tokyo - Berlin - Heidelberg - etc.: The Mathematical society of Japan, Springer-Verlag. Vol. 1: LXXXVII + 768 pages. Vol. 2: XIII + 916 pages. There were 18 sections. In Section 18 (invited 45-minute addresses, Vol. 2, History, teaching and the nature of mathematics, pp. 1639–1681), there is: –– Murakami, H., Teaching mathematics to students not majoring in mathematics Present situation and future prospects -, pp. 1673–1681. Abstract of the ICMI Lectures and summaries of the 22 short communications in Section 18 and of the 2 short communications in the postdeadline section were printed – if received in time – in the volume of abstracts issued to members during the Congress. The ICMI Lectures are –– De Guzmán, M., Games and puzzles and their role in the popularization of mathematics, p. 257. –– Fujita, H., Mathematical literacy and Japanese new mathematics curricula, pp. 258–259. –– Kim, Y., National mentality and mathematics education, pp. 260–261. –– Van Lint, J. H., Structuring discrete mathematics, p. 262. –– Hodgson, B. R., The ICMI Studies: Some personal views, p. 263.

8.22 Zurich (Switzerland), 3–11 August 1994 Chatterji, Srishti D. (ed.). 1995. Proceedings of the International Congress of Mathematicians. Basel  - Boston  - Berlin: Springer Verlag. Vol. 1: LXXI +  717 pages. Vol. 2: XIII + 887 pages.

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There were 1-hour addresses (plenary) and invited addresses (45-minute) presented in Sections 1–19. In addition, there were lectures organized by ICMI, which were scheduled along with the section lectures. The texts of these lectures and the abstracts of short communications were printed in two volumes given to the participants at the time of their registration at the Congress. In Section 18 (Teaching and popularization of mathematics, Vol. 2, pp. 1546–1567), there were invited lectures (45-minute) at the meetings of the sections: –– Hughes-Hallett, D., Changes in the teaching of undergraduate mathematics: The role of technology, pp. 1546–1550. –– Schneider, J., Issues for the popularization of mathematics, pp. 1551–1558. –– Stillwell, J., Number theory as a core mathematical discipline, pp. 1559–1567. The short communications listed in the volume of Abstracts are: –– Behboodian, J., Inner Pascal’s matrix, p. 263. (Abstract missing). –– Botana, F., Learning propositional logic through a computer game, p. 263. –– Burgin, M.  S., Problems of mathematical analysis teaching: from classical to neoclassical analysis, p. 263. –– Doraisany, L. and Munisany Doraisany, S., Do your students understand probability?, p. 264. –– Holme, A., A system for computer aided instruction in linear algebra, p. 264. –– Hughes, A., The Platonic solids, p. 264. –– Ito, Y., On the changes and the problems of mathematical education in Japan, p. 265. –– Kutzler, B., DERIVE – The future of teaching mathematics, p. 265. –– Lacko, M., On teaching mathematics at elementary schools through correspondent seminars, p. 265. –– Murakami, H., Matsumoto, S., and Yoshida, K., A new way of teaching mathematics at a distance, p. 266. –– Pourkazemi, M. H., A study of scientific situation of those licensed in Mathematics from different universities of Iran, p. 266. –– Shahvarani-Semani, A., The place of logic in Secondary School mathematics in Iran, p. 266. –– Siu, M.  K., What do we learn from the history of mathematics education in ancient China, p. 267. –– Andžāns, A. and Taimina, D., On teaching algorithmic methods to middle and high school students, p. 267. –– Yamashita, H., Takizawa, T., Nakashima, K., and Nishimura, K., Approximate analysis of Fuzzy Graph and its application, p. 267. In the volume Abstract of Plenary Addresses, Section Lectures, ICMI Lectures, ICHM Lectures, the following ICMI Lectures organized by ICMI are listed: –– Kilpatrick, J. and Sierpinska, A., What is research in mathematics education?: Preliminary outcomes of an ICMI Study, pp. 178–179. –– Mauduit, C., Challenging mathematical activities for young people, p. 180.

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–– Pollak, H., The role of applications in mathematics education, p. 181. –– Tall, D., Understanding the processes of advanced mathematical thinking, pp. 182–183.

8.23 Berlin (Germany), 18–27 August 1998 Fischer, Gerd and Ulf Rehmann (eds.). 1998. Proceedings of the International Congress of Mathematicians. Documenta Mathematica, Journal der Deutschen Mathematiker-Vereinigung, Extra volume ICM 1998. Vol. 1: 662 pages. Vol. 2: 881 pages. Vol. 3: 825 pages. Volumes containing abstracts of short communications and poster sessions were given at the moment of registration. They are now available at the address (Retrieved 9 June 2021) https://www.math.uni-­bielefeld.de/ICM98/. There were 18 sections. Section 18 (Teaching and popularization of mathematics, Vol. 3, pp. 717–786) contains 45-minute lectures and the panel at the section meetings: –– Andrew, G. E., Mathematics education: Reform or renewal?, pp. 719–721. –– Artigue, M., De la compréhension des processus d’apprentissage à la conception de processus d’enseignement, pp. 723–733. –– Bartolini Bussi, M. G., Drawing Instruments: Theories and practices from history to didactics, pp. 735–746. –– Panel: De Guzmán, M., Hodgson, B. R., Robert, A., and Villani, V., Difficulties in the passage from secondary to tertiary education, pp. 747–762. –– Lewis, D. J., Mathematics instruction in the twenty-first century, pp. 763–766. –– Niss, M., Aspects of the nature and state of research in mathematics education, pp. 767–776. –– Smith, D. A., Renewal in Collegiate mathematics education, pp. 777–786. –– Steiner, H.-G., Response to Michèle Artigue’s lecture titled “From the understanding of learning processes to the design of teaching processes” (not in the proceedings). –– Stigler, J., Classroom mathematics instruction in three cultures: An introduction to the TIMSS Video Study (not in the proceedings). The abstracts of short communications and poster sessions, which do not constitute a part of the published proceedings of ICM 1998, are: –– Alves, G., Modernization of basic disciplines in engineering courses at IME. –– Arnold, H.-J., Geometrische Relationen-Algebra und Piaget’sche Schematheorie. –– Asiedu-Addo, S.  K., Griffith, L.  K., Mullins, C.  W., Wiredu, E., and Bonsu, O. M., Perceptions and attitudes related to mathematics learning: A comparison between UCC Ghana and UCA Arkansas. –– Askin, S., Additional computer geometry courses for advanced students. –– Baidyk, T. N., Evaluation of mathematician achievements.

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–– Behrends, E., Nietzsch, J., and Stroth, G., Study of mathematics by high school students at the university. –– Bilotskiyi, N. N. and Khilchenko, L. Y., An algorithmic approach to the notion of an elementary function. –– Brecht, G., Do mathematical standards concern mathematics? –– Chiu, S.-Y., Pedagogical Mathematics Knowledge, PMK and P(PMK)K. –– Crespo Crespo, C., Pizzo, A., Ponteville, C., and Villella, J., Thinking about geometry teaching. –– Dimitrić, R., Using elements of mathematics in teaching (Poster). –– Eyles, J. W., Discovering R. L. Moore’s calculus course. –– Fjelstad, P., A picture for real arithmetic. –– Fritsch, R., Remarks on quadrangles. –– Fritzlar, T., Über Möglichkeiten der Sensibilisierung für die Komplexität von Mathematikunterricht. –– Graf, K.-D., Distance learning fördert die Kreativität im Mathematikunterricht. –– Kortesi, P., Mathematical Congress - Challenge for future Mathematicians. –– Kuragina, E., Methodical support for a programming-methodical complex of study methods of optimization. –– Leviatan, T., Misconceptions, fallacies, pitfalls and paradoxes in the teaching of Probability. –– Mori, M., Beginners confusion and terminology in mathematics. –– Morley, M., Bennett, R. and Quardt, D., Three response types for broadening the conception of mathematical problem solving in computer-graded tests. –– Moshnikova, J. M. and Sobolev, S. I., Correspondence School of Mathematics in Karelia. –– Ozaki, Y. and Sano, K., An approach in teaching by means of the personal computer software “Mathematica”. –– Paditz, L., Comparison of the statistic calculators TI-83, EL-9600 and CFX-­9850G PLUS. –– Pourkazemi, M.  H., Misconceptions, fallacies, pitfalls and paradoxes in the teaching of Probability. –– Rodríguez Del Río, R., Pacheco Esteban J. P., Rodríguez Trueba, M. I., Teaching Calculus with the computer. An Experience in a Science’s Faculty. –– Sahab, S., Mathematics basic skills for grades 1 to 3. –– Shahvarani-Semnani, A., The place of geometry in mathematics curriculum in secondary schools of Iran. –– Shahvarani-Semnani, A., The place of geometry in mathematics curriculum in Iran. –– Xu, Z.-L., The popularization of mathematical methods in China. –– Zimmermann, B., Al Sijzi on problem solving and creativity in elementary geometry. In Section 20 (Special activities of the ICM 1988), there were computer demonstrations of mathematical software.

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In Urania, an institution with a long tradition in the popularization of science, there were exhibitions and various activities for a nonspecialized audience and for high-school students.

8.24 Beijing (China), 20–28 August 2002 LI, Tatsien (LI Daqian) (ed.). 2002. Proceedings of the International Congress of Mathematicians. Beijing: Higher Education Press. Vol. 1: IX + 657 pages. Vol. 2: IV  + 823 pages. Vol. 3: IV + 960 pages. In Volume 1, among the 45-minute talks, there was the invited panel: –– Kaiser, G., Leung, F. K. S., Romberg, T., and Yaschenko, I., International comparisons in mathematics education: an overview, pp. 631–646. There were 19 sections. Section 18 (Mathematics education and popularization of mathematics, Vol. 3, pp. 873–920) contains: –– Loewenberg Ball, D., Hoyles, C., Jahnke, H. N., and Movshovitz-Hadar, N., The teaching of proof, pp. 907–920 (panel). –– Dorier, J. L., Teaching linear algebra at university, pp. 875–884. –– Hansen, V. L., Popularizing mathematics: From eight to infinity, pp. 885–895. –– Xiao, S., Reforms of the university mathematics education for non-mathematical specialties, pp. 897–906. In the book Abstracts of Short Communications and Poster Sessions, there are the abstracts of the following 25 contributions: –– Ai, S., The cultivation of creative thought and mathematics teaching, p. 391. –– Ashna, A., Standards in mathematics education for international comparisons, p. 391. –– Cassy, B., Mathematics teachers’ attitudes in Mozambican secondary classroom, p. 391. –– Constantinescu, G., Strategic mathematical education technology planning, p. 392. –– Dev Sarma, B.  K., Maths learning strategies among the FGL-STs of NE India, p. 392. –– Diansuy, M.  A. A., Reflective thinking through Journal writing: A strategy in enhancing learning in plane analytic geometry, p. 392. –– Emmer, M., Mathematics and culture: The role of education, p. 393. –– Gusev, V.  A., Modern state and changes in mathematics teacher training in Russian Federation, p. 393. –– Holte, J.  M., Properties of O-regularly varying sequences: Elementary proofs, pp. 393–394. –– Hu, C.-L., Higher mathematical education in Yi-Han languages, p. 394. –– Hughes, A. C., The poetry of mathematics, p. 394.

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–– Ito, Y., On lecture of mathematics at university, p. 394. –– Jin, G., Calculus Bible, the road learning to the reform of calculus teaching, pp. 394–395. –– Kaino, L. M., Mathematics curricular developments and change, challenges in the 21st century, p. 305. –– Katsap, A., Mathematics teacher education in a humanistic mathematics light: Process and products, pp. 395–396. –– Koenig, G., Electronic access to research literature in mathematics education, p. 396. –– Li, M. Types of personnel and types of textbook, p. 396. –– Michtchenko, T. M., Basic geometric configurations as a principle of methodical maintenance of the school geometry course, pp. 396–397. –– Pourkazemi, M. H., The necessity of using multiple-choice items in the Trans-­ Iranian admission test of mathematics, p. 397. –– Safuanov, I. S., Design of the system of genetic teaching of algebra at universities, p. 397. –– Shahvarani-Semnani, A., The place of trigonometry in secondary schools in Iran, p. 397–398. –– Tong, Z., To teach is to learn – East meets West, p. 398. –– Vasquez-Martinez, C. R., A methodology in the teaching process of calculus and its motivation, p. 398. –– Yang, D., The analysis of the Ravan reasoning test table, pp. 398–399. –– Ziegenbalg, J., Older and newer uses of computer technology in German mathematics education, p. 399.

8.25 Madrid (Spain), 22–30 August 2006 Sanz-Solé, Marta, Javier Soria, Juan Luis Varona, and Joan Verdera (eds.). 2007. Proceedings of the International Congress of Mathematicians. Zurich: European Mathematical Society. Vol. 1: 834 pages. Vol. 2: 1762 pages. Vol. 3: 1770 pages. There were 20 sections. Section 19 (Mathematics education and popularization of mathematics, Vol. 3, pp. 1583–1696), contains: –– Kenderov, P. S., Competitions and mathematics education, pp. 1583–1598. –– Siegel, A., Understanding and misunderstanding the Third International Mathematics and Science Study: what is at stake and why K-12 education studies matter, pp. 1599–1630. –– Stewart, I., Mathematics, the media, and the public, pp. 1631–1644. –– Panel A: Artigue, M. (moderator), de Shalit, E., and Ralston, A.: Controversial issues in K-12 mathematical education, pp. 1645–1661. –– Panel B: Lee, P. Y. (moderator), de Lange, J., and Schmidt, W., What are PISA and TIMSS? What do they tell us?, pp. 1663–1672.

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–– Panel C: Nebres, B. (moderator), Cheng, S.-Y., Osterwalder, K., and Wu, H.-H., The role of mathematicians in K-12 mathematics education, pp. 1673–1696. Special activity: –– Xambó, S. (moderator), Bass, H., Bolaños, G., Seiler, R., and Seppälä, M., e-learning mathematics, Vol. 3, pp. 1743–1768. Panel discussion organized by the European Mathematical Society: –– Bourguignon, J.  P. (moderator), Engquist, B., du Sautoy, M., Sossinsky, A., Tisseyre F., and Tondeur, F., Should mathematicians care about communicating to broad audiences? Theory and practice”, Vol. 1, pp. 737–756. In the book Abstracts. Poster. Short Communications. Mathematical software and other activities, there are the abstracts of the following contributions: Posters: –– Badillo, E., Deulofeu, J., and Figueiras, L., DIDIMA: dialogues on discrete mathematics, p. 179. –– Cañas, J. J. and Galo, J. R., Analysis of a constructivist experiment with ICT in mathematic education, pp. 179–180. –– Casaravilla, A., Castejón, A., and Gilsanz, M. CIMM: interactive course of mathematics with mathematics, p. 180. –– Castrillón López, M., ESTALMAT: A poster for selecting and nurturing talented children aged 13 to 15, p. 181. –– Carlavilla, J.  L. and Fernández, M., Is there mathematics in the Quixote?, pp. 181–182. –– Cuello Nebot, E., The self-knowledge applied to the study, p. 182. –– García Duarte Jr., G., Exploring dynamic movements in geometry with Cabri géomètre, p. 183. –– García Pineda, M. and Núñez del Prado, J. A., Mathematics in business administration, p. 183. –– González Manteiga, M. T., Lahoz-Beltra, R., Martínez Calvo, M. C., and Pérez de Vargas Luque, A., Application of the new information technologies to the decrease of the failure in the learning of mathematics in biological sciences undergraduate students, p. 184. –– Guerrero-García, P. and Santos-Palomo, Á., Motivational numerical examples for electrical/electronics engineers, pp. 184–185. –– Guirado-Granados, J. F. and Ramírez-Uclés, R., Mathematician waiting for the wedding cake and the cava at the reception, p. 185. –– Iglesias, M. T., Ríos, M., and Vidal, C., Some features about the convergence process of a genetic algorithm, pp. 185–186. –– Kapelou, E. and Kokalidis, S., The relation between the evaluation of primary and secondary school students in mathematics which the student’s choice of type of school (day school or evening school) and the role of students’ gender, pp. 186–187.

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–– Masferrer, C. and von Wutheanau, S., Reinventing the wheel: non-circular wheels, p. 187. –– Reyes, J. A., The influence of teaching the concept of function using graphing calculators. A Venezuelan experience, pp. 187–188. –– Roczen, M., Linear algebra on demand, p. 188–189. –– Volk, W., Monuments on mathematicians, p. 189. Short communications: –– Abramovitz, B. Berezina, M., Berman, A., and Shvartsman, L., How to make the best of multiple choice texts, p. 576. –– Alcolea-Banegas, J. and Santonja-Gómez, F. J., A Popperian characterization of mathematics: a reference for the teacher, pp. 576–577. –– Badillo, E., Deulofeu, J., and Figueiras, L., Mathematics meets the public: prizes, exhibitions and conferences in Catalonia and the rest of Spain, p. 577. –– Blasco, F., Maths, magic and media, pp. 577–578. –– Cassy, B., Is mathematics recognized as relevant in economics and management courses?, p. 578. –– Córcoles, C., Huertas, M. A., Juan A. A., Serrat, C., Steegmann, C., Math on-line education: state of the art, experiences and challenges, pp. 578–579. –– Eaton, P. and Kidd, S., What students think and why it matters – a survey of student teachers’ views of mathematics, p. 579. –– Epperson, J.  A. M., Towards improving secondary teachers’ understanding of mathematical problem solving via rich task design, p. 580. –– Ferrer, A., Hernando, B., Juan A.  A., Serrat, C., and Torrent, J.  A., Teaching applied statistics at UPC; integrating lectures, statistical software and e-learning, p. 580–581. –– Flores, R.  M., Lebrija Trejos, A., and Trejos Alvarado, M., Influences of the teachers’ beliefs and strategies in the teaching-learning process of Math: A Constructivist Solution Proposal, p. 581. –– Freiman, V., Understanding children talking mathematics: analysis of communication in the virtual problem solving environment CAMI, p. 582. –– García Valldecabres, M., Experience of didactic innovation in algebra. A didactic unit on the affine function, pp. 582–583. –– Ghenciu, P. I., Lesson study, p. 583. –– Gil Clemente, E. Teachers beliefs about proof in the Spanish education system, pp. 583–584. –– Hernández, S., Perea, C., Polo-Blanco, I., and Ramírez, C., A new generation of cubic surface models: retrieving the Clebsch, pp. 584–585. –– Holte, J. M., Discrete multifractals, p. 585. –– Jaramillo Quiceno, D., Pedagogical ideary of mathematics’ teachers: a historical-­ cultural perspective, pp. 585–586. –– Katsap, A., Take cognizance of comprehensive mathematics: ethnomathematics teacher education, p. 586. –– Klein, M. Various degrees of the number’s distinction, pp. 586–587.

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–– Pareja-Heredia, D., Urrea-Henao, J., Martin Gadner, Alice and the law of gravity, pp. 587–588. –– Nunes, G., Education issue on the dawning of randomness, p. 588. –– Tellechea Armenta, E., From the Riemann integral to the fundamental theorem of calculus: an approach with apple Descartes, pp. 588–589. –– Torres Hernández, R., Zúñiga Becerra, B., Some curiosities about Cabri: analytic geometry as a tool for teacher training, p. 589. Other activities: –– Ibáñez, R. and Ros, R. M., DivulgaMAT. Popularization of mathematics, p. 620. –– Ibáñez, R. and Ros, R.  M., Mathematics in action, an outreach strategy, pp. 621–622. –– Hernández, E., Nurturing the talent gifted children ages 13 to 15: A project of Miguel de Guzmán, p.  622. There was the presentation of mathematical software, which was mainly addressed to research, except the following: –– Eixarch Ferrer, R., Teaching and learning calculus with Wiris technology in Moodle environment, p. 150 of the program book.

Sources Furinghetti, Fulvia. 2007. Mathematics education and ICMI in the proceedings of the International Congresses of Mathematicians. In Revista Brasileira de História da Matemática Especial n. 1  - Festschrift Ubiratan D’Ambrosio  - (dezembro/2007) Publicação Oficial da Sociedade Brasileira de História da Matemática, 97–115. IMU Archive. Retrieved 9 June 2021. https://www.mathunion.org/outreach/imu-­archive. IMU Website. Retrieved 9 June 2021. https://www.mathunion.org/icm/proceedings. Publications such as books of abstracts and programs distributed during the congress, which are not part of the proceedings and have not ISBN.

Chapter 9

The Process of Internationalization of ICMI through Maps (1908–2008) Livia Giacardi

In this chapter, I propose to illustrate, using the maps created with Palladio software, the process of internationalization of ICMI over the first century of its life, taking into consideration the geographical origin of the ICMI national delegates of the member countries until 2002 and subsequently the member countries of the Commission until 2008. Actually, the Terms of Reference of 2002 introduced the following modification concerning the composition of the Commission: as for IMU, members of ICMI are countries, not individuals.1 The names of some countries have changed during the twentieth century; for this reason, I have kept the names as they appeared in L’Enseignement Mathématique (hereafter EM) or in ICMI Bulletin in the various periods considered. I have also included the map related to the participation in the First International Congress on Mathematical Education (ICME-1, Lyon 1969) because, as G. Howson affirms, it “was a landmark in the history of ICMI. Over six hundred mathematics educators from forty-two countries met in an unprecedented fashion” (ICMI Bulletin 16, 1984, p. 6).

 See Chap. 7 by Livia Giacardi in this volume.

1

L. Giacardi (*) University of Turin, Turin, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_9

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1908 The Commission consisted of delegates from countries that had participated in at least two International Congresses of Mathematicians with an average of at least two members: Austria, Belgium, Denmark, Spain, the USA, France, Germany, Greece, Holland, Hungary, the British Isles, Italy, Norway, Portugal, Romania, Russia, Sweden, and Switzerland. Besides, a number of “associated countries,” whose delegates were permitted to follow the activities, but were not entitled to vote, joined the Commission: Argentina, Australia, Brazil, Bulgaria, Canada, Chile, China, Cape Colony, Egypt, the Indian Raj, Japan, Mexico, Peru, Serbia, and Turkey (Les délégations, EM 10, 1908: 447–448). The delegates of 16 of the 18 countries to be represented on the Commission were appointed at a meeting in Karlsruhe (5–6 April 1909) (Constitution de la Commission, EM 11, 1909: 193–204). No delegate of the “associated countries” was appointed on this occasion. At the time of its creation, the composition of ICMI was purely Eurocentric, except for David E. Smith from the USA.

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1914 At the time of the ICMI Congress in Paris (1–4 April 1914), the countries of origin of the ICMI national delegates were the following: Australia, Austria, Belgium, Brazil, Bulgaria, Cape Colony, Denmark, Egypt, Spain, France, Germany, Greece, Holland, Hungary, the British Isles, Italy, Japan, Mexico, Norway, Portugal, Romania, Russia, Serbia, Sweden, Switzerland, and the USA (Liste des membres de la Commission au 1er Avril 1914, EM 1914, 16: 166). The membership of the first delegates from countries outside Europe, besides the USA already present in 1908, is shown in the map below.

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1932 The post-World War I period was very difficult for international scientific cooperation, which was compromised by the aftermath of the war. After its dissolution in 1920 (Dissolution de la Commission, EM 21, 1920–1921: 317–318), ICMI was reconstituted in 1928  in Bologna, during the International Congress of Mathematicians (ICM). However, this rebirth was ephemeral. The outbreak of World War II led to a second forced setback. At the time of the 1932 ICM held in Zurich from 5 to 12 September, the countries of origin of the ICMI national delegates were the following: Argentine, Australia, Austria, Belgium, Canada, Cape Colony, Czechoslovakia, Denmark, England, France, Germany, Hungary, Italy, Norway, Poland, Portugal, Romania, Sweden, Switzerland, the USA, and Yugoslavia (EM 31, 1932: 263). ICMI lost the national delegates from Brazil, Bulgaria, Egypt, Spain, Greece, Holland, Japan, Mexico, and Russia, but gained the membership of other countries.

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1958 In 1952, during the First General Assembly in Rome (6–8 March 1952) of the reconstituted International Mathematical Union (IMU), ICMI became a permanent subcommission of IMU. India, Japan, and Israel were among the adhering countries, but otherwise, the situation was quite similar to that of 1932,  the following countries being represented: Argentina, Australia, Austria, Belgium, Canada, Denmark, England, Finland, France, Germany, Greece, Hungary, India, Israel, Italy, Japan, Luxembourg, the Netherlands, Poland, Portugal, Sweden and Switzerland, the USA, and Yugoslavia (National Delegates of the Subcommissions, EM s.2, 4, 1958: 75–76). Stimulated by the new needs of society, interest in innovation in the field of mathematics education was starting to make its way. In addition, collaboration with UNESCO (United Nations Educational, Scientific and Cultural Organization)  – whose aim was to promote world peace and security through international cooperation in education, sciences, and culture  – further contributed to encouraging the process of internationalization of ICMI.

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1969 The First International Congress on Mathematical Education took place in Lyon (France) from 24 to 30 August 1969. The Congress was attended by 655 active participants from 42 countries (Internationale Mathematische Nachrichten 97, 1971: 4–5, ICMI Bulletin 5, 1975: 20–24).2

In this map, the size of the circles is proportional to the number of participants from each country.

 See Chap. 10 by Fulvia Furinghetti in this volume.

2

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1972 In October 1972 the ICMI Bulletin, which periodically would give information on the delegates from the various countries, started to be published. The first issue featured the following countries: Argentina, Australia, Austria, Belgium, Brazil, Bulgaria, Canada, Czechoslovakia, Denmark, Finland, France, Germany BDR,3 Germany DDR,4 Greece, Holland, Hungary, India, Ireland, Israel, Italy, Japan, Luxembourg, Malawi, Norway, Pakistan, Poland, Portugal, Romania, Senegal, Spain, Swaziland, Sweden, Switzerland, Tunisia, the UK, the USA, USSR, and Yugoslavia (Addresses of representatives, ICMI Bulletin 1, 1972: 5–7). The number of delegates from non-European countries, especially Asian ones, was increasing.

 BDR stands for Bundesrepublik Deutschland, that is, Federal Republic of Germany.  DDR stands for Deutsche Demokratische Republik, that is, German Democratic Republic.

3 4

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1984 Since the 1980s ICMI has continued to extend its action mainly in Africa and in Asia. In 1984 Mozambique and Portugal did not indicate their representatives, so the ICMI national delegates came from the following countries: Argentina, Australia, Austria, Bangladesh, Belgium, Botswana, Brazil, Bulgaria, Cameroon, Canada, China-Taiwan, Costa Rica, Czechoslovakia, Denmark, Egypt, Federal Republic of Germany, Finland, France, German Democratic Republic, Greece, Hungary, India, Iran, Ireland, Israel, Italy, Japan, Luxembourg, Malawi, Malaysia, the Netherlands, New Zealand, Nigeria, Norway, Pakistan, the Philippines, Poland, Romania, Senegal, Singapore, South Africa, South Korea, Spain, Swaziland, Sweden, Switzerland, Tunisia, the UK, the USA, USSR, Yugoslavia, and Zambia (National representatives, ICMI Bulletin 17, 1984: 11–14).

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1992 In this period, some countries of Central America and Central and Southern Africa joined ICMI. Greece and Pakistan did not indicate their representatives, so the ICMI national delegates came from the following countries: Argentina, Australia, Austria, Bangladesh, Belgium, Botswana, Brazil, Bulgaria, Cameroon, Canada, China, Costa Rica, Cuba, Czechoslovakia, Denmark, Egypt, Finland, France, Germany, Ghana, Hungary, Iceland, India, Iran, Ireland, Israel, Italy, Ivory Coast, Japan, Kuwait, Luxembourg, Malawi, Malaysia, Mexico, Mozambique, the Netherlands, New Zealand, Nigeria, Norway, the Philippines, Poland, Portugal, Romania, Senegal, Singapore, South Africa, South Korea, Spain, Swaziland, Sweden, Switzerland, Tunisia, the UK, the USA, USSR (ex), Yugoslavia, and Zambia (National Representatives, ICMI Bulletin 32, 1992: 34–39).

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2008 The Terms of Reference of 2002 stated that, as for IMU, members of ICMI are countries, not individuals. The ICMI Bulletin (62, 2008, p.  3) says: “All countries members of the International Mathematical Union, the mother organization of ICMI, are de facto members of ICMI. But it is possible for a country not a member of IMU to become a member of ICMI … There are currently 84 member countries of ICMI, 68 of which are also members of IMU and 2 are associate members (am) of IMU. In the following list, (*) indicates one of 14 members of ICMI that are not members of IMU.” From the following list and map, it appears that over 100 years, the membership of ICMI changed from Eurocentric in 1908 to truly international in 2008, thus expanding its network of contacts around the world, with the following member countries: Argentina, Armenia, Australia, Austria, Bangladesh*, Belgium, Bosnia and Herzegovina, Botswana*, Brazil, Brunei Darussalam*, Bulgaria, Cameroon, Canada, Chile, China, Colombia, Costa Rica*, Croatia, Cuba, Czech Republic, Denmark, Ecuador (am), Egypt, Estonia, Finland, France, Georgia, Germany, Ghana*, Greece, Hong Kong, Hungary, Iceland, India, Indonesia, Iran, Ireland, Israel, Italy, Ivory Coast, Japan, Kazakhstan, Republic of Korea, Kuwait*, Kyrgyzstan (am), Latvia, Lithuania, Luxembourg*, Malawi*, Malaysia*, Mexico, Mozambique*, the Netherlands, New Zealand, Nigeria, Norway, Pakistan, Peru, the Philippines, Poland, Portugal, Romania, Russia, Saudi Arabia, Senegal*, Serbia, Singapore, Slovakia, Slovenia, South Africa, Spain, Swaziland*, Sweden, Switzerland, Thailand*, Tunisia, Turkey, Ukraine, the UK, the USA, Uruguay, Venezuela, Vietnam, and Zambia* (The members of ICMI, ICMI Bulletin 62, 2008: 3).

Chapter 10

The Beginning of an Adventure: Glances at the First ICME (Lyon 1969) Fulvia Furinghetti

10.1 The Participants in ICME-1 (Lyon, 24–30 August 1969) As mentioned in Part I, the launch of the ICME congresses is a milestone in the evolution of ICMI. These congresses have changed the way of working on problems related to mathematics education, broadened the topics to be discussed, and, above all, involved a very large number of people compared to the small circle of participants in previous conferences dedicated to mathematics education. The proceedings of the first ICMEs do not contain the list of participants and their countries of origin, but Jerry Becker’s reports on ICME-1-2-3 give some interesting data; see (Becker 1970, 1975). At the first ICME of 1969, there were 655 participants from 42 countries. The data of participants and their country are given in Table 10.1.

F. Furinghetti (*) University of Genoa, Genoa, Italy e-mail: [email protected] © Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_10

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Table 10.1  Participants in ICME-1 and their countries ♯ of participants 38 6 3 9 28 425

% of participants 5.8% 0.9% 0.5% 1.4% 4.2% 64.9%

1 21 116 3 5

0.1% 3.2% 17.7% 0.5% 0.8%

Africa Asia (including Japan) Australia and New Zealand Central and South America (including Mexico) Eastern Europe and USSR Europe (Western including Great Britain and Ireland) Subcontinent (India) Middle East (including Turkey) North America Scandinavia “Unaccounted for participants”

To these data, we add the transcript of the list of participants and their country of origin mailed to working members of the congress. In principle, first and last names of the participants are given as they appear in the list. Despite some inaccuracies and omissions, the list represents a suggestive snapshot of the situation at the time of the establishment of ICME and the impact on the world of mathematics education of this event. Many new names which are present in the list will later become protagonists in the milieu of mathematics education; see Fig. 10.1. Abdeljaouad, Jouanidi Ackermans, Stan Aguado, Geneviève Aidi, Sadok Aizpun Lopez, Alberto Akkerhuis, Gérard Algra, Eit Almeido Costo, Antonio Altschul, Pierre Anka, Kjaja Apollodorus, Pierre Allard, J. C. and Françoise Audibert, Gérard Badesco, Radu Bajpai, Avinash Chandra Balle, Erling Barbey, Guy De Bary, Christiane Batier, Chantal, and Pierre Bazin, Simone Beauchant, Jeanini Begle, Edward

Tunisia Netherlands France Tunisia Spain Germany Netherlands Portugal France Yugoslavia Tunisia France France Romania UK Denmark France France France France France USA

Abrams, Francine Adda, Josette Ahmad, Salah Aiton, Éric Akel, Gilbert Akizuki, Yasuo Allen, Deryck Alonso Juaneda, Juan Anderson, Johnston Ansarov, Robert Arellano, Lilia Teresa Armitage, J. V. Augier, Jean Luis Bagby, James Ball, Evelyne Bandet, Émile Bardou, Geneviève Batdedat, André Baugh, Richard Beardsley, Leah Becques, Christian Behnke, Heinrich

USA France Syria UK Lebanon Japan UK Spain UK Yugoslavia Belgium UK Algeria USA UK France France France France USA France Germany

10  First ICME (Lyon 1969) Begnaud, Lurnice Bellaiche, Paul Bellot Rosado, Francisco Ben Jamaa, Mohsen Bernet, Théo Berri, Manuel Bhatnagar, P. L. Bigalke, Hanser Bishop, Alan Blanc, Émile Bercioux Blau, Peter Bodin, Antoine Boissier, Michèle Bonden, Herbert Bonnard, Colette Borier, Marguerite Bosveld, Derk Bourtot, Bernard Bouvier, Alain Bracewell, H. Bray, Suzanne Bright, Delores Bron, Alain Brousseau, Guy Bruley, Dominique Bruni, James Buisson, Pierre Burkhart, Sarah Byrne, Ciril Calame, Francis Calvet, Renée Cance, Henri Carrega, Jean Claude Carter, Brenda Castelnuovo, Emma Causier, Ariste Chaillou, Paul Chapon, Jacqueline Chatard, Michèle Chernick, Sada Chesney, Serge Chini Artusi, Liliana Cignetti, Alberto Cochrane, Margaret

337 USA Tunisia Spagna Tunisia Switzerland USA India Germany UK Switzerland – Switzerland Senegal France Netherlands Switzerland France Netherlands France France UK France USA Switzerland France France USA France USA Ireland Switzerland France France France UK Italy France France France France USA France Italy Italy UK

Bell, Alan W. Bellenger, Elisabeth Belouze, Bernard Bercioux, Albert Bernier, Daniel Berthault, Maurice Biemel, Rainer Biggs, Edith Blackburn, William Bollon, Louis Bied, Joseph Blonce, Jeanine Bogoslavov, Vene Bolon, Jeanne Bondesen, Aage Bonnetain, René Bosland, Pierre Bourret, Martine Bouteille, Maurice Boys, Geoffrey Bradley, Hugh Bressy, Georges Brkic, Jagoda Brossard, Roland Brumfied, Emalou Brun, Jean Brydegaard, Marguerite Bunt, Lucas Bury, Pauline Cabric, Miroljub Callagy, James Campogrande, Paolo Capitan, Jeanine Carroll, Edward Carter, Michael Casulleras, Juan Celanire, Yves Chamma’a, Samir Charnay, Roland Châtelet, François Cheroux, Paule Cheze, Jean Christiansen, Bent Clifford, Paul Coeckelberghs Bogaert, Léa

UK France Algeria Switzerland France France France UK UK (Scotland) France France France Yugoslavia France Denmark France Morocco France France UK USA Morocco Yugoslavia Canada USA Switzerland USA Netherlands USA Yugoslavia Ireland Italy France USA South Africa Spain France Lebanon France France France France Denmark USA Belgium

338 Cointet, Michel Colmez, François Combe, Marie-Louise Clifford, Paul Cote, Normand Courseau, Paulette Couty, Raymond Crawford, Douglas Croteau, Benoît Cuisance, Suzanne Damian, Brother Daumet, Gérard Deans, Edwina Decelles, Pierre Dedò, Modesto Delavault, Huguette Delessert, A. Denniss, John Desforge, Julien Despotovic, Radivoje Detry, Nicole Dienes, Zoltàn Docev, Kiril Dokic, Olivera Druel, Paulette Duby, Jean Jacques Duchène, Florence Duclos, Daniel Dumont, Marcel Dupont, Michel Durgan, Mary Duvert, Louis Egéa, Marcel Egsgard, John Eido, Rafik Elbaz, Roger Erard, Clotilde Esser, Albert Fang, Joong Fayala, Mohamed Félix, Lucienne Ferlay, Louis Ferreira, Jaime Fielker, David Fiou, Giulio Fleury, Françoise

F. Furinghetti France France France USA USA France France Canada Canada France UK Tunisia USA Canada Italy France Switzerland UK France Yugoslavia Belgium Canada Bulgaria Yugoslavia France France France France France Senegal USA France France Canada Lebanon France France Germany Germany Tunisia France France Portugal UK Italy France

Colmez, Jean Colomb, Jacky Cooper, Donald Cortial, Michel Courau, Dominique Courvoisier, André Coyle, Anne Crepin, Roger Crouth, Ralph Cvrkušić, Veselin Daugherty, J. Dwight Davalan, Jean-Paul Davidson, Louis Decombe, François Dehame, Édouard Delcourt, Pierre Delorme, Marianne Derwidue, Léon Desjardin, Christiane Desq, Roger Dhuin, Claudine Diserens, Robert Doicinov, Doicin Dravinac, Nevenka Dubrocard, Jacques Duceux, Ginette, and Pierre Ducker, Isadore Dulmage, Lloyd Dumousseau, Guy Dunkley, Mervin Duverger, Yves Dykman, Jacobus Edwards, C. Egyed, Andras Eiller, Robert Engel, Arthur Errecalde, Paule Exner, Robert Farmer, Harold Fehr, Howard Ferland, Yvette Ferrara, Ugo Fessel, Abraham Fierro, Manuel Fischbein, Efraim Folsom, Mary

France France Switzerland France France Switzerland USA France USA Yugoslavia USA France Cuba France France France France Belgium France France France Switzerland Bulgaria Yugoslavia France France USA Canada France Australia Syria Netherlands UK Romania France Germany France USA USA USA Canada Italy Israel USA Romania USA

10  First ICME (Lyon 1969) Fonteret, Alain Forde, John Fortin, Jacques Freudenthal, Hans Fried, Elisabeth Fuentes, Maria García Pradillo, Julio Gauthier, René Gentile, Maria Luisa Gianati, Colette Gibb, Elisabeth Glenadine Gilbert, H. Gilliespie, Robert P. Glaymann, Maurice Glass, Elisabeth Goeringer, Gabrièle González Roldan, Ismael Goullin, Chantal Gradelet, Simone Gravel, Hector Grootendorst, Albertus Wilhelmus Grossi Pillar, Esther Guillerault Astier, Michel Gutierres Trobajo, Ramon Haggmark, Per Marten Hamdi, Ali Haralambie, Jonesco P. Hartig, Klaus Harvey, John Grover Hayter, Raymond John Heddens, James Heinke, Clarence Henry, Michel Hight, Donald V. Holcomb, George Howson, Albert Hug, Colette Hutin, Raymond Inghilterra, Carlo Iyanaga, Shōkichi Jacquemier, Philippe Jeśmanowicz, Leon Jojon, Cécile Kanning, Eunice Kaufman, Burt

339 France Ireland Canada Netherlands Hungary Cuba Spain France Italy France USA France UK France USA France Spain France France Canada Netherlands

Fonvieille, Hélène Fort, Jacques Francis, Margaret Friant, Jean Friis, John Garbe, Jean Gaulin, Claude Geneva, L. Martin Gerard, Maurice Giannoni, Carla Gibellato Valabrega, Elda Gilis, Daniel Gillman, Leonard Glardon, Monique Glorian, Marie-Jeanne Goffree, Frederik Gorner, Frank Gouret, Alain Gradelet, Marcel Greig, Muriel Grolier, Yvette

France France USA France UK Algeria Canada USA France Italy Italy France USA Switzerland France Netherlands UK France France USA France

France France Cuba Sweden Tunisia Romania Germany USA Malawi USA USA France USA USA UK France Switzerland Tunisia Japan France Poland France USA USA

Guenoun, Yves Guillerault Astier, Mireille Haas, Raymond Hale, William T. Hansen, Ole Krag Hardgrove, Clarence Hartley, Elizabeth Mary Haugazeau, Denise Heffernan, Michael Heiede, Torkil Henry, Annie Herz, Jean-Claude Hlavaty, Julius H. Howe, Jennie A. Huau, Renée Hughes, David Hynes, Michael Ivanov, Petko Jacquemier, Yvette Jeffery, Gordon Bernard Jevremović, V. Branko Joksimović, Zivota Katz, Paul McKay, Maxwell

Algeria France USA USA Denmark USA UK France Ireland Denmark France France USA USA France UK USA Bulgaria France UK Yugoslavia Yugoslavia Israel UK

340 Kefuss, Josette Kieffer, Lucien Kindel, Marinette Klahn, Søren Kobeisse Kovacevic, Branka Kozlic, Frédéric Krygowska, Anna Zofia Kuntzmann, Jean Kyed, Thomas Lacondemine, Paul Ladoux, François Lamb, Phyllis Joyce Laub, Josef Laux, Josef Lavigne, Germaine Lefebvre, Huguette Lefebvre, Pierre Le Guern, Germaine Lehmann, Henri-L. Lemire, Lévis Lewis, Emma L’Hospitalier, Yvon Limoge, Mauricette Lingua, Pietro Lipson, Stanley Lopes, Antonio Augusto Lovett, C. James Lubuela, François MacNab, Donald McNicol, Shirley Mafrica, Demetrio Magers, Dexter Malme(?)g, Allan Manic, Mikka Manotte, Jeanne Markusevic Martin, William Ted Maslova, Galina Matthews, Geoffrey Maublanc de Chiseuil, Pierre Maxwell, Edwin Arthur Mazzi, Maria Teresa Meier, Fritz Mercier, Jean Merlan, Jacques

F. Furinghetti Germany Luxemburg France Denmark – Yugoslavia France Poland France Denmark Algeria France UK Austria Germany Belgium France France France Switzerland Canada USA France France Italy USA Portugal USA Belgium UK Canada Italy USA Denmark Yugoslavia France USSR USA USSR UK France UK Italy DDR France Ivory Coast

Ketterer, Henri Kies, Jakob Klaasse, Adrianus Koldijk, Albert Kooi, O. Kovacs, Geza Krstic, Slobodanka Kobeisse, Hafez Kurepa, Đuro Kyo, R. Jhin Lacourt, Marie Thérèse De Lagaye, François Lariccia, Giovanni Laurain, Hélène Lavabre, Simone Laymand, Roger Lefebvre, Jean Legris, Roberte Lehmann, Daniel Le Minous, Claude Le Pezron, Yves Lewis, Robert Lievens, Edna Mary Lindsay, Robert Liouville, Anne-Marie Lopata, Geneviève Loudot, Jeanne Lubbe, Adriaan Lyness, Robert MacNarb, William Mafrica Micotti, Carla Magarian, Elizabeth Malecamp, Eugénie Mancini Proia, Lina Manolov, Spas Marinkovic, Bogoljub Marques, Rubens Murillo Martin, Francette Mateev, Alipi Matthys, Jean-Claude Maurières, Marcel Mayor, René Meghannem, Mohamed Mellin, Eva Merigot, Michel Metenier, Jacqueline

Lebanon South Africa Netherlands Netherlands Netherlands Romania Yugoslavia Lebanon Yugoslavia USA France France Italy France France France France Canada France France France UK USA UK France France France South Africa UK USA Italy USA France Italy Bulgaria Yugoslavia Brazil France Bulgaria Belgium France Switzerland Tunisia Germany France France

341

10  First ICME (Lyon 1969) Meylan, Robert Michels, MArcel Miloskovic, Dusan Miyazaki, Katsuji Momcilo, Lalovic Morfin, Michel Morvan, Sister Irene Motto, Marie Munford, Donald Nano, Ada Narion, Yves Neumann, Bernhard Nicolas, Alain Nikolic, Milenko Noureddine Nunez Berro Maria O’Brien, Stephen Oleary, Sean O’Mahonq Ornstein, Avraham Osouf, Roger Page Becker, Jerry Palmer, Roberta Pann, Jacqueline Papy, Frédérique Pascual Xufre, Grisalde Penavin, Velimir Penfold Alec, Daisy Perdon, Christianne Perole, Yvette Pervine, Youri Peters, Nestor Picard, Philippe Pikaart, Leonard Pinchinat, Renée Platzker, Ovadia Plomp, Tjeerd Pollak, Henry Pool, Katherine Porcel, Nicole Pouget, Colette Powell, Marin J. Pradine, Anne-Marie Prevot, Yves Jean Quadling, Douglas Radovic, Milovan

Switzerland Luxemburg Yugoslavia Japan Yugoslavia France USA USA Canada Italy France Australia Morocco Yugoslavia Lebanon Cuba Ireland Ireland USA Israel France USA USA France Belgium Spain Yugoslavia UK France France France Belgium France USA Tunisia Israel Netherlands USA UK France Algeria UK France France UK Yugoslavia

Michelot, Georges Mijnlieff, Adrian Mison, Ginette Mohr, Thomas Moore, Charles Moris, Maurizio Motte, Magdelaine Mulder, Sophia Myx, André Neubert, Ernst Mar(?)us, Asher Nichols, Eugene Nicollerat, Marc Nikolic, Jagorka Novak, Josef Oberman, John O’Donnell, John De Oliveira Bender, Joana Orlov, Konstantin Osorio Dos Anjos, Alfredo Oulevay, Bernard Palmer, B. A. Henry Pampallona, Ugo Papazian, Marie-Jeanne Paquette, Gilbert Pélissier, Gérard Parish, George L. D. Pennaneach, Francis Perol, Charles Perret, Josette Pescarini, Angelo Petkantschin, Bojan Pickert, Günter Pinchinat, Joël Pissavin, Odette Pleijel, Ake Vilhlem Carl Poivey, Monick Pons Ballarin, Ricardo Popov, Koviljka Portron (twice) Poussin, Noëlle Prave, Marie-Madeleine Predescu, Valentina Pukades Duran, Roser Rade, Lennart Rageul, Lucien

France Netherlands France Lebanon USA Italy France Netherlands France Germany Israel USA Switzerland Yugoslavia Czechoslovakia Israel USA Brazil Lebanon Portugal Switzerland USA Italy France Canada France South Africa France France France Italy Bulgaria Germany Tunisia France Sweden France Spain Yugoslavia – France France Romania Spain Sweden France

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Rai, Ram Rakocevic, Stanko Ravizza, Paolo Revuz, André Riou, Alain Robert, Fabiola Ronveaux, André Rosenbloom, Paul C. Roumieu, Charles Rouquet, Gaston Roussel, Yves Roy, Monique Rueff, M. Sabbatiello, Elsa Elena Salas Palenzuela, Isidoro Samana, Guy Sarazin, Marie Louise Sauvy, Simone Schaack, François Schult, Veryl Schwartzman, Pauline Averil Semadeni, Zbigniew Shapira, Meir Shulman, Lee S. Sibille, Jean Louis Sinisa, Jerinic Slavka, Milicevic Smolec, Ignacije Soler, Argilaga José Sooy, John Milton Satoca, Garcia Manvel

Canada Yugoslavia Italy France France Canada Canada USA France France France France Switzerland Argentine Spain Tunisia France France Luxemburg USA South Africa Poland Israel USA Senegal Yugoslavia Yugoslavia Yugoslavia Spain USA Spain

Souza de, Dantas Martha Maria Steen, C. M. Brendan P. Steiner, Hans G. Stephens, Josephine Storer, Walter Owen Straszewicz, Stefan Sturgess, David Ahan Surányi, János Szendrei, János Tagg, Donovan Tradif, Mireille Therond, Chantal Thoft – Christensen, Palle Todorovic, Ljubisav

Brazil Ireland Germany UK UK Poland UK Hungary Hungary UK France France Denmark Yugoslavia

Raizen, Senta Rapley, Brooks Rennie, Basil Cameron Richard, Françoise Rivière Lanne, Paule Roditi, Édouard Room, Thoma G. Roumanet, André Rouquairol, Michel Rousseau, Robert Roveyaz, Giulietta Rucker, Isabelle P. Rouget, Jeanine Sacco, Maria Piera Salvano, Jacqueline Samson, Jean Sarrazin, Elzear Savalle, Albert Schramm, Ruben Schwamberger, Kurt Scroggie, George Servais, Willy Shuard, Hilary Bertha Sibagaki, Wasao Signetto, Fulvia Sitia, Candido Smith Sobolev Soos, Gyula Sorger, Peter Sousa Ventura, Manuel Joachim Sparks, Thelma Angelina Maclin Steffes, Pierre Steiner, Margaret Stewart, Shirley Ann Stosic, Radmila Studzinsky, Hermann Sulzer, Béar Suter, Anna K. Sabaud, Robert Tailleu, Gérard Tassy, Juliette Thille, P. M. Isabelle soeur Thwaites, Bryan Tosi, Armida

USA Canada UK France France France Australia France France France Italy USA France Italy Luxemburg France Canada France Israel France Canada Belgium UK Japan Italy Italy – USSR France Germany Portugal USA Luxemburg USA UK Yugoslavia Austria Switzerland USA Germany France France Canada UK Italy

10  First ICME (Lyon 1969) Touyarot, Marie Antoinette Valenza Antoine Vance, Irvin E. Van Den Briel, and Johan Kees Van Der Meiden Vanhamme, Jacqueline Van Speybroeck, James Omer Varga, Tamás Veit, Barbara Verhoef, Willem Verset, Claude Viggo, Hansen Vitali, Laurent Voisin, Pierre Vopni, Sylvia Vysin, Jan Wallbank, Sarah Anne Wansink, Johan Hendrick Watanabe, Renate Watson, John Robert Wells, Peter John White, Paul Whittington, Beverly Williams, Elizabeth May Wirszup, Izaak Wolff, Georg Wootten, Amy Yeshurun, Shraga Zammit, Christian Zaouli, M’hamed Zoll, Edward

343 France Tunisia USA Netherlands

Tumura, Yosiro Van Arsdel, Jean Vandenberghe, Roger Van Der Krogt, Bart

Japan USA Belgium Netherlands

Netherlands Belgium USA Hungary Italy South Africa France USA Tunisia Tunisia USA Czechoslovakia UK Netherlands Brazil South Africa UK USA USA UK USA Germany UK Israel France Tunisia USA

Van Dormolen, Joop Van Nieuwenhuysen, Roland Varenne, Gérard Vautravers Ventadoux, Madeleine Verset, Anfrée Vervoot, Gerardus Villeneuve, Jean Alain Vogeli, Bruce Vonk, Gustaaf Adolf Vuilleumier, Francis Vyvyan, Richard Walter, Marion I. Wasche, Hans Watson, Frank Richard Weidig, Ingo Westerthof, Binne Jan Whitney, Hassler Wilhelm, Aelette Wilson, John Wittmann, Erich Wolff, Leslie Wydeveld, Eduard Jan Zaller, Margaret A. Zammit, Jacqueline Zimorya, Dragied Zweng, Marylin J.

Netherlands Belgium France Switzerland France France USA France USA Netherlands Switzerland UK USA Germany UK Germany Netherlands USA France Turkey Germany USA Netherlands Italy France Yugoslavia USA

Fig. 10.1  Participants in ICME-1. In the picture there are (1) Shmuel Avital, (2) John Egsgard, (3) Arthur Engel, (4) Hans Georg Steiner whose face is almost completely hidden by the face of Hans Wäsche, (5) Willy Servais, (6) Janós Surányi, and (7) Erich Wittmann. (Courtesy of Erich Wittmann)

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10.2 The Conférences Libres at ICME-1 (Lyon, 1969) The proceedings of ICME-1 (The Editorial Board of Educational Studies in Mathematics 1969) contain only a selection of the contributions presented during the congress. ICME-1 had a program very rich, as evidenced by the following description in (Howson 2007, p.  1): “with its mornings occupied by twenty-one plenary talks on primary and secondary school education in Europe and North America, and the afternoons given over to fifteen-minute presentations by whoever wished to give them.” The list of the so-called Conférences libres (Free Lectures), which was distributed to the participants, gives further hints on what happened; see Table 10.2.

Table 10.2  Authors and titles of the Conférences libres Abeles, Francine, and Edward Zoll Aizpun, Alberto Haralambie, Jonesco, and Rado Badesco Becker, Jerry Bell, Alan W. Biggs, Edith E. Bishop, Alan Bright, Delores Brossard, Roland Colomb, Jacky Dupont, Pascal Egsgard, John Fang, J. Fehr, Howard Félix, Lucienne Fessel, Abraham Haimovici, Adolf Harey, John G., Thomas A. Romberg, and Harold J. Fletcher Bastad, Matte Beddens, James Hug, Colette

Networks, maps and Betti numbers: An eight year old’s thinking Initiation aux équations de premier degré dans l’école primaire Un schéma pour la modernisation de l’enseignement mathématique secondaire en Romanie Achievement testing in mathematics Mathematical activity in the context of teacher education Communication on primary education in mathematics practical work – for what purpose? A comparison of four common teaching methods Calculus in California High Schools L’emploi des structures parallèles comme outil pédagogique Une expérimentation dans le cycle élémentaire Recherche opérationnelle au niveau de l’enseignement secondaire. Combien? Comment? Pourquoi? Some ideas in geometry that can be taught in grades K-6 Les mathématiques en tant que science expérimentale The secondary school mathematics curriculum improvement study Géométrie, modèle privilégié pour l’enseignement de l’algèbre générale The aims and the role of teaching mathematics and their realization Sur l’idée d’approximation et sur l’approximation des racines d’une équation Analysis of mathematics instruction: A discussion and interim report In service training in Sweden using television, radio and correspondence material Experimental model for mathematics teacher preparation L’enfant et la mathématique (continued)

10  First ICME (Lyon 1969)

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Table 10.2 (continued) Hutin, Raymond Hynes, Michael Ilie, Ilasievici Jeffery, Gordon Kaufman, Burt A., Robert M. Exner, Hans G. Steiner, and Lennart Rade Kieffer, Lucien Kies, Jakob D. Klamkin, Murray S. Kurepa, Đuro Lewis, Robert Matthews, Geoffrey Nichols, Eugene Nikolic, Mikenko

O’Donnell, John Orlov, Konstantin Quadling, Douglas A. Rodrigues, Antonio Ronveaux, André O’Brien, Thomas, Bernard Shapiro Smolec, Ignacije De Souza Dantas, Martha Van Speybroeck, James Omer

Vogeli, Bruce Vopni, Sylvia Yeshurun, Shraga

Le renouvellement de l’enseignement de la mathématique dans les écoles primaires genevoises Teaching the mathematically disadvantaged—11–15 years old L’activité créatrice des élèves des classes spéciales de mathématiques (?) Reports on the comprehensive school mathematics program

Démonstrations vectorielles de quelques théorèmes de la géométrie dans l’espace Mathematization in the classroom: An example On the ideal role of an industrial mathematician and its educational implications La notion de vecteur dans l’enseignement Approaches to mathematics for non-specialists Primary problems Some aspects of algebra Contributions à l’étude de l’influence du transfert sur les résultats de l’enseignement des mathématiques chez les élèves de 11–15 ans Number, numeral, and Plato Balance mathématique aide visuelle pour l’enseignement de l’algèbre à l’école secondaire An organic approach to the mathematics curriculum for upper secondary students Sur une méthode pour l’enseignement de la géométrie plane euclidienne d’un point de vue actualisé Une expérience canadienne: Équations différentielles au secondaire The development of logical thinking in children-II Quelques dispositions techniques en faveur de meilleure compréhension de la mathématique Programme expérimental de mathématiques pour le premier cycle de l’enseignement secondaire A program of geometric constructions for the elementary grades requiring only a minimum of basic motor skills on the part of the child Sweep away of cows, ghosts, dragons and devils Mathematics education: Horizons and happenings The cognitive method

Note: The original documents, which were distributed to the participants in ICME-1, were kindly sent to me by Jerry Becker and Erich Wittmann.

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References and Sources Becker, Jerry P. 1970. Some notes on the first international congress on mathematical education. American Mathematical Monthly 77 (3): 299–302. Becker, Jerry. 1975. An analysis of participants in the first and second International Congress on Mathematical Education (ICME). ICMI Bulletin 5: 20-24. Howson, Geoffrey. 2007. Reflections on ICMEs. Papers from unpublished issues of the ICMI Bulletin. https://www.mathunion.org/fileadmin/ICMI/files/Digital_Library/ICMEs/Bulletin_ Howson_Studies.pdf (Retrieved 3 June 2021). The Editorial Board of Educational Studies in Mathematics. 1969. Proceedings of the first international congress on mathematical education. Dordrecht: Reidel Publishing Company. Also in Educational Studies in Mathematics 2, 1968: 135–418.

Part III

The Portraits of the Central/Executive Committee Members and Other Eminent Figures

1.1  Introductory Note This part of the volume provides information on important figures in the history of ICMI. Chapter 11 contains biographical portraits of the 54 members of the Central/ Executive Committee of ICMI who passed away in the first hundred years of ICMI. Almost all of them were professional mathematicians who were interested in problems concerning mathematics education and participated in various ways in the activities of ICMI. However, beginning in the 1950s, some mathematicians joined the ICMI Executive Committee as ex officio members for the sole reason that they occupied certain institutional positions – such as the IMU presidency. In some cases their involvement in ICMI activities was modest. Of course, for famous mathematicians many biographies are available; nevertheless, the portraits presented in this part of the volume add details on the actual involvement of the various people in the ICMI activities and, more generally, in mathematics education and dissemination of mathematical culture. Each of these portraits consists of a section of general information and a section dedicated to contributions to education and dissemination of mathematical culture. In the same vein, sources are divided into a part containing a selection of works on the person in question and of his mathematical works and a part including publications linked to mathematics education and dissemination of mathematical culture. Chapter 12 contains biographical portraits of the scholars awarded the title of Honorary Member of the Commission on the occasion of the International Congress of Mathematicians in Oslo (1936) and the portrait of Charles-Ange Laisant, one of the founders of L’Enseignement Mathématique, the official organ of ICMI. Among the scholars awarded the title in Oslo, there was Guido Castelnuovo, a former officer of ICMI, whose portrait is in Chap. 11. The portraits presented in this chapter follow the same criteria as the previous ones. The authors of the 63 portraits are 30 and belong to 18 countries.

Chapter 11

The Central/Executive Committee Members Masami Isoda, Man Keung Siu, Henrik Kragh Sørensen, Livia Giacardi, Jeremy Kilpatrick, Fulvia Furinghetti, Gert Schubring, Giorgio T. Bagni, Sébastien Gauthier, Catherine Goldstein, Michela Malpangotto, Jaime Carvalho e Silva, Margherita Barile, Éric Barbazo, Harm Jan Smid, Sten Kaijser, Adrian Rice, Hélène Gispert, Milosav M. Marjanović, Stevo Todorčević, Osmo Pekonen, Reinhard Siegmund-Schultze, Michèle Artigue, Aline Robert, Ewa Lakoma, and László Surányi

© Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_11

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11.1 Yasuo Akizuki (Wakayama Prefecture, 1902 – Kanagawa Prefecture, 1984): Member of the Executive Committee 1959–1966 Masami Isoda

Biography Yasuo Akizuki was born on 23 August 1902 at Wakayama Prefecture, near Osaka, and died on 11 July 1984 at Kanagawa Prefecture, near Tokyo. He graduated from Kyoto High School (1923) and Imperial University of Kyoto (1926). He obtained his PhD at Kyoto University in 1939. He was a professor at Kyoto High School (today, a part of the University of Kyoto) in 1929–1946, at University of Kyoto in 1947–1951 and at Tokyo University of Science and Literature (Tokyo Bunri University, renamed Tokyo University of Education, precursor of the University of Tsukuba) in 1951–1966. He was the president of Gunma University (1967–1971). He was decorated with the highest award for a scholar in Japan (1973). As a researcher in mathematics, Akizuki was known for the Krull-Akizuki theorem. He developed the Kyoto School of Algebraic Geometry and with Kunihiko

M. Isoda (*) University of Tsukuba, Tsukuba, Ibaraki, Japan e-mail: [email protected]

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Kodaira the Tokyo School of Algebraic Geometry. He was a key person in developing the ground for the current prosperity of Kyoto School in Mathematics, in supporting the enlargement of the mathematics department of the Tokyo University of Education after World War II and in managing and supporting international symposia in various areas on mathematics. He influenced world-famous mathematicians such as Heisuke Hironaka (Fields Prize in 1970) and educated the generation of those who contributed to the Japanese development at Kyoto High School. In 1957, he was invited as a visiting professor by the University of Chicago. During the first half of his career, Akizuki was a mathematician and a professor; in the second half, he mainly committed himself to mathematics and science education for the social context.

Contribution to Mathematics Education Akizuki was a great teacher for his whole life. At the same time, he contributed to bridge mathematics and mathematics education during his years at Tokyo University of Education. In Japan, he was well known for his contribution in three aspects of Japanese education, with the support of his colleagues. First, Akizuki supported the stepping up of Japanese Mathematics Education Society as a part of Academy. At Tokyo University of Education, he worked with researchers in mathematics education as well as with mathematicians. Even though the Japan Society of Mathematical Education (JSME) was founded in 1919 at Tokyo Higher Normal School (precursor of Tokyo University of Education), the members of the mathematics education committee in the mathematics board of the Science Council of Japan were limited to mathematicians. He changed this situation so that the committee included a math educator. He held or supported international meetings in mathematics education such as the research seminar of School Mathematics Study Group (1964),1 the Japan-US seminar of mathematics education (Tokyo 1971) and the ICMI-JSME International Conference of Mathematics Education (Tokyo 1974). In this later seminar, he delivered a talk entitled “How to Educate the Spirit of Mathematics”. Second, Akizuki worked for the modernization of school mathematics from the point of view of developing mathematical spirit as a part of human being. He chaired the committee of the mathematics curriculum standard in Japan (1971). Japanese modernization had developed in the tradition which enhanced mathematization (1943), mathematical activity (1947) and mathematical thinking (1951). He explained the importance of mathematical activity in education as follows: Defining what “Mathematical Thinking” is, is a necessary question when we learn and study mathematics, and educate via mathematics. Both mathematicians and math educators are human and we do not need to solve any question mathematically. “Mathematical

 See “Document” section, Research in Education. 1969. 4(1), p. 95.

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Thinking” is a human activity. What is known as Mathematical Thinking should be represented by mathematical activity, not only expressed by mathematics but also the activity to create mathematics and the thinking for creation, and it is experienced as intuitive feeling with application, not experienced by dividing each concept and way of reasoning. There are reasonings such as analogy, induction and deduction which are known to be powerful but if we focus on each mathematical concept and way of reasoning, we may lose many [things of] importance. Mathematical thinking is experienced in the mathematical activity in any level of students. (Mathematical thinking and its teaching. Journal of Educational Research, Elementary School of Tokyo University of Education: Elementary Education Research Institute 21(5): 8–9, 1966, translation by the author)

Akizuki enhanced the importance of value education in science (mathematics) and described the way to develop value as follows: Value is recognized through reflection. We can recognize what we are doing through reflections just as mirrors reflect each other recursively. Understanding the depth of value is a human ability. Science education should not only be training in technical skill. Recognizing Science and Technology as a part of human activity and creation, we should teach students to reflect on what is deeper. This is the principle of education for any level of students. (‘Lecture of Science and Humanity’, 1966, from the Memorial Book of Prof. Yasuo Akizuki, 1985, translation by the author)

Third, he collaborated with natural scientists to develop the science education based on his view of education and contributed to the development of “Science Education Area” in the Science Council of Japan, as well as in the inter-subjective bridge Science and Education, which already existed independently. He developed the “Akizuki” committee (1968–1976) funded by a grant from the Ministry of Education and gathered math educators and mathematicians to engage in the research of modernization, high school textbooks, and the integration of subjects. Many mathematics educators matured in his committee. Based on these fundamental activities of scientists and educators who worked for science education reform, Japan Society of Science Education (JSSE) was founded. In the meeting held in The Hague (1 September 1954), the General Assembly of IMU (International Mathematical Union) decided that the composition of ICMI has to be renewed starting on 1 January 1955. This assembly appointed ten members at large of ICMI, among them Akizuki (CIEM 1955). In 1959, he became member of the Executive Committee until 1966. After Ram Behari, he was the second member of the ICMI Committees (Central or Executive) not coming from Europe or North America. He actively participated in many international activities linked to mathematics education of that period: not only to the meetings organized in Japan mentioned above but also in other meetings such as the symposium on school mathematics teaching of 1962 organized with the approval of UNESCO in Budapest (see Hungarian National Commission for UNESCO. 1963. Report on the work of the international symposium on school mathematics teaching. Budapest: Akadémiai Nyomda) and ICME-1 in Lyon in 1969 (L’Enseignement Mathématique. 1970. 16: 117–122). He participated in the “Congress on science teaching and its role in economic progress”, held in Dakar (14–22 January 1965). In the ICMI-JSME Regional Conference (Tokyo, 5–9 November 1974), he delivered a plenary lecture entitled

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“How to Educate the Spirit of Mathematics” (ICMI Bulletin n. 4, 1974: 9). It is of particular interest Akizuki’s document annexed to the report of the ICMI meeting in Paris in 1959, since it makes explicit reference to cultural issues as a factor to be considered in mathematics education, and advocates the presence of the history of sciences, of mathematics in particular, at all curricular levels (see Akizuki 1959, p. 288).

Sources2 Akizuki Memorial Committee (ed.). 1985. Ikou. Tokyo: Kinokuniya Publisher (In Japanese). Akizuki, Yasuo and Yukio Kusunoki. 1973. Number theory, algebraic geometry and commutative algebra (In honor of Yasuo Akizuki). Tokyo: Kinokuniya Book-store Co. Akizuki, Yasuo. 1935. Einige Bemerkungen über primäre Integritätsubereiche mit Teilerkettensats. Proceedings of the Physico-Mathematical Society of Japan s. 3, 17: 327–336. Akizuki, Yasuo. 1941, 1947, 1970, 1973. Recent perspective of algebra. Tokyo: Kobundo (In Japanese). Desforge, Julien. 1955a. CIEM. L’Enseignement Mathématique s. 2, 1: 193–202. Kawaguchi, T. (ed.). 1971. Reports of U.S.-Japan seminar on mathematics education [Special issue]. Journal of Japan Society of Mathematics Education 53 (Suppl. issue). Proceedings of ICMI-JSME Regional Conference on Curriculum and Teacher Training for Mathematical Education, Tokyo, November 5–9, 1974. Tokyo: National Institute for Educational Research.

Publications Related to Mathematics Education Akizuki, Yasuo. 1959. Annexe I.  Proposal to I.C.M.I. L’Enseignement Mathématique s. 2, 5: 288–289. Akizuki, Yasuo. 1966. Principles in training school mathematics teachers. L’Enseignement Mathématique s. 2, 12: 111–117. Akizuki, Yasuo. 1968. Mathematical Thinking. Tokyo: Meijitosyo Publishers (In Japanese). Akizuki, Yasuo. 1968. Science and Humanity. National Education Center (edited by). Tokyo: Society of Imperial Local Administration Publisher (In Japanese).

Photo Author: Jacobs, Konrad. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

 In the library network of Japanese universities, more than 98 books are edited or written by him, most of them textbooks from the elementary school level to the university level. On the Zentralblatt Math database, there are 25 articles written by him. 2

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11.2 Aleksandr Danilovich Aleksandrov (Volyn, 1912 – St. Petersburg, 1999): Member of the Executive Committee 1959–1962 Man Keung Siu

Biography Aleksandr Danilovich Aleksandrov3 was born on 4 August 1912 in the village of Volyn and lived in St. Petersburg from a very young age. (St. Petersburg was renamed Petrograd in 1914, then Leningrad in 1924 and finally reverted to its old name in 1991.) He passed away in St. Petersburg on 27 July 1999. After finishing secondary school in 1929, he entered the Faculty of Physics of Leningrad State University, intending to specialize in theoretical physics. He studied under the leading theoretical physicist Vladimir Aleksandrovich Fok (1898–1974), and also learnt mathematics from the eminent geometer and algebraist Boris Nikolaevich Delone (Delaunay) (1890–1980), whose work on geometry of numbers and crystallography fascinated him. He graduated in 1933 and continued  His name has been transliterated into English as Alexandrov, Alexandroff and Aleksandrov. The last is preferred here. 3

M. K. Siu (*) University of Hong Kong, Hong Kong, SAR, China e-mail: [email protected]

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his research under the supervision of Fok and Delone, defending his PhD thesis in 1935 and his DSc thesis in 1937. In 1937, he was appointed a professor of geometry at Leningrad State University. In 1938, he joined the Leningrad Branch of the Steklov Institute of Mathematics of the USSR Academy of Sciences (now the Russian Academy of Sciences). During WWII, from 1942 to 1944, he moved to Kazan with the Institute, then returned to Leningrad State University in 1944. In 1964, at the invitation of Mikhail Alekseevich Lavrent’ev (1900–1980), he moved to Siberia and became the head of the Department of Geometry at Novosibirsk State University, as well as the head of Department of Geometry of the Mathematical Institute of the Siberian Branch of the USSR Academy of Sciences. He was known for his attentiveness and generosity in sharing ideas with the many graduate students that he supervised, thereby building up an impressive school of research. He continued his teaching and research at the Leningrad Branch of Steklov Institute of Mathematics from 1986 on. From 1952 to 1964, Aleksandrov was the rector of Leningrad State University. Based on the principles of universal humanity, of responsibility and of scientific excellence, he courageously and energetically supported colleagues against Lysenko’s pseudoscience4 in those days of persecution. It was recorded in an essay written on the occasion of his 80th birthday: In those difficult and terrible times when Soviet biology was completely dominated by T.D.  Lysenko, Leningrad State University with the active assistance of the Rector A.D.  Aleksandrov established a department of genetics, where the scientific theory of heredity and variability was taught, and not Lysenko’s ravings. Students of biology, sent down from other universities for attempting to study genetics illegally, were given the chance to continue their education within the walls of Leningrad State University. (Kutateladze et al. 1993, p. 258)

In a statement made by the Leningrad Mathematical Society on 28 March 1989, it was said: Leningrad scholars remember the numerous good deeds of A.D. Aleksandrov: in those difficult years his efforts helped to preserve science and individual scholars, and that required of him great personal courage. (ibid.)

In October 1990, for the contribution to the preservation and development of the study on genetics in Russia, Aleksandrov was honoured, together with a group of biologists, the Order of Labour of the Red Banner. Aleksandrov was a mathematician of international reputation, who made significant and diversified contributions to crystallography, the theory of convex bodies, the theory of functions of a real variable, measure theory, partial differential equations and the foundation of relativity theory. His contributions were recognized by many honours, including a USSR State Prize in 1942, the Lobachevsky Medal in

 Trofim Denisovich Lysenko was a Soviet agronomist who achieved crucial positions in soviet biology with Stalin’s support. He headed a campaign against genetics and neo-­Darwinism based on the ground of a neo-Lamarckian idea, claiming that in crop plants environmental influences are heritable via all cells of the organism. 4

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1951 and the First Euler Gold Medal in 1992. His monograph, Vypuklye mnogogranniki (Convex Polyhedra), published in 1950 has been translated into many languages, with the English translation published in 2005. Besides monographs and papers on mathematics, he was also noted for his numerous writings on the history and philosophy of science and mathematics. During the tenure as rector of Leningrad State University, he set up in the University studies on sciences that were both new and not yet recognized in his time, such as sociology and mathematical economics. In 1946, Aleksandrov was elected a corresponding member of the USSR Academy of Sciences (now the Russia Academy of Sciences) in the Division of Physical-Mathematical Sciences. In 1964, he was elected to a full member in the Division of Mathematics.

Contribution to Mathematics Education From the late 1970s on, Aleksandrov devoted his time and energy to the teaching of mathematics in school by writing both textbooks on geometry and on mathematics education in general. He served as the chairman of the Mathematics Section of the Methodological Council of the USSR Ministry of Education in the later 1980s. Aleksandrov was also active at the international level. He was one of the plenary speakers at the Conference on Mathematical Education in South Asia together with ICMI Vice-President Marshall Stone, Gustave Choquet, Hans Freudenthal and others (Report of a Conference on Mathematical Education in South Asia, The Mathematics Student, 24, 1956, 1–183). At ICME-7 in Québec, he delivered a talk (see Aleksandrov 1994). In 1959–1962, he was a member of the ICMI Executive Committee. He wrote many articles for mathematical journals for school and many entries in Bol’shaya Sovetskaya Entsiklopediya (Great Soviet Encyclopaedia). In 1956, together with Andrei Nikolaevich Kolmogorov (1903–1987) and Mikhail Alekseevich Lavrent’ev (1900–1980), he edited the famous three-­volume work Matematika, ee soderzhanie, metody i znaachenie (Mathematics: Its Content, Methods, and Meaning), which attempted to give an idea of the current state of mathematics (in the mid-twentieth century), its origins and probable future development. After half a century, some of the chapters are still considered to be excellent reading for someone who is seriously interested in mathematics looking for a friendly, but not superficial, overview of various important fields. In particular, Aleksandrov wrote Chapter I (A General View of Mathematics), Chapter VII (Curves and Surfaces) and Chapter XVII (Non-Euclidean Geometry), which are examples of excellent expository writing. (The book was translated into English by Sydney H. Gould and Tamas Bartha and published by the MIT Press in 1963/1964.) In Chapter I, to discuss the essential nature of mathematics, Aleksandrov concentrates on the following questions (p. 6): • What do these abstract mathematical concepts reflect? In other words, what is the actual subject matter of mathematics?

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• Why do the abstract results of mathematics appear so convincing, and its initial concepts so obvious? In other words, on what foundation do the methods of mathematics rest? • Why, in spite of its abstractness, does mathematics find such wide application and does not turn out to be merely idle play with abstractions? In other words, how is the significance of mathematics to be explained? • Finally, what forces lead to the further development of mathematic, allowing to unite abstractness with breadth of application? What is the basis for its continuing growth? Subsequently, in light of these questions, he reviews the various parts of mathematics with many historical references. In an obituary by Yurii Borisov that appears in the Russian Mathematical Surveys (1999), the following words summarize his life: Aleksandrov’s scientific ideas will live for a long time in the work of his students and successors. The unique charm, the combination of youthful spirit and experienced wisdom, fierce temperament and subtle intellect, the selflessness and tenderness of Aleksandr Danilovich remain happy memories of all those who had the good fortune to be with him. (Borisov 1999, p. 1017)

Sources Selected works. A.  D. Alexandrov Part I. 1996. ed. Samson S.  Kutateladze; translated from the Russian by P.S.V. Naidu. Amsterdam: Gordon and Breach Publishers. Selected works. A.  D. Alexandrov Part II. 2005. eds. Yurii G.  Reshetnyak, and Samson S. Kutateladze; translated from the Russian by Sergei A. Vakhrameyev. New York: Chapman and Hall/CRC. Aleksandrov, Pavel S., Nikolai V.  Efimov, Victor A.  Zalgaller, and Aleksei V.  Pogorelov. 1973. Aleksandr Danilovich Aleksandrov. (On his sixtieth birthday). Russian Mathematical Surveys 28(6): 225–230. Borisov, Yurii F. 1999. Aleksandr Danilovich Aleksandrov. Russian Mathematical Surveys 54(5): 1015–1018. Borisov, Yurii F., Samson S.  Kutateladze, Olga A.  Ladyzhenskaya, Sergei P.  Novikov, Aleksei V. Pogorelov, Yurii G. Reshetnyak, Sergei L. Sobolev, and Victor A. Zalgaller. 1988. Aleksandr Danilovich Aleksandrov (On his seventy-fifth birthday). Russian Mathematical Surveys 43(2): 191–199. Efimov, Nikolai V., Victor A.  Zalgaller, and Aleksei V.  Pogorelov. 1962. Aleksandr Danilovich Aleksandrov. (On his fiftieth birthday). Russian Mathematical Surveys 17(6): 127–141. Kutateladze, Samson S., Olga A. Ladyzhenskaya, Sergei P. Novikov, Aleksei V. Pogorelov, Yurii G.  Reshetnyak, and Victor A.  Zalgaller. 1993. Aleksandr Danilovich Aleksandrov (On his eightieth birthday), Russian Mathematical Surveys 48(4): 257–260. Reshetnyak, Yurii G. and Samson S. Kutateladze (eds.). 2012. Aleksandr Danilovich Aleksandrov: A Bibliography (in Russian). Novosibirsk: Akademiya Nauk SSSR, Sibirskoe. Otdelenie. Institut Matematiki].

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Publications Related to Mathematics Education Aleksandrov, Aleksandr D., Aleksei L, Verner, and Valerii I. Ryzhik. 1983. Geometriya: Probnyi. uchebnik dlya 9–10 klassov srednei shkoly [Geometry: Trial textbook for classes 9–10 of the secondary school]. Moscow: Prosveshchenie. Aleksandrov, Aleksandr D., Aleksei L. Verner, and Valerii. I. Ryzhik. 1984. Geometriya: dlya 9–10 klassov: Uchebnye posobie dlya uchashchikhsya shkol i classov s uglublennym. Izucheniem matematiki [Geometry: for the 9–10 classes: textbook for schools and classes with an advanced course of study in mathematics]. Moscow: Prosveshchenie. Aleksandrov, Aleksandr D., Aleksei L. Verner, and Valerii I. Ryzhik. 1984. Probnyi uchebnik dlya 6 klassa srednei shkoly [Trial textbook for class 6 of the secondary school]. Moscow: Prosveshchenie. Aleksandrov, Aleksandr D. 1986. Dialektika geometrii [The dialectics of geometry]. Matematika v Shkole 1: 12–19. Aleksandrov, Aleksandr D. 1988. Problemy nauki i pozitsiya uchenogo [Problems in science and the position of the scientist]. Leningrad: Nauka. Aleksandrov, Aleksandr D. 1989. O sushchnosti universiteta [On the essence of the university]. Vestnik Vysshey. Shkoly 5: 8–18. Aleksandrov, Aleksandr D. 1990. Geometriya [Geometry]. Moscow: Nauka. Aleksandrov, Aleksandr D. 1994. Geometry as an element of culture. In Proceedings of the 7th International Congress on Mathematical Education. Vol. 2 Selected Lectures, eds. Claude Gaulin, Bernard R. Hodgson, David H. Wheeler, and John C. Egsgard, 365–368. Sainte-Foy: Les Presses de l’Université de Laval.

Photo Source: https://www.mathedu.ru/indexes/authors/aleksandrov_a_d/

Aksel Frederik Andersen

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11.3 Aksel Frederik Andersen (Fodby, 1891 – Gentofte, 1972): Member of the Executive Committee 1952–1954 Henrik Kragh Sørensen

Biography Aksel Frederik Andersen was born on 10 February 1891 in Fodby (Denmark), the son of a farmer on Sealand (near Næstved5). He graduated (artium) from Sorø Akademi in 1909 and began studying at Københavns Universitet (the University of Copenhagen). From an early age, he was interested in mathematics. Andersen won a gold medal for solving a mathematical problem posed by the University of Copenhagen in 1913. The paper dealt with the behaviour of power series on the circle of convergence. In 1915, Andersen obtained his master’s degree; and in 1922, he defended his doctorate on Cesàro summability of infinite series (Andersen 1921). While still a student at the University, Andersen began his career as a teacher of mathematics at a Copenhagen gymnasium in 1912 and kept that position as a

 The following biographical facts are based on Bang (1979).

5

H. K. Sørensen (*) University of Copenhagen, København, Denmark e-mail: [email protected]

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supplement to his other engagements until 1934. After his graduation from the University, he became an assistant at the Polyteknisk Læreanstalt (Polytechnical College) in Copenhagen in 1916 and was promoted to associate professor (lektor) in 1923. From 1926 to 1930, he lectured at the Kongelig Veterinær og Landbohøjskole (Royal Veterinary and Agricultural College), before returning to the Polytechnic College as professor in 1930. Until his retirement in 1960, Andersen lectured on mathematics for the students of engineering and was responsible for the preparatory course on mathematics for future applicants to the Polytechnic College. Andersen was active in the Dansk Matematisk Forening (Danish Mathematical Society) giving a number of lectures, some of which were published. He also served as the secretary of the society for a few years after 1917 (see Ramskov 1995, pp. 228–229, and DMF 1973). Furthermore, Andersen partook in the organization of the Scandinavian congresses of mathematicians, and in 1932 he attended the International Congress of Mathematicians (ICM) in Zurich. Andersen’s research interests on Cesàro summability were in line with those of the mathematical group that was being established in the 1910s and 1920s in Copenhagen around Harald Bohr (1887–1951) and Niels Erik Nørlund (1885–1981). Infinite series, Cesàro summability and, more broadly, function theory remained Andersen’s main areas of mathematical research throughout his career, culminating in an article published in the proceedings of the London Mathematical Society (Andersen 1958). Andersen died on 18 February 1972 in Gentofte (Denmark) at the age of 81.

Contribution to Mathematics Education Andersen had a lifelong interest in the secondary and tertiary mathematics education and became an experienced teacher and author of influential textbooks. Through these occupations, his interest in the theory of functions also came to exert a substantial influence on the teaching of mathematics in Denmark. When just 21 years old and still a student at the University, Andersen began reviewing textbooks for the Danish journal of mathematics Nyt Tidsskrift for Matematik (New Journal of Mathematics), which in 1918 was renamed Matematisk Tidsskrift (Journal of Mathematics). Focused on rigorous analysis, Andersen was concerned with presenting mathematics with the utmost precision: With his calm disposition, Andersen was an eminent teacher who was able to provide his students with a firm foundation for mathematics while distancing himself from extreme axiomatic tendencies. (Bang 1979, p. 152)6

 The original text is: “Med sin rolige besindige optræden var. A. en udmærket lærer som hos sine elever evnede at lægge et fast grundlag for matematikken samtidig med at han tog afstand fra yderliggående aksiomatiske tendenser”. 6

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In a review of a new edition of one of the leading textbooks of mathematics for the gymnasium in 1935, Andersen elaborated on his views concerning axiomatics and the teaching in the gymnasium: When one notices how the axiomatic approach functions in this book, one is led directly to reflect upon whether it is at all appropriate to introduce axiomatics into the teaching in the gymnasium. According to my view, its inclusion cannot be dismissed right away. … Of course, one can settle for a looser description without complete logical coherence … that has been done previously and in other places … and try to be satisfied with what one accomplishes. But I certainly understand if one wants something more. And I consider it both possible and useful to give a logically satisfactory foundation for the calculus in the gymnasium. (Andersen 1935, p. 64)7

However, the axiomatics had to be presented in a way that connected it to the students’ previously acquired understanding. Andersen expressed himself similarly on the role of geometric intuition: I am well aware that in recent years a tendency has existed towards pure analytic geometry based on a completely arithmetic foundation and without appeal to concepts from intuition - but I doubt the appropriateness of such an approach. I think one is better served by, as is customary, connecting analytic geometry to the intuitive contents of the geometry that the students are already familiar with, even if something of the logical coherence is thereby lost. (Andersen 1935, p. 65)8

In 1934, the year before writing the previous review, Andersen was dissatisfied with the teaching of the foundations of functions in the Danish gymnasium and wrote an article for the Matematisk Tidsskrift addressed to the secondary schools (Andersen 1934). A colleague, Johannes Mollerup (1872–1937), reviewed it for the Jahrbuch über die Fortschritte der Mathematik (JFM): Because the instruction in mathematics in the Danish high schools is treating the finer parts of analysis with increasing care, the author - who as professor at the Polytechnical College is to bring the students of the gymnasium further – has taken it upon himself to present as simply as possible theorems concerning the exponential function (ex, ax), the logarithmic function (lx, logx), and the power function xn. (Mollerup, JFM 60.0861.05)9

 The original text is: “Naar man ser, hvorledes den aksiomatiske Fremgangsmaade virker i denne Bog, kommer man let til at reflektere over, hvorvidt det overhovedet er rimeligt at inddrage Aksiomatik i Gymnasiets Undervisning. Og efter min Formening kan man ikke uden videre afvise Betimeligheden heraf. […] Ganske vist kan man lade sig nøje med en løsere Beskrivelse uden fuldstændig logisk Sammenhæng, det kan man, det har man jo gjort tidligere og mange andre Steder end her i Landet — og søgt at være tilfreds med, hvad man opnaaede. Men jeg forstaar saa godt, om man vil noget mere. Og jeg anser det ogsaa baade for muligt og nyttigt at give et i logisk Henseende fuldt tilfredsstillende Grundlag for Infinitesimalregningen i Gymnasiet”. 8  The original text is: “Nu ved jeg vel, at der i de senere Aar har været en Tendens henimod den rene analytiske Geome- tri, der bygges helt talmæssigt op uden i ringeste Maade at hente sine Begreber fra Anskuelsen, men jeg tvivler på Hensigtsmæssigheden heraf. Jeg tror, man staar sig ved, som hidtil, at knytte den analytiske Geometri til det anskuelige Indhold af den Geometri, Eleverne allerede har levet sig ind i, selv om der ogsaa skulde gaa en Smule af den logiske Sammenhæng tabt derved”. 9  The original text is: “Da der Mathematikunterricht der dänischen Gymnasien sich stets s­ orgfältiger mit den feineren Teilen der Analysis beschäftigt, hat der Verf., der als Professor der Technischen 7

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From 1916, Mollerup, together with Bohr, taught mathematical analysis at the Polytechnic College based on a new textbook that they wrote together during the years 1915–1918. This new textbook in four volumes was a considerable improvement over previous systems and continued to be used in subsequent editions at the universities in Copenhagen and Aarhus until 1959, and even as late as 1973 at the Technical University (formerly the Polytechnic College; see Jørsboe 2000, p. 32). The first major reworking of the textbook was undertaken after Mollerup’s death by the new professors and teachers at Polytechnical College, of which Andersen was one. Together with Bohr and Richard Petersen (1894–1968), Andersen revised the system into Lærebog i matematisk Analyse (Andersen, Bohr and Petersen 1945–1949), bringing it up to date with changes in the curriculum while insisting on the precision and rigour that had marked those textbooks out from the beginning (see also Ramskov 1995, pp. 176–194, for a discussion of the textbook system and the changes it underwent at the hands of Andersen and Petersen). In an obituary of Bohr, Andersen commented upon the great impact of the textbook system throughout the Danish educational system in mathematics: For the Danish instruction in mathematics, the system “Bohr and Mollerup” was a milestone. Although the modern views were not unknown in this country, it was through this book that they managed to break through. The book did not only have a thorough impact on the teaching at the two institutions for which it was written [the Polytechnic College and the University of Copenhagen] but also sparked a renewal of the instruction in the gymnasium where the old traditions were fading away. (Ramskov 1995, p. 193)10

In the editing of the textbook, Andersen’s insistence on precision and rigour seems to sometimes have been rather pedantic and almost counterproductive. In a letter from Bohr to Børge Jessen (1907–1993), Bohr describes the work of the editors (Petersen and Andersen) as follows: I feel that Petersen - who has contributed 99% of the work +6% help from me - 5% anti-help from Andersen - deserves some acknowledgement after his long struggle with this book. (Bohr to Jessen, 28 April 1949, quoted in Ramskov 1995, p.  189; see also Jørsboe 2000, p. 35)11

Andersen’s interest in and concern for the teaching of mathematics in the gymnasium led him to write a textbook for secondary education together with Poul Mogensen (1895–1980). It appeared in its first edition in 1937–1940 and was revised as Lærebog i Matematik for Gymnasiets matematisk-­naturvidenskabelige Hochschule die Schüler der Gymnasien weiterführt, sich vorgenommen, die Sätze von der Exponentialfunction (ex, ax), von der Logarithmusfunktion (lx, logx) und von der Potenzfunktion xn so einfach wie nur möglich zu begründen”. 10  The original text is: “I den hjemlige matematikundervisning blev »Bohr og Mollerup« en ­skelsættende bog. Omend de moderne synspunkter ikke var. ukendte herhjemme, var. det først med denne bog, de formåede at trænge igennem. Bogen fik ikke alene gennemgribende betydning for undervisningen på de to institutioner, for hvilke den var. skrevet, men havde også en fornyende virkning på undervisningen i gymnasiet, hvor de gamle traditioner var. ved at miste deres glans”. 11  The original text is: “Jeg synes, at R. P., som har båret de 99% af arbejdet +6% hjælp fra min side −5% antihjælp fra A. F.’s side, fortjener lidt opmuntring efter sit mangeårige slid med bogen”.

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Linie (Textbook in Mathematics for the Upper Secondary School’s MathematicalNatural Sciences Line, Andersen and Mogensen 1942). As was to be expected of Andersen, the infinitesimal analysis was presented with some emphasis on rigour. In particular, the treatment of the elementary functions (powers, logarithms and exponentials) closely followed his approach of 1934 mentioned above. This required the treatment of infinite series, which was not customary in the gymnasium, but according to a review, its clarity and comprehensible nature attracted the interest of the students (see Duerlund 1942, pp. 73–74). The textbook was widely used until a reform of the curriculum in the gymnasium around 1960. Thus, by teaching in the gymnasium, teaching the introductory courses at the Polytechnic College and writing or editing influential textbooks for both the gymnasium and the Polytechnic College, Andersen was highly influential in forming the instruction in mathematics in Denmark in the period 1930–1960. After that in 1952 the International Commission on the Teaching of Mathematics was reconstituted in Rome, he was appointed as a member of the Executive Committee until 1954.

Sources Andersen, Aksel Frederik. 1921. Studier over Cesàro’s Summabilitetsmetode med særlig Anvendelse overfor Potensrækkernes Teori [Studies of Cesàro’s Method of Summability with Special Applications to the Theory of Power Series]. København: Jul. Gjellerups Forlag. Andersen, Aksel Frederik. 1934. Eksponential – og Logaritmefunktioner [Exponential and logarithmic functions]. Matematisk Tidsskrift, A 38–64. Andersen, Aksel Frederik. 1958. On the Extensions within the Theory of Cesàro Summability of a Classical Convergence theorem of Dedekind. Proceedings of the London Mathematical Society s. 3, 8: 1–52. Bang, Thøger. 1979. Aksel Frederik Andersen (1891–1972). Dansk Biografisk Leksikon [Danish Biographical Dictionary], 3rd Edn. (16 Vols.) Vol. 1, 152. Gyldendahl: København. DMF. 1973. Dansk Matematisk Forening 1923–1973 [Danish Mathematical Society, 1923–1973]. København: Dansk Matematisk Forening. Duerlund, Sigurd. 1942. Review of Aksel F. Andersen and Poul Mogensen, Lærebog i Matematik for Gymnasiets matematisk-naturvidenskabelige Linie I-IV [Textbook of mathematics for the mathematico-scientific line of the Gymnasium, Vols. I-IV]. København, Gyldendahl 1937–1940. Matematisk Tidsskrift A: 71–75. Jørsboe, Ole Groth. 2000. Undervisningen i Matematik på DTU 1829–2000 [The Teaching of Mathematics at DTU (Danish Technical University) 1829–2000]. Lyngby: Institut for Matematik. Bang, Thøger. 1979. Aksel Frederik Andersen (1891–1972). Dansk Biografisk Leksikon [Danish Biographical Dictionary], 3rd Edn. (16 Vols.) Vol. 1, 152. Gyldendahl: København. DMF. 1973. Dansk Matematisk Forening 1923–1973 [Danish Mathematical Society, 1923–1973]. København: Dansk Matematisk Forening. Ramskov, Kurt. 1995. Matematikeren Harald Bohr [The Mathematician Harald Bohr]. Aarhus: Institut for de eksakte videnskabers historie, Det naturvidenskabelige Fakultet, Licentiatafhandling.

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Publications Related to Mathematics Education Andersen, Aksel F. 1935. Review of Pihl, Kristensen and Rubinstein: Lærebog i Matematik for det matematisk-naturvidenskabelige Gymnasium I [Textbook in Mathematics for the Mathematicoscientific Gymnasium, Vol. I], Gyldendahl 1935, 220 pages, 59–66. Andersen, Aksel F., Harald Bohr, and Richard Petersen. 1945–1949. Lærebog i matematisk Analyse [Textbook in Mathematical Analysis] (4 Vols.). København: Jul. Gjellerups Forlag. Andersen, Aksel F. and Poul Mogensen. 1942. Lærebog i Matematik for Gymnasiets matematisknaturvidenskabelige Linie [Textbook of Mathematics for the Mathematico-Scientific Line of the Gymnasium] (4 Vols.), København, Gyldendalske Boghandel: Nordisk Forlag. First edition 1937–1940.

Photo Source: Archives of Fodby Parish.

Guido Ascoli

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11.4 Guido Ascoli (Livorno, 1887 – Turin, 1957): Treasurer of the Executive Committee 1952–1954 Livia Giacardi

Biography Guido Ascoli was born in Livorno on 12 December 1887 and studied at the University of Pisa, where he earned his degree in 1907 with a dissertation on the singularities of analytic functions, with Luigi Bianchi as his supervisor. After being awarded a Lavagna grant for a year of postgraduate studies in Pisa, he returned to Livorno for family reasons. His long career of teaching in high schools (the Italian scuole medie superiori) lasted from 1909 to 1932, with a 3-year interruption (1916–1918) for World War I.  As frequently happened at the time, he taught in schools throughout Italy: Spoleto, Cagliari, Caserta, Florence, Parma and finally Turin. During these years, Ascoli was almost completely absorbed with secondary teaching. In fact, in 1913 he published a book for secondary schools, Complementi di Geometria per gli Istituti Tecnici, and in 1924 a textbook of mathematical

L. Giacardi (*) University of Turin, Turin, Italy e-mail: [email protected]

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analysis for scientific high schools, Lezioni elementari di analisi matematica ad uso dei licei scientifici. After his arrival in Turin in 1920, contacts with the lively mathematical milieu of that city acted as a powerful stimulus for Ascoli to go back to research, above all in the field of mathematical analysis. Between 1926 and 1930, he published about ten important scientific works regarding partial differential equations, which led, in 1930, to his being one of three winners of a competition for a university chair. In 1932, he was called to Pisa to the chair of analysis, and in 1934, he transferred to the University of Milan, where he stayed from 1934 to 1938 and 1945 to 1948, with the long interruption due to the infamous anti-Jewish racial laws. In autumn 1948, he arrived at the University of Turin to hold the chair of “Complementary Mathematics” (Matematiche complementari), mainly aimed at the preparation of mathematics teachers of secondary schools; in his course, he was able to put to good use his own long experience in secondary education as well as his profound knowledge of mathematics. At the same time, he was assigned to teach courses in advanced analysis (from 1948/1949 to 1950/1951) and in the theory of functions (from 1951/1952 to 1954/1955). Guido Ascoli’s scientific production consists of only 82 works, but they are of excellent quality and generally concern questions that are fundamental for mathematical analysis. There are also interesting contributions to geometry and to education. Two among the works in analysis are particularly significant: the 1929 essay regarding Laplace’s equation in hyperbolic space, which provides significant results in the context of partial differential equations of mixed type studied in those years by Francesco Tricomi, and the 1935 monograph, in collaboration with Pietro Burgatti and Georges Giraud, on partial differential equations of elliptic and parabolic type, which received an award from the Scuola Normale Superiore in Pisa and remained a point of reference for scholars for a very long time. Other important papers are dedicated to examining the asymptotic behaviour of solutions, in a given domain, of ordinary or partial differential equations, research that was important for the advancement of physics, and in particular, for fluid dynamics. Also noteworthy are the works that concern the theory of abstract spaces (today called normed spaces) and the applications of this to linear functional analysis. One fruit of the lessons for the course of advanced analysis of 1950/1951 was the monograph Trasformazione di Laplace (1951), which explains the theory and the most interesting applications. Ascoli received numerous recognitions and honours. He received the Cross of War; he was a corresponding member of the Accademia dei Lincei, a member of Turin’s Accademia delle Scienze and of the Istituto Lombardo di Scienze e Lettere and president of the Turin section of the Associazione Mathesis (the Italian association of mathematics teachers), from 1950 until his death, and of the Italian Commission for Mathematics Teaching from 1954 to 1957. He died in Turin on 10 May 1957.

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Contribution to Mathematics Education Ascoli was treasurer of ICMI for the 1952–1954 period and member at large from 1955 to 1957 (L’Enseignement Mathématique 40, 1951–1954: 72; s. 2, 1 1955: 197). He retained a lifelong interest in educational aspects of mathematics and in problems of methodology related to its teaching. His student, Giovanni Zin, wrote: His great critical spirit was due to the character of his formation. If on the one hand it was always attentive to the latest progress in analysis, on the other hand it had its roots in the classic works of the great masters, and found nourishment in the history of mathematics, in logic, and in methodology, arriving at the great problems of science, that is, what are the place and value of mathematics in general, and of analysis in particular. (Zin 1956–1957, p. 14)12

The high value of mathematics was underlined by Ascoli in his talk on the role of mathematics and mathematicians in contemporary society during the International Congress of Mathematicians in Amsterdam in 1954: We cannot consider mathematics only as knowledge or as a tool; it also has an educational value, which gives it a place even in a humanist culture. … We think that the spread of the mathematical mentality in our time must have a beneficial influence on the law, on administrative language, on the conduct of discussions, on the spirit of tolerance. It is fair, in short, to claim for our science a role that is also a moral and human. (Ascoli 1957, p. 543)13

Ascoli’s talk originated from the enquiry promoted by ICMI on “The Part of Mathematics and the Mathematician in Contemporary Life” (L’Enseignement Mathématique 40, 1951–1954: 82). This was certainly a theme on which Ascoli reflected; in fact, in 1949, he had taught a course on the theory of analytic functions (Ascoli 1949) for the “Group for the study of physics” at the invitation of the Montecatini company for the mining and chemical industry. In the choice of the topics to be dealt with, he had considered what could be useful for the development of the mathematical physics, as well as those aspects related to research in the field of probability and statistics, that could be helpful for industry; in his lessons, he had also given space to the practical rules of thumb that often replace the theoretical solutions. In order to prepare for his speech in Amsterdam, Ascoli, with the help of Pietro Buzano, formulated a questionnaire that he sent to engineers, chemists, physicists,

 The original text is: “Il suo grande spirito critico si connetteva con il carattere della sua cultura. Essa se da una parte si rivolgeva ai sempre nuovi progressi dell’Analisi, dall’altra affondava le radici nelle trattazioni classiche dei grandi maestri, attingeva poi altro alimento alla storia delle matematiche, alla logica, alla metodologia, spingendosi fino ai grandi problemi della scienza, quali la posizione ed. il valore delle matematiche e dell’Analisi in particolare”. 13  The original text is: “On ne peut pas considérer la Mathématique seulement comme connaissance ou comme outil; elle a aussi une valeur éducative, qui lui donne une place même dans une culture humaniste. … Nous pensons que la diffusion de la mentalité mathématique dans notre temps doit avoir une influence bienfaisante sur le droit, sur le langage administratif, sur la conduite des discussions, sur l’esprit de tolérance. Il est juste, en somme, de revendiquer à notre science aussi un rôle moral et humain”. 12

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actuaries, biologists and mathematicians from all over Italy, including with it a letter in which he briefly explained the objectives of the enquiry.14 The last of the questions concerned the teaching of mathematics in high schools and asked if it was appropriate to the new needs of society. In the replies he received, the request to give more space to applications was very frequent, but there were also some extremely critical judgments, such as that of the physicist Eligio Perucca, who wrote: How many young people just leaving school understand the substance of mathematics? How many professors make any effort to teach it? How many professors reduce it to an arid game of logic, when the teacher should first of all take care to show how a concrete problem can be associated with a mathematical problem? (Ascoli Family Private Archives)15

It was precisely his belief in the importance of the applications that drove Ascoli to support the combined teaching of mathematics and physics in the Italian secondary schools at the time of the Giovanni Gentile Reform (1923). While “regretting the illogical, anti-juridical and vexatious way in which this measure had been applied” (Ascoli 1955b, p. 75), he had recognized in it the ideal antidote to the tendency to present a fossilized knowledge, to “re-mix old things or logical trifles”16 (ibidem) and to neglect the applications of mathematics. However, his personal experiences as a teacher, the results of the diploma exams (esami di maturità) and those of competitions for teaching positions led him to change his opinion, and in the report presented at the meeting of the Scientific Commission of the Italian Mathematical Union on 8 January 1955, he stated: Everyone can know what the results of mathematics teaching are. Critical sense is poor, as shown by the enormous diffusion of textbooks full of nonsense and errors, skilled only in easily glossing over the actual difficulties (but they have many exercises!); geometric culture is very scarce, analytic culture is uncertain and there is no interest in the foundations of mathematics … Then, having established that two cultures do not make a culture, there is nothing to do except go back to the old system. (Ascoli 1955b, p. 76)17

Therefore, Ascoli hoped that the teaching of mathematics would be again separated from that of physics in secondary schools, and on 17 April 1955, during the first official meeting of the Italian Commission for Mathematics Teaching, he proposed as a theme for reflection this question, which was also closely connected with the problem of teacher training (see Verbale 1955, pp. 301–302).  All the documents concerning this enquiry are kept in the Ascoli Family Private Archives.  The original text is: “Quanti giovani appena licenziati hanno capito la sostanza della matema­ tica? Quanti professori fanno qualche sforzo per insegnarla? Quanti professori la riducono a un giuoco arido di logica mentre dovrebbe anzitutto stare a cuore all’insegnante il mostrare come un problema concreto possa essere associato a un problema matematico?”. 16  The original text is: “deplorando il modo illogico, antigiuridico e vessatorio con cui il provvedimento era stato applicato”; “rimasticare vecchie cose o quisquilie logiche”. 17  The original text is: “Quali siano poi i risultati per l’insegnamento della matematica a tutti è dato di conoscere. Scarso il senso critico come mostra la enorme diffusione di libri di testo infarciti di non sensi ed. errori, abilissimi solo nello scivolare con disinvoltura sulle difficoltà effettive (ma hanno tanti esercizi!); scarsissima la cultura geometrica, incerta quella analitica e nessun interesse per le questioni sui fondamenti della matematica … Constatato allora che con due inculture non si fa una cultura, non resta che tornare all’antico”. 14 15

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Ascoli’s attention to educational aspects of mathematics can be clearly seen in the school textbooks that he wrote. For example, in the textbook of mathematical analysis, Lezioni elementari di analisi matematica ad uso dei licei scientifici (1924), written for the scientific secondary school (liceo scientifico) created in Italy by the Giovanni Gentile Reform, Ascoli does not limit himself to slavishly following the guidelines for university courses, but attempts to “discern, in the admirable edifice of concepts and results, that small bit that is essential in the very first study from what would be destined to remain a lifeless and inexpressive knowledge; to present to young people a simple and harmonic organism of the fundamental ideas that contribute effectively towards their intellectual formation”18 (Ascoli 1924, Prefazione, p. V). This is the reason why he introduces the concept of integral before that of derivative, because it is simpler and more intuitive when the point of departure is the calculation of areas. When Ascoli moved from teaching in secondary schools to teaching at university, he applied himself with true dedication to the training of secondary school teachers, not only by participating in the Associazione Mathesis activities, but also by ideating and creating a “course of mathematical culture”. This was a post-degree professional development course aimed at new graduates who had to participate in the competitions for teaching positions in secondary schools. The course lasted 3 hours a week and comprised lessons in mathematics aimed at completing or refreshing the basic preparation and to accustoming future teachers to look at elementary mathematics from a higher standpoint, as well as to take into consideration the history of mathematics; these lessons were flanked by exercises and discussions on the topics that would be addressed in the competitions. Ascoli’s lessons are collected in the volume Lezioni di Matematiche complementari (1952, 2nd ed. 1954) and in the books collecting the themes discussed during the “course of mathematical culture”, Svolgimento dei temi assegnati nel corso di cultura matematica dell’Università di Torino (1953, 1955, 1957). In the Lezioni di Matematiche complementari, Ascoli examines three topics: elements of number theory, integral rational functions and algebraic equations. He pays particular attention to those aspects that might be useful in the practice of teaching, in the attempt to mend the break between the knowledge of elementary and more advanced mathematics. The other methodological assumptions that animate his teaching are the importance of offering young people “a wide choice of complementary subjects, venturing as far as the field of physics and beyond, so as to encourage the collaboration between mathematicians and other specialists”19 (Ascoli 1955b, p.  77) and the attention to questions regarding the foundations, instilling respect not so much for rigour in the details, but above all for rigour in the logical ordering of the theories (Ascoli 1913, p. VI).  The original text is: “sceverare nel mirabile edificio di concetti e di risultati quel poco che è essenziale in un primissimo studio da ciò che sarebbe destinato a rimanere cognizione morta e inespressiva; di presentare ai giovani un organismo semplice e armonico di idee fondamentali che operino efficacemente alla loro formazione intellettuale”. 19  The original text is: “una larga scelta di materie complementari, spingendosi anche nel campo fisico e oltre, in modo da favorire la collaborazione fra matematici e altri specialisti”. 18

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Sources Ascoli Family Private Archives. Ascoli, Guido. 1929. Sull’equazione di Laplace dello spazio iperbolico. Mathematische Zeitschrift, 31: 45–96. Ascoli, Guido, Pietro Burgatti, and Georges Giraud. 1936. Equazioni alle derivate parziali dei tipi ellittico e parabolico. Firenze: G. Sansoni. Ascoli, Guido. 1949. Lezioni sulla teoria delle funzioni analitiche. Milano: Tipo-Lito P. Pasquetto. Ascoli, Guido. 1951. Trasformazione di Laplace. Torino: Gheroni. Buzano, Pietro. 1958. L’opera di Guido Ascoli nella Scuola e nella Mathesis, Commemorazione tenuta alla Sezione torinese Mathesis il 19 dicembre 1957, 3–12. Torino: Gheroni. Giacardi, Livia. 2009. The Italian contribution to the International Commission on Mathematical Instruction from its founding to the 1950s. In Dig where you stand. Proceedings of the Conference on On-going Research in the History of Mathematics Education (Garðabær, Iceland, June 21–23, 2009), eds. Kristín Bjarnadóttir, Fulvia Furinghetti, and Gert Schubring, 47–64. Reykjavik: University of Iceland. Picone, Mauro. 1958. Commemorazione del Socio Guido Ascoli. Atti della Accademia Nazionale dei Lincei. Rendiconti 24: 614–625 (with the list of publications). Skof, Fulvia. 1999. Guido Ascoli. In La Facoltà di Scienze Matematiche Fisiche Naturali di Torino, 1848–1998, Vol. II, ed. C. Silvia Roero, 575–578. Torino: Deputazione subalpina di storia patria. Tricomi, Francesco. 1957. Guido Ascoli. Bollettino della Unione Matematica Italiana s. 3, 12: 346–350. Tricomi, Francesco. 1957–58. Guido Ascoli (1887–1957). Atti dell’Accademia delle Scienze di Torino, Classe di Scienze Fisiche, Matematiche e Naturali s. 4, 92: 180–184. Verbale. 1955. Verbale della 1a riunione della Commissione Italiana per l’insegnamento matematico indetta per il 17 aprile in Bologna. Bollettino della Unione Matematica Italiana (3), X: 301–302. Virgopia, Nicola. 1962. Ascoli Guido. In Dizionario Biografico degli Italiani 4: 384–386. Roma: Istituto della Enciclopedia Italiana. Zin, Giovanni. 1956–57. Ricordo del prof. Guido Ascoli. Rendiconti del Seminario Matematico, Università e Politecnico di Torino 16: 11–35 (with an annotated list of publications).

Publications Related to Mathematics Education Ascoli, Guido. 1910a. Note sulla teoria dei poliedri. Periodico di matematica. Supplemento 13: 51–53. Ascoli, Guido. 1910b. Una questione algebrico-geometrica. Bollettino di Matematica 11: 11. Ascoli, Guido. 1913. Complementi di Geometria per gli Istituti Tecnici. Livorno: Giusti. Ascoli, Guido. 1914. Note di Geometria elementare. Periodico di matematica. Supplemento 17: 5–6. Ascoli, Guido. 1915. Sul metodo differenziale per la ricerca dei massimi e minimi nell’insegnamento medio. Periodico di Matematica 30: 142–144. Ascoli, Guido. 1924. Lezioni elementari di analisi matematica ad uso dei licei scientifici. Torino: G. Petrini. Ascoli, Guido. 1934. Sul principio di identità dei polinomi. Periodico di Matematiche s. 4, 14: 30–31 (abstract). Ascoli, Guido. 1952. Lezioni di Matematiche complementari. Torino: Gheroni (II ed. 1954).

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Ascoli, Guido. 1953. Corso di cultura matematica dell’Università di Torino. Svolgimento dei temi assegnati negli anni accademici 1949–50, 1950–51. Torino: Gheroni (in collaboration with Elda Valabrega Gibellato), II ed. 1954. Ascoli, Guido. 1955a. Corso di cultura matematica dell’Università di Torino. Svolgimento dei temi assegnati negli anni accademici 1951–52, 1952–53. Torino: Gheroni (in collaboration with Elda Valabrega Gibellato). Ascoli, Guido. 1955b. Sulla preparazione degli insegnanti delle scuole secondarie e sull’abbinamento della Matematica e della Fisica. Bollettino della Unione Matematica Italiana 3, 10: 75–77. Ascoli, Guido. 1955c. La funzione della matematica e del matematico nella vita contemporanea, L’Enseignement Mathématique s. 2, 1: 179–187 (A summary is found in Proceedings of the International Congress of Mathematicians 1954 Amsterdam September 2  – September 9. Groningen, Noordhoff – Amsterdam: North-Holland Publishing Co. I, 1957, 544). Ascoli, Guido. 1955d. Lezioni di Algebra. Torino: Gheroni. (This is the title that appeared on the third edition of Ascoli 1952). Ascoli, Guido. 1957. Corso di cultura matematica dell’Università di Torino. Svolgimento dei temi assegnati negli anni accademici 1953–54, 1954–55. Torino: Gheroni (in collaboration with Elda Valabrega Gibellato).

Photo Courtesy of the University of Turin.

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Edward Griffith Begle

11.5 Edward Griffith Begle (Saginaw, 1914 – Palo Alto, 1978): Member of the Executive Committee 1975–1978 Jeremy Kilpatrick

Biography Edward Griffith Begle was born in Saginaw, Michigan, on 27 November 1914. He attended the University of Michigan, where he studied under the topologist Raymond L.  Wilder, who had been the first doctoral student at the University of Texas of Robert Lee Moore, the founder of topology in the United States and the inventor of the Moore method for teaching mathematical proof. Begle received his AB in mathematics from Michigan in 1936 and his MA in 1937. Not surprisingly, given that Wilder was his mentor, Begle pursued the study of topology, going to Princeton University, where he obtained his PhD in 1940 under the direction of Solomon Lefschetz. Begle’s dissertation was on locally connected spaces and generalized manifolds. After teaching mathematics at Princeton during the 1940–1941 academic year and spending a year at Michigan as a National Research Council Fellow, Begle took a position in the mathematics department at Yale University in 1942 and remained there until 1961, rising through the ranks from instructor to associate professor.

J. Kilpatrick (1935–2022) University of Georgia, Athens, GA, USA

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While on the Yale faculty, he continued his research on generalized manifolds, publishing several important results such as a proof of the Vietoris-Begle mapping theorem. But he also took an interest in the improvement of undergraduate teaching, publishing an introductory calculus textbook in 1954. Unlike most such textbooks, Introductory Calculus, with Analytic Geometry is addressed to the students themselves and not to their instructors. It emphasizes the basic concepts of the subject, attempting to show students the logical structure of calculus and not simply to acquaint them with techniques. In the preface, Begle argues that students of science and engineering are best served by teaching for understanding: “The mathematical techniques employed [in science and engineering] have become so numerous and varied that no student can be expected to master more than a few of them without a framework of theory around which they can be organized” (p. v). In 1951, Begle was elected secretary of the American Mathematical Society, a position that brought him into the mainstream of the American mathematics community. He held the position for 6 years during a time when the AMS was coping with problems brought on by the post-war expansion of interest and activity in mathematics. Concern over the increased need for mathematicians led in 1958 to two conferences sponsored by the National Science Foundation, both of which called for a project to revamp the school mathematics curriculum. Begle was offered and accepted the post of director of the resulting project: the School Mathematics Study Group (SMSG). He served as director for the duration of the project from 1958 to 1972. In 1961, SMSG moved from Yale to Stanford University, where Begle was given a joint appointment as professor in the Department of Mathematics and the School of Education. Ultimately, however, his appointment was shifted entirely to the School of Education, a move that reflected his growing interest in educational topics such as curriculum development and evaluation, teacher education and research in mathematics education. He remained on the Stanford faculty until his death from emphysema on 2 March 1978. He was survived by his wife, six of his seven children and seven grandchildren. During his career, Begle was active in the National Council of Teachers of Mathematics, serving on its Board of Directors, Finance Committee and Research Advisory Committee. He was a member of the Committee on the Undergraduate Program of the Mathematical Association of America, chairman of the Conference Board of the Mathematical Sciences and trustee of the American Mathematical Society. In 1960, he was elected a fellow of the American Association for the Advancement of Science. In 1969, he received the Distinguished Service Award of the Mathematical Association of America, and in 1971, he received the Rosenberger Medal from the University of Chicago for his significant contributions to humanity.

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Contribution to Mathematics Education Begle served two terms on the United States Commission on Mathematical Instruction, from 1962 to 1966 and from 1970 to 1975. He chaired the commission from 1963 to 1966. He served on the Executive Committee of ICMI from 1975 to 1978. SMSG was the largest and most influential of the so-called new math curriculum projects in the United States. As its director, Begle organized teams of schoolteachers and mathematicians to write textbooks that better reflected and prepared students for the mathematics of the university, taking advantage of concepts such as set, relation and function to provide not only a more coherent discourse in the mathematics classroom but also a more meaningful structure for learning. Students would be drawn to mathematics by seeing how it fit together and, in particular, how the great ideas of modern mathematics brought order into a curriculum that was riddled with errors and imprecise language. Begle’s view of the school mathematics curriculum, as expressed in the first SMSG newsletter, reflected the goal of teaching for understanding that he had sought in his calculus textbook: Since no one can predict with certainty his future profession, much less foretell which mathematical skills will be required in the future by a given profession, it is important that mathematics be so taught that students will be able in later life to learn the new mathematical skills which the future will surely demand of many of them. (Quoted in Begle, 1968, p. 239)

SMSG prepared sample textbooks, including teachers editions, for all the grades from kindergarten to Grade 12. These textbooks included versions for some alternative courses such as an elementary algebra course to be given in 2 years instead of 1 and a coordinate geometry course to contrast with the more Euclidean version. Some of the textbooks were also published in Spanish editions. SMSG published and distributed extensive collections of books and films for teachers as well as a series of monographs for students, the New Mathematical Library. Begle was active at the international level. He was one of the guest speakers to the OEEC/OECE Seminar at Royaumont in 1959, which resulted in the publication of New Thinking in School Mathematics (1961, Paris: OEEC). His 1969 address at the First International Congress on Mathematics Education in Lyon, “The Role of Research in the Improvement of Mathematics Education”, was influential in putting research on the agenda of mathematics education internationally. He sought to give research an empirical footing: I see little hope for any further substantial improvements in mathematics education until we turn mathematics education into an experimental science. … We need to start with extensive, careful, empirical observations of mathematics learning. Any regularities noted in these observations will lead to the formulation of hypotheses. These hypotheses can then be checked against further observations, and refined and sharpened, and so on. To slight either the empirical observations or the theory building would be folly. They must be intertwined at all times. (Begle 1969, p. 242)

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After conducting the National Longitudinal Study of Mathematical Abilities – the largest study of factors affecting the learning of mathematics – Begle began a series of teaching units so that they could conduct studies in which a single dimension of instruction was varied and its effects measured. A posthumously published book (Begle 1979) contains his review of the research on many facets of mathematics education. Sources Begle, Edward G. 1942. Locally connected spaces and generalized manifolds. American Journal of Mathematics 64: 386–387. Begle, Edward G. 1950. The Vietoris mapping theorem for bicompact spaces. Annals of Mathematics 51: 534–543. Begle, Edward G. 1954. Introductory Calculus, with Analytic Geometry. New York: Holt, Rinehart & Winston. Demott, Benjamin. 1962 (Spring). The math wars. American Scholar 296–310. Reprinted in New curricula, ed. Robert W. Heath, 54–67. New York: Harper & Row, 1964. Goodman, Jr. George. 1978 (March 3). Prof. Edward G. Begle, chief proponent of ‘new math’. The New York Times, B2. A guide to the School Mathematics Study Group Records, 1958–1977. (n.d.). Retrieved January 30, 2020, from the Texas Archival Resources Online Web site: http://www.lib.utexas.edu/taro/ utcah/00284/cah-­00284.html Pettis, Billy J. 1969. Award for Distinguished Service to Professor Edward Griffith Begle, American Mathematical Monthly 76: 1–2. Wooten, William. 1965. SMSG: The Making of a Curriculum. New Haven, CT: Yale University Press. Iverson, William J., Elliot W. Eisner, and Richard G. Gross. 1978. Memorial resolution: Edward G.  Begle, 1914–1978. Retrieved from the Stanford Historical Society Web site January 30, 2020: https://web.archive.org/web/20060916225557, http://histsoc.stanford.edu/pdfmem/ BegleE.pdf Zelinka, Martha. 1978. Edward Griffith Begle, American Mathematical Monthly 85: 629–631.

Publications Related to Mathematics Education Begle, Edward G. 1958. The School Mathematics Study Group. Mathematics Teacher 51: 616–618. Begle, Edward G. 1961. Comments on a note on “variable”. Mathematics Teacher 54: 173–174. Begle, Edward G. 1962. Remarks on the memorandum “On the mathematics curriculum of the high school”. American Mathematical Monthly 69: 425–426; Mathematics Teacher 55: 74–75. Begle, Edward G. 1968. SMSG: The first decade. Mathematics Teacher 61: 239–245. Begle, Edward G. 1969. The role of research in the improvement of mathematics education. Educational Studies in Mathematics 2: 232–244. Begle, Edward G. (ed.). 1970. Mathematics Education (69th Yearbook of the National Society for the Study of Education, Part 1). Chicago: University of Chicago Press. Begle, Edward G. 1971. SMSG: Where we are today. In Confronting Curriculum Reform, ed. E. W. Eisner, 68–82. Boston: Little, Brown. Begle, Edward G. 1971. Time devoted to instruction and student achievement, Educational Studies in Mathematics 4: 220–224.

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Begle, Edward G. 1972. The Prediction of Mathematics Achievement (NSLMA Reports No. 27). Stanford, CA: School Mathematics Study Group. Begle, Edward G. 1973. Some lessons learned by SMSG. Mathematics Teacher 66: 207–214. Begle, Edward G. 1975. Basic skills in mathematics. In The NIE Conference on Basic Mathematical Skills and Learning: Vol. 1. Contributed position papers, 13–18. Washington, DC: National Institute of Education. Begle, Edward G. 1975. The Mathematics of the Elementary School. New York: McGraw-Hill. Begle, Edward G. 1979. Critical Variables in Mathematics Education: Findings from a Survey of the Empirical Literature. Washington, DC: Mathematical Association of America and National Council of Teachers of Mathematics.

Photo Source: Paul R. Halmos Photograph Collection, Archives of American Mathematics, Dolph Briscoe Center for American History, The University of Texas at Austin.

Ram Behari

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11.6 Ram Behari (Delhi, 1897 – Delhi, 1981): Member of the Executive Committee 1955–1958 Fulvia Furinghetti

Biography Ram Behari20 was born on 25 April 1897 in Delhi and died on 14 December 1981 in Delhi. He had his schooling at St. Stephen’s School and his college education mostly at St. Stephen’s College. In 1919 he obtained a MA in Applied Mathematics and in 1920 in Pure Mathematics at the Government College in Lahore in 1920. In 1925, he received the Central State scholarship for studying abroad. He chose to go to Cambridge (United Kingdom), where he joined the Sidney Sussex College. In 1927, he took the Mathematical Tripos of Cambridge University. Then he went to Trinity College in Dublin and obtained his PhD degree from the University of Dublin in 1932 under the supervision of Charles Henry Rowe. In 1945, he received

 The following biographical information is based on (Indian National Science Academy 1993; Prabook profiles; Fellows of the Indian Academy of Sciences; Monthly Notices …; Prakash 2008; Singal 1982). 20

F. Furinghetti (*) University of Genoa, Genoa, Italy e-mail: [email protected]

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his ScD at the University of Dublin (Ireland). After finishing his MA, he came back as a lecturer to St. Stephen’s College where he remained until 1947 as the head of the Department of Mathematics. In 1947, when the postgraduate department at the University of Delhi was set up, Behari was its first head. During his tenure, in 1957, the department also initiated an MA/MSc programme in Mathematical Statistics and the department was designated as the Department of Mathematics and Mathematical Statistics. Behari strongly contributed to the development of the high standard of the department. When he retired in 1962, he was appointed as founder director of Correspondence Courses started by Delhi University as a pilot project for the whole of India, which became a great success. In 1964–1966, he was the vice-chancellor of Jodhpur University. Behari can be credited with having started the tradition of research in differential geometry in India through his important contributions to the theory of ruled surfaces and rectilinear congruences, Riemannian spaces and generalized Riemannian spaces, complex manifolds and Einstein’s unified field theory. In 1942, he delivered a series of extension lectures on the differential geometry of ruled surfaces in threedimensional Euclidean space at the University of Lucknow. Based on these lectures, he produced the book Differential Geometry of Ruled Surfaces, a standard reference for anyone interested in the classical theory of surfaces (DeCicco 1949, p. 315). It is remarkable Behari’s interest in the valorization of the Indian mathematical culture. He was convenor of the Expert Committee in Mathematics for Scientific Terminology in Hindi appointed by the Government of India. Thanks to his strong involvement in this project, a dictionary of Hindi equivalents of mathematical terms was produced (Ministry of Education 1956). During the later years, with some of his students, he wrote articles on ancient Indian mathematics, including two survey articles on Al-Bõrønõ (Al-Biruni) and Bhåskaråcårya (Bhaskara). Among the many acknowledgements and honours of Behari, we remember that he was secretary of the Indian Mathematical Society in the years 1936–1943, treasurer in 1947–1953 and president in 1953–1957; in 1927 he was elected fellow of the Royal Astronomical Society and in 1941 Indian National Science Academy (INSA). In 1953, he represented the University of Delhi the Seventh Quinquennial Congress of the Universities of the Commonwealth held at Cambridge (United Kingdom). To honour Behari’s work at the University of Delhi, a Ram Behari Medal is awarded to the candidate who secures the highest percentage of marks among the successful candidates at the MA/MSc Examination in Mathematics obtaining first division. In 1958, Behari was invited to deliver lectures at the United States National Science Foundation Summer Institute at the University of Notre Dame and Oberlin College. He attended many national and international conferences; in particular, he was chairman of the differential geometry section at the International Congress of Mathematicians in 1958 (Singal 1982). In 1950, he was among the Indian delegates at the International Congress of Mathematicians in Cambridge (United States) (Proceedings of the International Congress of Mathematicians 1952, Vol. 1, p. 13 and p. 72). In 1958, he presented the short communication “Some Properties and Applications of Eisenhart’s Generalized Riemann Space” at the International

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Congress of Mathematicians in Edinburgh (Proceedings of the International Congress of Mathematicians 1960, p. xxxi).

Contribution to Mathematics Education In the field of education, the main efforts of Behari were addressed to the advanced level. He wrote more than ten books of undergraduate level, among them the treatise on differential geometry mentioned above. As convenor of the Review Committee in Mathematics appointed by the University Grants Commission, he produced a wide-ranging report containing, among other things, model syllabi in mathematics up to the highest level. This report influenced the restructuring of the syllabi at the undergraduate and postgraduate levels. In 1965, Behari was selected as the UNESCO observer from India to attend the Seventh Conference of the “International Council on Correspondence Education” held at Stockholm. Behari was the first member of the Executive Committee of ICMI (International Commission on Mathematical Instruction) from outside Europe and North America. His election happened in a period of reshaping the structure of ICMI. In the meeting held in The Hague (1 September 1954), the General Assembly of IMU (International Mathematical Union) decided to renew the composition of ICMI starting on 1 January 1955. This assembly appointed ten members at large of ICMI, among them Behari (CIEM 1955a). In the Geneva meeting (2 July 1955), Behari, not present, was elected a member of the Executive Committee of ICMI for the period 1955–1958 with 19 votes on 25 (CIEM 1955b). Behari’s election is a milestone in the history of the Commission since it paved the way for real internationalization. His presence in the Executive Committee fostered the promotion of connections between India and ICMI. On 22–28 February 1956, Komaravolu Chandrasekharan organized the Conference on Mathematical Education in South Asia at the Tata Institute of Fundamental Research of Bombay, held after the International Colloquium on Zeta Functions. Gustave Choquet, Hans Freudenthal, Thomas Arthur Alan Broadbent, Marshall Stone and Aleksandr Danilovich Aleksandrov were among the plenary speakers. This conference, organized in cooperation with UNESCO, “was the first of this kind and it [was] hoped that [it] may serve as a model for similar events in the future” (Report of the Executive … 1956, p. 9). The proceedings were published by one of the official journals of the Indian Mathematical Society (1956, The Mathematics Student 24: 1–183). Behari was a member of the panels on postgraduate teaching and research and on undergraduate instruction, created for preparing the Working Groups in the Conference. At the end of the conference, he supported the resolution proposing the constitution of a Committee for Mathematics in South Asia (The Mathematics Student, pp. 4–7). The mission and vision of Behari about mathematics teaching are well explained by the presidential address, presented before the Indian Mathematical Society at the University of Delhi in 1953. According to Behari, mathematics leads to a liberation of the mind, provides new ways of thinking and opens us to a different culture. He

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believes that a nation without modern mathematics is like a well-dressed man without character and that to neglect mathematics is to commit national suicide. He also gives some suggestions: raising mathematics teachers’ salaries, more time for study and standardization of symbols.

Sources Agrawal, Dharma Pal, Ram Behari, and Subbaiah Bidare V. Subbarayappa. 1975. Al-Bõrønõ. An Introduction to his life and writings on the Indian sciences. Indian Journal of History of Science 10(2): 98–110. Behari, Ram. 1940. A theorem on normal rectilinear congruences. Proceedings of the Indian Academy of Science Section A, 12: 205–207. Behari, Ram. 1946. The Differential Geometry of Ruled Surfaces. Lucknow University Studies, 18. Behari, Ram. 1977. Aryabhata as a mathematician. Indian Journal of History of Science 12(2): 147–49. Behari, Ram, and B. S. Jain. 1977. Some mathematical contributions of ancient Indian mathematicians as given in the Works of Bhåskaråcårya II (12th Cent. A.D.). Indian Journal of History of Science 12(1): 45–56. CIEM. 1955a. L’Enseignement Mathématique s. 2, 1: 193–202. CIEM. 1955b. Lettre circulaire du bureau de la Commission Internationale de l’Enseignement Mathématique aux dirigeants des sous-commissions nationales. L’Enseignement Mathématique s. 2, 1: 262–264. DeCicco, John. 1949. Book review: Ram Behari, The Differential Geometry of Ruled Surfaces. By Ram Behari. Bulletin of the American Mathematical Society 55(3): 313–315. Indian National Science Academy. 1993. Fellows of the Indian National Science Academy, 1935–1993. Biographical notes, 2: 654–655. Kapur, Jagat Narain. 1983. Some personal reminiscences of Professor Ram Behari. The Mathematics Student 51(1–4): 5–6. Prabook. Ram Behari mathematician, researcher. Retrieved on 10 March 2020 from https:// prabook.com/web/ram.behari/3751305 Prakash, Nirmala. 2008. Ram Behari. Biographical memoirs 34: 1–16. New Delhi: Indian National Science Academy. Ministry of Education. 1956. A provisional list of technical terms in Hindi mathematics II (Pub. n. 167 P), with foreword by Ram Behari. Delhi: Albion Press. Monthly Notices of the Royal Astronomical Society. 1927. 88(1): 1–2. Peak, Philip. 1955. Reviewed Work: “Presidential Address”, The Mathematics Student by Ram Behari. The Mathematics Teacher 48(5): 321. Proceedings of the International Congress of Mathematicians. 1952. Graves, Lawrence M., Paul A.  Smith, Einar Hille, and Oscar Zariski (eds.). Providence RI: American Mathematical Society. Proceedings of the International Congress of Mathematicians. 1960. Todd, J. A. (ed.). Cambridge: Cambridge University Press. Report of the Executive Committee to the national adhering organizations. Covering the period April 21, 1955  – May 31, 1956. 1956. Bulletin of the International Mathematical Union. Internationale Mathematische Nachrichten 47–48: 1–16. Singal, Mahendra Kumar. 1982. Professor Ram Behari founder President of the Delhi chapter of the Association. The Date with Mathematicians, 9–11. Delhi: Mathematical Association of India (Delhi chapter). Also published with the title “Professor Ram Behari: (1897-1981)” in The Mathematics Student 51(1–4): 1–4.

Ram Behari

Publications Related to Mathematics Education Behari, Ram. 1954. Presidential address. The Mathematics Student 22, January: 11–26.

Photo Source: Indian Academy of Sciences, Bengaluru.

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Heinrich Behnke

11.7 Heinrich Behnke (Hamburg, 1898 – Münster, 1979): Secretary 1952–1954, President 1955–1958, Vice-President 1959–1962, Member of the Executive Committee 1963–1970 Gert Schubring

Biography Heinrich Behnke was born on 9 October 1898  in Horn, near Hamburg, son of a manufacturing family. He passed his school years in Hamburg, first at a Vorschule and then at the Realgymnasium, renamed Oberrealschule St. Georg, hence without learning the classical languages. In the last three grades, from 1915 on, Behnke was fortunate to have as mathematics teacher a recent graduate from Göttingen who succeeded in interesting him in mathematics and physics, although until then he had shown no particular abilities. In March 1918, he passed the Abitur in Hamburg, despite the difficult conditions for living and learning at the time of World War I. Well prepared by his mathematics teacher, he began to study mathematics and physics at the University of Göttingen, in the summer term of 1918. Göttingen was G. Schubring (*) University of Bielefeld, Bielefeld, Germany e-mail: [email protected]

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then the centre for mathematics and Behnke was very lucky to have as his professors Edmund Landau, Constantin Caratheodory and Erich Hecke. The first term was interrupted by his being drafted into the army – happily, not to the front, but rather to a calculation service. He returned to Göttingen in November 1918, continuing his studies until the summer term 1919. His mentor Hecke then decided to accept the call to the newly created university in Hamburg, and Behnke followed him. There he became acquainted with Otto Toeplitz, then professor in Kiel, a relationship that would soon prove decisive. In the summer term 1922, Behnke presented his doctoral dissertation, Über analytische Funktionen und algebraische Zahlen, and passed the doctorate exam. In 1923, at a time of profound economic crisis, Behnke obtained a position as Assistent at Hamburg University. The next year, he passed the Habilitation exam, on function theory, and began lecturing in Hamburg. Extraordinarily early in his mathematical career, in 1927, he received the call for a full professorship at the University of Münster in Westphalia. Hesitating at first, due to the Catholic character of that region (Behnke was a Lutheran) and uncertainty about the perspectives of academic activities there, he decided to accept and succeeded in fact to develop mathematics at Münster into a highly acclaimed centre, in particular for function theory. He established close international relations, especially with France and there with Henri Cartan. Moreover, he became one of the editors of the Mathematische Annalen. A key work of the so-called funktionentheoretische school at Münster became the book, now a classic, Theorie der Funktionen mehrerer komplexer Veränderlichen (1934), elaborated jointly with his assistant Peter Thullen. After 1933, contrary to most of the other universities, no mathematicians were dismissed in Münster. Thullen, however, decided to emigrate to flee the Nazi system. Behnke was in danger, due to his liberal thinking and since his son from the first marriage was regarded as “half-Jewish”, but he managed to stay in his position and to maintain his international contacts. In the last period of World War II, some cooperation with Wilhelm Süss became established so that Behnke survived the end of the war in the Mathematisches Reichsinstitut in Oberwolfach. After Germany was liberated from Nazism, Behnke was appointed as dean at Münster, holding this position until 1949. In the post-war period, Behnke successfully continued to expand mathematics at Münster. At the same time, it was the period when he was most active in mathematics education. Retired in 1967, he remained a key figure in German and international mathematics and mathematics education.

Contribution to Mathematics Education Right from the beginning of his lecturing, Behnke invested much energy in effective teaching, in motivating students and in organizing tutoring activities. It proved enormously productive that Toeplitz, who had the same intention of turning the

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traditionally problematic lecture courses for beginners into effective studies, moved to Bonn in 1928, shortly after Behnke’s call to Münster. They cooperated closely in improving their teaching methods. For advanced students, they organized from 1928 on yearly joint seminars, called Sängerkriege. The great majority of Behnke’s mathematics students were future mathematics teachers at secondary schools. Contrary to other mathematicians, he did not marginalize this professional orientation; rather, he took it seriously and aimed at optimizing their studies in view of preparing to improved teaching abilities. In fact, until the establishment of the Diplom degree in 1942 for graduates, which should open careers in industry, the teaching profession had been basically the only professional career for graduates of mathematical studies. But even after the establishment of the two degrees, the Staatsexamen and the Diplom, and of two divergent careers, Behnke – and with him mathematics in Münster – remained committed to the Staatsexamen and to the formation of future teachers. As a consequence of his close cooperation with Toeplitz on teacher education, Behnke founded, in 1932, the journal Semesterberichte zur Pflege des Zusammenhangs von Universität und Schule aus den mathematischen Seminaren, which he edited together with Toeplitz. The title, promoting the connection between school and university, became emblematic for Behnke’s programme and for his emphasis on teacher education. In 1938, Behnke was forced to omit the name of Toeplitz in the subtitle and in 1940 the publication was interrupted, due to the war, but it resumed publication in 1950 and is still published today. After World War II, Behnke systematically expanded his agenda for mathematics teaching. Having revived the Semesterberichte journal, he now transformed the former joint yearly meetings of the mathematical seminars of Bonn and Münster into a yearly conference, officially called “Tagung zur Pflege des Zusammenhangs zwischen Höherer Schule und Universität”, but better known as the Pfingsttagung. And in 1951 he succeeded in promoting the establishment a ground-breaking institution, the “Seminar für Didaktik der Mathematik” – the first institutionalization of mathematics education at a German university. Seminare were traditional forms in German universities for specialized studies in given disciplines and for closer contact with the professors; they had been founded and extended since 1810 for the major university disciplines. While the former mathematical seminars had meanwhile mostly been transformed into mathematical institutes, no such form had ever before existed for mathematics education – the highest form of academic recognition had been two Habilitationen in Didaktik der Mathematik (taken by Rudolf Schimmack in 1911, Hugo Dingler in 1912) earlier that century. Hans-Georg Steiner became one of the first assistants at this seminar. In 1958, Behnke began editing a five-volume Grundzüge der Mathematik für Lehrer an höheren Schulen, presenting modern mathematics for teachers at secondary schools. Having in so many respects become a true successor of Felix Klein, he eventually followed Klein’s footsteps in organizational respect as well. Besides being active in IMU, he became, in 1951, a member of the committee that was to prepare the re-establishment of ICMI.  This was realized in 1952  in Geneva, and Behnke was the first secretary-­general of the renewed organization.

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Recent research of his correspondence during his periods of acting for ICMI reveals the amount of problems and obstacles impeding the revival. It proved to be decisive that he recalled well how all had functioned during Klein’s lifetime. His main concern was to focus the ICMI work on the problems of mathematics teaching and not having it dominated by university mathematicians without real knowledge of school teaching (Furinghetti, Giacardi and Menghini 2020, pp. 250–251).21 To enhance the ICMI work, it was essential for Behnke to revive the national subcommissions. This met at first a lot of resistance by the IMU officials who were only able to conceive all ICMI work as controlled by IMU. Behnke had to insist that the first national subcommissions had been nominated by the respective governments and not by mathematics associations; the primary criterion for commissions members should be their dedication to issues of teaching and not to be functionaries of IMU (ibidem, pp. 249ff). The IMU presidency even obstructed at first to have sections on mathematics teaching at the ICMs, not holding them “worthy enough”, so that Behnke had to remind them by earlier ICM Proceedings that such sections had always been an integral part (ibidem, p. 252). The major problem faced by Behnke was the elaboration of the “Terms of Reference”, defining the relation of IMU with ICMI, its new entity. The document decided in 1954 imposed a strict control by IMU on the central level of ICMI and on the national level that the delegates had to be named by the national mathematical societies (ibidem, p. 252). Behnke, as president, tried to modify the Terms of Reference by admitting other societies in countries where there were no IMU members and to broaden the activities by creating regional groups outside of Europe; yet, the new Terms of 1960 did not endorse his proposals (ibidem, p. 254). The first activity of the revived ICMI was to prepare national trend reports for the next IMU congress in 1954, though on a smaller scale than those of the first IMUK: on the state of mathematics teaching for 16–21-year-old students. Activating the German subcommittee of ICMI, Behnke succeeded in having that report ready by 1954. At the ICM in 1954 in Amsterdam, Behnke was elected president of ICMI for the period 1955–1958. For the next period, 1959–1962, he served as vice-president, and the two following periods, from 1963 to 1970, as a member of the Executive Committee. Behnke participated at ICME-1  in Lyon in 1969; for ICME-3, in Karlsruhe, in 1976, he acted as honorary president. Clearly, as regards mathematics teaching, his views were confined to secondary schools; and while focused on European countries, confessing frankly to not feeling competent for other parts of the world (Behnke 1978, p. 271), he strove to overcome the Eurocentric bias.

 In Furinghetti and Giacardi (2010, pp. 38–44), the following documents are transcribed: M. Stone to A. Châtelet and to H. Behnke, Rome, September 21,1954; Excerpt from Report of the president [H. Behnke] of the International Commission of Mathematical Instruction to the president of the International Mathematical Union. April 20, 1955 [Memorandum von Herrn Behnke über die Bildung von Gruppen]. 21

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Sources Behnke, Heinrich. 1947/1948. Vorlesungen über klassische Funktionentheorie, 2 Vols. Münster: Aschendorff. Behnke, Heinrich. 1956. Vorlesungen über allgemeine Zahlentheorie. Münster: Aschendorff. Behnke, Heinrich and Friedrich Sommer. 1955. Theorie der analytischen Funktionen einer komplexen Veränderlichen. Berlin: Springer. Behnke, Heinrich and Peter Thullen. 1934. Theorie der Funktionen mehrerer komplexer Veränderlichen. Berlin: Springer. Furinghetti, Fulvia and Livia Giacardi. 2010. People, events, and documents of ICMI’s first century. Actes d’Història de la Ciència i de la Tècnica, nova època 3(2), 11–50. Furinghetti, Fulvia, Livia Giacardi, and Marta Menghini. 2020. Actors in the changes of ICMI: Heinrich Behnke and Hans Freudenthal. In “Dig where you stand” 6. Proceedings of the Sixth International Conference on the History of Mathematics Education 2019, eds. Evelyne Barbin, Kristín Bjarnadóttir, Fulvia Furinghetti, Alexander Karp, Guillaume Moussard, Johan Prytz, and Gert Schubring, 247–260. Münster: WTM-Verlag. Segal, Sanford L. 2003. Mathematicians under the Nazis. Princeton. NJ: PUP. Grauert, Heinrich and Reinhold Remmert. 1981. In memoriam Heinrich Behnke. Mathematische Annalen 255 (1): 1–4. Griesel, Heinz. 1978. Zur Vollendung des 80. Lebensjahres von Prof. Dr. Dr. h.c. Dr. h.c. Heinrich Behnke (Münster) am 9. Oktober 1978. Praxis der Mathematik 20(1): 305–306. Krafft, Olaf. 1974. Heinrich Behnke  – 50 Jahre Doktor der Mathematik. MathematischPhysikalische Semesterberichte 21: 1–4. Remmert, Volker R. 2002. Ungleiche Partner in der Mathematik im ‘Dritten Reich’: Heinrich Behnke und Wilhelm Süss. Mathematische Semesterberichte 49(1): 11–27. Schubring, Gert. 1985. Das mathematische Seminar der Universität Münster, 1831/1875 bis 1951, Sudhoffs Archiv 69: 154–191. Tietz, Horst. 1980. Heinrich Behnke. Mathematisch-Physikalische Semesterberichte 27(1): 1–3.

Publications Related to Mathematics Education Behnke, Heinrich (ed.). 1954. Der mathematische Unterricht für die sechzehn- bis einundzwanzigjährige Jugend in der Bundesrepublik Deutschland. Göttingen: Vandenhoeck & Ruprecht. Behnke, Heinrich, Friedrich Bachmann, Kuno Fladt, and Wilhelm Süss. 1958–1971. Grundzüge der Mathematik für Lehrer an höheren Schulen sowie für Mathematiker in Industrie und Wirtschaft. 5 Vols. Göttingen: Vandenhoeck & Ruprecht. Behnke, Heinrich. 1973. Das Haus Still und seine Freunde aus der Wissenschaft. Recklinghausen: Bongers. Behnke, Heinrich. 1978. Semesterberichte – Ein Leben an deutschen Universitäten im Wandel der Zeit. Göttingen: Vandenhoeck & Ruprecht.

Photo Source: Wikimedia Commons.

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11.8 Evert Willem Beth (Almelo, 1908 – Amsterdam, 1964): Member of the Executive Committee 1952–1954 Giorgio T. Bagni

Biography Evert Willem Beth was born in Almelo (eastern Netherlands, near the DutchGerman border) on 7 July 1908. His father, Hermanus J. E. Beth, had studied mathematics and physics at Amsterdam University and worked as teacher in mathematics and physics in secondary schools (van Ulsen 2000); Evert studied mathematics and physics at Utrecht University and also studied both philosophy and psychology. His 1935 PhD was in philosophy (Faculty of Arts). In 1946, Beth became professor of Logic and Foundations of Mathematics in Amsterdam (more precisely, in 1946, he was appointed to a part-time professorship; in 1948, this became a full professorship; it is worth noting that he was the first professor of Logic and Foundations of Mathematics in the Netherlands). Apart from two brief interruptions (in 1952, he worked as a research associate at the University of California in Berkeley, with

G. T. Bagni (1958-2009) University of Udine, Udine, Italy

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Alfred Tarski, and in 1957–1958, he taught as a visiting professor at Johns Hopkins University in Baltimore), Beth worked in Amsterdam continuously until his death (Heyting 1966). Beth was a mathematician, logician and philosopher whose work principally concerned the foundations of mathematics. His name is often remembered with reference to semantic tableaux: the tableau method was devised independently by Beth (1955) and other researchers. Essentially, this important method is dual to Gentzen’s natural deduction (1934), and it is considered by many to be intuitively simple, particularly for students not acquainted with the study of logic. In fact, Gentzen’s method is a systematic search for proofs in tree form, while the tableau method is a systematic search for refutations in upside-down tree form. Beth himself underlined: At least three different methods of deduction are known today and are more or less currently applied in research: Hilbert-type deduction, Gentzen’s natural deduction, and Gentzen’s calculus of sequents. […] In point of fact the three methods must rather be considered as different presentations of one and the same method. (Beth 1962, p. XII)

With the semantic tableaux, Beth explored different areas (classical logic, modal logic and intuitionistic logic); in combination with this method, Beth made a proof-theoretic variant (the deductive tableaux) (Heyting 1966; van Ulsen 2000). More generally, Beth’s main contributions to logic were the definition theorem, semantic tableaux and the Beth models. The foundation of his work was Gentzen’s extended Hauptsatz, the subformula theorem and Tarskian model theory. During the last period of his life (1960–1964), Beth tried to make his logical research subservient to a wide range of applications, e.g. the study of language, theorem proving, mathematical heuristics and translation methods in natural languages (van Ulsen 2000). Beth was the main founder of the Netherlands Society for Logic and Philosophy of Science, and he was also active in the organization of the International Association for Logic and Philosophy of Science (Heyting 1966). He did the greater part of the editorial work for the series “Studies in Logic and the Philosophy of Mathematics” founded at his initiative. Beth’s merits were rewarded by his election in 1953 to the membership of the Royal Dutch Academy of Science and by an honorary doctorate in the University of Gent, conferred on him in 1964, when he was already too ill to travel to Gent in order to receive it (Heyting 1966). Beth died on 12 April 1964 in Amsterdam.

Contribution to Mathematics Education Beth was a member of the ICMI Executive Committee from 1952 to 1954. He collaborated to the preparation of the exhibition of didactical and pedagogical works in mathematics held in connection with Section VII of the International Congress of

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Mathematicians in Amsterdam (1954). In the early 1950s, Beth was one of the founders of CIEAEM (Commission Internationale pour l’étude et l’Amélioration de l’Enseignement des Mathématiques), a commission which gathered mathematicians, educators, teachers and epistemologists with the aim of improving the methods for teaching mathematics in light of the progress in mathematical epistemology and psychology. He is one of the authors in the first of the two books published by CIEAEM (Piaget et al. 1955). His chapter discusses the respective role of psychology and logic in mathematics teaching (see Beth 1955). Beth’s approach to research in mathematics education was very interesting and profound; let us quote Jean Piaget: A logician friend of mine, the late Evert W. Beth … for a very long time … was a strong adversary of psychology in general and the introduction of psychological observations into the field of epistemology, and by that token an adversary of my own work, since my work was based on psychology. Nonetheless, in the interests of an intellectual confrontation, Beth did us the honour of coming to one of our symposia on genetic epistemology and looking more closely at the questions that were concerning us. At the end of the symposium he agreed to co-author with me, in spite of his fear of psychologists, a work that we called Mathematical and psychological epistemology … In his conclusion to this volume, Beth wrote as follows: “The problem of epistemology is to explain how real human thought is capable of producing scientific knowledge. In order to do that we must establish a certain coordination between logic and psychology”. This declaration does not suggest that psychology ought to interfere directly in logic - that is of course not true - but it does maintain that in epistemology both logic and psychology should be taken into account, since it is important to deal with both the formal aspects and the empirical aspects of human knowledge. (Piaget 1970, p. 1)

It is worth noting that Beth and Piaget gave an important contribution to research in cognitive development; in their book (Beth and Piaget 1961), they stated that the problems posed by formalization can in some way correspond with mental mechanisms. So the logico-mathematical structures leading to formalization can be considered as the point of arrival of a long genetic process. In the “Preface” to the book (Beth 1962) Formal Methods: An Introduction to Symbolic Logic and to the Study of Effective Operations in Arithmetic and Logic, written in Amsterdam (October 1961), Beth tries to identify the factors on which the widespread lack of logical skills depends. He writes: Many philosophers have considered logical reasoning as an inborn ability of mankind and as a distinctive feature in the human mind; but we all know that the distribution of this capacity, or at any rate its development, is very unequal. Few people are able to set up a cogent argument; others are at least able to follow a logical argument and even to detect logical fallacies. Nevertheless, even among educated persons there are many who do not even attain this relatively modest level of development. According to my personal observations, lack of logical ability may be due to various circumstances. In the first place, I mention lack of general intelligence, insufficient power of concentration, and absence of formal education. Secondly, however, I have noticed that many people are unable, or sometimes rather unwilling, to argue ex hypothesi; such persons cannot, or will not, start from premises which they know or believe to be false or even from premises whose truth is not, in their opinion, sufficiently warranted. Or, if they agree to start from such premises, they sooner or later stray away from the argument into attempts first to settle the truth or falsehood of the

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premises. Presumably this attitude results either from lack of imagination or from undue moral rectitude. On the other hand, proficiency in logical reasoning is not in itself a guarantee for a clear theoretic insight into the principles and foundations of logic. Skill in logical argumentation is the result of congenital ability combined with practice; theoretic insight, however, can only arise from reflection and analysis. (Beth 1962, p. X)

He was convinced that the “lack of formal education can, of course, be remedied, but hardly by the study of logic alone” (Ibidem, p. X). Moreover: [The student] should become acquainted both with the semantic and with the purely formal approach to the notions, the problems, and the results of logical theory. A dogmatic attitude with respect to the different aspects of logic will easily result if the elements of logic are taught in a narrow spirit. … Each one-sided approach leaves part of the material more or less in the dark. … It should not be forgotten that later on it is extremely difficult to overcome the bad effects of a narrow-minded initiation. (Ibidem, p. XII)

The importance of historical aspects was frequently underlined by Beth; for instance: Recent discussion on the foundations of mathematics and physical science can not be fully understood without reference to their historical and philosophical background. These discussions for the greater part originate not merely from the results of contemporary scientific research in themselves, but rather from the incompatibility of these results with certain preconceived philosophical doctrines. (In: Critical Epochs in the Development of the Theory of Science, The British Journal for the Philosophy of Science 1(1), p. 27)

The contribution of Beth to mathematics education can be summarized taking into account two main aspects: the important reflection in the field of cognitive development carried out in the aforementioned research together with Piaget and the active contribution to the international collaboration.

Sources Beth, Evert W. 1940. Inleiding tot de wijsbegeerte der wiskunde (Introduction to the philosophy of mathematics). Amsterdam: Standaard Boekhandel. Beth, Evert W. 1946–1947. Logical and psychological aspects in the consideration of language. Synthèse 5: 542–544. Beth, Evert W. 1950. Les fondements logiques des mathématiques. Paris and Louvain: GauthierVillars and Nauwelaerts. Beth, Evert W. 1955. Remarks on natural deduction. Indagationes Mathematicae 17: 322–325. Beth, Evert W. 1955. Semantic entailment and formal derivability. Mededelingen Koninklijke Nederlandse Akademie van Wetenschappen, Nieuwe Reeks 18(13): 309–342. Beth, Evert W. 1957. La crise de la raison et la logique. Paris and Louvain: Gauthier-Villars and Nauwelaerts. Beth, Evert W. 1959. The foundations of mathematics, A study in the philosophy of sciences. Studies in Logic. Amsterdam: North Holland. Beth, Evert W. 1962. Formal methods: An introduction to symbolic logic and to the study of effective operations in arithmetic and logic. Dordrecht: Reidel. Beth, Evert W. 1963. The relationship between formalised languages and natural language. Synthese 15(1): 1–16.

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Beth, Evert W. 1965. Mathematical thought: an introduction to the philosophy of mathematics. Dordrecht: Reidel. Bagni, Giorgio T. 2008. Centenary birth anniversary of E.  W. Beth (1908–1964). Educational Studies in Mathematics 69: 73–76. Destouches, Jean-Louis (ed.). 1964. E. W. Beth memorial colloquium: Logic and Foundations of Science, Paris, Institut Henri Poincaré, May 19–21. Dordrecht: Reidel. Dyson, Verena Huber and George Kreisel. 1961. Analysis of Beth’s semantic construction of intuitionistic logic. Technical Report 3, Applied mathematics and statistical laboratories, Stanford, Stanford University. Franchella, Miriam. 2002. Evert Willem Beth’s Contributions to the Philosophy of Logic. Philosophical Writings 21: 1–23. Heyting, Arend. 1966. Evert Willem Beth: in memoriam. Notre Dame Journal of Formal Logic VII(4): 289–295. Piaget, Jean. 1966. In memory of E.  W. Beth (1908–1964). In Evert Beth and Jean Piaget. Mathematical epistemology and psychology (English translation), XI-XII. Dordrecht: Reidel. Piaget, Jean. 1970. Genetic epistemology. New York: Columbia University Press. Staal, Jan Frederik. 1965. E. W. Beth, 1908–1964. Dialectica 19: 158–179, with the complete list of Beth’s works. Van Ulsen, Paul. 2000. E. W. Beth als logicus. Proefschrift Universiteit van Amsterdam. https:// www.illc.uva.nl/Research/Publications/Dissertations/DS-­2000-­04.text.pdf Velthuys-Bechthold, Paula J. M. 1995. Inventory of the papers of Evert Willem Beth (1908–1964), philosopher, logician and mathematician, 1920–1964 (c. 1980), incorporating the finding-aid by J.C.A.P. Ribberink and P. van Ulsen. Haarlem: Inventarisreeks Rijksarchief in Noord-Holland.

Publications Related to Mathematics Education Beth, Evert W. 1937. L’évidence intuitive dans les mathématiques modernes. In Travaux du IXe Congrès international de philosophie, ed. Raymond Bayer, vol. VI, 161–165. Paris: Hermann. Beth, Evert W. 1955. Réflexions sur l’organisation et la méthode de l’enseignement mathématique. In J. Piaget et al. L’enseignement des mathématiques, 35–46. Neuchâtel: Delachaux et Niestlé. Beth, Evert W., Wolfe Mays, and Jean Piaget. 1957. Epistémologie génétique et recherche psychologique. Paris: Presses Universitaires de France. Beth, Evert W. and Jean Piaget. 1961. Epistémologie mathématique et psychologie. Paris: Presses Universitaires de France. Piaget, Jean, Evert W. Beth, Jean Dieudonné, André Lichnerowicz, Gustave Choquet, and Caleb Gattegno. 1955. L’enseignement des mathématiques. Neuchâtel, Switzerland: Delachaux et Niestlé.

Photo Courtesy of E.W. Beth Foundation.

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11.9 Henri Paul Cartan (Nancy, 1904 – Paris, 2008): Ex Officio Member of the Executive Committee 1967–1970 Fulvia Furinghetti

Biography Henri Paul Cartan was born in Nancy on 8 July 1904, the son of Élie Cartan, one of the most important mathematicians of the twentieth century, and Marie-Louise Bianconi. In 1909, when Élie, professor at the University of Nancy, was appointed as a lecturer at the Sorbonne, the family moved from Nancy to Paris. Then Henri studied at the Lycée Buffon and the Lycée Hoche in Versailles. In 1923, he was admitted at the École Normale Supérieure (ENS) in Paris. In those years, among the students at the ENS, there were some future participants in the early meetings of the Bourbaki group: André Weil and Jean Delsarte (admitted in 1922), René de Possel and Jean Coulomb (admitted in 1923), Jean Dieudonné and Charles Ehresmann F. Furinghetti (*) University of Genoa, Genoa, Italy e-mail: [email protected]

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(admitted in 1924) and Claude Chevalley and Jean Leray (admitted in 1926). Among Henri Cartan’s teachers at the ENS were Gaston Julia and his father Élie. The students at the ENS also had to attend general courses at the Sorbonne, so Henri studied there too. Admitted to the concours d’agrégation in 1926, in 1928, he defended his dissertation of Doctorat ès sciences entitled Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications under the supervision of Paul Montel. After being awarded his doctorate, Cartan taught at the Lycée Malherbe in Caen from 1928 to 1929, at the University of Lille from 1929 to 1931 and then at the University of Strasbourg. In 1935, he married Nicole Antoinette Weiss. They had two sons and three daughters. In 1940, Cartan joined the University of Sorbonne in Paris, with the exception of 2 years (1945–1947), when he was stationed at the University of Strasbourg. From 1940 to 1965, he taught at the ENS in rue d’Ulm, and after at the Faculté des Sciences de Paris until 1969 when he became a professor at the Faculté des Sciences d’Orsay, the embryo of the new Université Paris-Sud created in 1970. He retired in 1975 and died on 13 August 2008 in Paris. At the ENS, Cartan started the Séminaire Cartan. Fifteen ENS Seminars written by Cartan were published between 1948 and 1964; these publications became a fundamental reference. His research interest followed different directions: analytic functions, the theory of sheaves, homological theory, algebraic topology and potential theory. In 1953, he wrote the book Homological Algebra with Samuel Eilenberg (published in 1956 by Princeton University Press), which became a classic reference for the new algebraic methods used in topology, theory of functions of complex variables and algebraic geometry. Also, his university manuals became classical references for scholars all around the world. Cartan’s history as a mathematician is strongly linked to the life of the Bourbaki group. After a first informal meeting at the café Capoulade in Paris (10 December 1934) and successive encounters, the group was finalized in the meeting at Besseen-Chandesse (10–17 July 1935). The participants were Claude Chevalley, René de Possel, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, Szolem Mandelbrojt and André Weil. The main aim of this group was to renew French mathematics that suffered the loss of a generation of mathematicians dead during World War I. In his interview with the Notices of AMS (Jackson 1999), Cartan claims, “Almost all I know in mathematics I learned from and with the Bourbaki group” (p.  785). Nevertheless, he maintains that the Bourbaki style is no longer dominant in France, but in the past had an enormous influence in France and abroad. Through his role as teacher and mentor, Cartan contributed remarkably to the renaissance of French mathematics and created a generation of leading mathematicians. His students Jean-Pierre Serre and René Thom were each awarded the Fields Medal. In his interview with the Notices of AMS (Jackson 1999), Cartan describes his way of teaching at the ENS: “You have to respect [the student’s] personality, to help [the student] find his own personality, and certainly not impose somebody else’s ideas” (p. 786). His teaching was very individual. Cartan influenced the development of mathematics all around the world. He believed in international cooperation and was an active promoter of university

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exchanges. Already in May 1931, he was invited by Heinrich Behnke to visit Germany and delivered a series of lectures at Muenster University in Westphalia. Cartan’s younger brother, a member of the French Resistance fighting against the occupying German forces, was deported by Nazis and executed in 1943. Nevertheless, Cartan renewed contacts with his German colleagues in November 1946 when he visited the Research Institute in Oberwolfach and contributed to the reconstruction of mathematics research in Germany. In the years 1950 and 1960, Cartan was an active member of the French and international community of mathematicians: president of Société Mathématique de France in 1950, member of the Fields Medal Committee in 1954 and 1970 and president of IMU (International Mathematical Union) from 1967 to 1970. The final act of this presidency was the organization of the Sixteenth International Congress of Mathematicians in Nice (1–10 September 1970). In 1991, he was appointed as a president of the committee established in France to organize the first congress of the European Mathematical Society. Cartan’s strong interest in cooperation and internationalism was not confined to mathematics. He was involved in the European Federalist movement, and in 1984, he was a chief of an electoral list called Pour les États-Unis d’Europe, which had less than 1% of votes. In the interview mentioned above, he explains that he became a European Federalist by applying the mathematical way of reasoning to politics: You see, a mathematician thinks: “What is this question? What happens exactly? Why is it so, and not so? What is the reason? What is the logical consequence of all this?” I am applying this to politics. I have tried to analyze situations and to draw logical consequences. This was the way I became a European Federalist because I understood that there is no other way. (Jackson 1999, p. 788)

His political involvement was also devoted to supporting human rights. In 1974, together with Laurent Schwartz, Michel Broué and the American mathematician from Latvia Lipman Bers, he set up the Comité des mathématiciens, an informal organization aimed at supporting the dissident mathematicians persecuted in their countries for political reasons. For this activity in favour of dissidents, Cartan received the Pagels Award from the New York Academy of Sciences. Cartan was a member of the Académie des Sciences of Paris and of other academies in Europe, the United States and Japan. He has received honorary doctorates from several universities. He was awarded the Gold Medal of the National Centre for Scientific Research (1976) and the Wolf Prize in Mathematics in 1980 and was made Commandeur de la Légion d’Honneur in 1989. Cartan died on 13 August 2008 in Paris.

Contribution to Mathematics Education In Cartan’s works, a direct reference to the school world is very rare, as evidenced by the 100 titles in the three volumes edited by Remmert and Serre (1979). This is not surprising; in his paper on the new mathematics programmes,

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Cartan himself acknowledged that he had very little experience with secondary teaching: I have no title to take the floor, since it is the first time I am participating in the works of the C.I.E.M, and I really have very little experience in mathematics teaching at the secondary level.22 (Cartan, 1963, p. 84)

Mathematical Content Knowledge As we have seen in the biographical notes, Cartan taught as a secondary teacher only in 1928–1929. Afterwards, his entire career was devoted to mathematical research and university teaching. Nevertheless, he had some indirect influence in the school world of his times concerning both the mathematical contents to be taught and political issues. The changes in mathematical research taking place in the second half of the twentieth century made updating the mathematical knowledge of schoolteachers the main concern for both teachers and mathematicians. Important initiatives were organized in France for in-service teacher education (see Barbazo 2009; Dubreil 1959). At the beginning of 1956, the Société Mathématique de France (with Paul Dubreil as a president) and the Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP) planned an initiative consisting of cycles of talks delivered at the Institut Henri Poincaré by some important French mathematicians. The first cycle lasted from 9 February 1956 to 6 June 1957. The texts of the lectures were published in the Bulletin de Association des Professeurs de Mathématiques de l’Enseignement Public.23 Afterwards, the texts were gathered in a volume (see Cartan et  al. 1958). A second cycle of talks was organized by the Société Mathématique de France and the Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP) at the Institut Henri Poincaré from 15 November 1957 to 22 May 1958. The texts were published in the Bulletin de Association des Professeurs de Mathématiques de l’Enseignement Public.24 Afterwards, they were gathered in a volume (see Cartan, Dixmier, Dubreil, Lichnérowicz and Revuz, 1962). Cartan was one of the contributors in both cycles (see 1958a; 1958b; 1962). In her book on the history of CIEAEM, Lucienne Félix (1986, pp. 82–84) stresses the impact of this initiative on teachers’ mathematical knowledge. As a member of the Bourbaki group established in 1935, Cartan launched new methods in mathematics that after some decades affected the teaching in school  The original text is: “Je n’ai guère de titres à prendre la parole, car c’est la première fois que je participe aux travaux de la C.I.E.M, et j’ai vraiment très peu d’expérience de l’enseignement secondaire”. 23  The issues were n. 176 (March 1956), n. 177 (May 1956), n. 179 (October 1956), n. 180 (December 1956), n. 183 (January 1957), n. 184 (March 1957), n. 185 (June 1957), n. 187 (October 1957), n. 188 (January 1958), n. 191 (March 1958). 24  The issues were n. 192 (June 1958), n. 194 (October 1958), n. 196 (January 1959), n. 198 (March 1959), n. 199 (June 1959). 22

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through the movement of modern mathematics. One member of the Bourbaki group, Dieudonné, was an active promoter of the reform of mathematics teaching. He was a leading participant in the important meeting held in Royaumont (23 November to 4 December 1959) with the concrete aim of producing a curricular reform that was not limited to Europe. After Royaumont, other meetings addressed this issue. Cartan was invited to the conference promoted by ICMI in Bologna (4–8 October 1961) on the teaching of geometry in secondary schools. University professors, school inspectors and delegates from UNESCO and OEEC attended the meeting (see BUMI 1962). The texts of the contributions were published in L’Enseignement Mathématique, the official organ of ICMI (1963, Volume 9 of the second series). In his contribution, Cartan (1963) started by complaining that the results of the previous meetings were still under discussion. He suspected that some people were not really wishing to change current mathematics teaching. He advocated taking care of those students who will go to university as well as those who will stop their school careers. He also stressed the need to prepare teachers for the new mathematics contents. He was convinced that, when planning a radical reform, it was necessary to focus first on the mathematical content and after on educational objectives. About this point, he was polemic with Hans Freudenthal who in his report claimed that didactical problems have to be considered first (see Freudenthal 1963). Cartan advocated that axiomatics should be reduced to a minimum and that the algebraic ideas should be stressed. In mathematics, the teaching of algebra and geometry should not be separated, but instead, the two subjects should be taught as follows: Summarizing, I would hope that classical geometry (affine or Euclidean) is presented with the minimum of axiomatic, and the maximum of algebraic explanations. These algebraic explanations do not at all exclude geometric language; they justify it! Furthermore, they do not exclude the solution of problems using geometric means; it will always be interesting that a problem is approached two ways, geometrically and analytically.25 (Cartan 1963, p. 90, italic in the original)

In a message to Unione Matematica Italiana (1978, Notiziario della Unione Matematica Italiana 5(3), 23–28), Leray reported on a document prepared together with Cartan, Dieudonné and Lichnérowicz, published in the column Comité secret of Comptes Rendus de l’Académie des Sciences de Paris (1977, t. 285, Vie académique, deuxième semestre, 56–60). This document analyses some aspects of mathematics teaching in the grades fourth and third of the French Collèges (students’ ages 13–14 years).

 The original text is: “En résumé, je souhaiterais que la géométrie classique (affine ou euclidienne) fût exposée avec le minimum d’axiomatique, et le maximum d’explicitations algébriques. Ces explicitations algébriques n’excluent nullement le langage géométrique; elles le justifient! Elles n’excluent pas davantage la solution des problèmes par voie géométrique; il y aura toujours intérêt à ce qu’un même problème soit traité de deux manières, par voie géométrique et par voie analytique”. 25

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Political Issues in Mathematical Instruction The two contrasting positions about mathematics teaching held by Cartan and Freudenthal emerged also when political issues were to be managed in the years 1967–1970, with Freudenthal president of ICMI and Cartan president of IMU and as such ex officio member of the ICMI Executive Committee. The point was the controversial relationship between the bodies they were chairing. The article (Furinghetti and Giacardi, 2010) illustrates the struggle of the two bodies for finding agreement on this relationship. In the end, Freudenthal (without consulting IMU) launched two initiatives that marked an important step of ICMI for achieving its autonomy from IMU: a new journal on mathematics education, Educational Studies in Mathematics, was founded, and the tradition of holding ICMI congresses independent of the International Congresses of Mathematicians was established. In this story, Cartan’s and Freudenthal’s actions epitomize the spirit of an epoch of changes in society and reflect two different views of the relationship between school and mathematical research.

Sources Remmert, Reinhold and Jean-Pierre Serre (eds.). 1979. Henri Cartan. Oeuvres (3 vols.), BerlinHeidelberg-New York, Springer. Audin, Michèle. 2012. Henri Cartan & André Weil. Du vingtième siècle et de la topologie. In Henri Cartan and André Weil mathématiciens du XXe siècle. Journées mathématiques X-UPS, eds. Pascale Harinck, Alain Plagne, and Claude Sabbah, 1–61. Palaiseau: Éditions de l’École Polytechnique. Barbazo, Éric. 2009. La formation continue des enseignants: phénomène naturel?. In Actes en ligne du séminaire “Formation continue”, ed. F.  Plantevin. http://www.univ-­irem.fr/spip. php?rubrique106 (retrieved 12 July 2015). BUMI. 1962. Il convegno di Bologna promosso dalla Commissione internazionale dell’insegnamento matematico. Bollettino della Unione Matematica Italiana s. 3, 17: 199–214. Dubreil, Paul. 1959. Rapport sur les bases scientifiques des mathématiques dans l’enseignement du second degré. L’Enseignement Mathématique s. 2, 5: 273–277. Félix, Lucienne. 1986. Aperçu historique (1950–1984) sur la Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques. Bordeaux: IREM. Retrieved on 12 July 2015. http://www.cieaem.org/?q=node/18 Freudenthal, Hans. 1963. Enseignement des mathématiques modernes ou enseignement moderne. L’Enseignement Mathématique s. 2, 9: 28–44. Furinghetti, Fulvia and Livia Giacardi. 2010. People, events, and documents of ICMI’s first century. Actes d’Història de la Ciència i de la Tècnica, nova època 3(2): 11–50. D’Enfert, Renaud and Hélène Gispert. 2011. Une réforme à l’épreuve des réalités. Le cas des «mathématiques modernes» en France, au tournant des années 1960–1970. Histoire de l’Éducation 131: 27–49. Jackson, Allyn. 1999. Interview with Henri Cartan. Notices of AMS 46: 782–788. Serre, Jean-Pierre. 2009. Henri Paul Cartan. 8 July 1904 – 13 August 2008. Biographical Memoirs of the Fellows of the Royal Society 55: 37–44.

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Publications Related to Mathematics Education Cartan, Henri. 1958a. Structures algébriques. In Henri Cartan et  al., Structures algébriques et structures topologiques. Monographies de L’Enseignement Mathématique 7: 5–15. Cartan, Henri. 1958b. Sur la notion de dimension. In Henri Cartan et al., Structures algébriques et structures topologiques. Monographies de L’Enseignement Mathématique 7, 163–174. Cartan, Henri. 1962. Volume des polyèdres. In Henri Cartan et  al., eds. Problèmes de mesure. Monographies de L’Enseignement Mathématique 10: 8–20. Cartan, Henri. 1963. Réflexions sur les rapports d’Aarhus et Dubrovnik. L’Enseignement Mathématique s. 2, 9: 84–90. Cartan, Henri, Gustave Choquet, Jacques Dixmier, Paul Dubreil, Roger Godement, Pierre Lelong, Léonce Lesieur, André Lichnérowicz, Charles Pisot, André Revuz, Laurent Schwartz, and Jean-Pierre Serre. 1958. Structures algébriques et structures topologiques. Monographies de L’Enseignement Mathématique 7. Cartan, Henri, Jacques Dixmier, Paul Dubreil, André Lichnérowicz, and André Revuz. 1962. Problèmes de mesure. Monographies de L’Enseignement Mathématique 10.

Photo Author: Fischer, Gerd. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach. Signature Courtesy of Académie des sciences, Archives et patrimoine historique.

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11.10 Guido Castelnuovo (Venice, 1865 – Rome, 1952): Member of the Central Committee 1913–1920, Vice-President 1928–1932, Appointed Honorary Member in 1936 Livia Giacardi

Biography Guido Castelnuovo was born in Venice on 14 August 1865. Having been guided towards mathematics, during his secondary school studies, by Aureliano Faifofer, in 1886, he graduated from the University of Padua, where he was taught by Giuseppe Veronese, who introduced him to the study of hyperspatial geometry. After a year (1886/1887) of specialization in Rome, under the guidance of Luigi Cremona, Castelnuovo went to Turin. The Turin period (1887–1891) was extremely important for the orientation of his research, so much so that it was said that the new Italian geometry of algebraic curves was born of Castelnuovo’s conversations with his friend and mentor Corrado Segre in the course of their long walks under the arcades of Via Po. Thus, the Italian school of algebraic geometry saw the light in Turin and, for a number of decades, it would be one of the most important points of reference for scholars throughout Italy and elsewhere in Europe. He took up the chair of Analytic and Projective Geometry at the University of Rome and taught there for 44  years, also giving courses on complementary L. Giacardi (*) University of Turin, Turin, Italy e-mail: [email protected]

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mathematics, probability theory and, after Cremona’s death, higher geometry. Castelnuovo’s period of full scientific maturity and his fruitful collaboration with Federigo Enriques both started in Rome. This collaboration would lead to the publication of various works on algebraic surfaces including two articles (1908, 1914) for the Encyklopädie der mathematischen Wissenschaften, one devoted to the fundamental properties of algebraic surfaces in ordinary space from the projective point of view and the other on algebraic surfaces from the point of view of birational transformations. Castelnuovo recalls his method of working with Enriques thus: We had created ‘in an abstract sense, of course’ a large number of models of surfaces in our space or in higher spaces; and we had split these models, so to speak, between two display windows. One contained regular surfaces for which everything proceeded as it would in the best of all possible worlds; analogy allowed the most salient properties of plane curves to be transferred to these. When, however, we tried to check these properties on the surfaces in the other display, that is on the irregular ones, our troubles began, and exceptions of all kinds would crop up … With the aforementioned procedure, which can be likened to the type used in experimental sciences, we managed to establish some distinctive characters between the two surface families.26 (Castelnuovo 1928, p. 194)

The working partnership of over 20 years between the two mathematicians whose copious correspondence offers a vivid and precious testimony (Bottazzini, Conte, Gario 1996) was made possible by the combination of their different scientific qualities: while Enriques was endowed with an intellect of extraordinary power, it was, nonetheless, Castelnuovo who channelled its “torrential creativity” combining great breadth of vision and constructive criticism. As William Hodge writes, “their partnership was one of the happiest examples of collaboration in mathematics” (Hodge 1953, p. 121). Castelnuovo is one of the principal creators both of geometry on algebraic curves and of the theory of algebraic surfaces from the point of view of birational transformations. His scientific output consists of a hundred or so documents comprising notes, memoirs and treatises. In particular, the proof of the rationality of the plane involution (1893), the discovery of the necessary and sufficient condition for the rationality of a surface (1896) and the study of irregular surfaces (1905) are his most important contributions to algebraic geometry. From 1906 on, Castelnuovo’s scientific activity, except for some sporadic contributions, was focused on other questions which identify his interests of a methodological, didactic, historical and physical order. Particularly noteworthy are his

 The original text is: “Avevamo costruito - in senso astratto s’intende, un gran numero di modelli di superficie del nostro spazio o di spazi superiori; e questi modelli avevamo distribuito, per così dire, in due vetrine. Una conteneva le superficie regolari per le quali tutto procedeva come nel migliore dei mondi possibili; l’analogia permetteva di trasportare ad esse le proprietà più salienti delle curve piane. Ma quando cercavamo di verificare queste proprietà sulle superficie dell’altra vetrina, le irregolari, cominciavano i guai, e si presentavano eccezioni di ogni specie […] Col detto procedimento, che assomiglia a quello tenuto nelle scienze sperimentali, siamo riusciti a stabilire alcuni caratteri distintivi tra le due famiglie di superficie […]” Castelnuovo, Guido. 1928. La geometria algebrica e la scuola italiana. Atti del Congresso Internazionale dei Matematici, Bologna: Zanichelli, I, p. 194. 26

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studies on statistical methods in physics and on probability theory to which he devoted numerous notes and the treatise Calcolo delle probabilità e applicazioni (1st ed. 1918, 2nd ed. in 2 Vols. 1925–1928), and the text dedicated to Einstein’s theory of relativity (Spazio e tempo secondo le vedute di A. Einstein, 1923), as well as his volume on the history of infinitesimal calculus (Le origini del Calcolo infinitesimale nell’era moderna, 1938). His Lezioni di geometria analitica (1904) and his handwritten course notebooks (Gario 2010) stand as testimony to his university teaching. During the years of racial persecution, from 1938 to 1943, Castelnuovo organized a secret university for the politically and racially persecuted in Rome and, in the 9 months of the Nazi-Fascist occupation, he was hidden at the house of friends. Castelnuovo was a member of numerous academies and scientific societies both in Italy and abroad: in particular, from 1901, he was a member of the Accademia dei Lincei, of which he was also president (1946–1952). In 1949, he was appointed senator for life. He died in Rome on 27 April 1952.

Contribution to Mathematics Education Castelnuovo was one of the Italian delegates in the International Commission on the Teaching of Mathematics  – together with Enriques and Giovanni Vailati  – from 1909, a year after its foundation during the International Congress of Mathematicians (ICM) held in Rome from 6 to 11 April 1908. The choice of Castelnuovo was quite obvious: he was in contact with Felix Klein, the first president of the Commission, and shared his way of conceiving research, as well as his ideas on the teaching of mathematics. In 1913, he was co-opted as a member of the Central Committee of the Commission, which had decided to increase its membership (Schubring 2008, p. 14), and held this position until 1920, when the dissolution of the Commission became inevitable, due to the aftermath of WWI.  After the re-establishment of CIEM/IMUK in 1928 during the ICM in Bologna, Castelnuovo became vice-president and held this office until 1932. He was appointed honorary member of the Commission in 1936 during the ICM in Oslo. Castelnuovo’s commitment to education developed in a number of directions: in his activity in the International Commission on the Teaching of Mathematics; in his role as president of Mathesis (1911–1914), the Italian association of mathematics teachers founded in 1895–1896; in his university courses, some of which were expressly geared towards teacher training; and in various articles dedicated to the problems of the teaching of mathematics and in the drawing up of secondary school syllabi.

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The Involvement in Education as a Social Duty It is in 1907 that Castelnuovo began to turn his thoughts and efforts to the improvement of mathematics teaching in secondary schools, becoming involved also at an institutional level. That year, in fact, appeared his article entitled Il valore didattico della matematica e della fisica, which can be considered a manifesto of his thinking on education. That same year, Castelnuovo was involved in the organization of the ICM in Rome, which saw the establishment of the International Commission on the Teaching of Mathematics. He paid a great deal of attention to the section devoted to education, with the help of Vailati, who at the time was working with the Royal Commission on the project to reform Italian secondary schools (Giacardi 2019). So, it is natural that the contacts with Klein, whom Castelnuovo had already met in 1899, became more intense and his influence more evident. In 1909, during the Congress of the Mathesis Association in Padua, Castelnuovo explicitly proposed following Klein’s example with regard to teacher training: “At Klein’s suggestion, during the spring holidays a number of German universities hold short courses for Middle school teachers. Couldn’t we too set up similar courses in our universities?”27 (Castelnuovo 1909, p. 4). In the years immediately following, he himself began to include in his courses in higher geometry in Rome a number of topics designed specifically for the cultural training of future mathematics teachers following the examples of Klein and Segre. Of particular interest from this point of view are the following notebooks: Geometria non-euclidea (1910–1911), Matematica di precisione e matematica di approssimazione (1913–1914), Indirizzi geometrici (1915–1916), Equazioni algebriche (1918–1919) and Geometria non-euclidea (1919–1920). In the introduction to the 1913–1914 course on the relationship between precise and approximate mathematics, Castelnuovo explicitly discusses the various ways in which future teachers can be trained and refers to Klein: The educational value of mathematics would be much enriched if, in addition to the logical procedures needed to deduce theorems from postulates, teachers included brief digressions on how these postulates derive from experimental observations and indicated the coefficients with which theoretical results are verified in real ­experience … The relationship between problems pertaining to pure mathematics and those pertaining to applied mathematics is very interesting and instructive. Klein, who dedicated a series of lectures to the subject (1901), describes the first of these as problems of ‘precise mathematics’ and the second as problems of ‘approximate mathematics’.28

 The original text is: “Per impulso specialmente del Klein, si tengono durante le vacanze primaverili, dei corsi di poche lezioni, dedicati ai professori delle scuole medie. Non si potrebbero istituire dei corsi analoghi anche nelle nostre Università?”. 28  The original text is: “Il valore educativo della matematica sarebbe molto arricchito se, nell’insegnamento, accanto ai procedimenti logici che servono per ricavare i teoremi dai postulati, si accennasse mediante qualche digressione [che] questi si ricavano dall’osservazione, e d’altra parte con quali coefficienti i risultati teorici si verifichino in realtà […] Il rapporto tra i problemi delle matematiche pure, e quelle che interessano le applicazioni è molto interessante ed. istruttivo. Il Klein, che vi ha dedicato un corso di lezioni (1901), chiama gli uni, problemi della matematica di precisione, gli altri, problemi della matematica di approssimazione”. 27

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Thus, there is no doubt that Klein’s influence was important, but Castelnuovo’s interest in educational issues also arose from social concerns, as he himself affirmed in the lecture he gave in Paris (1914) during the congress of the International Commission on the Teaching of Mathematics in the stead of President Klein, at his express request: We sometimes wonder if the time we devote to questions concerning teaching would have been better spent in scientific research. Well, we answer that it is a social duty that forces us to deal with these problems … Should we not facilitate human beings in acquiring knowledge, which is a source of both power and happiness?29 (Castelnuovo 1914a, p. 191)

Castelnuovo’s great appreciation of Klein’s scientific and educational work is also evident on the occasion of WWI. Klein had signed the scandalous document, Aufruf an die Kulturwelt (1914), which denied the war crimes of the German army, and only much later did it become known that many of the scholars who signed that document, including Klein, were completely unaware of its real content. This fact made Klein’s position as president uncertain and caused a crisis within the Commission. On this occasion, Castelnuovo strongly supported Klein.30 “Break Down the Wall Separating Schools from the Real World” Castelnuovo’s approach to education grew out of a lucid critique of the Italian school system – and in particular of the teaching of mathematics – which, in his opinion, was too abstract and theoretical. All reference to practical application was neglected, and excessive specialization and an unnecessary compartmentalization of different areas led to a distorted cultural perspective: the general education which it [middle-school teaching] proposes to provide ought not to resemble a wild and mountainous land, whose peaks illuminated by the sun are separated from deep and unexplored abysses. It ought rather to be an already civilised domain, whose provinces are linked by bridges and roads.31 (Castelnuovo 1912–1913, pp. 16–17)

There are, according to Castelnuovo, two principal questions that those devising mathematics syllabi for secondary schools must ask themselves: who should middle school teaching address and what are the qualities which this teaching should develop. In his opinion, schools should cater above all for young people aiming to  The original text is: “Nous nous demandons parfois si le temps que nous consacrons aux questions d’enseignement n’aurait pas été mieux employé dans la recherche scientifique. Eh bien, nous répondons que c’est un devoir social qui nous force à traiter ces problèmes […] Ne devons-nous pas faciliter à nos semblables l’acquisition du savoir, qui est à la fois une puissance et un bonheur?”. 30  For further details, see Schubring (2008, pp. 19–21) and Part I.1 by Gert Schubring in this volume. See also the moving letter written by Castelnuovo to Klein, Rome, 10 March 1915 in Part I.4 by Livia Giacardi in this volume. 31  The original text is: “La cultura generale che esso (l’insegnamento medio) si propone di fornire non deve assomigliare ad un territorio selvaggio e montuoso, le cui vette illuminate dal sole sono separate da abissi profondi e inesplorati. Deve esser piuttosto un dominio già civilizzato, le cui provincie siano collegate da ponti e strade”. 29

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go into one of the so-called free professions and the qualities which teachers must foster and cultivate in their pupils are the creative imagination, the spirit of observation and the logical faculties: Middle schools should not furnish [their pupils] with knowledge, so much as with the desire and the need for knowledge. They should not seek to provide an encyclopaedic knowledge of everything, but must only offer a clear, although necessarily very limited, idea of the main questions of the various branches of knowledge, and of some of the methods which have been employed in tackling them. They should not be afraid of sacrificing profundity for range of ideas … Of course, this kind of teaching will not be sufficient to provide middle school students with preparation specific to one or another of the faculties of the university. However, this is not the aim of middle schools. They should only provide students with the aptitude to move on to more advanced studies.32 (Castelnuovo 1910, p. 24)

Three aspects of education therefore were very important for Castelnuovo: to form a “cultured democracy”, where sciences were considered as important as humanities; to connect mathematics teaching with the real world; and to develop the “creative imagination” of students. In his article, Il valore didattico della matematica e della fisica, the placing together of mathematics and physics is no coincidence. Indeed, Castelnuovo here upholds the importance of observation and experimental activities, the usefulness of constantly comparing abstraction with reality and the importance of applications as a means of shedding light on the value of science. He, moreover, affirms that heuristic procedures should be favoured for two reasons: “the first, and the most important reason, is that this type of reasoning is the best way to attain the truth, not just in experimental sciences, but also in mathematics itself … The other reason is in the fact that this is the only kind of logical procedure that is applicable in everyday life and in all the knowledge involved with it”33 (Castelnuovo 1907, p. 336). To illustrate more incisively his idea of mathematics teaching, Castelnuovo often introduced veritable slogans in his speeches and in his articles: “Rehabilitate the senses”, “Break down the wall separating schools from the real world” or “Teaching should walk hand in hand with nature and with life”.

 The original text is: “La scuola media deve dare non la sapienza, ma il desiderio, il bisogno della sapienza; non la cultura enciclopedica, ma un’idea chiara, per quanto necessariamente molto limitata, delle principali questioni che i vari rami di conoscenza prendono in esame, e di qualcuno dei metodi che furono impiegati per trattarle. Non si tema di sacrificare la profondità alla larghezza di idee […] Certo, un insegnamento così inteso non potrà fornire al licenziato la preparazione specifica per l’una o per l’altra Facoltà universitaria. Ma non deve esser questo il compito delle scuole medie. Esse devono dare solo l’attitudine a seguire studi più elevati”. 33  The original text is: “la prima, e più elevata, è che quel tipo di ragionamento costituisce il più efficace mezzo per giungere alla verità, non solo nelle scienze sperimentali, ma nella stessa matematica […] L’altra ragione sta nell’esser quella l’unica forma di procedimento logico, che sia applicabile nella vita quotidiana ed. in tutte le conoscenze che con questa hanno rapporti”. 32

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Various Directions of Castelnuovo’s Commitment to Education As a delegate and later as a member of the Central Committee of the International Commission on the Teaching of Mathematics, Guido Castelnuovo promoted the exchange of information about the new movements for reform in Europe, in particular that proposed by Klein, whose methodological approach he wholeheartedly endorsed (see Castelnuovo 1909, 1911a, b, 1912b, 1914a, b, c).34 He guided the work of the Italian subcommission encouraging its members not to “occupy themselves only with statistical data” but to “turn the investigation to more elevated fields and to treat pedagogical and psychological questions”35 (Castelnuovo 1909, p. 2). As president of the Mathesis Association, he inserted into the Bollettino, the journal of the Association, summaries of the activities of the subcommission, translations of lectures and inquiries on problems concerning mathematics teaching in the various kinds of schools. He also encouraged debates regarding method. For example, he wrote to Giovanni Vacca: Almost unexpectedly and against my will, I have been elected president of the Mathesis. I accept the nomination only because I think that it might be helpful for the affairs of the Italian Commission for mathematics teaching, for which the Bollettino of the Math[esis] will become the publishing organ. I would like keep the level of the Bollettino high, reducing to a minimum the Byzantine discussions in which secondary teachers too often delight. I am therefore very much counting on your cooperation.36

In 1911, the minister Luigi Credaro instituted a liceo moderno which differed from the liceo classico from the second class onwards. The new curriculum replaced Greek with a modern language (German or English), dedicated more time to the sciences and incorporated elements of economics and law. Castelnuovo drew up the mathematics syllabus, putting a number of Klein’s proposals into practice by introducing the notion of function and the concepts of derivative and integral into the Italian curricula and attaching a greater importance to numerical approximations. In the instructions for teachers accompanying this new syllabus, he also implemented the founding principles of his didactic theory: the importance of coordinating mathematics and physics teaching, the need to avoid the excessive subtlety of modern criticism, without falling into an over-crude empiricism, the need to constantly bear in mind the historical process whereby problems arose and their solutions were found and above all the importance of interesting pupils in mathematics by making them understand its important role in modern society. He writes:

 For further details, see Giacardi (2019).  The original text is: “occuparsi soltanto di dati statistici”, “rivolgere l’indagine in campi più elevati e trattare le questioni pedagogiche e psicologiche”. 36  See the letter of G. Castelnuovo to G. Vacca, Roma, 2 January 1911, in Nastasi and Scimone (1995, p. 46). The original text is: “Quasi di sorpresa, e contro le mie volontà, mi capita la nomina a presidente della Mathesis. La accetto solo pensando che può giovare agli interessi della Commissione italiana per l’insegnamento matematico, di cui il Bollettino della Math. Diventerà l’organo. Vorrei tener alto il livello di quel Bollettino, riducendo al minimo le discussioni bizantine di cui i professori secondari troppo spesso si compiacciono. Faccio perciò molto assegnamento sulla sua cooperazione”. 34 35

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But if we truly wish the middle school students to feel an inspiring breeze in this modern mathematics, and perceive something of the grandeur of its whole structure, we much speak to them of the concept of function and show them, even summarily, the two operations that constitute the foundation of infinitesimal calculus. In this way, if the pupil has a scientific spirit, he will acquire a more correct and balanced idea of the exact sciences nowadays … If the pupil’s mind is more disposed towards other subjects, he will at least find mathematics to be, instead of a logical drudge, a set of tools and results which can be easily applied to concrete problems.37 (Castelnuovo 1919, p. 5)

On that occasion, Castelnuovo wrote to Klein: Regarding education, certain that the news will please you, I will tell you that the (modern) programs of mathematical teaching that I have adopted in modern high schools have been so well received that the Ministry of Public Instruction is now thinking of introducing them even in classical lycées and technical institutes, by further developing, in these latter schools, the course in infinitesimal calculus.38

In the same year Castelnuovo presented a report on rigor in secondary schools in various European countries at the CIEM/IMUK congress in Milan (Castelnuovo 1911b). Always keen to establish and reinforce international contacts, Castelnuovo presented the new syllabi to the Conférence internationale de l’enseignement mathématique which was held in Paris in 1914 (Castelnuovo 1914b).39 Unfortunately, the liceo moderno was short-lived. The reorganization of secondary schools was introduced in 1923 by the Gentile Reform in completely different terms: the liberal democratic culture was defeated by new political trends  – firmly opposed by Castelnuovo – and by the triumph of Neo-Idealism. Castelnuovo’s daughter Emma would take up her father’s legacy.

Sources Castelnuovo, Guido. 1937. Memorie scelte, Bologna, Zanichelli.

 The original text is: “Ma se si vuole che l’allievo delle scuole medie senta di questa matematica moderna il soffio ispiratore ed. intravveda la grandezza dell’edifizio, occorre parlargli del concetto di funzione ed. indicargli, sia pure sommariamente, le due operazioni che costituiscono il fondamento del Calcolo infinitesimale. Così egli se avrà spirito scientifico, acquisterà un’idea più corretta ed. equilibrata dell’organismo odierno delle scienze esatte […] Se poi la mente dell’allievo sarà portata verso altre discipline, egli almeno troverà nella matematica, anziché un esercizio logico a lui penoso, una raccolta di metodi e risultati che hanno facili applicazioni in problemi concreti”. 38  The original text is: “A propos de l’enseignement, certain que vous agréerez la nouvelle, je vais vous communiquer que les programmes (modernes) de l’enseignement mathématique que j’ai fait adopter dans les lycées modernes, ont été si bien accueillis que le Ministère de L’Instr. P. pense maintenant de les introduire même dans les lycées classiques et dans les instituts techniques, en développant davantage, dans ces dernières écoles, le programme de calcul infinitésimal”. 39  The programmes for the liceo moderno and the instructions regarding methodology were also translated into French: see G. Loria, Les Gymnases-lycées “modernes” en Italie, Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht aller Schulgattungen, 1914: 188–193. 37

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Castelnuovo, Guido. 2002–2007. Opere matematiche. Memorie e Note. eds. Enrico Arbarello, Umberto Bottazzini, Maurizio Cornalba, Paola Gario, and Edoardo Vesentini. Roma: Accademia Nazionale dei Lincei, Vols. I-IV. Bottazzini, Umberto, Alberto Conte, and Paola Gario. 1996. Riposte armonie Lettere di Federigo Enriques a Guido Castelnuovo. Torino: Boringhieri. Brigaglia, Aldo. 2006. Da Cremona a Castelnuovo. Continuità e discontinuità nella visione della scuola. In Da Casati a Gentile. Momenti di storia dell’insegnamento secondario della matematica in Italia, ed. Livia Giacardi, 159–179. Pubblicazioni del Centro Studi Enriques, La Spezia: Agorà Edizioni. Brigaglia, Aldo, and Ciro Ciliberto. 1995. Italian Algebraic Geometry between the Two World Wars. Kingston: Queen’s University, pp. 24–32. Gario, Paola. 1991. Singolarità e geometria sopra una superficie nella corrispondenza di C. Segre a G. Castelnuovo. Archives for History of Exact Sciences 43: 145–188. Gario, Paola. 2004. Guido Castelnuovo e il problema della formazione di docenti di matematica. In Studies in the History of Modern Mathematics, V, Suppl. Rendiconti Circolo Matematico 74: 103–121. Gario, Paola. 2006. I corsi di Guido Castelnuovo per la formazione degli insegnanti. In Da Casati a Gentile Momenti di storia dell’insegnamento secondario della matematica in Italia, ed. Livia Giacardi, 239–268. Pubblicazioni del Centro Studi Enriques, La Spezia: Agorà Edizioni. Gario, Paola (ed.). 2010. Lettere e Quaderni dell’archivio di Guido Castelnuovo. Roma: Accademia N. dei Lincei: http://operedigitali.lincei.it/Castelnuovo/Lettere_E_Quaderni/menu.htm (Retrieved June 2021). Gario, Paola. 2016. Segre, Castelnuovo, Enriques: Missing Links. In From Classical to Modern Algebraic Geometry. Corrado Segre’s Mastership and Legacy, eds. Gianfranco Casnati, Alberto Conte, Letterio Gatto, Livia Giacardi, Marina Marchisio, and Alessandro Verra, 289–323. Cham, Switzerland: Springer. Giacardi, Livia. 2015. The Italian School of Algebraic Geometry and the Teaching of Mathematics in Secondary Schools: Motivations, Assumptions and Strategies. In Geometry of Algebraic Varieties in Honor of Alberto Conte, Rendiconti del Seminario Matematico. Università e Politecnico, Torino 71 (3–4): 421–461. Giacardi, Livia. 2019. The Italian Subcommission of the International Commission on the Teaching of Mathematics (1908–1920). Organizational and Scientific Contributions. In National Subcommissions of ICMI and their Role in the Reform of Mathematics Education, ed. Alexander Karp, 119–147. Cham, Switzerland: Springer. Nastasi, Pietro and Aldo Scimone (eds.). 1995. Lettere a Giovanni Vacca. Palermo: Quaderni PRISTEM 5. Schubring, Gert. 2008. The origins and the early history of ICMI. International Journal for the History of Mathematics Education 3(2): 3–33. Carruccio, Ettore. 1971. Castelnuovo Guido. In Dictionary of Scientific Biography 3: 117. Conte, Alberto and Livia Giacardi. 1999. Guido Castelnuovo. In La Facoltà di Scienze Matematiche Fisiche Naturali di Torino, 1848–1998, Vol. II, ed. C.  Silvia Roero, 539–545. Torino: Deputazione subalpina di storia patria. Garnier, René. 1952. Notice Nécrologique sur M.  Guido Castelnuovo. Comptes Rendus de l’Académie des Sciences Paris 234: 2241–2244. Godeaux, Lucien. 1953. Guido Castelnuovo, Federigo Enriques et la géométrie. Revue Générale des Sciences Pures et Appliquées 60: 8–14. Hodge, William. 1953. Guido Castelnuovo. Journal of the London Mathematical Society 28: 120–125. Onoranze alla memoria of Guido Castelnuovo. Rendiconti di Matematica e delle sue Applicazioni 5, 13: 1–50 (with contributions by Enrico Bompiani, Francesco Severi, Ugo Papi, Sabato Visco, Beniamino Segre, Francesco P. Cantelli and with the list of publications). Terracini, Alessandro. 1951–52. Guido Castelnuovo. In Atti della Reale Accademia delle Scienze di Torino 86: 366–377.

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Publications Related to Mathematics Education Castelnuovo, Guido. 1907. Il valore didattico della matematica e della fisica. Rivista di Scienza 1: 329–337. Castelnuovo, Guido. 1909. Sui lavori della Commissione Internazionale pel Congresso of Cambridge. Relation del prof. G. Castelnuovo della R. Università di Roma. In Atti II Congresso della Mathesis, Padova, 20–23 Settembre 1909, Allegato F, 1–4. Castelnuovo, Guido. 1910. La scuola media e le attitudini che essa deve svegliare nei giovani. Federazione Nazionale Insegnanti Medi: 33–47. Castelnuovo, Guido. 1911a. Commissione internazionale dell’insegnamento matematico  – Congresso di Milano. Bollettino della Mathesis III: 172–184. Castelnuovo, Guido. 1911b. La rigueur dans l’enseignement mathématique dans les écoles moyennes. L’Enseignement Mathématique 13: 461–464. Castelnuovo, Guido. 1912a. I programmi di matematica proposti per il liceo moderno. Bollettino della Mathesis IV: 120–130. Castelnuovo, Guido. 1912b. La commissione internazionale dell’insegnamento matematico al Congresso di Cambridge. Bollettino della Mathesis IV: 131–139. Castelnuovo, Guido. 1912–1913. La scuola nei suoi rapporti colla vita e colla Scienza moderna. In Atti III Congresso della Mathesis, Genova, 21–24 Ottobre 1912, 15–21. Roma: Cooperativa Tip. Manuzio. Castelnuovo, Guido. 1913a. I programmi di matematica del liceo moderno. Bollettino della Mathesis V: 86–94. See also http://www.associazionesubalpinamathesis.it/wp-­content/ uploads/2017/12/pl24.pdf (Retrieved June 2021). Castelnuovo, Guido. 1913b. Risposta ad un’osservazione del Prof. Catania. Bollettino della Mathesis V: 119–120. Castelnuovo, Guido. 1913c. Osservazioni all’articolo precedente. Bollettino della Mathesis V: 143–145. Castelnuovo, Guido. 1914a. Discours de M. G. Castelnuovo. In Publications du Comité central II s., III, Compte Rendu de la Conférence Internationale de l’Enseignement Mathématique, Paris, 1–4 avril 1914, L’Enseignement Mathématique 16: 188–191. Castelnuovo, Guido. 1914b. Italie. L’Enseignement Mathématique 16: 295. Castelnuovo, Guido. 1914c. La riunione di Parigi della Commissione internazionale dell’insegnamento matematico. Bollettino della Mathesis VI: 85–88. Castelnuovo, Guido. 1918a. Il calcolo delle probabilità e le scienze di osservazione. Rivista di Scienza 23: 176–184. Castelnuovo, Guido. 1918b, Questioni di metodo nel calcolo delle probabilità. Rivista di Scienza 23: 241–248. Castelnuovo, Guido. 1919. La riforma dell’insegnamento matematico secondario nei riguardi dell’Italia, Bollettino della Mathesis XI: 1–5. Castelnuovo, Guido. 1920. Sull’insegnamento medio delle matematiche in Italia dal 1867 ad oggi. Bollettino della Mathesis XII: 17–21. Castelnuovo, Guido (rapporteur). 1923. Sopra i problemi dell’insegnamento superiore e medio a proposito delle attuali riforme. Rome: Tip. della R. Accademia Nazionale dei Lincei, pp. 1–12.

Photo Courtesy of the University of Turin.

Albert Châtelet

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11.11 Albert Châtelet (Valhuon, 1883 – Paris, 1960): President 1952–1954 Sébastien Gauthier and Catherine Goldstein

Biography Albert Châtelet was born on 24 October 1883 in Valhuon, a village of 600 inhabitants in the north of France. His father, François Châtelet, was the local elementary school teacher; he also received prizes for farming courses and for his reports to local societies for the improvement of agriculture (Condette 2009, pp. 40–41). From the Valhuon school, Albert Châtelet went to the Saint-Pol secondary school, and

S. Gauthier (*) Université Claude Bernard Lyon 1, CNRS UMR5208, Institut Camille Jordan, Villeurbanne, France e-mail: [email protected] C. Goldstein (*) CNRS, Institut de mathématiques de Jussieu-Paris Rive Gauche, Sorbonne Université, Université de Paris, Paris, France e-mail: [email protected]

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then to the Douai lycée, where, thanks to a public grant, he prepared for the highly selective competitive examination for the École Normale Supérieure (ENS) in Paris, where he was accepted in 1904. After a year of military service, he followed the usual curriculum of the ENS, which culminated with a second place at the Agrégation de mathématiques, just behind Georges Valiron. During this period, he met Marguerite Brey, a friend of his sister and a student in mathematics: they were married in August 1909, and she would leave her own career to take care of their nine children (Condette 2009, pp. 96–101). Scholarships from the Commercy Foundation then allowed Châtelet to prepare a thesis (initially supervised by Jules Tannery, the scientific director of the ENS), Sur certains ensembles de tableaux et leur application à la théorie des nombres, which he defended on 27 April 1911 (after Tannery’s death) (Condette 2007). Châtelet also selected the same themes for the prestigious Foundation Peccot-funded lectures he was chosen to give at the Collège de France between April and June 1912 and that he published under the title Leçons sur la théorie des nombres (Châtelet 1913). In these first mathematical works, Châtelet studied continued fractions and algebraic numbers in the arithmetical tradition created by Charles Hermite; he also systematized the number-theoretical use of matrices (then called tableaux) and used recent German developments in number theory, such as the theory of ideals and Hermann Minkowski’s geometry of numbers – a rare case in the French mathematics of the time – obtaining in particular new explicit results for cubic fields (Gauthier 2021). Just before and after his thesis, Châtelet was employed as a teacher in classes préparatoires (special classes preparing the competitive examinations for the ENS or the Ecole polytechnique) in Paris and then in Tours (Brasseur 2010). After these first experiences, he was charged with the lectures in rational and applied mechanics at the Faculty of Sciences and at the Electrotechnical Institute of Toulouse in the south of France. He had just been given a position back in his home region, at the University of Lille, in 1914, when World War I broke out (Condette 2007). Mobilized on 2 August 1914, in the health services of the army, he finally joined the Commission d’expériences d’artillerie navale de Gâvre40 in April 1916. He worked there (like other mathematicians such as Jules Haag or Arnaud Denjoy) on a variety of ballistic questions, including “the loading, priming and efficiency of projectiles, … personally directing with great competency the preparation and execution of firings and experiments of all kinds”,41 as is stated in the post-war testimony of satisfaction from the War Ministry, which also praised Châtelet for his “practical sense, a great clarity of judgment and a remarkable spirit of implementation” (quoted in Gauthier 2021).42 Châtelet finally took up his position in Lille in February 1919 (although he still acted for years as adviser for the Artillery Commission). From this moment,  Commission for Experiments in Naval Artillery, near Lorient in Brittany  The original text is: “des questions de chargement, d’amorçage et d’efficacité des projectiles, …, dirigeant enfin personnellement avec une grande compétence la préparation et l’exécution de tirs et d’expériences de toutes sortes”. 42  The original text is: “sens pratique, une clarté de jugement et un esprit de réalisation remarquable”. 40 41

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although he still published a few research articles on number theory and algebraic structures, his career took a new direction; he took on increasing administrative responsibilities, first as Doyen de la faculté des sciences de Lille (Dean of the Faculty of Sciences of Lille) from 1921 to June 1924 and then as Recteur de l’académie de Lille43 until January 1937 (Châtelet 1953; Condette 2007; Gauthier 2021). The whole region, and in particular the university, had been severely affected by the war, and Châtelet was at the origin of many projects for its reconstruction during this period. Among these, we may mention the establishment of an institute of fluid mechanics, a laboratory of astronomy, an observatory, an institute of social sciences, the construction of a new medical faculty, housing and a restaurant for students, etc. (Condette 2009; Gauthier 2021). In 1937, Châtelet was appointed director of Secondary Education at the Ministry of Education – meaning that he was in charge of the organization of teaching in France from elementary school to high school – and launched, with the support of the minister, Jean Zay, a modernization and democratization of the school system, unifying in particular the programmes of the various school tracks. In 1940, he was fired by the Vichy government and found himself for a time without a position, but the Faculty of Sciences of Paris decided to create a chair for higher arithmetic and proposed Châtelet as its candidate. Although this proposal was rejected by the government and Châtelet was officially posted to another university, he nevertheless lectured on higher arithmetic in Paris. In 1945, this position became permanent: Châtelet was hired as associate professor in April, and then full professor in October, in the newly created chair of Arithmetic and Number Theory of the Faculty of Sciences of Paris; he would become doyen (dean) of the faculty in 1949 until his retirement in 1954. He worked actively for the renovation of the university and its buildings. It was also during this period, from 1947 on, that Châtelet organized with Paul Dubreil a regular seminar on algebra and number theory which has lasted, under various organizers, until the present day. He also participated in the edition of Georges Halphen’s and Henri Poincaré’s complete works (Pérès 1960; Condette 2009). Châtelet was very active at a high administrative and political level. From July 1945 to August 1946, he was director of the Mouvements de jeunesse et de l’éducation populaire (Youth and Popular Education Movements) at the Ministry of Education. He wrote various reports for the government about education, research and universities and became vice-president of the Commission de la recherche scientifique et technique (Commission for Scientific and Technical Research), in charge of planning a modernization and equipment programme for theoretical and applied research. He was also engaged in the development of cultural relations with other countries: for example, in December 1949, he was sent to establish cultural agreements with Vietnam, and later, during the Cold War period, he promoted scientific and cultural links with the Soviet Union, East Germany and China, as a means to further international peace. He also belonged to several commissions of UNESCO;

43

 State Superintendent of Education for the Lille region.

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he chaired the International Commission for the Teaching of Mathematics from 1951 to 1954 and was the first president of the renovated ICMI. From 1955 on, he was also president of the Union rationaliste (Rationalist Union), an association created by Paul Langevin in the 1930s to promote rationality and science in public debate. The high point of Châtelet’s political engagement was probably his participation in the Union des Forces Démocratiques (Union of Democratic Forces), a political party created in 1958 around personalities such as Alfred Kastler, Pierre Mendès-France, François Mitterrand and Laurent Schwartz; it was a left-wing, noncommunist party, opposed to Charles De Gaulle’s policy, in particular with respect to colonization. Châtelet, a militant against the Algerian War, accepted the candidacy of this party against De Gaulle (and the Communist Party candidate, Georges Marrane) in the French presidential election of 1958, obtaining 8.4% of the votes of the electoral college (Condette 2009, Chaps. 6 and 7). Albert Châtelet died in Paris on 30 June 1960.

Contribution to Mathematics Education Châtelet expressed his interest in mathematical education as early as 1909 (the same year as his first number theory papers), when he published a study of the principles of geometry and their impact on elementary teaching for a pedagogical journal. From then until his death, he maintained this interest; due to his various responsibilities, both at a local and a national level, Châtelet was able to renovate the French educational system, extensively planning new sites, facilities and buildings. As rector, he promoted and inaugurated in 1926 a pilot school in the northern town of Saint-­Amand-­les-Eaux. While the French system then sharply segregated after primary school (often on a financial and social basis) between the pupils going to the free, local, practically oriented higher primary schools and those going to the longer curriculum of the fee-charging high schools, Saint-­Amand’s new (and free) school offered a 4-year common programme, with possible options and bridges, which partly inspired the reforms at a national level of the 1930s and later (Condette 2012). Châtelet also organized or participated in several conferences on education (Condette and Savoye 2011). As for the ICMI, Châtelet’s correspondence allows us a glimpse of the difficulties confronted by its first officials: to decide the conditions for admission and representation of new countries (when mathematical life in the 1950s could vary considerably from one country to another), to solve delicate issues of independence with respect to the different national mathematical societies and to the International Mathematical Union (particularly explicit on the occasion of the Amsterdam 1954 International Congress of Mathematicians and the subsequent administrative meeting of the IMU in The Hague) or simply to obtain recognition for pedagogical work. The exchanges between Heinrich Behnke, Marshall Stone and Châtelet clearly display these issues. For instance, on 26 August 1954, an aerogramme from Stone to Châtelet stated:

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Account must be taken of the real difficulties that formally complicate relations between the [International Mathematical] Union, ICSU [International Council of Scientific Unions], the member organizations, the national commissions and the commissions, especially ICMI, which in my opinion should enjoy the fullest possible autonomy. The trust and goodwill on which one can certainly rely within a scientific organization are not always effective substitutes for consultations on the means to achieve the intended ends.44 (Fonds Albert Châtelet, 81J 21, Archives départementales du Pas-de-Calais, Arras, France)

During the same month, Heinrich Behnke wrote to Châtelet: It is a very difficult matter to engage mathematicians, well-known for their research work, into problems of instruction. Most of our colleagues refuse to be active for our commission because they regard this kind of work of little value, and then even neglect to forward circulars. Twice I have sent out a letter asking to set up sub-­commissions and to designate delegates, to all adhering national organizations, using a list with addressees which I had received from Prof. Bompiani. There were only a few answers. … It was not so easy to convince the organization committee in Amsterdam of the value of our efforts.45 (Fonds Albert Châtelet, 81J 21, Archives départementales du Pas-de-Calais, Arras, France)

The journal L’Enseignement Mathématique, which had become the official journal of the ICMI, also suffered from various problems (in particular financial), and Châtelet accepted its direction after Behnke had replaced him as president of the ICMI. Châtelet was also active at a pedagogical level. He developed new university courses, for instance, on vector calculus (Châtelet and Kampé de Feriet 1924) or on algebra (Châtelet 1954–1966). More unusually, he collaborated with publishers to develop various series of schoolbooks. Following a change in the French programme for the teaching of arithmetic at an elementary level, he directed for the publishers Eugène Chimènes and Michel Bourrelier a series of books for nursery and primary schools, written together with schoolteachers and with the elementary school inspector Georges Condevaux (with whom he had already collaborated in the SaintAmand project), and which went through numerous editions (Radtka 2017). Châtelet also launched a collection of scientific textbooks for secondary teaching with the publisher Albert Baillière (including his own Géométrie et algèbre, written in 1935 with Roger Ferrieu, and an Arithmétique in 1943). His involvement went from choosing the authors to writing or supervising in detail the contents to discussing how to introduce mathematical concepts more efficiently and more naturally to proofreading and commenting on illustrations and layouts (Radtka 2018). Like Jean Zay, but also like the psychologist Jean Piaget with whom he collaborated, Châtelet favoured an active and inductive pedagogy, developing the critical

 The original text is: “Il faut tenir compte des difficultés réelles qui compliquent d’une façon formelle les relations entre l’Union, l’ICSU, les organisations adhérentes, les commissions nationales et les commissions, surtout l’ICI qui à mon avis doit jouir d’une autonomie aussi complète que possible. La confiance et la bonne volonté sur lesquelles on peut certainement compter au sein d’une organisation scientifique, ne sont pas toujours des substituts effectifs pour les consultations relatives aux moyens aptes à arriver aux fins envisagées”. 45  The letter is in English. Behnke sent an identical letter to Stone (Behnke to Stone, Oberwolfach, 11 August 1954, in IA, 14A, 1952–1954). 44

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and reflexive minds of the pupils. One of his arithmetic books for 10-year-olds (Condevaux and Châtelet 1949) has the title I Learn to Reason.46 “The various textbooks should be linked by their very content”, Châtelet wrote, and for this, “it suffices to accept the need for a humanist learning. e.g. in physics, not to be satisfied with mere technical study, but to also present the problems with their philosophical, historical, etc., aspects” (Fonds Albert Châtelet, 81J 45, Archives départementales du Pas-de-Calais, Arras, France, quoted in Radtka 2018, p. 155).47 Methods should be based on concrete manipulation of material objects, or on personal observation, and teachers should encourage collaboration among pupils, as well as among themselves. The issue was “simplifying and invigorating, teaching arithmetic by appearance and by action” (Radtka 2017, p. 184).48 Châtelet’s books for elementary classes thus promoted the use of dominos, sticks and marbles, and his correspondence with his publisher shows him discussing details of colour illustrations of hens and farmers, as well as whether or not one can legitimately write a product such as “100 m × 100 m” in a chapter on areas. Châtelet however insisted above all on a non-dogmatic approach, as in his own research, a “renovation without a revolution” (Radtka 2017, p. 177).49 In an address to the Congress of Childhood in 1931 on number learning, he concluded: The good method, the true method, the unique method, is the one [the schoolteacher] knows how to handle and to apply. The best way of teaching is that which each teacher practices in her class, provided that she does so confidently and joyfully.50 (Châtelet 1932, préface)

Sources Châtelet, Albert. 1911. Sur certains ensembles de tableaux et leur application à la théorie des nombres. Annales Scientifiques de l’École Normale Supérieure 28: 105–202. Châtelet, Albert. 1913. Leçons sur la théorie des nombres. Paris: Gauthier-Villars. Châtelet, Albert. 1953. Notice sur les titres et travaux scientifiques. Archives de l’Académie des Sciences, Dossier Châtelet, Paris. Brasseur, Roland. 2010. Quelques scientifiques ayant enseigné en classes préparatoires aux grandes écoles (saison 2). Bulletin de l’UPS 230 (83e année): 16–19. [Coll.] 1963. Hommage à Albert Châtelet, plaquette éditée à l’occasion de l’inauguration du centre universitaire Albert Châtelet le 6 juin 1963, sous la présidence du ministre de l’Education nationale.

 The original text is: “J’apprends à raisonner”.  The original text is: “Il faut donc que les divers ouvrages soient liés par leur substance même. Il suffit d’admettre la nécessité d’une étude humanisante. Ex. en physique ne pas se contenter d’une étude technique mais présenter les problèmes sous leur jour historique, philosophique, etc.” 48  The original text is: “Simplifier et vivifier, enseigner le calcul par l’aspect et par l’action”. 49  The original text is: “Renouveler sans révolutionner”. 50  The original text is: “La bonne méthode, la vraie méthode, l’unique méthode, est celle que chacune d’elles sait manier et appliquer. La meilleure façon d’enseigner est celle que chacune d’elles pratique dans sa classe, à condition qu’elle le fasse avec confiance et avec joie”. 46 47

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Condette, Jean-François. 2007. Albert Châtelet. In Dictionnaire biographique du mouvement ouvrier et du mouvement social: Période 1940 à 1968, ed. Claude Pennetier, Vol. 3: CA-COR, 254–257. Paris: L’Atelier. Condette, Jean-François. 2009. Albert Châtelet. La République par l’école (1883–1960). Arras: Artois Presses Université. Condette Jean-François. 2012. Une innovation pédagogique septentrionale: l’École unique de Saint-Amand-les-Eaux. Un « modèle » pour la réforme de l’enseignement moyen dans l’entredeux-guerres? Revue du Nord 394: 91–123. Condette, Jean-François and Antoine Savoye. 2011. Le congrès du Havre (31 mai-4 juin 1936): Albert Châtelet et la réforme de l’enseignement du Second degré. Carrefours de l’Education 31 (janvier-juin 2011): 61–88. Gauthier, Sébastien. 2021. Albert Châtelet: de la théorie des nombres à la politique universitaire. In La Grande Guerre des mathématiciens français, eds. Catherine Goldstein and David Aubin, to appear. Radtka, Catherine. 2017. Renouveler l’enseignement des mathématiques au primaire dans les années 1930 en France: le Cours d’Arithmétique Albert Châtelet aux éditions Bourrelier et son élaboration. In L’enseignement des mathématiques à l’école primaire (1880–1970): Etudes Brésil-France, eds. Renaud d’Enfert, Marc Moyon and Wagner Valente, 167–185. Limoges: PULIM. Radtka, Catherine. 2018. Aspects d’une trajectoire mathématique dans la France d’entre-deuxguerres: l’édition et le tournant pédagogique d’Albert Châtelet. Philosophia Scientiæ 22(1): 143–161. Sename, Céline. 2001. Répertoire numérique détaillé du Fonds Albert Châtelet, 81 J 1–128. ArrasDainville: Archives départementales de Pas-de-Calais. Pariselle, Henri. 1961. Notice Albert Châtelet. Bulletin de l’Association Amicale de Secours des Anciens Élèves de l’École Normale Supérieure: 31–32. Pérès, Joseph. 1960. Nécrologie: Albert Châtelet. Annales de l’Université de Paris 4 (octobredécembre): 578–582. Nécrologie. 1960. Le Courrier Rationaliste (24 juillet): 154–156. Nécrologie: Albert Châtelet. 1960. L’Enseignement Mathématique 2e s. 6: 1–2.

Publications Related to Mathematics Education Châtelet, Albert and Joseph Kampé de Feriet. 1924. Calcul vectoriel. Théorie, applications géométriques et cinématiques destinés aux élèves des classes de mathématiques spéciales et aux étudiants en sciences mathématiques et physiques. Paris: Gauthier-Villars. Châtelet, Albert. 1928–1929. La théorie des nombres positifs et négatifs dans l’enseignement du second degré. L’Enseignement Scientifique, nov. 1928: 40–48; dec. 1928: 70–76; jan. 1929:107–115; feb. 1929: 136–140; mar. 1929: 169–174. Repr. as: 1929. La théorie des nombres positifs et négatifs dans le second degré. Paris: Eyrolles. Châtelet, Albert. 1929. Les modifications essentielles de l’enseignement mathématique dans les principaux pays depuis 1910: La France. L’Enseignement Mathématique 28: 6–12. Châtelet, Albert. 1932. L’apprentissage des nombres: examen de quelques méthodes d’initiation arithmétique pour les écoles maternelles et les cours préparatoires des écoles primaires: conférence faite au Congrès de l’Enfance le 30 juillet 1931. Paris: Bourrelier-Chimènes. Châtelet, Albert and Roger Ferrieu. 1935. Géométrie et  algèbre, classe de 3e. Paris: Baillière et fils. Châtelet, Albert, with the coll. of L. Blanquet and E. Crépin and illustrations by H. Lerailler. 1947. Pour apprendre les nombres, à l’usage des maîtres de l’école maternelle et des cours préparatoires des écoles primaires, suivi de J’apprends les nombres. Paris: Bourrelier.

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Condevaux, Georges and Albert Châtelet. 1949. J’apprends à raisonner, arithmétique établie pour le cours moyen et les classes de 8e et 7e des lycées et collèges, application du programme 1945, examen d’entrée en 6e. Paris: Bourrelier. Châtelet, Albert. 1954–1966. Arithmétique et algèbre modernes, 3 Vols. Paris: PUF.

Photo Source: Wikimedia Commons.

Bent Christiansen

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11.12 Bent Christiansen (Aalborg, 1921 – Copenhagen, 1996): Vice-President 1975–1986, Ex Officio Member of the Executive Committee 1979–1982 Gert Schubring

Biography Bent Christiansen was born on 7 May 1921 in Aalborg (Denmark). He studied, from 1939 to 1944, mathematics, physics, chemistry and astronomy at the University of Copenhagen. After his university studies, he became a mathematics teacher committed to improving teaching methods and thus giving more students competence in, and an appreciation of, mathematics. From 1944 to 1957, he was a teacher at the Gymnasium in Holte, today a suburb of Copenhagen, but from 1949, he was himself also training other teachers at the Copenhagen State College of Education. The year 1958 brought a decisive change to his life: commissioned by UNESCO, he taught as a professor of mathematics at an African university, in Monrovia/Liberia, until 1960, and experienced the G. Schubring (*) University of Bielefeld, Bielefeld, Germany e-mail: [email protected]

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challenges from other cultural contexts. Upon his return, from 1961 on, he was called to the Royal Danish School of Educational Studies (Danmarks Lærerhøjskole) as a professor of mathematics and mathematics education. He remained in this position until his retirement in 1991. As Mogens Niss wrote in his obituary: Bent was a legend in mathematics education in Denmark and the Nordic countries. His impact on the development of the teaching and learning of mathematics in primary and lower secondary education can hardly be over-estimated. He wrote textbooks and books on mathematics education, especially the very influential ‘Goals and means in basic mathematics education’ (‘Mål og midler I den elementære matematik-undervisning’, 1967). He gave innumerable in-service courses and invited lectures at meetings and conferences. Naturally, he also served on hosts of national committees, including the Danish National Sub-Commission of ICMI (1961-1972). All this earned him a reputation as a charismatic, enthusiastic and extremely energetic mentor for generations of mathematics teachers, teacher trainers and colleagues. (Howson and Niss 1996, p. 23)

During the 1970s, Christiansen developed from a teacher trainer and a curriculum specialist to a didactician of mathematics who took a serious interest in all aspects of mathematics education. The focus of his development and research activities was at first the elaboration and practical experimentation of a great number of teaching materials and textbooks for students and also for teacher training and in-service training. A first such work was on combinatorics and probability for teachers, followed by an axiomatic development of geometry and a textbook “Mathematics 65”, for teachers.

Contribution to Mathematics Education Since the late 1950s, Christiansen came into ever closer contact with international developments for a modernization of mathematics instruction. It was first through his participation at the seminars for mathematics teachers at Arlon, Belgium, organized by the Centre Belge de Pédagogie de la Mathématique and Georges Papy and Willy Servais, and later on at meetings of the CIEAEM (Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques) and at regional conferences of ICMI, that he became acquainted with the innovators and their ideas. A telling experience was his participation at a regional ICMI conference in Aarhus in 1960 on “Modern Teaching of Geometry” where Jean Dieudonné repeated in a more provocative manner his approach “Euclid must go!”, but which was sharply criticized by Hans Freudenthal then. The outbreak of this bitter controversy about the goals for reforms was shocking for most of the participants. Christiansen engaged actively in modernizing mathematics teaching not only in Denmark but also in the Nordic countries. He became a key member of the Nordic Committee for the modernization of mathematics education, 1960–1967. While

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being an active propagator of the so-called new mathematics, he later on defended it against thoughtless misinterpreters. Since the late 1960s, Christiansen reactivated his relations with UNESCO.  In October 1968, he organized a working group within the International Seminar on In-service Training of Mathematics Teachers, which took place at the UNESCO Institute for Education in Hamburg; in March 1969, he gave a paper at the meeting in Cairo initiating the “UNESCO Mathematics Project for the Arab States” and continued to be involved in this project. Eventually, in 1972, he became the programme specialist for mathematics of UNESCO – at a time, when UNESCO was a force in education and contributed considerably to promote mathematics education. Particularly memorable initiatives in this period, until 1974  – on leave from the University of Copenhagen during this time – were the Conference on Language and Mathematics, a topic of crucial interest to developing countries (Nairobi 1974) and UNESCO’s series New Trends in School Mathematics, in particular its Volume 4 (1979) giving key results of ICME-3 at Karlsruhe. His special concern was to promote mathematics teaching in the developing countries. After the period of work for UNESCO, Christiansen joined the Executive Committee of ICMI in 1975 and was re-elected two times so that he served there for 12 years, until 1986. In 1979–1982, he was also ex officio member of the Executive Committee as representative of CTS/ICSU. His particular merit is the preparation and organization of ICME-3 in 1976, together with Hans-Georg Steiner. As a member of the International Program Committee, he prepared ICME-4 (Berkeley) and ICME-5 (Adelaide) too. Besides the international congresses, he was enormously active in preparing and organizing regional and thematic ICMI seminars and congresses. Two initiatives in launching new directions in research into mathematics education should be mentioned in particular. Starting from ICME-4, an international working group “Systematic Cooperation between Theory and Practice in Mathematics Education” (SCTP), co-chaired by him, and gathering active members like Piet Verstappen, Guy Brousseau, Tom Cooney, Alan Bell, Leone Burton, John Mason, Anna Sierpinska, Luciana Bazzini, Erich Wittmann, Heinz Steinbring and Falk Seeger, developed important contributions. The other initiative is the BACOMET group, started in 1978 by Bent Christiansen, Geoffrey Howson and Michael Otte: Basic Components of Mathematics Education for Teachers, elaborating well-known research for improving teacher education.

Sources Howson, Geoffrey and Niss Mogens. 1996. Obituary: Bent Christiansen 1921–1996. ICMI Bulletin 41: 21–24. Steiner, Hans-Georg. 1997. In Memoriam Bent Christiansen. ZDM 29: 97–101.

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Publications Related to Mathematics Education Christiansen, Bent. 1964. Elementær kombinatorik og sandsynlighedsregning [Elementary combinatorics and probability calculus]. Copenhagen: Munksgaard. Christiansen, Bent and Jonas Lichtenberg. 1965. Matematik 65: en elementær lærebog [Mathematics 65: an elementary textbook]. Copenhagen: Munksgaard. Christiansen, Bent. 1967. Mål og midler i den elementære matematikundervisning [Goals and means in basic mathematics education]. Copenhagen: Munksgaard. Christiansen, Bent. ca. 1962. Geometri for realafdelingen. Faglig tilrettelaeggelse og udarbejdelse af haefter [Geometry for the lower secondary school department. Professional organization and preparation of issues]. Copenhagen: DR, Danmarks Skoleradio/TV. Christiansen, Bent et  al. 1984. Mini-Conference at ICME 5: systematic co-operation between theory and practice in mathematics education: topic area research and teaching, Adelaide, Australia, August 25–29, 1984. Copenhagen: Royal Danish School of Educational Studies, Dept. of Mathematics. Christiansen, Bent, Albert Geoffrey Howson, and Michael Otte (eds.). 1986. Perspectives on mathematics education. Papers submitted by members of the Bacomet Group. Dordrecht: Reidel.

Photo Courtesy of Hanne Christiansen (Holte/Denmark).

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11.13 Emanuel Czuber (Prague, 1851 – Gnigl, 1925): Member of the Central Committee 1913–1920 Michela Malpangotto

Biography Emanuel Czuber was born in Prague on 19 January 1851. He graduated from the Realschule, a German secondary school, in 1869 and continued his studies at the German Technical University in Prague. Here, he took an active role in the Association for Free Lectures on Mathematics: the student association which will become the Union of Czech Mathematicians and Physicists. In spite of still being a student, Czuber became an assistant to Karel Koristka from 1872 to 1875, and in 1976, he obtained the right to lecture with a thesis on practical geometry (geodesy) submitted to the Technical University of Prague. From 1875 to 1886, he taught at the Second German Realschule in Prague, and in 1886, he was appointed ordinary professor at the German Technical University in Brno, becoming rector of that university in 1890–1891.

M. Malpangotto (*) CNRS, Centre Jean Pépin, Ecole Normale Supérieure-Ulm, Paris, France e-mail: [email protected]

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After this year, he was appointed ordinary professor at the Technical University in Vienna, becoming rector in 1894–1845, and holding the position of professor at the same University of Vienna until he retired in 1919. In 1909, he became related to the Austrian royal family, since his daughter Berta married the younger brother of Franz Ferdinand d’Este, the successor to Austrian throne. Hykšová (2008) stresses the role of Czuber on the development of probability theory in the Czech region. He studied the fundamentals of probability as far as related areas and applications such as the theory of errors. In this field, he contributed in 1900 to the Encyklopädie der mathematischen Wissenschaften with the essay Wahrscheinlichkeitsrechnung and wrote several books in German on probability and its philosophical foundations and on statistics such as Czuber (1884, 1891, 1899, 1903, 1923). Czuber died in Gnigl (Austria) on 22 August 1925.

Contributions to Mathematics Education Czuber’s interest in teaching is already shown by the textbooks he wrote before the foundation of ICMI: Lehrbuch über Differential-­und Integralrechnung (1898) and Einführung in die höhere Mathematik (1909). He also translated a textbook of probability theory by A. Meyer. As one of the directors of the Zeitschrift für das Realschulwesen, Czuber enlarged the collaborations of this review, by opening to mathematicians. When the international journal L’Enseignement Mathématique was founded, Czuber joined its Comité de Patronage, a kind of Editorial Board which existed until 1914; in 1909, this journal became the official organ of the Commission on the Teaching of Mathematics. He was delegate of Austria on the International Commission on the Teaching of Mathematics from 1909 and was a member of the Central Committee from 1913 until 1920, when the Commission dissolved. Czuber firmly led as president the Austrian subcommission (Klein and Fehr 1910, p. 127). In this sense, he took an active part in the development of a report about the teaching of mathematics in the principal kind of school structures. During the first meeting of the ICMI Central Committee at Basel on 28 December 1909, the Austrian section presented the first results of this study: three booklets, introduced by an article of Czuber himself, had already been published as supplements of the Zeitschrift für das Realschulwesen and, within the works announced for future publication, there was a study on polytechnic schools developed by Czuber himself and Emil Müller (Fehr 1910, p. 361). Czuber has always shown a great interest in scientific teaching in secondary school. During the Paris Congress of 1914, he chaired the session on the teaching of

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mathematics in the schools for engineers on 3 April 1914 and the session of 4 April dedicated to the discussion on the inquiries launched by the International Commission on the Teaching of Mathematics about the results obtained by the introduction of differential and integral calculus into the upper years of middle school and about the place and role of mathematics in higher technical instruction (Fehr 1914, pp. 171–172). He was personally involved in this second inquiry, taking part as well in the discussion following Paul Staeckel’s general report (Fehr 1920, p. 315).

Sources Czuber, Emanuel. 1884. Geometrische Wahrscheinlichkeiten und Mittelwerte. Lepzig: Teubner. Czuber, Emanuel. 1891. Theorie der Beobachtungsfehler. Leipzig: Teubner. Czuber, Emanuel. 1899. Die Entwicklung der Wahrscheinlichkeitstheorie und ihrer Anwendungen. Jahresbericht der Deutschen Mathematiker-Vereinigung 7: 1–271. Czuber, Emanuel. 1903. Die Wahrscheinlichkeitsrechnung und ihre Anwendungen auf Fehlerausgleichung. Statistik und Lebensversicherung. Leipzig: Teubner. (Second edition 1908, reprinted 1968). Czuber, Emanuel. 1923. Die philosophischen Grundlagen der Wahrscheinlichkeitsrechnung, Lepzig: Teubner. Fehr, Henri. 1910. Compte rendu des séances de la commission et des conférences sur l’enseignement scientifique et sur l’enseignement technique moyen. L’Enseignement Mathématique 12: 353–415. Fehr, Henri. 1913. Commission Internationale de l’Enseignement Mathématique. L’Enseignement Mathématique 15: 489–490. Fehr, Henri. 1914. Compte rendu de la conférence internationale de l’enseignement mathématique, Paris 1–4 Avril 1914. L’Enseignement Mathématique 16: 165–177. Fehr, Henri. 1920. La Commission Internationale de l’Enseignement Mathématique de 1908 à 1920. L’Enseignement Mathématique 21: 305–318. Hykšová, Magdalena. 2008. Contribution of Czech mathematicians to probability theory. In History and Epistemology in Mathematics Education, Proceedings of the Fifth European Summer University, eds. Evelyne Barbin, Naďa Stehliková, and Constantinos Tzanakis, 829–840. Plzeň, Czech Republic: Vydavatelsky servis. Klein, Felix and Henri Fehr. 1910. Commission Internationale de l’Enseignement Mathématique. Circulaire n. 2. L’Enseignement Mathématique 12: 124–139. Doležal, Eduard. 1928, Emanuel Czuber. Jahresbericht der Deutschen Mathematiker-Vereinigung 37: 287–297 (with the list of publications). Radon, Johann. 1951. Emanuel Czuber zum Gedächtnis. Nachrichten der Oesterreichischen Mathematischen Gesellschaft 5: 11–13.

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Publications Related to Mathematics Education Czuber, Emanuel. 1898. Lehrbuch über Differential-und Integralrechnung. Leipzig: B. G. Teubner. Czuber, Emanuel. 1909. Einführung in die höhere Mathematik. Leipzig: B. G. Teubner. Czuber, Emanuel. 1915. Die Lehrkanzeln der Mathematik. Gedenkschr. Die K.  K. Technische Hochschule in Wien 1815–1915, ed. Joseph Neuwirth, 356–360. Vienna: Selbstverlag der K. K. Technischen Hochschule in Wien. Czuber, Emanuel. 1915. Mathematik und Technik. Zum hundertjährigen Jubiläum der Wiener Technischen Hochschule. Jahresbericht der Deutschen Mathematiker-Vereinigung 24: 461–467.

Photo Source: Wikimedia Commons.

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11.14 Miguel de Guzmán (Cartagena, 1936 – Madrid, 2004): President 1991–1998, Ex Officio Member of the Executive Committee 1999–2002 Jaime Carvalho e Silva

Biography Miguel de Guzmán Ozámiz was born on 12 January 1936 in Cartagena (Murcia, Spain) into a family of sailors surrounded by tragic events: his father was executed with 157 other navy officers during the Spanish Civil War, when Miguel de Guzmán was only 6 months old. He studied industrial engineering from 1952 to 1954  in Bilbao (Vizcaya), but before completing his studies, he entered the Jesuit order and studied humanities and classical arts from 1954 to 1958 in Orduña (Vizcaya); he then went to Germany, where he studied philosophy in Munich until 1961, in the Berchmanskolleg. De Guzmán returned to Spain where he decided to study mathematics in order to achieve the solid knowledge that gives a sure domination over the world and the nature, and also to find a sense of beauty that attracts. As he wrote in a text posted

J. Carvalho e Silva (*) University of Coimbra, Coimbra, Portugal e-mail: [email protected]

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in his website,51 “Elementary geometry was the great hobby that attracted me to study mathematics and has remained with me throughout my life” (de Guzman 2000, p. 44). De Guzmán completed studies both in mathematics and statistics at the University of Madrid in 1965 and also became interested in the didactics of mathematics because of the humanistic and communicational aspect linked to mathematics. He went to study analysis to Chicago after a visit of Alberto Calderón to Spain in that year, and in 1968, he obtained a PhD in mathematics (his doctoral dissertation was on singular integrals with generalized homogeneity) at the University of Chicago under Calderón’s supervision. After earning his PhD, he became a professor at the Washington University in St. Louis (1968–1969). He considered this an immensely productive year because he took contact with many active colleagues in the field of analysis. In 1969, de Guzmán returned to Spain and became a professor of mathematical analysis at the Universidad Complutense de Madrid where he remained from then on, except for a period of two academic years (1982–1984) when he held a position at the Universidad Autónoma de Madrid. By this time, he had left the Jesuit order because he thought he could do better what he wanted to do outside the order, but he maintained excellent relations with other Jesuits. In 1971, he married Mayte García Monge and had two children: Miguel, an architect, and Mayte, a physician. In 1982, he was named a member of the Spanish Royal Academy of Mathematical, Physical and Natural Sciences, an institution where he started an ambitious programme aimed at the detection and stimulation of mathematical talent in students from elementary and secondary schools. According to Spanish mathematicians, de Guzmán was a key figure in Spanish mathematics in the twentieth century. Eugenio Hernández and Fernando Soria wrote in the ICMI Bulletin (n. 54, June 2004, p. 72) that Miguel de Guzmán was a central figure in the development of harmonic analysis in Spain and …captivated the enthusiasm of several generations of mathematicians. He was an extraordinary teacher and communicator and his ideas in mathematical education have had a profound influence on the teaching of mathematics in Spain and in the world. His books, translated into several languages, have made accessible to a large audience that extraordinary activity of the human spirit known as Mathematics.

To maintain alive the ideas of Miguel de Guzmán, in September 2007, the Faculty of Mathematical Sciences of the Universidad Complutense de Madrid created a chair (cátedra) named after him. Its activities address issues on mathematics education in Spain and internationally, through research projects, seminars and advanced courses. He died in Madrid on 14 April 2004.

 His Internet page, built immediately after his department opened its server, contains numerous texts – mostly in Spanish, a few in English. Most of the texts available in his former webpage are accessible at the server of the Cátedra Miguel de Guzmán (retrieved 16 May 2020): http://blogs. mat.ucm.es/catedramdeguzman/legado-2/ (Retrieved June 2021). 51

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Contribution to Mathematics Education De Guzmán was president of ICMI from 1991 to 1998. During his mandates, he initiated the ICMI Solidarity Fund, a very important solidarity programme established by ICMI in 1992 during ICME-7 held in Québec, Canada. The purpose of this programme is to help the development of mathematics education in countries in which there is a need of international assistance. The Solidarity Fund is based on private contributions by individuals, associations, etc. In its first years, it has contributed to different projects in Central America and Africa (reports have appeared in the ICMI Bulletin).52 The commitment of de Guzmán to education began very early and permeated all his activities. He not only said that “teaching in any form is very attractive”53 but also that the nature of the mathematical task makes it capable of stimulating important ethical aspects that we should foster in ourselves and that we should try to instil in any healthy educational system of our times. According to de Guzmán, one should have a broad vision of what mathematics is and reject the somewhat empty routine that mathematics so many times seems to entail in the classroom. He believed that mathematics is a way to understand the harmony of the universe, a science that seeks the truth, a tool that other sciences use and a creative activity with a beauty that, to use Plato’s words, can only be seen with the eyes of the soul. These facets of mathematics are profoundly human and should put mathematics into one of the great axes of our educational system, if teachers are well prepared for that task; he also believed that we are failing in the preparation of mathematics teachers at all levels. In (de Guzmán 1989), he presented aspects that should be considered in order to change the situation, such as the exploration of applications, games, etc., and he also discussed the impact of calculators and computers, and of new areas like discrete mathematics, etc. One of the aspects he emphasized the most is the role of history of mathematics. He maintained that history of mathematics is an important aide in: • Giving an idea of how peculiar the surge of mathematical ideas is • Fixing the period and place of significant ideas and problems, along with their motivation • Identifying the open problems in each period, their evolution and how they stand at present • Pointing out the historical connections of mathematics with other sciences, whose interaction traditionally produced a large number of important ideas One of Miguel de Guzmán’s main ideas was that mathematics teaching should pay particular attention to problem solving, placing the emphasis on the processes of  At present, the Solidarity Programme has relaunched its activities according to new lines suitable to the changed context and better coordinated with those of the IMU Commission for Developing Countries. 53  The original text is: “la enseñanza en cualquiera de sus formas es muy atractiva”. 52

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thinking, allowing the student to manipulate the mathematical objects, thus activating his mental capacity. He was very critical of textbooks for merely containing exercises and not real problems (unlike the textbooks he co-authored, which were magnificent and, fortunately, had a positive influence on others). De Guzmán claimed that problem solving is an important element in creating a passion for discovery and thought of it as an important element in the attempt to change the attitude of students towards mathematics. As a mathematician, de Guzmán tried to convince all other mathematicians to become involved in the troubled waters of mathematics education. At ICME-7, he, besides the presidential address, gave a lecture in which he discussed the right way to introduce young people to mathematics research. He tries to answer the following questions: “How should they be introduced to mathematical content and to mathematical theories? What is the attitude we should try to foster in them? What do those who are most successful in preparing young mathematicians actually do?” (de Guzmán 1994, p. 147). In 1996, he gave a talk at ICME-8 on the role of the mathematician in mathematics education, in which, admitting that mathematical education is a rather complex task, he claimed that all mathematicians should collaborate together in order to face its many difficult problems with efficiency promoting a global vision of mathematics in human culture (de Guzmán 1998). In this respect, mathematicians should devote their effort to several projects, namely: • • • •

Pre- and in-service preparation of teachers Research in mathematics education Mathematics education Educational treatment of young talent in mathematics

His efforts of popularizing mathematics can be seen in several of his books, for example, in Aventuras matematicas (1986), which was translated into several languages, including French (Aventures mathématiques) and English (The Countingbury Tales: Fun with Mathematics). This book also tries to convince the reader that mathematics problems should be dealt with following a certain heuristics similar to the one of Pólya. These ideas are well developed in another book, Para pensar mejor (To Think Better, 1991). In the paper Juegos matemáticos en la enseñanza (“Mathematical Games in Teaching”, 1984), he proposes another heuristics best suited to the use of mathematical games in teaching. His main ideas can be found in de Guzmán (2007). In order to keep the ideas and the example of de Guzmán alive, the RSME, the Spanish Royal Mathematics Society, in 2005 launched a Summer School of Education aimed at secondary school teachers of mathematics.54  The first such school was held in 2005 in El Pazo de Mariñán (Bergondo, La Coruña) with the title “Computers and Mathematics Education”. It is presently organized by the RSME and one of the societies affiliated to the FESPM, the Spanish Federation of Mathematics Teacher Societies. The tenth was organized in 2018 in the Facultad de Ciencias de la Universidad de La Laguna, Tenerife, with the title “Problems Solving as an Essential Part of Mathematical Activity” (https:// fespm.es/index.php/2018/02/26/x-escuela-de-educacion-matematica-miguel-de-guzman-2018/) 54

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In 2014, his family decided to publish de Guzmán’s personal journal where his passion for teaching is visible, with phrases like Lord, help me to communicate with all generosity what with effort I learn and give me to rejoice with all enthusiasm how much my students achieve beyond what I helped them to learn.55 (de Guzmán 2016, p. 174)

Sources Andradas, Carlos et al. 2004. Miguel de Guzmán Ozámiz, matemático y humanista. La Gaceta de la RSME, Suplemento 7(3). Asemil, J. (ed.)2005. Matemáticas: Investigación y Educación. Un homenaje a Miguel de Guzmán. Madrid: Anaya. De Guzmán, Miguel. 1975. Differentiation of Integrals in Rn. Berlin-Heidelberg-New York: Springer-Verlag. De Guzmán, Miguel. 1975. Ecuaciones diferenciales ordinarias: teoría de Estabilidad y Control [Ordinary differential equations: stability theory and control]. Madrid: Alhambra. De Guzmán, Miguel, Ireneo Peral, and Magdalena Walias. 1978. Problemas de ecuaciones diferenciales ordinaries [Problems of ordinary differential equations]. Madrid: Alhambra. De Guzmán, Miguel and Baldomero Rubio. 1979. Integración: Teoría y técnicas (Integration: theory and techniques). Madrid: Alhambra. De Guzmán, Miguel. 1981. Real Variable Methods in Fourier Analysis. Amsterdam-New YorkOxford: North-Holland. De Guzmán, Miguel. 1983. Impactos del Análisis Armónico (Impact of harmonic analysis), Discurso de ingreso en la Real Academia Española. De Guzmán, Miguel and Baldomero Rubio. 1990–93. Problemas, conceptos y métodos del Análisis Matemático (Problems, concepts and methods of Mathematical Analysis) (3 Vols). Madrid: Alhambra. De Guzmán, Miguel, Miguel Á. Martín, Manuel Morán, and Miguel Reyes. 1993. Estructuras fractales y sus aplicaciones (Fractal structures and its applications). Madrid: Labor. Andradas, Carlos. 2004. Miguel de Guzmán Obituary. ICMI Bulletin n. 54: 80–81. Rubio Segovia, Baldomero. 2005. Como homenaje a Miguel de Guzmán: Algunas reflexiones sobre educación matemática. Boletín de la Sociedad ‘Puig’ Adam de Profesores de Matemáticas n. 70: 34–46. Carvalho e Silva, Jaime. 2004. In memoriam Miguel de Guzmán (1936–2004), Educação e Matemática. 78: 17–19. Carvalho e Silva, Jaime. 2004. In memoriam: Miguel de Guzmán (1936–2004). HPM Newsletter n. 56: 5–7. Castrillón, Marco and María Gaspar. 2004. On Miguel’s project ESTALMAT. ICMI Bulletin n. 54: 79. Fernández Fernández, Santiago et al. 2004. Homenaje a Don Miguel de Guzmán. SIGMA. Revista de matemáticas 25 (Noviembre): 7–67. Herederos de Miguel de Guzmán Ozámiz (eds.). 2016. Miguel de Guzmán Ozámiz. Un legado de Fe. Madrid: Grand Guignol Ediciones.

 The original text is: “Señor, ayúdame para comunicar con toda generosidad lo que con esfuerzo aprendo y dame que me alegre con todo entusiasmo de lo mucho que mis alumnos logran más allá de lo que yo les he ayudado a aprender”. 55

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Hernández, Eugenio and Fernando Soria. 2004. Miguel de Guzmán Ozámiz (January 12, 1936 – April 14, 2004). ICMI Bulletin n. 54: 72–78. Hernández, Eugenio and Fernando Soria. 2006. A walk among the mathematical interests of Miguel de Guzmán (1936–2004). Newsletter of the EMS 61: 23–25. Pérez Sanz, Antonio. Miguel de Guzmán. El último pitagórico. DivulgaMAT. Retrieved on 19 May 2020 from http://platea.pntic.mec.es/~aperez4/miguel/Miguel%20de%20Guzm%E1n.htm Recio, Tomás. 2004. In Memoriam – Miguel de Guzmán Ozámiz (ICMI president 1991–1998). ICMI Bulletin n. 54: 70–72. Rico, Luis. 2015. Spanish Heritage in Mathematics and Mathematics Education. In The Proceedings of the 12th International Congress on Mathematical Education, ed. Sung Je Cho, 331–341. Cham Etc.: Springer. Rubio Segovia, Baldomero. Guzmán Ozámiz, Miguel de (1936–2004), DivulgaMAT. Retrieved on 19 May 2020 from http://www.divulgamat.net/divulgamat15/index.php?option=com_content &view=article&id=3405&directory=67&showall=1 Sánchez Rodríguez, Gerardo, Agustín Cuenca de la Villa, Alfonsa García López, Francisco García Mazarío. 2005. La visualización en la obra de Miguel de Guzmán. Boletín de la Sociedad Puig Adam de Profesores de Matemáticas n. 71: 12–30. Vázquez, Juan Luis, Jesús Ildefonso Díaz, Claudi Alsina, Jesús García, Baldonero Rubio, and Mª Luz Callejo de la Vega. 2004. Miguel de Guzmán (1936–2004). SUMA, Revista Sobre la Enseñanza y el Aprendizaje de las Matemáticas 46 (Junio): 5–22. Textos de Miguel de Guzmán 2005, Monografía 02 de SUMA.

 ublications Related to Mathematics Education and Popularization P of Mathematics De Guzmán, Miguel. 1977. Mirar y ver [Look and see]. Madrid: Alhambra (2004. Madrid: Nivola). De Guzmán, Miguel. 1983. Sobre la educación matemática [About mathematics education]. Revista de Occidente n. 26: 37–48. De Guzmán, Miguel. 1984. Juegos matemáticos en la enseñanza [Mathematical games in teaching]. In Actas de las IV Jornadas sobre Aprendizaje y Enseñanza de las Matemáticas, IV JAEM, 49–85. Sociedad Canaria de Profesores de Matemáticas “Isaac Newton”: Sociedad Canaria de Profesores de Matemáticas “Isaac Newton”. De Guzmán, Miguel. 1985a. Enfoque heurístico de la enseñanza de la matemática [Heuristic focus of mathematics teaching]. Aspectos didácticos de matemáticas. Publicaciones del Instituto de Ciencias de la Educación de la Universidad de Zaragoza 1: 31–46. De Guzmán, Miguel. 1985b. Cuentos con cuentas (Stories with calculations), Barcelona: Labor (2004. Madrid: Nivola). De Guzmán, Miguel. 1986. Aventuras matemáticas [Mathematical adventures]. Barcelona: Labor. De Guzmán, Miguel. 1987. Enseñanza de la matemática a través de la resolución de problemas. Esquema de un curso inicial de preparación [Mathematics teaching through problem solving; scheme of a course of preservice preparation]. Aspectos didácticos de matemáticas. Publicaciones del Instituto de Ciencias de la Educación de la Universidad de Zaragoza 2: 52–75. De Guzmán, Miguel. 1989. Tendencias actuales de la enseñanza de la matemática [Present tendencies of mathematics teaching]. Studia Paedagogica. Revista de Ciencias de la Educación 21: 19–26. De Guzmán, Miguel. 1991. Para pensar mejor [To think better]. Barcelona: Labor (1994. Madrid: Pirámide). De Guzmán, Miguel. 1994. The origin and evolution of mathematical theories. In Proceedings of the 7th International Congress on Mathematical Education. Vol. 2 Selected Lectures, eds.

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Claude Gaulin, Bernard R.  Hodgson, David H.  Wheeler, and John C.  Egsgard, 147–155. Sainte-Foy: Les Presses de l’Université de Laval. De Guzmán, Miguel. 1995. Aventuras matemáticas: una ventana hacia el caos y otros episodios [Mathematical adventures: a window to the chaos and other episodes]. Madrid: Pirámide. De Guzmán, Miguel and Claudi Alsina. 1996. Los matemáticos no son gente seria [Mathematicians are not serious people]. Barcelona: Rubes. De Guzmán, Miguel. 1997. El rincón de la pizarra: ensayos de visualización en Análisis Matemático [The corner of the blackboard: essays on visualization of mathematical analysis]. Madrid: Pirámide. De Guzmán, Miguel. 1997. Del lenguaje cotidiano al lenguaje matemático [From everyday language to mathematical language]. Epsilon: Revista de la Sociedad Andaluza de Educación Matemática “Thales” n. 38: 19–36. De Guzmán, Miguel. 1998. El papel del Matemático en la educación matemática [The role of the mathematician in mathematics eduction]. In Proceedings of the 8th International Congress on Mathematical Education, eds. Claudi Alsina, José María Alvarez, Mogens Niss, Antonio Pérez, Luis Rico, and Anna Sfard, 47–63. Sevilla: SAEM Thales. De Guzmán, Miguel. 2000a. Some ethical aspects in the mathematical activity. Round Table “Shaping the 21st Century”, 3ECM, 11–14 July 2000. Barcelona. http://blogs.mat.ucm.es/catedramdeguzman/some-­ethical-­aspects-­in-­the-mathematical-activity/ De Guzmán, Miguel. 2000b. El sentido del ICMI hoy [The sense of ICMI today]. Números n. 43–44: 445–448. De Guzmán, Miguel. 2002. La experiencia de descubrir en Geometría [The experience of discovering in geometry]. Madrid: Nivola. De Guzmán, Miguel. 2003. Un programa para detectar y estimular el talento matemático precoz en la Comunidad de Madrid [A program to detect and stimulate the precocious mathematical talent in the Madrid community]. Uno: Revista de didáctica de las matematicas n. 33: 20–33. De Guzmán, Miguel. 2004. Cómo hablar, demostrar y resolver en Matemáticas [How to speak, prove and solve in Mathematics]. Madrid: Anaya. De Guzmán, Miguel. et al. 2004. Laboratorio de Matemáticas [Mathematics Laboratory]. Facultad de Matemáticas de la Universidad Complutense de Madrid. De Guzmán, Miguel. 2007. Enseñanza de las ciencias y la matemática [Teaching of Sciences and Mathematics]. Revista Iberoamericana de Educación 43: 19–58.

Photo Source: https://tesoricosdemurcia.com/people/ciencias/miguel-de-guzman-6/

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Georges de Rham

11.15 Georges de Rham (Roche, 1903 – Lausanne, 1990): Ex Officio Member of the Executive Committee 1963–1966 Margherita Barile

Biography Georges de Rham was born in Roche in the district of Aigle in Canton Vaud, Switzerland, on 10 September 1903. After attending a secondary school specializing in classical studies in Lausanne, and though feeling inclined, since childhood, towards the arts (watercolour and drawing), and fascinated by philosophy and literature, in 1921, he entered the Faculty of Sciences at the University of Lausanne. Many years later, he said that “the feeling of a gap in my knowledge, the curiosity and the charm of mystery”56 (Burlet 2004, p. 5) had been responsible for this choice. After five semesters, he gave up chemistry, biology and physics to devote himself 56

 The original text is: “Le sentiment d’une lacune, la curiosité et l’attrait du mystère”.

M. Barile (*) University of Bari “Aldo Moro”, Bari, Italy e-mail: [email protected]

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totally to mathematics. This subject, which he once used to consider “a closed domain, where it seemed impossible to do anything new”57 (ibidem, and, similarly, de Rham 1980, p.  19), now appeared to him as an “immense domain” (ibidem). With the support and advice of his two good teachers (De Rham 1980, p.  19), Gustave Dumas and Dmitry Mirimanoff, he graduated in the autumn of 1925. In June 1931, he received his doctoral degree in Paris, with a dissertation on the topology (analysis situs) of differentiable manifolds; his interest for this topic had originated in Poincaré’s memoirs, where he had run into his famous conjecture (which he had also tried to prove), and which, in November 1926, motivated him to move to Paris to attend the lectures delivered by Henri Lebesgue at the Collège de France. There he would meet Élie Cartan, who was to become his thesis adviser. “The chance of my life”58 (ibidem, p.  24): with these words, de Rham described the moment when he came across the note59 by Élie Cartan, published in 1928, which suggested a connection between the theory of multiple integrals on a closed manifold and its topological invariants, but without giving a proof. This goal was achieved by the result which is nowadays known as “de Rham’s theorem” and can be formulated in terms of isomorphisms between cohomology groups. With its many consequences, it marked a milestone in contemporary mathematics. Its implications reach far into sheaf theory, complex geometry, algebraic geometry, algebraic topology and even non-­commutative differential geometry. In physics, it is related to many aspects of modern field theory. It is the emblem of de Rham’s unifying view  – focused on general, abstract structures – which culminated in his theory of “currents”, born as an intuition inspired by electromagnetism and further developed in the framework of the distributions introduced by Laurent Schwartz in 1945. Other important contributions of de Rham concern Riemannian manifolds, for which he gave a beautiful reducibility theorem and studied the harmonic differential forms. His textbook, Variétés différentiables, first published in 1955, was reprinted in 1960, in 1973 and in 1982 and translated into Russian (1956) and English (1984). De Rham spent his whole academic career at the University of Lausanne, where he had been working as Dumas’ assistant since his graduation, and at the University of Geneva: at both institutions, he was appointed extraordinary professor in 1936, then became full professor and, finally, the title of emeritus professor was conferred on him when he retired, in the early 1970s. He visited the University of Göttingen (1930/1931), Harvard (1949/1950), the Institute for Advanced Study at Princeton (1950, 1957/1958) and the Tata Institute of Fundamental Research in Bombay (1966). In August 1960, he gave a course at the Centro Internazionale Matematico Estivo in Saltino, near Florence. These were his only long stays abroad; in fact, he distinguished himself as a scientist devotedly attached to his country. While teaching a great number of university courses in analysis and geometry, he committed  The original text is: “un domaine fermé où je ne concevais pas qu’on puisse faire quelque chose de nouveau”. 58  The original text is: “ce fut la chance de ma vie”. 59  Cartan, Élie. 1928. Sur les nombres de Betti des espaces de groupes clos. Comptes Rendus 187: 196–198. 57

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himself to promote mathematical culture in Switzerland. From 1932 onwards, he took part in the organizing committee of the international colloquia at the Cercle Mathématique of the French-speaking Swiss universities. For many years, he was editor of the journals Commentarii Mathematici Helvetici (1950–1966) and L’Enseignement Mathématique (1967–1978). Furthermore, he held leading positions in several national organizations: he was member of the Commission Fédérale pour l’Encouragement des Recherches Scientifiques (1950/1965) and of the Conseil National de la Recherche (1956–1969) and president of the Swiss Mathematical Society (1944–1945), which awarded him the status of honorary member in 1960. His efforts were constantly directed towards enhancing international exchange, especially in the aftermath of World War II. He succeeded in maintaining his relations with many prominent colleagues from all over Europe, whom he regularly invited to give seminars; he also supported the participation of Switzerland in the project of creation of the Institut des Hautes Études Scientifiques in Bures-sur-Yvette, France. As a recognition of de Rham’s lifelong engagement, the Swiss foundation named after Marcel Benoist awarded him its annual prize in 1965 – so far, the only case where this honour has fallen upon a “pure” mathematician.60 Earlier he had received doctorates honoris causa from the universities of Strasbourg (1954), Grenoble (1955) and Lyon (1959) and from the École Polytechnique Fédérale of Zurich (1961). De Rham was elected foreign member of the Accademia Nazionale dei Lincei in Rome (1962), corresponding member of the Akademie der Naturwissenschaften in Göttingen (1974) and foreign associate member of the Académie des Sciences de l’Institut de France (1978). On the occasion of the mathematical colloquium in honour of de Rham held in Geneva in March 1969, Komaravolu Chandrasekharan wrote: He did as much as any one man could do to bring mathematicians together, young and old, classical and modern, from the East and West. He found a special joy in the spectacle of younger colleagues adding to the old heritage, and did as much as he could to encourage them.61

Georges de Rham died in Lausanne on 7 October 1990.

Contribution to Mathematics Education In one of the hardest periods of the Cold War (1963–1966), de Rham was president of the International Mathematical Union and, as such, member ex officio of the Executive Committee of the International Commission on Mathematical Instruction

 The list of laureates also includes the mathematician Jürg Martin Fröhlich (1997), who is also a theoretical physicist. 61  Letter to the editors published in Essays on Topology and Related Topics, Mémoires dediés à Georges de Rham, eds. André Haefliger and Raghavan Narasimhan, viii, Berlin: Springer, 1970. 60

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(ICMI). In this commission, he was national delegate of Switzerland from 1955. He was one of the organizers of the seminar on “The teaching of analysis in the secondary school and the university” (26–29 June 1961, Lausanne), jointly sponsored by ICMI and the Swiss Mathematical Society. According to the testimonies of those who have known de Rham, it seems hard to separate, in his personality and his public engagement, the loyal colleague from the generous friend, the passionate researcher from the fatherly teacher and the brilliant scientist from the brave alpinist. He regarded proving a theorem, delivering a lecture or reaching the top of a mountain as a personal endeavour, requiring a full involvement of the deepest and most precious faculties of the individual, in the name of some kind of transcendental beauty. On the occasion of the centenary of de Rham’s birth, his former doctoral student, Oscar Burlet, remembers that he once explained to him: For alpinism is not only a physical exercise, but it also a task for the mind, and it allows one to create a marvellous harmony between nature, soul and body. (Burlet 2004, p. 6)62

Climbing higher to attain a larger view could be considered as a metaphor for de Rham’s constant attempt to advance in abstraction, to gain generality by looking at single problems and objects from above. The development set off by his first proof of the theorem bearing his name moved exactly in the direction of embracing specific questions in a more and more general framework. His scientific approach is based on the comparison of structures, which, unlike in the so-called abstract nonsense, should not be treated as empty buildings, but employed as containers that enable us to grasp different particular cases at the same time. He thus described his first encounter with this method at school: I remember a mathematics professor who made me understand the idea of algebra, which consists in making computations with an unknown x, and produces a great simplification in solving arithmetical problems. (Ibidem, p. 3)63

Clarity and conciseness in exposition were the most evident effects of this principle in his teaching, which, in his global vision of scientific culture, had to be closely intertwined with research. For him, both activities required a concrete effort of interpretation and nourished one another: the joy of discovering surprising connections between distant areas could not be detached from the joy of sharing these “nice things” (Unknown author 1990, p. 207)64 with others. Essentiality, linearity of thought and calmness made up the elegance of his lectures and seminars, where, according to Henri Cartan, he was able to “suggest a lot of things in few words” (Cartan 1970, p. 1).65 His approach to the audience was always full of benevolence  The original text is: “Car l’alpinisme n’est pas seulement un exercice physique, c’est aussi un travail de l’esprit”. 63  The original text is: “Je me souviens pourtant d’un des nombreux professeurs de mathématiques qui m’avait fait comprendre l’idée de l’algèbre, qui consiste à calculer avec une inconnue x, et la grande simplification qui en résulte pour résoudre les problèmes d’arithmétique”. 64  The original text is: “de belles choses”. 65  The original text is: “suggérer beaucoup de choses en peu de mots”. 62

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and care. In de Rham’s obituary in the Notices of the American Mathematical Society, Raoul Bott thus recalls the profound impression that de Rham left on him as a graduate student: De Rham had a subtle charm which drew younger people to him immediately. In those early days at Princeton he would easily mingle with the boisterous postdocs, his exquisite manners contrasting amusingly with our rude ways. (Bott 1991, p. 115)

De Rham would reply that he just intended to follow the example of Dumas, who had initiated him to mathematics in Lausanne, and whom he admired for his ability in conveying his passion to others, to make them think autonomously, by “never stopping at the purely formal aspect of a question, but always seeking the deep general idea that enlightens all things from inside, from their centre” (De Rham 1955a, p. 121).66 This is what he writes, in 1955, in the memorial note published in Elemente der Mathematik. Beyond this, de Rham considered moral straightness, altruism and respect as the very reasons of Dumas’ success in teaching. “Never forget that you must love your pupils” (ibidem, p. 122)67 was his favourite advice to his assistants. And de Rham cannot refrain from mentioning that his communication talents and human qualities were certainly rooted in his devotion to philosophy and literature. While treasuring the crucial role of teachers, at all levels, from school to university, de Rham firmly condemned the passive attitude of certain students, who only act as recipients, and are not willing to elaborate the subject by themselves (as he had been forced to do, when, at the age of 15, the Spanish flu epidemic had kept him away from the collège for several months). He once addressed this recommendation to a class of pupils: The courses, the books should only be suggestions and inspirations to work: a mathematician must judge on his own account, he must be critical and must not admit anything which he has not clearly recognized himself as well-founded. (Burlet 2004, p. 4)68

He, indeed, used to carefully verify the content of all the articles that he quoted in his papers; in his opinion, this belonged to scientific rigour and intellectual integrity, two values which he obstinately defended throughout his life.

Sources De Rham, Georges. 1980. Quelques souvenirs des années 1925–1950. Cahiers du séminaire d’histoire des mathématiques 1: 19–36. De Rham, Georges. 1981. Œuvres mathématiques. Université de Genève: L’Enseignement Mathématique.

 The original text is: “Ne s’arrêtant jamais à l’aspect purement formel des questions, il cherchait toujours l’idée profonde et générale qui éclaire les choses de l’intérieur, du centre”. 67  The original text is: “n’oubliez jamais qu’il faut aimer ses élèves”. 68  The original text is: “Les cours, les livres ne devraient en somme être que des suggestions et des excitations au travail, des invitations: le mathématicien doit tout juger par lui-même, il doit être critique et ne rien admettre qu’il n’ait clairement reconnu lui-même comme fondé”. 66

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Audin, Michèle. 2012. Cartan, Lebesgue, de Rham et l’analysis situs dans les années 1920. Scènes de la vie parisienne. Gazette des Mathématiciens 134: 49–75. Cartan, Henri. 1970. Les travaux de Georges de Rham sur les variétés différentiables. In Essays on Topology and Related Topics, Mémoires dediés à Georges de Rham, eds. André Haefliger and Raghavan Narasimhan, 1–11. Berlin: Springer. Cartan, Henri. 1992. La vie et l’œuvre de Georges de Rham. La Vie des Sciences, Comptes Rendus, série générale 9: 453–455. Chatterji, Srishti and Manuel Ojanguren. 2013. A glimpse of the de Rham era. Notices of the International Congress of Chinese Mathematicians 1: 117–137. Chenaux, Jean-Philippe. 1978. Un mathématicien suisse élu à l’Académie des Sciences de Paris. Georges de Rham ou la passion de la simplicité. Interview with Georges de Rham. Journal de Genève: September 14. Ojanguren, Manuel and Jérôme Scherer. 2010. Georges de Rham et le problème d’homéomorphie des rotations. Elemente der Mathematik 65: 1–7. Weil, André. 1952. Sur les théorèmes de de Rham. Commentarii Mathematici Helvetici 26: 119–145. Bott, Raoul. 1991. Georges de Rham 1901–1990. Notices of the American Mathematical Society 38: 114–115. Burlet, Oscar. 2004. Souvenirs de Georges de Rham. École Polytechnique Fédérale de Lausanne, 22 pp. Retrieved on 26 April 2020 from https://math.ch/cuso/activities/deRham/derham.pdf Eckmann, Beno. 1992. Georges de Rham 1903–1990. Elemente der Mathematik 47: 118–122. Unknown author. 1990. Georges de Rham 1903–1990. L’Enseignement Mathématique s. 2, 36: 207–214.

Publications Related to Mathematics Education De Rham, Georges. 1936. Sur la méthode d’abstraction en mathématiques. Leçon Inaugurale. In Georges de Rham 1903–1990, eds. Daniel Bach, Oscar Burlet, and Pierre de la Harpe, 47–57. Le Brassus: Imprimerie Dupuis SA. De Rham, Georges. 1947a. Un peu de mathématiques à propos d’une courbe plane. Elemente der Mathematik 2: 73–76. De Rham, Georges. 1947b. Un peu de mathématiques à propos d’une courbe plane (suite). Elemente der Mathematik 2: 89–97. De Rham, Georges. 1955a. Gustave Dumas. Elemente der Mathematik 10: 121–122. De Rham, Georges. 1955b. Henri Fehr (1870–1954). Verhandlungen der schweizerischen naturforschenden Gesellschaft 135: 334–339 (with a portrait). De Rham, Georges. 1957. Sur un exemple de fonction continue sans dérivée. L’Enseignement Mathématique s. 2, 3: 71–72. De Rham, Georges. 1976. L’Enseignement Mathématique – Revue internationale et la Commission Internationale de l’Enseignement Mathématique (CIEM). ICMI Bulletin 7: 29–34.

Photo Source: Wikimedia Commons.

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Julien Desforge

11.16 Julien Desforge (Calvi, 1891 – Bourg-la-Reine, 1984): Secretary 1955–1958 Éric Barbazo

Biography Julien Desforge was born in Calvi (Corsica) on 30 November 1891, the son of an army officer. He studied in the lycée of Briançon on Hautes-Alpes and later in the lycée of Toulon (south-east of France). He was a brilliant student both in scientific and literary domains. Because of these school skills, he hesitated in choosing the sector allowing him to enter the École Normale Supérieure. In the end, in 1908, he decided to enter the Lycée Saint Louis in Paris, where he followed the courses of mathématiques spéciales delivered by Arthur Tresse. In the following years, he kept in constant contact with his old mathematics teacher. In 1910, Desforge passed both exams for the admittance to the École Polytechnique and to the Ecole Normale Supérieure. He chose the latter. Among his teachers à la Sorbonne were Émile Borel, Henri Lebesgue and Émile Picard. He obtained the Agrégation de mathématiques for becoming a mathematics teacher in 1914, the year when World War I has broken out.

É. Barbazo (*) Établissement International Français, Houston, TX, USA

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The life of Desforge, an officer’s son, is closely linked to the two World Wars of the twentieth century. Just after the Agrégation, he was mobilized in the battalion of the Chasseurs alpins in Alsace. He was seriously injured, but, after his convalescence, he came back to the front and participated in many military actions for which he was repeatedly honoured. In 1939, he was again mobilized. He was taken prisoner by the Germans, and during his captivity, he was promoted to Inspecteur général de l’Instruction publique of the Vichy government. In 1941, when the captivity ended, Desforge started his new job as an inspector. His main task was to care for the connection between the Ministry of Public Instruction and the “universities of captivity”, which were special universities created in the prison camps. His action allowed the official recognition of the studies carried out in these universities and, after the war, the validation of the degrees which were resulting from the studies in those conditions. Desforge was first of all a teacher of the classes préparatoires to the grandes écoles. His first position was at the lycée of Nantes (mathématiques spéciales). In 1925, he was transferred to Paris at the Lycée Saint Louis (mathématiques spéciales) until his appointment as inspecteur générale. He retired in 1963 and passed away in Bourg-la-Reine (Hauts-de-Seine) on 24 July 1984.

Contributions to Mathematics Education Desforge was a teacher of mathématiques spéciales whose pedagogical qualities were unanimously recognized, both by his colleagues and his former students. He was therefore naturally interested in both higher and secondary education. His relocation in Paris fostered contact with different institutions and scientific societies which were involved in the development of mathematics teaching. He became president of the Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP) from April 1931 until February 1932 and again from April 1934 until March 1937. Desforge’s presidency took place in the context of important reforms in the French secondary school in the aftermath of the scientific equality principle (principe d’égalité scientifique) launched in the programmes by the Ministry of Public Instruction in 1925. APMEP was radically opposed to this principle, which standardizes the mathematical content of the programmes from college to the classe de première. For mathematics teachers, the supposed equality was a pretext for reducing the number of hours of science introduced by the reform of 1902 in the secondary education cycles. The consequences of a general lowering of scientific knowledge were harshly felt and denounced for many years by the entire scientific community and by successive presidents of APMEP.  Desforge’s deep acquaintance of the classes preparing for the French scientific grandes écoles made him one of the main opponents to the scientific equality principle. The presidency of APMEP enabled Desforge to grasp the problems both mathematical and pedagogical which affect secondary teaching, e.g. the standardization

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of notations and of definitions of a mathematical dictionary. The discussion on these themes, which started at the beginning of World War I, continued until the 1960s. Desforge’s interest in pedagogical issues did not stop after his appointment as a general inspector. Then he wrote many reports prompting teachers to use certain notations rather than others and certain formulations of a definition rather than others. In 1931, Desforge was presented by Turmel and Michel became a member of the Société Mathématique de France (SMF). He became secretary of SMF in 1937 and remained a member of its council until 1942. It was within the SMF that he rubbed shoulders with leading mathematical personalities such as Maurice Fréchet, with whom he will continuously maintain close relations. Being neither a researcher nor an academic, he thus kept abreast of developments in contemporary mathematics. His various responsibilities and his excellent knowledge of the French secondary and higher education systems gave him national and international stature. In 1932, the Commission Internationale de l’Enseignement Mathématique (CIEM) under the presidency of David E.  Smith resumed the inquiry launched before World War I on the theoretical and practical preparation of mathematics teachers in the various countries. In France, Desforge gathered the observations and the answers to the inquiry of the French mathematics teachers. His report was presented at the International Congress of Mathematicians held in Zurich in 1932. The report dealt with the pre-service and in-service teacher training, scientific teaching, mathematical and pedagogical knowledge for teaching, professional development, laws concerning teachers and the role of mathematics in the teaching. In these subjects, he was unanimously considered an expert. The report by Desforge and Ghidale Iliovici was published with the title “Rapport sur la préparation théorique et pratique des professeurs de l’enseignement secondaire” in L’Enseignement Mathématique. Desforge was secretary of CIEM from 1955 to 1958 under the presidency of Heinrich Behnke. He was also a delegate to ICMI (L’Enseignement Mathématique. 1955. s 2, 1: 195–198) and a member of the commission appointed by IMU for revising the statute of ICMI (ICMI Archives 14A 1955–1957, 33). In 1936, Desforge took on major responsibilities in France. He was elected at the Conseil supérieur de l’Instruction Publique, a body under the direct authority of the French Minister of Education. He sat as the representative of the teachers agrégés de mathématiques in replacement of Pierre Chenevier who has been promoted to the rank of inspector general. In his letter of nomination, he stated the reasons for his willingness to be elected: The serious problems which affect the field of national education for many years (formation and recruitment of teachers, scientific equality, programs, assessment, and evaluation of pupils, organization of the teaching) are particularly urgent.69 (Bulletin de l’Association des Professeurs de Mathématiques 1936, 97, p. 61)

 The original text is: “Les graves problèmes qui se posent depuis plusieurs années dans le domaine de l’éducation nationale (formation et recrutement des professeurs, égalité scientifique, programmes, sélection des élèves, organisation de l’enseignement), sont particulièrement pressants”. 69

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From 1941, when he was promoted to the position of inspector general of National Education, Desforge was constantly interested and involved in the development of mathematics teaching and the evolution of the curricula that took place after World War II. The subject of preparing future teachers continued to be one of his major concerns. He was president of the jury of the Certificat d’Aptitude à l’Enseignement dans les Collèges (CAEC) which later became the Certificat d’Aptitude au Professorat de l’Enseignement du Second degree (CAPES) in 1951, a competitive examination for teachers of French secondary education. He was also a member of the jury of the Agrégation feminine for several years. Moreover, in the aftermath of World War II, the curricula and methods of teaching mathematics have continued to evolve. Desforge contributed to the development of the instructions that accompanied the 1946 curricula, the complementary instructions of 1957 that preceded the changes to the curricula of college and lycée from 1960 to 1963. He also took part in the development of the curricula for the Classes Préparatoires aux Grandes Écoles, which are specific to France and to which Desforge remained particularly attentive throughout his career.

Sources Brasseur, Rolland. 2020. Julien Desforge. Dictionnaire des professeurs de mathématiques en classe de mathématiques de spéciales. Retrieved on 9 February 2020 from https://drive.google. com/file/d/0B71JfRYrV2lYRmVfbTBqTHhsRlE/view Magnier, André. 1986. Julien Desforge. Association Amicale de Secours des Anciens Élèves de l’École Normale Supérieure: 38–41.

Publications Related to Mathematics Education Desforge, Julien. 1940. Géométrie. Paris: Hatier. Desforge, Julien. 1950. Rapport sur l’agrégation de mathématiques agrégation féminine, session de 1950. Paris: Imprimerie nationale. Desforge, Julien. 1952. Préface. In André Fouché, La pédagogie des mathématiques. V-VIII. Paris: Presses universitaires de France. Desforge, Julien, Ghidale Iliovici, and Paul Robert. 1936. L’œuvre de M. Jacques Hadamard et l’enseignement secondaire. L’Enseignement Scientifique 9 (84, janvier): 97–117. Zadou-Naïsky, Georges. 1954, Les sciences physico-mathématiques dans l’enseignement. Paris: Presses universitaires de France. Préface de Marc Bruhat et Julien Desforge. Iliovici, Ghidale and Julien Desforge. 1933. Rapport sur la préparation théorique et pratique des professeurs de l’enseignement secondaire français. L’Enseignement Mathématique 32: 208–239.

Photo Source: Roland Brasseur. Dictionnaire des professeurs de mathématiques en classe de mathématiques spéciales 1914–39.

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Stanley Herbert Erlwanger

11.17 Stanley Herbert Erlwanger (Bulawayo, 1934 – Harare, 2003): Member of the Executive Committee 1979–1982 Harm Jan Smid

Biography Stanley Herbert Erlwanger was born on 17 July 1934 in Bulawayo, the second town in what was then called Southern Rhodesia, a British colony, and since its independence known as Zimbabwe. He completed his undergraduate studies in mathematics and got a teaching diploma at the University of Cape Town in the mid-1950s. For 10 years, Erlwanger was a mathematics and science teacher in Zimbabwe, a secondary school principal and a senior inspector of schools in Zambia. In the 1970s, he moved to the United States and got his MA (Mathematics) and his PhD (Mathematics Education, 1974) with Robert B. Davis at the University of Illinois, where he was also involved with the Plato Computer Project (see Hillel 2003). For 2  years, he worked as a deputy director on the project “Basic Research on How Children Learn Mathematics”, launched in 1974 at the Institute for Advanced Study at Princeton by Hassler Whitney. This project had a duration of 20 months and a twofold purpose: to understand how children learn mathematics, observing children at work in the classroom, and to support teachers in developing more refined methods for helping

H. J. Smid (*) Delft University of Technology, Delft, The Netherlands

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them learn.70 He was also engaged in major projects on the mathematical development of children at the Florida State University. Then Erlwanger returned to Africa, where he had a rich career as high school math teacher, a senior inspector of schools in Zimbabwe, head of the Department of Science Education and, subsequently, dean of the Faculty of Education at the University of Botswana and Swaziland. In 1980, he became an associate professor in the Department of Mathematics and Statistics of Concordia University until 1991. Afterwards, he came back to Africa. Stanley Erlwanger died in Harare, the capital of Zimbabwe, on 18 June 2003.

Contribution to Mathematics Education In 1973, Erlwanger published an article under the title of “Benny’s Conception of Rules and Answers in IPI Mathematics”. IPI stands for individual programmed instruction. It became a famous article, still known and cited today, and included in the National Council of Teachers of Mathematics (NCTM) publication Classics in Mathematics Education Research. John Mason (2005) wrote about Erlwanger’s article: For example, Stanley Erlwanger made revelations that are as fresh now as they were startling then (…). His article shocked the developers of programmed learning, and it will shock those who try to engineer teacher-independent materials. (p. 468)

Erlwanger was one of the pioneers of the case studies, research not focusing on learning results in large groups, but on the learning processes in individual learners. In the early 1970s, he was working on his doctoral dissertation with Davis at the University of Illinois, who used the new insights of the cognitive psychology in his research on mathematics education. Their studies showed that cognitive processing and processes in learning mathematics were much more complicated and important than was thought and that research based on global learning results in large groups as in the case of IPI could be very misleading. In Erlwanger’s doctoral dissertation and other publications, several more case studies were described. In later years, he published with Victor Byers, a colleague from Concordia University where he was then working, some more general articles on mathematics education, and around 1990 with Joseph Brody, another colleague from Concordia, articles on the use of technology in mathematics education. At Concordia, Erlwanger was involved in a long-term research on children’s geometric notions while working with LOGO. He also was actively involved with math teachers, and he played a role in initiating curricular reforms. From 1975 until 1977, he was a contributing editor of the Journal of Children’s Mathematical Behavior. But no doubt, Erlwanger will be remembered most for his first article that he published and that made him famous: “Benny’s conception”.  See Development Projects in Science Education: Precollege, Higher Education, Continuing Education. National Science Foundation 1977, pp. 189–190. 70

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Erlwanger was involved in many international bodies of mathematics education. He was a member of the Executive Committee of ICMI in the period 1979–1982, under the presidency of Hassler Whitney. In 1984, he was a member of the International Program Committee for ICME-5 at Adelaide. He was also a member of the National Council on Teacher Education of Botswana and of the Regional Math Panel of Botswana, Lesotho and Swaziland and a consultant to UNESCO on Science/Mathematics Teaching in Africa.

Sources Hillel, Joel. 2006. Personal communication. In Memoriam Stanley H. Erlwanger. 2003. Concordia Thursday Report, 28, 1, Sept. 11. Montreal: Concordia University. Mason, John. 2005. Review: Coming of age in mathematics education: 17 Characters in search of a direction?. Journal for Research in Mathematics Education 36(5): 467–473. Speiser, Bob and Walter Chuck. 2004. Remembering Stanley Erlwanger. For the Learning of Mathematics 24(3): 33–39.

Publications Related to Mathematics Education Behr, Merlyn, Stanley H. Erlwanger, and Eugene Nicholls. 1976. How children view equality sentences. PMDC Technical Report n. 3. Tallahassee: Florida State University. (ERIC Document Reproduction Service n. ED 144802). Behr, Merlyn, Stanley H. Erlwanger, and Eugene Nicholls. 1980. How children view the equal sign. Mathematics Teaching 92: 13–15. Byers, Victor and Stanley H.  Erlwanger. 1984. Content and form in mathematics. Educational Studies in Mathematics 15: 259–275. Byers, Victor and Stanley H.  Erlwanger. 1985. Memory in mathematical understanding. Educational Studies in Mathematics 16: 259–281. Erlwanger, Stanley H. 1973. Benny’s conception of rules and answers in IPI mathematics. Journal of Children’s Mathematical Behaviour, 1(2): 7–26. (reprinted in Classics in Mathematics Education Research, eds. Thomas P. Carpenter, John A. Dossey, and Julie L. Koehler, 49–59. Reston VA: NCTM, 2004). Erlwanger, Stanley H. 1975. Case studies of children’s conceptions of mathematics. Journal of Children’s Mathematical Behavior 1(3): 157–283. Erlwanger, Stanley H. 1975. Case studies of children’s conceptions of mathematics. Thesis (Ph.D.) University of Illinois at Urbana-­Champaign. http://hdl.handle.net/2142/62885. Erlwanger, Stanley H. 1975. The observation-interview method and some case studies. In Proceedings of the Conference on the Future of Mathematics Education (held in Tallahassee, September 11–12, 1973), 125–142. Tallahassee: Florida State University. Erlwanger, Stanley H. 1978. The African university in development. Botswana Journal of African Studies 1(1): 189–192. Erlwanger, Stanley H. and Maurice Berlanger. 1983. Interpretations of the equal sign among elementary school children. In Proceedings of the 5th North American Chapter of the International Group for the Psychology of Mathematics Education, eds. Jacques C. Bergeron and Nicolas Herscovics, Vol. 1, 250–258. Montréal: Université de Montréal.

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Erlwanger, Stanley H. and Joseph Brody. 1987. A calculator-based computational approach. Part 1. Linear functions and equations. International Journal of Mathematical Education in Science and Technology 18: 393–402. Erlwanger, Stanley H. and Joseph Brody. 1988. A calculator-based computational approach. Part 2. Quadratic functions and equations. International Journal of Mathematical Education in Science and Technology 19: 691–703. Erlwanger, Stanley H., and Joseph Brody. 1990. A calculator-based computational approach. Part 3. The differentiation. International Journal of Mathematical Education in Science and Technology 21: 645–659. Erlwanger, Stanley H., Joseph Brody, and Steven Rosenfield. 1991. A calculator-based computational approach Part 4. On Differences and Graphs. International Journal of Mathematical Education in Science and Technology 22: 833–842.

Photo Courtesy of Andre and Gladys Erlwanger.

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Henri Fehr

11.18 Henri Fehr (Zürich, 1870 – Geneva, 1954): SecretaryGeneral 1908–1920, 1932–1936,71 Honorary President 1952–1954 Gert Schubring

Biography Henri Fehr was born in Zürich on 2 February 1870. He attended secondary school in Geneva and began studying mathematics in Geneva as well. He continued these studies at the Eidgenössische Technische Hochschule in Zürich, but finished them in Paris, where he obtained the teachers’ licence, the licence ès sciences   During the  ICM 1936 in  Oslo, “The Congress requests the  International Commission on  the  Teaching of  Mathematics to  continue its work, prosecuting such investigations as  shall be  determined by the  Central Committee” (L’Enseignement Mathématique 35, 1936, p.  388), but  because of  WWII, the  Commission remained inactive until 1952 when it is transformed in a permanent subcommission of IMU. 71

G. Schubring (*) University of Bielefeld, Bielefeld, Germany e-mail: [email protected]

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mathématiques in 1892. Fehr began his career as a mathematics teacher; he taught in Geneva, at the École professionnelle and the Collège, but also lectured, from 1895 to 1900, at the university as a Privatdozent, introducing courses on projective geometry and infinitesimal geometry. He succeeded in 1899  in obtaining the doctoral degree at the University of Geneva, with a thesis on Graßmann’s vectorial analysis. Just a year after, in 1900, he became professor for algebra and higher geometry at the Faculté des Sciences of the University of Geneva. Dedicated to teaching, he proved to be a pedagogue of exceptional qualities and enjoyed remarkable successes in teaching. In addition, he published numerous research papers. Fehr maintained his position until his retirement in 1945. Several times he was chosen to act as dean of the faculty and as vice-rector and rector of the university. He was one of the founders of the Swiss Mathematical Society, in 1910, and became its second president, from 1913 to 1915. Likewise, in 1928, he was one of the founders of the journal Commentarii Mathematici Helvetici. For 20  years, as part of the Euler Commission, he contributed to the publication of Euler’s Opera Omnia. Fehr died in Geneva on 2 November 1954. His Nachlass is extant in Geneva, in the Bibliothèque de Genève, in their Département des manuscrits et archives privées. In 2012, an excellent inventory was elaborated.72 Unfortunately, it contains no material regarding his extended activities and correspondence for the Internationale Mathematische Unterrichtskommission (IMUK) and later ICMI. Also, his descendants do not have this part of his Nachlass.

Contribution to Mathematics Education Fehr’s chef d’œuvre is the journal L’Enseignement Mathématique. Together with his friend Charles-Ange Laisant (1841–1920), he founded it in 1899 as the first international journal emphasizing communication about issues of mathematics teaching, across the respective national borders. The motivation of the two founders for this dedication to teaching was tied to ideas of internationalism and solidarity based on social and political ideals. The journal succeeded not only in establishing an effective international communication about mathematics teaching  – actually concentrated on secondary schools – but even contributed to the progress of international cooperation. The establishment of the first ICMI (IMUK/CIEM) in Rome in 1908, during the Fourth International Congress of Mathematicians (ICM), would not have been possible without the work of this journal. In fact, when the Comité Central of that Commission was constituted, Fehr became the secretary-general of IMUK. Together with Felix Klein and George Greenhill, he acted highly effectively to realize the goals of IMUK.

  The inventory can be accessed at https://archives.bge-geneve.ch/archive/fonds/fehr_henri (Retrieved June 2021). 72

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The former unity and cooperation within the Comité Central ended almost immediately when World War I began on 1 August 1914. Fehr, citizen of a neutral state (but by his socialization sympathetic to French culture and scientists), acted on the side of the Western allies. During and after the war, he was an active promoter to aligning the Commission’s policy with that of the later victorious powers. Surely, one has to consider that the German army invaded Belgium, a neutral state, right from the start and effected terrible destructions. In the first months since the outbreak of WWI, Fehr expressed optimism and a willingness to continue the work as far as possible, only cancelling the conference planned for 1915. But from December 1914 on, he required that all the work had to be stopped, until better times arrived (Schubring 2008, p. 17). Fehr published without consultation that all the work should stop and asked Klein to step back as IMUK president, referring to Klein’s signature of the ominous Manifest (Schubring 2008, pp. 18ff.). In fact, thereafter, almost all IMUK work stagnated. By the end of the war, Fehr backed the policy of the victorious allies to ban cooperation with scientists of the defeated countries, to exclude these scientists from international meetings and to dissolve organizations with members from these countries. Fehr urged, from March 1920, in the wake of the next – now no more really international – ICM in Strasbourg, the liquidation of IMUK. There was no other way and the dissolution was published by Fehr in July 1920 (ibidem, pp. 23ff.). Fehr maintained the position as secretary-general when the organization was revived in 1928, so that he represented its continuity, until after WWII. When ICMI was reconstituted in 1952, Fehr was chosen as one of the members of the committee of five appointed to draw up the plan for resuming the work. When this committee co-opted new members to form an ICMI Executive Committee, Fehr was elected its honorary president. After the constitution of IMUK/CIEM, L’Enseignement Mathématique had become the official organ of the committee. When the impetus for reforms faded away, since the interwar period, it became, however, essentially a journal for pure mathematics. Beyond the journal and the international work, Fehr was active as well in the Swiss Association of Mathematics Teachers; from 1905 to 1909, he acted as its president.

Sources Fehr, Henri. 1895. Sur l’emploi de la multiplication extérieure en Algèbre. Nouvelles Annales de Mathématiques 14: 74–79. Fehr, Henri. 1899. Application de la méthode vectorielle de Grassmann à la géométrie infinitésimale. Paris: Carré et Naud. Anonymous. 1955. Henri Fehr 1870–1954. Sa vie et son œuvre. L’Enseignement Mathématique s. 2, 1: 5–14. de Rham, Georges. 1955. Henri Fehr. Verhandlungen der Schweizerischen Naturforschenden Gesellschaft 135: 334–339. Ruffet, Jean 1955. Henri Fehr. Elemente der Mathematik 10: 1–4.

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Wilson, Edwin Bidwell. 1901. [Review of the book by Henry Fehr] Application de la méthode vectorielle de Grassmann à la géométrie infinitésimale”. Bulletin of the New York Mathematical Society 7: 231–233. Coray, Daniel, Fulvia Furinghetti, Hélène Gispert, Bernard R.  Hodgson, and Gert Schubring (eds.). 2003. One Hundred Years of L’Enseignement Mathématique, Moments of Mathematics Education in the Twentieth Century. Monographie n. 39 de L’Enseignement Mathématique. Furinghetti, Fulvia. 2003. Mathematical instruction in an international perspective: the contribution of the journal L’Enseignement Mathématique. In eds. Coray, Daniel et al. 2003, 19–46. Furinghetti, Fulvia. 2009. The evolution of the journal L’Enseignement Mathématique from its initial aims to new trends. In “Dig where you stand”. Proceedings of the conference on “On-going research in the History of Mathematics Education, eds. Kristín Bjarnadóttir, Fulvia Furinghetti, and Gert Schubring, 31–45. Reykjavik: University of Iceland  – School of Education. Schubring, Gert. 2003. L’Enseignement Mathématique and the first International Commission (IMUK): The emergence of international communication and cooperation In eds. Daniel Coray et al., 47–65. Schubring, Gert. 2008. The origins and the early history of ICMI. International Journal for the History of Mathematics Education 3(2): 3–33.

Publications Related to Mathematics Education Fehr, Henri. 1902. L’extension de la notion de nombre dans leur développement logique et historique. L’Enseignement Mathématique 4:16–27. Fehr, Henri. 1902. Elementare Mathematik. Allgemeine Übersicht über die verschiedenen Gebiete nebst kurzen Notizen über die historische Entwicklung der Mathematik. Genève. Fehr, Henri. 1905. Der Funktionsbegriff im mathematischen Unterricht der Mittelschule. L’Enseignement Mathématique 7: 177–187. Fehr, Henri and Théodore Flournoy. 1906. Enquête sur la méthode de travail des mathématiciens. Genève: Kundig. L’enseignement mathématique en Suisse. 1912. Rapport publié sous la direction de Henri Fehr. Bâle et Genève: Georg & Cie. Compte rendu de la Conférence internationale de l’enseignement mathématique, tenue à Paris du 1er au 4 avril 1914; publication du Comité Central rédigées par Henri Fehr. Paris: GauthierVillars, 1914.

Photo Source: L’Enseignement Mathématique. 1955, s. 2, 1 (Photo at p. 4). https://archives.bge-­geneve.ch/archive/fonds/fehr_henri

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Hans Freudenthal

11.19 Hans Freudenthal (Luckenwalde, 1905 – Utrecht, 1990): Member of the Executive Committee 1963–1966, President 1967–1970, Ex Officio Member 1971–1978 Gert Schubring

Biography Hans Freudenthal was born in Luckenwalde, a small town in the Prussian province of Brandenburg, between Berlin and Frankfurt/Oder, on 17 September 1905. His parents were Elisabeth Ehmann and Joseph Freudenthal. His father was the precentor and religious teacher of the small local Jewish community. In fact, the house where he was born served as the synagogue. He attended the local state school, Friedrichsschule, which evolved into a Reformrealgymnasium and passed his Abitur there. Although highly interested in the humanities as well as in literature, he decided to study mathematics and physics and entered Berlin University in 1923. The mathematics professors there were Erhard Schmidt, Richard von Mises, Issai Schur and Ludwig Bieberbach, and among the younger staff were John von Neumann, Heinz Hopf and Karl Löwner, who introduced Freudenthal into G. Schubring (*) University of Bielefeld, Bielefeld, Germany e-mail: [email protected]

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intuitionism. A visit by Luitzen E.J.  Brouwer in the winter semester 1926–1927 brought him into closer contact with intuitionism. The following term, summer 1927, Freudenthal spent in Paris, assisting the lectures by Jacques Hadamard, Gaston Julia and Émile Picard. After his return to Berlin, he obtained a position as Hilfsassistent at the mathematical seminar, allowing him to concentrate on elaborating a doctoral thesis. The subject of the thesis was topology, at that time rapidly developing as a new sub-discipline. The adviser was Hopf, while Bieberbach acted as the formal director of the thesis. He passed the exam in 1930; the thesis itself, on the theory of ends of topological spaces and topological groups, made research on these ends popular. Right after passing the doctoral exam, Freudenthal accepted an offer by Brouwer to come to Amsterdam University, as his assistant in mathematics. This was due more to interest in topology than in intuitionism, although Brouwer’s interest in topology was no longer as intense as earlier on. He met there as a fellow assistant Witold Hurewicz; both lectured  – after having obtained the status of Privaatdocenten, in 1931  – on algebraic fields, group theory, measure theory, complex analysis, topology and linear operators. In 1936, Hurewicz left for the United States and in 1937 two of the older professors retired, so that Arend Heyting and Freudenthal took over their teaching duties and were somewhat promoted in status, yet not to professorships. In 1932, Freudenthal married Suzanne (Suus) Johanna Lutter, who had studied German language and literature at the University of Amsterdam. During the Nazi occupation of the Netherlands, she succeeded in having him surviving. After WWII, she became interested in reforms of school pedagogy and introduced the ideas of the German pedagogue Peter Petersen and of the “Jenaplan” schools to Dutch schools. In the 1930s, Freudenthal became involved for the first time in editing a journal. Brouwer, dismissed by David Hilbert in 1929 from the Editorial Board of Mathematische Annalen, due to several deep conflicts, decided to establish his own international journal, the Compositio Mathematica. When the new journal finally began to be published, in 1934, it was Freudenthal, who first under the guidance of Brouwer and gradually on his own, conducted the managing of the journal: “Eventually Freudenthal just submitted each complete issue to Brouwer for his fiat. … Although Brouwer was the responsible editor, most of the work was done swiftly and competently by Freudenthal” (van Dalen and Remmert 2006, p. 1088, p. 1090). He became never a member of the Board, however, probably due to his lacking the title of professor. In 1940, after the occupation of the Netherlands by the Nazi army, the journal had to cease publication. After the war, when Compositio was reorganized  – now without Brouwer  – Freudenthal became a member of its Editorial Board (see van Dalen and Remmert 2006). World War II meant a decisive rupture and turnabout. Following the invasion by German troops in May 1940, Freudenthal, like many others, was suspended from his duties at the university in December. Thanks to a lack of bureaucratic attention on the part of the German occupiers, the Dutch functionaries continued to pay the reduced salary during the entire period of war. While at first the suspension meant having plenty of time for personal work, the ongoing oppression made the situation

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ever more threatening. For a certain time, although regarded as non-Aryan, as Jewish, he was protected by being married with “Suus”, his Dutch wife. In 1942, he suffered 6 weeks in a Gestapo prison when the German occupiers detected that he had a German passport, even without a “J” stamped into it. His fragile health prevented him from being recruited for working camps a few times, but in May 1944, he was interned in such a camp in Havelte, in the east of the Netherlands, for hard labour. In September 1944, his wife succeeded in helping him escape from that camp, and he returned to Amsterdam where he was eventually able to experience the liberation. After the end of WWII, in May 1945, Freudenthal resumed his activities at the University of Amsterdam. In 1946, the Faculty of Sciences at Utrecht State University offered him the full professorship for geometry, which he accepted, moving to Utrecht. He held this chair until his retirement in 1976. The teaching assignment to geometry made him change his research directions, and his interest now turned to the connection between geometries and their symmetry groups. It was here that his research into Lie groups began. Eventually, in 1972, he became the founder of yet another journal (see below): the journal Geometriae Dedicata; until 1981, Freudenthal served as its chief editor. The number of his publications is enormous; below we will indicate just a small selection. Major achievements in mathematics were the theory of ends in topology, the suspension theorems, a spectral theorem for Riesz spaces, the algebraic characterization of the topology of the real semi-simple Lie groups, work on the characters of the semi-simple Lie groups, octonion planes and other geometries connected with the exceptional Lie groups. He contributed not only to mathematics but also to philosophy and to the history of mathematics. Yet, his primary interest was not mathematical, but literary. He was an avid reader of the classics of literature; he kept diaries and wrote poetry, novellas, librettos and children’s stories; most of these productions remained unpublished. He even participated in several literary contests. Josette Adda has called him a “homo universalis”. Freudenthal died on 13 October 1990 in Utrecht.

Contribution to Mathematics Education It is said that Freudenthal developed an interest in the teaching of mathematics as early as the days of his studies in Berlin. It turned out to be his major field of work, however, only from the 1950s. He became the energetic opponent of the structuralist “new math” and thanks to the foundation of the Instituut voor de Ontwikkeling van het Wiskunde Onderwijs (IOWO, Institute for the Development of Mathematics Education) in Utrecht, the founder of a new direction in mathematics education, the so-called realistic mathematics teaching, which would eventually gain international dominance, in particular with and after “PISA”.

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In fact, since the re-establishment of ICMI in 1952, Freudenthal occupied increasingly influential positions, first on the national – Dutch – and then on the international scene. He became a leading member of the national Dutch subcommittee of ICMI. He organized and published several of the national thematic reports. Freudenthal has often expressed his regret not to have participated at the Congress in Royaumont in 1959, organized by the OEEC (later OECD), which became the founding event for the “new math” movement. While he hoped that he might have been successful there in preventing its emergence, he later on showed highly polemical qualities in attacking the nefarious effects for mathematics teaching, which he attributed to new math. In 1963, he entered the Executive Committee of ICMI, as a member until 1966, and from 1967 to 1970 as its president (and thereafter, from 1971 to 1978, as ex officio member). It was in particular during the period of his presidency where he succeeded in launching far-reaching and decisive innovations for the emerging international community of mathematics educators. Unsatisfied with the journal L’Enseignement Mathématique, since 1908 the official organ of IMUK/ICMI, but since 1920 essentially a journal for pure mathematics, Freudenthal initiated, in 1968, a new journal, Educational Studies in Mathematics (ESM), which was genuinely international and genuinely devoted to research into mathematics education. Published by a Dutch publisher  – Reidel, becoming later Kluwer (and today Springer, New York) – the journal established itself successfully and is still today one of the leading international journals in the field. The other decisive innovation was brought about by the small venue provided by what was traditionally the only section on mathematics teaching within the International Congresses of Mathematicians: it had become inadequate for the communication of problems and ideas in the now rapidly expanding field. This led Freudenthal to organize the first International Congress on Mathematical Education (ICME) in Lyon in 1969. These ICMEs have since then evolved into quadrennial congresses and into being the discipline’s major events. The first proceedings were published as a special issue of ESM. Actually, all these innovations were put into action by Freudenthal against the advice of IMU or even disregarding the explicit refusals by IMU officers. He acted in this manner, due to his bad experience with IMU rejecting his proposals to improve the functioning of ICMI. The first initiatives, in particular for a permanent secretary, were rejected (Furinghetti, Giacardi and Menghini 2020, p.  255). Freudenthal developed his reform programme at the meeting of the ICMI Executive Committee in August 1967. While well accepted there, IMU tried to dissuade – and since not accepted by them, he simply realized the programme, without informing IMU and by securing other financial resources (ibidem). Even his last initiatives and decisions irritated the IMU president Henri Cartan (ibidem). Right after the end of his presidency, Freudenthal was able to realize another important innovation, first of all relevant in the Dutch context, but soon to become of international relevance. In 1971, as mentioned above, the Instituut voor Ontwikkeling van het Wiskunde Onderwijs (IOWO, Institute for the Development of

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Mathematics Teaching) had been established at Utrecht University, with Freudenthal as its director. The founding of such an institute for research and development had been prepared since 1961 by the CMLW  – Commissie Modernisering Leerplan Wiskunde (Mathematics Curriculum Modernization Committee). Despite many bureaucratic obstacles, the institute was able to continue functioning; Freudenthal remained its director until 1980. While it was then at first dissolved, it was later transformed into the group OW&OC, and this became eventually, in 1991, the Freudenthal Institute in Utrecht. Freudenthal even helped inaugurate another new development within mathematics education, the more specialized psychological studies on learning mathematics for which the group PME (International Group for the Psychology of Mathematics Education) established at ICME-3 in Karlsruhe began to organize its yearly congresses. The first such congress was organized by Freudenthal in Utrecht in 1977. Practically as Freudenthal’s legacy, the institute’s work became decisive in disseminating the particular approach to mathematics teaching, which Freudenthal had at first developed by working with his grandson Bastiaan, the so-called realistic math education, at an unexpectedly broad international level: due to widely shared criticism of the conceptual assumptions of the TIMSS evaluation study, a new conception was developed for the successor PISA. And this conception, mathematical literacy, was based on the work of the Freudenthal Institute and on its approach called realistic mathematics teaching. In 2002, the ICMI Executive Committee created two awards in mathematics education research, one of them being the Hans Freudenthal Award, aimed at acknowledging the outstanding contributions of an individual’s theoretically well-conceived and highly coherent research programme in mathematics education. The award is conferred every 2 years (Fig. 11.1).

Fig. 11.1  The Hans Freudenthal Medal for the Hans Freudenthal Award. Source: https://www. mathunion.org/icmi/awards/icmi-emmacastelnuovo-felix-klein-and-hans-freudenthal-medals

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Sources Freudenthal, Hans. 1958. Logique mathématique appliquée. Paris: Gauthier-Villars. Freudenthal, Hans (ed.). 1961. The Concept and the Role of the Model in Mathematics and Natural and Social Sciences: Proceedings of the Colloquium … / organized at Utrecht, January 1960. Dordrecht: Reidel. Freudenthal, Hans (ed.). 1962. Algebraical and Topological Foundations of Geometry. Proceedings of a Colloquium held in Utrecht, August 1959. Oxford: Pergamon Press. Freudenthal, Hans. 1966. The language of logic. Amsterdam: Elsevier. Dutch original: Exacte logica. Freudenthal, Hans. 1965. Probability and statistics. Amsterdam: Elsevier. Dutch original: Waarschijnlijkheid en statistiek. Freudenthal, Hans. 1968. Mathematik in Wissenschaft und Alltag. München: Kindler. Freudenthal, Hans and Hendrik de Vries. 1969. Linear Lie groups. New York: Academic Press. Freudenthal, Hans. 1987. Berlin 1923–1930: Studienerinnerungen. Berlin: de Gruyter. Freudenthal, Hans. 1987. Schrijf dat op, Hans. Knipsels uit een leven. Amsterdam: Meulenhoff.

Essential Secondary Bibliography Dalen, Dirk van. 1991. Freudenthal and the foundations of mathematics. Nieuw Archief voor Wiskunde (4), 9 (2): 137–143. Dalen, Dirk van and Volker Remmert. 2006. The birth and youth of Compositio Mathematica: ‘Ce périodique foncièrement international’. Compositio Mathematica, 142: 1083–1102. Furinghetti, Fulvia, Livia Giacardi, and Marta Menghini. 2020. Actors in the changes of ICMI: Heinrich Behnke and Hans Freudenthal. In “Dig where you stand” 6. Proceedings of the Sixth International Conference on the History of Mathematics Education 2019, eds. Evelyne Barbin et al., 247–260. Münster: WTM-Verlag. La Bastide-van Gemert, Sacha (2015). All Positive Action Starts with Criticism. Hans Freudenthal and the Didactics of Mathematics: Dordrecht: Springer. Heege, Hans Ter (ed.). 2005. Freudenthal 100: speciale editie ter gelegenheid van de honderdste geboortedag van Professor Hans Freudenthal [Freudenthal 100: special edition on the occasion of the hundredth birthday of Professor Hans Freudenthal]. Utrecht: Freudenthal Institut, Universiteit. (Nieuwe wiskrant; 25,1; Panama-Post; 24,3). Streefland, Leen (ed.). 1993. The legacy of Hans Freudenthal. Dordrecht: Kluwer. Bos, Henk J. M. 1993. ‘The bond with reality is cut’ – Freudenthal on the foundations of geometry around 1900. In The legacy of Hans Freudenthal, ed. Leen Streefland, 51–58. Dordrecht: Kluwer. Veldkamp, Ferdinand D. 1985. In honor of Hans Freudenthal on his eightieth birthday. Geometriae Dedicata 19 (1): 2–5. Veldkamp, Ferdinand D. 1991. Freudenthal and the octonions. Nieuw Archief voor Wiskunde (4), 9(2): 145–162.

Obituaries Adda, Josette. 1993. Une lumière s’est éteinte: H. Freudenthal – homo universalis. Educational Studies in Mathematics 25: 9–19. Bos, Henk J.  M. 1992. In memoriam: Hans Freudenthal (1905–1990). Historia Mathematica 19(1): 106–108. Hogendijk, Jan P. 1991. In memoriam: Hans Freudenthal (1905–1990). Archives Internationales d’Histoire des Sciences 41(127): 353–354.

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Monna, Antonie F. 1992. Werner Fenchel and Hans Freudenthal. Nieuw Arch. Wiskunde s. 4. 10(1–2): 111–114. Monna, Antonie F. 1993. Supplementary note: ‘Werner Fenchel and Hans Freudenthal’. Nieuw Arch. Wiskunde s. 4, 11 2): 171. Strambach, Karl and Ferdinand D. Veldkamp. 1991. In memoriam Hans Freudenthal. Geometriae Dedicata 37 (2): 119. Est, Willem T. van. 1993. Hans Freudenthal (17 September 1905 – 13 October 1990). Educational Studies in Mathematics 25: 59–69. Est, Willem T. van. 1999. Hans Freudenthal: 17 September 1905 – 13 October 1990. In History of topology, ed. Ioan M. James, 1009–1019. Amsterdam: North Holland. Veldkamp, Ferdinand D. 1991. Hans Freudenthal: 1905–1990. Notices of the American Mathematical Society 38(2): 113–114.

Publications Related to Mathematics Education Freudenthal, Hans. (ed.). 1958. Report on Methods of Initiation into Geometry. Groningen: Wolters. ([Reports] / International Commission on Mathematical Instruction / Subcommittee for the Netherlands; 3). Freudenthal, Hans (ed.). 1962. Report on the Relations between Arithmetic and Algebra in Mathematical Education up to the Age of 15 [fifteen] Years. Groningen: Wolters. ([Reports] / International Commission on Mathematical Instruction / Subcommittee for the Netherlands; 5). Freudenthal, Hans. 1973. Mathematics as an educational task. Dordrecht: Reidel. Freudenthal, Hans (ed.). 1975. Les applications nouvelles des mathématiques et l’enseignement secondaire: conférences du 3me séminaire organisé par la C.I.E.M. à Echternach, juin 1973. Esch-sur-Alzette (Gr. D. de Luxembourg): Victor. Freudenthal, Hans. 1978. Weeding and sowing: preface to a science of mathematical education. Dordrecht: Reidel. Freudenthal, Hans. 1983. Didactical phenomenology of mathematical structures. Dordrecht: Reidel. Freudenthal, Hans. 1991. Revisiting mathematics education: China lectures. Dordrecht: Kluwer.

Photo Source: Noord-Hollands Archief (Haarlem/NL), Archief Freudenthal, inv.nr. 1914.

Otto Frostman

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11.20 Otto Frostman (Munkarp, 1907 – Djursholm, 1977): Member of the Executive Committee 1959–1962, Ex Officio Member 1971–1974 Sten Kaijser

Biography Otto Albin Frostman was born in Munkarp (Sweden) on 3 January 1907 and died in Djursholm (Sweden) on 29 December 1977. In 1926, he entered the University of Lund, where after receiving his BSc he commenced graduate studies under the guidance of Marcel Riesz, the younger of the famous Hungarian brothers Frigyes and Marcel. He defended his thesis Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions (Frostman 1935), which was to become a classic in potential theory, in 1935. Using modern measure theory, he could complete and considerably extend Gauss’s ideas from the 1820s to kernels of a very general type. Several good papers followed the thesis (Widman 2002, p. 19). He was given a docent position in Lund the same year, but this kind of position was time limited, and since no chair in mathematics became vacant during his years as docent, he worked as a schoolteacher for

S. Kaijser (*) University of Uppsala, Uppsala, Sweden e-mail: [email protected]

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10  years. In 1952, he became professor of mathematics at Stockholms Högskola (now Stockholm University), where he remained until his retirement in 1973. Frostman’s most important contributions are in potential theory and complex analysis. He was a member of numerous academies and scientific societies and held posts of great responsibility. He was elected a member of the Royal Academy of Sciences’ class for Mathematics on 28 May 1952, and his administrative capability was soon discovered which led to a successful career in scientific boards and committees. He was director of the Mittag-Leffler Institute from 1952 to 1967 and held the position of president of the Royal Swedish Academy of Sciences from 1965 to 1967. He was a member of the Committee of the Nobel Foundation from 1965 to 1973. When the International Congress of Mathematicians was held in Stockholm (15–22 August 1962), he chaired the organizing committee. In 1966, he became secretary of IMU and held this position until 1974. In his capacity of secretary of IMU, he was involved in the editing of the IMU Bulletin, first issued in January 1971.

Contribution to Mathematics Education Frostman was a member of the Executive Committee of ICMI from 1959 to 1962 and again (ex officio as secretary of IMU) from 1971 to 1974; in the period 1963–1966, he was member at large of ICMI. In 1955, he was designated by his national committee as a national delegate of Sweden to ICMI. He was the rapporteur on the mathematical instruction in Sweden at the International Congress of Mathematicians in Amsterdam in 1954 (Frostman 1957) and one of the Swedish delegates to the 1959 seminar at Royaumont. He presented a contribution at the International Symposium on the Coordination of Instruction in Mathematics and Physics, organized with the collaboration of ICMI in Belgrade (19–24 September 1960) (see Frostman 1962). He was mainly interested in the teaching of mathematics in the Swedish “gymnasium” attended by pupils of about 16–19  years old. In particular, he brought attention to a typical feature of the Swedish education and examination system “which places the problems in the foreground and the theory as a background” (Frostman 1956, p. 253). He writes: But a good idea alone does not secure progress. Solving a problem applying step by step the laws of arithmetic, the geometric intuition or the knowledge of some simple functions. A talented youth could do all this by simple reasoning without much practice, but very few will succeed in this way. Most pupils will need practice, again and yet again, and the more of it the more complicated the problems are. Even the best pupils need much practice in order to be able to solve all the problems in a test, the time being limited. Then what sort of problems should the teacher select for training of his pupils? (Ibidem)

Subsequently, Frostman highlights some defects of the examination tests and presents some ideas to remedy these defects. In particular, he believes that Swedish

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teachers should give more weight to the theoretical part in the teaching, and not only on training students to solve problems, also leaving room for a thorough examination of some subjects. However, he is aware that, in secondary education, it is not possible to make all the demonstrations necessary for a rigorous treatment of the subject. Frostman states: There is a principle that must be adopted in any case: It must clearly said which propositions cannot be demonstrated and are left to higher education. Above all, one should never give a wrong demonstration which looks like a correct demonstration.73 (Frostman 1962, p. 159)

Sources Frostman, Otto. 1935. Potentiel d’équilibre et capacité des ensembles avec quelques applications à la théorie des functions, Doctoral thesis, Lund: http://www.stat.ualberta.ca/people/schmu/ preprints/frostman.pdf Frostman, Otto. 1936. La méthode de variation de Gauss et les fonctions sous-harmoniques. Acta litterarum ac scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae. Sectio scientiarum mathematicarum 8: 149–159. Frostman, Otto. 1950. Sur les distributions vectorielles de masses. K. Fysiografiska Sällskapets i Lund Förhandlingar [Proceedings of the Royal Physiographical Society in Lund] 20: 192–198. Frostman, Otto. 1950. Potentiel de masses à somme algébrique nulle. K. Fysiografiska Sällskapets i Lund Förhandlingar [Proceedings of the Royal Physiographical Society in Lund] 20: 1–21. Frostman, Otto and Hans Wallin. 1979. Riesz potentials on manifolds. In Potential theory, 106–120. Lecture Notes in Mathematics 787. Berlin: Springer. Lech, Christer, 1978. Otto Frostman död. Dagens Nyheter. January 4. Widman, Kjell-Ove. 2004. Household names in Swedish mathematics. Some who should be and some others. EMS Newsletter 52:19.

Publications Related to Mathematics Education Frostman, Otto. 1953. En Sats Av Fáry Med Elementära Tillämpningar [A theorem by Fàry with elementary applications]. Nordisk Matematisk Tidskrift 1: 25–32. Frostman, Otto. 1956, Range of mathematical education in Swedish grammar Schools with regard to the examination-paper. L’Enseignement Mathématique s. 2, 2: 250–256. Frostman, Otto. 1957. Summary of a report on the mathematical instruction in Sweden for students between 16 and 21 years of age. In Proceedings of the International Congress of Mathematicians, eds. J. C. H. Gerretsen, and J. de Groot, Vol. 1, 552–553. Groningen: E. P. Noordhoff N. V.; Amsterdam: North-Holland. Frostman, Otto. 1962. La notion de convexité dans l’enseignement élémentaire. L’Enseignement Mathématique s. 2, 8: 158–162. Frostman, Otto. 1962. Mathematics and physics in Swedish secondary schools. In Proceedings of the International Symposium on the Coordination of Instruction in Mathematics and in Physics,

 The original text is: “Or il y a un principe qu’on doit adopter dans tous les cas: Il faut dire clai­ rement ce qu’on ne peut pas démonter et qu’on laisse à l’enseignement supérieur. Surtout, on ne doit jamais faire une démonstration fausse qui ressemble à une démonstration correcte”. 73

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eds. Yugoslav Union of Societies of Mathematicians and Physicists, 33–34. Belgrade: Yugoslav Union of Societies of Mathematicians and Physicists.

Photo Source: Lehto, Olli. 1998. Mathematics without Borders: A History of the International Mathematical Union. New York: Springer Verlag (Photo at p. 165).

Alfred George Greenhill

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11.21 Alfred George Greenhill (London, 1847 – London, 1927): Vice-President 1908–1920 Adrian Rice

Biography Alfred George Greenhill was born in London on 29 November 1847. Educated at Christ’s Hospital in London, he entered St. John’s College, Cambridge, in 1866, graduating as second wrangler in 1870. After being elected to a fellowship at St. John’s, he briefly taught at the Royal Indian Engineering College at Cowper’s Hill, before returning to Cambridge in 1873 as a fellow and lecturer at Emmanuel College. In 1876, he left Cambridge to assume the position in which he would spend the rest of his career: he was appointed professor of mathematics at the Artillery College in Woolwich, where he would stay for over 30 years. The majority of Greenhill’s research was concerned with elliptic functions or, more specifically, their applications in applied mathematics. He also contributed to the pure theory of elliptic functions, introducing useful simplifications in papers on “Complex multiplication” (1887) and “Pseudo-­elliptic integrals” (1895). But we are told, he always valued the practical applications more highly than the pure mathematics behind them, since “analysis was for him a means rather than an end” (Love 1928, p. 28). Indeed, he regarded a pure mathematical result “as of no real value until its correspondence with phenomena shown by some inorganic ‘corpus vile’ had been investigated” (Nicholson 1928–1929, p. 418). To this end, he investigated A. Rice (*) Randolph-Macon College, Ashland, VA, USA e-mail: [email protected]

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the uses of elliptic functions in dynamics, hydrodynamics, electrostatics and elasticity theory, using them to treat problems such as the motion of a symmetrical top, various kinds of fluid motion and the distortion of a circular wire under pressure. Towards the end of his career, he also supervised the computation of a set of elliptic function tables, which were published by the Smithsonian Institution in 1922. Not surprisingly, considering his employment in a military institution, his research often had military applications, such as ballistics and aeronautics. For example, a paper on hydrodynamics from 1879 concerned the ballistic implications of the theory of the motion of a solid in a fluid: Greenhill applied this theory to give an account of the steadiness of flight conferred upon an elongated projectile by rifling. He determined the least angular velocity about its axis for which steady motion of a solid of revolution, moving in the direction of its axis, can be stable. … This practical application of what was regarded as a recondite mathematical theory earned for him much renown at Woolwich. (Love 1928, p. 29)

In the theory of elasticity, Greenhill’s most important work was probably a paper from 1883 concerning the maximum length possible for an upright cylinder before it is bent under its own weight. Once he had solved the problem, Greenhill applied it to the computation of the greatest height to which a tree can grow, collecting data on the heights of particular trees to evaluate his result. Throughout his long scientific career, Greenhill received many honours. He was elected a fellow of the Royal Society in 1888, serving on its council in 1896 and 1897, and receiving its Royal Medal in 1906. He was also a prominent member of the London Mathematical Society, acting as its president from 1890 to 1892 and receiving its De Morgan Medal in 1902. He was also known and respected overseas, being on friendly terms with several continental mathematicians, most notably Felix Klein. A corresponding member of the Académie des Sciences and an honorary foreign member of the Accademia dei Lincei, Greenhill was also the first British scholar to give a plenary address at an International Congress of Mathematicians, which he did at Heidelberg in 1904. Finally, on his retirement from the Artillery College, he was knighted by King Edward VII in 1908. He continued to work on mathematics during his retirement, publishing papers and attending conferences, including a meeting of the British Association for the Advancement of Science in Canada in 1925. But his health gradually declined throughout the 1920s, and he died on 10 February 1927.

Contribution to Mathematics Education To British mathematics students, Greenhill would have been known principally for his university-level textbooks. The first of these, Differential and Integral Calculus (1886), was an introductory treatise which, in a departure from the usual approach, introduced integration alongside differential calculus, and developed the two subjects in parallel, instead of keeping them separate. His most famous book, Applications of Elliptic Functions (1892), used problems from mechanics and physics to motivate and illustrate a more or less systematic treatment of elliptic function

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theory. Greenhill observed that, since the subject was considered too difficult to be included in elementary British textbooks, “many of the most interesting problems in Dynamics are left unfinished, because the complete solution requires the use of the Elliptic Functions” (Greenhill 1892, p. viii). It was Greenhill’s intention with this book “to make the elliptic functions a familiar and trusted resource for completing the solution of problems that require them, just as the elementary functions are” (Love 1928, p. 28). According to the applied mathematician A. E. H. Love, a still more original book was his Treatise on hydrostatics (1894). In this he developed a thoroughly physical presentation of the subject, using such mathematics, whether elementary or not, as might be needed, and constructing examples from data gleaned from such technical publications as the Transactions of the Institute of Naval Architects. (Love 1928, pp. 30–31)

In his later years, Greenhill’s style of applied mathematics became increasingly unfashionable when compared to the new rigorous approach to pure mathematics, as practised by such early twentieth-century British mathematicians as Godfrey H. Hardy and John E. Littlewood, which he disparagingly referred to as “the morbid pathology of the mathematical function” (Love 1928, p.  31). His old-fashioned views were accentuated by such a high admiration of Newton’s Principia that, he said, “I should prefer to see the whole three books prescribed in the Cambridge course, to be studied in the original Latin” (Greenhill 1892–1893, p. 13). However, Love conjectures that “in making such assertions he was not more than half serious” (Love 1928, p. 31). On his retirement from his Woolwich professorship, Greenhill devoted his mathematical activities more explicitly to matters concerning pedagogy. When the international journal L’Enseignement Mathématique was founded, he joined its Comité de Patronage, a kind of Editorial Board which existed until 1914; in 1909, this journal became the official organ of the Commission on the Teaching of Mathematics. At the 1908 International Congress of Mathematicians at Rome, he was appointed one of the founding three members of ICMI, the other two being Klein and Fehr (Lehto 1998, p.  13; Howson 1982, p.  164). He served as vice-president of the Commission until 1920. Interestingly, when Greenhill requested funding from the British government, “an application to our Board of Education for patronage met a very curt refusal - No funds!” (Greenhill 1913–1914, p. 259). Financial and organizational help was eventually provided by another body with which Greenhill was involved: the Mathematical Association, Britain’s oldest professional society for mathematics teachers. Greenhill took an active interest in its affairs and contributed a number of mathematical articles to its journal, The Mathematical Gazette. He also served as the Association’s president from 1913 to 1914, delivering a valedictory address on “The Use of Mathematics”, a rambling miscellany of no fixed theme which can only be described as eccentric. But despite his unconventionality, Greenhill was apparently a successful and highly respected teacher. “In all his work, as writer, as examiner, as teacher, with a ruthless insistence on reality, he combined such remorseless pertinacity and such painstaking thoroughness as should make his memory an inspiration to future generations of mathematicians” (Love 1928, p.  32). Indeed, as one of his former

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students reports, “Greenhill was loved by his old pupils to a degree which few Professors can have enjoyed” (Nicholson 1928–1929, p. 417).

Sources Greenhill, Alfred George. 1886. Differential and integral calculus, with applications. London: Macmillan. 2nd edition, 1891. 3rd edition. 1896. Greenhill, Alfred George. 1892. The applications of elliptic functions. London: Macmillan. Greenhill, Alfred George. 1892–1893. Collaboration in mathematics. Proceedings of the London Mathematical Society 24: 5–16. Greenhill, Alfred George. 1894. A treatise on hydrostatics. London: Macmillan. Greenhill, Alfred George. 1912. The dynamics of mechanical flight. London: Constable. Howson, A.  Geoffrey. 1982. A History of Mathematics Education in England. Cambridge: University Press. Lehto, Olli. 1998. Mathematics without Borders: A History of the International Mathematical Union, New York: Springer. Fehr, Henri. 1927. Sir George Greenhill. 1847–1927. L’Enseignement Mathématique 26: 141–142. Love, Augustus E. H. 1928. Alfred George Greenhill. Journal of the London Mathematical Society 3: 27–32. Nicholson, John W. 1928–1929. Sir George Greenhill. The Mathematical Gazette 14: 417–420.

Publications Related to Mathematics Education Greenhill, Alfred George. 1911–1912. Presidential address to the London branch of the Mathematical Association. The Mathematical Gazette 6: 104–108. Greenhill, Alfred George. 1913–1914. The use of mathematics. The Mathematical Gazette 7: 253–259.

Photo Source: Wikimedia Commons.

Jacques Hadamard

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11.22 Jacques Hadamard (Versailles, 1865 – Paris, 1963): Vice-President 1928–1932, President 1932–1936, 1936–74 Hélène Gispert

  During the  ICM 1936 in  Oslo, “The Congress requests the  International Commission on  the  Teaching of  Mathematics to  continue its work, prosecuting such investigations as  shall be  determined by the  Central Committee” (L’Enseignement Mathématique 35, 1936, p.  388), but because of World War II, the Commission remained inactive until 1952, when it was transformed in a permanent subcommission of IMU. 74

H. Gispert (*) GHDSO, EST, Université Paris Saclay, Paris, France e-mail: [email protected]

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Biography A Man of Science Jacques Hadamard was born on 8 December 1865 in Versailles. His father was a teacher of Latin in an important high school in Paris, the Lycée Louis-le-Grand, and his mother gave piano lessons. The eldest of two children, he carried out very brilliant classical secondary studies but did not seem particularly destined for science. In his speech on the occasion of the jubilee celebration of his 50 years in science, he declared: How did I devote myself to mathematics? It would have surprised my family and myself if someone had predicted it in my first years of high school. I can dedicate my example to parents who despair of their children’s inability to overcome the first problems of arithmetic… I can still see myself asking my father … a question about the École normale: “Do they do mathematics there?” “Yes”, he replied, “at the École normale, science section, they do mathematics”. And I immediately drew the conclusion, “Oh, then, it is not in there that I will go”. (Jubilée scientifique 1937, pp. 51–52)75

Hadamard ended up meeting professors who introduced him “to the beauty of scientific things”76 (Jubilée scientifique 1937, p.  52) and, after the baccalauréat (diploma at the end of secondary school), he turned to science. In 1884, he ranked first in the entrance exams to the École Polytechnique and the École Normale Supérieure (ENS) (section of science!), having obtained in the first of these competitions a number of points that no candidate had ever achieved. Hadamard opted for the ENS. There, “a light lit up his life, as happened for all the normaliens of his age  – added Hadamard  – … the luminous figure of Jules Tannery … scientific, intellectual, moral guide” (Jubilée scientifique 1937, p. 52).77 His professors were Émile Picard, Charles Hermite, J.  Gaston Darboux, Paul Émile Appell and Édouard Goursat. Agrégé in 1887, he was, like many young mathematicians of that time, appointed professor in classical high schools for a few years before having a position of maître de conférence (lecturer) in the provinces. He taught at Lycée Buffon in Paris from 1890 to 1893. His beginnings as a secondary school teacher were difficult as made evident in his biography by Vladimir Maz’ya and Tatyana Shaposhnikova (1998), which includes a letter addressed to the Minister of Education by the vice-rector of the Paris Academy:

 The original text is: “Comment me suis-je consacré aux mathématiques? On aurait bien étonné ma famille et moi-même en me le prédisant dans mes premières années de lycée. Je puis dédier mon exemple aux parents que désespère l’inaptitude de leurs enfants à triompher de premiers problèmes d’arithmétique. … Je me vois encore posant à mon père … une question sur l’Ecole normale: “Y fait-on des mathématiques?” Oui, m’a-t-il répondu, à l’Ecole normale, section ­sciences, on fait des mathématiques”. Et moi de tirer tout de suite la conclusion “Oh, alors, ce n’est pas là-dedans que j’irai”. 76  The original text is: “à la beauté des choses scientifiques”. 77  The original text is: “une lumière éclaira sa vie … la lumineuse figure de Jules Tannery … guide scientifique, intellectuel, moral”. 75

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Mr. Minister, the last bimonthly report of the Headmaster of Lycée Buffon contains the following note about Mr. Hadamard: Mr. Hadamard’s classes leave increasingly to be desired. No concern for the moral interests of students young and old. No authority over them. A brittle and capricious discipline. … No practical preparation of the classes. Mr. Hadamard believes himself relieved of everything by his remarkable mathematical skills. The further we go, the more we sacrifice the public good for the personal convenience of this young scholar. I invite Inspector Piéron to do an inspection of Mr. Hadamard’s class. (Maz’ya and Shaposhnikova 1998, p. 46)78

The reasons for these difficulties might have been various, either lack of attention or interest, or inability to judge the level of students. What is certain, however, is that Hadamard devoted himself above all to mathematics. He prepared a thesis of analysis  – Essai sur l’étude des fonctions données par le développement de Taylor  – which he defended in 1892. In particular, it was in this thesis that the so-called Hadamard formula for the calculation of the radius of convergence of a power series appeared. That same year, he published a paper on power series and in particular the zeta function of Riemann, which was awarded the Grand prix des sciences mathématiques of the Académie des Sciences, and married Louise Trénel, with whom he had five children. It was in 1893 that Hadamard obtained his first university position, at the University of Bordeaux, starting as a lecturer and then, in 1896, becoming a professor of astronomy and rational mechanics. In 1896, the theory of analytic functions which he himself helped to develop was a resounding success: Hadamard demonstrated, at the same time as Charles de la Vallée-Poussin, that the number of prime numbers less than or equal to n is equivalent to n/lnn when n tends to positive infinity. It is a far-reaching result. The following year, he was appointed to the Faculty of Sciences in Paris but – as was the tradition at that time – initially as a lecturer; in 1900, he became an assistant professor. In 1909, he left the Sorbonne for the Collège de France where he obtained the professorship of mechanics. According to the French tradition of the multiple positions, in 1912, Hadamard also became professor of analysis at the École Polytechnique where he succeeded Camille Jordan when this latter retired. That same year, Hadamard was elected member of the geometry section of the Académie des Sciences following the death of Henri Poincaré; he was 47 years old and was the youngest of the academicians. This nomination crowned a corpus of works and results of considerable importance and variety in almost all areas of mathematics.79 Following Lévy (1967), we can distinguish four main centres of

 The original text is: “Monsieur le Ministre, le dernier rapport bimensuel de Monsieur le Proviseur du lycée Buffon contient au sujet de M. Hadamard la note suivante: Les classes de M. Hadamard laissent de plus en plus à désirer. Aucun souci des intérêts moraux des élèves petits et grands. Aucune autorité sur eux. Une discipline cassante et capricieuse. … Nulle préparation pratique des classes. M.  Hadamard se croit dispensé de tout par ses remarquables aptitudes mathématiques. Plus nous allons, plus nous sacrifions le bien public aux convenances personnelles de ce jeune savant. J’invite M. l’Inspecteur Piéron à voir la classe de M. Hadamard”. 79  Vladimir Maz’ya and Tatyana Shaposhnikova (1998) list 184 publications (articles or books) between 1888 (the first note in the Comptes rendus de l’Académie des sciences) and 1914. 78

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interest: analytic or quasi-analytic functions and their applications to arithmetic; differential equations and their applications both to mechanics and to the study of geodesics of a surface; partial differential equations and their applications to mathematical physics; and the calculus of variations and functional calculus. World War I brought a dramatic end to these first 50 years of Hadamard’s life. In 1916, two of his sons, like so many young polytechniciens and normaliens sent to the front, were killed. In 1920, alongside his teaching at the École Polytechnique and the Collège de France, Hadamard took over from Paul Appell as professor at the École Centrale des Arts et Manufactures. He continued his mathematical work,80 but his post-war activity was above all marked by the direction of the seminar he created at the Collège de France in 1913 and which dominated the French mathematical scene during the interwar period. During all these years, he also continued his political commitment to human rights and peace (see Sect. 1.2). The aim of Hadamard, as he recalls during his jubilee in 1936, was “that our young workers do not remain strangers to anything that is happening in the field of mathematics” (Jubilée scientifique 1937, p. 56).81 But, well beyond the young students he contributed to train, the seminar brought together the academic mathematical elite. Moreover, foreign mathematicians on study trips to France felt almost obliged to come and present their latest research there. However, it should be noted that, despite Hadamard’s objective of completeness and the variety of the subjects covered, this seminar did not cover all of the mathematics being done at that time in France or abroad. Analysis was dominant, as it was in research in France, and certain areas of mathematics, such as geometry or German algebra, were treated scantly or not at all. Throughout this period, Hadamard played a decisive role in the relationships of French mathematicians with other countries. On the one hand, with his seminar, he tried to publicize foreign work in France; on the other hand, he helped to “export” French mathematics to other nations and fostered dialogues between different mathematicians from France and abroad, through his rich correspondence, the trips he made in many countries around the world, and his participation in the International Congresses of Mathematicians (ICMs). In 1937, Hadamard retired. World War II began soon after. After the French defeat of 1940, Hadamard, of Jewish origin, sought to flee anti-Semitic persecution; he took refuge in the south of France and then in the United States. This war was no kinder to him than the first: his third son, who had enlisted in the free French forces, was killed in 1942. During his stay in the United States, Hadamard gave a course at Columbia University and published, among other things, his work on the psychology of invention in the mathematical field, the French edition of which appeared only in 1959.

 Hadamard published more than 160 articles or works between the two wars (Maz’ya and Shaposhnikova 1998). 81  The original text is: “que nos jeunes travailleurs ne restent étrangers à rien de ce qui se passe d’important dans [le] domaine des mathématiques”. 80

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Hadamard returned to Paris in 1945. He was 80  years old, but continued his mathematical activity. He continued to be invited abroad: the Soviet Union in 1945; India in 1947; Harvard in 1950 for the first ICM after the war; Romania in 1956. The participation in the ICM 1950 at Harvard was jeopardized because the State Department of the United States refused to issue a visa for Hadamard. It was indeed the middle of the Cold War and McCarthyism. Hadamard – as well as another mathematician from the French delegation, Laurent Schwartz  – was criticized for his left-wing political commitments (see Sect. 1.2). It was only under the joint pressure of the French delegation and the American mathematicians that Hadamard obtained his visa and was able to attend the congress, where he was elected honorary president jointly with Guido Castelnuovo and de la Vallée-Poussin. Hadamard died on 19 October 1962 in Paris. All the French press honoured his memory. The titles are enlightening: mixing the two types of commitments that Hadamard cultivated during all his life, they celebrate the “great scientist, man of progress”, the “illustrious mathematician” and “a life of science and conscience”.82 During his lifetime, he received countless honours, prizes and awards. The list is so long that already in 1936, during Hadamard’s jubilee, Lebesgue declared at the conclusion of his opening address: “I would have liked, in closing, to let you know about these honors; but Hadamard was incapable of helping me to draw up a truly complete list: this is the only point where I have ever discovered deficiencies in his knowledge” (Lebesgue 1937, p. 14).83 A Man of Peace and Progress Hadamard’s commitment to human rights, one of the main aspects of his long life, began with the Dreyfus Affair in the mid-1890s. Alfred Dreyfus, a captain in the French army, was accused of spying, judged, wrongly condemned and deported following a first trial. Several intellectuals – including Henri Poincaré – demonstrated the errors of the trial and denounced the climate of violent anti-Semitism that surrounded the whole affair. Dreyfus’s supporters then created the “League for Human Rights”. Hadamard became a member of its central committee in 1909 and remained so until his last years. This commitment was further solidified after World War I.  Hadamard then wrote numerous articles on social and political questions and on peace. In the 1930s, he defended the leftist, anti-Fascist positions and supported the government of the Popular Front of 1936 in France. Among his battles during the  The original titles are: “Jacques Hadamard: Un grand savant, un homme de progrès” (L’Humanité dimanche, 20 October 1963); “L’Illustre mathématicien Jacques Hadamard” (France Nouvelle, 22 October 1963); “Jacques Hadamard. Une vie de science et de conscience” (L’Humanité, 18 October 1963). 83  The original text is: “J’aurais voulu, en terminant, vous donner connaissance de ces distinctions; mais Hadamard s’est révélé incapable de m’aider à établir une liste vraiment complète: c’est le seul point sur lequel j’ai jamais pris son savoir en défaut”. 82

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1920s and 1930s, the defence of science education and scientific culture was one of the most constant (see below), and he spoke on this subject repeatedly. Despite his age, Hadamard continued his battles after World War II and into the 1950s. He took part in numerous meetings against the French colonial wars in Indochina and Algeria, against wars and human rights abuses all over the world, and was always ready to defend what seemed to him humane and right.

Contribution to Mathematics Education In 1928, at the ICM in Bologna, Hadamard was elected vice-president of the Commission Internationale de l’Enseignement Mathématique (CIEM). The Commission was relaunched on this very occasion after the strong tensions following World War I that had been created in the scientific world between the allies and other countries. At the next congress, in Zurich in 1932, he was elected president. When the Commission’s mandate was renewed during the ICM in Oslo (1936), he was an invited lecturer in China.84 Among the production so important and copious of Hadamard – more than 400 articles or works – the part devoted to teaching is not very large: 30 articles and 5 treaties published between 1898 and 1964. Quantity aside, it should be noted that, on the one hand, some of his works had a remarkable success and influence and, on the other hand, one of the battles that he considered top priority was the improvement of teaching. His first work, Leçons de géométrie élementaire, which appeared in 1898 in a collection of textbooks directed by Gaston Darboux for the classe de mathématiques, the final year of the French lycées, had indeed a strong influence on generations of mathematicians. This book, which Henri Lebesgue, generally stingy with compliments, called “an unequaled masterpiece”85 (Lebesgue 1937, p.  10), was already in its eighth edition in 1922; it continued to be published and, after World War II, in 1947, the thirteenth edition appeared. One of the strengths of this book is certainly the extraordinary variety and abundance of problems offered. Before World War I, this book had been followed by the texts of the lessons Hadamard taught at the Collège de France and then, at the end of the 1920s, those of his course of analysis at the École Polytechnique. Moreover, he participated in the Congress of the CIEM held in Paris in 1914 and contributed to the discussion that followed the speech by Charles Bioche on the teaching of infinitesimal calculus in the French lycées. Hadamard declared: “In the teaching practice automatism should be avoided; as often as possible, common sense should be used. The teacher must make sure that the student knows how to study the elementary functions for themselves, by direct discussion and observation, before introducing the derivative”

 Comptes rendus du Congrès International des Mathématiciens, Oslo 1936, Oslo, A. W. Brøggers Boktrykkeri, 1937, Vol. 2, p. 287. 85  The original text is: “chef d’œuvre inégalé”. 84

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(L’Enseignement Mathématique, 16, 1914: 301)86; he illustrated his statement with his memories of having been an examiner. But it was between the two World Wars that Hadamard intervened publicly and insistently on questions of secondary education. He devoted some 30 articles to this, published in the educational as well as in the general press. In a first type of articles, often polemic, Hadamard dealt with scientific secondary education and its crucial role in the general culture provided to young people educated in high schools. This vehement defence of scientific education took place in the context of the post-war period in France when the 1902 reform was called into question. This reform had established in French secondary education an alternative to the purely classical humanist training of the elite, giving a greater place to sciences including mathematics. Criticizing the “bath of realism”87 brought about by this reform, those supporting a step back advocated the affirmation of a Latin and classic French tradition against a Germanic tradition hated during these years of war and its immediate aftermath. This anti-scientific campaign led to a new reform called égalité scientifique. Hadamard fought against it and affirmed that, beyond their usefulness, the experimental sciences have an irreplaceable educational value; indeed, it is only with the practice of the scientific method – which is acquired with the sciences – that one can fully understand and master Descartes and his discourse on the method. While his interventions related mainly to the physical and natural sciences, more threatened than mathematics, Hadamard also dealt with the latter. As he had done at the beginning of the century when the new programmes of mathematics had been designed by the 1902 reform, he developed the idea – then shared by several mathematicians including Poincaré and Émile Borel – that geometry is a physical science and that its teaching must have recourse to “topographic operations, to the carpentry workshop, an essential auxiliary  – as paradoxical as it may seem to some – of general culture”88 (Hadamard 1934, p. 247).89 Hadamard also wrote articles in these interwar years more specifically devoted to the teaching of particular concepts of mathematics and to the precise design of special mathematics class programmes. Let us note in particular a lengthy paper that appeared in 1925  in the Nouvelles Annales de Mathématiques (Hadamard 1925) filled with concrete reflections and precise examples, in which Hadamard, again polemic, is astonished that so little consideration has been given to the suggestions

 The original text is: “Dans l’enseignement il faut éviter l’automatisme; il faut le plus souvent possible, faire appel au bon sens. Le professeur doit s’assurer que l’élève sait étudier les fonctions élémentaires pour elles-mêmes, par la discussion directe et l’observation, avant de faire intervenir la dérivée”. 87  The original text is: “bain de réalisme”. 88  The original text is: “aux opérations topographiques, à l’atelier de menuiserie, auxiliaire essentiel – si paradoxal que cela puisse paraître à certains – de la culture Générale”. 89  See also Borel, Émile. 1904. Les exercices pratiques de mathématiques dans l’enseignement secondaire. Revue générale des sciences pures et appliquées, 14: 431–440; Poincaré, Henri. 1904. Les définitions générales en mathématiques. L’Enseignement Mathématique, 5: 257–283. 86

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presented by him on behalf of his colleagues mathematicians at the École Polytechnique; their experience in the matter should have been better taken into account.

Sources90 Primary Bibliography Hadamard, Jacques. 1909. Notice sur les travaux scientifiques. Autograph manuscript (Archives École polytechnique). Œuvres de Jacques Hadamard. 1968. Vols 1–4, Paris: Centre national de la recherche scientifique.

Secondary Bibliography Desforge, Julien, Ghidale Iliovici, and Paul Robert. 1936. L’œuvre de M. Jacques Hadamard et l’enseignement secondaire. L’Enseignement Scientifique 9(84): 97–117. Kahane, Jean-Pierre. 1991. Jacques Hadamard. Mathematical Intelligencer 13(1): 23–29. Lebesgue, Henri. 1937. Allocution. In Jubilé scientifique de M.  Jacques Hadamard, pp.  7–14. Paris: Gauthier-Villars. Lévy, Paul. 1967. Jacques Hadamard, sa vie et son œuvre. Calcul fonctionnel et questions diverses. In Paul Lévy, Bernard Malgrange, Paul Malliavin, and Szolem Mandelbrojt. La vie et l’œuvre de Jacques Hadamard (1865–1963), pp. 1–24. Monographie de L’Enseignement Mathématique: Université Genève. Maz’ya, Vladimir and Tatyana Shaposhnikova. 1998. Jacques Hadamard, a Universal Mathematician. Providence: American Mathematical Society/London Mathematical Society. Scientific Jubilee of M. Jacques Hadamard. 1937. Paris: Gauthier-Villars.

Publications Related to Mathematics Education Hadamard, Jacques. 1898. Leçons de géométrie élémentaire, I. Géométrie plane. Cours de Gaston Darboux pour la classe de Mathématiques. Paris: Armand Colin. Hadamard, Jacques. 1901. Leçons de géométrie élémentaire, II. Géométrie dans l’espace. Cours de Gaston Darboux pour la classe de Mathématiques. Paris: Armand Colin. Hadamard, Jacques. 1903. Les sciences dans l’enseignement secondaire. Conférence à l’EHES. Paris: Alcan. Hadamard, Jacques. 1905. A propos d’enseignement. Revue Générale des Sciences Pures et Appliquées 16: 192–194.  A detailed bibliography can be found in Maz’ya, Vladimir and Shaposhnikova, Tatyana. 1998. Jacques Hadamard, a  Universal Mathematician. American Mathematical Society/London Mathematical Society at pages 511–531. In this chronologically established bibliography which includes 406 titles, the authors list all of Hadamard’s publications, whether they are strictly mathematical or concern his civic commitment. At pages 533–535, the authors list 62 titles of secondary bibliography, to which their own book should be added; one of the merits of this list of titles is that it contains press articles written about Hadamard throughout his life. 90

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Hadamard, Jacques. 1921. A propos de l’enseignement secondaire. Revue Internationale de l’Enseignement 75: 289–294 (also in Œuvres, vol. 4, pp. 2201–2206). Hadamard, Jacques. 1925. À propos du nouveau programme de mathématiques spéciales. Les Nouvelles Annales de Mathématiques s. 6, 1: 257–276 et 391–393 (also in Œuvres, vol. 4, pp. 2207–2226). Hadamard, Jacques. 1927. Les méthodes d’enseignement des sciences expérimentales. Revue Internationale de l’Enseignement 81: 355–356. Hadamard, Jacques. 1931. A propos des lettres de Monsieur Léon Blum. Réflexions générales sur un cas particulier. L’Enseignement Scientifique 4(40): 309–311. Hadamard, Jacques. 1934. Réponse à l’enquête sur les bases de l’enseignement des mathématiques.

Photo Source: Wikimedia Commons.

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Poul Heegaard

11.23 Poul Heegaard (Copenhagen, 1871 – Oslo, 1948): Vice-President 1932–1936, 1936–91 Henrik Kragh Sørensen

Biography Poul Heegaard (1871–1948) was born on 2 November 1871 in Copenhagen.92 His father was a professor of philosophy at Københavns Universitet (the University of Copenhagen) with an interest in mathematics and the sciences. According to his

  During the  ICM 1936 in  Oslo, “The Congress requests the  International Commission on  the  Teaching of  Mathematics to  continue its work, prosecuting such investigations as  shall be  determined by the  Central Committee” (L’Enseignement Mathématique 35, 1936, p.  388), but because of WWII, the Commission remains inactive until 1952 when it is transformed in a permanent subcommission of the IMU. 92  The following biographical facts are based on Jørgensen and Nielsen (1980) and Munkholm and Munkholm (1998) and Munkholm and Munkholm (1999) and the obituary (Johansson 1949), which also contains an extensive bibliography for Heegaard. 91

H. K. Sørensen (*) University of Copenhagen, Copenhagen, Denmark e-mail: [email protected]

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later recollections, Heegaard’s way into mathematics was indirect and highly curious. Because he had trouble with mental arithmetic but showed more aptitude for algebra, his teachers had given him more advanced algebra problems to solve – and he had felt the success of doing mathematics. In retrospect, he reflected that this aptitude “let my surroundings … drive me towards an occupation with mathematics for which I did have a talent, but for which I do not have the burning interest that I have met in others” (Munkholm and Munkholm 1999, p. 927). Later in his mathematical career, his many other interests would often distract him from doing mathematical research, and his career as a research mathematician was based on less than a handful of truly original works. Instead, he spent much of his time lecturing on mathematics and the sciences to the public and on affairs related to the teaching of mathematics (to be described below). Heegaard attended the gymnasium (Metropolitanskolen in Copenhagen) from which he obtained his Examen artium in 1889. He then began studying at the University of Copenhagen, but not full-time because the death of his father had left him in a position where he had to support himself. He did so by tutoring other students and functioning as an examiner at the Polyteknisk Læreanstalt (Polytechnical College). While a student, Heegaard attended the lectures of the three dominant Danish mathematicians of the second half of the nineteenth century: Hieronymus G. Zeuthen (1839–1920), Julius Petersen (1839–1910) and T. N. Thiele (1838–1910). In 1893, Heegaard graduated after writing a thesis on algebraic curves in the tradition of Michel Chasles (1793–1880) under the supervision of Zeuthen. After his graduation, Heegaard wanted to go abroad to continue his studies. Following the advice from the Francophile Zeuthen, Heegaard went to Paris in 1893 with letters of recommendation from Zeuthen. However, Heegaard’s reception in Paris was cool, and he never had any really fruitful contacts with Paris mathematicians. In particular, it does not seem that Heegaard ever met Henri Poincaré (1854–1912) – either in Paris or later in life – although much of Heegaard’s fame was connected to Poincaré’s research (Munkholm and Munkholm 1999, p.  931). Frustrated with his situation in Paris, Heegaard decided to follow an earlier idea of his to go to Göttingen and study with Felix Klein (1849–1925). The stay with Klein proved to be more productive than Heegaard’s time in Paris (Munkholm and Munkholm 1999, p. 931). When he returned from his study tour in 1894, Heegaard began work on his dissertation while earning a living teaching mathematics at two or more gymnasia. The dissertation proceeded from discussions with Klein in Göttingen, and in 1897, Heegaard became aware of a recent publication by Poincaré on Analysis situs. In Poincaré’s paper, Heegaard found that one of the theorems (the duality theorem) was problematic, and he started searching for a counterexample. The analysis of Poincaré’s duality theorem and Heegaard’s counterexample made up the larger part of Heegaard’s doctoral dissertation, which he defended in 1898 (Heegaard 1898). Its contents were also of great interest to the international community, and Heegaard communicated with Poincaré and others. A French translation of the dissertation

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was later published in Heegaard (1916).93 Heegaard’s quick rise to fame inspired the editors of the Encyklopädie der mathematischen Wissenschaften to ask him to write a chapter on the new Analysis situs together with Max Dehn (1878–1952), resulting in Dehn and Heegaard (1907). This collaboration initiated a lasting connection between Dehn and Scandinavian mathematicians. After his return from his European tour and his graduation, Heegaard extended his teaching obligations to even more secondary and tertiary institutions in Denmark. Although these teaching jobs left him little or no time for continued mathematical research, Heegaard later remembered the period as peaceful and happy (e.g. see Munkholm and Munkholm 1999, p. 932). In particular, Heegaard was relieved from the tensions that he had felt in the group of Danish mathematicians at the University of Copenhagen and the Polytechnical College. Despite his relatively meagre list of mathematical research publications, Heegaard was appointed professor of mathematics at the University of Copenhagen in 1910 after a rather bizarre turn of events. Heegaard had been persuaded to apply for the vacant position after Zeuthen, and there is evidence that he was reluctant to accept the position: it would mean a reduction in his salary (his salary at the University of Copenhagen would be only 1/5 of what he was earning teaching multiple jobs in gymnasia and other institutions; see Munkholm and Munkholm 1999, p. 936), he would have to focus more on research, and it would force him to deal with some of the tensions and conflicts that he had been happy to avoid previously. Despite his own reservations, Heegaard was appointed professor and held the professorship until 1917 when he suddenly retired claiming overwork and collegial problems as his reasons (see Munkholm and Munkholm 1999, pp. 935–936). Heegaard’s strained relations with his colleagues in Copenhagen, in particular with Harald Bohr (1887–1951), played out both at the University and in the Danish Matematisk Forening (Mathematical Society) until they were instrumental in bringing about Heegaard’s resignation in 1917. The following year, Heegaard was called to a chair in geometry at Universitetet i Oslo (the University of Oslo) – the city was then still called Kristiania – and he left Denmark for a new career in Norway. Heegaard’s resignation and subsequent move to Oslo made the headlines of Danish newspapers and tabloids. The character of Heegaard’s conflicts with his colleagues in Copenhagen has been analysed in Ramskov (1998) and Ramskov (2004).

Contributions to Mathematics Education From 1907 to 1911, Heegaard served as a consultant to the ministerial inspection of the secondary schools (undervisningsinspektionen), and he was teaching at a number of institutions in Denmark (see Johansson 1949, p. 38). In 1908, in connection

 For a description of the contents of Heegaard’s dissertation, see Munkholm and Munkholm (1999) and the references therein. 93

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with the Fourth International Congress of Mathematicians (ICM) in Rome, the Internationale Mathematische Unterrichtskommission (IMUK, International Commission on the Teaching of Mathematics) was established, and Heegaard was appointed the Danish delegate to the commission. He was thus, originally, a representative of the ministerial inspection and an experienced teacher in the gymnasium and subsequently (after 1910) a representative of the academic mathematical milieu. In his capacity as a delegate to the International Commission on Mathematical Instruction, Heegaard wrote the first comprehensive report on the instruction of mathematics in Denmark (Heegaard 1912). In that report, Heegaard described the organization of mathematics teaching in Denmark at all levels of education. In 1915, a foreigner’s account of the Danish mathematical teaching was published as Roherberg (1915). After the initial setting up of the International Commission on Mathematical Instruction and Heegaard’s appointment as delegate for Denmark, he took it upon himself to inform his Danish colleagues of the commission in the Matematisk Tidsskrift (Journal of Mathematics) by translating the commission’s mandate (Heegaard 1909). A Danish subcommission was formed headed by Heegaard and involving many of the centrally positioned mathematicians. Nevertheless, the work of the IMUK received only scant attention, and in 1918, Heegaard published a short notice in the Nyt Tidsskrift for Matematik in which he drew attention to the work of the IMUK (Heegaard 1918). The notice begins by a story that perhaps illustrates how the Commission was thought of by Danish mathematicians: A German recently told in a newspaper of how he had felt when a small child whom he had wanted to give an orange had declined the offer. The child was a Kriegskind [child of war] who had never seen an orange before. Similarly, many young mathematicians are likely to have grown up without knowing what is hidden behind the four letters that caption these lines [I.M.U.K.]. (Heegaard 1918, p. 108)94

Heegaard went on to describe how the proposed plans for a report from the IMUK at the planned ICM in Stockholm 1916 had been abandoned. Nevertheless, he wanted to draw the attention of “all those interested in these matters” to recent publications on the state and history of mathematics instruction in Germany and the United States (Heegaard 1918, p.  109). Similar attempts to draw attention to the work of the IMUK and its publications were also made in the Nyt Tidsskrift for Matematik by, for example, Heckscher (1912a), Heckscher (1912b), Heckscher (1913), IMUK (1914) and Trier (1915). When Heegaard arrived in Oslo, he soon felt much more at ease and at home with the mathematical community there than he had done in Copenhagen. Soon, he organized the Norsk Matematisk Forening (Norwegian Mathematical Society), which began its meetings in 1918, and he served as the first editor of the society’s journal, the Norsk Matematisk Tidsskrift (Norwegian Journal of Mathematics) from 1919 to  The original text is: “En Tysker fortalte nylig i en Avis om den Stemning, der havde grebet ham, da et lille Barn, som han vilde give en Appelsin, afslog Tilbuddet: Barnet var. et »Kriegskind«, som aldrig havde set Appelsiner. Således er der vel også nu vokset op mange unge Matematikere, der ikke ved, hvad der skjuler sig bag de 4 mystiske Bogstaver, som står over disse Linjer”. 94

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1923 and as the chairman of the Norwegian Mathematical Society from 1929 to 1934 (see Johansson 1949, p.  39). He also maintained his contacts with the International Commission on the Teaching of Mathematics and wrote two reports on the teaching of mathematics in Scandinavia after reforms had introduced differential calculus into the curriculum in 1910 and on the training of future teachers of mathematics in Norway (Heegaard 1930; Heegaard 1933). Heegaard held the chair as professor of geometry in Oslo until his retirement in 1941. He died on 7 February 1948 in Oslo. Heegaard devoted a considerable part of his time and energy to popularizing the sciences and to topics concerning the teaching of mathematics. Thus, he wrote popular books on astronomy and physics, travelled the vast Norwegian country, lecturing in the small communities, and lectured on the great “chiefs of the sciences” on the radio (see Munkholm and Munkholm 1999, pp.  942–944). These latter radio shows were produced during the German occupation of Norway and were later considered politically problematic because the access to radio receivers was restricted to families in good standing with the occupation forces and their ideology (Munkholm and Munkholm 1999, p. 942). He also published papers on the history of mathematics and on “elementary” subjects in the Norwegian Journal of Mathematics. Nevertheless, Heegaard only wrote two textbooks on mathematics – one on mathematics for the naval technical education (Heegaard 1905) and one together with Olaf M.  Thalberg on mathematical geography for the gymnasium (Heegaard and Thalberg 1927). In the preface to the first of these textbooks, Heegaard described his presentational style by quoting – in German – from Klein’s Elementary Mathematics from a Higher Standpoint: The teaching for beginners and for those students who want to use mathematics only as an auxiliary means for other studies has to make a naïve use of the intuitive approaches. (Heegaard 1905, i–ii)95

Heegaard often favoured the process of intuition over that of rigorous proof. In his lectures, he is reported to have made good use of geometrically intuitive illustrations and of using stories to make the mathematics more interesting (Johansson 1948, p. 40). As professor of geometry, Heegaard became involved with the publication of the collected works of Sophus Lie (1842–1899) as a co-editor together with Friedrich Engel (1861–1941). These activities occupied Heegaard to the extent that he produced few research papers of international importance after his dissertation and the chapter in the Encyklopädie der mathematischen Wissenschaften. Indeed, it seems

 The original text is: “Der Unterricht derjenigen, welche die Mathematik nur als ein Hülfsmittel [sic] gebrauchen wollen, soll von den anschauungsmässigen Momenten einen naiven Gebrauch machen”. Its full context in Klein’s preface as translated in Klein (2016, p. ix) is: “My opinion is still that the teaching for beginners and for those students who want to use mathematics only as an auxiliary means for other studies has to make a naïve use of the intuitive approaches; the conviction that this is necessary for pedagogic reasons, considering the disposition of the majority of students, has become noticeably stronger in the last few years, here and abroad”. 95

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that he saw himself more as an “educator” in a broad sense than as a research mathematician.

Sources Dehn, Max and Poul Heegaard. 1907. Analysis situs. In Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen [Encyclopedia of mathematical sciences including their applications], Vol. III.AB, Cchapter 3, 153–220. Leipzig: B. G. Teubner. Heegaard, Poul. 1898. Forstudier til en topologisk Teori for de algebraiske Fladers Sammenhæng [Preliminary studies for a topological theory for the algebraic surfaces consistency]. København: Det Nordiske Forlag – Ernst Bojesen. Heegaard, Poul. 1916. Sur l’ “analysis situs”. Bulletin de la Société Mathématique de France 44: 161–242. Heckscher, Ivar. 1912b, Fra I.M.U.K. [From I.M.U.K.] Nyt Tidsskrift for Matematik. Afdeling A 23: 105–109. Heckscher, Ivar. 1913, Fra I.M.U.K. [From I.M.U.K.] Nyt Tidsskrift for Matematik. Afdeling A 24: 82–85. I.M.U.K. 1914. Nyt Tidsskrift for Matematik Afdeling A 25: 48. Johansson, Ingebrigt. 1949. Minnetale over Professor Poul Heegaard holdt i den mat.-naturv. klasses møte den 1. november 1948 [Memorial speech about Professor Poul Heegaard held in the meeting of the math.-nat. sciences class [of the Academy] on November 1, 1948]. Det norske Videnskaps-Akademi i Oslo, Årbok 1948, 36–45. Oslo: Jacob Dybwad. Jørgensen, V.T. and Jakob Nielsen. 1980. Poul Heegaard (1871–1948). Dansk Biografisk Leksikon, 3rd Edn. (16 vols.), Vol. 6, 134. København: Gyldendahl. Klein, Felix. 2016. Elementary Mathematics from a Higher Standpoint. Volume III: Precision Mathematics and Approximation Mathematics. Trans. by Marta Menghini. Berlin and Heidelberg: Springer. Munkholm, Ellen S. and Hans J. Munkholm. 1999. Poul Heegaard. In History of Topology, ed. Ioan M. James, Cchapter 34, 925–946. Amsterdam: North Holland. Munkholm, Ellen S. and Hans J.  Munkholm. 1998. Poul Heegaard (1871–1948), dansk-norsk topolog [Poul Heegaard (1871–1948), Danish-Norwegian topologist]. Normat 46(4): 145–169. Ramskov, Kurt. 1998. Fra de akademiske kulisser: universitetets matematiske stillinger 1909–17 [From behind the academic scenes: the mathematical positions of the university 1909–17]. KU Preprint series, 10. Ramskov, Kurt. 2004. Matematiske stillinger og karrierepleje i første tredjedel af det 20.århundrede i København [Mathematical positions and careers in the first third of the 20th century in Copenhagen]. HOSTA, 16. Roherberg, Albert. 1915. Der mathematische Unterricht in Dänemark, n. 2. In Beihefte zur Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht. Berichte und Mitteilungen veranlasst durch die Internationale Mathematische Unterrichtskommission, Zweite Folge. Leipzig/Berlin: B. G. Teubner. Trier, Ved V. 1915. Fra I.M.U.K. [From I.M.U.K.]. Nyt Tidsskrift for Matematik, Afdeling A 26: 53–77.

Publications Related to Mathematics Education Heegaard, Poul. 1905. Lærebog i Mathematik udarbejdet til Brug ved Forberedelsen til Maskinisteksamens anden Afdeling og i Maskinistskolens yngste Klasse [Mathematical textbook for Preparation for the Second Department of the Mechanical Examination and in the Youngest Class of the Mechanical School]. København: G. E. C. Gad’s Universitetsboghandel.

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Heegaard, Poul. 1909. Den Internationale Matematikundervisningskommission. Foreløbig Meddelelse om Kommissionens Organisation og almindelige Arbejdsplan [The International Commission on The Teaching of Mathematics. Preliminary communication on the Commission’s organization and general work Plan]. Nyt Tidsskrift for Matematik, Afdeling A, Oversættelse: 37–50. Heegaard, Poul (ed.). 1912. Der Mathematikunterricht in Dänemark. København: Gyldendalske Boghandel – Nordisk Forlag. Heegaard, Poul (1918). I.M.U.K. Nyt Tidsskrift for Matematik, Afdeling A, 108–110. Heegaard, Poul. 1929. La représentation des points imaginaires de Sophus Lie et sa valeur didactique. In Atti del Congresso Internazionale dei Matematici, ed. Guido Castelnuovo, Vol. 3, 421–423. Bologna: N. Zanichelli. Heegaard, Poul. 1930. Scandinavie. L’Enseignement Mathématique 29: 307–314. Heegaard, Poul. 1933. Norvège: préparation théorique et pratique des professeurs demathématiques de l’enseignement secondaire. L’Enseignement Mathématique 32: 360–364. Heegaard, Poul and Thalberg, Olaf M. 1927. Matematisk geografi for gymnasiet (Mathematical geography for gymnasiet) Oslo: Gyldendal.

Photo Source: Wikimedia Commons.

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11.24 Heinz Hopf (Breslau, Now Wrocław, 1894 – Zollikon, 1971): Ex Officio Member 1955–1958 Giorgio T. Bagni

Biography Heinz Hopf was born on 19 November 1894 in Breslau, Germany (Wrocław, now in Poland); from 1901 until 1904, he attended the school of Dr. Karl Mittelhaus, and until 1913 the König-Wilhelm Gymnasium in Breslau. In 1913, he entered the Silesian Friedrich Wilhelm University in Breslau to study mathematics. Hopf’s teachers were Adolf Kneser (1862–1930), Erhard Schmidt (1876–1959) and Rudolf Sturm (1841–1919); he also attended lectures by Max Dehn (1878–1952) and Ernst Steinitz (1871–1928), teachers at the Polytechnic in Breslau. His studies were interrupted by the outbreak of World War I, and Hopf fought on the Western front as a lieutenant. However, in 1917, he attended a class by Schmidt on set theory at the University of Breslau. After the end of the war, he returned to his studies both in Breslau and at the University of Heidelberg. In 1920, he went to study for his doctorate at the University of Berlin, where Erhard Schmidt was teaching. Hopf attended some courses by Issai Schur (1875–1941) in Berlin and received his doctorate in 1925 with a thesis, supervised by Erhard Schmidt, studying the topology of manifolds (Freudenthal 1972). His dissertation “Über Zusammenhänge zwischen Topologie und Metrik von Mannigfaltigkeiten” was published in two parts in the Mathematischen Annalen in G. T. Bagni (1958–2009) University of Udine, Udine, Italy

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1925 (“Zum Clifford-Kleinschen Raumproblem” and “Über die Curvatura integra geschlossener Hyperflächen”). This was the basis for further results by himself and a group of students. Then Hopf went to Göttingen, where he met both Emmy Noether (1882–1935) and Pavel Sergeyevich Alexandroff (1896–1982). In 1926, with Alexandroff, he was in the south of France, and in the academic year 1927–1928, they were in Princeton in the United States, where they collaborated with Solomon Lefschetz, Oswald Veblen and Alexandroff, and planned a great work on topology. The first volume appeared in 1935 (Alexandroff and Hopf 1935). The greater part of Hopf’s mathematical work was in algebraic topology, a sector where his contribution can be interpreted as a continuation of Luitzen Brouwer’s work. As Hans Freudenthal wrote, “his initially crude and too directly geometric methods underwent gradual refinement, first by Emmy Noether’s abstract algebraic influence, then through the combinatorial ideas of the American school” (Freudenthal 1972, p. 496). In 1931, Hopf defined what is now known as the “Hopf invariant” in his important work on maps between spheres into spheres of lower dimension. The main ideas which he introduced in this investigation led him to define what is today called a “Hopf algebra”. Finally, it is worthwhile mentioning the great achievements of Hopf in global differential geometry. In Hopf’s mathematical work, three aspects can be highlighted: besides an extraordinary geometric intuition, the continuous confrontation of the algebraic and topological aspects of the problem; the fact that “Hopf always concerned himself with particular and explicit problems, which in the context of his ideas must be considered ‘concrete’ (vector fields on spheres and manifolds, prime ends of spaces and groups, essential mappings, etc.) …he always gave the solution to the particular problem and simultaneously created a method to deal with it, which clarified the guiding idea, the deep reason behind it and ulterior possibilities” (Eckmann 1971, p. 6); his ability to interact with other scholars and with his many students with an extremely fruitful exchange of ideas. With reference to Hopf’s Collected Papers (Hopf 2001), Beno Eckmann wrote: It is indeed astonishing to realise that this oeuvre of a whole scientific life consists of only about 70 writings, Comptes Rendus Notes and survey articles included, and of course the book “Topology I” written jointly with Paul Aleksandrov. Astonishing also the transparent and clear style, the concreteness of the problems, and how abstract and far-reaching the methods Hopf invented to master them  – abstract, but without unnecessary generalities. (Eckmann 2001, p. V)

From 1955 until 1958, Hopf was president of the International Mathematical Union (IMU) and member ex officio of the ICMI Executive Committee in the same period. He received honorary doctorates from Princeton, Freiburg, Manchester, Sorbonne (Paris), Brussels and Lausanne University, and was awarded several important prizes, including the Gauss-Weber Medal and the Lobachevsky Prize. Heinz Hopf died on 3 June 1971 in Zollikon, Switzerland.

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Contribution to Mathematics Education When Hopf became president of IMU, the Union had been reconstituted from 1952, and he was involved in the mission of shaping its new status and strengthening the mathematical community. The expansion of the Union was favoured by his mathematical stature and his non-political nature. In 1958, he advocated that the Union took part in the planning of the mathematical programme of the International Congresses of Mathematicians (Lehto 1998, p. 116). In the meeting of ICMI held in Paris on 29 December 1956, Hopf, then ex officio member of ICMI and president of IMU, proposed to the Executive Committee a study to be carried out by ICMI concerning the difficulties that arise in recruiting secondary teachers in mathematics and sciences, a problem linked to the topic presented by Kurepa at the International Congress of Mathematicians of 1954  in Amsterdam (see L’Enseignement Mathématique. 1957. s. 2, 3: 79). The questionnaire for the inquiry was prepared by Howard Fehr, Đuro Kurepa, Willy Servais and David van Dantzig (L’Enseignement Mathématique. 1958. s. 2, 4: 220–223). Hopf was mainly a researcher, but in his academic work, he was also an excellent teacher. Gianfranco Arrigo (Lugano, Switzerland), who attended his lectures at ETH in Zürich, epitomizes his role in the transmission of mathematical knowledge with the following words: Heinz Hopf influenced me deeply and lastingly. He was my professor during the last period of his career: he was rather old, small and slim, but his mind was formidable, bright, and really lighted up all the students’ minds. His lessons were unforgettable mathematical journeys. He taught us to build up our own mathematics. All his students understood him without particular difficulties, he always took care to make sure that we were able to follow his arguments: in the first part of every lesson, he used to ask us explicitly to remember all the elements needed in order to understand his ideas. He was very kind with all the students, always encouraging them to do better and better. He let us know that true mathematics is not something locked up into formulas or into books, but rather incited us to find out the mathematical ideas that we were able to construct little by little. (Private communication to the author)

Sources Hopf, Heinz. 1964. Selecta Heinz Hopf. Berlin: Springer Verlag. Hopf, Heinz. 2001. Heinz Hopf Collected Papers (edited by Beno Eckmann). Berlin: Springer Verlag. Alexandroff, Pavel and Heinz Hopf. 1935. Topologie I. Berlin: Julius Springer. Alexandroff, Pavel Sergeyevich. 1976–1977, Heinz Hopf zum Gedenken I: Einige Erinnerungen an Heinz Hopf. Jahresberichte der Deutschen Mathematiker–Vereinigung 78(3): 113–125. Alexandroff, Pavel Sergeyevich. 1977. In memory of Heinz Hopf (Russian). Uspekhi Matematicheskikh Nauk 32(3) (195): 203–208. Behnke, Heinrich and Friedrich Hirzebruch. 1972. In memoriam Heinz Hopf. Mathematische Annalen 196: 1–7.

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Cartan, Henri. 1972. Heinz Hopf (1894–1971). IMU Bulletin 3: 7–10. Eckmann, Beno. 1971. In memory of Heinz Hopf. Boletim da Sociedade Brasileira de Matemática [Bulletin of the Brazilian Mathematical Society] 2(2): 1–7. Eckmann, Beno. 1972. Zum Gedenken an Heinz Hopf. L’Enseignement Mathématique s. 2, 18: 105–112. Eckmann, Beno. 1994. Zum 100. Geburtstag von Heinz Hopf. Elemente der Mathematik 49(4): 133–136. Eckmann, Beno. 2001. Editor’s preface. In Heinz Hopf Collected Papers (edited by Beno Eckmann), 5–6. Berlin: Springer Verlag. Frei, Günther and Urs Stammbach. 1999. Heinz Hopf. In History of topology, ed. I. M. James, 991–1008. Amsterdam: Elsevier. Freudenthal, Hans. 1972. Hopf, Heinz. In Dictionary of Scientific Biography, ed. C. Gillispie, Vol. 6, pp. 496–497. New York: Charles Scribner’s Sons. Hilton, Peter J. 1972. Heinz Hopf. Bulletin of the London Mathematical Society 4: 202–217. Samelson, Hans. 1976–1977, Heinz Hopf zum Gedenken II: Zum wissenschaftlichen Werk von Heinz Hopf. Jahresberichte der Deutschen Mathematiker–Vereinigung 78(3): 126–146. Voss, Konrad. 1973. In memoriam Heinz Hopf. Elemente der Mathematik 28(4): 81–83.

Photo Source: Wikimedia Commons.

Shōkichi Iyanaga

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11.25 Shōkichi Iyanaga (Tokyo, 1906 – Tokyo, 2006): VicePresident 1971–1974, President 1975–1978, Ex Officio Member of the Executive Committee 1979–1982 Fulvia Furinghetti

Biography Shōkichi Iyanaga was born on 2 April 1906 in Tokyo, as a son of a banker. He had his schooling in Tokyo. From 1926 to 1929, he studied at the (Imperial) University of Tokyo, where he attended the algebra lectures by Teiji Takagi, a well-known scholar in class field theory. In 1928, he published two papers in the Japanese Journal of Mathematics and the Proceedings of the Imperial Academy of Tokyo, before obtaining his undergraduate degree in 1929. In 1931, he completed his PhD in mathematics under the supervision of Takagi. In the same year, he went to Paris with a scholarship from the French government and to Hamburg, where he studied with the Austrian mathematician Emil Artin. In the 3 years spent in Europe, he met top mathematicians such as Claude Chevalley, Henri Cartan, Jean Dieudonné and André Weil. In 1932, he attended the International Congress of Mathematicians in Zurich. F. Furinghetti (*) University of Genoa, Genoa, Italy e-mail: [email protected]

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In 1934, Iyanaga returned to the University of Tokyo, where he was appointed assistant professor until 1942 when he was promoted to full professor in this University until 1967. He spent the year 1961–1962 as a visiting professor at the University of Chicago. After he retired from the University of Tokyo in 1967, he was a professor at Gakushūin University in Tokyo until 1977 and was a visiting professor at Nancy in France from 1967 to 1968. Iyanaga wrote many research papers in German, French and English. His works on mathematics were assembled in the book entitled Collected Papers, published by Iwanami Publisher in 1994. Besides his main scientific production in algebraic geometry, functional analysis and number theory, Iyanaga wrote some notes on the history of mathematics published in Historia Scientiarum, and in 1999 and 2002, he published two books on Evariste Galois. In 1954, he edited the Encyclopedic Dictionary of Mathematics (in Japanese), which had many editions, including the English translations published by the MIT Press in 1977 and 1980 (with co-editor Yukiyosi Kawada). Bass and Hodgson (2006) mention Iyanaga’s autobiography published by the Iwanami Publisher in 2004. Iyanaga had many honours and relevant positions. He was a member of the Science Council of Japan in 1947, and in this capacity, he was the main organizer of an International Symposium on Algebraic Number Theory held in Japan in 1955 where the Taniyama-Shimura conjecture originated. From 1965 to 1967, he was dean of the Faculty of Science at Tokyo University. Several times between 1950 and 1971, he was elected president of the Mathematical Society of Japan. From 1952 to 1954, he was a member of the Executive Committee of the International Mathematical Union. In 1976, he was awarded the Rising Sun of Second Class from Japan. In 1978, he was elected a member of the Japan Academy. He received the Ordre des Palmes Académiques from France in 1979 and the Ordre de la Légion d’Honneur from France in 1980. In 1990, when the International Congress of Mathematicians was held in Kyoto, the honorary president of the Congress Kiyosi Itō acknowledged that as early as the 1960s, Kawada and Iyanaga began campaigning for the Congress to be held in Japan. Iyanaga was appointed honorary member of the Organizing Committee of the Congress. Until the age of 98, Iyanaga attended the seminar on number theory at Gakushūin University. His last mathematical paper was published in 2006 (Travaux de Claude Chevalley sur la théorie du corps de classes: Introduction. Japanese Journal of Mathematics 1: 25–85). He died on 1 June 2006 in Tokyo.

Contribution to Mathematics Education Iyanaga spent the final period of WWII in the countryside to escape the frequent bombardments of Japanese cities. At the end of the war, when Japan was having a hard time, he contributed to rebuilding the science and the culture of his country in a better form. He delivered courses and organized seminars for younger researchers.

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Moreover, he was the editor-in-chief of many mathematical textbooks in Japanese for schools, such as those on new arithmetic (I–VI, for primary schools, 1950–1974), new mathematics (I–III, textbooks for junior secondary schools, 1951–1975), general mathematics, etc. (textbooks for senior secondary schools, 1956–1961), published by the publisher Tokyo Shoseki. At the international level, Iyanaga was one of the delegates of the Japan subcommission of ICMI (L’Enseignement Mathématique. 1957. s. 2, 3, pp. 308–309). In 1971–1974, he was vice-president of ICMI, president in 1975–1978 with his former student Kawada as ICMI secretary and ex officio member in 1979–1982. Iyanaga’s appointment is noteworthy because he was the first president from outside Europe and North America. This fact explains the perplexities about this appointment expressed by Otto Frostman, secretary of the International Mathematical Union (IMU), in a letter to ICMI President James Lighthill (7 December 1974): I am a bit more dubious about Iyanaga, but I admit I don’t know him very well. Now it is time for the Japanese to show that they are active and generous and prepared to pay a lot of money at least for their travel! You can’t have all the meetings in Japan.

In his reply to Frostman (19 December 1974), Lighthill recalled that He has made two important visits to Europe during this period, one to the ICMI Symposium in Bielefeld which was held in September when he took a lot of trouble to discuss our plans for the Third International Congress to be held in Karlsruhe in 1976, and the second visit in November when a large number of us who will be concerned with the Karlsruhe Congress met together at UNESCO Headquarters (at the expense of UNESCO) to take the detailed planning of that Congress a stage further. Iyanaga’s contribution during both these visits was important. In particular, he showed admirable tact in dealing with all the personalities from all the different countries and this is a quality of great value in the President of an international organization. I believe that you need have no anxiety and that he will prove to have been an excellent choice.

Iyanaga, as Japan delegate for long years (L’Enseignement Mathématique s. 2, 3: 308–309, 1957), was not new in the ICMI milieu. According to the list of participants provided to us by Erich Wittmann, he attended ICME-1 of 1969 in Lyon. He was vice-­president of ICMI in 1971–1974. Iyanaga was aware of the different situations of mathematics education in Japan and other countries in Europe and America; in his report, written at the beginning of his mandate, he expressed the wish of continuing the work carried out in those countries (see Iyanaga 1975). In Iyanaga (2001), he expressed some caution about the new math movement. In 1972 at ICME-2 in Exeter, he chaired the Working Group on “Algebra at School Level” (see Howson, Albert Geoffrey. 1973. Developments in Mathematical Education. Cambridge: Cambridge University Press, p. 300). During Iyanaga’s presidency, two important meetings were organized: ICME-3  in Karlsruhe (1976) and the ICMI Symposium organized during ICM-1978  in Helsinki with the cooperation of UNESCO and IDM (Institut für Didaktik der Mathematik) on “What Knowledge, Experience and Understanding of Mathematics Should a Mathematics Teacher Have?” (1978. ICMI Bulletin 11: 13–15). Moreover, ICMI-related symposia took place in Africa, Europe, India, Latin America and Southeast Asia, as reported in the ICMI Bulletin issued in that period. Iyanaga delivered a plenary lecture at the first

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“Southeast Asian Conference on Mathematical Education” (1978. ICMI Bulletin 11: 5–10). The mathematicians and mathematics educators in the Philippines and Southeast Asia owe much to Iyanaga and Kawada. They connected them not only to the Japanese mathematics community but because of their position in ICMI and IMU, also to the international mathematics and mathematics education community. As for the policy of ICMI, in his address, Iyanaga (1975) referred to the Terms of Reference of ICMI adopted at the Paris meeting of the Executive Committee of IMU in April 1960. He was trusting in communication, and one of the points of his programme was to improve the issue of ICMI Bulletin. In his remembrances from the period of his presidency (Iyanaga 2001), he mentioned his visits to Bent Christiansen in Denmark, the invitation to meetings in India and the Philippines and to the Institute of Bielefeld on the occasion of the ICMI-IDM Symposium on the teaching of geometry organized by Hans-Georg Steiner (Bielefeld, 16–20 September 1974), where he delivered a plenary entitled “Combined Geometry  - Algebra Curriculum for Japanese Secondary Schools”. Iyanaga acted as co-chairman with Bernhard Neumann of the ICMI-JSME Regional Conference of Mathematics Education (Tokyo, 5–6 November 1974). A valuable achievement of Iyanaga, together with the ICMI secretary Kawada, was giving visibility to ICMI at the International Congress of Mathematicians in Helsinki (1978). In the introductory chapter of the Proceedings of this Congress we read “Unofficial mathematical activities also included a three-day symposium organized by the International Commission on Mathematical Instruction” (Lehto 1980, p. 7, Vol. 1). ICMI was also mentioned in the chapter “Closing Ceremonies” (ibidem, p. 13), by stressing the fact that it has “a history antedating that of our [of mathematicians] Union. Under the able guidance of its chairman, Professor Iyanaga, and its secretary, Professor Kawada, it has continued and expanded its valuable role. Its activities … are too varied to be described here: it is sufficient to remind you of the successful conference at Karlsruhe [he refers to the third ICME] two years ago”.

Sources Iyanaga, Shōkichi. 1994. Collected works. Tokyo: Iwanami. Iyanaga, Shōkichi, and Kunihiko Kodaira. (1961). Gendai Sugaku Gaisetsu I [Introduction to Modern Mathematics I]. Tokyo: Iwanami. Iyanaga, Shōkichi. 1968. Kikagaku Josetsu [Introduction to Geometry]. Tokyo: Iwanami. Iyanaga, Shōkichi. 1954. (ed.), Sugaku Jiten [Encyclopedic dictionary of mathematics], first edition. Tokyo: Iwanami Shoten. Iyanaga, Shōkichi. 2001. Memories of Professor Teiji Takagi. In Class Field Theory – Its Centenary and Prospect, ed. Katsuya Miyake, 1–11. Advanced Studies in Pure Mathematics 30. Tokyo: Mathematical Society of Japan. Bauersfeld, Heinrich, Michael Otte, and Hans-Georg Steiner (eds.). Proceedings of the ICMI-IDM Regional Conference on the Teaching of Geometry. IDM-Schriftenreihe, Vol. 3. Bielefeld: IDM. Lehto, Olli (Ed.) (1980). Proceedings of the International Congress of Mathematicians. Helsinki: Academia Scientiarum Fennica. Vol. 1.

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Proceedings of ICMI-JSME Regional Conference on Curriculum and Teacher Training for Mathematical Education, Tokyo, November 5–9, 1974. Tokyo: National Institute for Educational Research. Bass, Hyman, Bernard R. Hodgson, and Shigeru Iitaka. 2006. In memoriam. Shokichi Iyanaga. ICMI Bulletin 58: 45–49. Fujita, Hiroshi. 2006. In Memory of Professor Shōkichi Iyanaga Mathematical Sciences and Professor Shōkichi Iyanaga. Japan Journal of Mathematics Education and Related Fields 47(3–4): 1–7 (In Japanese). Kota, Osamu. 2006. In Memory of Professor Shōkichi Iyanaga. Memories of Professor Shōkichi Iyanaga. Japan Journal of Mathematics Education and Related Fields 47(3–4): 9–13 (In Japanese). Iitaka, Shigeru. 2008. Shōkichi Iyanaga. Retrieved January 30, 2020, from Fulvia Furinghetti and Livia Giacardi. 2008. The first century of the International Commission on Mathematical Instruction (1908–2008): https://www.icmihistory.unito.it/portrait/iyanaga.php

Publications Related to Mathematics Education Iyanaga, Shōkichi. 1965. Sugaku no Manabikata [How to Learn Mathematics]. Tokyo: Diamond. Iyanaga, Shōkichi. 1975. Report of the President. ICMI Bulletin 5: 3–4. Iyanaga, Shōkichi. 1978. Report of the President. ICMI Bulletin 11: 3–4. Iyanaga, Shōkichi. 1979. The role of Mathematical societies in the development of mathematics. In Proceedings of the First Southeast Asian Conference on Mathematical, eds. Iluminada C.  Coronel, Josefina C.  Fonacier, Bienvenido F.  Nebres, and Norman F.  Quimpo, 11–16. Manila: Mathematical Society of the Philippines, Inc. Iyanaga, Shōkichi. 2000. Sugakusha no Nijisseiki [A mathematician in the twentieth Century]. Tokyo: Iwanami. It is a collection of Iyanaga’s works that contains some essays concerning mathematics education. Iyanaga, Shōkichi. 2001. My remembrances from the period 1975–1978. ICMI Bulletin 50: 6–7.

Photo Author: Jacobs, Konrad. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

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Ralph Lent Jeffery

11.26 Ralph Lent Jeffery (Overton, 1889 – Wolfville, 1975): Member of the Executive Committee 1952–1954 Harm Jan Smid

Biography Ralph Lent Jeffery was born on 3 October 1889 in the small village of Overton in Nova Scotia, Canada. In his childhood, nothing seemed more unlikely than an academic career in mathematics. In the middle of the eighth grade, he left school and joined his father, Frank Jeffery, to become a fisherman. In 1910, however, he decided that he did not want to remain a fisherman all his life. That was not because he was unhappy with his work but as he told much later, because he was not certain that he would still like it being 41 instead of 21. He attended Yarmouth Academy and then Nova Scotia Normal College, completing his studies there in 1915. He then became principal of Port Maitland High School. On 23 August 1916, he married Nellie Huntington Churchill from his native village Overton. She persuaded him to continue with his education, and he enrolled in Acadia University in Wolfville, Nova Scotia. Although he took a degree in economics, he also took two mathematics courses, one in calculus and one in analytic geometry, and this was decisive for his future life. He fell in love with mathematics and went to Cornell University to undertake graduate studies in math. Two undergraduate courses may seem a rather poor foundation for graduating in math in Cornell, but he did. He continued his studies for a year in Harvard, and in H. J. Smid (*) Delft University of Technology, Delft, The Netherlands

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1924, he was appointed as head of mathematics of Acadia University. In 1928, he stayed for 1 year in Cornell to obtain his doctorate. Besides his doctoral thesis, he published several research articles on analysis in those years. Jeffery remained in Acadia until 1942, with the exception of 1938 when he stayed in Saskatchewan. In 1942, he moved to Queen’s University in Kingston, Ontario, where he became head of mathematics. After his retirement in 1960, he came back to Acadia, where he kept teaching until his death in 1975.

Contribution to Mathematics Education In Queen’s University, Jeffery built a strong mathematics department, with fine undergraduate and graduate teaching and research programmes. Queen’s honoured his achievements by naming the new building that houses the math department after him. Jeffery was involved in the founding of the Canadian Mathematical Society and was its fourth president in 1957–1961. Perhaps his most important achievement was the establishment of the Summer Research Institute (SRI) in 1950. Suitable accommodation was found in Queen’s, and Jeffery raised the money for the support of the fellows. The SRI became a success and an important factor in fostering mathematical research in Canada. Jeffery served the SRI as director for 15 years. In the same period, 1952–1954, he was a member of the Executive Committee of the International Mathematical Instruction Commission (IMIC), and in 1955–1958, 1963–1966 member at large (free member) of ICMI. When he was 71, Jeffery retired from Queen’s and returned to Acadia, accepting a full teaching load. However, this did not use up all of his energy. Before long he came involved with the Nova Scotia Department of Education. He persuaded them to set up a Mathematics Curriculum Committee (CMC) that played an important role in modernizing the math curriculum. Of course, Jeffery was a member of the committee, with a number of prominent teachers of the province. Another important initiative he was involved in was the Nova Scotia Summer School for Teachers, in cooperation with CMC Summer School. Jeffery received many honours. He was awarded honorary degrees by Acadia University, Dalhousie University, St. Mary’s University, Memorial University, McMaster University, Windsor University and Queen’s University. He was a fellow of the Royal Canadian Society and served it as president of Section III in 1953. The Canadian Mathematical Society honoured him by setting up the Jeffery-Williams Prize, and several mathematical scholarships bear his name.

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Sources Jeffery, Ralph L. 1931. The uniform approximation of a sequence of integrals and the sequence of functions which define a definite integral containing a parameter. Ithaca: PhD thesis Cornell 1928. Jeffery, Ralph L. 1946. The future of mathematics in Canada. In Proceedings of the First Canadian Mathematical Congress (Montreal 1945), 64–74. Toronto: University of Toronto Press. Jeffery, Ralph L. 1951. The theory of functions of a real variable. Toronto: University of Toronto Press. (reprinted by Dover Publications 1985). Jeffery, Ralph L. 1954. Calculus. Toronto: University of Toronto Press. Jeffery, Ralph L. 1956. Trigonometric series: a survey. Presidential address to section III of the Royal Canadian Society, Montreal 1953. Toronto: University of Toronto Press. Jeffery, Ralph L., Irving Kaplansky, Alexander Grothendieck, and Albert John Coleman. 1971. A survey of operator algebras. Kingston: Queen’s University. De Beauregard Robinson, Gilbert. 1976. Ralph L. Jeffery 1889–1975. Proceedings of the Royal Society of Canada 14(4): 68–70. Tingley, Arnold J. 1995. Random Memories of Ralph Jeffery. In CMS • SMC 1945–1995, Mathematics in Canada, 321–325. Ottawa: Canadian Mathematical Society.

Publications Related to Mathematics Education Jeffery, Ralph L. 1966. Algebra, grade 10: a preliminary teaching guide: Halifax, Department of Education. Jeffery, Ralph L. 1969. Coordinate geometry and trigonometry. Halifax: Department of Education. Jeffery, Ralph L. 1970. Matrices, determinants and their applications to linear equations: comments and solutions to problems. Halifax: Department of Education.

Photo Courtesy of the Canadian Mathematical Society.

Yukiyosi Kawada

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11.27 Yukiyosi Kawada (Tokyo, 1916 – Tokyo, 1993): Secretary 1975–1978 Fulvia Furinghetti

Biography Yukiyosi Kawada was born in Tokyo on 15 January 1916. In 1935, after graduating from Tokyo High School, he entered the (Imperial) University of Tokyo, where he studied number theory, especially class field theory under Zyoiti Suetuna. There he met Shōkichi Iyanaga, who was an assistant to Teiji Takagi. Suetuna studied also probability, which, later on, was a topic approached also by Kawada. In 1938, he graduated from the University of Tokyo, and in 1945, he completed his PhD in mathematics at the University of Tokyo under the supervision of Iyanaga. From 1941 to 1950, he was an assistant professor at the University of Tokyo and associate professor at Tokyo Bunri University. In 1950, he became a professor at the University of Tokyo until his retirement in 1976. Then from 1976 to 1986, he taught mathematics as a professor at Sophia University in Tokyo, a private university founded by Jesuits. In summer 1962, he was a visiting professor in Salvador (Brazil) and delivered a course on algebraic geometry (see Mattedi Dias 2008). At the same time, between 1971 and 1973, Kawada was the director of the Institute of Statistical Mathematics in Tokyo University. He held other prestigious F. Furinghetti (*) University of Genoa, Genoa, Italy e-mail: [email protected]

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positions, such as the presidency of the Mathematical Society of Japan. In 1971–1974, he was a member of the Executive Committee of the International Mathematical Union (IMU), and in 1974–1978, he was appointed by IMU as one of the seven members of the Consultative Committee of the International Congress of Mathematicians held in Helsinki in 1978. In 1990, when the International Congress of Mathematicians was held in Kyoto, the honorary president of the Congress Kiyosi Itō acknowledged that as early as the 1960s, Kawada and Shōkichi Iyanaga began campaigning for the Congress to be held in Japan. Kawada was a member of the General Committee. Kawada’s main theme of research was on the theory of class formations related to the theory of class field, but he studied also algebraic functions, group theory, probability, operator rings, cohomology theory, topological groups, infinite extension of fields, analytic line bundles and theory of cosheaves. Iitaka (2008) reports that, according to the president of IMU Komaravolu Chandrasekharan, Kawada was as quiet as he was efficient and it was a joy to be with him. He died on 28 October 1993 in Tokyo.

Contribution to Mathematics Education With Iyanaga, Kawada edited the two volumes of the English version of the Encyclopedic Dictionary of Mathematics. According to the review by Peak (1978, p. 786), These two volumes represent an outstanding, scholarly production … [which] provides broad coverage in both subjects and terminology. It concisely represents the history of mathematics as well as its cognitive aspects. … They should be in the libraries of those secondary and postsecondary schools where serious mathematics students are enrolled.

Between 1967 and 1969, Kawada was a member of the Evaluating Committee of Mathematics textbooks, nominated by the Ministry of Education, and between 1973 and 1981, he was a supervisor of mathematical education, nominated by the Ministry of Education. In the 1970s, he proposed to include contents on Euclidean geometry, calculus and axiomatics in mathematics curricula for Japanese secondary school, in some form to match the learners’ level, when the wave of the new math movement reached Japan. In fact, Kawada’s proposal had been realized to a significant extent in the Japanese national mathematics curricula for some years (see Fujita 2004, p. 28). Kawada was secretary of the International Commission on Mathematical Instruction (ICMI) in 1975–1978 under the presidency of his former supervisor. As secretary of ICMI, Kawada edited the ICMI Bulletin and reported on the important events of the years 1975–1978: ICME-3 in 1976, the Symposium on the mathematical education of mathematics teachers during the International Congress of Mathematicians in 1978 and other ICMI-related symposia that took place in Africa,

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Europe, India, Latin America or Southeast Asia. In particular, as Nebres (2008) points out, Kawada strengthened the impact of ICMI on East-Southeast Asian mathematics education. He was a member of the Advisory Committee of Southeast Asian Mathematical Society. In April 1976, he recommended organizing a Regional Conference in Mathematics Education, which would be co-sponsored by ICMI. The member countries of the Southeast Asian Mathematical Society (SEAMS), founded in 1972, were invited to apply for membership with ICMI. Manila accepted to host the First Southeast Asian Conference on Mathematics Education (SEACME) in 1978. In the words of Suat Khoh Lim-Teo (2008, p. 248), “This first regional conference was a resounding success judging by the learning that took place through a series of activities leading up to the conference and the follow-up actions after the conference”. The mathematicians and mathematics educators in the Philippines and Southeast Asia owe much to Iyanaga and Kawada. They connected them not only to the Japanese mathematics community but because of their position in ICMI and IMU, also to the international mathematics and mathematics education community. In 1984, Kawada was a member of the International Program Committee of ICME-5 in Adelaide.

Sources Kawada, Yukiyosi and Kiyosi Itō. 1940. On the probability distributions on a compact group, I. Proceedings of the Physico-Mathematical Society of Japan s. 3, 22: 977–998. Kawada, Yukiyosi. 1953. On the ramification theory of infinite algebraic extensions. Annals of Mathematics 58: 24–47. Kawada, Yukiyosi, John T. 1955. On the Galois cohomology of unramified extensions of function fields in one variable. American Journal of Mathematics 77: 197–217. Kawada, Yukiyosi. 1955. Class formations. Duke Mathematical Journal 22: 165–178. Dieudonné, Jean. 1979. Reviews: Encyclopedic Dictionary of Mathematics. The American Mathematical Monthly 86(3): 232–233. Fujita, Hiroshi. (2004). Goals of mathematical education and methodology of applied mathematics. In Proceedings of the Ninth International Congress on Mathematical Education, eds. Hiroshi Fujita, Yoshihiko Hashimoto, Bernard R. Hodgson, Peng Yee Lee, Stephen Lerman, and Toshio Sawada, 19–36. Springer, Dordrecht. Iitaka, Shigeru. 2008. Yukiyosi Kawada. Retrieved January 30, 2020, from Fulvia Furinghetti and Livia Giacardi. 2008. The First Century of the International Commission on Mathematical Instruction (1908–2008) https://www.icmihistory.unito.it/portrait/kawada.php Mattedi Dias, André Luís. 2008. Instituto de Matemática e Física da Universidade da Bahia: atividades matemáticas (1960–1968). História, Ciências, Saúde – Manguinhos 15(4): 1049–1075. Nebres, Bienvenido F. 2008. Centers and peripheries in mathematics education. In The first century of the International Commission on Mathematical Instruction (1908–2008). Reflecting and shaping the world of mathematics education, eds. Marta Menghini, Fulvia Furinghetti, Livia Giacardi, and Ferdinando Arzarello, 149–163. Rome: Istituto della Enciclopedia Italiana. Peak, Philip. 1978. Review of EDM – Encyclopedic Dictionary of Mathematics (L), by Shōkichi Iyanaga and Yukiyosi Kawada, and Kenneth O. May. The Mathematics Teacher 71: 785–786.

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Suat Khoh, Lim-Teo. 2008. ICMI activities in East and Southeast Asia: Thirty years of academic discourse and deliberations. In The first century of the International Commission on Mathematical Instruction (1908–2008). Reflecting and shaping the world of mathematics education, eds. Marta Menghini, Fulvia Furinghetti, Livia Giacardi, and Ferdinando Arzarello, 247–252. Rome: Istituto della Enciclopedia Italiana.

Publications Related to Mathematics Education Iyanaga, Shōkichi and Yukiyosi Kawada. (eds.) 1980. Encyclopedic Dictionary of Mathematics. Translation from the 2nd Japanese edition reviewed by Kenneth O.  May (by Mathematical Society of Japan with the cooperation of the American Mathematical Society). Cambridge, MA and London: MIT Press. Two volumes. First edition: 1977. MIT Press. Kawada, Yukiyosi. 1975. Commission Internationale de l’Enseignement Mathématique. Information for the period 1975–1978. L’Enseignement Mathématique s. 2, 21: 331–335.

Photo Author: Jacobs, Konrad. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

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11.28 Felix Klein (Düsseldorf, 1849 – Göttingen, 1925): President 1908–1920 Gert Schubring

Biography Youth and University Studies Felix Christian Klein was born in Düsseldorf on 25 April 1849. Düsseldorf, seat of the provincial government of the Prussian Rhineland and a centre of industrialization in Prussia, was  – contrary to other parts of the Rhineland  – of dominantly Protestant religion. His family was Protestant too. Caspar Klein, Felix’s father, acted as secretary to the provincial governor. His mother, Sophie Elise Klein, descended from a draper’s family in Aachen. After elementary instruction at home by his mother, the 6-year-old boy entered a private school. He left it already two and a half years later and entered in 1857 the Gymnasium in Düsseldorf. Traditionally, one uses to ascribe to German Gymnasia a one-sided cultivation of classical education. This is not correct, however, for the Gymnasia in Prussia since their consistent reform in the 1810s and 1820s and in particular not for the Rhineland – at least until G. Schubring (*) University of Bielefeld, Bielefeld, Germany e-mail: [email protected]

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the 1850s. In fact, mathematics teaching was well represented at the Gymnasium in Düsseldorf and always taught by several qualified teachers. His solutions of the mathematics problems in the Abitur exam showed him as well instructed (Fischer 1985). And the Abitur-Zeugnis attributed to him equal qualities in the humanities and in mathematics and the sciences (Lorey 1926, p. 149). The original curricular conception of the Prussian educational reforms had been that of a complementary unity of three basic components of human knowledge: languages, history and geography, and mathematics and the sciences. Klein seems to have been profoundly moulded by this organic view; he always strove to overcome one-sided or dichotomic conceptions and to unravel essential correspondences and relations. His studies at the University of Bonn, which he began at the unusually early age of 16 and a half years, in 1865, document this desire of an all-embracing knowledge: he studied not only mathematics but also all the natural sciences. This was possible thanks to the extraordinarily well-organized studies at Bonn for the sciences: the Seminar für diegesamten Naturwissenschaften with its five sections for physics, chemistry, geology, botany and zoology offered a coherent and systematic programme of theoretical and practical studies (Schubring 1989a). Klein regularly participated for four terms in all the five sections; he was particularly praised by Julius Plücker (1801–1868) for his abilities in theoretical and experimental physics and served as an assistant in the physical Praktikum; but he won also distinctions in zoology and botany (Schubring 1989b, p. 208ff.). At the same time, he studied mathematics, mainly with Plücker, but also with his analytically minded rival Rudolf Lipschitz and participated actively in both sections of the Mathematisches Seminar, run by the two professors. From the winter term 1867–1968 on, Klein concentrated on mathematics and became mainly influenced by Plücker and his analytic-geometric approach. Besides Plücker’s lectures, Klein participated in Lipschitz’s lectures on analytic geometry, number theory, mechanics, statics, differential equations and variational calculus. As a result of Plücker’s premature death in May 1868, Klein, who had closely assisted him in elaborating his new work on Liniengeometrie, was charged by Alfred Clebsch to elaborate the missing parts of its second volume. This new cooperation with Clebsch helped him produce the conception for a doctoral thesis, on a general form of Linienkomplexe of the second degree. The Early Academic Career In December 1868, less than 20 years old, he obtained the doctoral degree, examined by Lipschitz – and without a previous exam for a Gymnasium teacher licence, as was still the custom. To perfect his mathematical qualifications, Klein moved to Göttingen and studied there from January to August 1869 with Clebsch who was about to establish a mathematical school rivalling that of Berlin. Klein was keen, however, to study in Berlin as well, and spent there the next months, until April 1870. He attended the

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highly specialized Mathematisches Seminar96 and a lecture by Kronecker, but did not want to attend a lecture by Weierstraß, not liking his dominant position. In Berlin, Klein got to know Sophus Lie and their cooperation became highly important for him. In April 1870, the two young men set up to Paris to study there new developments in French mathematics. It was the first time since the 1820s and since Dirichlet and Plücker that a German mathematician thought it worthwhile to study in France. The outbreak of the German-French war of 1870–1871 in July ended, however, this research stay. Klein, patriotically minded, volunteered for the army, but served in a medical corps. Infected by typhus, he had to leave the army and used the time of recovery to prepare his Habilitation. He obtained this degree in January 1871 and acted in the following time in Göttingen in the characteristically German position of a Privatdozent. He did not have to wait long in this intermediary position. Distinguished by already impressing geometric publications, he obtained the position of full professor in October 1872, when only 23  years old, at the University of Erlangen. Mathematics had become modernized in Bavaria only from the late 1840s on, and there were but a few students in mathematics by that time. Klein would, however, learn the differing status of mathematics in the various institutional situations which it experienced in the numerous German states. His first position is for always tied with one of his major achievements, the famous Erlanger Programm – innovating and generalizing the study of geometric transformations. It was unnoticed for a long time, however, that Klein set out at the same time a second Erlanger Programm: it was traditional practice not only to deliver a scientific paper upon assuming a chair as full professor but also to deliver an inaugural address, the Antrittsrede. In this speech, Klein demonstrated his profound interest in educational reform: he propagated a vision of uniting classical and modern (“Real-”) secondary schools, which were organized in a particularly dichotomous manner in Bavaria (Rowe 1985). Clearly, this vision corresponded to the original ideal of Prussian neo-humanist education. Moreover, Klein developed in Erlangen skilful managerial qualities which should become characteristic of his professional activities: due to the untimely death of his second supervisor, Clebsch, in November 1872, he not only entered the editorial team of Mathematische Annalen, the journal established by Clebsch rivalling the Journal für die reine und angewandte Mathematik, but had also to take over the leadership of the Clebsch school, so that several of the members rallied to stay with him in Erlangen (e.g. Aurel Voss, Ferdinand Lindemann, Axel Harnack). In 1875, Klein accepted the challenge to move to the Technical College of Munich, with a considerable number of students studying mathematics, albeit as a service subject. He devoted himself intensely to assure a sound formation in the fundamentals of mathematics for these students; in addition, he also looked after the few students specializing in mathematics, since there was the special situation in

 For the importance of the institutional form of the Seminar for the development of mathematical research in Prussia, see Schubring (2000b). 96

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Bavaria where future teachers for the technical and realist schools were trained at the technical college. And Klein enjoyed coming into contact with technical sciences and their practitioners, in particular Carl von Linde. Scientist and “Fachpolitiker” In 1880, Klein accepted still another appointment, as professor of geometry at Leipzig University, in Saxonia. Mathematics at Leipzig had always maintained an independent and serious position within German universities. It was in Leipzig that Klein devoted himself to even more systematic efforts for improving the quality of studies for all mathematics students and succeeded in establishing decisive innovations: an independent building for mathematics with lecture halls, rooms for Übungen, for the collection of models, and for the newly established Mathematisches Seminar and – most notably – for a specialized library (König 1981). It was in Leipzig, too, that Klein – hitherto known mainly for his works on geometric problems, in particular the icosahedron – worked intensely on analytic problems, prompted by elliptic functions involved in treating the icosahedron. Developing what he called “automorphe Funktionen”, he entered into a fierce competition with Poincaré who worked on them, too, calling them “fonctions fuchsiennes”. Although he was able to elaborate a key theorem for this theory, Klein suffered in 1882a nervous breakdown and needed so long to recover that Poincaré surpassed him, making it impossible to catch up with him again and so achieving definite leadership. Klein continued up to the 1890s in productive research on the theory of functions, but he engaged more and more in activities as “Fachpolitiker” (see Klein 1977). The platform for such activities was provided to him by his call in 1886 to Göttingen – hence, as a successor to Riemann and to his supervisor Clebsch – since Göttingen now belonged to Prussia, and Prussia as the dominant state within the confederation Deutsches Reich offered ample opportunities for policymaking, including those outside the realm of education. It were calls to new, research-oriented US universities and the alliance which he forged with Friedrich Althoff – the almost omnipotent Ministerialdirektor in the Prussian ministry – which resulted in his obtaining the dominant position in the German mathematical community. The foundation of the Deutsche Mathematiker-Vereinigung (DMV) in 1890 still met resistance from the opposing Berlin school, but in the long run the Berlin school was not able to resist integration into the dynamic of the DMV (Hashagen 2003). In 1896, Klein succeeded in having David Hilbert appointed to the other mathematical chair in Göttingen. Together, Hilbert and Klein achieved the transformation of their setting into the world centre of mathematics. Students and scientists from all the countries made the pilgrimage to this mathematical Mecca. Numerous new positions for mathematicians were established at this quasi department, and in 1904 the first chair for applied mathematics was created, with Carl Runge as its first professor. On 22 June 1925, Klein died in Göttingen.

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Contribution to Mathematics Education The Reform Agenda One key issue in Klein’s agenda was the strengthening of relations of mathematics to its applications in the sciences and in technology. A first issue in this was the promotion of applied mathematics within the mathematical community, but also institutionally. This implied a redefinition of the mathematics to be taught at the technical colleges, which in the meantime aspired to become equivalent in status to the universities – a problematic he knew well since his own teaching at the Munich Polytechnic. A next step taken for this issue was unprecedented: Klein forged an alliance with leading German industrialists to promote his programme. They established in 1898 the Göttinger Vereinigung für angewandte Physik, broadened in 1901 to und Mathematik. The association not only sponsored and co-financed research activities in Göttingen but also helped establish new buildings and institutes, most notably the new Physics Institute in 1904. Klein’s dream was to use the means of the association to erect an analogous building and institute for mathematics. Resistance from the Ministry of Finance prevented its realization at first, then followed by WWI and, eventually, inflation. It was only after Klein’s death and thanks to the Rockefeller Foundation that the first impressive building for a Mathematical Institute was inaugurated in 1929 in Göttingen (Schubring 2000, pp. 282ff.). The major issue was, however, to modernize mathematics teaching. Klein deplored what he used to call the “double gap”: the discontinuity between school mathematics and university mathematics and the double forgetting of the respective knowledge – first one had to forget school mathematics upon beginning one’s university studies and later as a teacher one had to forget university mathematics and return to school mathematics (see Klein 2016, p. 1). In the 1890s, Klein developed many initiatives to improve teacher education, such as establishing summer schools as in-service training and by cooperating with the teachers’ association, the Deutscher Verein zur Förderung des Mathematischen und Naturwissenschaftlichen Unterrichts. His lectures, Elementary Mathematics from an Advanced Viewpoint, published in three volumes, became a universally adopted textbook for methodological formation of mathematics teachers, still serving as a model today. Klein followed through his insight that a reform of university mathematics necessitated taking an even more extensive system into consideration: the school system as the basic foundation for higher education. He began, therefore, to interest himself in the improvement of teacher education. By so doing, he hoped to reverse the trend towards one-sidedly formal, abstract approaches to mathematics instruction by promoting practical instruction and the development of spatial intuition. An early breakthrough in this pursuit, although still realized by the traditional method of administrative decrees from “on high”, took place in 1898. In that year, new regulations for the examination of prospective teachers in Prussia were introduced. These created a second teaching licence for applied mathematics, thus complementing the

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traditional one, which was thereafter referred to as the examination in “pure” mathematics. The new regulations also enabled the Prussian technical colleges to introduce disciplinary studies in mathematics for the first time. Henceforth, future mathematics teachers were allowed to spend three terms studying at these colleges. At the same time, the enlarged university curriculum made it possible to establish positions for applied mathematics at some of these institutions, thereby breaking up the monopoly that pure mathematics once enjoyed there. The new university-like status attained by the technical colleges in 1899 as well as the imminent recognition of all three types of 9-year secondary schools as entitled to have their graduates admitted to institutions of higher education posed new structural problems, however, for mathematics within these institutions. Should the graduates of the modernist schools (Realgymnasien and Oberrealschulen) be directed only towards the Technische Hochschulen or to the universities as well? If the former course were to be adopted, this would imply acceptance of the Bavarian model with its bifurcated educational system and consequent split approach to mathematics instruction (see Schubring 1989a). Instead of assimilating and integrating the technical schools (Realschulen) into the Gymnasien and the technical colleges into the universities, as he had proposed earlier, Klein now favoured the full and independent development of the Realschulen (which had undergone a remarkable process of transformation throughout the nineteenth century since their beginnings as vocational schools of low status) and the Technische Hochschulen (whose transformation had been almost as profound). Another consequence was that the mathematical curricula in all the various secondary and tertiary institutions would have to be redefined in order to make the free transition from each school type to both higher education types. Thus, by 1900, Klein’s views had evolved to the point that he now recognized that an effective and stable reform of the subsystem of higher education required changes that went beyond this institutional plane. A new approach to teacher education was needed; but beyond this the link to the underlying school system had to be reformed, in particular, by reforming the mathematical curricula in the secondary schools (Schubring 1989a). Klein proposed a radical reform. Since the mathematical courses at the technical colleges consisted of a basic preparatory “general” part and an advanced or higher part, he recommended that the basic studies be transferred to the preparatory schools, i.e. the secondary schools, and that only the advanced studies should remain as part of the college curriculum. However, they would be reformed and taught not as independent branches of knowledge, but rather “in permanent touch with the requirements of the students of mechanical and construction engineering […]. Mathematics at the colleges can thus become once again what it ought to be: a specific power pervading the whole” (Schubring 1989a, p.  219). Transferring the courses to secondary school would help create a firm mathematical basis for the pursuit of college studies, due, in particular, to the “benevolent coercion of school”, as opposed to a university environment where academic freedom reigned. But which subjects were to be considered as part of the basic core of this new secondary mathematical curriculum? According to Klein, this core would include analytic

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geometry, but primarily differential and integral calculus! In private letters to close friends and co-workers, Klein in fact always stressed that the key to his reform plans was to solve the question of mathematical preparation for the technical colleges (Schubring 1989a, p. 187). It is evident that the most complicated part of his proposal concerned the changes that would be required in school mathematics, particularly since the secondary schools had dropped analytic geometry and calculus from their curriculum more than a half-century earlier. The Ministry refused to decree the desired curricular changes from above. Instead, they advised Klein to organize the introduction of the curricular changes “from below” by enlisting the support of appropriately trained teachers who could act as agents for the implementation of the reforms in selected schools (Schubring 1989a, p. 187). From 1902 on, Klein began an intense and thorough study of the state of mathematics education in order to discover the pivotal issues that might activate mathematics teachers. Having familiarized himself with some of the main problems facing mathematics teaching, in the schools, he proceeded to coin the key phrase that would hereinafter serve as the slogan for his reform programme, but which also helped to convey the impression this programme was motivated exclusively by a desire to improve education within the schools. This was the famous notion functional reasoning, or – to say it more concretely – the idea that the function concept should pervade all parts of the mathematics curriculum. Walther Lietzmann (1880–1959), Klein’s principal assistant in organizing the reform movement, later noted that the success of the reform movement depended on finding a fundamental idea that would serve as a rallying point and which at the same time would automatically carry the calculus into the Gymnasium curriculum. That pivotal rallying point was the concept of function, which according to Klein’s programme would already be introduced in the lower grades (Lietzmann 1930, p. 255). This slogan of functional thinking in hand, Klein began gathering support for this reform movement from below. He succeeded in triggering an enormously broad and dynamic reform movement among mathematics teachers of all school types and German states (Schubring 1989a, pp. 188ff.). Its most visible and seminal document is known as the “Meraner Reform” or the Meran syllabus, named after the syllabus proposed by a committee for all the three school types, presented at a meeting of the DMV in the then Austrian town Meran in 1905. The Work for IMUK Since the foundation of L’Enseignement Mathématique, Klein was a member of its Comité de Patronage, a kind of Editorial Board which existed until 1914. When in 1908 the Fourth International Congress of Mathematics decided to establish, upon the proposal of David Eugene Smith, the Internationale Mathematische Unterrichtskommission (IMUK), Klein who had proved as an energetic and dynamic reformer of mathematics instruction was the obvious person to be elected as president. Klein expanded his policy agenda, now, to the international level and realized an enormous network of national commissions on all the continents cooperating

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first on reports on the state of mathematics instruction, but then on thematic issues as well. In fact, Klein succeeded in linking the compilatory task of the IMUK with a reformist agenda, and the organization began to function as an agent for curricular change. At the Fifth International Congress of Mathematicians in Cambridge, 1912, it was decided that because the statistical work of IMUK was still in progress, the Commission – originally restricted to a mandate of 4 years – should be prolonged until 1916. Klein invested restlessly an enormous activity to meet this deadline. Being forced to stay 1911–1912, for 1 year, in a sanatorium, due to serious health problems, in Hahnenklee in the Harz mountains, he managed to have meetings with his German assistants and with foreign colleagues. During the stay in Hahnenklee, Max Liebermann executed his famous painting of Klein, now located in the Göttingen Mathematical Institute. The culmination of IMUK work was reached at the 1914 IMUK session in Paris where the two topics presented corresponded to the two main issues in Klein’s reform agenda: the mathematical training of the engineers and “The Evaluation of the Introduction of Differential and Integral Calculus into the Secondary Schools”. This was the topic that Klein prepared more carefully than any other. He not only helped design the international questionnaire that dealt with this matter, but he also chose its coordinator and reporter: Emanuel Beke, a Hungarian scholar. The Paris meeting took place shortly before the outbreak of WWI, which should change everything. Neither the projected Congress of 1916 nor the last international thematic study on the training of mathematics teachers was realized. In October 1914, Klein signed, together with 92 other German scholars and intellectuals, the disastrous “Aufruf an die Kulturwelt” which denied any war crimes committed by the German army in Belgium, claimed a German cultural mission in Europe and backed German militarism. As was revealed only very much later, none of the signatories except the initiators had seen the text – they had just been asked telegraphically whether they would agree to sign (Von Ungern-Sternberg 1996). For the foreigners, the text was a horrible document of German chauvinism. Klein was excluded as member of the Paris Academy, and the Western allies inflicted a boycott on German scientists. The work of IMUK came to a standstill, and after the end of WWI, Klein had to suffer the dissolution of this first IMUK in 1920 (Schubring 2008). The national subcommissions continued to work, however (Karp 2019). Klein invested an enormous energy, from 1908 on, as president also of the German subcommission, to establish the structure of the reports and to forge a group of collaborators to elaborate in close cooperation the intended reports. This was a complicated task since the reports had to take account of the highly differentiated German educational systems. Due to this complexity, only a part of the reports was ready for the 1912 Congress in Cambridge. Particularly in the period of prolongation of IMUK after 1912, the various innovative thematic reports were elaborated, in particular on psychology of mathematics education, on history in mathematics teaching and on the history of higher education in mathematics. After the outbreak of WWI, Klein strove to speed up the finishing of the missing texts. Eventually, in 1916, all the reports were ready: 5 volumes, with 38 contributions (Schubring 2019).

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Fig. 11.2  The Felix Klein Medal for the Felix Klein Award. Source: https://www.mathunion. org/icmi/awards/icmi-emma-castelnuovofelix-klein-and-hans-freudenthal-medals

Klein retired in 1913, after his long illness, but not only did he remained active in the IMUK work; he even engaged in new fields: having earlier on promoted research on history of mathematics and related publications, he gave now lectures on the history of mathematics in the nineteenth century. Later, these lectures became published: the seminal Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert (1926/1927). Together with Hilbert, he had worked on the mathematical aspects of the theory of special and general relativity. In 2002, the ICMI Executive Committee created two awards in mathematics education research, one of them being the Felix Klein Award for lifetime achievement in mathematics education research. The award is conferred every 2  years (see Fig. 11.2).

Sources Essential Primary Sources Klein, Felix. 1921–1923. Gesammelte mathematische Abhandlungen. Hrsg. von Robert Fricke et al. Berlin: Springer, Vol. 1, Liniengeometrie. Grundlegung der Geometrie zum Erlanger Programm. Hrsg. von R. Frickeu. A. Ostrowski. 1921. Vol. 2, Anschauliche Geometrie. Substitutionsgruppen u. Gleichungstheorie zur mathematischen Physik. Hrsg. von R. Fricke u. H. Vermeil. 1922. Vol. 3, Elliptische Funktionen, insbes. Modulfunktionen. Hyperelliptische und Abelsche Funktionen. Riemannsche Funktionentheorie und automorphe Funktionen. Anh. Hrsg. von R. Fricke, H. Vermeil u. E. Bessel-Hagen. 1923. Klein, Felix. 1974. Das Erlanger Programm: Vergleichende Betrachtungen über neuere geometrische Forschungen. Leipzig: Akad. Verlagsges. [French and Italian transl.]

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Klein, Felix. 1987. Funktionentheorie in geometrischer Behandlungsweise. Vorlesung, gehalten in Leipzig 1880/81; mit 2 Original-­Arbeiten von F. Klein aus dem Jahre 1882. Hrsg., bearb. u. komm. von F. König. Leipzig: Teubner. Klein, Felix. 1884 and 1993. Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade [Reprogr. Nachdr. der Ausg. Leipzig: Teubner, 1884]. Hrsg. mit einer Einf. und mit Kommentarenvon Peter Slodowy. Basel: Birkhäuser, 1993. [Engl. transl.] Klein, Felix. 1991. Einleitung in die analytische Mechanik: Vorlesung, gehalten in Goettingen 1886/87. Hrsg.und bearb. von E.  Dietzel u. M.  Geisler. Mit einem Beitrag von F.  König. Stuttgart: Teubner. Klein, Felix. 1986. Riemannsche Flächen. Vorlesungen, gehalten in Göttingen 1891/92. Hrsg. Und kommentiert von G. Eisenreich u. W. Purkert. Leipzig: Teubner Springer [in Komm.]. Klein, Felix. 1892, 1893. Einleitung in die höhere Geometrie. Vorlesung, [Bearb.] Friedrich Schilling, Göttingen. Faksimile-Druck: Vol. 1, Wintersem. 1892/93. 1893. Vol. 2, Sommersem. 1893. Klein, Felix. 1893. Vorlesungen über die hypergeometrische Funktion: [gehalten an d. Univ. Göttingen im Wintersem. 1893/94]. Ausgearbeitet von Ernst Ritter. Hrsg. u. mit Anm. vers. von Otto Haupt]. –Reprint [d. Ausg. Berlin 1933]. Berlin: Springer. Klein, Felix. 1966. Vorlesungen über die Theorie der elliptischen Modulfunktionen. [Bearb.] Robert Fricke (New York: Johnson Teubner). Repr. of Leipzig 1890–1892: Vol. 1, Grundlegung der Theorie. 1966. Vol. 2, Fortbildung und Anwendung der Theorie. 1966. Klein, Felix. 1894. Lectures on mathematics: delivered from Aug. 28 to Sept. 9, 1893, before members of the congress of mathematics held in connection with the world’s fair in Chicago at Northwestern Univ., Evanston, Ill. Reported by Alexander Ziwet. New York: Macmillan. Klein, Felix. 1894. Über lineare Differentialgleichungen der zweiten Ordnung. Vorlesung, gehalten im Sommersem. 1894. Ausgearb. von F. Ritter. Göttingen. Klein, Felix. 1895. Über Arithmetisierung der Mathematik. Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Geschäftliche Mitteilungen, Heft 2. Klein, Felix. 1965. Vorlesungen über die Theorie der automorphen Funktionen. Bearb. Robert Fricke. New York: Johnson Teubner. Repr. of Leipzig 1897–1912. Vol. 1, Die gruppentheoretischen Grundlagen. 1965. Vol. 2, Die funktionentheoretischen Ausführungen und Anwendungen. 1965. Klein, Felix and Sommerfeld Arnold. 1910, 1921, 1923. Über die Theorie des Kreisels. Hefte 1–4. Leipzig: Teubner [Engl.transl.]: Vol. 1, Die kinematischen u. kinetischen Grundlagen der Theorie. 3. Aufl. 1923. Vol. 2, Durchführung der Theorie im Falle des schweren symmetrischen Kreisels. 2. Aufl. 1921. Vol. 3, Die störenden Einflüsse. Astronomische und geophysikalische Anwendungen. 2. Aufl. 1923. Vol. 4, Die technischen Anwendungen der Kreiseltheorie. 1910. Klein, Felix. 1968. Vorlesungen über nicht-euklidische Geometrie. Für den Druck neu bearb. von W. Rosemann. Reprint Berlin 1928. Berlin: Springer. Klein, Felix. 1926, 1927. Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Berlin: Springer: Vol. 1 (1926) [English translation: 1979]. Vol. 2 Die Grundbegriffe der Invariantentheorie und ihr Eindringen in die mathematische Physik. 1927.

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Klein, Felix. 1977. Handschriftlicher Nachlaß. Hrsg. Konrad Jacobs. Erlangen: Mathematisches Institut der Friedrich Alexander Universität. Materialien für eine wissenschaftliche Biographie von Gauss. Gesammelt von F. Klein et al. 5 Vol. Berlin: Weidmann, 1911–1917. Der Briefwechsel David Hilbert  – Felix Klein: (1886–1918). Mit Anm. hrsg. von Günther Frei (Göttingen: Vandenhoeck & Ruprecht, 1985). Korrespondenz Felix Klein – Adolph Mayer: Auswahl aus den Jahren 1871–1907. Herausgegeben, eingeleitet und kommentiert von Renate Tobies et al. Leipzig: Teubner, 1990.

Essential Secondary Sources Fischer, Gerd. 1985. Abitur 1865: Reifeprüungsarbeit in Mathematik von Felix Klein. Der mathematische und naturwissenschaftliche Unterricht 38: 459–465. Hashagen, Ulf. 2003. Walther von Dyck: (1856–1934); Mathematik, Technik und Wissenschaftsorganisation an der TH München. Stuttgart: Steiner. Hensel, Susann. 1989. Die Auseinandersetzungen um die mathematische Ausbildung der Ingenieure an den Technischen Hochschulen Deutschlands Ende des 19. Jahrhunderts“. In Susann Hensel; Karl-Norbert Ihmig; Michael Otte (Hrsg.), Mathematik und Technik im 19. Jahrhundert in Deutschland: soziale Auseinandersetzung und philosophische Problematik. Vandenhoeck & Ruprecht: Göttingen 1989, 1–111. Karp, Alexander (ed.). 2019. National Subcommissions of ICMI and their Role in the Reform of Mathematics Education. Cham: Springer. König, Fritz. 1981. Die Gründung des “Mathematischen Seminars” der Universität Leipzit. In 100 Jahre Mathematisches Seminar der Karl-Marx-­Universität Leipzig, eds. Herbert Beckert and Horst Schumann, 41–71. Berlin: Berlin VEB Deutscher Verlag der Wissenschaften. Manegold, Karl-Heinz. 1968. Felix Klein als Wissenschaftsorganisator. Technikgeschichte 35,177–204. Manegold, Karl-Heinz. 1970. Universität, Technische Hochschule und Industrie: ein Beitragzur Emanzipation der Technik im 19. Jahrhundert unter besonderer Berüksichtigung der Bestrebungen Felix Kleins. Berlin: Duncker & Humblot. Mehrtens, Herbert. 1990. Moderne – Sprache – Mathematik: eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme. Frankfurt am Main: Suhrkamp. Reid, Constance. 1978. The road not taken. Mathematical Intelligencer 1(1): 21–23. Rowe, David E. 1985. Felix Klein’s “Erlanger Antrittsrede”, a transcription with English translation and commentary. Historia Mathematica 12: 123–141. Rowe, David E. 1985. Essay Review: Felix Klein (Renate Tobies with Fritz König; Karl-Heinz Manegold; Lewis Pyenson). Historia Mathematica 12: 278–291. Rowe, David E. 1986. Jewish mathematics’ at Göttingen in the era of Felix Klein. Isis 77: 422–449. Rowe, David E. 1989. The early geometrical works of Sophus Lie and Felix Klein. In The history of modern mathematics, Vol. I, eds. D.  E. Rowe and J.  McCleary, 209–273. Boston, MA: Academic Press. Rowe, David E. 1989. Klein, Hilbert, and the Göttingen mathematical tradition. Osiris s. 2, 5: 186–213. Rowe, David E. 1992. Felix Klein, David Hilbert, and the Göttingen mathematical tradition. PhD thesis. New York, N.Y.: City University of New York. Rowe, David E. 1994. The philosophical views of Klein and Hilbert. In The intersection of history and mathematics, eds. C, Sasaki, M, Sugiura, J. W. Dauben, 187–202. Basel: Birkhäuser. Schubring, Gert. 1989a. Pure and Applied Mathematics in Divergent Institutional Settings in Germany: the Role and Impact of Felix Klein. In The History of Modern Mathematics. Vol. II: Institutions and Applications, eds. D.  E. Rowe and J.  McCleary, 171–220. Boston: Academic Press.

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Schubring, Gert. 1989b. The Rise and Decline of the Bonn Natural Sciences Seminar (Conflicts between Teacher Education and Disciplinary Differentiation), Science in Germany. The Intersection of Institutional and Intellectual Issues. ed. K.M. Olesko, Osiris (Second series) 5: 56–93. Schubring, Gert. 1999. O primeiro movimento internacional de reforma curricular em matemática e o papel da Alemanha: um estudo de caso na transmissão de conceitos. Zetetiké 7(11): 29–50. Schubring, Gert. 2000a. Felix Kleins Gutachten zur Schulkonferenz 1900: Initiativen für den Systemzusammenhang von Schule und Hochschule, von Curriculum und Studium. Felix Klein und die Berliner Schulkonferenz des Jahres 1900, Der Mathematikunterricht 46: 3, 62–76. Schubring, Gert. 2000b. Kabinett  – Seminar  – Institut: Raum und Rahmen des forschenden Lernens. Berichte zur Wissenschaftsgeschichte 23: 269–285. Schubring, Gert. 2007. Der Aufbruch zum ‘funktionalen Denken’: Geschichte des Mathematikunterrichts im Kaiserreich. N.T. M. 15: 1–17. Schubring, Gert. 2008. The Origins and the Early History of ICMI“, International Journal for the History of Mathematics Education, 2008, 3: 2, 3–33. Schubring, Gert. 2019. “The German IMUK subcommission”. In: Alexander Karp (ed.). National Subcommissions of ICMI and their Role in the Reform of Mathematics Education (Cham: Springer, 2019), 65–91. Shields, Allen. 1998. Klein and Bieberbach: mathematics, race, and biology. The Mathematical Intelligencer 10 (3): 7–11. Tobies, Renate. 1981. Felix Klein. Leipzig: Teubner. Tobies, Renate. 1992. Felix Klein in Erlangen und München: Ein Beitrag zur Biographie. In Amphora: Festschrift for Hans Wussing on the occasion of his 65th birthday, eds. Sergey Sergei Demidov et al., 751–772. Basel- Boston- Berlin: Birkhäuser. Tobies, Renate. 2019. Felix Klein  – Visionen für Mathematik, Anwendungen und Unterricht. Berlin: Springer-Spektrum. Von Ungern-Sternberg, Jürgen, and Wolfgang Von Ungern-Sternberg. 1996. Der Aufruf “Andie Kulturwelt!”: das Manifest der 93 und die Anfänge der Kriegspropaganda im Ersten Weltkrieg; mit einer Dokumentation. Stuttgart: Steiner.

Obituaries Courant, Richard. 1925. Felix Klein. Jahresberichte der Deutschen Mathematiker-Vereinigung 34: 197–213. Courant, Richard. 1925. Felix Klein. Die Naturwissenschaften 134: 765–772. Lorey, Wilhelm. 1926. Felix Klein. Leopoldina, Berichte der Kaiserlichen Deutschen Akademie der Naturforscher in Halle 1: 136–151. Weber, M. 1925. Felix Klein. Zeitschrift des Vereins Deutscher Ingenieure 69: 1118–1119.

Publications Related to Mathematics Education Klein, Felix. 1895. Vorträge über ausgewählte Fragen der Elementargeometrie. Ausgearb. von F. Taegert. Eine Festschrift zu der Pfingsten 1895 in Göttingen stattfindenden 3. Versammlung d. Vereins zur Förderung des math. u. naturwiss. Unterrichts.Leipzig: Teubner. Klein, Felix. 1899. Über Aufgabe und Methode des mathematischen Unterrichts an den Universitäten. Jahresbericht der Deutschen Mathematiker Vereinigung 7: 126–138. Klein, Felix. 1900. Über angewandte Mathematik und Physik in ihrer Bedeutung für den Unterricht an den höheren Schulen: nebst Erläuterung der bezülichen Göttinger Universitätseinrichtungen. Vorträge, gehalten in Göttingen, Ostern 1900, bei Gelegenheit des Feriencurses für Oberlehrer der Mathematikund Physik; mit einem Wiederabdruck verschiedener einschlägiger Aufsätze von F. Klein. Leipzig: Teubner.

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Klein, Felix. 1902. Der Unterricht in der Mathematik. In Die Reform des Höheren Schulwesens, ed. Wilhelm. Lexis, 254–264. Halle: Buchhandlung des Waisenhauses. Klein, Felix. 1904. Neue Beiträge zur Frage des mathematischen und physikalischen Unterrichts an den höheren Schulen: Vorträge gehalten bei Gelegenheit d. Ferienkurses für Oberlehrer d. Mathematik und Physik, Göttingen, Ostern 1904. Gesammelt und hrsg. von F. Klein. Leipzig: Teubner. Klein, Felix. 1925 1928, 1933, 1968. Elementarmathematik vom höheren Standpunkte aus (Leipzig: Teubner/reprints: Springer, Berlin) [English and Spanish transl.] Vol. 1, Arithmetik; Algebra; Analysis. Reprint 4. Aufl. 1933. 1968. Vol. 2, Geometrie. Reprint 3. Aufl. 1925, 1968. Vol. 3, Präzisions- und Approximationsmathematik. Reprint 3. Aufl. 1928, 1968. Klein, Felix. 1907. Vorträge über den mathematischen Unterricht an den höheren Schulen. [Bearb.] Rudolf Schimmack. Leipzig: Teubner: Vol. 1, Von der Organisation des mathematischen Unterrichts. 1907. Klein, Felix. 1907. Universität und Schule: Vorträge auf der Versammlung Deutscher Philologen und Schulmänner am 25. September 1907 zu Basel. Leipzig: Teubner. Klein, Felix. 1911. Aktuelle Probleme der Lehrerbildung: Vortrag auf der Versammlung d. Vereins zur Förderung d. mathemat. u. naturwiss. Unterrichts am 6. Juni 1911 zu Münster; Mit verschiedenen Anlagen. Leipzig: Teubner.

Photo Courtesy of the University of Turin.

510

Ðuro Kurepa

11.29 Ðuro Kurepa (Majske Poljane, 1907 – Belgrade, 1993): Vice-President 1952–1962 Michela Malpangotto, Milosav M. Marjanović and Stevo Todorčević

Biography Ðuro Kurepa was born on 16 August 1907 at Majske Poljane (a town in today’s Croatia), being the 14th child in his father’s family. His parents were Rade and Andelija Kurepa. He studied pure mathematics and physics at the University of Zagreb, where he graduated in 1931. M. Malpangotto (*) CNRS, Centre Jean Pépin, Ecole Normale Supérieure-Ulm, Paris, France e-mail: [email protected] M. M. Marjanović (*) Serbian Academy of Sciences and Arts, Beograd, Serbia e-mail: [email protected] S. Todorčević (*) CNRS, Paris, France University of Toronto, Toronto, ON, Canada e-mail: [email protected]; [email protected]

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In 1932, Kurepa entered Collège de France, where his guide and maître was the outstanding French mathematician Maurice Fréchet. From the beginning of his stay in Paris, Kurepa focused his attention on the famous Suslin’s problem, which asks for a particular characterization of the real line. He introduced some special classes of partially ordered sets and used them to formulate several statements equivalent to Suslin’s hypothesis. In 1935, at the Sorbonne, he defended his doctoral thesis, Ensembles ordonnés et ramifiés, before a committee consisting of Paul Montel as president, Fréchet as supervisor and Arnaud Denjoy as its third member. Suslin’s problem, which suggests the equality of cellularity and cardinality of minimal everywhere dense sets in order continua, was a long-term preoccupation of Kurepa and led to some of his best-known contributions to the theory of partially ordered sets. For example, the standard notions such as Kurepa’s trees and Kurepa’s branching hypothesis honour his name and are widely used today. Kurepa was the first to express the idea that Suslin’s hypothesis could be a new postulate of axiomatic set theory. This conjecture was later confirmed by Jech, Solovay and Tennenbaum (see Jech 1967 and Solovay and Tennenbaum 1971) shortly after the invention of the method of forcing by Paul Cohen in the early 1960s. During the same early period of his research career, Kurepa generalized Fréchet’s distance function, opening thereby an approach to the theory of uniform spaces different from that of André Weil (see Colmez 1947). The book Theory of Sets written in Serbo-Croatian by Kurepa and published in Zagreb in 1951 illustrates well the scientific interests that stem from the early stages of his career. During the latter parts of his scientific life, Kurepa turned his attention towards other areas of mathematics, most notably algebra and number theory. In 1971, Kurepa made a particularly significant contribution in number theory with the introduction of the left factorial function. In the discrete case, it is defined by !n = 0! + 1 ! + 2! + ⋯ + (n − 1)!. He conjectured that the greatest common divisor of !n and n! is 2, for each n > 1, and this conjecture became quite famous. This hypothesis still seems to remain open in spite of attempts by Barsky and Benzaghou (2004; 2011). Kurepa was a professor at the University of Zagreb from 1937 to 1965 and at the University of Belgrade from 1965 until his retirement in 1977. He frequently visited high-ranking scientific institutions in France. He spent half a year as a member of the Institute for Advanced Study in Princeton. He also visited and delivered lectures at many other institutions in the world, including the University of Warsaw, Harvard University, University of Chicago and Universities of California at Berkeley and at Los Angeles. Kurepa played a prominent role in the mathematical community of the former Yugoslavia. He founded the Society of Mathematicians and Physicists of Croatia and was its first president. From 1954 to 1960, he held the position of president of the Union of Yugoslav Societies of Mathematicians, Physicists and Astronomers, and he was also the president of the Balkan Mathematical Society. From 1970 to 1980, he was director of Mathematical Institute of Serbian Academy of Sciences. Kurepa was a full member of Serbian Academy of Sciences and Arts, as well as the member of all national academies of the other former Yugoslav republics.

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Being a Serb born in Croatia, Kurepa lived through all the tragic South Slav’s conflicts, always believing in reconciliation as the only way to the future. He died on 2 November 1993 in Belgrade.

Contribution to Mathematics Education In 1952, as the delegate of Yugoslavia, Kurepa attended the First General Assembly of the International Mathematical Union (IMU) in Rome, when ICMI was reconstituted. In this occasion, the new ICMI Executive Committee was formed: Albert Châtelet (France), president; Kurepa (Yugoslavia), vice-president; Ralph L. Jeffery (Canada), vice-president; and Heinrich Behnke (Germany), secretary (including Henri Fehr as a lifelong honorary president). Afterwards, he was representative of Yugoslavia (1972. ICMI Bulletin 1: 5–7). As evidenced by the minutes of the meetings of the following years, Kurepa actively participated in the activities of the commission. At the first meeting of ICMI Commission (Geneva, October 1952), as a plan of work scheduled for the coming International Congress of Mathematicians (Amsterdam 1954), two main themes were fixed, one of which, “The Role of Mathematics and Mathematician at the Present Time”, was entirely assigned to Kurepa for elaboration. He conducted an international poll on this theme and elaboration of it was assessed as very satisfactory at the Amsterdam Congress (see Kurepa’s reports at the II Congress of Mathematicians and Physicists of Yugoslavia, Zagreb, 1954, as well as Kurepa 1955a). At the International Congress of Mathematicians in Edinburgh (1958), Kurepa was invited to deliver a half-hour address in Section VIII (History and Education) (see Kurepa 1960a). On the suggestion of and in cooperation with ICMI, an International Symposium on the Coordination of Instruction in Mathematics and Physics was held in Belgrade, 19–24 September 1960. Following the Symposium, a book was published by the Union of Societies of Mathematicians and Physicists of Yugoslavia, Belgrade, containing all contributed papers (of Gustave Choquet, Richard Courant, Otto Frostman, Kurepa, Paul Libois, Giovanni Sansone, Marshall Stone and others). The proceedings were also reprinted in Serbo-Croat as a special issue of the journal Nastava Matematike i Fizike (Teaching Mathematics and Physics. 1961. 10(1–4)). On that occasion, Kurepa delivered a welcome speech and relating mathematics and physics where he said (Kurepa 1962a, p. 6), “… every phenomenon we encounter contains intrinsically something of Mathematics and something of Physics and is a vivid specimen of ties Mathematics  - Physics”. His contribution “Some Principles on Coordination in the Instruction of Mathematics-­Physics” is split into 16 paragraphs, each having a title indicating main idea. Let us cite some of those titles that are also suggestive of the depth of his considerations: Physics – Mathematics; Particular – General; Autonomy and Interdependence; Functional and Relational Standpoint as Link Between Mathematics and Other Fields; Infinitesimal, Vectorial and Statistical Methods; Hierarchy of Methods, Nature as a Whole and Mathematics as Its

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Language; The Fundamental Tetrad: Action  – Perception  – Intelligence  – Imagination (Kurepa 1962b). Kurepa was a man of wide-ranging mathematical knowledge with an evident aptitude for generalizations. He played a prominent part in the propagation of the new math. He considered the concept of set to be basic at all levels of teaching and learning mathematics, but that the examples of sets should vary from sensory level (collections of isolated, concrete objects) going up to more complex abstractions (see Kurepa 1955b). From 1953 until 1971, Kurepa was the author or co-author of a number of textbooks for secondary schools and for university (altogether 147 books with 53 first editions). Among these books, we will mention his Higher Algebra (Zagreb, 1965; Belgrade, 1971) in two volumes (1519 pages). Let us also single out a number of lines characteristic for his views on education (Kurepa 1959a): Teaching is a process in which the social environment, the teacher and the taught influence each other. (p. 203) Education at all levels must be intimately linked to the creative efforts of the Society and in particular to new points of view and to the progress of science. (p. 204) Teaching must be current … In particular, it is better … to bring out new points of view and applications than to lose oneself pedantically in the logical finesse of long and complex demonstrations. (p. 205) We are facing a radical reform of mathematics education. More specifically, it is the notions of set, transformation and structure that must play an active role in teaching.97

Kurepa also emphasized the importance of using various methods in teaching and of involving all of the senses in the teaching process: an active interpenetration of methods, domains, topics, is of great importance and usefulness… In the teaching process, the hands are to be active (writing, showing), the tongue, ears, brain, i.e. all organs are more or less in active interdependence and co-operation. Let us remember that for a long time, even in instruction of geometry, and still more in arithmetic, the factors action and perception were either eliminated or at least neglected. (Kurepa 1960a, p. 570)

Kurepa is held in high regard by those who were his students. His lectures were very interesting, full of historical references and anecdotes and when proving theorems he touched only main details. When finishing a proof, he used to ask the audience “have you seen why the theorem holds true?” and the audience would answer in one voice “yes, we have” (and only pedants thought it was proof by acclamation). At the  The original text is: “L’enseignement est un processus dans lequel le milieu social, l’enseignant et l’enseigné s’influencent mutuellement. (p.  203) A tous ses degrés l’enseignement doit être intimement lié aux efforts créateurs de la Société et en particulier aux points de vue nouveaux et aux progrès des sciences. (p.  204) L’enseignement doit être actuel […]. En particulier, il vaut mieux […] faire ressortir les points de vue nouveaux et les applications que se perdre avec pédantisme dans les finesses logiques de démonstrations longues et complexes; (p.  205) Nous nous trouvons devant une réforme radicale de l’enseignement des mathématiques. Plus précisément ce sont les notions d’ensemble, de transformation et de structure qui doivent jouer un rôle actif dans l’enseignement. (p. 212)” 97

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end, let us say that the influence of Kurepa on the development of mathematics and mathematical education was significant and great in all parts of the former Yugoslavia.

Sources Ivić, Aleksandar, Zlatko Mamuzić, Žarko Mijajlocić, and Stevo Todorčević (eds.). 1996. Selected papers of Duro Kurepa. Matematicki Institut. Belgrade: SANU. Barsky, Daniel and Bénali Benzaghou. 2004. Nombres de Bell et somme de factorielles. Journal de Théorie des Nombres de Bordeaux 16 (1): 1–17. Barsky, Daniel and Bénali Benzaghou. 2011. Erratum à l’article Nombres de Bell et somme de factorielles. Journal de Théorie des Nombres de Bordeaux 23 (2): 527. Behnke, Heinrich. 1951–1954. Commission internationale de l’enseignement mathématique. Sa participation au Congrès d’Amsterdam 1954, Travaux préparatoires. L’Enseignement mathématique 40: 72–74. Cohen, Paul J. 1963. The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences of the United States of America 50: 1143–1148. Cohen, Paul J. 1964. The independence of the continuum hypothesis, II. Proceedings of the National Academy of Sciences of the United States of America 51: 105–110. Colmez, Jean. 1947. Sur divers problèmes concernant les espaces topologiques. Portugaliae Mathematica 6: 119–244. Grulović, Milan Z. 1995. The work of Professor Djuro Kurepa in the set theory and the number theory. Zbornik Radova Prirodno-Matematichkog Fakulteta. Serija za Matematiku [Review of Research, Faculty of Science, Mathematics Series] 25(1): 211–223. Jech, Tomáš. 1967. Non-provability of Souslin’s hypothesis. Commentationes Mathematicae Universitatis Carolinae 8: 291–305. Kočinac, Ljubiša D. 1994. Djuro R. Kurepa (1907–1993). Filomat 8: 115–127. Kovačević-Vujčić, Vera and Žarko Mijajlocić. 1996. Symposium Dedicated to the Memory of Ðuro Kurepa: Papers from the International Mathematical Symposium held at the University of Belgrade, Belgrade, May 27–28, 1996. Science Review Series, Science & Engineering 19–20. Belgrade: Serbian Scientific Society. Solovay, Robert M. and Stanley Tennenbaum. 1971. Iterated Cohen extensions and Souslin’s problem. Annals of Mathematics s. 2, 94: 201–245. Tennenbaum, Stanley. 1968. Souslin’s problem. Proceedings of the National Academy of Sciences of the United States of America 59: 60–63. Mijajlović, Žarko. 1995. Đuro Kurepa. Publications de l’Institut Mathématique (Đuro Kurepa memorial volume) 57: 13–18. Dimitrić, Radoslav. 1994. Academician, Professor Ðjuro Kurepa (1907–1993). The Review of Modern Logic 4: 401–403. Mićić, Vladimir, Zoran Kadelburg, and Branislav Popović. 2008. The mathematical society of Serbia – 60 years. The Teaching of Mathematics 11(1): 1–19.

Publications Related to Mathematics Education Kurepa, Ðuro. 1950. On the character of mathematics. In Proceeedings of the First Congress of Mathematicians and Physicists Yugoslavia, Bled 1949, eds. Yugoslav Union of Societies of Mathematicians and Physicists, 182–183. Belgrade: Yugoslav Union of Societies of Mathematicians and Physicists.

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Kurepa, Ðuro. 1955a. Le rôle des mathématiques et du mathématicien à l’époque contemporaine. L’Enseignement Mathématique s. 2, 1: 93–111. The English version was published in 1956 in Proceedings of the International Congress of Mathematicians, eds. J. C. H. Gerretsen, and J. de Groot, Vol. 3, 305–317. Groningen: E. P. Noordhoff N. V.; Amsterdam: North-Holland, with the title “International inquiry of the International Mathematical Instruction Commission (IMIC) on the role of mathematics and mathematician at present time”. Kurepa, Ðuro. 1955b. Quelques aspects de l’importance de la théorie des ensembles. Glasnik Matematicko-Fizicki i Astronomski s. 2, 10: 255–257. Kurepa, Ðuro. 1959a. Des principes de l’enseignement mathématique. L’Enseignement Mathématique s. 2, 5: 203–212. Kurepa, Ðuro. 1959b. Scientific foundation of school mathematics. L’Enseignement Mathématique s. 2, 5: 196–202. Kurepa, Ðuro. 1959c. Alcuni aspetti internazionali della riforma dell’insegnamento matematico. Bollettino della Unione Matematica Italiana s. 3, 14: 226–236. Kurepa, Ðuro. 1960a. Some principles of mathematical education. In Proceedings of the International Congress of Mathematicians, ed. John Arthur Todd, 567–572. Cambridge: Cambridge University Press. Kurepa, Ðuro. 1960b. On the teaching of geometry in secondary schools. L’Enseignement Mathématique s. 2, 6: 69–80 and 313–320. Kurepa, Ðuro. 1962a. Welcome speech on the opening of the Third Congress of mathematicians and physicists of Yugoslavia and of the International Symposium on the Coordination of Instruction in Mathematics and in Physics, Belgrade, September 19–24, 1960. In Proceedings of the International Symposium on the Coordination of Instruction in Mathematics and in Physics, eds. Yugoslav Union of Societies of Mathematicians and Physicists, 5–6. Belgrade: Yugoslav Union of Societies of Mathematicians and Physicists. Kurepa, Ðuro. 1962b. Some principles on the coordination in the instruction of mathematics and physics. In Proceedings of the International Symposium on the Coordination of Instruction in Mathematics and in Physics, eds. Yugoslav Union of Societies of Mathematicians and Physicists, 51–57. Belgrade: Yugoslav Union of Societies of Mathematicians and Physicists.

Photo Author: Jacobs, Konrad. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

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André Lichnerowicz

11.30 André Lichnerowicz (Bourbon-l’Archambault, 1915 – Paris, 1998): President 1963–1966, Ex Officio Member of the Executive Committee 1971–1974 Hélène Gispert

Biography André Lichnerowicz was born on 21 January 1915 in Bourbon-l’Archambault in Auvergne. An only child, he grew up in a family in which his father, professor of letters, and his mother, professor of mathematics, believed that “the general culture was really general and included not only the Past but also the Present, physics and mathematics too”.98 At the dinner table, he discussed mathematics with his parents. His secondary studies, both classic and scientific, were carried out at the Lycée Louis-le-Grand in Paris.  The original text is: “la culture générale était vraiment générale et ne comportait pas seulement les Anciens mais aussi le Présent, la physique et les mathématiques également”. See the interview with André Lichnérowicz by Jacques Nimier (http://pedagopsy.eu/entretien_lichnerowicz.html) 98

H. Gispert (*) GHDSO, EST, Université Paris Saclay, Paris, France e-mail: [email protected]

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A student at the École Normale Supérieure, he continued his scientific studies at the Sorbonne in the Faculty of Sciences in Paris; in particular, he had as a professor Élie Cartan, who taught differential geometry. In 1939, he defended a dissertation in mathematics on general relativity from the point of view of global differential geometry, under the advisement of Georges Darmois. First attaché, then research fellow at the Centre National de la Recherche Scientifique (CNRS) from 1937 to 1941, in 1941 André Lichnerowicz was appointed lecturer in rational mechanics at the University of Strasbourg, moved to Clermont-Ferrand and then was promoted to professor. He was arrested during the November 1943 roundup, the objective of which was to destroy the University of Strasbourg, and narrowly escaped. In 1949, he was appointed professor at the Faculty of Science in Paris, where he created the “certificate of mathematical methods in physics”. In 1952, at the Collège de France, he assumed the chair of mathematical physics, where he taught until 1986. His most important scientific work was in differential geometry, general relativity and symplectic geometry. Let us mention the study of the global problems related to the system of Einstein’s equations; the application of the Cauchy problem to the equations of general relativity; the study of the mathematical instruments “propagators” and “commutators”; and the demonstration of a solution to the problem of the quantification of the gravitational fields. Elected member of the Académie des Sciences in 1963, Lichnerowicz was also a member of many other academies. He was elected president of the World Institute of Science in 1991. Doctor “honoris causa” from multiple universities, Lichnerowicz earned numerous awards. He was president of the Société Mathématique de France in 1959 and president of the Comité National Français d’Histoire et de Philosophie des Sciences from 1985 to 1993. Starting in the 1950s, Lichnerowicz intervened at the political level on questions of teaching and research. In 1954, he was the scientific adviser to the president of the Council of Ministers Pierre Mendès-France and in 1956 was one of the founding fathers of the colloquium of Caen (1956) which established a policy and a strategy for French scientific research. Until the 1960s, he was consulted by the French government on issues concerning research and technology.

Contribution to Mathematics Education Lichnerowicz also intervened in the more specific field of mathematics education and its necessary renovation at the university and secondary school levels both nationally and internationally. Internationally, from 1962 to 1966, he was president of ICMI and, for the period 1971–1974, ex officio member as a representative of the Committee on the Teaching of Science of the International Council of Scientific Unions (CTS/ICSU). During his term as president, Lichnerowicz especially promoted collaboration with UNESCO. UNESCO representatives were officially invited to the congresses organized by ICMI; ICMI members were consulted as experts by UNESCO; they were

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often sent on missions in various countries.99 Conferences were organized jointly, joint reports were written and published, including the collection of Nouvelles Tendances de l’Enseignement Mathématique, which appeared in French, English and Spanish and disseminated throughout the world the experiences and things of this like (Lichnerowicz 1966). Lichnerowicz was a member – with Enrico Bompiani and Bruno de Finetti  – of the organizing committee of the international meeting held in Frascati (8–11 October 1964) under the aegis of ICMI and UNESCO on the theme “Mathematical Preparation for Admission to University: Actual Situation and Desirable Situation”. At the end of his mandate as president, he hoped for the creation of a permanent secretariat to deal with the expansion of the Commission’s activities. Nationally, in France, from 1966 to 1973, he chaired the Ministerial Commission on the teaching of mathematics, known as the Commission Lichnerowicz, responsible for the renovation of mathematics education at primary and secondary levels.  he Renovation of Mathematics Education “from Kindergarten T to University” At the university, first of all, Lichnerowicz was a professor who left a mark on his students thanks to the clarity of his presentation, his computational virtuosity and the perfect organization of difficult demonstrations (Marle 1999). In its homage to him, the Académie des Sciences underlined “his teaching, always of remarkable clarity, [which] will remain as a model of elegance and distinction”.100 He was a tireless activist for the modernization of university teaching of mathematics, which in France, even after World War II, had not integrated many of the modern developments that had occurred at the beginning of the century and during the interwar period, particularly in algebra. His numerous courses and the book in which these were collected, Algèbre et analyse linéaires (translated into English as Linear Algebra and Analysis), provide ample evidence of his commitment to disseminating modern writings on vector spaces and tensorial calculus, the notion of variety and that of exterior differential form, Hilbert space, series and Fourier transformation and integral equations. These subjects were all part of his teaching in Strasbourg, and his book had a considerable influence. Lichnerowicz was also interested in the renovation of secondary education. He participated in the very first meetings of the Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques (CIEAEM) in the early 1950s. He was one of the mathematicians, along with Gustave Choquet and Jean Dieudonné, who participated in the meetings of the CIEAEM that took place  Lichnerowicz, André. 1966. Rapport sur la période 1963–1966. L’Enseignement Mathématique 12: 132–138. 100  The original text is: “son enseignement, toujours d’une remarquable clarté, [qui] restera comme un modèle d’élégance et de distinction”. 99

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in La Rochette in 1952 alongside the philosopher Ferdinand Gonseth and the psychologist Jean Piaget. These were the first of a whole series of meetings, whose theme “Mathematical structures and mental structures” illustrates well the orientations of this movement of renovation and modernization of mathematics education. In France, in the years 1955–1965, Lichnerowicz participated in the work of training teachers in “modern mathematics” supported by the Association des Professeurs de Mathématiques (APMEP). This association was one of the protagonists in the renovation of the teaching of mathematics in France, and in 1956 he published in its Bulletin an article entitled “Applications Linéaires et Matrices”.101 As the head of the Lichnerowicz Commission, mentioned above, he played a decisive role in the design and implementation of the so-called Réforme des mathématiques modernes in French primary and secondary education. In a preliminary report published in the APMEP Bulletin, the commission specified its objectives and its method. Starting from the observation that “Mathematics plays a privileged role for the understanding of what we call reality, both physical reality and social reality”, the report declares: the problem of mathematics and its teaching has perhaps become the first of the international problems of education … We must now prepare our children and our students to understand and use what mathematics have become today … [that is, mathematics] which have become radically different over the past half century, following a real intellectual transformation which has occurred at a rate far exceeding the renewal of human generations.102 (Lichnerowicz 1967, p. 247)

The commission was responsible for developing the new curricula for the different classes from primary to baccalauréat, reflecting on methods and designing the IREMs, the institutes for teacher training. These programmes caused tension within both the scientific community and civil society in the early 1970s, and in 1973 Lichnerowicz resigned from the presidency of the Commission, which did not continue its work.

Sources Lichnerowicz, André. 1939. Problèmes globaux en mécanique relativiste. Paris: Hermann. Lichnerowicz, André. 1947. Algèbre et analyse linéaires. Paris: Masson. Lichnerowicz, André. 1950. Elements de calcul tensorial. Paris: Armand Colin.

 See https://www.apmep.fr/IMG/pdf/Histoire-brochures_VCAVa.pdf  The original text is: “La mathématique joue un rôle privilégié́ pour l’intelligence de ce que nous nommons le réel, réel physique comme réel social”, “le problème des mathématiques et de leur enseignement est devenu le premier, peut-être, des problèmes mondiaux de l’éducation”; “Il nous faut désormais préparer nos enfants et nos étudiants à comprendre et à utiliser ce que sont devenues les mathématiques de notre temps”; “devenues radicalement différentes depuis un demi siècle à la suite d’une véritable mutation intellectuelle qui s’est produite à un rythme dépassant de fort loin le renouvellement des générations humaines”. 101 102

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Lichnerowicz, André. 1954. Les théories relativistes de la gravitation et de l’électromagnétisme. Paris: Masson. Lichnerowicz, André. 1955. Théorie globale des connexions et des groupes d’holonomie. Rome: Cremonese. Lichnerowicz, André. 1958. Géométrie des groupes de transformations. Paris: Dunod. Lichnerowicz, André. 1961. Propagateurs et commutateurs en relativité générale. Paris: P.U.F. Lichnerowicz, André. 1962. Les spineurs en relativité générale. Bologna: Zanichelli. Lichnerowicz, André. 1967. Relativistic Hydrodynamics and Magnetohydrodynamics: Lectures on the Existence of Solutions. New York-Amsterdam: W. A. Benjamin. Lichnerowicz, André. 1970. Ondes et radiations électromagnétiques et gravitationnelles en relativité générale. Paris: CNRS. Berger, Marcel. 1999. Lichnerowicz et la géométrie différentielle. Gazette des Mathématiciens 82: 93–98. Choquet-Bruhat, Yvonne. 1999. Lichnerowicz, et la relativité générale. Gazette des Mathématiciens 82: 99–101. Flato, Moshé and Cahen, Michel. 1976. Differential Geometry and Relativity: A Volume in Honour of André Lichnerowicz on His 60th Birthday. Dordrecht: D. Reidel. Hennequin, Paul-Louis. 1999. André Lichnerowicz, Bulletin de l’APMEP 421: 133–136. Marle, Charles-Michel. 1999. L’Œuvre d’André Lichnerowicz, en géométrie symplectique. Gazette des Mathématiciens 82: 102–108. Revuz, André. 1999. Lichnerowicz et la réforme des mathématiques. Gazette des Mathématiciens 82: 90–92.

Interviews Entretien avec le professeur André Lichnerowicz, Jacques Nimier: Retrieved on 21 September 2020 from http://pedagopsy.eu/entretien_lichnerowicz.html Entretien avec Jean-François Picard et Antoine Prost (14 mai 1986): Archives nationales, CNRS (1984–1989), Répertoire (19970478/1-19970478/107): cote 19970478/65, transcription cote 20040108 art.2.

Publications Related to Mathematics Education Lichnerowicz, André. 1966. Rapport sur la période 1963–1966. l’Enseignement Mathématique 12: 131–138. Lichnerowicz, André. 1967. Rapport préliminaire de la commission ministérielle. Bulletin de l’APMEP 258: 245–282. Lichnerowicz, André. 1970. Les mathématiques et leur enseignement. Bulletin de l’APMEP 275–276: 405–412. Lichnerowicz, André. 1971. Eduquer c’est conquérir constamment. Entretien avec A. Lichnérowicz. L’école et la nation 195 (Janvier 1971). Lichnerowicz, André. 1972. Communication à l’Académie des sciences. Bulletin de l’APMEP 283: 370–374. Lichnerowicz, André. 1972. Analyse critique du rapport de J. Leray. Bulletin de l’APMEP 286: 1016–1018. Lichnerowicz, André. 1972. Table ronde de Caen (11 mai 1972) sur la finalité des mathématiques. Bulletin de l’APMEP 286:1043–1050.

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Lichnerowicz, André. 1972. Mathématique et Transdisciplinarité. In L’Interdisciplinarité. Problèmes d’enseignement et de recherche dans les universités, Paris: Centre pour la recherche et l’innovation dans l’enseignement.

Photo Courtesy of the University of Turin.

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Walther Lietzmann

11.31 Walther Lietzmann (Drossen, 1880 – Göttingen, 1959): Member of the Central Committee 1928–1932, Vice-President 1932–1936103 Gert Schubring

Biography Youth and Studies Walther Lietzmann was born on 7 August 1880 in Drossen, a small town in Prussia, east of the Oder river, the son of an accountant. After elementary school in his native town, he moved at 11 years old to the Gymnasium in nearby Frankfurt/Oder. The G. Schubring (*) University of Bielefeld, Bielefeld, Germany e-mail: [email protected]  During the  ICM 1936 in  Oslo, “The Congress requests the  International Commission on  the  Teaching of  Mathematics to  continue its work, prosecuting such investigations as  shall be  determined by the  Central Committee” (L’Enseignement Mathématique 35, 1936, p.  388), but because of WWII, the Commission remains inactive until 1952 when it is transformed in a permanent subcommission of the IMU. 103

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spirit of this school favoured the classical languages, but the young student became more interested in mathematics, physics and astronomy. Passing the Abitur exam in 1899, he decided to study astronomy, but arriving at the University of Berlin, he learned that he should begin by studying mathematics and physics. Not very satisfied with his studies in Berlin, he moved after three semesters to Göttingen and became absorbed in mathematics. Studying primarily with Hilbert, he elaborated a PhD thesis on the theory of algebraic number fields and passed the doctoral exam in 1903. After this, in 1904, he passed the Staatsexamen for the profession of secondary school teacher in mathematics, physics and biology. Despite his academic qualifications, he decided to follow the career of teacher. In fact, this orientation would make him the exponent of the reform of mathematics teaching in Germany. Pedagogical Career Lietzmann became a devoted teacher. After 1 year as a Referendar at a Realschule in Landsberg a. W., in the Prussian province Brandenburg, he moved as Probekandidat to Berlin at the Prinz-Heinrich-Gymnasium. His abilities as a teacher were such that he was chosen, even before the end of the probationary year, as an Oberlehrer – for mathematics and physics  – at the Oberrealschule in Barmen (today a part of Wuppertal). In 1914, he was called to Jena, as director of the Oberrealschule. Eventually, in 1919, he moved to Göttingen, as director of the Oberrealschule, combined with a Reformgymnasium  – today, the school is named Felix-­Klein-­ Gymnasium. Lietzmann remained in this position until his retirement in 1946. The range of his activities can hardly be exaggerated. Besides his commitment to IMUK and DAMNU (see next section), he held a Lehrauftrag for the didactics of mathematics and the sciences at the University of Göttingen from 1920 on. He was a member of the teacher examination body Wissenschaftliches Prüfungsamt in Göttingen. Having been active since his student days in editing publications, from 1905 to 1910, he was the editor of the journal Mathematisch-naturwissenschaftliche Blätter – a journal for students’ mathematical associations at German universities. From 1914 until it had to cease publication in 1943 due to WWII, he was editor of the Zeitschrift für Mathematischen und Naturwissenschaftlichen Unterricht, the ZfmnU, the major journal for teachers of mathematics and the sciences in Germany. From 1911 on, together with Alexander Witting, he was the editor of the Mathematisch-physikalische Bibliothek, a series of booklets for teachers. From 1924 to 1930, he was president of the influential Deutscher Verein für die Förderung des Mathematischen und Naturwissenschaftlichen Unterrichts, the Förderverein. His influence extended even to the community of mathematicians: for a short time, during the Nazi period, from 1936 to 1937, he was the president of the Deutsche Mathematiker-Vereinigung; and in 1936, he was the head (“Führer”) of the delegation of German mathematicians participating at the ICM in Oslo (Hollings et  al. 2020, pp. 76ff.). Without exaggerating, one can state that Lietzmann was the dominant personality in the German mathematics education community over decades – nobody else

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before and after him has ever had an analogous impact and influence. And he was during this period almost the only German representative with personal relations to other countries and an international standing. Lietzmann was also a highly prolific writer, not only of numerous reports on issues of mathematics teaching in various journals, but especially of textbooks. The register of the journal ZfmnU for the years 1908 to 1930 lists 99 papers published by Lietzmann on an enormous range of subjects. His publication of books is even more surprising; the catalogue of the Northern regional libraries GBV lists 360 monographies! There are numerous books for mathematics teachers: on issues of elementary mathematics, on recreations, on pedagogy and didactics and on history. His major book publications were, however, schoolbooks. The first such publishing activity, between 1911 and 1921, was re-editing and revising an already existing series of geometry textbooks by an otherwise not so well-known teacher, Max Schuster. The next step was a more profound revision of a classic text, an Aufgabensammlung, a collection of exercises used as a complement to a proper textbook, a very common textbook form in Germany. He revised the classic “Bardey” – one of the nineteenth-century German bestsellers, by Ernst Bardey – transforming it into the “Reform-Bardey”, that is, remaking it according to the ideas of the reform movement. Published since 1913, in a Gymnasium and in a Realschule version, it experienced up to 17 editions. However, the bestseller was the textbook of his own, the Leitfaden der Mathematik, published from 1917 on, also in different versions according to Gymnasien and Realschulen, and complemented from 1924 by a version for girls’ secondary schools. Lietzmann estimated that between one and two million copies were used in schools, not only in Germany, but also in Austria, Switzerland and the Baltic states (Lietzmann 1960, p. 90). During the Nazi period, Lietzmann published books according to its racist Aryan spirit  – most notably a book on early geometry on Germanic territories (Lietzmann 1940). The culmination of his commitment to improve teachers’ performance in teaching became his Methodik, a methodology for teaching mathematics (in secondary schools). It was first published in 1916, originally in two volumes; the second edition comprised three volumes. Many chapters were revised in content and orientation; the third edition even appeared with a revised title during the Nazi period – and thoroughly revised contents, expressing the Germanic racism. After WWII, an earlier version of the Methodik was re-edited. Lietzmann died in Göttingen on 12 July 1959. His autobiography was published after his death, “cleaned” by his friend Kuno Fladt who eliminated a chapter apparently too sympathetic with the Nazi period.

Contribution to Mathematics Education and the Work in IMUK Although studying in Göttingen, Lietzmann had not been in contact with Felix Klein, since he was only a student of Hilbert. At the time, he had no interest in questions of teaching. This changed during the time of his teaching apprenticeship in

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Berlin: Klein had been invited there to report on the Meran reform syllabus, and Lietzmann was asked to give a review of Klein’s talk in the Mathematischnaturwissenschaftliche Blätter. Meeting Klein on this occasion, he became impressed with the reform movement. Following Klein’s references, Lietzmann began to study the French textbooks to which he referred for his reform programme. The paper that he wrote as his evaluation was his first didactic publication. However, his association with the reform movement itself proved to be the side effect of another development: since about 1906, due to his own teaching, he had been interested in the foundations of geometry for schools. Since Italian authors had substantially contributed to this issue, in particular with several outstanding textbooks, Lietzmann began to study Italian publications (after studying the Italian language) and corresponded with Italian mathematicians. The paper that he wrote about Italian visions of the teaching of foundations of geometry was published in the ZfmnU in 1908, and should become fundamental for his future life, without him having imagined it. In fact, Klein’s attention had been called to this paper and to its author, since the foundations of geometry proved to be the key obstacle for integrating Italian mathematicians into the reform movement. Due to his own private interest, and in order to speak personally with the mathematicians with whom he had corresponded, Lietzmann travelled to Italy at Easter time in 1908 and participated at the Fourth International Congress of Mathematicians in Rome. Being present in the section on mathematics teaching  – where he met Giovanni Vailati – he happened to assist at the founding of IMUK. Klein, after his election as president of IMUK, approached Lietzmann and proposed that he participate in IMUK work. In fact, he practically became Klein’s secretary for IMUK matters. Right from the first meeting of Klein with Fehr and Greenhill, Lietzmann assisted Klein in German as well as in international IMUK work. This was the beginning of an intense involvement into the first international reform movement for mathematics education. Regarding the German scene, Lietzmann not only wrote some of the highly substantial reports within the remarkable series of German reports on the state of mathematics teaching, he also supervised, together with Klein, the production of the other reports (Schubring 2019). His insertion into the inner kernel of IMUK had the unintended effect that Lietzmann, who was still a teacher and later on a school director, worked in his own teaching to realize the intentions of the reform movement and was thus able to disseminate them more effectively among teacher colleagues and, by means of his teaching materials, much more widely than he would have been able to do through personal contacts alone. There was, on the German side, another institutional means for promoting the reform of mathematics teaching: as a successor to the Breslauer Unterrichtskommission of 1904–1905, which had elaborated the Meran reform syllabus, the Deutscher Ausschuß für Mathematischen und Naturwissenschaftlichen Unterricht (DAMNU), a committee constituted of numerous German professional associations and societies active in mathematics and the sciences, had been established, with the aim of promoting jointly the reform of teaching mathematics and the sciences in German schools. In 1910, Lietzmann was elected the general secretary of DAMNU and was thus once more in a situation where he was able to further

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promote Klein’s ideas. When, in 1925, not only the proposals of the Meran syllabus but also more far-reaching concepts of Klein eventually became part of the official syllabus for the secondary schools in Prussia, this was not only due to the broad dissemination of the reform ideas among the mathematics teachers and their practising by them but also by the institutional activities of DAMNU. Regarding the international scene, Lietzmann prepared together with Klein all the meetings of the Comité Central and of the IMUK itself, the international meetings and Congresses, first in Brussels in 1910, then in Milan in 1911, in Cambridge in 1912 and eventually in Paris in 1914. Some of the thematic reports were prepared by him, for instance, that on “rigour”, due to his studies on the foundations of geometry in Italy. WWI and the dissolution of the first IMUK put a stop to these international activities. When they were resumed in 1928, Lietzmann was elected member of the Comité Central and in 1932 vice-president of IMUK. In 1936, he participated at the ICM in Oslo not only as the head of the German delegation but was also reelected as vice-president. The agenda of IMUK for this Congress had been that the national delegates should present reports on the state of mathematics education in their respective countries. For Germany, Lietzmann had delivered that report (Lietzmann 1937), which was well received (Hollings et al. 2020, pp. 232ff.). In his internal report about the Congress for the Reichs-Ministerium, Lietzmann emphasized also the importance of the contributions for IMUK to the pedagogy section VIII (Hollings et al. 2020, p. 277). While there were only few IMUK activities from 1928 on, and even less after 1932, there existed but a nominal body from 1936 on. Lietzmann was fortunate, however, in seeing the re-establishment of IMUK as ICMI in 1952 and the reconstitution of the German subcommittee, with Behnke as its president. In fact, after Klein’s death, it had been Lietzmann who had presided over the German subcommittee. Klein had passed his files on IMUK activities, which he deemed important to preserve, to Lietzmann; nowadays, they constitute a part of the Lietzmann Nachlass in the Stadtarchiv Göttingen.

Sources Fladt, Kuno. 1940. Walter Lietzmann, 25 Jahre Herausgeber der Zeitschrift für Mathematischen und Naturwissenschaftlichen Unterricht. ZfmnU 70: 1–2. Hollings, Christopher, Reinhard Siegmund-Schultze, and Henrik Kragh Sørensen. 2020. Meeting under the integral sign?: the Oslo Congress of Mathematicians on the eve of the Second World War. Providence (Rhode Island): American Mathematical Society. Schubring, Gert. 2019. The German IMUK subcommission. In National subcommissions of ICMI and their Role in the Reform of Mathematics Education, ed. Alexander Karp, 65–91. Cham: Springer.

Obituaries Wolff, Georg. 1959. Nachruf. Praxis der Mathematik 1: 127.

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Proksch, R. 1959–1960. Walter Lietzmann, Der Mathematische und Naturwissenschaftliche Unterricht 12: 227–228. Schoen. 1960. Nachruf. Praxis der Mathematik 2: 19. Stender, Richard. 1960. Walther Lietzmann zum Gedächtnis, Mathematisch-physikalische Semesterberichte 7: 1.

Publications Related to Mathematics Education Lietzmann, Walther. 1908. Die Grundlagen der Geometrie im Unterricht (mit besonderer Berücksichtigung der Schulen Italiens)”, Zeitschrift für Mathematischen und Naturwissenschaftlichen Unterricht 39: 177–191. Lietzmann, Walther. 1909. Stoff und Methode im mathematischen Unterricht der norddeutschen höheren Schulen. Leipzig: Teubner. Lietzmann, Walther. 1910. Die Organisation des mathematischen Unterrichts an den höheren Knabenschulen in Preußen Leipzig: Teubner. Lietzmann, Walther. 1912. Stoff und Methode des Raumlehreunterrichts in Deutschland. Leipzig: Teubner. Lietzmann, Walther. 1912. Stoff und Methode des Rechenunterrichts in Deutschland Leipzig: Teubner. Lietzmann, Walther. 1912. Der pythagoreische Lehrsatz: mit einem Ausblick auf das Fermatsche Problem Leipzig: Teubner. (Mathematische Bibliothek; 3). Lietzmann, Walther. 1914. Die Organisation des mathematischen Unterrichts an den preußischen Volks- und Mittelschulen. Leipzig: Teubner. Lietzmann, Walther. 1916. Riesen und Zwerge im Zahlenreich: Plaudereien für kleine und grosse Freunde der Rechenkunst Leipzig: Teubner. (Mathematische Bibliothek; 25). Lietzmann, Walther & V.  Trier. 1917. Wo steckt der Fehler?: Trugschlüsse und Schülerfehler Leipzig: Teubner. (Mathematisch-­physikalische Bibliothek: [Reihe 1]; 10). Lietzmann, Walther (ed.). 1917. Berichte und Mitteilungen, veranlaßt durch die Internationale Mathematische Unterrichtskommission. Leipzig; Berlin: Teubner. Lietzmann, Walther. 1919. Methodik des mathematischen Unterrichts (Leipzig: Quelle und Meyer): Band 1. Organisation, allgemeine Methode und Technik des Unterrichts, 1919; Band 2. Didaktik der einzelnen Unterrichtsgebiete, 1916; Band 3. Didaktik der angewandten Mathematik. Lietzmann, Walther. 1941. Mathematik in Erziehung und Unterricht. Unter Mitarbeit von U. Graf (Leipzig: Quelle & Meyer): Band 1. Ziel und Weg; Band 2. Lehrstoff. Lietzmann, Walther. 1935. Altes und Neues vom Kreis. Leipzig: Teubner. (Mathematischphysikalische Bibliothek. Reihe 1; 87). Lietzmann, Walther. 1935. Theorie und Praxis der geometrischen Konstruktionsaufgaben. Darmstadt: Schlapp. (Aus der Praxis; 16). Lietzmann, W. (1937). Die gegenwärtigen Bestrebungen im mathematischen Unterricht der höheren Schulen Deutschlands. Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht 68: 19–22. Lietzmann, Walther. 1940. Frühgeschichte der Geometrie auf germanischem Boden. Breslau: Hirt. Lietzmann, Walter. 1960. Aus meinen Lebenserinnerungen. Im Auftrag von Walter und Käthe Lietzmann hrsg. von Kuno Fladt. Göttingen: Vandenhoeck & Ruprecht.

Photo and Signature Source: Lietzmann, Walter. 1960. Aus meinen Lebenserinnerungen. Im Auftrag von Walter un Käthe Lietzmann hrsg. von Kuno Fladt. Göttingen: Vandenhoeck & Ruprecht.

528

Michael James Lighthill

11.32 Michael James Lighthill (Paris, 1924 – Sark, 1998): President 1971–1974, Ex Officio Member 1975–1978 Adrian Rice

Biography James Lighthill was born in Paris on 23 January 1924. After schooling at Winchester College, he won a scholarship to Cambridge, entering Trinity College in 1941. Due to World War II, his degree course was shortened by a year, so he graduated after just 2  years in 1943. He immediately joined the Aerodynamics Division of the National Physical Laboratory in Teddington as a junior scientific officer, staying until the end of the war, before returning to Cambridge as a fellow of Trinity College in 1945. The following year, he was appointed senior lecturer at the University of Manchester, being promoted to the Beyer professorship of applied mathematics in 1950. During his time in Manchester, he established a research group in fluid dynamics which, through its creativity and innovative work, soon exerted a dominant influence on hydrodynamic research worldwide. He left Manchester in 1959 to become the director of the Royal Aircraft Establishment at Farnborough.

A. Rice (*) Randolph-Macon College, Ashland, VA, USA e-mail: [email protected]

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The 1960s were among the most productive years of a very productive life. He was instrumental in the development of television and communications satellites, plans for manned spacecraft able to return to earth and work on the joint BritishFrench project which resulted in the Concorde supersonic passenger jet. Also, during this time, in response to the lack of adequate government support for applied mathematics, he was the chief founder of the Institute of Mathematics and its Applications (IMA), serving as its first president from 1964 to 1966. In 1964, he returned to academia as a Royal Society research professor based at Imperial College London. In 1969, he replaced Paul Dirac as Lucasian professor of mathematics at Cambridge, being succeeded by Stephen Hawking in 1979 when he accepted the administrative position of Provost of University College London, which he also held for 10 years. His mathematical research continued throughout his time as Provost, and when he finally retired in 1989, he remained active scientifically, holding an honorary research fellowship in the mathematics department at University College until his death in 1998. Lighthill began to publish mathematical research almost immediately on joining the National Physical Laboratory, where his work concentrated on supersonic flight. His first paper, on “Two-Dimensional Supersonic Aerofoil Theory”, was published early in 1944, 2 days before his 20th birthday. This early work already provided evidence of “the thoroughness of the author’s grasp of the then existing literature and of his uncanny ability to present a unified view of the subject” (Hussaini 1997, p. xxiii). While at Manchester, he was responsible for launching two new major fields of fluid mechanics. The first of these, aeroacoustics, was introduced in a two-part paper of 1952, entitled “On Sound Generated Aerodynamically”. It was directly responsible for subsequent work in the field, proving to be of immense technological utility. In particular, “Lighthill’s Law”, which states that “the acoustic power radiated by a jet is proportional to the eighth power of the jet speed” (Hussaini 1997, p. xxv), was of immediate importance in reducing jet-engine noise. He initiated the second area, non-­linear acoustics, in a famous 100-page paper in 1956, resulting in a host of subsequent research papers and applications in the study of traffic flow and flood waves in rivers. During his time at Imperial College and at Cambridge, he totally transformed the subject of biological fluid dynamics to create what was, in effect, a new sub-discipline: mathematical biofluid dynamics. In this he developed new techniques to mathematically analyse blood flow in cardiovascular systems, respiratory flow, aquatic locomotion and ciliary and flagellar propulsion. As with all of his work on fluid dynamics, this research was characterized by great insight, originality and interdisciplinary scope. Widely regarded as one of the greatest applied mathematicians of the twentieth century, Lighthill received an abundance of honours and awards from all parts of the world. Following his election to the Royal Society at the early age of 29, he was awarded honorary foreign membership by 11 learned societies in Europe, America and Asia and was the recipient of 22 honorary doctorates. He received the Royal

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Society’s Royal Medal in 1964, served as chairman of the Oceanography and Fisheries Research Committee from 1965 to 1970 and was president of the International Union of Theoretical and Applied Mechanics from 1984 to 1988. He was knighted by the Queen in 1971. An enthusiastic and expert long-distance swimmer, in which his detailed knowledge of tides and local currents was used to good effect, in 1973 Sir James became the first person to swim the entire coast of Sark in the Channel Islands, a feat he subsequently repeated several times. On 17 July 1998, towards the end of another attempt to swim round the island, he suffered a fatal heart attack in rough seas and died close to the shore. He was 74.

Contribution to Mathematics Education Sir James was once accurately called “an amazingly extrovert and engaging personality” (Thwaites 1998, p. 497), and these qualities were undoubtedly reflected in his teaching style when lecturing on mathematics. Described as “a truly great teacher”, for Lighthill, “the lecture-room’s dais was a stage on which to perform with word and gesture alike so as to entrance the audience” (Thwaites 1998, p.  497). As Lucasian professor at Cambridge, “he taught indefatigably and with enormous gusto six days of the week at nine in the morning” (Crighton & Pedley 1999, p. 1228), and it would appear that he enjoyed every minute of it. For example, when teaching biofluid dynamics, “he revelled in lectures, not only in the articulation of all the Latin names, but in his ability to perform the appropriate gymnastics to illustrate certain flying characteristics” (Crighton & Pedley 1999, p. 1227). His first book, Introduction to Fourier Analysis and Generalised Functions (1958), grew out of undergraduate lectures he gave at the University of Manchester in the 1950s and was motivated by the need for more powerful mathematical techniques in his research on wave generation and propagation in fluids. Building on previous work by Paul Dirac, Laurent Schwartz and George Temple, Lighthill employed the Dirac delta function in his treatment of Fourier transforms, “making available a technique for their asymptotic estimation which seems superior to previous techniques” (Lighthill 1958, p.  1). The book thus contained both a valuable high-level introduction to Fourier analysis and an original development of the theory. Sir James had a great interest in mathematical education at all levels. In the 1960s, he was involved in an advisory capacity in the creation of the School Mathematics Project, of which he was a trustee for many years. In 1970, he was elected president of Britain’s Mathematical Association, being subsequently described as “the obvious choice for the Presidency of our Association in its Centenary Year” (Thwaites 1998, p.  497). His presidential address, delivered in 1971 at a meeting to mark 100 years of the association’s activity, dealt with a subject of great interest to him: “The art of teaching the art of applying mathematics”. One of the key problems when teaching applied mathematics, he said, is how to communicate the material to the students effectively.

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On the one hand, mathematics teaching should be permeated with concrete examples which give an impression of how widely and diversely mathematical ideas penetrate into human problems generally, including everyday matters and technical and scientific matters. On the other hand, it is necessary to tell at least one lengthy connected story of the application of mathematics in real depth, and to communicate the message that no-one can expect to solve the whole of any problem mathematically. (Lighthill 1971, p. 256)

In short, he said, to best convey the practice of applying mathematics, teachers must make integrated use of experiment, observation and theory in their classes. Not surprisingly, given the interdisciplinary nature of his research, he bemoaned the obstacles to closer connections between teachers of mathematics and subjects like physics, chemistry and biology caused by excessive departmentalization. Despite this, he drew attention to what he believed to be a very positive feature of British mathematical education, namely “the very close association between pure and applied mathematics, and a general predilection for teaching mathematics in a way that emphasised at least some of its applications” (Lighthill 1971, p.  250). Characterizing British mathematics teaching by its “flexibility”, i.e. its focus on applications in a variety of practical and real-life situations, Lighthill observed that Now that the requirement for effective application in mathematics is found in so many fields, and the demand for people able and willing to apply mathematics has become so great, Britain’s long experience in the field of integrated pure and applied mathematics education has become invaluable. (Lighthill 1971, p. 251)

He believed that the practical consequence of this would be that In a century of rapid change, when the lifetime of a particular technology may be less than a person’s working life, a flexibility in his education that prepares him to some extent for a possible vocational switch in middle life may be particularly valuable. (Lighthill 1971, p. 254)

But Lighthill’s activities in mathematics education were not focused on Britain alone. “As in so many other fields, he played on the world stage” (Thwaites 1998, p. 497), and his presidency of the Mathematical Association was immediately followed by an even greater honour. In 1971, presumably in light of his high reputation in mathematics educational circles in Britain, he was elected to serve as the first British president of ICMI. In this capacity, he helped organize and chair the Second International Congress on Mathematical Education, held at Exeter in September 1972. After his presidential term expired in 1974, he remained an ex officio member of the ICMI Executive Committee until 1978.

Sources Hussaini, Mohammed Yousuff (ed.). 1997. Collected Papers of Sir James Lighthill. 4 vols. Oxford/ New York: Oxford University Press. Lighthill, Michael James. 1952–54. On sound generated aerodynamically, Parts I and II. Proceedings of the Royal Society, s. A 211 (1952): 564–587, and 222 (1954): 1–32.

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Lighthill, Michael James. 1956. Viscosity effects in sound waves of finite amplitude. In Surveys in mechanics, ed. George Keith Batchelor and Rhisiart Morgan Davies, 250–351. Cambridge: Cambridge University Press. Lighthill, Michael James. 1958. Introduction to Fourier analysis and generalised functions. Cambridge: Cambridge University Press. Lighthill, Michael James. 1975. Mathematical biofluiddynamics. Philadelphia: Society for Industrial and Applied Mathematics. Lighthill, Michael James. 1978. Waves in fluids. Cambridge/New York: Cambridge University Press. Crighton, David George and Timothy John Pedley. 1999. Michael James Lighthill (1924–1998). Notices of the American Mathematical Society 46: 1226–1229. Thwaites, Bryan. 1998. Sir James Lighthill F.R.S. 1924–1998. The Mathematical Gazette 82: 496–497.

Publications Related to Mathematics Education Lighthill, Michael James. 1971. The art of teaching the art of applied mathematics. The Mathematical Gazette 55: 249–270. Lighthill, Michael James. 1973. Presidential address. In Developments in Mathematical Education. Proceedings of the Second International Congress on Mathematical Education, ed. A. Geoffrey Howson, 88–100. Cambridge: Cambridge University Press. Lighthill, Michael James. 1973. The interaction between mathematics and society. In Proceedings of the Third International Congress on Mathematical Education, ed. Hermann Athen and Heinz Kunle, 27–60. Karlsruhe: Organising Committee, University of Karlsruhe.

Photo Source: Pedley, Timothy J. 2001. Sir (Michael) James Lighthill. Biographical Memoirs of Fellows of the Royal Society 47: 334–356 (Photo at n. n. p.).

Jacques-Louis Lions

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11.33 Jacques-Louis Lions (Grasse, 1928 – Paris, 2001): Ex Officio Member of the Executive Committee 1975– 1982, 1991–1994 Margherita Barile

Biography Jacques-Louis Lions was born in Grasse, in the district of Alpes-Maritimes, in south-eastern France, on 3 May 1928. There he attended primary and secondary school. In 1943, at the young age of 15, he joined the Forces Françaises de l’Intérieur, where he served as a resistant to the German occupation for about 1 year. In 1946, he left his beloved hometown to study at the Lycée Félix-Faure of Nice, and in 1947, he entered the École Normale Supérieure of Paris. He obtained the doctorate in 1954 at the University of Nancy, where he had studied with a grant from the Centre National de la Recherche Scientifique. His dissertation, entitled “Problèmes aux limites en théorie des distributions”, prepared under the supervision of Fields Medallist Laurent Schwartz, was published in Acta Mathematica. In the same year, he was appointed professor. He remained in Nancy until 1962; then he moved to M. Barile (*) University of Bari “Aldo Moro”, Bari, Italy e-mail: [email protected]

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Paris, where he taught numerical analysis at the Faculty of Science of the University of Paris (later, at the Université Paris VI “Pierre et Marie Curie”) until 1973. There he also initiated a weekly seminar at the Institut Blaise Pascal, created a postgraduate diploma in numerical analysis and set up a laboratory that still bears his name. At the same time, he was a part-­time professor at the École Polytechnique from 1966 to 1986, the year when he was appointed honorary professor. In 1973, he was named by the Collège de France to the chair of “Mathematical Analysis of Systems and Their Control”, which he retained until 1998, when he retired and received professor emeritus status. In 1973, he was also elected to the Académie des Sciences, where he was to become vice-president in 1995 and then president in the term 1996–1998. He held high-level managing and counselling positions in several scientific organizations, governmental institutions and industrial companies, among which the Société d’Économie et de Mathématiques Appliquées (from 1958 on), the Institut National de Recherche en Informatique et en Automatique (INRIA) (1980–1984) of which he founded the main centre in Paris, and later the new sites at Nice-Sophia Antipolis and Rennes, the Ministère de l’Industrie et de la Recherche (1983–1984), the Centre National d’Études Spatiales (CNES) (1984–1992), the Conseil Scientifique d’Électricité de France (1986–1996), the Comité Scientifique de la Météorologie Nationale (1990–2001), the Conseil Scientifique de Gaz de France (1993) and the Conseil Scientifique de France Télécom (1998–2001). He was president of the International Mathematical Union from 1991 to 1994, and in this capacity, he proclaimed, at the Instituto de Matemática Pura e Aplicada of Rio de Janeiro, on 6 May 1992, the year 2000 as the World Mathematical Year (WMY). This year should also be recalled as the one where, during an official ceremony at the Elysée Palace, with the members of the Académie des Sciences as invited guests, Lions handed to President Jacques Chirac, as the result of a 3-year teamwork, a detailed report on the most recent scientific achievements concerning the following socially relevant issues: “access to knowledge for all and electronic processing of information; knowledge of our planet and ways of life; understanding life systems and improving health care of all”. The most remarkable feature of Lions’ scientific profile is, indeed, the great effort towards theoretical progress combined with a great interest in applications in economics and finance, management, engineering, energy production, meteorology, environmental problems and life sciences in general. Following the original approach of Laurent Schwartz, based on distributions, Lions succeeded in enhancing the study of partial differential equations in a groundbreaking way, not only improving its understanding in the context of variational and quasi-­variational calculus, but also investigating its connections with interdisciplinary areas such as optimization problems, fluid dynamics, electromagnetism and material physics. During the 1960s, his activity became more and more focused on deterministic and stochastic system control: this was the subject of his inaugural lecture held at the Collège de France on 4 December 1973. Working in this direction, he strongly endorsed the use of computer-based numerical techniques and the implementation of symbolic algorithms, for the elaboration of models to be verified by experimental results. He considered an efficient

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collection and completion of data as the crucial starting point towards a better knowledge of the physical phenomena, a goal that could be pursued both from the mathematical point of view (see the concept of “sentinels” for the control and detection of pollution introduced and studied in one of his books on control, 1988) and from the technological side (see his contribution to the French-American satellite TOPEX/Poséidon, 1992). Approximation should step in where exactness was out of reach, as in non-linear problems, a global view should make up for the insufficiency of local observations, and reality should be regarded as a network of interrelated subsystems, such as the coupled atmosphere and ocean, or the Mediterranean region, the monsoon, El Niño, cloud physics (Lions 1992, p. 13): mastering such a complexity was a visionary’s104 dream, that the emerging powerful ICT tools were making come true. Lions expressed this firm conviction when, in his invited conference on 28 January 1998 at the Real Academia de Ciencias in Madrid, he thoroughly described the background of the Japanese project of an “Earth Simulation” (Lions 1998). In his balanced, comprehensive view of scientific progress, the machines should not be called to replace the researcher’s mind, but be employed as a source of motivation and enrichment. Their historical role in the problems of understanding, predicting, designing and managing the behaviour of systems is best outlined by Lions’ own illuminating overview: For these questions, a ‘divine help’ (‘help’ is different from ‘solution’) came from the computer. The Universal machine of A. Turing (1937) became ‘real’ (and not only ‘virtual’). The ‘inextricable’ computations (in the D’Alembert words) for Euler equations became possibly accessible. The ‘dream’ of L.F. Richardson (1910-1920) for weather prediction could be envisioned. This was indeed the J. von Neumann program stated in the period 1948-1950.105

The underlying idea of what Lions called the universal trilogy, formed by modelling, simulation and system control, would be further developed in his public address at the Spanish Congreso de los Diputados (21 January 2000). His answer to the question posed in the title (Is it possible to describe the nonliving and the living world in the languages of mathematics and computer science?106) presents the scientific and cultural achievements of the past as the natural foundations of the ongoing technological advances. Lions wrote around 500 scientific papers and authored or edited nearly 60 expository books, among which the nine volumes of the monumental treatise “Analyse mathématique et calcul numérique pour les sciences et les techniques” (1984–1985), written with Robert Dautray, which is considered as his masterpiece and lifework. Lions is nowadays regarded as the father of applied mathematics in France. At a time when, in his country, the structural approach of the Bourbaki group was the  See Ciarlet 2002, p. 284. The same word appears in Yoccoz (2001, p. 3).  This is the closing paragraph in the abstract of Lions’ talk From Function Spaces to Action, held in Brussels, at the Symposium de la Fondation Jacques et Yvonne Ochs-Lefebvre “Reflections on XXth century sciences”, on 16 April 1999. See pp. 41–42 of the document available at http://www. academieroyale.be/Academie/documents/OchsReflections13648.pdf. 106  Lions 2000. For an extended, detailed presentation of the topic, see the monograph (Lions 1990). 104 105

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predominant trend, he had the sagacity and braveness of moving against the tide, and search for interactions with the real world. Lions collected a huge number of prestigious international awards. Degrees honoris causa were bestowed on him by 19 universities from the whole world. Moreover, he won the Prix Cognacq-Jay (1972), the John Von Neumann Prize (1986), the Japan Prize in Applied Mathematics (1991), the Harvey Prize from Technion, Haïfa, Israel (1991), the Daedalon Gold Medal for Science and Technology from Greece (1991) and the W.T. and Idalia Reid Prize of the Society for Industrial and Applied Mathematics (1998), the Lagrange Prize of the International Council for Industrial and Applied Mathematics (1999) and the Hilbert Medal of the International Conference on Computational & Experimental Engineering and Sciences (2000). He also received the following decorations: Commandeur de la Légion d’Honneur (1993), Grand Officier dans l’Ordre National du Mérite (1998) from France and the Order of the Rising Sun, Gold and Silver Star (1998) from Japan. As a result of the international character of his research activity, he became member of over 20 foreign academies. His innumerable scientific visits abroad left a lasting mark, e.g. in Italy, Spain and the former Soviet Union: the Russian Gury Marchuk and Lev Pontryagin were, together with Lions and the Indo-American Alampallam Venkatachalaiyer Balakrishnan, the co-founders of the journal Applied Mathematics & Optimization. Lions is also credited with being the initiator of the joint French-Russian manned space missions Soyouz in 1982 and 1992. Lions’ organizing activities in developing countries are one of his most remarkable merits; he played a major role in the creation of an “applied” branch of the Tata Institute in Bangalore (India), of the Laboratoire Franco-Chinois d’Informatique, d’Automatique et de Mathématiques Appliquées in Beijing and of the Institut SinoFrançais de Mathématiques Appliquées in Shanghai and thoroughly supported the Third World Academy of Sciences in its promotion of mathematical research in Africa. Lions was an influential personality, whose outstanding professional and moral qualities earned him worldwide recognition. The former French minister and president of the European Space Agency Hubert Curien thus explains the reasons of his success: “He was not a man of diktat, but of contact, discussion and decision” (Bermúdez de Castro et al., p. 355). The French Societé de Mathématiques Appliquées et Industrielles, in collaboration with INRIA and CNES, has created the annual Prix Jacques-­Louis Lions, which was awarded for the first time in 2003, to Roger M. Temam. Jacques-Louis Lions died in Paris on 17 May 2001.

Contributions to Mathematics Education In addition to the prestigious awards listed above, Lions also obtained the positions of secretary and then of president of IMU. Thanks to these institutional positions, he was part of the Executive Committee of ICMI as an ex officio member in the periods

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1975–1978, 1979–1982 and 1991–1994. Although he was not directly involved in mathematical education, a relevant part of his mission was raising awareness of mathematics, at any level and everywhere. In the framework of the activities of general interest planned for WMY 2000, he chaired the scientific committee of the conference “Mathematics and Its Role in Civilization” (11–14 January 2000). The meeting took place in Macau, soon after its reincorporation into China. The programme was centred on interdisciplinary, historical and multicultural aspects. His Chinese student Li Tatsien (Li Daqian) points out that the initiative “added a piece of bright landscape for new-returned Macao” (Tatsien 2004, p. 23). Lions’ versatile, far-reaching gifts and his intensive engagement for the innovation of mathematics as a lively component of modernity earned him not only worldwide fame but also the devoted admiration of his colleagues and students (at least 46, with a total of over 3300 descendants, spread over 6 scientific generations). Enrico Magenes, with whom Lions co-authored a three-volume book on nonhomogeneous boundary value problems between 1968 and 1970, says: I had the opportunity to fully appreciate the intellectual and human qualities of Lions: his unaffected manners; his commitment and energy in work; his rapidity of intuition and decision; his openness to new ideas and new problems in a body of knowledge that increased more and more over time, even outside mathematics; his love of freedom; and his respect for the opinions of others. (Lax et al. 2002, p. 1321)

According to Peter D. Lax, “Lions had an open, friendly, generous personality, with a light touch and a subtle sense of humour” (ibid., p. 1320). Lions’ collaborator Roland Glowinski also mentions his sense of essentiality along with his inborn sense of theatre, which made his lectures and written works so attractive to a broad audience. All his presentations were “remarkable for their clarity, their style, their conciseness and their organization” (Glowinski 2003, pp. 2–4). And these are the words of gratitude expressed by Lions’ student Roger M. Temam in his commemorative article (Temam 2001): All his students were delighted and amazed at his quick reading of their drafts, and at his availability to each of them. Also a factor in his success as an adviser was his ability to determine very quickly which research would suit a new student, and then to tailor new problems to the student’s abilities. Always careful not to influence his students too much, he described himself as a counsellor, striving to help each student develop the best of his/ her possibilities. … Jacques-Louis Lions was an exceptional person in many respects. Charismatic, generous, open, and accessible, he avoided conflict whenever possible. Among the most striking aspects of his personality was his long-term vision; he was able to see and pursue ideas that would come to maturity only five, ten, or twenty years later. He had many good ideas, and he had the mathematical talent, the physical strength, and the understanding of people needed to implement them.

In a written multivoiced testimony from Lions’ numerous visits to Spain, Enrique Fernández Cara recalls his farsightedness and optimism, which enabled him to overcome contentious situations and simplify difficult tasks (Bermúdez de Castro et al., p. 361). Nevertheless, he “would claim that he was learning from his students”, as Pierre Bernhard reports in his editorial (Bernhard 2002, p. 570). As an evidence of Lions’ expectant, confident attitude towards his auditory, Magenes (Magenes 2001,

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p. 193) reports how his written and oral presentations always ended with the statement of some unsolved problems. These personal memories, together with the extraordinary talent he passed on to his son Pierre-Louis (who won the Fields Medal in 1994), are Jacques-Louis Lions’ invaluable human legacy.

Sources Lions, Jacques-Louis. 2003. Œuvres choisies de Jacques-Louis Lions, eds. Alain Bensoussan, Philippe G. Ciarlet, Roland Glowinski, François Murat, Jean-Pierre Puel and Roger Temam, Vol. I-III. Paris: EDP Sciences, Les Ulis, Société de Mathématiques Appliquées et Industrielles. Bermúdez de Castro, Alfredo, Eduardo Casas Rentería, Enrique Fernández Cara, and Antonio Valle Sánchez. 2002. Testimonios. La Gaceta de la Real Sociedad Matemática Española 5: 354–367. Bernhard, Pierre. 2002. Jacques-Louis Lions. Automatica 38: 569–570. Ciarlet, Philippe G. 2002. Jacques-Louis Lions (2 May 1928  – 17 May 2001). Biographical Memoirs of Fellows of the Royal Society. 48: 275–287. This is an English translation of the French original text published in MATAPLI 66 (2001) and reprinted in Revue Roumaine des Mathématiques Pures et Appliquées 47: 259–269. For the English version, see also Chinese Annals of Mathematics 23B: 1–12, Computational and Applied Mathematics 21: 5–21. European Mathematical Society Newsletter 42: 21–25. For a Spanish version, see La Gaceta de la Real Sociedad Matemática Española 5: 311–322. Dahan-Dalmedico, Amy. 2005. Jacques-Louis Lions, un mathématicien d’exception entre recherche, industrie et politique. Paris: Éditions La Découverte. Díaz, Jesús Ildefonso. 2002. El legado de Jacques-Louis Lions (1928–2001) a través de sus libros: mi limitada visión. La Gaceta de la Real Sociedad Matemática Española 5: 330–353. Glowinski, Roland. 2003. Mathématiques appliqueés et calcul scientifique: la vision de JacquesLouis Lions. In Jornadas Científicas. La obra científica de J. L. Lions y su influencia en España (Embajada de Francia, 15–16 octubre 2003), 8  pp. Madrid: Real Academia de Ciencias Exactas, Físicas y Naturales. Electronic publication: http://www.rac.es/ficheros/doc/00193.pdf Kallianpur, Gopinath, and Roberto Triggiani. 2002. Jacques-Louis Lions. Applied Mathematics & Optimization 46: 79–80. Lax, Peter D., Enrico Magenes, and Roger Temam. 2001. Jacques-Louis Lions (1928–2001). Notices of the American Mathematical Society 48: 1315–1321. Li, Tatsien (Li Daqian). 2004. Always remembered. In Frontiers in mathematical analysis and numerical methods. In memory of Jacques-Louis Lions, ed. Li Tatsien. 19–24. Singapore: World Scientific. Magenes, Enrico. 2001. Ricordo di Jacques-Louis Lions. Bollettino della Unione Matematica Italiana s. 8, 4-A: 185–198. Magenes, Enrico. 2005. The collaboration between Guido Stampacchia and Jacques-Louis Lions on variational inequalities. In Variational analysis and applications (Nonconvex Optimization and its applications 79), eds. Franco Giannessi and Antonino Maugeri, 33–38. New  York: Springer. Mawhin, Jean. 2001. Éloge. Jacques-Louis Lions (1928–2001). Académie Royale de Belgique, Bulletin de la Classe de Sciences s. 6, 12: 167–172. Mawhin, Jean. 2005. Les histoires belges de Jacques-Louis Lions. Académie Royale de Belgique, Bulletin de la Classe de Sciences s. 6, 16: 219–226. Temam, Roger M. 2001. Obituaries: Jacques-Louis Lions. SIAM News 34. Electronic publication: http://archive.siam.org/news/news.php?id=560

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Valle, Antonio. 2003. Jacques-Louis Lions y sus discípulos españoles. In Jornadas Científicas La obra científica de J. L. Lions y su influencia en España (Embajada de Francia, 15–16 octubre 2003), 8 pp. Madrid: Real Academia de Ciencias Exactas, Físicas y Naturales. Electronic publication: http://www.rac.es/ficheros/doc/00192.pdf Yoccoz, Jean-Christophe. 2001. Jacques-Louis Lions (1928–2001). Collège de France, 3  pp. Electronic publication: http://www.college-­de-­france.fr/media/professeurs-­disparus/ UPL53967_necrolions.pdf Zuazua, Enrique. 2002. Jacques-Louis Lions: Hasta siempre. La Gaceta de la Real Sociedad Matemática Española 5: 323–329.

Publications Related to Science Popularization Lions, Jacques-Louis. 1990. El planeta Tierra. El papel de las matemáticas y de los super ordenadores. Madrid: Instituto de España, Espasa-Calpe. Lions, Jacques-Louis. 1992. World Mathematical Year 2000 and computer sciences. In Future tendencies in computer science, control and applied mathematics. Proceedings of the international conference on the occasion of the 25th anniversary of INRIA, Paris, France, December 8–11, 1992, eds. Alain Bensoussan and Jean-Pierre Verjus, 3–16. Berlin: Springer. Lions, Jacques-Louis. 1998. Le simulateur de la Terre. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales 92: 67–81. Lions, Jacques-Louis. 2000. ¿Es posible describir el mundo de lo inanimado y del ser vivo con los lenguajes matemático e informático? In Jornada Matemática. 21 de enero de 2000, eds. Jesús Ildefonso Díaz Díaz, José Luis Fernández Pérez, Antonio Martinón Cejas and Teresa Riera Madurell, 63–75. Madrid: Congreso de los Diputados.

Photo Author: Jacobs, Konrad. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

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Saunders Mac Lane

11.34 Saunders Mac Lane (Norwich, 1909 – San Francisco, 2005): Vice-President 1952–1954 Jeremy Kilpatrick

[email protected]

Biography Saunders Leslie Mac Lane was born in Norwich, Connecticut, on 4 August 1909. He graduated from high school at the age of 15 and entered Yale University that year, receiving his BA from Yale in 1930 and his master’s degree from the University of Chicago in 1931. At the urging of Eliakim H. Moore at Chicago, who was retired but still teaching, Mac Lane went to Göttingen, Germany, for his PhD, which he obtained in 1934 after studying logic and mathematics with Paul Bernays, Emmy Noether and Hermann Weyl. In his dissertation, he gave a formal means of abbreviating proofs, arguing that every mathematical proof has a leading idea that determines its steps and provides a plan. Once back in the United States, Mac Lane found it difficult to get a job in logic, so he turned his attention to algebra. From 1934 to 1938, he was a mathematics instructor at Harvard University, Cornell University and the University of Chicago. He returned to Harvard in 1938, holding a tenure-­track position there until 1947. In J. Kilpatrick (1935–2022) University of Georgia, Athens, GA, USA

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1944 and 1945, he also directed Columbia University’s Applied Mathematics Group, which was one of the largest groups of mathematicians involved in the war effort. In 1947, he went back to Chicago, where he spent the rest of his career. He chaired the mathematics department from 1952 to 1958. He was appointed the Max Mason Distinguished Service Professor in Mathematics in 1963 and became professor emeritus in 1982. He moved to San Francisco in 2002, where he died on 14 April 2005 after a long illness. He was survived by his second wife, two children, three stepchildren, one grandchild and five step-grandchildren. Mac Lane did his early mathematical work in field theory and valuation theory. In 1943, he began his influential collaboration with Samuel Eilenberg on what are now termed Eilenberg-Mac Lane spaces K(G,n), which have a single nontrivial homotopy group G in dimension n and which are building blocks of homotopy theory. Mac Lane’s central achievement, with Eilenberg, was the creation of category theory in 1945. Best known of Mac Lane’s many publications is the famous introductory textbook written with Garrett Birkhoff in 1941, A Survey of Modern Algebra, a book that was for years the leading textbook in its field. Mac Lane was one of only four people to have been elected president of both the Mathematical Association of America and the American Mathematical Society. He was elected to the National Academy of Sciences in 1949. In 1989, he received the nation’s highest award for scientific achievement, the National Medal of Science. He received two Guggenheim Fellowships and visited Australia as a Fulbright Scholar. He also received honorary degrees from Purdue University, Yale University and the University of Glasgow, among others. Other honours include both the Chauvenet Prize and the Distinguished Service Award of the Mathematical Association of America, the Steele Career Prize of the American Mathematical Society and honorary fellowship in the Royal Society of Edinburgh.

Contribution to Mathematics Education From 1952 to 1954, Mac Lane was a vice-president of ICMI.  He served on the Executive Committee of the International Mathematical Union from 1955 to 1958. While president of the Mathematical Association of America in the early 1950s, Mac Lane worked to update mathematics teaching. He helped establish the Committee on the Undergraduate Program in Mathematics, which focused in particular on improving the education of mathematics teachers. Also begun during his presidency were the Hedrick Lectures (a series of lectures accessible to those who teach college mathematics), the Employment Register and the Combined MAAAMS Membership list. Retiring from the presidency in December 1953, Mac Lane catalogued the many troubles besetting collegiate mathematics education and then addressed what he termed its “real objectives”: We must contrive ever anew to expose our students – be they general students or specialized students – to the beauty and excitement and relevance of mathematical ideas. We must set forth the extraordinary way in which mathematics, springing from the soil of basic human

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experience with numbers and data and space and motion, builds up a far-flung architectural structure composed of theorems which reveal insights into the reasons behind appearances and of concepts which relate totally disparate concrete ideas. (Mac Lane, 1954, p. 152)

He proceeded to describe in some detail the curriculum in mathematics at the University of Chicago as an illustration of the integrated structure he sought for college mathematics courses. Writing in the Bulletin of the National Association of Secondary-School Principals in May 1954, Mac Lane argued that “the lively modern development of mathematics has had no impact on the content or on the presentation of secondaryschool mathematics” (p. 66). He noted how the growing applications of mathematics in science and technology together with the severe shortage of new mathematical talent demanded both an improved school mathematics curriculum but also a new generation of teachers trained in modern mathematics. He proceeded to sketch some of the ideas of modern mathematics that should be dealt with in secondary school, concluding with a call for change: The mathematical subjects now taught [in secondary school] need drastic overhauling. It no longer suffices to continue to teach the old ideas; rather, one must start afresh to determine which ideas should be taught and in what perspective. In this way, school mathematics could become fully relevant to the modern world. (p. 70)

Mac Lane (1994) offered intuition, trial, error, speculation, conjecture, proof as the sequence in which we come to understand mathematics. His point was that although mathematicians may do mathematics in many different ways, the path to understanding it always ends in rigorous proof. In his own teaching and in his writings on education, he advocated careful attention to abstraction and proof treated carefully and in recognition of the soil from which mathematics springs.

Sources Kaplansky, Irving (ed.). 1979. Saunders Mac Lane: Selected papers. New  York & Berlin: Springer-Verlag. Birkhoff, Garrett and Saunders L.  Mac Lane. 1941. A survey of modern algebra. New  York: Macmillan. Mac Lane, Saunders L. and Garrett Birkhoff. 1967. Algebra. New York: Macmillan. Mac Lane, Saunders L. 1971. Categories for the working mathematician. Berlin: Springer-Verlag. Mac Lane, Saunders L. 1986. Mathematics: Form and function. Berlin: Springer-Verlag. Mac Lane, Saunders L. 1994. Response to “Theoretical mathematics: Toward a cultural synthesis of mathematics and theoretical physics”, by A. Jaffe and F. Quinn. Bulletin of the American Mathematical Society New Series, 30: 190–193. Mac Lane, Saunders L. 2005. A mathematical autobiography, Wellesley, MA: AK Peters. Alexanderson, Gerald L. and Saunders L. Mac Lane. 1989. A conversation with Saunders Mac Lane, College Mathematics Journal 20: 2–25. Boas, Ralph P.  Jr. 1975. Award for Distinguished Service to Professor Saunders Mac Lane. American Mathematical Monthly 82: 107–108.

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Mc Larty, Colin. 2005. Saunders Mac Lane (1909–2005): His mathematical life and philosophical works. Philosophia Mathematica 13(3): 237–251. Pearce, Jeremy. 2005. Saunders Mac Lane, 95, pioneer of algebra’s category theory. The New York Times A21.

Publications Related to Mathematics Education Mac Lane, Saunders L. 1954. Of course and courses. American Mathematical Monthly 61, 151–157. Mac Lane, Saunders L. 1954 (May). The impact of modern mathematics on secondary schools. Bulletin of the National Association of Secondary-School Principals 38: 66–70, reprinted in 1956 in the Mathematics Teacher 49: 66–69. Mac Lane, Saunders L. 1957. Algebra. In Insights into modern mathematics (23rd Yearbook of the National Council of Teachers of Mathematics, ed. F.L. Wren, 100–144. Washington, DC: The Council. Mac Lane, Saunders L. 1989. The education of Ph.D.s in mathematics. In A century of mathematics in America, ed. P. Duren, Part 3, 517–523. Providence, RI: American Mathematical Society.

Photo Author: Jacobs, Konrad. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

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Edwin Maxwell

11.35 Edwin Arthur Maxwell (Aberdeen, 1907 – Cambridge, 1897): Member of the Executive Committee 1952–1958, Secretary 1971–1974 Adrian Rice

Biography Edwin Arthur Maxwell was born in Aberdeen on 12 January 1907. After attending school and then university in his home city, he moved to Cambridge to study mathematics there. During the nineteenth century, and the early part of the twentieth century, due to the great influence and dominance of Cambridge mathematics, it was quite common for students who had already performed well in mathematics at other British institutions to subsequently enrol for a further degree at the University of Cambridge. This is what Maxwell did. Staying on to study for a doctorate under the supervision of Henry Frederick Baker, he was awarded a PhD in 1935 for a thesis entitled “An Examination of Particular Surfaces with Regard to Their Invariants”. He was quickly elected to a fellowship of Queens’ College, Cambridge, where he was to spend the rest of his career. During his time at Queens’, Maxwell served the College in a number of different capacities, first as the assistant director and then the director of studies in A. Rice (*) Randolph-Macon College, Ashland, VA, USA e-mail: [email protected]

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mathematics, and later as praelector, keeper of the records and junior and senior bursar. His output of original mathematics was quite small, but, inspired by his work with Baker, his chief interests were largely geometric. His main contribution was the development and use of general homogeneous coordinates in three-dimensional space. These coordinates had been originally invented, independently, by Möbius and Feuerbach in the nineteenth century, but in his Methods of Plane Projective Geometry (1946), Maxwell introduced them to a British undergraduate audience. The book was explicitly written “as a study of methods and not as a catalogue of theorems; and I hope that a student reading it will have nothing to unlearn as he proceeds to apply these methods to study the geometry of figures in three dimensions or in higher space” (Maxwell 1946, p. xiii). His General Homogeneous Coordinates in Space of Three Dimensions (1951) continued his work on the subject, using matrix algebra to provide clear proofs of standard results. During the 1940s and 1950s, his research-level interest in geometry gave way to the composition of originally presented geometric textbooks such as his Geometry for Advanced Pupils (1949), wherein “configurations” rather than theorems are taken as the fundamental component. These geometric textbooks were to be his most original contribution to mathematics. Maxwell also dabbled, successfully, in writing “popular” works on mathematics. The most famous, Fallacies in Mathematics (1959), successfully used humorous examples of fallacious reasoning to instil in its readers an appreciation of the importance of logical and mathematical rigour. As he wrote in the book’s preface, his aim was to instruct through entertainment: “The general theory is that a wrong idea may often be exposed more convincingly by following it to its absurd conclusion than by merely announcing the error and starting again. Thus a number of by-ways appear which, it is hoped, may amuse the professional, and help to tempt back to the subject those who thought they were losing interest” (Maxwell 1959, p. 7). After retiring from teaching in the early 1970s, he was made a Life Fellow of Queens’ College in 1974. He died in Cambridge on 27 August 1987.

Contribution to Mathematics Education Throughout his career, Maxwell’s prime mathematical activity was in the field of education. Indeed, not only did he love teaching, but “he had the gift of being able to adapt his approach to a wide range of audiences” (Quadling 1988, p. 51). His skill at mathematical exposition was reflected in his production of ten textbooks for various levels, some of which have already been mentioned. In addition to several books on his special interest of geometry, he also wrote a couple on abstract algebra and An Analytic Calculus for School and University (1954–1957), “a masterpiece of lucidity which takes the subject in four volumes from its beginnings into the realms of convergence and partial differential equations” (Quadling 1988, p. 51).

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As a mathematician with a keen interest in pedagogy, Maxwell was an active member of the Mathematical Association from the 1930s, becoming a frequent contributor to The Mathematical Gazette. During the 1950s, he became involved with educational developments overseas, serving on the Executive Committee of ICMI from 1952 to 1958 and as its secretary from 1971 to 1974, under the presidency of Sir James Lighthill. He was also one of the guest speakers to the OEEC/OECE Seminar at Royaumont in 1959, which resulted in the publication of New Thinking in School Mathematics (1961. Paris: OEEC). An active supporter of the School Mathematics Project in its early years, an SMP book on Geometry by Transformations (1975) was one of his final publications. In 1960, he was elected president of the Mathematical Association. His presidential address, “Pastors and Masters”, referred to the contemporary “crisis in the teaching of mathematics” (Maxwell 1961, p. 174) concerning the number of schoolteachers available to teach the subject. “In most parts of the world today there is a serious shortage of teachers”, he said. “This shortage is particularly acute in mathematics, and urgent steps are necessary if the shortage is not to become catastrophic” (Maxwell 1961, p. 168). Dividing human professions into those that deal with “body”, i.e. physical wellbeing, “mind” and “spirit”, Maxwell observed that those in the first category are almost always valued higher by society than those concerned with learning or spiritual matters. The upshot, he said, was that twentieth-century man, with all his progress and technical achievements, has reached an attitude that will give high reward to those who look after his physical well-being, much more grudging reward to those whose concern is his mind, and, to be honest, disgracefully low reward to those who care for his spirit. (Maxwell 1961, p. 170)

He acknowledged that the job satisfaction of many teachers is completely independent of financial remuneration and that those entering the profession are well aware that they could earn more money in other jobs of comparable difficulty. But, he said, “that is no reason why the community should not face its own responsibilities and insist that its pastors and masters are properly looked after” (Maxwell 1961, p. 170). This message, together with his overall theme, “the worthwhile-ness of teaching and the proper status of the profession” (Maxwell 1961, p. 174), remains as compelling today as when it was first delivered. Maxwell’s work for the Mathematical Association also included a term as editor of The Mathematical Gazette between 1963 and 1971, during which time “the journal earned a reputation for lively and relevant articles, and a view that mathematics could be serious without being solemn” (Quadling 1988, p. 52). He also took over as the Mathematical Association’s treasurer in 1976, when the previous incumbent died suddenly, serving in that position for 3 years. So strong was his affection for the Association that he even donated some of the royalties from his book on Fallacies in Mathematics in “gratitude for much that I have learned and for many friendships that I have made” (Quadling 1988, p. 52).

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Sources Maxwell, Edwin A. 1946. The methods of plane projective geometry based on the use of general homogenous coordinates. Cambridge: Cambridge University Press. Maxwell, Edwin A. 1949. Geometry for advanced pupils. London: Clarendon Press. Maxwell, Edwin A. 1951. General homogeneous coordinates in space of three dimensions. Cambridge: Cambridge University Press. Maxwell, Edwin A. 1954–57. An analytical calculus for school and university. 4 vols. Cambridge: Cambridge University Press. Maxwell, Edwin A. 1959. Fallacies in mathematics. Cambridge: Cambridge University Press. Maxwell, Edwin A. 1965. A gateway to abstract mathematics. Cambridge: Cambridge University Press. Maxwell, Edwin A. 1965. Algebraic structure and matrices [= Advanced algebra, pt. 2]. Cambridge: Cambridge University Press. Quadling, Douglas. 1988. Edwin Arthur Maxwell. The Mathematical Gazette 72: 51–52.

Publications Related to Mathematics Education Maxwell, Edwin A. 1961. Pastors and masters. The Mathematical Gazette 45: 167–174. Maxwell, Edwin A. 1975. Geometry by transformations. Cambridge: Cambridge University Press. Extracts from Professor Maxwell’s paper. In New thinking in School Mathematics, 83–90. Paris: OEEC. 1961.

Photo Source: Maxwell, Edwin A. 1961. Pastors and masters. The Mathematical Gazette 45: 167–174 (Photo at n.n.p.).

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11.36 Edwin Evariste Moise (New Orleans, 1918 – New York, 1998): Vice-President 1963–1970 Jeremy Kilpatrick

Biography Edwin Evariste Moise was born in New Orleans, Louisiana, on 22 December 1918. He received his BA from Tulane University in 1940. During World War II, he served in the US Navy as a Japanese translator and cryptanalyst in the Office of the Chief of Naval Operations. His doctoral work was done at the University of Texas under the direction of Robert L. Moore, the founder of topology in the United States and the inventor of the Moore method for teaching mathematical proof. Moise earned his PhD in 1947 with a dissertation on continuum theory in which he constructed the pseudo-arc, a term he coined, to solve an old problem posed by Bronislaw Knaster. From 1947 to 1960, Moise taught at the University of Michigan, where he rose to the rank of professor. He began his important work on 3-­manifolds at Michigan. That work culminated in his proof, completed at the Institute for Advanced Study, that every 3-manifold can be triangulated. From 1960 to 1971, Moise was James B.  Conant professor of education and mathematics at Harvard University. From

J. Kilpatrick (1935–2022) University of Georgia, Athens, GA, USA

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1971 to 1987, he held a distinguished professorship at Queens College of the City University of New York. After becoming an emeritus professor in 1987, he turned to the study of nineteenth-century English poetry. He was living in Manhattan when he died, on 25 December 1998, from complications following heart surgery. He was survived by his wife, two children and one grandchild.

Contribution to Mathematics Education Moise wrote a number of successful school and college mathematics textbooks and a treatise on geometric topology. In 1961, he was elected a fellow of the American Academy of Arts and Sciences. He served as vice-president of the American Mathematical Society in 1973–1974 and as president of the Mathematical Association of America in 1967–1968. From 1963 to 1970, he was vice-president of ICMI. Moise was a member of the geometry writing team when the School Mathematics Study Group, with a grant from the US National Science Foundation, started its work in the summer of 1958 at Yale University. The team produced a course outline and some sample pages for a tenth-grade geometry textbook. Moise took a leave of absence from the University of Michigan during the spring semester of 1959 to write the first draft of the textbook (Wooten, 1965, p. 55). In the summer of 1959, the SMSG writing teams met at the University of Colorado to complete a set of textbooks for trial in secondary schools, and they met during the summer of 1960 at Stanford University to revise the books in light of the tryout. Moise noted that SMSG had begun what he called a different sort of work. Its “crash program” is finished; and it is expected that its highschool books will be withdrawn from circulation in another two or three years, when similar books become available through commercial publishers. From now on, the main job of the SMSG will be long-­range experimentation with courses and programs that are not necessarily suitable for wide use in the near future. (Moise 1962, p. 90)

SMSG’s policy was to withdraw textbooks from circulation as soon as at least two comparable commercial versions were published. Moise and Floyd Downs, a high school teacher who was also a geometry-writing-team member, published a geometry textbook in 1964 that reflected the approach used in the SMSG Geometry and that eventually captured a large share of the market. A second such book, however, never appeared, so the SMSG textbook was not withdrawn from circulation. Writing about the new programmes in mathematics that were appearing during the early 1960s, Moise (Moise, Calandra, Davis, Kline and Bacon 1965) praised the textbooks that SMSG had produced: One thing was obvious … as soon as the books were written, and before they were tried: the improvement in intellectual content was so great that they surely would produce either an educational improvement or a collapse of classroom morale. The latter has not occurred; the new programs in general are far more popular than the old ones. (p. 3)

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Under Moise’s influence, the SMSG Geometry textbook had replaced Euclid’s postulates with metric postulates so as to take advantage of students’ knowledge of the real number system and the algebraic treatment of proportionality, an action very much supported by mathematicians such as Saunders Mac Lane (1959) but criticized by others such as Alexander Wittenberg (1963) and Morris Kline (1973). Moise (1962), however, remained satisfied with the progress he and his colleagues had made in reforming school mathematics: I cannot believe that anybody has found the final answer to any of our problems; I cannot even believe that such final answers exist. But the progress made in the past few years forms the basis of a long overdue revolution in mathematical education, and I am convinced that even better work is soon to come. (p. 100)

Sources Moise, Edwin E. 1952. Affine structures in 3-manifolds: V.  The triangulation theorem and Hauptvermutung. Annals of Mathematics s. 2, 56: 96–114. Moise, Edwin E. 1977. Geometric topology in dimensions 2 and 3. Berlin: Springer-Verlag. Moise, Edwin E. 1982. Introductory problem courses in analysis and topology. Berlin: Springer-Verlag. Fitzpatrick, Benjamin D. 2005, The legacy of R. L. Moore: The students of R. L. Moore. Retrieved on 2 June 2020 from the Legacy of the R. L. Moore Project web site: http://www.discovery. utexas.edu/rlm/reference/fitzpatrick.html Kline, Morris. 1973. Why Johnny can’t add: The failure of the new math. New  York: St. Martin’s Press. Mac Lane, Saunders L. 1959. Metric postulates for plane geometry. American Mathematical Monthly 66: 543–555. Saxon, W. 1998 (December 28). Edwin Evariste Moise, 79, mathematics scholar. The New York Times, B8. Wittenberg, Alexander I. 1963. Sampling a sample mathematical text. American Mathematical Monthly 70: 452–459. Wooten, William. 1965. SMSG: The making of a curriculum. New Haven, CT: Yale University Press.

Publications Related to Mathematics Education Halmos, Paul R., Edwin E. Moise, and George Piranian. 1975. The problem of learning to teach. American Mathematical Monthly 82: 466–474. Moise, Edwin E. 1960. The SMSG Geometry Program: A description of its development. Mathematics Teacher 53: 437–442. Moise, Edwin E. 1962. The new mathematics programs. School Review 70: 82–101, reprinted in 1964. In Modern viewpoints in the curriculum, ed. P.  C. Rosenbloom, 73–87. New  York: McGraw-Hill. Moise, Edwin E. 1963a. Elementary geometry from an advanced standpoint, Reading, MA: Addison-Wesley. Moise, Edwin E. 1963b. Some reflections on the teaching of area and volume. American Mathematical Monthly 70: 459–466. Moise, Edwin E. and Floyd L. Downs. 1964. Geometry. Reading, MA: Addison-Wesley.

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Moise, Edwin E. 1965. Activity and motivation in mathematics. American Mathematical Monthly 72: 407–412. Moise, Edwin E. 1975. The meaning of Euclidean geometry in school mathematics. Mathematics Teacher 68: 472–477. Moise, Edwin E. 1984. Mathematics, computation, and psychic intelligence. In Computers in mathematics education (1984 Yearbook of the National Council of Teachers of Mathematics, eds. Viggo P. Hansen and Marilyn J. Zweng, 35–42. Reston, VA: NCTM. Moise, Edwin E., Alexander Calandra, Robert B.  Davis, Morris Kline, and Harold M.  Bacon. 1965. Five views of the “new math” (Occasional Paper n. 8), Washington, DC: Council for Basic Education.

Photo Source: Wikimedia Commons.

552

Deane Montgomery

11.37 Deane Montgomery (Weaver, 1909 – Chapel Hill, 1992): Ex Officio Member of the Executive Committee 1975–1978 Jeremy Kilpatrick

Biography Deane Montgomery was born in Weaver, Minnesota, on 2 September 1909 and grew up on a farm. He went to Hamline University, Saint Paul, Minnesota, the first institution of higher education in Minnesota. He received a BS degree from Hamline in 1929 and then went to the University of Iowa, where he received his MS in 1930. He obtained his PhD there in 1933 under the direction of Edward W. Chittenden. Montgomery’s thesis was on point-set topology. He then went to Harvard University, where he was a National Research Council (NRC) fellow in 1933–1934. The following year, he was again an NRC fellow, this time at Princeton University, where the Institute for Advanced Study had just been established. In 1935, Montgomery was appointed assistant professor at Smith College. He was promoted to associate professor in 1938 and then full professor in 1941. J. Kilpatrick (1935–2022) University of Georgia, Athens, GA, USA

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In 1941–1942, he was a Guggenheim fellow at the Institute for Advanced Study. After a year back at Smith College in 1942, he returned to Princeton in 1943 to teach Army students. He undertook other war work at that time, working with John von Neumann on numerical analysis. In 1946, he left Smith to accept an appointment as associate professor at Yale University but then returned to the Institute for Advanced Study in 1948 as a permanent member, becoming a professor in 1951. At the Institute, Montgomery was at the centre of activity in topology, running the topology seminar for many years. Early in his career, his research interests had shifted from point-set topology to topological transformation groups. For example, in one of his early papers (Montgomery 1937), he proved that a pointwise periodic self-map of a manifold is periodic. In collaboration with Leo Zippin, he published a series of papers on the linearizability of group actions on low-dimensional manifolds (e.g. Montgomery and Zippin 1940). Montgomery and Zippin were especially interested in Hilbert’s fifth problem: A connected locally compact group G is a projective limit of a sequence of Lie groups; and, if G has no small subgroups, then it is a Lie group.

In 1948, Montgomery and Zippin solved the problem for three dimensions and by 1952 had solved it under the assumption of finite dimensionality. Later that year, Montgomery’s assistant Hidehiko Yamabe was able to remove that restriction, and in 1955, in collaboration with Zippin, Montgomery published the monograph Topological Transformation Groups. He continued to work on transformation groups, studying with a variety of collaborators such groups and their relations to surgery theory. Until he retired in 1980, Montgomery remained at the Institute. In 1988, he and his wife moved to Chapel Hill, North Carolina, to be near their daughter and their two granddaughters, all of whom survived him when he died on 15 March 1992. Montgomery received honorary doctorates from Hamline University (1954), Yeshiva University (1961), Tulane University (1967), the University of Illinois (1977) and the University of Michigan (1986). He was vice-president of the American Mathematical Society in 1952–1953 and president in 1961–1962. In 1988, he received the Leroy P.  Steele Prize for Lifetime Achievement from the American Mathematical Society for his lasting impact on mathematics, particularly mathematics in the United States. In 1955, he was elected a member of the US National Academy of Sciences and in 1958 was elected to the American Academy of Arts and Sciences.

Contribution to Mathematics Education Montgomery served on the Executive Committee of the International Mathematical Union (IMU) from 1963 to 1966. He was vice-­president of IMU from 1967 to 1970 and president from 1975 to 1978. From 1975 to 1978, as IMU president, he was also an ex officio member of the Executive Committee of the International Commission on Mathematical Instruction.

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As a teacher and mentor, Montgomery was known for his efforts to help young topologists and for his interest in the careers of students from graduate programmes at smaller institutions. He would seek out and encourage visitors to the Institute, particularly young mathematicians, conducting seminars in his office to acquaint them with recent developments in the field. Montgomery was one of 65 mathematicians who signed the 1962 article, “On the Mathematics Curriculum of the High School”, usually attributed to Lipman Bers, Morris Kline, George Pólya and Max Schiffer (with Kline as a main inspirator), that criticized some of the new math curriculum efforts for being too abstract and formal: There are several levels of rigor. –they write – The student should learn to appreciate, to find and to criticize proofs on the level corresponding to his experience and back ground. If pushed prematurely to a too formal level he may get discouraged and disgusted. Moreover the feeling for rigor can be much better learned from examples wherein the proof settles genuine difficult ties than from hairsplitting or endless harping on trivialities. (p. 190)

The article offered “fundamental principles and practical guidelines” for reforming the school mathematics curriculum, arguing that anyone attempting reform needed to link school mathematics more closely to its history and to concrete applications so that future non-mathematicians would not be turned away. He was not one of the writers of the article, but he obviously supported the argument.

Sources Bibliography of Deane Montgomery. 1985. Contemporary Mathematics 36: 13–16. Montgomery, Deane. 1937. Pointwise periodic homeomorphisms. American Journal of Mathematics: 59: 118–120. Montgomery, Deane and Leo Zippin. 1940. Topological group foundations of rigid space geometry. Transactions of the American Mathematical Society 48: 21–49. Montgomery, Deane and Leo Zippin. 1955. Topological transformation groups. New  York: Interscience. Borel, Armand. 1992. Deane Montgomery (1909–1992). Notices of the American Mathematical Society 39(7): 684–686, reprinted in 1993 in the Proceedings of the American Philosophical Society 137(3): 452–456. Fintushel, Ronald. 2005 (March). A tribute to Deane Montgomery. Notices of the American Mathematical Society 52(3): 348–349. 1988 Steele Prizes awarded at centennial celebration in Providence. 1988. Notices of the American Mathematical Society 35(7): 965–970. Raymond, Frank and Reinhard Schultz. 1985. The work and influence of Deane Montgomery. Contemporary Mathematics 36: 1–11. Saxon, Wolfgang. 1992 (March 18). Deane Montgomery is dead at 82; taught theoretical mathematics. The New York Times, p. D22.

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Publications Related to Mathematics Education On the mathematics curriculum of the high school. 1962. American Mathematical Monthly 69: 189–193. This paper also appeared in 1962. Mathematics Teacher 55: 191–195.

Photo Author: Jacobs, Konrad. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

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Jürgen Kurt Moser

11.38 Jürgen Moser (Königsberg, Now Kaliningrad, 1928 – Zurich, 1999): Ex Officio Member 1983–1986 Giorgio T. Bagni

Biography Jürgen Kurt Moser was born on 4 July 1928 in Königsberg, then an eastern outpost of Germany, now the Russian city of Kaliningrad. He attended the gymnasium in Königsberg, but when he was 15 years old, he was forced into a military auxiliary force. After the end of the war, he studied at the University of Göttingen. In Göttingen, Moser studied the spectral theory of differential equations with the South-Tyrolian mathematician Franz Rellich (1906–1955). Moreover, Carl Ludwig Siegel (1896–1981), who returned to Göttingen in 1951, became a major influence on Moser, who acquired an interest in astronomy and number theory through him. In 1952, Moser was awarded his doctorate from Göttingen University. Moser remained in Göttingen until 1953 when he went to the United States to spend a year at New York University on a Fulbright Scholarship. After working for a year at the Courant Institute, he returned to Germany, where he was an assistant to Siegel at Göttingen University in the academic year 1954–1955. During this period, G. T. Bagni (1958–2009) University of Udine, Udine, Italy

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he took notes of Siegel’s lectures, which first became the basis for a book by Siegel published in 1956, then later became the basis of a joint book Lectures on Celestial Mechanics (1971). In 1955, Moser emigrated to the United States, where he married Gertrude Courant on 10 September 1955. Moser was appointed to New York University as an assistant professor in 1955. Then he went to the Massachusetts Institute of Technology in 1957. He spent 1961 as a Sloan Fellow and, since 1967 to 1970, he was director of the Courant Institute. In 1980, he left the United States and accepted a position at the Eidgenössische Technische Hochschule (ETH) in Zurich, where he was director of its Research Institute for Mathematics from 1984 to 1995, when he was made professor emeritus. Moser’s mathematical interests included the theory of partial differential equations, spectral theory, dynamical systems, complete integrability, differential geometry and complex analysis (Lax 2002). He introduced some techniques which could be applied to almost any dynamical system of Hamiltonian type and the “Moser twist stability theorem”. When combined with the work of Vladimir Igorevich Arnold (born in 1937), this led to the so-called KAM theory. Based on ideas by Andrei Nikolaevich Kolmogorov (1903–1987), this important theory provided a new approach to stability problems in celestial mechanics. Moser wrote several books, including Lectures on Hamiltonian Systems (1968) which examines problems of the stability of solutions, the convergence of power series expansions and integrals for Hamiltonian systems near a critical point, and Stable and Random Motions in Dynamical Systems (1973, reprinted in 2001) which describes how stable behaviour and statistical behaviour take place together in analytic conservative systems of differential equations. Integrable Hamiltonian Systems and Spectral Theory (1983) arises from a course of lectures which Moser gave at the Scuola Normale Superiore in Pisa (1981). A course of lectures that Moser gave at ETH in the spring of 1988 became the basis for Selected Chapters in the Calculus of Variations (2003). In 1979–1980, Moser and the Swiss mathematician Eduard Zehnder (born in 1940) began to write a book on Hamiltonian dynamical systems, but never finished, only writing the first three of five planned chapters. These three chapters were published in 2005. Moser was invited to give the Gibbs lecture of the American Mathematical Society in Dallas in 1973, the Pauli lectures at ETH in 1975, the American Mathematical Society Colloquium lectures in Toronto in 1976, the Hardy lectures in Cambridge in 1977, the Fermi lectures in Pisa in 1981 and the John von Neumann Lecture of the Society for Industrial and Applied Mathematics in Seattle in 1984. Elected to the National Academy of Sciences in 1973, he had been awarded its Craig Watson Medal in 1969 for his fundamental contributions to dynamic astronomy. He was president of the International Mathematical Union and member ex officio of the ICMI Executive Committee, from 1983 to 1986, and was awarded many prizes, including AMS/SIAM Birkhoff Prize in Applied Mathematics (1968), Dutch Mathematical Society Brouwer Medal (1984), DVR Medal (1992), Wolf Prize (1994–1995) and London Mathematical Society Honorary Membership (1996). In

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1998, he was invited to deliver a plenary lecture at the International Congress of Mathematicians in Berlin. Moser, “one of the leading mathematicians of the post-war era”, in Konrad Osterwalder’s words, a colleague at Swiss Federal Institute of Technology, died on Friday 19 December 1999 in Zurich, Switzerland.

Contribution to Mathematics Education The short biographical notes reported above show that Moser’s professional life was completely dedicated to the academic career and mathematical research. Nevertheless, we would like to quote the testimonies of his friends or pupils that show his great commitment to educating his mathematics students and researchers. Louis Nirenberg wrote: Moser was one of the most profound analysts of the last half century. His work ranged over different fields of mathematics, both pure and applied. Much of his deepest work was concerned with dynamical systems, especially small divisor problems and relations with celestial mechanics. He also did fundamental work in functional analysis  - the Nash-Moser theory - and partial differential equations, and he made deep contributions in completely integrable systems, geometry, and complex analysis. Moser was also a master of exposition. His papers and published lectures are beautifully written. (Mather, McKean, Nirenberg and Rabinowitz 2000, p. 1398)

The words of his student Paul H. Rabinowitz evidence the esteem and affection he enjoyed in the mathematical community: He was a lifelong music lover - he played the cello - and amateur astronomer. He enjoyed biking and swimming. At the age of sixty, he took up paragliding. With his untimely death at seventy-one, many of us in the mathematics community lost a hero, friend, and mentor. (Ibidem, 1393)

Paul H. Rabinowitz wrote: To those who knew him Moser exemplified a creative scientist and, perhaps even more important, a human being. His standards were high and his taste impeccable. His papers were elegantly written. Not merely focused on his own research, he worked successfully for the well-being of mathematics in many ways. He stimulated several generations of younger people by his penetrating insights into their problems, scientific and otherwise, and by his warm and wise counsel, concern, and encouragement. My experience as his student was typical: then and afterwards I was made to feel like a member of his family. (Ibidem, pp. 1392–1393)

Sources Moser, Jürgen Kurt. 2009. Brief biography and list of publications. Nelineinaya Dinamika 5(1): 5–10. Moser, Jürgen Kurt. 1968. Lectures on Hamiltonian Systems 61:1–60.

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Moser, Jürgen Kurt and Carl Ludwig Siegel. 1971. Lectures on celestial mechanics. New York: Springer. Moser, Jürgen Kurt. 1983. Integrable Hamiltonian Systems and Spectral theory (Lezioni Fermiane, Pisa 1981). Pisa: Publications of the Scuola Normale Superiore. Moser, Jürgen Kurt. 2003. Selected chapters in the calculus of variations. Lectures in Mathematics. ETH Zürich. Basel: Birkhäuser. Moser, Jürgen Kurt. 1973. Stable and Random Motions in Dynamical Systems: With Special Emphasis on Celestial Mechanics. Princeton: Princeton University Press. AA. VV. 2003. Special issue dedicated to the memory of Jürgen K. Moser. Communications on Pure and Applied Mathematics 56(7): i–ii; 813–1028. AA. VV. 2003. Special issue dedicated to the memory of Jürgen K. Moser. Communications on Pure and Applied Mathematics 56(8): i–ii; 1029–1245. Arnold, Vladimir I. 2000. J.  Moser (1928–1999). Déclin des mathématiques (après la mort de Jürgen Moser). La Gazette des Mathématiciens 84: 92–95. Giorgilli, Antonio. 2000, A tribute to Jürgen Moser (1928–1999). Celestial Mechanics and Dynamical Astronomy 77(3): 153–155. Hasselblatt, Boris and Anatole Katok. 2002. The development of dynamics in the 20th century and the contribution of Jürgen Moser. Ergodic Theory and Dynamical Systems 22(5): 1343–1364. Lax, Peter D. 2002. Jürgen Moser. 1928–1999. Ergodic Theory and Dynamical Systems 22(5): 1337–1342. Mather, John N., Henry P.  Mckean, Louis Nirenberg, and Paul H.  Rabinowitz. 2000. Jürgen K. Moser (1928–1999). Notices of the American Mathematical Society 47(11):1392–1405. Zehnder, Eduard J. 1993. Cantor-Medaille für Jürgen Moser. Jahresbericht der Deutschen Mathematiker-Vereinigung 95(2): 85–94. Zehnder, Eduard J. 1994. Hommage à Jürgen Moser. La Gazette des Mathématiciens 59: 57–66.

Photo Author: Fischer, Gerd. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

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Bernhard Hermann Neumann

11.39 Bernhard Hermann Neumann (Berlin, 1909 – Canberra, 2002): Member of the Executive Committee 1979–1982 Michela Malpangotto

Biography Bernhard Hermann was born in Berlin on 15 October 1909. After studying at the Herderschule in Berlin and at the University of Freiburg, he obtained his doctorate in philosophy from the University of Berlin in 1932 with a dissertation developed with the support of Issai Schur and devoted to what was later called the wreath product of groups (now known as combinatorial group theory). Being Jewish, he emigrated to Britain in 1933 and continued his studies at Cambridge University, where he was awarded a second PhD in 1935 for a dissertation in which he laid the foundations for the theory of varieties of groups. His university career started as a temporary assistant lecturer in mathematics at Cardiff, where he spent 3  years. He was there in June 1940 when World War II started. Now Neumann’s position as a German in England became a difficult one. M. Malpangotto (*) CNRS, Centre Jean Pépin, Ecole Normale Supérieure-Ulm, Paris, France e-mail: [email protected]

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He was briefly interned as an enemy alien; upon his release, he joined the British Army, serving in the Intelligence Corps from 1940 to 1945. In 1946, he was appointed as a lecturer at the University of Hull, and in 1948, he accepted a position at the University of Manchester, where he stayed for the next 13 years. He served on the Council of the London Mathematical Society from 1954 to 1961 and was its vice-president from 1957 to 1959. In 1959, he was elected fellow of the Royal Society of London for his research in abstract algebra, and in particular for his numerous and influential contributions to the theory of infinite groups. In 1962, Neumann accepted an invitation to set up a research department of mathematics in the Institute of Advanced Studies at the Australian National University. He was appointed as professor and head of the department. From the first beginning, Neumann was strongly engaged in improving and promoting Australian mathematics. The first two international conferences on a mathematics topic in Australia in 1965 and 1973 were held under his leadership and devoted to the theory of groups. Both were notable for the quality of the main speakers and the range of countries from which they came. The first was also notable because Neumann was able to arrange for young people from overseas to earn their way by teaching at a university in Australia. A third international conference on the theory of groups was held in 1989 to mark his 80th birthday. Neumann was also active in mathematical circles in Australia. He served on the Council of the Australian Mathematical Society for over 15 years, including three terms as its vice-president and one term as its president (1964–1966), and in 1969 he founded the Bulletin of the Australian Mathematical Society, acting as its editor from then until 1979. He was elected an honorary member of the Australian Mathematical Society in 1981 and was further honoured by having a prize named after him for the most outstanding talk by a student at the Annual Meeting of the Society. He was chairman of the National Committee for Mathematics of the Australian Academy of Science from 1966 to 1975 and a member of the Council of the Australian Academy of Science from 1968 to 1971. On retiring as professor and head of the Department of Mathematics at the end of 1974, he was made professor emeritus and an honorary fellow of Australian National University. Even following his retirement, he has maintained a close association with organizations such as these. In 1994, he was made a Companion of the Order of Australia for service to the advancement of research and teaching in mathematics. He was awarded honorary doctorates from a number of universities: the University of Newcastle (New South Wales), Monash University, the University of Western Australia, the University of Hull, the ANU, Waterloo University and the Humboldt University of Berlin. He was also appointed a senior research fellow at CSIRO for 3 years and then became an honorary research fellow, reappointed annually until his death on 21 October 2002. Neumann’s contributions to the mathematical community during his long career are remarkable. He has published over 100 research papers, supervised and gave

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valuable advice to numerous students and fellow workers and lectured at many conferences and in universities all around the world. Some of Neumann’s contributions to mathematics are recognized because they bear his name, such as the Higman-Neumann-­Neumann construction for groups, the Douglas-Neumann theorem in geometry and the Mal’cev-Neumann construction for division rings.

Contribution to Mathematics Education Neumann was actively involved in the work of ICMI. He was the Australian representative to ICMI from 1968 to 1975, a member at large of ICMI from 1975 to 1982 and a member of the ICMI Executive Committee from 1979 to 1982. According to Newman (2002), he attended seven ICMEs, from the first held in Lyon in 1969 through to the seventh held in Québec in 1992.107 He was a member of the International Program Committee for ICME-3 (Karlsruhe, 1976) and ICME-5 (Adelaide, 1984). In 1972, he chaired the Working Group entitled “Mathematics for Specialists at University and College Level” (see Howson 1973, p. 301). It was in Manchester (1948) that he started his active involvement with mathematics education and more broadly through the Manchester Branch of the Mathematical Association of Great Britain. In 1960, Neumann was invited to become the foundation professor of mathematics at the Australian National University with the special goal of establishing a PhD programme. He held this position from 1962 until his retirement in 1974. Within days of his permanent arrival on 2 October 1962, Neumann became involved in activities supporting the teaching of mathematics in schools. The following year, he played a key role in forming a local association of teachers of mathematics, the Canberra Mathematical Association, and became its first president. He was part of a group which met in Adelaide in 1964 and decided to push for the formation of a national association. The Australian Association of Mathematics Teachers (AAMT) was founded in 1966 with Neumann as its first president. He became a regular participant in AAMT biennial conferences. “Indeed, the memory most members have of him is that of a keen auditor, and his making pertinent contributions to subsequent discussion” (Neumann 2002, p. 60). He served an extended term on the National Committee for Mathematics (1963–1975) and as part of Australian delegations to many meetings of the International Mathematical Union (IMU), held in conjunction with International Congresses of Mathematicians (ICMS): he attended 13 of these congresses. Neumann also served on the Exchange Commission of the IMU (1975–1979), and he continued to serve the worldwide mathematical community with his regular edition of the IMU Canberra Circular.

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 For some of these ICMEs, official lists of participants do not exist.

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Neumann was present at the Lyon meeting. He acted as co-chairman with Iyanaga of the ICMI-JSME International Conference of Mathematics Education (Tokyo, 5–6 November 1974) and delivered a talk entitled “Teaching Teachers of Teachers”. In 1978, he chaired the panel discussion at the ICMI Symposium on the education of mathematics teachers at the ICM held in Helsinki. His visibility at the early ICMEs led to the suggestion (at Karlsruhe) that Australia consider hosting a forthcoming ICME (Hilton 1981, p. 350). He and other members of AAMT were able to respond with enthusiasm, and intensive work led to the successful bid to hold the 1984 Congress in Adelaide. There followed an unprecedented period of cooperative work across all sectors of mathematics education and mathematics, involving many Australians and a number of significant international figures, which resulted in the extremely successful Adelaide congress and much greater ongoing international recognition for the work of Australian mathematics educators. In 1975, he also became an honorary member of the Canberra Mathematical Association, the Australian Association of Mathematics Teachers and the New Zealand Mathematical Society. In “retirement” he gave considerable encouragement to Peter O’Halloran and his colleagues involved in the formative stages of what is now the internationally known Australian Mathematics Competition and maintained an active interest in it. Through the ICMI, he also became involved in mathematical Olympiad activities, chairing the Australian Mathematical Olympiad Committee from its inception in 1980 until 1986. A better structure and operation of the International Mathematical Olympiads resulted from a Site Committee he chaired (1981–1983). The holding of the 1988 International Mathematical Olympiads in Australia, in the bicentenary year, owed a great deal to him (see Newman 2002, p. 61).

Sources Neumann, Bernhard and Hanna Neumann. 1988. Selected works of B H Neumann and Hanna Neumann, eds. D. S. Meek and Ralph G. Stanton. Winnipeg: Charles Babbage Research Center. Neumann, Bernhard H. 1973. Byron’s daughter. The Mathematical Gazette, 57(400), 94–97. Neumann, Bernhard H. 2003. [Review] Notable women in mathematics, a biographical dictionary, edited by Charlene Morrow and Teri Perl. 1998. (Greenwood Press, Westport, Connecticut).  – Women becoming mathematicians. Creating a professional identity in post World War II America, by Margaret A.  M. Murray. 2000. (The MIT Press, Cambridge, Massachusetts). The Mathematical Gazette, 87(508), 183–184. Hilton, Peter John. 1981. International Commission on Mathematical Instruction, L’Enseignement Mathématique s. 2, 27: 347–350. Howson, Albert Geoffrey. 1973. Developments in mathematics education, Cambridge: Cambridge University Press. Proceedings of ICMI-JSME Regional Conference on Curriculum and Teacher Training for Mathematical Education, Tokyo, November 5–9, 1974. Tokyo: National Institute for Educational Research.

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Conder, Marston Donald Edward. 1989. Bernhard Hermann Neumann. New Zealand Mathematical Society Newsletter 47: 20–22. Crompton, Bob. 1998. Professor Bernhard Neumann (1909–2002) mathematician, interviewed by Professor Bob Crompton in 1998. Retrieved on 28 May 2020 from https://www.science.org.au/ learning/general-­a udience/history/interviews-­a ustralian-­s cientists/professor-­b ernhard­neumann-­1909 Newman, Michael F. 2002. In memoriam of Bernhard Neumann (1909–2002). ICMI Bulletin 51: 59–62. Praeger, Cheryl E. 2010. Bernhard Hermann Neumann 1909–2002. Historical Records of Australian Science 21: 253–282.

Publications Related to Mathematics Education Neumann, Bernhard H. 1952. (Mathematical notes) 2276. On some interesting sets of circles. The Mathematical Gazette 36(316): 121–122. Neumann, Bernhard H. 2000. [Review] Memoirs of a maverick mathematician, by Zoltan Paul Dienes. 1999 (Minerva Press Atlanta, London, Sydney). The Mathematical Gazette 84(500): 348–350. Neumann, Bernhard H. 2001. Calls from the past, Zoltan Paul Dienes. 2000. (Minerva Press). The Mathematical Gazette 85(504): 534.

Photo Author: Jacobs, Konrad. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

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11.40 Rolf Nevanlinna (Joensuu, 1895 – Helsinki, 1980): Ex Officio Member 1959–1962 Osmo Pekonen

Biography Rolf Herman Nevanlinna was born on 22 October 1895 in Joensuu, in the Grand Duchy of Finland, which was then part of the Russian Empire. The extraordinary talent of the Nevanlinna family (earlier also called Neovius) has abounded through centuries. The Neovius-Nevanlinna lineage has produced dozens of mathematicians and schoolteachers of mathematical sciences, several of whom have also been known as authors of successful mathematics textbooks. Rolf Nevanlinna’s interest in the teaching of mathematics certainly stems from the family legacy. Rolf Nevanlinna’s grandfather, Major General Edvard Engelbert Neovius, taught mathematics and topography at a Russian military academy and conceived a pioneering approach to SETI (Search for Extraterrestrial Intelligence). Rolf’s father, Otto Neovius (from 1906 Nevanlinna), was a physicist and mathematician. He married Margarete Romberg, daughter of Herman Romberg, the German chief astronomer of the Pulkovo Observatory in St. Petersburg, so that Rolf was of German, and indeed Prussian, descent on his mother’s side. Rolf’s elder brother Frithiof also O. Pekonen (1960-2022) University of Jyväskylä, Jyväskylä, Finland

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became a well-known mathematician. Moreover, Otto Nevanlinna’s cousin was Ernst Lindelöf, an eminent professor of mathematics at the University of Helsinki and the would-be mentor of both Rolf and Frithiof. The young doctor Otto started his career as a schoolteacher of physics and mathematics at Joensuu, a rural town far away from the capital. In 1903, the family, with four children, returned to Helsinki where Rolf entered a grammar school which emphasized Greek and Latin. He was hesitant at first as to whether he should pursue classical studies at the university, but Ernst Lindelöf’s example soon convinced him of his true calling, which was to be analysis. In December 1917, in the aftermath of the Russian revolution, Finland declared independence. The following spring, the country was torn by a bloody Civil War which opposed Red and White Guards. The Reds were supported by revolutionary Russia, whereas the Whites received military assistance from Germany. The young Rolf joined the White Guards but did not see much military action. Frithiof Nevanlinna received his doctorate in 1918, and his brother Rolf received his in 1919, both in the field of complex analysis. Immediately after completing his dissertation, Rolf married his cousin Mary Selin, but this marriage ended in divorce. In 1958, he married his second wife Sinikka Kallio. He had five children, four from Mary and one from Sinikka. Nevanlinna established his worldwide fame as a leading complex analyst during a 3-year period of intensive creation in 1922–1925. Together with Frithiof, he created the “Nevanlinna theory”, or the value distribution theory of meromorphic functions. Despite their eminent research contributions, neither of the Nevanlinna brothers held an academic position. Rolf was a schoolteacher of mathematics at a school for about 20 hours a week, whereas Frithiof was employed by an insurance company where he later hired Rolf as well. In his memoirs, Nevanlinna states that his school teaching took up most of his time but the evenings, weekends and holidays were devoted to research. In 1926, Nevanlinna, then 31, finally received a professorship at the University of Helsinki. His best student, Lars Ahlfors, created a sensation in the mathematical world in 1928 when he proved, at the age of 21, the Denjoy hypothesis. Ahlfors was awarded the Fields Medal in 1936. Nevanlinna visited Zürich and Paris together with Ahlfors. During the term 1936–1937, he visited Göttingen, where Oswald Teichmüller attended his lectures. The so-called Winter War broke out when the Soviet Union invaded Finland in November 1939. A curious episode of this period is André Weil’s visit to Rolf Nevanlinna. Weil, who carried visiting cards signed N. Bourbaki, was suspected of being a Soviet spy and was arrested by the military police. After Nevanlinna and other mathematicians had testified in his favour, Weil was released and expelled to Sweden. During the war years, Nevanlinna did research in ballistics for the improvement of field artillery firing tables. He was awarded the Cross of Liberty, Second Class, for his services. Finland having mounted a heroic resistance to the Red Army, a precarious truce was signed between Finland and the Soviet Union in March 1940. Faithful to his Prussian origins and a certain military aspect of his upbringing, Rolf Nevanlinna played a political role during the truce, campaigning for a

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Finnish-German alliance in case of a new war. In 1941, the year of Operation Barbarossa, which Finland joined, Rolf Nevanlinna was elected rector of the University of Helsinki, an eminent public position which marked the end of his most fruitful research period. On the insistence of Germany, Finland had to recruit a volunteer corps for the German armed forces, which, moreover, was to be incorporated into the SS and not into the Wehrmacht, as the Finns would have preferred. The issue was politically delicate because, on the diplomatic front, the Finnish government – which remained democratic throughout the war – did everything possible to avoid full collaboration with the Nazis, aiming to wage a separate war with the Soviet Union, but not with the Western allies. A suitable private person was needed to handle the recruitment of the politically burdensome SS troops. The Finnish government decided to put Rolf Nevanlinna’s well-known German sympathies to good use. In 1942, at the request of the Foreign Minister, Nevanlinna made himself available as chairman of the Finnish SS Volunteer Committee. In this role, Nevanlinna rubbed shoulders with SS brass and was awarded the German Eagle with diamonds. Having successfully opposed an armed resistance to the Soviet onslaught in the summer of 1944, Finland opted out of the World War and again signed a truce with the Soviet Union in September 1944, losing some territories. At this point, the political winds turned and Nevanlinna was portrayed as a Nazi collaborator by the Finnish left-wingers. On the Prime Minister’s advice, he had to step down from the Rector’s office. He accepted a position at the University of Zürich instead, occupying a chair where he succeeded his former student Ahlfors, who had moved on to Harvard. Official Finland did not abandon Nevanlinna, however. When the Academy of Finland was founded in 1948, Nevanlinna was elected a member despite fierce opposition from leftist quarters. The government justified the election pointing out that Nevanlinna’s wartime action had been in accordance with Finland’s official policy and the interests of the state. In particular, anti-Semitism had not been part of Nevanlinna’s pro-German ideology. The Soviets were quick to rehabilitate Nevanlinna; they did not try to block his election to the presidency of the International Mathematical Union (IMU) for the 4-year term 1959–1962. Nevanlinna was the president of the International Congress of Mathematicians (ICM) held in Stockholm in 1962, and he also chaired the programme committee of the Moscow ICM of 1966. Nevanlinna mainly stayed in Zürich until his retirement in 1963. For decades after the war, most new Finnish doctors in mathematics were his academic descendants, and many of them came to stay in Zürich. After his retirement, Nevanlinna continued to play a public role. He was, for instance, chancellor of the University of Turku from 1965 to 1970. He played a certain political role in the massive enlargement of tertiary education in Finland after the war. Nevanlinna was the honorary president of the Helsinki ICM of 1978. He passed away in Helsinki on 28 May 1980.

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In the ICM of Warsaw in 1983, the IMU established the Rolf Nevanlinna Prize to recognize work on mathematical aspects of information science.108 The FinnishSwiss Rolf Nevanlinna Colloquia also perpetuate his memory.

Contribution to Mathematics Education In his quality of president of the IMU, Rolf Nevanlinna served as an ex officio member of the Executive Committee of the ICMI in 1959–1962. As we have earlier stated, teaching of mathematics was close to Nevanlinna’s heart because of his family’s long-­standing commitment to this field. He was known as an inspiring lecturer, and he had a special ability to convey the sense of aesthetics of the mathematical experience. In an increasingly specialized academic world, Nevanlinna insisted much on the unity of Western culture and the usefulness of practising arts and sciences together. “There are no Two Cultures; there is only one”, he commented on C.  P. Snow’s essay titled The Two Cultures. His public lectures on topics such as general relativity or the foundations of geometry were sometimes attended by cultural figures, including poets and composers. Nevanlinna often cited the view of Jacob Burckhardt, who claimed that, besides a civilized person’s main profession, his life should involve another interest that was more than just a hobby. In the case of Nevanlinna, music was such a passion. He was an amateur of the violin and a somewhat fanatic aficionado of Jean Sibelius’s music; if he was not happy with the performance of an orchestra, he would go and tell the conductors. He also served as the chairman of the board of Sibelius Academy. Nevanlinna was first of all a research mathematician. He wrote a number of textbooks but did not publish theoretical reflections on mathematics education. Occasionally, he addressed the annual meeting of the Finnish Union of Teachers of Mathematical Sciences. At the advent of the new math movement, Nevanlinna took a conservative stand. One of his addresses was widely noticed when it appeared successively in Swedish, German, English and Russian (Nevanlinna 1966). Nevanlinna pleaded for the maintenance of Euclidean geometry in the school curriculum, in opposition to the ravages of Bourbaki-inspired enthusiasm about set theory: We can learn at least one thing from all that has happened in mathematics during these decades of vast expansion: this period proves, if anything, the great significance, both in principle and substance, of the Euclidean elementary system. This theory has always stood out as the ideal of exact science. But the present century has experienced a peculiar revival of this basic mathematical theory, which has shown enormous vitality in two different main directions: it has given impetus to the revolution in mathematics represented by the break-

 In 2018, the General Assembly of the IMU decided to remove the name of Rolf Nevanlinna from the prize. In 2019, the Executive Committee of IMU agreed that the prize would be named the IMU Abacus Medal (see https://www.mathunion.org/imu-awards/rolf-nevanlinna-prize) 108

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through of the axiomatic mode of thought, and this has had fruitful effects in the most varied branches of mathematical and logical research. But it has also, in a more substantial respect, undergone an unforeseen revival through the generalizations it has allowed in infinite-dimensional Euclidean geometry, Hilbert-space and its applications. In view of all this it is amazing that a group of enthusiasts supporting a radical reform of school mathematics have mustered around the slogan ‘Down with Euclid!’ How can anything as narrow as this ever be suggested and, furthermore, by people who without doubt deserve credit for notable contributions to contemporary mathematical science? The matter cannot be dismissed merely as a joke in bad taste, even on the assumption that the slogan has been coined mainly with a view to creating a sensation. (Nevanlinna 1966, p. 456)

Politically, Nevanlinna was a staunch conservative and a right-winger throughout his life. When student radicalism spread to the campuses in the 1960s, Nevanlinna developed a grudge against the “abstract theoreticians of pedagogy” who had allowed this to happen. His constructive response to the demands of the youth movement was the creation of the so-called Presidential Working Group in Finland. Under the leadership of President Urho Kekkonen, this working group prepared a famous report, on the basis of which a law for the development of Finnish tertiary education was enacted in 1966. The new legislation was far-reaching in its effects, leading to the creation of several new universities. Since the 1950s, Nevanlinna also was one of those who advocated equipping the Finnish universities with computers, thus heralding the birth of Information Society whose successful champion today’s Finland has become.

Sources Louhivaara, Ilppo Simo. 1976. List of Rolf Nevanlinna’s publications. Annales Academiæ Scientiarum Fennicæ s. A I Mathematica 2: vii–xxv. Louhivaara, Ilppo Simo. 1982. List of Rolf Nevanlinna’s publications, Annales Academiæ Scientiarum Fennicæ s. A I Mathematica 7(1): 111–112. Nevanlinna, Rolf. 1925. Zur Theorie der meromorphen Funktionen. Acta Mathematica 46: 1–99. Nevanlinna, Rolf. 1929. Le théorème de Picard-Borel et la théorie des fonctions méromorphes. Paris: Gauthier-Villars et Cie. Nevanlinna, Rolf. 1936. Eindeutige analytische Funktionen. Berlin: Springer. Nevanlinna, Rolf. 1953. Uniformisierung. Berlin: Springer. Nevanlinna, Rolf and Frithiof Nevanlinna. 1959. Absolute Analysis. Berlin: Springer. Nevanlinna, Rolf. 1976. Muisteltua. Helsinki: Otava. (Nevanlinna’s memoirs in Finnish). Cartan, Henri. 1980. Notice nécrologique sur Rolf Nevanlinna. Comptes rendus de l’Académie des Sciences Paris. Vie Académique 291 (5–8): 56–57. Lehto, Olli. 1980. Rolf Nevanlinna 22.10.1895–28.5.1980. Arkhimedes 32(3): 134–138. (In Finnish). Lehto, Olli. 1981. Rolf Nevanlinna. Normat 29(1):1–6. (In Swedish). Lehto, Olli. 1998. Mathematics without borders: A history of the International Mathematical Union. New York: Springer. Lehto, Olli. 2001. Erhabene Welten. Das Leben Rolf Nevanlinnas. Basel: Birkhäuser. Pekonen, Osmo. 2012. Review of Olli Lehto’s book “Erhabene Welten: Das Leben Rolf Nevanlinnas”. The Mathematical Intelligencer 1: 57–62.

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Publications Related to Mathematics Education Nevanlinna, Rolf. 1966. Reform in teaching mathematics. The American Mathematical Monthly 73: 451–464.

Photo Source: Wikimedia Commons.

Eric Harold Neville

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11.41 Eric Harold Neville (London, 1889 – Reading, 1961): Member of the Central Committee 1932– 1936, 1936–109 Adrian Rice

Biography Eric Harold Neville was born in London on 1 January 1889. Attending the William Ellis School, his mathematical abilities were recognized and encouraged by his mathematics teacher, Thomas Percy Nunn. In 1907, he entered Trinity College, Cambridge, graduating as second wrangler 2 years later and subsequently winning  During the  ICM 1936 in  Oslo, “The Congress requests the  International Commission on  the  Teaching of  Mathematics to  continue its work, prosecuting such investigations as  shall be  determined by the  Central Committee” (L’Enseignement Mathématique 35, 1936, p.  388), but because of WWII, the Commission remains inactive until 1952 when it is transformed in a permanent subcommission of the IMU. 109

A. Rice (*) Randolph-Macon College, Ashland, VA, USA e-mail: [email protected]

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a Trinity fellowship. While there he became acquainted with other Cambridge fellows, most notably Bertrand Russell and Godfrey Harold Hardy. The story of Hardy and his Indian protégé Srinivasa Ramanujan is one of the most famous in the history of modern mathematics. But Neville’s role in it is less well known. In 1914, as a visiting lecturer, he travelled to India, where, in response to a request from Hardy, he managed to persuade the cautious Ramanujan to accompany him back to England, thus playing a vital role in the initiation of one of the most celebrated mathematical collaborations of the last hundred years. Despite being eligible to serve during World War I, Neville did not join the army when hostilities erupted in the summer of 1914. Although poor eyesight would have prevented him from active service, he openly declared his opposition to the conflict and refused to fight. While no reason was ever given, it was probably this pacifist declaration that resulted in the non-renewal of his Trinity fellowship in 1919. On leaving Cambridge, he was appointed to the chair of mathematics at the small University College, Reading. In a few years, he had built up the mathematics department there, working vigorously to enable the institution to receive a university charter and award its own degrees from 1926. Neville had a wide variety of mathematical interests, but his principal areas of expertise were geometric, with differential geometry dominating much of his early work. Early on in his Trinity fellowship, in a dissertation on moving axes, he extended Darboux’s method of the moving triad and coefficients of spin by removing the restriction of the orthogonal frame. He later wrote an introductory tract on how to generalize concepts and operations of 3-space into the four-dimensional arena. But his ambition to write a comprehensive treatise on differential geometry was never realized. During his time in Cambridge, he had been greatly influenced by Bertrand Russell’s work on the logical foundations of mathematics, and in 1922, he published his Prolegomena to Analytic Geometry prompted by the absence, in his view, of adequate foundational treatments of the topic. Highly influenced by Russell’s work, it is a detailed and logical investigation of the foundations of analytic geometry, providing an axiomatic development of the subject. Neville had long held a keen interest in elliptic functions, having taught the subject to postgraduate students at Reading since the 1920s. He believed that the subject’s recent decline in popularity was due to its dependence on a mass of complicated formulae, a variety of differing and confusing notations and an artificial definition relying on a familiarity with theta functions. A period of recuperation from an illness in 1940 gave him the opportunity to put several years of lecture notes into publishable form. The result was his best-known, and perhaps most original, work: Jacobian elliptic functions (1944). By starting with the Weierstrass P-function and associating with it a group of doubly periodic functions with two simple poles, he was able to give a simple derivation of the Jacobian elliptic functions, as well as modifying the existing notation to provide a more systematic approach to the subject. Like all of Neville’s books, Jacobian Elliptic Functions, while intricate and not easy to read, is expertly crafted and painstakingly thorough. Unfortunately, it failed to achieve its author’s stated

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intention “to restore the Jacobian functions to the elementary curriculum” (Neville 1951, p. vi), and its appearance came too late to have any real effect on the dominance of the classical approach to elliptic functions. Neville was an active member of several mathematical and scientific bodies. Elected to membership of the London Mathematical Society in 1913, he served on its council from 1926 to 1931. He regularly attended meetings of the British Association for the Advancement of Science, being president of Section A (Mathematics and Physics) in 1950. He also chaired its Mathematical Tables Committee from 1931 to 1947, and, when it came under the auspices of the Royal Society, he contributed two sets of tables, on Farey Series of Order 1025 (1950) and Rectangular-Polar Conversion Tables (1956). Neville published many papers, but the vast majority were short items, focusing on concise and succinctly solved problems, often in The Mathematical Gazette, to which he was a frequent contributor. As with all of his writings, they were focused and highly polished, yet, as one obituary says with regret, “so brilliant and versatile a talent could have been harnessed to some major mathematical investigation” (Broadbent 1962, p. 482). Indeed, a former student at Reading could “never understand why his published work of substance was so small in quantity” (Langford 1964, p. 133). Neville retired from the University of Reading in 1954, after which he continued to publish papers in The Mathematical Gazette. He was working on a sequel to his book on elliptic functions when he died on 22 August 1961.

Contribution to Mathematics Education Neville’s work in mathematical education manifested itself in three main forums: his teaching at the University of Reading, his work for the Mathematical Association and his membership of the Executive Committee of ICMI (1932–1936). By all accounts, in his lectures at Reading, Neville was a skilful and considerate teacher, possessing a “unique gift of handling problems with superb technical skill and economy” (Langford 1964, p. 132). From the perspective of one of his students from the 1920s, “as a teacher he was an inspiring guide (though sometimes so far ahead as to be almost out of sight) but with the small classes of those days––there were never more than three of us in the honours group––a lecture could always become a seminar if we wished, and he delighted in the arguments which could develop” (Langford 1964, p. 134). However, according to one of his colleagues at Reading, the sharpness of his mind and the depth of his knowledge could often leave the less able students feeling rather baffled: Honours students were inspired by the brilliance of his lectures and the immensity of his erudition; and if the pass degree pupils sometimes found him above their heads, this was never from any failure of his sympathy, but because he could often modestly forget how fast his own mind worked. (Broadbent 1962, p. 479)

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The challenges of teaching mathematics at Reading prompted the beginning of Neville’s active involvement in issues concerning mathematical education, and his membership of the chief organization in Britain devoted to mathematical pedagogy, the Mathematical Association. We are told: “In the steady growth of the Association from 1920 onwards, in the widening of its interests, in the spreading of its influence, Neville played a part second to none” (Broadbent 1964, p. 139). Introduced to its activities by his former teacher, Thomas Percy Nunn, in 1922 he chaired a subcommittee of the Association’s General Teaching Committee charged with reporting on the teaching of geometry in British schools. The resulting report recommended dividing school geometry into stages: experimental, deductive, systematizing and advanced. Later described as “revolutionary”, one committee member later wrote: “It is perhaps not giving away a secret to say that T.P.N. and E.H.N. were the two principally responsible for the Report, which has been a bestseller ever since” (Broadbent 1964, p. 137). His involvement with the Association’s chief publication, The Mathematical Gazette, began while he was still at Cambridge. For over four decades, Neville contributed a multitude of articles, classroom notes and book reviews on a wide variety of mathematical topics. He also briefly edited the journal in the late 1920s, following the illness and death of the then editor, until a replacement was appointed. He was the Association’s librarian for over 30 years, from 1923 to 1954, and served as its president in 1934. His presidential address, “The Food of the Gods”, drew attention to the widening gap between school and university mathematics. He began by noting that, in the quarter-century since he took his degree, British university mathematics courses had grown considerably, featuring subjects (such as matrices, vectors, Lebesgue integration, tensor calculus, statistics, relativity and wave mechanics) that were all but unknown to British undergraduates 25 years before. His central argument was that “the university builds a different mathematical structure, but is content to build it on foundations which have not changed since the beginning of the century” (Neville 1935, p. 7). He continued: The burden of my plea this afternoon is that changes in emphasis in creative mathematics, which have now a direct influence on teaching at the university, ought to have a greater and a far more rapid influence on teaching at the school than they seem to have. (Neville 1935, p. 16)

It was his opinion that, while the more experienced teachers have greater authority and influence, it is younger teachers who are more in touch with what will best equip a student for university studies. Twenty years ago we did know what were the best current methods of presentation, where emphasis had to be placed to serve most efficiently the needs of those who were soon to be undergraduate students of mathematics. How many of us who are engaged in teaching rather than research can make the same boast to-day? (Neville 1935, p. 16)

Urging his colleagues to consider changes to the secondary school curriculum, he asked:

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Is it absolutely certain that the [school] curriculum is perfect, that there is nothing which could be postponed in favour of some subject now acquired at a later stage? Is it quite indisputable that none of the teaching is wasteful, that nowhere would better methods enable us to explain in one hour a principle over which we have got into the habit of spending two? (Neville 1935, p. 7)

The theme is as pertinent today as when it was delivered in 1935. Neville’s work for the Mathematical Association and British mathematical education generally brought him recognition from overseas. The high regard in which he was held by the British pedagogical community was reflected in his appointment as a member of the Central Committee of the International Commission on the Teaching of Mathematics, on which he served with Hadamard as president and Fehr as secretary-general, in 1932 and in 1936. In 1933, he presented a long report on “The Training of Mathematics Teachers in England” (Neville 1933b), and in the Oslo ICM (1936), he was the rapporteur on the present trends in the teaching of mathematics in his country, but he made only a few comments on the report presented in 1929 by George St. Lawrence Carson (L’Enseignement Mathématique 36, 1937, p. 237).

Sources Neville, Eric Harold. 1921a. Multilinear functions of direction, and their uses in differential geometry. Cambridge: University Press. Neville, Eric Harold. 1921b. The fourth dimension. Cambridge: University Press. Neville, Eric Harold. 1922. Prolegomena to analytical geometry in anisotropic Euclidean space of three dimensions. Cambridge: University Press. Neville, Eric Harold. 1944. Jacobian elliptic functions. Oxford: Clarendon Press. Second edition, 1951. Neville, Eric Harold. 1950. The Farey series of order 1025. Displaying solutions of the Diophantine equation by – ax = 1. Cambridge: University Press. Neville, Eric Harold. 1956. Rectangular-polar conversion tables. Cambridge: University Press. Broadbent, Thomas A. A. 1962. Eric Harold Neville. Journal of the London Mathematical Society 37: 479–482. Broadbent, Thomas A.  A. 1964. On the Teaching Committee. The Mathematical Gazette 48: 136–139. Langford, Walter J. 1964. Professor Eric Harold Neville, M.A., B.Sc.: The man. The Mathematical Gazette 48: 131–136.

Publications Related to Mathematics Education Neville, Eric Harold. 1919. Notes for lessons introductory to differential geometry. The Mathematical Gazette 9: 369–371. Neville, Eric Harold. 1930. Higher trigonometry for schools. The Mathematical Gazette 15: 180. Neville, Eric Harold. 1933a. The teaching of geometry. The Mathematical Gazette 17: 307–312. Neville, Eric Harold. 1933b. The training of mathematics teachers in England. L’Enseignement Mathématique 32: 178–183.

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Neville, Eric Harold. 1935. The food of the gods. The Mathematical Gazette 19: 5–17. Neville, Eric Harold. 1937. The influence of the university on school geometry. The Mathematical Gazette 21: 339–343. Neville, Eric Harold. 1964. Mathematical notation. The Mathematical Gazette 48: 145–163.

Photo Source: Langford, Walter J. 1964. Professor Eric Harold Neville, M.A., B.Sc.: The man. The Mathematical Gazette 48: 131–136 (Photo at n. n. p.).

Kay Waldemar Kielland Piene

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11.42 Kay Waldemar Kielland Piene (Kristiania, 1904 – Oslo, 1968): Member of the Executive Committee 1955–1958 Reinhard Siegmund-Schultze

Biography Kay Waldemar Kielland Piene was born on 7 May 1904 in Kristiania (since 1925 Oslo), the son of the artist, painter and art teacher Johannes (Jo) Bugge Piene and his wife, Christiane (Jannik) Mathilde, born Waage. The latter was the daughter of the famous chemist Peter Waage and his wife Mathilde Sophie, the sister of the applied mathematician Cato M. Guldberg. Waage and Guldberg had discovered the chemical mass action law in 1864. After attending the old Cathedral School in Kristiania – the school where Abel had once been a pupil – Piene studied mathematics at the universities in Kristiania and Copenhagen. His minor subjects were astronomy, mechanics and physics. He passed his final exam in 1929, with outstanding results. In 1931–1932, he studied mathematics in Göttingen. R. Siegmund-Schultze (*) University of Agder, Kristiansand, Norway e-mail: [email protected]

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Although originally interested in mathematical research, the poor job prospects in the late 1920s induced Piene to take the state exam for teachers in 1929 and look for a teaching job, which he found first in Hamar, then from 1936 in Oslo. As a teacher, his interest in general pedagogical problems developed. While still in Hamar, he even reported on a permanent exhibit for children’s books which was shown by the Bureau International d’Éducation in Geneva (Piene 1935). Soon thereafter, Piene became a member of the “Norwegian Carnegie Committee for Exams” where he collaborated with Einar Høigård (a Norwegian educator and politician, who died in the resistance against Germany). Piene’s work, which was partly funded by the Americans, was interrupted by the war and by the German occupation of Norway in 1940. Piene – together with other so-called Kirkenes teachers – was arrested by the Nazis in 1942 and deported to the far north, to Kirkenes at the border with Russia, for 8 months.

Contributions to Mathematics Education In 1946, Piene became “inspector”, from 1953 rector of the “pedagogical seminar” of the University of Oslo, founded in 1907, the biggest and at that time the only institute for secondary school teachers’ education in Norway. There Piene continued his work from the late 1930s on the evaluation of students’ results in mathematics with the help of marks and grades. He worked as chairman in the “Commission for the System of Marks” (Karaktersystemutvalget). His book, Exam Marks (Grades) and Temporary Marks (Eksamenskarakterer og forhåndskarakterer), of 1961 had much importance also for other commissions which discussed similar questions. Piene wrote in the introduction (p. 11): We understand as evaluation all judgment which can be done in a school. … School organisation, the subjects taught, the teachers, the students (physically and psychologically), the buildings, the equipment, the students’ performance, the motivations and adaptations of the students. All this not only can be evaluated, it must be evaluated. Such a comprehensive evaluation, however, has never taken place at Norwegian schools. Here we shall restrict our aim to one fundamental point in the evaluation, namely the one which is based on student marks (grades).110

For the purpose of his book, Piene explored the historical records from the Norwegian secondary school (gymnas) collected by him between 1935 and 1939, using information from about 12,000 index cards. Not least in this context he  The original text is: “La oss da med evaluering forstå all den vurdering som blir foretatt eller kan foretas i en skole. Det er ikke lett a gi en uttømmende fortegnelse her, men evalueringen må kunne komme inn på en rekke punkter i skoleprosessen. Skoleorganisasjon, lærestoff, lærere, elever (fysisk og psykisk), bygninger, utstyr, elevprestasjoner, elevinnstilling, elevtilpasning osv. Alt dette verdsettes og vurderes, ja det ikke bare kan, men må faktisk vurderes dersom en vil nå det mål som ble stilt opp ovenfor.En så omfattende evaluering har aldri vært foretatt i norsk skole. Det er vesentlig på ett punkt vi har foretatt en evaluering, og det er den som skjer ved hjelp av. karakterer”. 110

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revealed his interest in the history of mathematics and of mathematical education, which is shown by several of his articles such as the two parts (Piene 1937; 1938). During his work on the evaluation of exams, Piene resumed his contacts with the Carnegie Foundation and the International Institute of Teachers College, Columbia University. Piene stayed for one semester as a guest professor at Columbia University in New York City; with his great language skills, he was well suited to international collaboration in the field of mathematical education. The one area which most captivated Piene was the improvement of mathematics teaching in the secondary school. When, in 1959, the Organisation for Economic Co-operation and Development (OECD) organized a seminar for the modernization of mathematics teaching in Royaumont near Paris, Piene’s participation was a matter of course. In the subsequent “Nordic committee for the modernisation of mathematics teaching”, Piene was one of the most active members in all meetings which took place between 1960 and 1967. During his life, Piene was a member and partly chairman of many committees of different kinds. He was a member of the ICMI Executive Committee from 1955 to 1958. He took part in the International Congresses of Mathematicians in Oslo (1936), Amsterdam (1954), Edinburgh (1958) and Stockholm (1962). In Amsterdam, he gave an address by invitation of the Organizing Committee (Piene 1956). Among mathematicians, Piene is best known as an author of textbooks for high schools (gymnas), as editor of Norsk Matematisk Tidsskrift (1945–1952) and of its successor, the Nordisk Matematisk Tidskrift (1953–1961). Piene was a very friendly and tolerant person. He could discuss everything with everybody without the danger of serious conflicts arising. His love for mathematics was passed on to his children. His son, Jo Piene, born in 1945, is a computer scientist. His daughter, Ragni Piene, born in 1947, is a mathematician specializing in algebraic geometry. She has been a professor at the University of Oslo since 1987 and was the first woman appointed as member of the Executive Committee of the International Mathematical Union (2003–2010). Piene died on 16 April 1968  in Oslo at only 63 years old.

Sources Johansson, Ingebrigt. 1968. Kay Piene in memoriam. Nordisk Matematisk Tidsskrift 16: 129–130. Website (Retrieved 14 February 2020): http://freepages.genealogy.rootsweb.com/~kielland/slekt/ per02027.htm#0

Publications Related to Mathematics Education Piene, Kay. 1935. Den permanente barnebokutstilling i Genf [The permanent children’s book exhibition in Geneva], English translation. Bok og Bibliotek 2: 173–176. Piene, Kay. 1937, 1938. Matematikkens stilling i den høiere skole i Norge efter 1800 [The position of mathematics in the high school in Norway after 1800]. Norsk Matematisk Tidsskrift 19(2): 52–68 and 20 (2): 33–58.

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Piene, Kay. 1953. Karakterfordelingsnormer [Breakdown of marks norms]. In Festskrift til B. Ribsskog 25 Januar 1953, eds. Ruth Nielsen, Hans Bergersen, and Trygve Dokk, 145–160. Oslo: Gyldendal. Piene, Kay. 1956. School mathematics for universities and for life. In Proceedings of the International Congress of Mathematicians, Vol. 3, eds. Johan C. H. Gerretsen and Johannes de Groot, 318–324. Groningen: E. P. Noordhoff N. V.; Amsterdam: North-Holland Publishing Co. Piene, Kay (1960a). Kronikk. OEEC seminar i matematikk [Chronicle. OEEC seminar in mathematics]. Nordisk Matematisk Tidskrift, 8(1), 53–62. Piene, Kay (1960b). Nye vejer i skolematematikken [New paths in school mathematics]. Nordisk Matematisk Tidskrift, 8(2), 65–71. Piene, Kay. 1961. Eksamenskarakterer og forhåndskarakterer [Exam marks and temporary marks]. Oslo: Cappelen.

Photo Courtesy of Ragni Piene.

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11.43 André Revuz (Paris, 1914 – Créteil, 2008): Member of the Executive Committee 1967–1970 Michèle Artigue and Aline Robert

Biography André Revuz was born in a modest family (his father was an accountant and his mother an embroiderer) in Paris on 15 March 1914. Being a brilliant student, after his primary schooling at the communal school of the Rue de Vaugirard, he passed the entrance exam for secondary school and entered the Lycée Buffon (Buffon High School) with a scholarship. He had his secondary education there and then entered the specific French system of CPGE (Classes Préparatoires aux Grandes Écoles – Higher School Preparatory Classes).111  CPGE are part of the French postsecondary system. In these classes located in high schools, students prepare the highly selective competitions for enrolment in the prestigious tertiary institutions called “Grandes Écoles”, the most famous ones being the École Polytechnique and the École Normale Supérieure (ENS), created at the time of the French Revolution. 111

M. Artigue (*) · A. Robert (*) LDAR & IREM de Paris, Université Paris Diderot – Paris 7, Paris, France e-mail: [email protected]; [email protected]

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In 1934, he was admitted both at the École Polytechnique and the École Normale Supérieure (ENS Ulm). He chose the ENS and also the same year married Germaine Chazottes, whom he had met at the Buffon High School. They had six children. In 1937, after graduating in mathematics and succeeding at the highly selective competition called “Agrégation”,112 he did his military service. Released from military service in March 1939, he was appointed as “Elève-agrégé”113 at the ENS. In August 1939, due to the general mobilization, he could not benefit from the “ArconatiVisconti” research grant he had obtained, and in May 1940, taken prisoner of war near Boulogne, he was transferred to an officers’ camp in Lower Saxony. In captivity, Revuz taught mathematics to a few fellow captive students and gave them exams that would be validated after the war. At the beginning of 1943, he was repatriated for medical reason. He joined his family in Poitiers and finished the school year as a lecturer at the Faculty of Sciences of Poitiers. In October 1943, he was appointed to the Lycée of Poitiers in the class of “Mathématiques élémentaires” (elementary mathematics)114 and joined the local resistance to the German occupation. One year later, in September 1944, he was sent to the Lycée Montaigne in Bordeaux to take in charge a Math-Sup115 class. After the end of World War II, in October 1945, André Revuz and his family left France for Istanbul where he taught 5 years at the Technical University (“Holidays in a wonderful country that ignored the ‘restrictions’ from which Europe suffered”, as he wrote in Revuz (2006).116 Returning to France in January 1950, he was appointed “chef de travaux” (lecturer) at the Faculty of Sciences in Paris. During this period, he also taught at the ENS of Saint-Cloud117 (until 1955) and began his thesis under the supervision of his classmate at the ENS, Gustave Choquet (“Every week, three days devoted to teaching and four to research” (Revuz 2006)). On 8 April 1954, he defended his thesis entitled “Fonctions croissantes et mesures sur les espaces topologiques ordonnés” (Increasing functions and measure on ordered topological spaces) in Paris, before a jury composed of the mathematicians Arnaud Denjoy, Paul Dubreil and Gustave Choquet. The thesis was published in the journal Annales de l’Institut Fourier (Revuz 1956). André Revuz’s research work was partly correlated with some results of Gustave Choquet’s “Theory of Capacities”,  In France, traditionally, teachers are civil servants and competitions are organized by the Ministry of Education for their recruitment. For the recruitment of high school mathematics teachers, two national competitions co-exist, respectively named CAPES and Agrégation, the second one being much more selective than the first one and giving a higher status. 113  This specific status was given to some brilliant students of the ENS, after they had got the Agrégation. They could thus begin to develop research activity. 114  The “Mathématiques élémentaires” class was the grade 12 last class of general high school with specific orientation towards mathematics, physics and chemistry at the time. 115  Math-Sup class corresponds to the first year of scientific CPGE. 116  All quotations included in this text are taken from an unpublished CV written by Revuz in French (Revuz 2006). The translation and notes added are ours. 117  Created respectively in 1880 for girls and in 1882 for boys, the ENS of Fontenay-aux-Roses and the ENS of Saint-Cloud were, at the time, the higher education institutions preparing teachers for the Écoles Normales in charge of the training of primary teachers. 112

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published in 1954, in the same Annales de l’Institut Fourier. This work dealt with increasing set functions – whose capacities were a particular case – and increasing functionals. In particular, one of the unicity results proved by André Revuz had a significant impact on Gustave Choquet’s own reflection. In October 1956, Revuz was appointed associate professor at the Faculty of Sciences of Bordeaux and, in October 1957, full professor at the Faculty of Sciences of Poitiers where he will take the direction of the mathematics department. From 1955 to 1967, he also gave courses at the ENS of Sèvres,118 and between 1958 and 1961, he was an examiner in the entrance examination to the ENS Ulm. In October 1967, he was appointed professor at the Faculty of Sciences of Paris, and, in 1969, when the University of Paris was split, he chose the University Paris 7 (then renamed University Paris Diderot – Paris 7) and actively participated in its creation with the mathematician François Bruhat. In 1969, he founded the IREM of Paris (Institut de Recherche sur l’Enseignement des Mathématiques) in this university, of which he remained director until 1979. He also took over the direction of the CPR (Centre Pédagogique Régional  – Pedagogical Regional Center) of the Académie of Paris,119 a position he held until 1980. In October 1982, he officially retired. Revuz was not only a mathematician and a teacher. He was also a mountaineer and talented alpinist, very attached to the Vallorcine valley where he owned a chalet and to the Mont Blanc Massif, from which he climbed several important peaks. He appreciated the physical effort and, for instance, at the time of the great strikes of the May 1968 movement in France, he came by bicycle from Les Essarts-le-Roi where he lived in the South of Paris to the Institut Henri Poincaré, particularly to participate in the assemblies of candidates for the agrégation of mathematics which were decisive in the creation of the IREMs. Long after his retirement, he continued to play tennis.

Contribution to Mathematics Education This short review of Revuz’s career does not reveal the intense activity he has developed towards mathematics education since 1950, an activity first linked to his engagement in the APMEP (Association des Professeurs de Mathématiques de l’Enseignement Public) created in 1910.

 The ENS Sèvres was the equivalent of ENS Ulm for girls.  France is divided into Académies (currently 31 Académies) for the administration of the educational system. Before the creation of the IUFM (Instituts Universitaires de Formation des Maîtres – University Institutes for Teacher Education) in 1990, the CPR were in charge of the professional training of future secondary teachers. 118 119

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André Revuz’s Engagement in the APMEP The article (Zehren, Biratelle and Hennequin 2009) published in the APMEP Newsletter of February 2009 details the different facets of this engagement. He himself recounted the story in these terms (Revuz 2006):120 There was a widely shared feeling that a modernization of content and teaching methods was absolutely necessary. Modernizing content at the university level was achieved very easily. The problem was more difficult at the high school level. But since the 1950s, it has been the subject of reflection within the APM121: working groups, small conferences in which I participated. An acceleration occurred thanks to the organization by Gustave Choquet and Gilbert Walusinski, Secretary of the APM, in 58-59 and 59-60, of SMF122APM lectures followed by the ‘APM Course’ that I gave from 60 to 63, at the instigation of many APM members, and by the ‘Chantiers Mathématiques’ (Mathematical Worksites) television program organized by the INRP123 and presented by Georges-Théodule Guilbaud and myself.124

From 1958, André Revuz was indeed very much involved in the APMEP and elected several times to its National Committee. He was president of the association for 2 years, from May 1960 to May 1962. Between December 1956 and October 1959, the Bulletin of the APMEP published an important series of articles he authored on projective spaces (Revuz 1956), Euclidean and metric spaces (Revuz 1957) and integration theory (Revuz 1959a, 1959b, 1959c). One year later, the Bulletin also published an influential text of him on the language of modern mathematics (Revuz 1960). Jointly with Germaine Revuz, his wife, he also wrote down the series of lectures on algebraic structures (groups, rings and fields), vector spaces and elements of topology he had given in Paris, and these were published in three volumes by the APMEP (Revuz and Revuz 1962; 1963; 1966). These books immediately became reference books for teacher education.  We acknowledge Joëlle Pichaud who made us accessible the unpublished text from which Revuz’s quotations in this chapter come, and the SMF, this chapter being an adaptation and translation into English of the biographical notice co-authored by Michèle Artigue, Aline Robert and François Colmez, who unfortunately passed away in 2012, published in the Gazette de la SMF, in 2009. 121  APM is often used as an abbreviation of APMEP. 122  SMF: Société Mathématique de France (French Mathematical Society) 123  INRP: Institut National de Recherche Pédagogique (National Institute of Pedagogical Research), now IFÉ (French Institute of Education) 124  The original text is: “Il y avait un sentiment majoritairement partagé qu’une modernisation des contenus et des méthodes d’enseignement était indispensable. Moderniser les contenus au niveau des Universités se fit très facilement. Le problème était plus difficile au niveau des lycées. Mais dès les années 50, il fit l’objet de réflexions au sein de l’APM: groupes de travail, petits colloques auxquels je participais. Une accélération se produisit grâce à l’organisation par Gustave Choquet et Gilbert Walusinski, secrétaire de l’APM, en 58–59 et 59–60 des conférences SMF-APM, suivies par le « Cours de l’APM » que je donnais de 60 à 63, à l’instigation de nombreux membres de l’APM, et par les « Chantiers mathématiques », émissions télévisées organisées par l’INRP et animées par Georges Théodule Guilbaud et moi-même”. 120

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Revuz was also keen to make modern mathematics accessible to a very wide audience, beyond mathematics teachers alone, and this was the aim of the book (Revuz 1963), which was widely disseminated. He also directed two collections of mathematics textbooks: from 1969, with Michel Queysanne, his classmate at the ENS, a collection of textbooks for secondary mathematics education (Publisher Fernand Nathan) and, at the university level, the “Mathematics Series” of the “U Collection” (Publisher Armand Colin). In 1966, he was elected president of the SMF and made efforts to strengthen the links between SMF and APMEP. As he wrote in Revuz (2006), these efforts were not successful: I was hoping, on this occasion, to create closer links between SMF and APM, but I ran into an impassable wall. Forty years later, the problem is recurring with perhaps more chance of finding a solution.125

André Revuz, Member of the “Lichnerowicz” Commission It was also in 1966 that the Commission ministérielle d’étude pour l’enseignement des mathématiques (Ministerial Commission of Study for Mathematics Education), known as Commission Lichnerowicz from the name of his president, André Lichnerowicz, was created. This commission that will work until 1974 is, in the collective memory, associated with the new math reform, its successes but also its failures. André Revuz was one of the most active members of this commission. In particular, he closely followed the work carried out in the hundred or so experimental middle school classes where volunteer teachers were testing the Commission’s curricular proposals. Regarding these classes, he wrote (Revuz 2006): For me, these classes remain a model of what teaching should be: students' activity, responsibility assumed by teachers to distinguish what works and what does not and to know how to modify the approach accordingly.126

As shown by history, these experimental successes did not make it possible to predict what the full-scale implementation of the reform would bring, and the difficulties encountered reinforced André Revuz’s conviction that the key to any change in the education system is teacher education: initial preparation and in-service training. As for initial preparation, as director of the CPR, he enriched the training proposed to future teachers during their probational year, organizing for them mathematics courses oriented towards mathematics teaching, and also courses in cognitive psychology taken in charge by the psychologist Pierre Gréco. Regarding

 The original text is: “j’espérais à cette occasion créer des liens plus étroits entre la SMF et l’APM, mais je me suis heurté à un mur infranchissable. Quarante ans plus tard, le problème se repose avec peut-être plus de chances de trouver une solution”. 126  The original text is: “Ces classes restent pour moi un modèle de ce devrait être l’enseignement: activité des élèves, responsabilité assumée par les professeurs pour distinguer ce qui marche de ce qui ne marche pas et savoir modifier la démarche en conséquence”. 125

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in-­service training, from the 1970s onwards, his action deployed in the IREM structure, the initial project of which he had contributed to elaborate inside APMEP. André Revuz, Director of the IREM of Paris In 1969, Revuz was appointed director of the IREM of Paris, one of the first three IREMs created. He would make this institution, which in its early years was allocated significant material and human resources (for instance, 6 academic positions to be distributed among university mathematicians working part-time at the IREM, 20 half-time positions for secondary teachers, 6 positions for administrative and technical staff), a particularly dynamic institution, organizing a multiplicity of training courses for thousands of teachers and publishing many educational resources. At that time of implementation of the new math reform, in-service teacher education, called recyclage (retraining), tended to mobilize all IREM forces. However, Revuz was well aware that quality training must be based on research, and he set up structures to support research development at the IREM and at the University Paris 7. Thematic working groups were progressively created in the IREM and also, in 1973, an experimental elementary school was attached to this structure. At the university level, in collaboration with the logician Daniel Lacombe, he created the Didactic Department and obtained the accreditation of a DEA (Diploma of Advanced Studies) in mathematics didactics in 1975, one of the first three created in France. At a time when such an approach was not obvious in the mathematical community (see Dorier 2012), he also encouraged university mathematicians interested in educational issues, especially those involved in IREM activities, to carry out didactic research and prepare doctorates in this new domain. This was the case first for Aline Robert in 1982, then Michèle Artigue, Régine Douady, Jacqueline Robinet in 1984 and Janine Rogalski in 1985 and then many others. Revuz was also open to other disciplines. In the early years of IREM, when resources were still substantial, he made colleagues from different disciplines benefit from these resources, to support the good functioning of various interdisciplinary IREM working groups (mathematics and physical sciences, mathematics and biology, mathematics and technology, mathematics and French, etc.). In the 4 years preceding his retirement, with the physicist Jean Matricon, he was the driving force behind the creation of an experimental section of DEUG127 at the University Paris 7, offering integrated teaching of mathematics and physics, an original and very innovative structure that would work successfully until 1985. He also took part in juries of doctoral theses related to mathematics, especially in the history of sciences and linguistics. After his retirement, Revuz continued to participate in many seminars and other meetings, but, above all, when he was almost 90 years old, in 2002, he was at the

 At the time, the DEUG (Diplôme d’Études Universitaires Générales) was the first university diploma, obtained after a 2-year cursus. 127

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origin of the creation of the collective action sciences, bringing together 14 associations to defend the teaching of all sciences. And it remained, until his death, one of its pillars. André Revuz, an Exceptional Teacher This notice would be incomplete if we did not mention the passionate and fascinating teacher that André Revuz was. He has left a deep impression on us, as he did on generations of teachers and students. His teaching skills were due in particular to his desire to make as many students as possible understand, beyond symbols and definitions, what was at stake in his classes: images, comments, questions, small drawings and even jokes, punctuated his interventions, which were always very lively. He knew how to convey his deep conviction that the mathematics he loved so much was accessible to all. The same dynamism and passion for the intelligent transmission of mathematics inhabited him at IREM, which he directed with contagious pleasure and so many initiatives to involve all actors. The same passion animated him when he was experimenting in schools. When we read them today, his articles on the teaching of continuity in high school, for instance, Revuz (1972a; 1972b), seem to express an ambition beyond reach. However, these were not only the result of theoretical reflection; they expressed a real experience made possible by his exceptional qualities as a teacher. For André Revuz, nevertheless, teaching was not something simple – it is not because we know mathematics well that we know how to teach it well, even if he recognized mathematics knowledge to be an essential prerequisite. Teaching was a permanent source of questions for him (see, for instance, the reflection developed in Revuz 1980), which was quite exceptional at the time, and his enthusiasm for all scientific attempts to better understand the phenomena related to mathematics teaching has never been put in default. Moreover, he always trusted researchers, in their diversity, and he decisively encouraged anything that he believed could, in the near or distant future, improve this mathematics education that was so dear to him. André Revuz’s International Engagement The above may suggest that André Revuz’s activities were limited to the national sphere. This was by no means the case. His strong involvement in the new math reform movement led him to establish contacts with those who in other countries were working along the same line from the 1950s. He participated in the work of the CIEAEM (Commission Internationale pour l’Etude et l’Amélioration de l’Enseignement des Mathématiques – International Commission for the Study and Improvement of Mathematics Education) created in 1952 and in the Royaumont Seminar in 1959.

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In 1967, he became a member of the Executive Committee of the International Commission on Mathematics Instruction (ICMI). His term of office, from 1967 to 1970, coincided with a particularly important time for this institution, that of its profound renovation under the presidency of Hans Freudenthal, as shown in Furinghetti, Menghini, Arzarello and Giacardi (2008). He would be his closest collaborator within the ICMI Executive Committee. As he recalled in the interview conducted for the ICMI Centennial (Revuz Interview 2007), the aim was for ICMI to assert itself with regard to its mother institution, the International Mathematical Union, and to make the field of mathematical education recognized as a field of research and practice in its own right. He worked for it with strong conviction. It was during his term of office that the first ICME congress was held in France, in Lyon, in 1969. During an ICMI meeting held in Utrecht in August 1967, he pointed out that the journal L’Enseignement Mathématique, the official organ of ICMI, was too high level and proposed to create a new journal more accessible to secondary school teachers (Commission Internationale de l’Enseignement Mathématique 1967). This new journal, Educational Studies in Mathematics, was created in 1968. He was a member of its Editorial Board until 1990 and published several articles in it (Behnke et al. 1968; Revuz 1968; 1969a; 1969b; 1971; 1972; 1976; 1978). From its creation in the early 1970s until 1992, he was also a member of the Scientific Council of the Institut für Didaktik der Mathematik in Bielefeld, an institution that played an essential role in the development of didactic research in Germany, and contributed to the establishment of fruitful collaborations between French and German researchers (see Schubring 2010 for more details). Despite his impressive activity at the service of mathematics education (see Colmez, de Hosson, Pichaud and Robert 2010 for more details and the complete list of his publications), André Revuz looked at this activity with modesty. In Revuz (2006), he wrote: “All this represents a lot of work for a very modest outcome. However, I think that this work had to be done and that it must be continued. In truth, teaching is a permanent problem”. He has been and remains a model for all those who have wanted and are willing to contribute to the improvement of mathematics education, leaving no one behind, for all those who know the crucial role of teacher education, for all those who know the fragility of education systems and are convinced of the need to rebel against the arbitrariness and unconsciousness with which they are often governed. To honour his memory and acknowledge the importance of his contribution to mathematics education, his name was given to our research laboratory in the didactics of mathematics and sciences in 2009, a laboratory that has always been closely linked to the IREM of Paris.

Sources Artigue, Michèle, François Colmez, and Aline Robert. 2009. André Revuz (1914–2008), Gazette de la Société Mathématique de France 120: 97–102.

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Behnke, Heinrich, John M.  Hammersley, Anna Krygovska, Henry Pollak, André Revuz, Willy Servais, Sergei Sobolev, and Hans Freudenthal (Discussion leader). 1968. Panel Discussion. Educational Studies in Mathematics 1: 61–79. Colmez, François, Cécile de Hosson, Joëlle Pichaud, and Aline Robert (eds.). 2010. Hommage à André Revuz. L’engagement universitaire, l’héritage didactique. Paris: Université Paris Diderot – Paris 7. https://hal.archives-­ouvertes.fr/hal-­02345760 (Retrieved on 28/06/2020). Commission Internationale de l’Enseignement Mathématique. 1967. Compte rendu de la séance de la C.I.E.M. tenue à Utrecht, le 26 août 1967, L’Enseignement Mathématique, 13: 243–246. Dorier, Jean-Luc. 2012. La didactique des mathématiques: émergence d’un champ autonome au carrefour des mathématiques, de la psychologie et des sciences de l’éducation. In Les didactiques en question(s). Etat des lieux et perspectives pour la recherche et la formation, eds. Marie-Laure Elalouf, Aline Robert, Anissa Belhadjin, and Marie-France Bishop, 42–48. Bruxelles: De Boeck. Furinghetti, Fulvia, Marta Menghini, Ferdinando Arzarello, and Livia Giacardi. 2008. ICMI Renaissance: the emergence of new issues in mathematics education, In The first century of the International Commission on Mathematical Instruction (1908–2008). Reflecting and shaping the world of mathematics education, eds. Marta Menghini, Fulvia Furinghetti, Livia Giacardi, and Ferdinando Arzarello, 131–147. Rome: Instituto della Enciclopedia Italiana, Schubring, Gert (2010). André Revuz, l’Allemagne Fédérale et l’« Institut für Didaktik der Mathematik » (IDM). In François Colmez, Cécile de Hosson, Joëlle Pichaud, and Aline Robert (eds.). Hommage à André Revuz. L’engagement universitaire, l’héritage didactique. 159–163. Paris: Université Paris Diderot  – Paris 7. https://hal.archives-­ouvertes.fr/hal-­02345760 (Retrieved on 28/06/2020). Zehren, Christiane, Hugues Biratelle, and Paul-Louis Hennequin. 2009. Hommage à André Revuz, Bulletin de l’APMEP 480: 5–11, 67.

Publications Related to Mathematics Education Works Addressed to Teacher Training Revuz, André. 1956. Espaces projectifs. Bulletin de l’APM 180: 100–106. Revuz, André. 1957. Espaces euclidiens et espaces métriques. Bulletin de l’APM 183: 177–184. Revuz, André. 1959a. Théorie de l’intégration (I). Bulletin de l’APM 196: 95–106. Revuz, André. 1959b. Théorie de l’intégration (II). Bulletin de l’APM 198: 173–181. Revuz, André. 1959c. Théorie de l’intégration (III). Bulletin de l’APM 199: 254–263. Revuz, André. 1956. Fonctions croissants et mesures sur les espaces topologiques. Annales de l’Institut Fourier 6: 187–269. Revuz, André, and Germaine Revuz. 1962. Groupes, Anneaux, Corps. Paris: APMEP. Revuz, André, and Germaine Revuz. 1963. Espaces vectoriels. Paris: APMEP. Revuz, André, and Germaine Revuz. 1966. Eléments de topologie. Paris: APMEP.

Papers on Mathematics Education Revuz, André. 1960. Le langage simple et précis des mathématiques modernes. Bulletin de l’APM 201: 59–68. Revuz, André. 1963. Mathématique Moderne mathématique vivante. Paris: OCDL. Revuz, André. 1968. Les pièges de l’enseignement des mathématiques. Educational Studies in Mathematics 1: 31–36. Revuz, André. 1972. La notion de continuité dans l’enseignement du second degré: compte-rendu d’une expérience. Bulletin de l’APMEP 293: 287–304.

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Revuz, André. 1969a. Further training of mathematical teachers of secondary level. Educational Studies in Mathematics 1: 493–498. Revuz, André. 1969b. Les prémiers pas en analyse. Educational Studies in Mathematic, 2: 270–278. Revuz, André. 1971. The position of geometry in mathematical education. Educational Studies in Mathematics 4: 48–52. Revuz, André. 1972. La notion de continuité dans l’enseignement du second degré. Educational Studies in Mathematics 4: 281–298. Revuz, André. 1976. Stratégies pour une approche de Z. Educational Studies in Mathematics 7: 113–120. Revuz, André. 1978. Change in mathematics education since the late 1950’s. Ideas and realization in France. Educational Studies in Mathematics 9: 171–181. Revuz, André. 1980. Est-il impossible d’enseigner les Mathématiques? Paris: Presses Universitaires de France. Revuz, André. 2006. Unpublished text. Revuz, André. 2007. Entretien avec André Revuz [by Michèle Artigue]. ICMI Historical Website https://www.icmihistory.unito.it/clips.php (Retrieved June 2021).

Photo Courtesy of Joëlle Pichaud.

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11.44 Gaetano Scorza (Morano Calabro, 1876 – Rome, 1939): Vice-President 1932–1936, 1936–128 Livia Giacardi

Biography Bernardino Gaetano Scorza was born in Morano Calabro (Cosenza) on 29 September 1876. He completed his studies at the Collegio Nazareno in Rome and at the Pious Schools of the Piarist Fathers in Florence and then enrolled at the University of Pisa. Among his teachers were Eugenio Bertini, Ulisse Dini and Luigi Bianchi. In 1898,  During the  ICM 1936 in  Oslo, “The Congress requests the  International Commission on  the  Teaching of  Mathematics to  continue its work, prosecuting such investigations as  shall be  determined by the  Central Committee” (L’Enseignement Mathématique 35, 1936, p.  388), but because of World War II, the Commission remained inactive until 1952, when it was transformed in a permanent subcommission of IMU. 128

L. Giacardi (*) University of Turin, Turin, Italy e-mail: [email protected]

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he received his degree (laurea) in mathematics. He was immediately assigned a position as an assistant in analytic and projective geometry at the University of Pisa, and in 1899–1900, through an exchange of assistants agreed upon between Bertini and Corrado Segre, he was an assistant in projective and descriptive geometry at the University of Turin. Once back in Pisa, he reassumed his position as assistant and earned his qualification (abilitazione) to teach in secondary schools at the prestigious Scuola Normale Superiore in Pisa. From 1902 to 1912, he taught in secondary schools, more precisely, in technical institutes in Terni, Bari and Palermo. On 9 September 1907, Scorza married Angiola Dragoni (1875–1934), with whom he had four children: Giuseppe, Bernardino, Maria and Giovannino. In that same year, he earned his university teaching qualification (libera docenza), and in 1912, he won the competition for a professorship in projective and descriptive geometry at the University of Cagliari. From here, he went to Parma (1913), to Catania (1916), to Naples (1920) and finally to Rome (1934), where he stayed until his death. In the various universities, he taught several other courses as well, including that of complementary mathematics, mainly aimed at the preparation of mathematics teachers in secondary schools. The nucleus of Scorza’s scientific production regards hyperspatial projective geometry, Abelian functions, the matrices that he called Riemann matrices, the theory of algebras and groups. The works of hyperspatial projective geometry fall into the more ample context of the geometry of birational transformations and, extending the results of Federigo Enriques and Guido Castelnuovo, arrive at the complete classification of the multidimensional varieties with elliptical curve sections. But the area in which Scorza obtained his most significant results is that of Abelian functions, particularly of singular Abelian functions. In his research, he was guided by a particular geometric representation with which all the problems of the existence of singular Abelian functions are reduced to problems of a more elementary type relative to the existence of linear systems of rational homographies of a hyperspace. His masterpiece is a substantial article of 1916 entitled “Intorno alla teoria generale delle matrici di Riemann e ad alcune sue applicazioni” (Rendiconti del Circolo MAtematico di Palermo, 1916, 41: 263–380, also in Opere scelte, II, pp. 127–275). Beginning in 1921, the main object of his research was the general theory of algebras, which led him to grapple with problems in number theory and the theory of finite groups. In that year, he published his authoritative treatise, Corpi numerici e algebre, in which the general theory of algebras is presented elegantly and systematically, with original contributions. In 1942, the volume Gruppi astratti was published posthumously, edited by Giuseppe Scorza Dragoni and Guido Zappa. He was also occupied with questions of political economy and theoretical photogrammetry. A rich group of his publications is closely connected with teaching in secondary schools. Scorza was a member of many Italian scientific academies and institutes, held numerous offices and received various awards. In particular, in 1922, he was awarded the Gold Medal of the Academy of XL, and in 1926, he became a member of the Accademia dei Lincei. He was a member of the Consiglio Superiore della Pubblica Istruzione, the High Council for Public Instruction, from 1923 to 1932; he

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was the director of the journal Rendiconti del seminario matematico della R.  Università di Roma; he was president of the mathematics committee of the Consiglio Nazionale delle Ricerche, the Italian Research Council, from 1928 to 1931. In June 1939, a few days before his death, he was appointed senator of the Kingdom. Scorza died in Rome on 6 August 1939.

Contribution to Mathematics Education When Giovanni Vailati died in 1909, Gaetano Scorza took over his place as Italian delegate, along with Castelnuovo and Enriques, to the International Commission on the Teaching of Mathematics. He became vice-president of the Commission in 1932 and held this position until his death. His commitment to the Commission, connected with that within the Associazione Mathesis, the Italian association of mathematics teachers, is apparent from the two long, accurate reports on the teaching of mathematics in technical schools and institutes (Scorza 1911b) and on geometry textbooks for upper secondary schools (Scorza 1912a). From these reports, as well as from later articles that touch on education, emerges his vision of mathematics teaching, a vision that is shaped by the influence of Felix Klein (Scorza 1911b, pp. 76–80), whose book Elementarmathematik vom höheren Standpunkte aus was reviewed by Scorza (Scorza 1910, 1911a). Scorza admired Klein for his research method, his mastery of the most disparate branches of mathematics, his capacity for grasping the even most hidden relationships between them and for having outlined vast programmes of work (Scorza 1931, p. 171). Scorza’s vision of mathematics teaching can be summarized in the following points: • Bridge the gap between secondary teaching and university teaching by introducing elements of differential and integral calculus and “the highest theories of elementary geometry … in conformance with the views that underlie modern geometry and constitute its greatest value and most compelling fascination”129 (Scorza 1914, p. VII). • Avoid fanatical purism; in fact, he affirms: Klein himself, speaking about the evolution of the teaching of geometry in France, Germany, England and Italy, identifies the peculiar character of our didactic movement in the high scientific value of our best textbooks of elementary geometry. However, if our love for logical rigor shows that we have not remained passive in face of the progress of the research on

 The original text is: “le teorie più elevate della geometria elementare […] in conformità delle vedute che soggiacciono alla geometria moderna e ne costituiscono il pregio migliore e il fascino più suggestivo”. 129

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foundations, it is also, today, one of the strongest causes of that discontinuity [between secondary and university teaching] which was alluded to above.130 (Scorza 1911b, p. 77)

• Coordinate the teaching of mathematics with that of physics and other sciences. • Do not use only one method so as to “close oneself in it as behind iron gates”131 (Scorza 1931, p. 170). • Do not be too tied to textbooks; he writes: A lesson in mathematics can be a work of art and not a tired repetition of postulates, theorems and corollaries strung one after another with lifeless mechanicalness as the disconnected words of a child’s nursery rhyme; but provided that who is giving the lesson appears to the student … as a very secure master of what he has to say … so that he presents [each argument] to the students as if it sprang … bright, spontaneous and new, from the living flow of the discussion.132 (Scorza 1931, p. 174)

When, in 1923, the Neo-Idealist philosopher Giovanni Gentile, then minister of public instruction, taking advantage of the full powers granted him by the first Mussolini government, put into action in a single year a complete and organic reform of the Italian school system, Scorza was his consultant for the formulation of the mathematics programme for secondary schools. Gentile separated secondary education into two main branches, of which the classical-humanistic one (ginnasioliceo) was destined for the formation of the ruling class and which prevailed absolutely over the technical-scientific branch (liceo-scientifico) to which was conceded only limited access to university faculties. Barely 2 years earlier, at the lecture given for the inauguration of the Mathematics Circle of Catania, Scorza expressed his doubts about the Neo-­Idealistic philosophical movement, behind which he seemed to “glimpse a hardly seductive nihilism”133 (Scorza 1921, p. 19), and passionately defended the educational, ethical and aesthetic value of mathematics. Taking up the same theme once again in 1923, he explicitly stated that “the philosopher who hasn’t had a good education in mathematics pays bitterly for his deficiency”134 (Scorza 1923, p. 97), and he maintained

 The original text is: “Lo stesso Klein, discorrendo dell’evoluzione dell’insegnamento ­geome­trico in Francia, in Germania, in Inghilterra e in Italia, trova nell’alto valore scientifico dei nostri migliori libri di geometria elementare il carattere peculiare del nostro movimento didattico. Se non che, codesto nostro amore per il rigore logico, se dimostra che non siamo rimasti inerti di fronte ai  progressi della critica dei fondamenti, è pure, oggi, una delle più energiche cause di quella ­solu­zione di continuità cui si alludeva più sopra”. 131  The original text is: “rinchiudervisi come tra cancelli di ferro”. 132  The original text is: “Una lezione di matematica può essere un’opera d’arte e non uno stanco ripetio di postulati, teoremi e corollari infilzati l’un dietro l’altro con smorta meccanicità quasi si trattasse delle parole irrelate di una filastrocca da bambini; ma a patto che chi l’impartisce non si presenti dinanzi alla scolaresca senza essersi reso […] tanto sicuro padrone di ciò che ha da dire da […] presentarlo ai discenti come balzante […] leggero, spontaneo e nuovo, dal vivo fluir del discorso”. 133  The original text is: “intravedere un poco seducente nullismo”. 134  The original text is: “Il filosofo che non abbia avuta una buona educazione matematica sconta amaramente il suo difetto”. 130

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the importance of a mathematics teaching which was not passive or dogmatic but which availed itself of an active and heuristic method, developing both the creative imagination and an acute critical sense. As an example, Scorza cites the German schools where the pupil is constantly stimulated by the teacher to answer questions and to actively participate in the lesson. In laying out the new mathematics programmes, he was in any case strongly conditioned by the general framework of the reform. The directives of Fascism and the Neo-Idealistic ideology were opposed to an ample diffusion of scientific culture and above all to its interaction with other areas of culture: humanistic culture was to constitute the cultural axis of national life and especially of the schools. The old programmes, which provided the teacher with the detailed outline of the subject and with helpful instructions on methodology, were substituted by programmes of examinations that indicated the objective, but not the way to achieve it. During the Fascist regime, Scorza was also the author of the part concerning mathematics of the State textbook for the third, fourth and fifth grades of the elementary schools (Scorza 1930–1933). Here, he tries to begin with the concrete, making reference to the child’s limited world; he is attentive to the precision and clarity of the language; he introduces new subjects by starting with one or two examples and only in a second moment does he give, if it is necessary, the rule; he proposes simple problems taken from everyday life. Unlike textbooks by other authors, these do not present signs of subjection to Fascism.

Sources Scorza, Gaetano. 1960, 1961, 1962. Opere scelte, Roma, Ed. Cremonese, 3 Vols. Scorza, Gaetano. 1921. Corpi numerici e algebre. Messina: Principato. Scorza, Gaetano. 1942. Gruppi astratti, eds. Giuseppe Scorza and Guido Zappa. Roma: Cremonese. Berzolari, Luigi. 1939. Gaetano Scorza, Bollettino della Unione Matematica italiana, s. 2, 1: 401–408. Giacardi, Livia. 2019. The Italian Subcommission of the International Commission on the Teaching of Mathematics (1908–1920). Organizational and Scientific Contributions. In A. Karp, National Subcommissions of ICMI and their Role in the Reform of Mathematics Education, 119–147. Springer Series: International Studies in the History of Mathematics and its Teaching. Rogora, Enrico. 2018. Scorza, Bernardino Gaetano. In Dizionario Biografico degli Italiani. Roma: Istituto della Enciclopedia Italiana, Vol. 91: 617–619. Sansone. Giovanni. 1977, Scorza Gaetano. In Geometri algebristi, ex-normalisti del periodo 1860–1929, pp. 13–15, 31. Pisa: Scuola Normale Superiore di Pisa. Severi, Francesco. 1941, L’opera scientifica di Gaetano Scorza. Annali di Matematica pura ed applicata, s. 4, 20: 1–20. Zappa, Guido. 1991. I contributi di Gaetano Scorza alla Teoria dei Gruppi. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, s. 9, 2.2: 95–101.

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Publications Related to Mathematics Education Scorza, Gaetano. 1910. [Review of] F. Klein’s Elementarmathematik vom höheren Standpunkte aus. 2 Teile (Teubner). Bollettino della Mathesis II: 130–146. Scorza, Gaetano. 1910–1911. Sulla teoria dei numeri reali. Pitagora 1: 25–28; written under the pseudonym ex-Langham. Scorza, Gaetano. 1911a. [Review of] F. Klein’s Elementarmathematik vom höheren Standpunkte aus. 2 Teile (Teubner). Bollettino della Mathesis a. III: 48–52. Scorza, Gaetano. 1911b. L’insegnamento della matematica nelle Scuole e negli Istituti tecnici. Bollettino della Mathesis, a. III (Atti della Sottocommissione italiana per l’insegnamento matematico): 49–80; abstract in L’Enseignement Mathématique 14, 1912: 416–420. Scorza, Gaetano. 1911c. Logica e matematica. Le geometrie non-euclidee in rapporto alla teoria della conoscenza. Annuario della Biblioteca filosofica 1: 226–227. Scorza, Gaetano. 1912a. Sui libri di testo di geometria per le scuole secondarie superiori. Bollettino della Mathesis a. IV (Atti della Sottocommissione italiana per l’insegnamento matematico): 235–247; abstract in L’Enseignement Mathématique 15, 1913: 427–430. Scorza, Gaetano. 1912b. Prefazione. In C. de Freycinet, Dell’esperienza in geometria (Italian translation by G. Fazzari). Palermo: Reber. Scorza, Gaetano. 1914. Complementi di geometria, Vol. I. Bari: Laterza. Scorza, Gaetano. 1921. Essenza e valore della matematica. Esercitazioni matematiche, Catania 1: 1–25; also in Opere scelte, Vol. III, pp. 1–23. Scorza, Gaetano. 1922. Intorno al principio di causalità e alle applicazioni della matematica alle scienze sociali. Periodico di matematiche, s. 4, 2: 1–20; also in Opere scelte, Vol. III, pp. 43–60. Scorza, Gaetano. 1923. Il valore educativo della matematica. Esercitazioni matematiche, Catania, 3: 251–273; also in Opere scelte, Vol. III, pp. 77–97. Scorza, Gaetano. 1931. La matematica come arte. In Atti della XIX riunione della Società Italiana per il Progresso delle Scienze, Vol I, pp. 130–146. Roma: SIPS; also in Opere scelte, Vol. III, pp. 158–175. Gaetano Scorza is also the author of numerous reviews of textbooks for secondary schools or books concerned with the training of teachers.

Photo Source: Scorza Gaetano. Opere scelte. Vol. I, Roma, Ed. Cremonese, 1960, n. n. p.

Igor Fedorovich Sharygin

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11.45 Igor Fedorovich Sharygin (Moscow, 1937 – Moscow, 2004): Member of the Executive Committee 1999–2002 Man Keung Siu

Biography Igor Fedorovich Sharygin was born in Moscow on 13 February 1937. He lived in Moscow all his life, save 1 year during World War II when he evacuated to Kazan in 1942. He passed away on 12 March 2004. As a high school pupil, Sharygin took part in various mathematical Olympiads and joined the activities of the mathematical circle run by some professors of Moscow State University. His ability and talent in mathematics had already been noticed by that time by Nikolai Sergeevich Bakhvalov (1934–2005), who was to become his PhD thesis supervisor. Bakhvalov said of him: Igor Fedorovich Sharygin was a member of a mathematical circle which I ran for high school pupils. The circle produced a lot of known mathematicians. Igor was distinguished among other participants by a large modesty coefficient: the ratio of someone’s real mathematical abilities to his opinion about them. (Bakhvalov 2004, p. 69)

In 1954, Sharygin entered the Department of Mathematics and Mechanics of Moscow State University. After completion of his study there with honours in 1959, he continued his PhD (candidate of science) study at the same university. In 1965, M. K. Siu (*) University of Hong Kong, Hong Kong, SAR, China e-mail: [email protected]

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he defended his PhD thesis, “The lower bounds in the theory of integration and approximations on certain functional classes”, written under the supervision of N.S. Bakhvalov. Until 1972, Sharygin worked in the Department of Mathematics and Mechanics and the Department of Applied Mathematics and Cybernetics of Moscow State University. Later, he taught mathematics in various institutions of higher learning in Moscow. In 1985, he became a senior researcher in the Moscow Institute of Educational Systems and Methods of the Russian Academy of Education (formerly the Academy of Pedagogical Sciences) and was later promoted to the position of head researcher. Until the very end of his life, he continued writing books and struggling for the betterment of mathematics education in Russia, over which he harboured deep concerns about what he saw as ill effects brought about by “globalization” of education and society.

Contribution to Mathematics Education Sharygin wrote over 30 books for school pupils, especially on geometry. His book, Problems in Plane Geometry (in Russian, 1981, revised version 1982), followed by its sequel, Problems in Solid Geometry (in Russian, 1983), quickly became very popular and has been translated into several languages. He continued to publish from 1989 to 1999 the books Solving Problems in Mathematics, Volumes 1 and 2 (in Russian), Visual Geometry (in Russian) and Mathematical Medley (in Russian) and some textbooks on plane geometry and solid geometry for various grades. In 1984, he became editor-in-chief of the problem section of the journal Mathematics in School. In his many books on mathematical problems, the level of complexity ranges from that of ordinary school problems to that which requires really creative thinking. Moreover, the majority of the problems are original problems he himself composed. As an acknowledgement of Sharygin’s strong passion for geometric teaching in 1985, the “Geometric Olympiad in honour of I.  F. Sharygin” was launched, an international competition for high school students, which is held annually. In his obituary of Sharygin, Tabov (2003, p. 68) wrote: Add also that Igor Sharygin fostered the creativity and the talent not only of the students, but also of the school teachers. His attention was turned to the teacher training and he was one of the founders of the All Union Association of the School Teachers.

In 1970 at the initiative of Andrei Nikolaevich Kolmogorov (1903–1987) and Isaak Kostantinovich Kikoin (1908–1984), the famous magazine on elementary mathematics for school pupils, Kvant, was founded. Since the very first days of the existence of Kvant, Sharygin worked actively on it, writing articles and composing problems regularly. As a long-time contributor to the magazine, he later became the Russian head of the mathematics department of its sister magazine, Quantum, published in English in the United States. He took part in the activities of mathematical

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Olympiads and helped to prepare the Russian teams for the International Mathematical Olympiads. In 1999–2002, he was a member of the ICMI Executive Committee. He had been the national representative of Russia to ICMI from 1997. Bernard R. Hodgson, who served as secretary-general of the ICMI in those years, recalled an incident that revealed the lively and witty mind of Sharygin. In the last committee meeting held in Paris, Sharygin brought an amusing drawing which he arranged that depicted all executive members, each associated with a national icon! Sharygin had a special passion for geometry because “it possesses the widest spectrum of educational and cultivating possibilities” (Sharygin 2002, p. 66). It is worth mentioning that he also launched a project in Russia aimed at producing electronic textbooks for geometry (ibidem). He once wrote: Geometry is a phenomenon of human culture. … Geometry, as well as mathematics in general, helps in moral and ethical education of children. … Geometry develops mathematical intuition, introduces a person to independent mathematical creativity. … Geometry is a point of minimum for the distance between school mathematics and the mathematics of high level. (Quotation from Tikhomirov 2004, p. 69) Native language and literature, physical training, and mathematics are three crucial components of secondary education. Of all these subjects, it is mathematics, and especially Geometry, that is concerned with the widest range of long- and short-term educational goals. (Sharygin 2004, p. 43)

Particularly noteworthy is the emphasis he placed on the moral value and civic importance of learning mathematics in general and of geometry in particular: Learning mathematics builds up our virtues, sharpens our sense of justice and our dignity, strengthens our innate honesty and our principles. The life of mathematical society is based on the idea of proof, one of the most highly moral ideas in the world. (Sharygin 2004, p. 45)

He was of the opinion that people who are mathematically literate and understand what proof means cannot easily be manipulated and that although mathematics and state authority are two incompatible things, rational sovereigns often turn to mathematicians for assistance in difficult moments.

Sources Biographical information on Igor Fedorovich Sharygin. 1999. ICMI Bulletin n. 47: 6–7. Sharygin, Georgii, Vladimir Tikhomirov, Nikolai Bakhvalov, and Bernard R. Hodgson. 2004. In memoriam: Igor F. Sharygin (1937–2004). ICMI Bulletin n. 55: 67–72. Tabov, Jordan. 2003. Igor Fedorovich Sharygin 1937–2004. The Teaching of Mathematics 6: 67–68. [In the article there is the following footnote: “This fascicle of “The Teaching of Mathematics” has the date of the volume VI for 2003”.] Karp, Alexander and Bruce Vogeli. 2011. Russian mathematics education: Programs and practices. Singapore: World scientific.

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Publications Related to Mathematics Education Many papers of Sharygin on mathematics education in Russian can be found online. Sharygin, Igor. 2002. A project in Russia: Electronic textbooks for geometry. ICMI Bulletin. 51: 66–67. Sharygin, Igor F. 2004. On the concepts of school geometry. In Trends and Challenges in Mathematics Education, eds. J.  Wang and B.  Xu, 43–51, Shanghai: East China Normal University Press.

Photo Source: Sharygin, Georgii, Vladimir Tikhomirov, Nikolai Bakhvalov, and Bernard R. Hodgson. 2004. In memoriam: Igor F. Sharygin (1937–2004). ICMI Bulletin n. 55: 67–72 (Photo at p. 68).

David Eugene Smith

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11.46 David Eugene Smith (Cortland, 1860 – New York, 1944): Vice-President 1912–1920, President 1928–1932 Gert Schubring

Biography David Eugene Smith was born on 21 January 1860 in Cortland, New York, into a financially comfortable family. His father, Abram, had studied at Albany Normal School and, after graduating in 1853, had been a teacher for 2 years at Marathon, New York. He changed, however, to pursue a career in law. He was admitted to the bar in 1856, and in 1868 he became surrogate judge of Cortland County. His mother, Mary Elizabeth, had learned classical languages and natural sciences and made her son accompany her on many excursions into nature and to art expositions. In 1869, Smith enrolled in the practice school of the newly opened normal school in Cortland. He completed intermediate studies in 1876 and gained admission to the academic department. This enabled him to take those courses in the normal school that were of a non-professional nature. From 1877 on, he studied at Syracuse University. During his undergraduate years there, his interests were marked by G. Schubring (*) University of Bielefeld, Bielefeld, Germany e-mail: [email protected]

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travelling, collecting objects and presenting his ideas. In the summer of 1879, he undertook a 2-month trip to Europe – the first of his overseas travels. Thereafter, he organized an exhibition of 90 “specimens”, including money, shells and “antique relics”. In fact, among the broad range of undergraduate courses were elocution and rhetoric. Smith was still far from mathematics education: after receiving a Bachelor of Philosophy degree in July 1881, he entered his father’s second profession: he apprenticed in his father’s law office. There he served as clerk and notary public while preparing for the bar. Smith combined this professional work with travels to Central and South America. He also continued his academic studies, by travelling twice a week to Syracuse and being advised there in graduate work in the disciplines of history, modern languages and mathematics. Eventually, in 1884, he was admitted to the bar and awarded a Master of Philosophy degree. A promising career as a lawyer seemed to be open for him. In that same year, however, an event changed his life entirely. He began to teach mathematics at the Cortland Normal School, at first by chance, in order to “help out” by substituting a missing teacher. Since he had studied enough mathematics at Syracuse to be effective as a teacher, the principal asked him to accept the position. Finding the law profession not especially agreeable, Smith accepted and began thus his pioneering work for mathematics education in the United States. During the next 3 years, he continued not only his engagements as a lawyer but also his academic studies. Eventually, in 1887, he was granted the PhD degree by Syracuse University in the history of fine arts. While the courses he taught at Cortland had been standard  – arithmetic, algebra, plane and solid geometry and trigonometry  – from 1887 on, he introduced courses on history of mathematics. After 7 years of teaching as a mathematics professor at Cortland Normal School, he was granted a leave of absence to study at Göttingen. Before he could embark for this centre of mathematics, he obtained the offer of the position of mathematics professor at the Michigan State Normal School at Ypsilanti. Thus, Smith came into direct contact with Felix Klein only some years later. At Ypsilanti, from 1891 on, Smith developed the kernel of his programme for mathematics education. The normal school there, affiliated with Michigan University, had expanded to provide teacher education for all types of public schools – not only common schools but also secondary schools. Smith, becoming head of the mathematics department, assured the academic level of teacher education, balancing the professional and the academic sides of the formation. He began to publish and became known as a mathematics educator not only nationally but also internationally. He even published a review of a German textbook in the key German journal Zeitschrift für Mathematischen und Naturwissenschaftlichen Unterricht. During a trip in 1894 to Germany and France, he met the historian of mathematics Moritz Cantor in Heidelberg and the mathematician Rudolf Lipschitz in Bonn. In 1904, he joined the Comité de Patronage of the journal L’Enseignement Mathématique, the first international journal for mathematics education. Wishing to exert an administrative position, Smith moved in 1898 to Brockport, in the state of New York, as principal of the Normal School. While not teaching mathematics there, he published his first seminal contribution to mathematics

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education: the book The Teaching of Elementary Mathematics (1900), a methodology for mathematics teachers. After 3 years, in 1901, he returned to training mathematics teachers himself: at Teachers College, New  York, the most prestigious institution in the United States, rivalled only by the School of Education at the University of Chicago. Originally just somewhat associated to Columbia University, Teachers College had evolved to be a professional school of university rank. Its students had to be college graduates or experienced teachers. By 1910, Teachers College had raised its status even more and constituted a graduate college for professional education within Columbia University. Smith had been called to Teachers College in order to raise in particular the mathematics department to this level of quality. In fact, in 1906, the first PhD degrees were conferred on two of Smith’s students. Distinctive of his programme was that teacher students should gain a historical perspective on teaching their subject. Smith retired in 1926. Due to his good relations with the mathematical community in the States, he served as an effective link between the demands of mathematicians and the needs of professional teacher training. From 1902 to 1920, Smith served as an associate editor of the Bulletin of the American Mathematical Society. He also served as an associate editor of the journal of the Mathematical Association of America, The American Mathematical Monthly, from 1916 on. He was elected president of the MAA for the term 1920 to 1921. He helped to organize the Association of Teachers of Mathematics and was elected its first president. His publications were decisive in shaping mathematics education in the United States. His handbook of 1900 was followed by The Teaching of Arithmetic in 1909 and by The Teaching of Geometry in 1911. His textbooks in arithmetic, algebra and geometry and accompanying handbooks, published since 1904, were dominant during the 1910s. He died at his home in New York City on 29 July 1944 after a long illness. The D.E. Smith Collection When Smith began to teach mathematics, he turned his attention to mathematical artefacts. By the time Smith donated his collection to Columbia in 1931, he had gathered more than 3000 portraits and autographed letters of famous mathematicians and approximately 300 rare astronomical instruments and ancient counting devices. Today, his enormous and unique collection of books, manuscripts, letters, medals and instruments constitutes a part of the Rare Book and Manuscript Library of the Butler Library at Columbia University, New York. Smith enjoyed collecting for himself, but he also delighted in sharing his broad knowledge and insight with others. Beginning in 1901, Smith advised George A.  Plimpton, head of the New  York office of the publishing house Ginn and Company, on acquisitions for his collection of mathematical textbooks. By 1908, Plimpton had assembled the most complete library of arithmetic books printed before 1601. To make the collection more widely known, Smith prepared a richly

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illustrated catalogue, Rara Arithmetica (1908). His strong interest in the history of mathematics resulted in publishing a proper textbook on it in 1923/1925.

Commitment to IMUK Upon the proposal of Smith, the Fourth International Congress of Mathematicians (ICM), Rome 1908, decided to establish an international committee, the Internationale Mathematische Unterrichtskommission (IMUK), to study the situation of mathematics instruction internationally. He had published such a proposition first in 1905 (Smith 1905). Also, upon his proposal, Felix Klein, Sir George Greenhill and Henri Fehr were elected in Rome as the core of this committee, later named its Comité Central. As a matter of fact, Smith having already travelled so much in various European countries and encountered numerous pertinent persons, he acted as midwife for putting the elected members into contact and to initiate them taking up the task in the months following the Rome Congress (Schubring 2008, pp. 5ff.). Moreover, thanks to Smith’s excellent contacts with mathematicians interested in education in many countries, it was also largely due to his advice that the national subcommittees were constituted soon. His merits were acknowledged in 1912, when the next ICM in Cambridge elected him as an additional member of the IMUK Comité Central. There his membership was upgraded to the second vice-president (ibidem, p. 14). After the outbreak of WWI, Smith played an important role regarding Fehr’s acting against Klein. While he supported Klein to maintain as far as possible the IMUK work, he denied Fehr’s proposals to replace Klein as president (ibidem, p. 20). After the war, Smith disagreed with the policy to exclude scientists from the defeated countries, endorsed by Fehr (ibidem, p. 27). When the Commission was re-established, in 1928, upon cancelling the ban of German and Austrian scientists, Smith was even elected its president. In 1932, he retired from IMUK activities too. Evidently, he was also the dynamic element in the work of the US national subcommission, constituted early in 1909. Smith took the initiative to nominate its three delegates: besides himself, William Fogg Osgood (Harvard) and Jacob William Albert Young (Chicago). He succeeded, too, in securing financing by the Federal Government (Karp 2019, pp. 197ff. and 208ff.). And, moreover, he succeeded in a manner somewhat analogous to Klein’s acting in Germany, to establish an enormous set of commissions for elaborating the required reports on the state of mathematics teaching – finally it were 12 committees, some even with subcommittees – and an enormous net of hundreds of mathematics educators collaborating in the writing of the reports. The reports were ready and printed as originally planned for the 1912 Cambridge Mathematics Congress (ibidem, pp. 218ff.). During the period extended there for the IMUK work, the American subcommission elaborated a comparative international evaluation (ibidem, pp. 226ff.).

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Sources Essential Primary Sources Smith, David Eugene. 1908. Rara Arithmetica. Boston etc.: Ginn. Smith, David Eugene. 1923/1925. History of Mathematics, 2 Vols. Boston etc.: Ginn.

Essential Secondary Sources Donoghue, Eileen F. 1987. The origins of a professional mathematics education program at Teachers College. Ed.D. thesis, Columbia University New York, Teachers College. Donoghue, Eileen F. 1998. In Search of Mathematical Treasures: David Eugene Smith and George Arthur Plimpton. Historia Mathematica 25: 359–365. Donoghue Eileen F. 2001. Mathematics education in the United States: Origins of the field and the development of early graduate programs. In One Field, Many Paths: U. S. Doctoral Programs in Mathematics Education, ed. R. E. Reys and Jeremy Kilpatrick, 3–17. Providence: American Mathematical Society/Mathematical Association of America. Karp, Alexander. 2019. The American National Subcommission of ICMI.  In National Subcommissions of ICMI and their Role in the Reform of Mathematics Education, ed. Alexander Karp, 193–234. Cham: Springer. Sarton, George (ed.). 1936. The David Eugene Smith Presentation Volume (Jan. 21, 1936). Osiris 1.

Obituaries Fite, W. Benjamin. 1945. David Eugene Smith. American Mathematical Monthly 52: 237–238. Simons, Lao Genevra. 1945. David Eugene Smith – In Memoriam. Bulletin of the AMS 51: 40–50.

Publications Related to Mathematics Education Smith, David Eugene. 1900. The Teaching of Elementary Mathematics. Teachers’ Professional Library, ed. Nicholas Murray Butler. New York: Macmillan. Smith, David Eugene. 1901. L’enseignement des mathématiques aux États Unies. L’Enseignement Mathématique 3: 157–171. Smith, David Eugene. 1905. Opinion. L’Enseignement Mathématique 7: 469–471. Smith, David Eugene. 1909. The Teaching of Arithmetic. New York: Teachers College, Columbia University. Smith, David Eugene. 1911. The Teaching of Geometry. Boston etc.: Ginn. Smith, David Eugene. 1912. Report of the American Commissioners of the International Commission on the Teaching of Mathematics. Washington, DC: Government Printing Office.

Photo Source: Sarton, George. 1936. Dedication to David Eugene Smith. Osiris 1: 4–8 (Photo at p. 4).

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11.47 Sergei L’vovich Sobolev (St. Petersburg, 1908 – Moscow, 1989): Vice-President 1967–1970, Member of the Executive Committee 1971–1974 Man Keung Siu

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Biography Sergei L’vovich Sobolev was born in St. Petersburg on 6 October 1908. He passed away in Moscow on 3 January 1989. He was bereaved of his father at 14 and was raised by his mother, a very educated woman who played an important role in his upbringing. She cultivated in the young Sobolev such qualities as adherence to principles, honesty and purposefulness, which characterized him as a scholar and a person throughout his life. Sobolev studied the course of secondary school largely on his own, already taking a great interest in mathematics. At 15, he entered Leningrad School No. 190 and graduated with distinction a year later. After M. K. Siu (*) University of Hong Kong, Hong Kong, SAR, China e-mail: [email protected]

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graduation, he was still too young to be admitted to a university, so he took up studying piano at the First State Art School. In 1925, he entered the Faculty of Physics and Mathematics of Leningrad State University, meanwhile continuing his study at the First State Art School. Sobolev was a truly worthy descendant of the St. Petersburg School, which is famous in the Russian history of mathematics through illustrious names such as Pafnuty Lvovich Chebyshev (1821–1894), Andrei Andreyevich Markov (1856–1922) and Aleksandr Mikhailovich Lyapunov (1857–1918). Indeed, at Leningrad State University and later at the Seismological Institute of the USSR Academy of Sciences (now the Russian Academy of Sciences) where he worked after graduation in 1929, Sobolev studied and did research work under the supervision of Nikolai Maksimovich Gjunter (1871–1941) and Vladimir Ivanovich Smirnov (1887–1974), who were themselves academic descendants of Chebyshev, Markov and Lyapunov. Sobolev’s first paper, which was published in 1929 in Doklady Akademii Nauk SSSR, spoke for his talent in mathematics, which was apparent at an early age. During a lecture in the course on partial differential equations given by Gjunter, the second-year student Sobolev raised doubts about a theorem taught in class. Following the advice of Gjunter, he read the original paper, discovered that the proof was incorrect and constructed a counterexample for the theorem. Much impressed by this piece of work, Gjunter recommended that Sobolev write it up and submit it. From then on, Sobolev went on to make fundamental contributions to many branches of mathematics, both pure and applied. He created new theories and trends in mathematics, most notably the origin of the theory of generalized functions (distributions) and the concept which has come to be known as Sobolev space. In 1933, he was elected a corresponding member in the Division of PhysicalNatural Sciences of the USSR Academy of Sciences. In 1939, at the age of 31, he was elected a full member in the same division and was for a long time the youngest full member ever elected to the Academy. His book, Nekotorye primeneniya funktsional’nogo analiza v matematicheskoi fizike (Applications of Functional Analysis in Mathematical Physics), of 1950 played an extremely important role in the development of functional analysis and partial differential equations and has educated several generations of mathematicians in these fields in Russia and abroad. It was translated into English and published by the American Mathematical Society in 1963, with a third edition published in 1991, four decades after it first appeared. From 1932, Sobolev worked at the Steklov Mathematical Institute which in 1934 was moved to Moscow. During World War II, he was the director of the Institute in evacuation to Kazan and after the re-evacuation (in 1941–1944). Later, he moved to the Institute of Atomic Energy (originally known as Laboratory No. 2). For the main task of investigating complex systems to obtain nuclear fuel, he worked with physicists and was involved with a lot of numerical computation at a time when computer was not yet available as a calculating tool. Sobolev gave nearly all his time, strength and energy to this task and made use of his unusual mathematical intuition and ingenuity to solve complex problems within the assigned time period. For the work he did at the Institute of Atomic Energy, Sobolev was twice awarded a First-Degree

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State Prize, and in 1951 he was awarded the title of a Hero of Socialist Labour. Sobolev later related that he developed a taste for computational mathematics through working at the Institute of Atomic Energy, so that he accepted an invitation to head the first Department of Computational Mathematics in Russia established at Moscow State University. Serving in that capacity from 1952 to 1959, he set the scientific and pedagogical direction of the department. He was one of the first Soviet mathematicians to use computers. Sobolev was among the initiators of the creation of new scientific centre in Siberia. When following the decision of the Soviet government the Siberian Branch of the USSR Academy of Sciences was formed, Sobolev was named the director of the Institute of Mathematics of the Siberian Division set up at Novosibirsk Akademgorodok. Responding to friends who asked him why he left a strong department at Moscow State University in favour of Siberia, which was essentially virgin scientific soil, to take on the complex and troublesome project of building an institution from scratch, Sobolev answered: “The natural desire of mankind is to live several lives and to begin something new” (Bakhvalov et al. 1988, p. 10). Above all, he took on the difficult task in Siberia because he considered this development to be one of the most important national problems. During the subsequent decade, under his leadership, the Institute of Mathematics (now the Sobolev Institute of Mathematics) became an internationally acclaimed centre of mathematical sciences. Although he himself had never studied cybernetics or mathematical economics, he recognized their importance and did everything he could for their development at the Institute. He was also one of the organizers of Novosibirsk State University, founded in 1958 to nurture young scientists for the Siberian Branch of the USSR Academy of Sciences. A new journal, Siberian Mathematical Journal, was founded, with Sobolev as the editor-­in-­chief from 1968 till 1988. In 1984, he went back to Moscow because of a deterioration in his health but continued to work at the Steklov Institute of Mathematics of the USSR Academy of Sciences. With his many doctoral students in the theory of partial differential equations, computational mathematics and mathematical physics, he was rightly one of the founders of the Soviet school in these areas. In 1981, he was awarded the Bolzano Gold Medal, and in 1989 the Lomonosov Gold Medal, and was decorated with honours by many foreign national academies.

Contribution to Mathematics Education Since his graduation from Leningrad State University, Sobolev was continually involved in pedagogical work. The lectures he gave at Moscow State University for the course Equations of Mathematical Physics were turned into a popular textbook that was translated into many foreign languages. He wrote many articles on the education of young people and was concerned about school education. In one article, he wrote:

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What is the most important thing that a scholar must cultivate in himself? He must rid himself of unnecessary ambition. He must not think that only geniuses can be happy. He must train himself to value even small achievements, to be glad, and never overestimate himself. He must develop in himself a love of work. He must try to foster in himself the joy of knowledge, which is almost the same as the joy of life itself. Happiness is in the fact that something in one's life was needed by people. (Cited in Bakhvalov et al. 1988, p. 14)

Sobolev supported the reform of mathematics education (known as the Kolmogorov reform) and objected to its critique by Lev Pontryagin (however, his objections were not published at this time). In 1967–1970 Sobolev was a vice-president and then in 1971–1974 a member of the Executive Committee of ICMI.  In 1970, at the International Congress of Mathematicians in Nice, he gave a talk in Section F2 Enseignement des mathématiques. In 1972, he was a plenary speaker at ICME-2 in Exeter. During the last year of his membership in the Executive Committee, Sobolev participated in the ICMIJSME Regional Conference (Tokyo, 5–9 November 1974) and delivered a plenary lecture entitled “The Experience of Siberian Mathematical Olympiads 1962-73” (ICMI Bulletin 4, 1974: 9). Sobolev once said: I live with the sensation that much was given to me earlier on credit: all my life I have strived to prove (although to myself) that all this was given to me for a purpose. (Bakhvalov et al. 1988, p. 14)

Sources Sobolev, Sergei L. 2006. Selected Works of S. L. Sobolev. Vol 1, eds. Gennadii V. Demidenko and Vladimir L. Vaskevich. New York: Springer. Bakhvalov, Nikolai S., Andrey A. Gonchar, Lev D. Kudryavtsev, Vyacheslav I. Lebedev, Sergei M.  Nikol’skii, Sergei P.  Novikov, Olga A.  Oleinik, Yurii G.  Reshetnyak, and Vasily S. Vladimirov. 1988. Sergei L’vovich Sobolev (On his eightieth birthday). Russian Mathematical Surveys 43(5): 1–18. Bitsadze, Andrei V., Leonid V.  Kantorovich, and Mikhail A.  Lavrent’ev. 1968. Sergei L’vovich Sobolev (On his sixtieth birthday). Russian Mathematical Surveys 23(5): 131–140. Maz’ya, Vladimir (ed.). 2009. Sobolev Spaces in Mathematics II: Applications in Analysis and Partial Differential Equations. New York: Springer.

Publications Related to Mathematics Education Sobolev, Sergei L. 1971. Quelques aspects de l’enseignement des mathématiques en U.R.S.S. In Actes du Congrès International des Mathématiciens, eds. Marcel Berger, Jean Dieudonné, Jean Leray, Jacques-Louis Lions, Paul Malliavin, & Jean-Pierre Serre, Vol. 3, 359–367. Paris: Gauthier-Villars. Sobolev, Sergei L. 1973. Some questions of mathematical education in the USSR. In Developments in mathematical education. Proceedings of the Second International Congress on Mathematical Education, ed. Albert Geoffrey Howson, 181–193. Cambridge: Cambridge University Press.

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Sobolev, Sergei L. 1978. Teaching mathematics in the Soviet Union (in Russian), in Na putyakh obnovleniya shkol’nogo kursa matematiki [Ways of renewing the school mathematics course], Moscow, 100–111. Sobolev, Sergei L. and Leonid V.  Kantorovich. 1979. Mathematics in the modern school (in Russian). Matematika v Shkole [Mathematics in School] 4: 6–10.

Photo Author: Jacobs, Konrad. Source: Archives of the Mathematisches Forschungsinstitut Oberwolfach.

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11.48 Hans-Georg Steiner (Witten a.d. Ruhr, 1928 – Bielefeld, 2004): Vice-President 1975–1978 Gert Schubring

Biography A Life Devoted to Mathematics Education Hans-Georg Steiner was one of the pioneers of the modern development of mathematics education135 and its establishment as a scientific discipline. Born on 21 November 1928 in Witten a.d. Ruhr, Steiner was 16 years old when World War II ended in Germany. He had been forced by the Nazi regime to become a member of the anti-aircraft protection forces in the last stages of the war. After the war, he completed his school education in the difficult conditions of post-war

 In Germany – as in France and Italy – Didaktik means the scientific investigation of teaching and learning processes. However, in the Anglo-Saxon world, “didactics” often has negative connotations and “mathematics education” is preferred. Since not all meanings of the German terms would be thus translated, we shall maintain the use of didactics to express the German meaning. 135

G. Schubring (*) University of Bielefeld, Bielefeld, Germany e-mail: [email protected]

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Germany.136 In 1949, he enrolled in mathematics and physics at the University of Münster. From his first days at university onwards, he showed a great interest in other subjects as well and was an active participant in lectures and seminars on linguistics and literature, philosophy and pedagogy. In 1955, he graduated with his first teacher state exam, followed by the second teacher state exam in 1957. In the ensuing 2 years, he worked as a research assistant of Heinrich Behnke at the Institute of Mathematics in Münster where he was responsible for the programme and content of the seminars on the didactics of mathematics held every Tuesday afternoon. Heinrich Behnke was from 1952  – the year of re-establishment of ICMI  – to 1966 a member of the ICMI Executive Committee, its president from 1955 to 1958 and later its vice-president. Within Germany, Behnke had succeeded in establishing in 1951 at Münster – the traditional home of high-quality mathematics teacher education – the first “seminar for didactics of mathematics” at a German university.137 Behnke soon realized the potential of the young Steiner and involved him in the organization of conferences while he still was an undergraduate student. The series was called Tagung zur Pflege des Zusammenhangs zwischen Höherer Schule und Universität (Münster Conferences for the Promotion of the Connection between School and University). Upon invitation by Behnke, Steiner delivered his first paper at the 1957 Zusammenhangs-Conference, on “The introduction of modern mathematical concepts to the mathematics classroom”. The lecture was published as an essay (Steiner 1959), which was later considered a key contribution to the reform of secondary mathematics education in Germany (Knoche 1988). It provided the starting point for an extensive sequence of papers and publications on theoretical as well as classroom-based topics in the field of mathematics education. Steiner’s extensive international involvement already started when he still was an undergraduate student. The international guests invited to Münster by Behnke, such as Howard Fehr and James Lighthill, brought Steiner quickly into contact with international developments. While collaborating with Behnke, Steiner became involved in the ICMI activities, and in 1962 he became a member of the German ICMI subcommittee himself. In the course of the ICMI subcommittee meetings, which took place under the leadership of Behnke at the University of Münster, Steiner met teachers and researchers interested in pedagogy, philosophy and didactics – the basis for intensive working contacts as well as personal friendships, for example, with Heinz Griesel, Günter Pickert and Hans Freudenthal. While working with Behnke, Steiner attended his first international congress in 1958  – the International Congress of Mathematicians (ICM) in Edinburgh, Scotland, where he met a number of leading US American curriculum reformers. These initial contacts were further established at the ICM 1962 in Stockholm and the ICM 1966 in Moscow. In 1960, Hans-Georg Steiner undertook his first visit to the United States where he established contacts within the rapidly developing curriculum reform movement.

 For a detailed account, see Steiner (1998).  Seminare were traditional forms in German universities for specialized studies in given disciplines and for closer contact with the professors. 136 137

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He was invited to give lectures in a summer institute for mathematics teachers at Columbia University in New  York. This first professional journey to the United States was followed by another series of invited lectures and workshops at several teacher colleges and universities across the country sponsored by the American Association for the Advancement of Science (AAAS). The result of these journeys was a developing cooperation with US American curriculum projects: the “Secondary School Mathematics Curriculum Improvement Study” (SSMCIS) founded at Columbia University and the “Comprehensive School Mathematics Program” (CSMP) initiated by Burt Kaufman and supported by the “Central Midwestern Regional Educational Laboratory” (CEMREL) in St. Louis, Missouri. Steiner became the European co-director of the CSMP, which was dedicated to the development of an enhancement programme for mathematically gifted students. A number of publications between the years 1967 and 1973 document the intensive and fruitful work of these two projects. Steiner’s key talents – initiating, fostering and extending international relationships in the field of didactics of mathematics – were already becoming evident. At the same time, he became acquainted with educational structures in which elementary and secondary schools were not separated from each other socially and conceptually. Indeed, the curriculum work of CSMP also involved elementary schools. School-related publications during this time address: –– –– –– ––

The teaching of logic in middle and high school The role of algebraic games for the understanding of algebraic structures The teaching of equations and functions The teaching of geometry

Several of these publications were fundamental analyses of lasting value. In 1964 and 1965, Steiner published two papers on a correspondence between Gottlob Frege and David Hilbert, which he commented on in great detail. Teaching experiments and didactical analyses on the topic of mathematization of political structures were a major focus of his mathematical as well as didactical work in the late 1960s and 1970s. Steiner’s “Mathematical Theory of Voting Bodies” (Eine mathematische Theorie von Abstimmungsgebilden) was also the core of this doctoral thesis completed in 1969 at the University of Darmstadt with Detlef Laugwitz (Steiner 1969). After his return from the United States, Steiner founded, together with Heinz Kunle, in 1968 the Zentrum für Didaktik der Mathematik in Karlsruhe. Together with the annual federal conferences on didactics of mathematics that began in 1967, the Zentrum provided a decisive means for bringing together the separate traditions of Volksschul-Methodiker and Gymnasialdidaktiker.138 Later, in 1969, Steiner founded, based on the Zentrum, the Zentralblatt für Didaktik der Mathematik (ZDM), the first truly international journal on the didactics of mathematics to be established in Germany. The journal was formed with two main purposes in

 The Volksschule, Realschule and Gymnasium are the components of the tripartite system of secondary education still to be found in most regions of Germany (see Schubring 2016). 138

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mind – to propagate research and to review international publications – key means for forging the scientific development of the new discipline. This probably constitutes the most salient achievement of Steiner both for the German and the international community. In May 1970, Steiner was appointed professor for didactics of mathematics at the Pädagogische Hochschule Bayreuth and was thus able to apply there his concepts in teacher education. Among other projects, he was involved in a teaching project for elementary schools, “Modern Mathematics in the First and Second School Years”. This did not aim to teach set theory, but rather to develop mathematical thinking with numbers and calculations. For the teacher trainee students, accompanying lectures offered reflections on practice and theory. Eventually, in 1973, he was appointed as one of the three foundation professors at the IDM – the Institut für Didaktik der Mathematik (IDM) at the University of Bielefeld, together with Michael Otte and Heinrich Bauersfeld, and developed there a comprehensive activity in which he was able, in particular, to further the programme in which he had been engaged so far  – namely establishing a scientific didactics with internationalization as an essential element of this process. In the structure of the IDM based on the division of labour between three main groups, Steiner led the group that concentrated upon curriculum development in the upper secondary grades. As in other aspects of the work of the IDM, this group was interdisciplinary and included mathematicians, didacticians of mathematics, sociologists and psychologists. Major projects that were undertaken included the reorganization of the upper grades of the Gymnasien, vocational education, statistics and the social role of mathematics. Following the introduction of pocket calculators and computers in the classrooms, the group’s research and developments involving the new media and technologies were both innovative and original (Schubring 2018). The Third International Congress on Mathematics Education (ICME-3, Karlsruhe, 1976) became an effective landmark in the process of the national and international disciplinary stabilization of the didactics of mathematics, following its predecessors in Lyon (1969) and Exeter (1972). Steiner, who from 1975 to 1978 was vice-president of ICMI and chair of the programme committee, succeeded in bringing together there the different, hitherto also institutionally separated strands and to engage them in lasting cooperation. An excellent means for realizing this was the new organization of the congress. This was organized into 13 sections which investigated central issues of mathematics instruction and which benefited from the careful preparation of the teams which were composed internationally and had worked together well before the congress. Steiner also succeeded, by means of the extensive integration of various competencies, to diminish the initial obvious tensions between the Bielefeld institute and the didacticians of mathematics working elsewhere in the Federal Republic. Besides his work on research and development at the IDM, it was always a central concern for Steiner to give conceptual stimulation and to help spread new ideas by organizing national and international colloquia and conferences. This stimulating function is certainly one of his most important contributions to the development

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of the didactics of mathematics, in particular for the Federal Republic. The series of these influential conferences first began in 1974 with the Regional ICMI-IDM Conference on the Teaching of Geometry. Of the other conferences, one should name the ICMI Symposium of 1978, the Education of Mathematics Teachers, and the larger conference of 1979 Comparative Studies of Mathematics Curricula  – Change and Stability. Steiner also gave innovative suggestions via smaller-scale working conferences, such as a series of colloquia devoted to the relation of educational history and history of mathematics. Other valuable forms for the exchange of ideas were the numerous bilateral conferences organized by him: for example, the German-Italian and the German-French symposia on recent developments in the didactics of mathematics. The conference series that proved in the long term to be probably the most important for the development of research was the series of symposia on TME: Theories of Mathematics Education established by him in 1985. TME had originated from his conviction that – thanks to the scientific level achieved – the ulterior major task lay in the advancement of theoretical approaches. At ICME-10  in Copenhagen, many participants still spoke to me about TME and told me how essential their participation in TME conferences had been for their scientific development. Finally, Steiner merits particular respect for having organized two conferences with participants drawn from the old Federal Republic and the GDR immediately after the “Wende” in the GDR and to have thus once again initiated communication and cooperation. Unfortunately, it was not granted to Steiner, after having become professor emeritus in 1993, to continue and to conclude his numerous projects, because, increasingly, a protracted serious illness impaired his working possibilities. He died on 14 December 2004 in Bielefeld.

Sources139 Essential Secondary Sources Biehler, Rolf, Roland Scholz, Rudolf Sträßer, and Bernhard Winkelmann (eds.). 1994. Didactics of mathematics as a scientific discipline. Dordrecht: Kluwer. Griesel, Heinz and Roland Fischer. 1988. Fachdidaktische Grundfragen des Mathematikunterrichts. Special Issue for Hans-Georg Steiner’s 60th birthday. Zentralblatt für Didaktik der Mathematik 20(5). Henning, Herbert and Peter Bender (eds.). 2003. Didaktik der Mathematik in den alten Bundesländern – Methodik des Mathematikunterichts in der DDR. Magdeburg & Paderborn:

 I am thanking Erika-Luise Steiner for her kind authorization to use the pictures from her late husband’s photo collection published in the website on the history of ICMI. 139

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Otto-von-Guericke-Universität Magdeburg & Universität Paderborn. http://www.math.uni-­ magdeburg.de/private/henning/tagung.pdf Knoche, Norbert. 1988. Hans-Georg Steiner 60 Jahre. Mathematische Semesterberichte 35(2): 147–161. Winkelmann, Bernhard (ed.). 1988. Wissenschaftliches Kolloquium Hans-Georg Steiner zu Ehren. IDM Occasional Paper 116. Bielefeld: Universität Bielefeld, IDM. Schubring, Gert. 2016. Die Entwicklung der Mathematikdidaktik in Deutschland. Mathematische Semesterberichte 63(1): 3–18. Schubring, Gert. 2018. Die Geschichte des IDM Bielefeld als Lehrstück. Ein Forschungsinstitut in einer Universität. Aachen: Shaker Verlag, 2018.

Obituaries Biehler, Rolf and Andrea, Peter-Koop. 2007. Hans-Georg Steiner: a life dedicated to the development of didactics of mathematics as a scientific discipline. ZDM – The International Journal on Mathematics Education 39(1): 3–30. Howson, A.  Geoffrey. 2004. Hans-Georg Steiner: some personal reminiscences. ICMI-Bulletin 55: 78–80. Schubring, Gert. 2004. Hans-Georg Steiner (21.11.1928–14.12.2004)  – A life for mathematics education. ICMI-Bulletin 55: 73–77. Schubring, Gert. 2004. Hans-Georg Steiner. Ein Leben für die Mathematik-Didaktik. Mitteilungen der Gesellschaft für Didaktik der Mathematik 79 (Dezember 2004): 94–98.

Publications Related to Mathematics Education A complete list of his publications is given in Biehler/Peter-Koop 2007. Some selected publications are mentioned here: Christiansen, Bent and Hans-Georg Steiner (eds.) (1979). New Trends in Mathematics Teaching, IV. Paris: UNESCO. Steiner, Hans-Georg. 1959. Das moderne mathematische Denken und die Schulmathematik. Der Mathematikunterricht 5(4), 5–79. Steiner, Hans-Georg. 1962. Die Verbindung von Logik und Mathematik im mathematischen Unterricht. Mathematisch-­Physikalische Semesterberichte 9 (1), 74–95. Steiner, Hans-Georg. 1962. Die Behandlung des Funktionsbegriffs in der höheren Schule. L’Enseignement Mathématique s. 2, 8: 62–92. Steiner, Hans-Georg. 1963. Explizite Verwendung der reellen Zahlen in der Axiomatisierung der Geometrie. Der Mathematikunterricht 9(4): 66–87. Steiner, Hans-Georg. 1964. Frege und die Grundlagen der Geometrie I. MathematischPhysikalische Semesterberichte 10(1): 35–47. Steiner, Hans-Georg. 1964. Moderne begriffliche Methoden bei der Behandlung der komplexen Zahlen. Der Mathematikunterricht 10(2): 5–35. Steiner, Hans-Georg. 1964c. Elementare Beweise zum Fundamentalsatz der Algebra. Der Mathematikunterricht 10(2): 60–93. Steiner, Hans-Georg. 1965. Frege und die Grundlagen der Geometrie II. MathematischPhysikalische Semesterberichte 10(2): 175–186. Steiner, Hans-Georg. 1965. Wie steht es mit der Modernisierung unseres Mathematikunterrichts? Mathematisch-Physikalische Semesterberichte 11(2): 186–200. Steiner, Hans-Georg. 1965. Menge, Struktur, Abbildung als Leitbegriffe für den modernen mathematischen Unterricht. Der Mathematikunterricht 11(1): 5–19.

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Steiner, Hans-Georg. 1969. Eine mathematische Theorie der Abstimmungsgremien [Dissertation]. Fakultät für Mathematik und Physik, Technische Universität Darmstadt. Steiner, Hans-Georg. 1977. ICMI and congress recommendations. Conferences 1972–1976. In Proceedings of the 3rd International Congress on Mathematical Education, eds. Hermann Athen and Heinz Kunle, 383–388. Karlsruhe: ZDM. Steiner, Hans-Georg. 1978. Zur Entwicklung der Didaktik der Mathematik. In: H.-G. Steiner (ed.), Didaktik der Mathematik. Wege der Forschung (pp. ix–xlviii). Darmstadt: Wissenschaftliche Buchgesellschaft. Steiner, Hans-Georg. 1979. The present situation in the development of mathematics curricula for the primary school. A critical survey. In L’Insegnamento integrato delle Scienze nelle Scuola Primaria, 245–269. Roma: Accademia Nazionale dei Lincei. Steiner, Hans-Georg. 1985. Theory of Mathematics Education (TME): An Introduction. For the Learning of Mathematics 5(2): 11–17. Steiner, Hans-Georg. 1992. Recent and coming activities of the international study group on theory of mathematics education (TME). Zentralblatt für Didaktik der Mathematik 24(2): 67–68. Steiner, Hans-Georg. (1998). Didactics of mathematics as a scientific discipline—A sketch of its development from a personal point of view. Zentralblatt für Didaktik der Mathematik 30(6), 224–232. Steiner, Hans-Georg and Alfred Vermandel (eds.). 1988. Foundations and Methodology of the Discipline Mathematics Education (Didactics of Mathematics). Proceedings of the 2nd international TME-conference, Bielefeld, July 15–19, 1985. Antwerpen/Bielefeld: Universiteit Antwerpen.

Photo Courtesy of Gert Schubring.

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Marshall Harvey Stone

11.49 Marshall Harvey Stone (New York, 1903 – Madras, 1989): Ex Officio Member 1952–1954, Vice-President 1955–1958, President 1959–1962 Jeremy Kilpatrick

Biography Marshall Harvey Stone was born on 8 April 1903 in New York City and attended public schools in Englewood, New Jersey. He was the son of Harlan Fiske Stone, who was professor and dean at the Columbia University Law School and later the Chief Justice of the United States (1941–1946). Stone’s family expected him to become a lawyer like his father, but he developed an interest in mathematics while an undergraduate at Harvard University, receiving his AB when he was 18. He completed a Harvard PhD in 1926, with a thesis on differential equations supervised by George Birkhoff. Between 1925 and 1937, he taught at Harvard, Yale and Columbia. He was promoted to full professor at Harvard in 1937. During World War II, Stone did classified research as part of the Office of Naval Operations and the Office of the Chief of Staff of the War Department. In 1946, he became chairman of the mathematics department at the University of Chicago, a post he held until 1952. He remained on the faculty at Chicago until his retirement in 1968. J. Kilpatrick (1935–2022) University of Georgia, Athens, GA, USA

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The department that Stone joined in 1946 was in the doldrums, after having been at the turn of the twentieth century arguably the best American mathematics department, thanks to the leadership of Eliakim Hastings Moore. Stone did an outstanding job of making the Chicago department eminent again, mainly by hiring Paul Halmos, André Weil, Saunders Mac Lane, Antoni Zygmund and Shiing-Shen Chern. The years of Stone’s chairmanship became known as “the Stone Age” at Chicago. Stone did much of his important work in mathematics during the 1930s, including his 1930 proof of the celebrated Stone-von Neumann uniqueness theorem, his definitive 1932 monograph on linear transformations in Hilbert space, his work in 1934 on what is now called Stone-Čech compactification theory and his 1937 proof of the famous Stone-Weierstrass theorem. After Stone retired from Chicago, he accepted the newly created George David Birkhoff Professorship of Mathematics at the University of Massachusetts in Amherst. He occupied that position full-time until 1973 and half-time thereafter until his second retirement in 1980. He was an avid traveller who served as a visiting professor at many universities around the world. He died suddenly on 9 January 1989  in Madras, India, where he had gone to attend a music festival. He was survived by his second wife, three children, one stepchild, six grandchildren and two great-grandchildren. Stone received the US National Medal of Science in 1983. He held seven honorary degrees, was a member of the National Research Council and had been elected to the National Academy of Sciences in 1938. He was president of the American Mathematical Society in 1943–1944. From 1948 to 1950, Stone directed the worldwide work that led to the re-establishment of the International Mathematical Union after World War II. He served on the Interim Executive Committee of the new IMU from 1950 to 1952 and was its first president from 1952 to 1954.

Contribution to Mathematics Education During his term as ICMI president, Stone initiated a number of important activities in education. In 1959, he chaired an influential conference of mathematicians and educators at Royaumont, France, that was sponsored by the Organisation for European Economic Co-operation (OEEC). In his introductory address, Stone, after listing the signs of change and highlighting the need for a modern spirit in mathematics education, addressed the following aspects: problems of mass education at secondary level; mounting pressure on teachers; coordination needed with science teaching; making mathematics attractive to young people; introducing a new curriculum and coordinating it with science subjects; use of films and television; broadening research in the teaching of mathematics. Stone also helped initiate the First Inter-American Conference on Mathematics Education, which was held in Bogotá, Colombia, in 1961. Prestigious mathematicians from the United States and Europe attended the conference, as well as representatives from almost every country of the

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Americas. As a result, the Inter-American Committee on Mathematics Education (Comité Interamericano de Educación Matemática (CIAEM)) was founded, with Stone as its first president (from 1961 to 1972). In the United States, Stone was an articulate, unabashed advocate of reforming school and college curricula to include modern mathematics (Demott 1962; Zitarelli 2001). In a 1961 article addressed to US college teachers, he sketched the revolutionary developments in mathematics during the twentieth century and argued that the teaching of mathematics needed to be modernized accordingly: The college can hardly undertake a richer and more advanced program in mathematics unless it is willing to eliminate a considerable amount of deadwood from the traditional college curriculum. There are a good many obvious candidates for elimination, subjects which should either be taught in high school or not be taught at all. Let us cite college algebra, solid geometry, most of numerical trigonometry, descriptive geometry, and some topics in the calculus. Since a great many-perhaps a majority-of our colleges are still teaching these subjects, the work of reform has to begin by banishing them from the college curriculum. Only when this has been accomplished can the college mathematics program be given its proper scope and brought to the proper level of quality. (Stone 1961a, p. 732)

Reviewing the Cambridge Conference Report Goals for School Mathematics, seen by many as radical, Stone (1965) was disappointed that the conferees had been so timid in setting curricular goals: I am in no doubt whatsoever that the Report by implication rejects many of the bolder, more imaginative, and more profound modifications of the school mathematics curriculum that are beginning to win acceptance in Europe, both in the west and in the east. (Stone 1965, p. 353)

He saw the report not as daring, comprehensive or progressive, but rather as superficial, confused and full of wishful thinking: It is superficial in its treatment of the content and organization of the mathematics curriculum; confused in the presence of the very deep reasons for anew, modern approach to algebra and geometry at the school level; and willful in its refusal to face up to the pedagogical difficulties involved in the sweeping changes it proposes. (Ibidem, p. 354)

Stone ended his critique by pointing to the “outmoded and inadequate preparation we are still giving to our future mathematics teachers” (Ibidem, p. 360) as the real obstacle to reforming school mathematics.

Sources Stone, Marshall H. 1932. Linear transformations in Hilbert space and their applications to analysis. American Mathematical Society Colloquium Publications 15. Providence, RI: American Mathematical Society. Stone, Marshall H. 1936. The theory of representations for Boolean algebras. Transactions of the American Mathematical Society 40: 37–111. Stone, Marshall H. 1937. Applications of the theory of Boolean rings to general topology. Transactions of the American Mathematical Society 41, 375–481.

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Stone, Marshall H. 1967. Mathematics and the future of science. Bulletin of the American Mathematical Society 63: 61–76. Stone, Marshall H. 1976. A reminiscence on the extension of the Weierstrass approximation theorem. Historia Mathematica 3: 328. Browder, Felix E. 1989. The Stone Age of mathematics on the Midway. Mathematical Intelligencer 11(3): 20–25. De Mott, Benjamin H. 1962 (Spring). The math wars. American Scholar 31: 296–311, reprinted in New curricula, ed. R. W. Heath, 54–67. 1964. New York: Harper & Row. Kolata, Gina. 1989 (January 11). M. H. Stone, acclaimed mathematician, dies at 85. The New York Times B10. Mackey, George W. 1989. Marshall Harvey Stone. Notices of the American Mathematical Society 36: 221–223. Parshall, Karen. 2009. Marshall Stone and the internationalization of the American mathematical research community. Bulletin of the American Mathematical Society n. s., 46: 459–482. Zitarelli, David E. 2001. Towering figures in American mathematics, 1890–1950. American Mathematical Monthly 108: 606–635. Zund, Joseph D. 1999, Stone, Marshall Harvey, American National Biography, 20, 864–865, retrieved from the American National Biography web site: http://www.anb.org/articles/13/13­02509.html

Publications Related to Mathematics Education Stone, Marshall H. 1961a. The revolution in mathematics. American Mathematical Monthly 68: 715–734. Stone, Marshall H. 1961b. Reform in school mathematics. In New Thinking in School Mathematics. 14–29. Paris: OEEC. Stone, Marshall H. 1963. La choix d’axiomes pour la géométrie à l’école. L’Enseignement Mathématique s.2, 9: 45–55. Stone, Marshall H. 1964 (July–December). Reflections on the teaching of mechanics. Mathematics Student (Indian Mathematical Society) 32, nos. 3 and 4. Stone, Marshall H. 1965. Review of Goals for school mathematics: The report of the Cambridge Conference on School Mathematics. Mathematics Teacher 58: 353–360.

Photo Source: Celebratio Mathematica. Marshall Harvey Stone. Mathematical Sciences Publishers. Design & Software © 2012–2022 MSP.

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Stefan Straszewicz

11.50 Stefan Straszewicz (Warsaw, 1889 – Warsaw, 1983): Vice-President 1963–1966 Ewa Lakoma

Biography Stefan Straszewicz born in Warsaw on 9 December 1889 was an outstanding Polish mathematician, a member of the Warsaw branch of the Polish School of Mathematics (a group belonging to the most important and creative mathematicians of the twentieth century) and a propagator of mathematics and significant educator. He was born and grew up during a difficult period in the history of Poland: the country was not independent and had been divided up between Austria, Prussia and Russia. Warsaw was occupied by the Russians, who tried to suppress Polish patriotism by strict injunctions against the use of Polish language. With regard to education, for Poles who lived in the part of the country under Russian occupation, the only possibility to be taught in a secondary school or at a university (the University of Warsaw in the years 1870–1915) was an education in Russian language (Karp and Vogeli 2010; Kitzwalter 2016). Polish society resisted these methods of domination and regarded the Polish education of young people as the most important E. Lakoma (*) Military University of Technology, Warsaw, Poland e-mail: [email protected]

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means for preserving the Polish nationality. Support for talented students was organized by establishing private schools and courses at the university level, as well as by providing funds for their education in leading scientific centres abroad (an idea that was strongly developed by Samuel Dickstein). In 1915, during World War I, in Warsaw political conditions permitted the founding of the Polish University of Warsaw (Kitzwalter 2016). In November 1918, Poland regained its independence. Straszewicz first attended the secondary school in the town Skierniewice, then continued his education in Warsaw, finally finishing the gymnasium in 1906 in the city Białystok (Zaleski 1989). In the years 1906–1907, he studied mathematics at the Mathematics and Science Faculty of the Society of Scientific Courses (Towarzystwo Kursów Naukowych) in Warsaw, the only school at university level, with Polish language. This Society was founded by Poles (at the initiative of Samuel Dickstein) after the Main School (Szkoła Główna) in Warsaw had been closed by Russian occupants (1869). Straszewicz, as a gifted student of the faculty, left Warsaw in 1907 to study in Zurich (Switzerland) at the University and at the Institute of Technology (1907–1911). He attended lectures of Adolf Hurwitz (1859–1919) and Ernst Zermelo (1871–1953) in mathematics and of Albert Einstein (1879–1955) in physics. In 1911, he returned to Warsaw and worked there for the next 2 years as a mathematics teacher in a secondary school. In 1913, he went to Switzerland again in order to continue his scientific research in Zurich as a student of Ernst Zermelo. In 1914, Straszewicz published his doctoral thesis, entitled “Beiträge zur Theorie der Convexen Punktmengen” (Zürich: J.J.  Meier) and obtained a PhD from the University of Zurich, Faculty of Mathematics and Science. He worked at that university until 1919. At the same time, he was a mathematics teacher in a secondary school in Zurich. After his return to Warsaw in 1919, Straszewicz worked as a teacher of mathematics in a gymnasium and as a lecturer at the Warsaw University of Technology. In 1921, he obtained a scientific position at the University of Warsaw and continued his research. In 1925, he published the work entitled “Über die Zerschneidung der Ebene durch abgeschlossene Mengen”,140 which was the basis for his qualification as a professor. In 1928, Straszewicz obtained a position as a professor in mathematics at the Warsaw University of Technology in the Engineering Faculty. From 1932 to 1935 he was a dean of the faculty and in the years 1938–1939 a vice-­president of that university. In September 1939, at the beginning of World War II, Warsaw was under Nazi occupation and the University of Technology was closed by the Germans. Straszewicz was forced to leave Warsaw. He moved to Vilnius and worked there as secondary school teacher. In 1942, he returned to Warsaw and worked at the State Higher School of Technology (1942–1944), which recruited professors from the University of Technology for its staff and gave to students an education in Polish language, but only at postsecondary level, due to the strong restrictions of German

 Straszewicz, Stefan. 1925. Ueber die Zerschneidung der Ebene durch abgeschlossene Mengen. Fundamenta Mathematicae 7: 159–187. Warsaw: Polish Mathematical Society. 140

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occupants. From 1942 to 1945, Straszewicz was actively involved in organizing and giving clandestine teaching at the university level. He was the president of the clandestine University of Technology. He was also actively engaged in saving the property of the University of Technology and its scientific achievements as widely as possible that time. After World War II, Straszewicz was very active in rebuilding the Warsaw University of Technology and restoring it as leading scientific and educational centre. From 1945 to 1960 he was a head of the mathematics department and from 1948 to 1951 was a vice-president of that university. In 1974, he was awarded an honorary doctorate degree by the Warsaw University of Technology. From 1945 to 1951, Straszewicz also gave lectures on geometry and mathematical analysis at the University of Warsaw. As a scientist, Straszewicz is known due his work, entitled “Über exponierte Punkte abgeschlossener Punktmengen” (1935).141 His scientific research concerned mainly topology (from 1913 on), set theory and geometry. A generalization of the thesis of Zygmunt Janiszewski (1888–1920) on cutting the plane was one of the most important results of his scientific research. Moreover, he was an author of the method concerning drawing tangent lines to curves, elaborating on the example of the ellipse. He also gave a new definition of the notion of an arc and worked out the basic equations of spherical trigonometry. The main results of Straszewicz scientific work were the points of departure for outstanding mathematicians as Kazimierz Kuratowski (1896–1980), Stefan Mazurkiewicz (1988–1945) and Bronisław Knaster (1893–1980). Straszewicz was also involved in creating new Polish terminology of concepts appearing in mathematics in the twentieth century, especially in elementary geometry and differential geometry. He translated the work of Richard Dedekind142 into Polish and was a co-translator into Polish of the work of Edouard Goursat.143 Straszewicz was also strongly involved in editing scientific and pedagogical journals. He was a co-creator and editor of Przegląd Matematyczny i Fizyczny (Mathematical and Physical Review, 1924). Together with Antoni Marian Rusiecki, he edited the journal Parametr (Parameter) (1930–1939, published in Warsaw by Księgarnia św. Wojciecha), which was dedicated to secondary school mathematics teachers and included an appendix, titled Młody Matematyk (Young Mathematician, 1931–1932). Another journal, Matematyka i Szkoła (Mathematics and School), was edited by Straszewicz in the years 1937–1939. After World War II, in 1948, Straszewicz co-created the journal Matematyka (Mathematics) and was a member of its editorial committee until 1970.

 Straszewicz, Stefan. 1935. Ueber exponierte Punkte abgeschlossener Punktmengen. Fundamenta Mathematicae 24, 139–143. Warsaw: Polish Mathematical Society. 142  Dedekind, Richard. 1914. Ciągłość i liczby niewymierne [Stetigkeit und irrationale Zahlen]. Warsaw: Gebethner i Wolff. 143  Goursat, Edouard. 1914. Kurs analizy matematycznej [Cours d’analyse mathematique]. Warsaw: Kasa im. Mianowskiego Instytutu Popierania Nauki. 141

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Starting from the 1930s until the late 1970s, Straszewicz wrote many textbooks on mathematics, algebra, geometry for lyceums, technical and vocational schools, that had numerous editions. Straszewicz died in Warsaw on 10 December 1983. His tomb is at the Bródno Cemetery in Warsaw.

Contribution to Mathematics Education In addition to the significant achievements in mathematical research, Straszewicz rendered great services in the domain of the methodology of mathematics teaching and in improving curricula for elementary and secondary levels, which were used to many generations of students, raising the level of their mathematical knowledge. The textbooks written by Straszewicz included new didactical ideas in school mathematics curriculum, for example, the idea of introducing the concepts of geometric transformations: symmetries and homotheties as a basis for elementary geometry teaching. He was also a lecturer and mentor to many generations of mathematicians and mathematics teachers. He was an active member of the Polish Mathematical Society (from 1920 on): its president from 1953 to 1957 and an honorary member from 1969 on. In 1979, he was awarded the prestigious prize named after Samuel Dickstein from the Polish Mathematical Society for his achievements for the benefit of mathematical culture. Straszewicz was also a chair of the Polish Commission for Mathematical Curricula and Textbooks for Elementary and Secondary Schools in Poland from 1957 to 1975. He is perhaps best remembered for having founded in 1949 the Polish Mathematical Olympic Games, an annual competition for students especially talented in mathematics. Of all his activities, this was his “baby”: he elaborated the methodological basis and the rules of competition, was a head and co-organizer of all Mathematical Olympiads from 1949 to 1969. It is worth noting that the majority of the laureates of these Olympiads became outstanding mathematicians, now famous throughout the scientific world. As a mathematician and educator, Straszewicz was well known in international scientific bodies. He was an active member of ICMI as the national delegate of Poland to the commission and as a vice-president from 1962 to 1966. He attended many international meetings: from 1932 on and after World War II in 1957–1972, he was a delegate of Poland and also a chair of sections concerning mathematics teaching. In particular, he participated together with Samuel Dickstein in the International Congress of Mathematicians in 1932  in Zurich as a delegate of the Ministry of Polish Education, University of Warsaw and Polytechnic School. He prepared the report entitled “Pologne. La préparation des professeurs des mathématiques de l’enseignement secondaire”, which was published anonymously in L’Enseignement Mathématique (1933. 32: 365–374). In ICM 1936 in Oslo, he presented the report “Die gegenwärtigen Entwicklungstendenzen im mathematischen

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Unterricht in Polen” (Ibidem, 1937. 36: 377–384), and in ICM 1962 in Stockholm, he reported on the “Relations entre l’arithmétique et l’algèbre dans l’enseignement des mathématiques pour les enfants jusqu’à l’age de quinze ans” (Ibidem, 1964. s. 2, 10: 271–293). In this last report, Straszewicz examines the reports submitted by the national subcommissions of 11 countries, giving particular attention to some of the topics addressed: algebraic notation, equations, sets, functions and relationships and the concept of number. He notes that the general tendency is the following: to bring school mathematics education - even in the lower classes – closer to contemporary science and new applications by gradually introducing modern mathematical language. For example, it is proposed to introduce the simplest notions of algebra of sets and propositional logic, to better highlight the structural properties of the different number systems considered … (p. 272)144

Sources145 Straszewicz, Stefan. 1925. Über die Zerschneidung der Ebene durch abgeschlossene Mengen (On the cutting of the plane by the closed sets). Fundamenta Mathematicae. Vol. 7. Warsaw: Polish Mathematical Society. Straszewicz, Stefan and Kazimierz Kuratowski. 1928. Generalization d’une theorie de Janiszewski. Fundamenta Mathematicae 12:19–28. Straszewicz, Stefan. 1935. Über exponierte Punkte abgeschlossener Punktmengen. Fundamenta Mathematicae. Warsaw: Polish Mathematical Society. Vol. 24, 139–143. Straszewicz, Stefan. 1957. Sur la trigonométrie de Lobatschevsky. Annales Polonici Mathematici 3: 225–239. Straszewicz, Stefan. 1974. Geometria wykreślna (Descriptive Geometry). Warsaw: WSiP. (Also: 1971. Warsaw: PZWS). Karp, Alexander and Vogeli, Bruce (eds.). (2010). Russian mathematics education: History and world significance (Vol. 4). London-New Jersey-­Singapore: World Scientific. Kitzwalter, Tomasz (ed.). 2016. History of the University of Warsaw 1816–2016. Warsaw: University of Warsaw Publishers. Kowalski, Janusz. 2004. The Polish Mathematical Society (PTM). European Mathematical Society Newsletter 54: 24–29. Kuratowski, Kazimierz. 1973. Pół wieku matematyki polskiej 1920–1970 [A Half Century of Polish Mathematics 1920–1970]. Warsaw: Książka i Wiedza. English version: 1980. A Half Century of Polish Mathematics: remembrances and reflections. Oxford: Pergamon Press Oxford. Warsaw: PWN.

 The original text is: “rapprocher l’enseignement scolaire des mathématiques - même dans les classes inférieures - de la science contemporaine et des nouvelles applications en faisant introduire progressivement le langage mathématique moderne. On propose, par exemple, d’introduire assez tôt les plus simples notions de l’algèbre des ensembles et de la logique propositionnelle, de faire mieux ressortir les propriétés structurales des différents systèmes de nombres considérés …”. 145  Most books and works of Straszewicz are available in Poland at the National Library in Warsaw, the  Library of  the  University of  Warsaw, the  Jagiellonian Library in  Cracow, the  Library of the Nicolaus Copernicus University in Toruń, the Library of the Adam Mickiewicz University in  Poznań, the  Library of  the  University of  Gdańsk and  the  Central Mathematical Library of the Polish Academy of Sciences in Warsaw. 144

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Rapport du Cercle mathematico-physique de Varsovie. 1911. L’Enseignement des mathématiques et de la physique dans les écoles privées de la Pologne, L’Enseignement Mathématique 13: 299–319. Zaleski, Jan. 1989, Stefan Straszewicz 1889–1983. Warszawa, Pracownia Historyczna BGPW. s. 100. Available at http://bcpw.bg.pw.edu.pl/dlibra/docmetadata?id=886&from=&dirids=1

Publications Related to Mathematics Education Straszewicz, Stefan. 1924. O wielobokach [On Polygons]. Lvov: Książnica – Atlas. Dickstein, Samuel and Stefan Straszewicz. 1933. POLOGNE. La préparation des professeurs de mathématiques de l’enseignement secondaire. L’Enseignement Mathématique 32: 365–374. The report is not signed, but attributed to Straszewicz in the list of authors at the end of the issue; Samuel Dickstein named in the same list without page numbers is probably his co-author. Straszewicz, Stefan. 1934. Geometrja dla II klasy gimnazjalnej [Geometry for the 2nd Grade of Gymnasium]. Lvov: Książnica – Atlas. Straszewicz, Stefan. 1937. Die gegenwärtigen Entwicklungstendenzen in mathematischen Unterricht in Polen. L’Enseignement Mathématique 36: 377–384. Straszewicz, Stefan. 1938. Geometria analityczna [Analytical Geometry]. Lvov: Książnica – Atlas (second edition: Wrocław 1947). Straszewicz, Stefan. 1937. Matematyka. Algebra, trygonometria i geometria wykreślna [Mathematics. Algebra, trigonometry and descriptive geometry]. Lvov: Książnica  – Atlas (other edition: Wrocław 1949). Straszewicz, Stefan. 1947. Algebra [Algebra] Wrocław: Książnica – Atlas. Straszewicz, Stefan. 1947. Geometria [Geometry]. Warsaw: PZWS. Other edition 1948. Wrocław: Książnica – Atlas. Straszewicz, Stefan. 1948, Repetytorium elementów matematyki [Mathematics Element Repository]. Warsaw: Trzaska, Evert, Michalski. Straszewicz, Stefan. 1951. Zadania z I Olimpiady Matematycznej [Mathematical problems from the first Polish Mathematical Olympiad]. Warsaw: PZWS.  In the years 1952, 1953, 1954 Straszewicz published analogous books for the second, third, and fourth Olympiads. Straszewicz, Stefan. 1956. Zadania z Olimpiad Matematycznych [Mathematical Problems and Puzzles from the Polish Mathematical Olympiads]. Warsaw: PZWS (other editions: 1960, 1961, 1967, 1972). English version: volume 12 of the series Popular Lectures in Mathematics. Oxford: Pergamon Press, 1965. Straszewicz, Stefan. 1964. Relations entre l’arithmétique et l’algèbre dans l’enseignement des mathématiques pour les enfants jusqu’à l’âge de quinze ans, L’Enseignement Mathématique s. 2, 10: 271–293. Straszewicz, Stefan, Stefan Kulczycki, and Zofia Krygowska. 1957. Nauczanie geometrii w klasach licealnych szkoły ogólnokształcącej [Teaching geometry in high school grades of general education]. Warsaw: PZWS. Straszewicz, Stefan, Jerzy Browkin, and Jan Rempała. 1975. Dwadzieścia pięć lat olimpiady matematycznej [Twenty-five years of the Polish Mathematical Olympiads]. Warsaw: WSiP.

Photo Source: https://www.ptm.org.pl/galeria/prezesi-­ptm/stefan-­straszewicz-­11 (Polskie Towarzystwo Matematyczne).

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János Surányi

11.51 János Surányi (Budapest, 1918 – Budapest, 2006): Vice-President 1971–1974 László Surányi

Biography János Surányi was born on 29 May 1918 and passed away on 8 December 2006 in Budapest, Hungary. His family background is interesting to note: his parents met in the Galileo Circle, a group formed in the 1910s by progressive youths mainly from the capital. One of his aunts was the known painter Margit Gráber; his cousin was the biochemist Endre Bíró, a student of Albert Szent-Györgyi, later a founding Head of the Department of Biochemistry of Eötvös Loránd University in Budapest, also the first translator into Hungarian of parts of Joyce’s Finnegans Wake. Through another cousin, he came into contact with the thinker and philosopher Lajos Szabó, who became one of his mentors during his youth; in turn, his own letter to Szabó on Gödel’s Incompleteness Theorem influenced Szabó’s thinking on “language mathesis”; see http://lajosszabo.com/SZL/indexSZLangol.htm (retrieved on 27 March 2021). L. Surányi (*) Independent Researcher, Budapest, Hungary

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Surányi studied at the University of Szeged from 1936, where he encountered mathematical logic, linking his interests in mathematics and philosophy. He became first a student, then an assistant of László Kalmár, the leading Hungarian authority on the subject. He completed a doctoral dissertation in 1943 on the reduction theory of the Decision Problem: as one cannot give an algorithm to decide whether a given logical formula is universally valid, it becomes an interesting question to look for classes of decidable formulae, as well as formulae to which all others reduce. In 1953, he completed his kandidatúra (candidature, a form of higher doctorate), and in 1957 he became a Doctor of the Hungarian Academy, submitting dissertations on this general subject. His results, which essentially close this area of research, are summarized in his book (Surányi 1959). Besides mathematical logic, he worked also on number theory and graph theory. Until 1948, he was employed at the University of Szeged; in 1951, he joined the Department of Algebra and Number Theory of Eötvös Loránd University (as mathematical logic did not have its own department at this point in Budapest). In 1960, he was promoted to a Professorial Chair, and from 1976 till his retirement in 1988, he was the Head of Department. Besides books and publications, he presented his results at national and international congresses and conferences. Alongside his research activities, he was an active reformer and organizer of the teaching of mathematics in Hungary at Secondary School level. He was VicePresident of ICMI (the International Commission on Mathematical Instruction) in the years 1971–1974, and member at large in 1975–1978. He was also an elected member of CIEAEM (Commission Internationale pour L’Etude et L’Amélioration de L’Enseignement des Mathématiques, International Commission for the Study and Improvement of Mathematics Teaching). He was President of two IMOs (International Mathematical Olympiads) held in Hungary.

Contribution to Mathematics Education  athematical Community Organization and Contributions M to the Modernization of the Teaching of Mathematics During World War II, János Surányi spent more than 2 years in forced labour camps, then was captured by German troops, and finally on his way home after the Allied victory became severely ill and spent time in a makeshift Hungarian hospital. Having made a full recovery, he threw himself into mathematics. He became a community organizer: he was one of the drivers of the re-establishment of the János Bolyai Mathematical Society, the society of all Hungarian mathematicians and mathematics teachers. He was its General Secretary and then President for 15 years. His idea was to create a real “society”. For him, the institutional structure was only a means to the creation of a friendly environment where researchers, lecturers, teachers could feel at home and discuss their problems; and a context where forward-looking initiatives could be realized. On the other hand, the Society gave a

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significant degree of protection to the mathematical community against the consolidating one-party dictatorship. It allowed those working in particular on the modernization of mathematics teaching to form a kind of movement, thereby enabling them to fight the strong political and societal opposition against their plans. Surányi himself carried out a large share of this work. In 1947, together with Paula Soós, he restarted the legendary KöMaL (Mathematical and Physical Journal for High Schools) that had already played an important role in the development of mathematics in Hungary in the early twentieth century. Surányi’s generation had grown up solving the problems and reading the articles of this journal, and during his Editorship, that only ended in 1970, he nurtured and extended this tradition. He carried on publishing in this journal late into his career. He also helped restart secondary school mathematics competitions, such as the Loránd Eötvös (later József Kürschák) Competition, that had been organized every year since 1894 apart from War years; he chaired its Organising Committee, contributing to its direction and outlook. His work is reflected in the thorough but at the same time easy-to-follow collection of competition problems and solutions, the Hungarian Problem Book (Kürschák, Hajós, Neukomm and Surányi, 1955); after his fellow authors had passed away, he himself carried on writing this series from Volume III (Surányi, 1992). These collections were translated into several languages, including English, Japanese and Russian. His book Topics in the Theory of Numbers (Erdős and Surányi 1960, English version 2003), written with the collaboration of Pál Erdős, had a major impact.146 The detailed list of topics was drawn up by the two authors working together, but Surányi wrote a large part of the text himself. He appeared to have found the right language to present some substantial theorems of number theory to interested senior secondary school students, who may yet lack a deeper understanding of mathematics. “I read my copy of this book so much it fell apart” says the Academician Miklós Laczkovich in an interview given to his son László. In his speech given at the burial of Surányi’, the Wolf Prize and Abel Prize-winning László Lovász summarizes what many feel: “I only really understood what is mathematics from the beautiful proofs of this book.” He goes on to say: “Later, as a student, I often went to his [of Surányi] apartment, where he taught me number theory, presented some problems, gave me things to read. In his explanations, he always concentrated on the essence of the argument; he showed us how proofs emerge, how one could have discovered them”; see (Surányi, László 2017) for both quotes. It was in Surányi’s nature not to find it demeaning to occupy himself with problems of secondary school students; he approached them in the same way as everybody else: his university students, members of his department or schoolteachers. This derived from his innate democratic spirit, as did the fact that the popularization of mathematics was also important for him; he wrote popular articles in several

  In 1971, Surányi participated to the seminar on Zahlentheorie at the Mathematisches Forschungsinstitut in Oberwolfach. 146

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languages (Surányi 1968; 1969; 1981; 1993). His many elementary writings on mathematics are mostly in Hungarian; these are not recorded in the bibliography below. The most important issue for him was to turn mathematics into a positive experience for students. He saw very clearly that this requires the modernization of the teaching of the subject. For him, the experience of mathematics consisted of the unity of logical rigor and a playful approach. Thus he regarded the conveying of this unity as the main purpose of the teaching of mathematics, as well as the development of independent thinking in students; for this, he saw it as essential that lessons should be conducted in a spirit of freedom, moving away from the frontal teaching method, popular at the time. On the other hand, he sought to move away from the traditional development of the subject, largely based on numeracy and routine exercises, towards a thinking- and discovery-­based approach, one that allows for the emergence of mathematical notions. With time, more and more of his effort was spent on developing these principles, and the pedagogical methods that accompany them. The necessary teaching experiments were started by Tamás Varga at the elementary school level. Alongside these, the secondary school experiments were designed and led by Surányi himself. For a long while, these had an informal form, as there was a lot of political and societal resistance, alongside resistance by the teachers. But eventually, he managed to organize a small but enthusiastic and active community of teachers who were willing to try the new methods based on discovery and the arousal and development of learners’ interest in the subject. The Bolyai Society helped here too: at their request, weekly-fortnightly seminars were held by Surányi and others at the Mathematical Institute of the Hungarian Academy, led by Alfréd Rényi, which soon took on the shape of a movement. Later, a Group of Didactics was also founded at the Institute with Surányi’s leadership, which provided teaching materials for the experimenting teachers. Finally in 1962, it suddenly became possible to start a special mathematics secondary school stream (until then, only the Russian stream had been allowed to deviate from the centrally determined curriculum), and a special mathematics syllabus needed to be put together; Surányi oversaw this process too, and advised the teachers who met a host of new problems in its delivery, organizing training sessions for them. These “spec math” classes contributed in a major way to the emergence of a new generation of mathematicians in the country, as well as enabling an education based on independent thinking. The International Background A major impetus to the movement was provided by Zoltán Dienes, a mathematics educator of Hungarian origin who held a free-­spirited, playful math club for elementary school pupils in Budapest in the summer of 1962. He stated explicitly the radical idea that the primary aim of mathematics teaching (and, we should add, all teaching) is to develop the personality. His views were also incorporated into the Hungarian experiment. Both Tamás Varga and Surányi spent a year at the Université de Sherbrooke with Dienes; Surányi was there in 1970–1971. A further important

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role was played by another Hungarian émigré, the well-known mathematician and mathematics educator György Pólya. His book How to solve it had been translated into Hungarian already in 1957. Later Pólya visited Budapest several times at Surányi’s invitation, holding successful presentations; the two were related through their families also. The links between the Hungarian and international “New Math” movements became very strong with time; Surányi was particularly close to Emma Castelnuovo, Anna Zofia Krygowska, Willy Servais and Klaus Härtig (who he was connected with also through logic and number theory). Around the time of Surányi’s vice-presidency, there was an explosion of the international events that shaped the future developments of ICMI. He was an active participant. He took part in Symposium on The Co-ordination of the Teaching of Mathematics and Physics, sponsored by the ICMI and the Yugoslav Association of Mathematicians and Physicists, held in Belgrade from 19 to 24 September 1960, giving the talk Bemerkungen zum Zusammenhang zwischen der Mathematik und anderen Unterrichtsfächern (Internationale Mathematische Nachrichten. 1961. 66, p. 5). Surányi was the chairman of the Organizing Committee of the international symposium held in Budapest (31 August  – 8 September 1962) on mathematics teaching with the support of UNESCO. At the Colloquium “How to teach mathematics so as to be useful” (Utrecht, 21–25 August 1967), he was the national delegate and took part at the important meeting of ICMI held at the end of the colloquium, where the key decisions for the future of the Commission were taken, say the foundation of a new journal (Educational Studies in Mathematics) and the organization of a conference dedicated to mathematics education (ICME).147 In this occasion, André Revuz proposed an editorial Board for the Series “New trends” published by UNESCO, which included him. Surányi makes the following proposals: to organize exchanges of secondary teachers, to specify the role of ICMI in outlining the guiding principles for research and reform concerning mathematics education and to organize documentation. Surányi participated in the International Symposium held in Bucharest from 23 September to 2 October 1968 on “Modernisation of Mathematics Teaching in European Countries” and took part in the discussion during the sessions VIII, X, XI (see Colloque International UNESCO Modernization of Mathematics Teaching in European countries. Bucharest: Editions didactiques et pédagogiques, 1968, pp.  399–400). He attended the International Symposium on the Coordination of Instruction in Mathematics and Physics held in Belgrade, 19–24 September 1960 (see Proceedings of the International Symposium on the Coordination of Instruction in Mathematics and in Physics, eds. Yugoslav Union of Societies of Mathematicians and Physicists. Belgrade: Yugoslav Union of Societies of Mathematicians and Physicists). We know that he participated in ICME-1 (1969) and ICME-2 (1972).

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 See L’Enseignement Mathématique. 1967. 13, pp. 245–246.

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Sources Surányi, János. 1959. Reduktionstheorie des Entscheidungsproblems in Prädikationskalkül der ersten Stufe. Budapest: Akadémiai Kiadó; Berlin: VEB Deutscher Verlag der Wissenschaften. Surányi, János. 1949. Reduction of the decision problem to formulas containing a bounded number of quantifiers only. In Proceedings of the Tenth International Congress of Philosophy, Vol. II, Evert W.  Beth and Hendrik J.  Pos, eds., and Jan H.  A. Hollak, assistant ed., 769–772. Amsterdam: North-Holland Publishing Co. Surányi, János. 1971. Reduction of the decision problem of first order predicate calculus to reflexive and symmetrical binary predicates. Periodica Mathematica Hungarica 1: 97–106. Hajnal, András, and János Surányi. 1958. Über die Auflösung von Graphen in vollständige Teilgraphen. Annales Univ. Sci. Bp. Eötvös Loránd, Sectio Math. Annales Universitatis Scientiarium Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica 1: 113–121. Erdős, Pál, and János Surányi. 1960. Válogatott fejezetek a számelméletből [Selected Chapters in Number Theory]. Budapest: Tankönyvkiadó. (Revised editions 1996, and 2004). Szeged: Polygon 6; in English: Topics in the Theory of Numbers, Springer, Heidelberg, 2003. Szabó, Péter Gábor (ed.). 2008. The correspondence of László Kalmár and János Surányi (Hungarian) in Kalmárium II, 272–368. Szeged: Polygon. Fried, Ervin. 2007. Surányi János (1918–2006). Természet Világa 3, March. http://lajosszabo.com/ SL/Suranyi_Janosrol_Termeszet_Vilaga_03_2007.pdf Surányi, László. 2017. Sohasem azt tanítjuk, amit tanítunk – 100 éve született Surányi János [What we really teach is not what we actually teach – The 100th anniversary of János Surányi’s birth] http://www.ematlap.hu/index.php/interju-­portre-­2018-­06/736-­sohasem-­azt-­tanitjuk-­amit-­ tanitunk-­100-­eve-­szuletett-­suranyi-­janos also in Természet világa 2017/1. I-IX.

Publications Related to Mathematics Education Surányi, János. 1949. Szempontok az új középiskolai matematikatanításhoz [Some Aspects of the New Way of Teaching Mathematics]. I. Megtanítható-­e a matematika? [Can Mathematics be Taught Efficiently?] II. Tudomány és gyakorlat egysége [The Unity of Science and Praxis]. Köznevelés 5: 443–444; 469–470. Surányi, János. 1968. Remarques sur les taches de l’enseignement des mathématiques et ses obstacles. In Modernisation de l’enseignement mathématique dans les pays Européens, ed. N. Teodorescu, 104–109. Paris: UNESCO. Surányi, János. 1972. A matematikaoktatás korszerűsítésének néhány alapelve [Some Basic Ideas in the Modernization of Teaching Mathematics]. In Néhány hazai és külföldi kísérlet [Some National and International Experiments], 7–53. Budapest: Tankönyvkiadó, Gádor, Endréné and János Surányi. 1974. Célok és lehetőségek az iskolai matematikai nevelésben [Aims and ways in teaching Mathematics at schools], Magyar Tudomány: 570–576. note: Magyar Tudomány is a periodica, not a Publishing House. Hódi, Endre and János Surányi. 1977. Tendance de renouvellement de l’enseignement de la mathématique dans les école hongroises. In Evaluation et enseignement mathématique, Proceedings of the 29th CIEAEM meeting, ed. Jacques le Roy, with collaboration of Pierre Jaquier and Alain Schori, 106–109. Lausanne. Halmos, Mária, and János Surányi. 1978. The evolution of Modern Mathematics education in Hungary. In Socialist Mathematics Education, ed. Frank Swetz, 253–300. Southampton: Burgundy Press. Bartal, Andrea, Sarolta Pálfalvi, and János Surányi. 1978. The “Work-Textbook” for Secondary School Mathematics teaching. In Osnabrücker Schriften zur Mathematik, Reihe D Mathematishc-didaktische Manuskripte, Bd. I. Proceedings of the 2nd International Conference

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for the PME, eds. E.  Cohors-Fresenborg and I.  Wachsmuth, 42–46. Osnabrück: Universität Osnabrück. Surányi, János. 1985. Mathematics Teachers Associations. In Studies in Mathematics Education, Vol 4, ed. R. Morris, 133–142. Paris: UNESCO. French: 1986. Les Associations de professeurs de mathématique. In Études sur les l’enseignement des mathématiques. La Formation des professeurs de mathématiques de l’enseignement secondaire, Vol. 4, ed. R.  Morris, 161–170. Paris: UNESCO. Surányi, János. 1989. A short account on mathematics teaching in Hungary. L’insegnamento della Matematica e delle Scienze Integrate 12: 578–583. Kürschák, József, György Hajós, Gyula Neukomm, and János Surányi. 1955. Matematikai versenytételek [Hungarian Problem Book] Vol. I-II.  Budapest: Tankönyvkiadó. Revised editions: 1965 and 1987–88; in English: Hungarian Problem Book, Vol. I (in two volumes) transl. by Elvira Rapaport. New  York: Random House. 1963. Vol. II. transl. by Andy Liu, The Mathematical Association of America, 2001. In Russian: Vengerskie matematicseskie Olimpiadi, Mir, Moskau, 1976. Vol. I-II. Also published in Japanese and Rumanian. Surányi, János. 1992, Matematikai versenytételek [Hungarian Problem Book] Vol. III. 1998. Budapest: Tankönyvkiadó; Vol. IV. Budapest: Typotex. Surányi, János. 1969. Mathematical contests in Hungary. Educational Studies in Mathematics 2: 80–82, 99–100. Surányi, János. 1983. Some Ideas on Mathematical Competitions. In Proceedings of the Fourth International Congress on Mathematical Education, eds. Zweng, M., Green, T., Kilpatrick, J., Pollak, H., & Suydam, M., 565–566. Boston: Birkhäuser. Surányi, János. 1948. Hasonlóság és szerkesztés [Similarity and straight edge-and-compass construction]. Budapest: Országos Neveléstudományi Intézet. Herczeg, János, and János Surányi. 1966. Valós számok [Real Numbers]. In Mathematics for special classes in Secondary Schools (textbook). Vol. II. 3–134. Budapest: Tankönyvkiadó. Surányi, János. 1977. Polinomok és egyenletek az iskolában [Polynomials and Equations in Secondary School]. Budapest: FPI.

Contributions to the Popularization of Mathematics Surányi, János. 1968. A számkör felépítése [Setting up the number system]. In Élő matematika [Living Mathematics, studies], 179–224. Budapest: Tankönyvkiadó. Surányi, János. 1969. (In Hungarian, German and Bulgarian): Tud Ön fejben ötödik gyököt vonni? In. Matematikai érdekességek, 45–61. Budapest: Gondolat, second edition: Typotex, Budapest, 1999. 50–66. German translation: Können Sie fünfte Wurzeln im Kopf ziehen? in: Mathematisches Mosaik Urania Verlag Leipzig-Berlin, 1977. 50–68. Bulgarian translation: Mozsete li da izvlicsati peti koren? In Matematicseszkaja Mozajka Sophia, 1980. 44–60. Surányi, János. 1981. (in Hungarian and German): Már az ókori görögök is… In Nagy pillanatok a matematika történetéből. Budapest. Gondolat, 9–49. Schon die alten Griechen haben das gewußt. In Grosse Augenblicke aus der Geschichte der Mathematik. Budapest, R. Freud ed.: Akadémiai Kiadó, 1990. 3–50. Surányi, János. 1993. Some problems memorable to me. Mathematics Competitions 64: 56–59.

Photo Courtesy of László Surányi.

Jacobus Hendricus van Lint

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11.52 Jacobus Hendricus Van Lint (Bandung, 1932 – Nuenen, 2004): Ex Officio Member 1987–1994 Sten Kaijser

Biography Jacobus Hendricus (Jack) van Lint was born in Bandung on 1 September 1932. In Bandung – a town on the island of Java in what was then the Dutch East Indies and today is Indonesia – his father was a teacher of mathematics at a secondary school. During World War II, his family first escaped from Java to Australia and then moved on to the United States. After the war, he spent a year in Australia until the family finally returned to the Netherlands in 1946. As a result, his school years were rather chaotic. After a couple of years in Djakarta, there were a few years in the United States, then a year in Australia, then 4 years in Arnhem until he finished school in Zwolle in 1950. He entered Utrecht University the same year, where he soon became an assistant to Hans Freudenthal. He received his PhD from the same university in 1957 and became professor at the Technical University at Eindhoven in 1959. At that time, his main interest was in number theory, and in fact, his thesis was about automorphic forms. When his S. Kaijser (*) University of Uppsala, Uppsala, Sweden e-mail: [email protected]

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university wanted to start a programme for “mathematical engineers”, he was asked to specialize in “combinatorics”. In order to get well into the new field, he spent half a year at the Bell Laboratories in Murray Hill, New Jersey. He became fascinated by the rapidly emerging field of “discrete mathematics” and soon became one of its strongest proponents. His earliest work in this field was in coding theory. Early in his new career, he wrote some important articles and an influential textbook on this theory. In the University of Eindhoven, van Lint has been dean of the Department of Mathematics and Computer Science and from 1991 to 1994 Rector Magnificus of the University. He became a member of the Royal Netherlands Academy of Arts and Sciences in 1972 and took on important tasks in several international committees, such as the European Mathematical Council from 1978 to 1990 and Committee on the Teaching of Science (CTS) of the International Council for Scientific Unions (ICSU) from 1986 to 1994. He received several rewards, such as becoming Knight in the Order of the Lion in the Netherlands and became honorary doctor at five universities. He was an ex officio member of the Executive Committee of ICMI from 1987 to 1994 as the representative of the International Mathematical Union on the CTS of ICSU. Van Lint died in Nuenen, a town close to Eindhoven, on 28 September 2004.

Contribution to Mathematics Education Van Lint was an ex officio member of the Executive Committee of ICMI from 1987 to 1994, as the representative of the International Mathematical Union on the then existing CTS of ICSU. Besides being an eminent scientist, van Lint was also a brilliant lecturer. He was invited to give talks all over the world, altogether more than 300 at more than 100 places. With his love for communicating mathematics, it was quite natural that he also took a great interest in the teaching of mathematics, especially at the university level. He wrote several papers both for high school teachers and for colleagues teaching university students or engineering students. As a result of his interest and his innovative ideas about the teaching of mathematics (in particular, discrete mathematics), he was a speaker at the ICME conferences in Karlsruhe (1976), Berkeley (1980), Adelaide (1984) and Quebec (1992). It is worth mentioning, for example, the long paper he wrote after the ICME-3  in Karlsruhe, based on the report he presented at Section A 4 entitled “Mathematics Education at University Level (Excluding Teacher Training)” (Van Lint 1976a), which was published as a chapter of the volume New Trends in Mathematics Teaching (see Van Lint 1979). Van Lint prepared this paper very carefully by sending an outline of it to 200 universities all over the world in November 1975. Many questions were also included with the request for answers and general information. The numerous suggestions and comments sent to him were useful to write the final version of the chapter. The main points taken into consideration are the following:

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curriculum trends (contents and objectives); structure of the programme (educational methods); mathematics as a minor subject; the role of university mathematics teachers. Within each section, he first discusses trends in objectives and then turns to trends concerning the actual achievement of these objectives. Van Lint concludes his paper saying that Another universal trend is an increasing amount of administrative work, committee work, etc. Some countries are suffering the difficulties of a parliamentary system in universities, which has made the process of administration and of taking decisions excessively slow. It is strange that so many complain bitterly about such waste of time and yet nobody actually does much to remedy the difficulties. (Van Lint 1979, p. 82)

Sources Cameron, Peter and Jacobus Hendricus Van Lint. 1975. Graph Theory, Coding Theory and Block Designs. London Mathematical Society Lecture Notes 19. Cambridge, New York: Cambridge University Press. Cameron, Peter J. and Jacobus Hendricus Van Lint. 1980. Graphs, Codes and Designs, London Mathematical Society Lecture Note Series 43. Cambridge, New  York: Cambridge University Press. Cameron, Peter J. and Jacobus Hendricus Van Lint. 1991. Designs, Graphs, Codes and their Links, London Math. Soc. Student Texts 22. Cambridge, New York: Cambridge University Press. Van Lint, Jacobus Hendricus. 1971. Coding Theory. Lecture Notes in Mathematics 201. Berlin: Springer Verlag. Van Lint, Jacobus Hendricus. 1982. Introduction to Coding Theory, Graduate Texts in Mathematics 86. New York: Springer Verlag, second edition 1992, third (enlarged) edition 1998. Van Lint, Jacobus Hendricus and Richard M.  Wilson. 1991. A Course in Combinatorics. Cambridge: Cambridge University Press. De Bruijn, Nicolaas Govert G. 2005. In memoriam Jack van Lint (1932–2004). NAW 5/6(2): 106–109. Pollak, Henry O., Mogens Niss, and Jean-Pierre Kahane. 2004. In Memoriam — Jacobus H. van Lint (1932–2004). Bulletin of the International Commission on Mathematical Instruction n. 55: 80–84. Van Asch, Bram, Aart Blokhuis, Henk Hollmann, William Kantor, and Henk van Tilborg (eds.). 2006. Jack van Lint (1932–2004): A survey of his scientific work. Journal of Combinatorial Theory s. A, 113(8), 1594–1613.

Publications Related to Mathematics Education Churchhouse, Robert Francis, Bernard Cornu, Andrey Petrovych Ershov, Albert Geoffrey Howson, Jean-Pierre Kahane, Jacobus Hendricus van Lint, François Pluvinage, Anthony Ralston, and Masaya Yamaguti. 1984. The influence of computers and informatics on mathematics and its teaching. An ICMI Discussion Document. L’Enseignement Mathématique s. 2, 30: 161–172. Churchhouse, Robert Francis, Bernard Cornu, Andrey Petrovych Ershov, Albert Geoffrey Howson, Jean-Pierre Kahane, Jacobus Hendricus van Lint, François Pluvinage, Anthony Ralston, and Masaya Yamaguti (eds.). 1986. The influence of computers and informatics on mathematics and its teaching: Proceedings from a symposium held in Strasbourg, France in March 1985 and

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sponsored by the International Commission on Mathematical Instruction. Cambridge, UK: Cambridge University Press. ICMI Study Series. Griffiths, Brian, Jack van Lint, and Jane Pitman (organizers). 1986. Action Group 5: Tertiary (postsecondary) Academic Institution (18+). In Proceedings of the Fifth International Congress on Mathematical Education, ed. Marjorie Carss, 95–110. Boston: Birkhäuser. Van Lint, Jacobus Hendricus. 1972–1973. De 13e internationale Wiskunde Olympiade. Euclides: Maandblad voor de Didactiek van de Wiskunde 48: 47–58. Van Lint, Jacobus Hendricus, and H. J. L. Kamps. 1975. A comparison of a classical calculus test and a similar multiple choice test. Educational Studies in Mathematics 6: 259–271. Van Lint, Jacobus Hendricus. 1976a. Mathematics education at university level. In Proceedings of the Third International Congress on Mathematics Education, eds. Hermann Athen and Heinz Kunle, 174–184. Universität (West) Karlsruhe: Zentrallblatt für Didaktik der Mathematik. Van Lint, Jacobus Hendricus. 1976b. Educacion matematica a nivel universitario, excluyendo la preparacion de profesores. Boletin Informativo ICMI-CIAEM 3: 31–39. Van Lint, Jacobus Hendricus. 1979. Mathematics education at university level. In New Trends in Mathematics Teaching, ed. the International Commission on Mathematical Instruction (ICMI). The Teaching of Basic Sciences 4, 66–84. Paris: Unesco. (Translated into French, Japanese, and Spanish). Van Lint, Jacobus Hendricus. 1983. Algebraic coding theory. In Proceedings of the Fourth International Congress on Mathematical Education, eds. Marilyn Zweng, Thomas Green, Jeremy Kilpatrick, Henri Pollack, and Marilyn Suydam, 299–303. Boston: Birkhäuser. Van Lint, Jacobus Hendricus. 1985. On geometry and discrete mathematics. Nieuwe Wiskrant 5(1): 45–47. Van Lint, Jacobus Hendricus. 1988. Discrete Mathematics: some personal thoughts. In Mathematics as a service subject, eds. Albert Geoffrey Howson; Jean-Pierre Kahane; Pierre Lauginie, and Elisabeth de Turckheim, 58–62. Cambridge, UK: Cambridge University Press, ICMI Study Series. Van Lint, Jacobus Hendricus. 1988. Chief guest’s address at the annual conference of A.I.M.T. on 30 January 1988. Indian Journal of Mathematics Teaching 14: 31–32. Van Lint, Jacobus Hendricus. 1993. A European view of a course on discrete mathematics. New England Mathematics Journal 25: 18–19. Van Lint, Jacobus Hendricus. 1994. What is Discrete Mathematics and how should we teach it?. In Proceedings of the 7th International Congress on Mathematical Education. Vol. 2 Selected Lectures, eds. Claude Gaulin, Bernard R. Hodgson, David H. Wheeler, and John C. Egsgard, 263–270. Sainte-Foy: Les Presses de l’Université de Laval.

Photo Source: https://www.win.tue.nl/~jlint/ (Personal website. Technische Universiteit Eindhoven).

Gilbert Walusinski

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11.53 Gilbert Walusinski (Paris, 1915 – Brou, 2006): Secretary 1959–1962 Éric Barbazo

Biography Gilbert Walusinski was born on 31 January 1915 in Paris, the son of a Polish immigrant who moved to Paris at the end of the nineteenth century and obtained French nationality in 1893. He studied at Lycée Charlemagne in the Marais district of Paris, where he obtained his baccalaureate. His math teacher Francisque Marotte, who was one of the authors reacting to Smith’s paper about the establishment of an international commission published in 1905 on the journal L’Enseignement Mathématique, communicated to him his passion for mathematics and its teaching to him. Gilbert Walusinski continued his studies at the Sorbonne. He fervently followed the courses of Élie Cartan who confirmed his vocation to teach mathematics. He obtained his Licence ès sciences in 1935 and then the Agrégation de mathématiques in 1941 after having been a free auditor at the École Normale Supérieure. For almost half a century, Walusinski devoted his life to the development of mathematics education. He was convinced that the teaching must be constantly evolving and in line with contemporary scientific news. It is with this state of mind that Walusinski engaged in numerous national and international scientific associations and societies. He was president of the Association des Professeurs de Mathématiques (APMEP) É. Barbazo (*) Établissement International Français, Houston, TX, USA

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from 1955 to 1958, a member of the Societé Mathématique de France (SMF), secretary of ICMI from 1959 to 1962, then a national delegate from 1963.148 Botella and Dalançon (2019) describe the intense participation in the activities of the national trade unions of teachers and report that he refused the prestigious Légion d’honneur, but in 1976 he accepted the title of honorary president of APMEP. Walusinski died in Brou (Department Eure et Loire) on 13 January 2006. Scientist and Man of Letters Though Professeur agrégé of mathematics, Walusinski was also fond of astronomy. In 1976, he is one of the founders of the Comité de Liaison Enseignement et Astronomie (CLEA – Education and Astronomy Liaison Committee), whose main task is the training of mathematics teachers in astronomy. For 20 years, he contributed to the development of Cahiers Clairaut, a quarterly publication of CLEA. Even if astronomy was just a passion for Gilbert Walusinski, it allowed him to contribute to the improvement and renewal of mathematics education. For his action in favour of the teaching of astronomy, on 26 November 2001, he received the Grand prix de l’Académie des Sciences. Besides a solid scientific background, Gilbert Walusinski had remarkable literary qualities. He was the author of numerous articles in various scientific journals. In particular, he gave the full measure of his literary talent in the Bulletin de l’APMEP in which he wrote from 1950 to 1990. In L’Enseignement Mathématique, the official organ of ICMI, he published one of his most famous articles (Walusinski, 1957), which discusses the new educational problems raised by the development of modern mathematics. The writing was Walusinski’s second passion. His articles from the Cahiers Clairaut or the Bulletin de l’APMEP constitute a work that measures the evolution of mathematics education in France over half a century after World War II. Walusinski mainly devoted his articles to the evolution of mathematics education and the initial and in-service education of mathematics teachers in French collèges and lycées. His national activity as president of APMEP together with its international presence as secretary of ICMI gave him a leading role in the transformation of secondary mathematics education in the second half of the twentieth century. Friend of Albert Camus, he also collaborated in the literary magazine La Quinzaine Littéraire in which he gave the full measure of his editorial qualities. Between 1966 and 2001, he wrote more than 150 reports of works devoted to philosophy, history, political or social human sciences without forgetting the thoughts of the great scientists of the last three centuries. Scientist and man of letters, Walusinski signed (with different pseudonyms such as the initials of Leibniz G.W., K.  Mizar, Evariste Dupont, E.  Duponsky,

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 Délégués nationaux. L’Enseignement Mathématique. 1964. 10, p. 298.

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W. Mountebank) a literary work under different pseudonyms, which sheds light on the second half of the twentieth century.

Contributions to Mathematics Education An Outstanding Teacher Walusinski was above all an educator appreciated by his students of the collèges and lycées to whom he mainly devoted his career. The numerous testimonies of his former students reveal a very available professor, who punctuated his lessons with parentheses on astronomical news. Pedagogical reflections have marked the entire career of Walusinski. Since the 1950s, he advocated a reform of the teaching of mathematics both in terms of programmes and the pedagogy used: The height, said Alain, would be that peace is founded on the world and that it is not ready to receive it. Would it be the same for the educational reform, always announced by the Minister but never carried out by his successor? For our part, we are constantly preparing for a reform whose urgency, both from the social point of view and the point of view of science and pedagogy, is increasing every day. (Walusinski 1956, p. 76)149

Enthused by the plan Langevin-Wallon and favourable to the Classes Nouvelles de la Libération (New Classes of Liberation) launched at the end of World War II by the high state officer and educator Gustave Monod, Walusinski never ceased to promote new teaching methods. He did not hesitate to present the experiments carried out in other countries as models. Thus, he translated the book Construction des Mathematics by Zoltán Pál Dienes from English, devoted long articles to the experiences of Georges Papy in Belgium and Trevor J. Fletcher in England. For Walusinski, the renewal of pedagogy was closely linked to the training of teachers. Continuing education was non-existent in France until the emergence of the Instituts de Recherche sur l’Enseignement des Mathématiques (Research Institutes for the Teaching of Mathematics, IREMs). These institutes quickly became veritable educational and didactic research laboratories within French universities by pooling the skills of college, high school and university teachers.

 The original text is: “Le comble, disait Alain, serait que la paix fonde sur le monde et qu’il ne soit pas prêt à la recevoir. En serait-il de même pour la réforme de l’enseignement, toujours annoncée par le Ministre mais jamais réalisée par son successeur? Pour notre part, nous n’arrêtons pas de nous préparer à une reforme dont l’urgence, tant du point de vue social que du point de vue de la science et de la pédagogie, augmente tous les jours”. 149

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Walusinski’s Effective Activism Walusinski actively participated in the transformations that the teaching of French mathematics underwent between 1945 and 1980. When he was president of the APMEP from 1955 to 1958, with his friend Gustave Choquet launched a series of conferences carried out with the support of the Société Mathématique de France (SMF). From 1956 to 1963, more than 40 conferences were held, half of which were devoted to modern mathematics, a theme dear to Walusinski. These conferences fully printed in the Bulletins de APMEP thus constituted the beginnings of the continuing education desired by him. Building on the success encountered, Walusinski offered André Revuz, professor at the Faculty of Sciences of Poitiers, a new cycle of lectures on algebraic structures. A series of courses, entitled Cours de l’APMEP, on groups, rings and fields, vector spaces and elements of topology finally were published in three collections by APMEP between 1963 and 1967. Finally, Walusinski actively participated within the APMEP in the development of an internal Charter to the Association, concerning teacher education and the reform of French secondary education programmes. The conclusions of this Charter were taken up by the ministerial study commission for the teaching of mathematics (known as the Commission Lichnérowicz) set up by the Minister of National Education in 1967. The work of this commission led to the creation in 1968 of the IREMs and completely transformed the curricula of collèges and lycées. Secretary of ICMI from 1959 to 1962, and then national delegate, he participated actively in the initiatives of the Commission. For example, in 1960, at the Symposium on the Co-ordination of the Teaching of Mathematics and Physics, sponsored by the ICMI and the Yugoslav Association of Mathematicians and Physicist (Belgrade, 19–24 September 1960), he delivered a talk entitled “Comment l’enseignement de l’astronomie peut-il servir la coordination de l’enseignement des mathématiques et de la physique” (How can the teaching of astronomy serve to coordinate the teaching of mathematics and physics),150 and at the international congress held in Frascati in Italy on 8–10 October 1964, he delivered a talk entitled “Sur les connaissance requises à l’entrée de l’Université pour les professeurs de l’enseignement élémentaire ou moyen” (On the knowledge required at the entrance to the University for teachers of elementary or intermediate education).151 He was a member of the Program Committee of the conference organized in Bologna (14–18 October 1961) with the aim of discussing the results of the meeting of Aarhus and Dubrovnik on the teaching of geometry.152

 See 1961. Internationale Mathematische Nachrichten 66, 1961, p. 5.  R. G. 1964. Seminario matematico internazionale. Archimede 16: 314–320. 152   BUMI. 1962. Il convegno di Bologna promosso dalla Commissione internazionale dell’insegnamento matematico. Bollettino della Unione Matematica Italiana s. 3, 17: 199–214. 150 151

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Sources Walusinski, Gilbert. 1981. Ciel, passé, présent: les grandes découvertes de l’Astronomie, Paris: Etudes Vivantes. Walusinski, Gilbert. 1988. La lune et ses satellites. Paris: Edition Epigone. Walusinski, Gilbert. 1987. Planètes et comètes. Paris: Edition Epigone. Walusinski, Gilbert. 1990. Astronomes et observatoires. Paris: Edition Epigone. APMEP. 2007. Hommage à Gilbert Walusinski. Supplément au Bulletin Vert 471. Barbazo, Éric. 2005, L’influence de Gilbert Walusinski dans la création des IREM. Mémoire de DEA dirigé par Jean Dhombres. Paris: EHESS. Botella, Louis and Alain Dalançon. https://maitron.fr/spip.php?article136499, notice WALUSINSKI Gilbert. Version mise en ligne le 1er avril 2011, dernière modification le 19 janvier 2022. Acessed on 9 February 2022. Dhombres, Jean. 2006. Walusinski, professeur de mathématiques. Le Monde, January 19.

Publications Related to Mathematics Education Walusinski, Gilbert. 1956. La rentrée à l’heure de la réforme. Bulletin de l’APMEP 179 (October): 73–79. Walusinski, Gilbert. 1957. Au pays de Clairaut et de Bourbaki. L’Enseignement Mathématique s. 2, 3: 289–297. Walusinski, Gilbert. 1971. Guide blanc. Pourquoi une mathématique moderne. Paris: Armand Colin. Walusinski, Gilber. 1986. L’instructive histoire d’un « échec»: les mathématiques modernes (1955–1972). Bulletin de l’APMEP n. 353: 141–155. Dienes, Zoltan Paul. 1966. Construction des mathématiques, Traduit par G. Walusinski. Vendôme: Presses Universitaires de France.

Photo Courtesy of APMEP (Association des Professeurs de Mathématiques de l’Enseignement Public).

644

Hassler Whitney

11.54 Hassler Whitney (New York, 1907 – Princeton, 1989): President 1979–1982, Ex Officio Member 1983–1986 Jeremy Kilpatrick

Biography Hassler Whitney was born on 23 March 1907 in New York City. Both of his grandfathers, the philologist William D. Whitney and the astronomer Simon Newcomb, were members of the US National Academy of Sciences, to which Whitney was elected in 1945. Whitney received bachelor’s degrees in philosophy (1928) and music (1929) from Yale University. A keen mountaineer all his life, he made a famous climb while still an undergraduate, with his cousin Bradley Gilman, of a cliff in New Hampshire. The cliff was later named the Whitney-Gilman Ridge. For graduate studies in mathematics, Whitney went to Harvard University, where he earned his PhD under George David Birkhoff in 1932 with a dissertation entitled The Coloring of Graphs, in which he found an equivalent graph-theoretic formulation of the four-colour problem. He taught mathematics at Harvard for a year and then was a National Research Council Fellow at Harvard and Princeton. Returning J. Kilpatrick (1935–2022) University of Georgia, Athens, GA, USA

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to the Harvard faculty in 1933, he rose through the ranks, becoming an assistant professor in 1935, associate professor in 1940 and full professor in 1946. From 1943 to 1945, he was a member of the Mathematics Panel of the National Defense Research Committee. In 1952, Whitney joined the Institute of Advanced Study at Princeton as professor of mathematics, officially retiring in 1977 but continuing as an emeritus professor until his death on 10 May 1989 in Princeton, New Jersey, 2 weeks after suffering a stroke. He was survived by his third wife, five children and six grandchildren. Whitney’s work in topology was wide-ranging and innovative. His first research, a by-product of his dissertation, was in graph theory; he introduced the notion of duality, which he used to characterize planar graphs (1933), and developed a theory of linear dependence that led to the theory of matroids (1935). He also did early work on the singularities of mappings between n-­dimensional Euclidean spaces that turned out to be important for catastrophe theory. His most famous work concerned his theory of differentiable manifolds and its related machinery in algebraic and differential topology (see Zund, 1999, for details). After World War II, Whitney turned his attention to the interaction of algebraic topology with integration theory (1957) and then to complex global analysis (1972), making major contributions in both fields. Among his many activities in mathematics, Whitney was editor of the American Journal of Mathematics from 1944 to 1949 and editor of Mathematical Reviews from 1949 to 1954. Among many honours, he received the US National Medal of Science in 1976, a Wolf Prize in 1982 and a Steele Prize for Lifetime Achievement in 1985.

Contribution to Mathematics Education From 1979 to 1982, Whitney was president of ICMI from 1979 to 1982 and an ex officio member from 1983 to 1986. Several years before his retirement in 1977, Whitney developed an interest in mathematics education, especially elementary school mathematics, which occupied him for the last two decades of his life (D’Ambrosio 1989; Lax 1989). In 1974, he launched the project “Basic Research on How Children Learn Mathematics”, connected with the Institute for Advanced Study in Princeton. The project had a duration of 20  months and a twofold purpose: to understand how children learn mathematics, observing children at work in the classroom, and to support teachers in developing more refined methods for helping them learn.153 He gave a number of lectures on education at national and international meetings, conducted summer courses for teachers and once spent 4  months teaching

 See Development Projects in Science Education: Precollege, Higher Education, Continuing Education. National Science Foundation 1977, pp. 189–190. 153

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pre-algebra mathematics to a class of seventh graders (Zund, 1999). He opposed formal instruction in arithmetic in the early grades, repeatedly citing a little-known study by Louis P.  Bénézet (1935a, 1935b, 1936), superintendent of schools in Manchester, New Hampshire, who managed to get several schools in his system to abandon all formal instructions in arithmetic prior to seventh grade. After a year’s instruction, the students’ arithmetic test scores were at the level of those of comparable students who had undergone regular instruction. Whitney saw the Bénézet study as justifying his argument that too many mathematics teachers were focusing on the passing of tests rather than what he called “meaningful goals”. He was particularly disturbed by national reports calling for more mathematics to be taught earlier in school: The most pressing need I see is for us to face fully the consequences of interventions we make, and hold up on those with bad results. I speak, of course, of mandating more work in mathematics for failing students, raising standards for these without helping them toward meeting the standards, and starting mathematics teaching at an earlier age. It is unthinkable to market drugs without a thorough study of all effects; in education I see no parallel concern, though there should be. (Whitney 1985b, p. 233)

Whitney (1986) saw the purposes of learning mathematics as being distorted in many classrooms: Mathematical reasoning is becoming steadily more important in many domains; hence a basic need is for students to group in power in such reasoning. In addition, they must gain control over their work, see interrelations with other aspects of the subject, and communicate well with colleagues and others about these matters. … In the usual classroom, however, the student's focus is elsewhere. … The student's aim is to try to remember particular patterns of thought coming from reasoning, but to disregard the reasoning from which they came. Thus the essence of the process is lost. (pp. 129–130)

Using illustrations from his work with children, Whitney (1986) argued for a gradual, unforced, positive change in the way mathematics is taught so that students would have more opportunities for exploration, discussion and discovery: One must find practical ways to move towards reform, leaving most of the present teaching unchanged (at least at the start). Not only the students, but also the professors and others must be considered. We push at the students to make them learn; it does not work. Pushing at the professors to teach differently will also fail. Pressures have too much that is negative. We need positive ways; and these must grow by themselves, not be forced. Humans grow through care and love during long periods; the same is true throughout nature. (p. 139)

Sources Eells, James and Domingo Toledo (eds.). 1992. Hassler Whitney: Collected papers, 1–3. Boston: Birkhäuser. Whitney, Hassler. 1933 Planar graphs. Fundamenta Mathematicae 21: 73–84.

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Whitney, Hassler. 1935 On the abstract properties of linear dependence. American Journal of Mathematics 57: 509–533. Whitney, Hassler. 1936. Differentiable manifolds. Annals of Mathematics s. 2, 37: 645–680. Whitney, Hassler. 1957, Geometric integration theory, Princeton, NJ, Princeton University Press. Whitney, Hassler. 1968a. The mathematics of physical quantities: Part 1: Mathematical models for measurement. American Mathematical Monthly 75: 115–138. Whitney, Hassler. 1968b. The mathematics of physical quantities: Part 2: Quantity structures and dimensional analysis. American Mathematical Monthly 75: 227–256. Whitney, Hassler. 1972. Complex analytic varieties. Reading, MA: Addison-Wesley. Whitney, Hassler. 1985a. Letting research come naturally. Mathematical Chronicle 14: 1–19. D’Ambrosio, Ubiratan. 1989. The visits of Hassler Whitney to Brazil: Hassler Whitney, in Memoriam. Humanistic Mathematics Newsletter 4: 8. Bénézet, Louis P. 1935a. The teaching of arithmetic 1: The story of an experiment. Journal of the National Education Association 24: 241–244. Bénézet, Louis P. 1935b. The teaching of arithmetic 2: The story of an experiment. Journal of the National Education Association 24: 301–305. Bénézet, Louis P. 1936. The teaching of arithmetic 3: The story of an experiment. Journal of the National Education Association 25: 7–8. Dieudonné, Jean A. 1989. A history of algebraic and differential topology: 1900–1960. Boston: Birkhäuser. Fowler, Glenn. 1989 (May 12), Hassler Whitney, geometrician: He eased ‘mathematics anxiety’. The New York Times, B10. Hechinger, Fred M. 1986 (June 10). Learning math by thinking. The New York Times. C1. Lax, Anneli. 1989. Hassler Whitney 1907–1989: Some recollections 1979–1989. Humanistic Mathematics Newsletter 4: 2–7. Thom, René. 1990. La vie et l’oeuvre de Hassler Whitney. Comptes Rendus de l’Académie des Sciences Paris Sér. Gén. La Vie des sciences 7: 473–476. Zund, Joseph D. 1999. Whitney, Hassler. American National Biography 23: 303–304. Retrieved February 9, 2020, from the American National Biography web site: http://www.anb.org/ articles/13/13-­02523.html (Retrieved June 2021).

Publications Related to Mathematics Education Whitney, Hassler. 1973. Are we off the track in teaching mathematical concepts? In Developments in mathematical education: Proceedings of the Second International Congress on Mathematical Education, ed. A. Geoffrey Howson, 283–296. Cambridge: Cambridge University Press. Whitney, Hassler. 1985b. Taking responsibility in school mathematics education. Journal of Mathematical Behavior 4: 219–235. Whitney, Hassler. 1986. Coming alive in school math and beyond. Journal of Mathematical Behavior 5: 129–140, also published in 1987  in Educational Studies in Mathematics 18: 229–242. Whitney, Hassler. 1989. Education is for the students’ future (draft). Humanistic Mathematics Newsletter 4: 9–12.

Photo Source: Celebratio Mathematica. Hassler Whitney. Mathematical Sciences Publishers. Design & Software © 2012–2022 MSP.

Chapter 12

Other Eminent Figures Gert Schubring, Sándorné Kántor Tünde Varga, Éric Barbazo, Gregg De Young, Ewa Lakoma, Livia Giacardi, Eduardo L. Ortiz, Fulvia Furinghetti, and Snezana Lawrence

© Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0_12

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Emanuel Beke

12.1 Emanuel [Manó] Beke (Pápa, 24 April 1862–Budapest, 27 June 1946): Appointed Honorary Member in 1936 Gert Schubring and Sándorné Kántor Tünde Varga

Biography Emanuel (Manó) Beke was born in Pápa (Hungary) on 24 April 1862. He began his studies at the Technical University of Budapest but soon switched to the University of Budapest. He obtained his teacher’s diploma in 1883 and his doctorate in 1884. He taught at a secondary school in Budapest until 1895. Thanks to a scholarship, Beke was able to study during the years 1892–1893 in Göttingen, where he participated in the mathematics seminar and became known to Felix Klein. He became interested in

G. Schubring (*) University of Bielefeld, Bielefeld, Germany e-mail: [email protected] S. Kántor Tünde Varga Debrecen University, Debrecen, Hungary e-mail: [email protected]

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Klein’s activities for reforming the teaching of mathematics, and after his return he worked actively for reforming mathematics teaching in Hungary, applying Klein’s conceptions. In 1895 he started teaching at the “Mintagimnázium” (Gymnasium, Budapest, Trefort Street). In 1900 he was appointed professor at the University of Budapest. He became a corresponding member of the Hungarian Academy of Sciences in 1914. After World War I, the Council of the University started a disciplinary investigation against him because of his support for democratic structures in the university, during the short period of the revolutionary republic of 1919, and in 1922 he was stripped of his position at the university and of his membership in the Academy. Consequently, Beke changed to work for a publisher. His book on Differential and Integral Calculus was the textbook for several generations of mathematics students. His major areas of scientific interest were linear differential equations, determinants, problems from physics and the teaching of mathematics. In 1951 the Hungarian Bolyai János Mathematical Society inaugurated the Beke Manó (Emanuel) Prize for outstanding results in teaching and popularisation of mathematics. Beke died on 27 June 1946 in Budapest.

Contribution to Mathematics Education Between 1906 and 1909, Beke was the chairman of the Commission to reform the teaching of mathematics in secondary schools in Hungary. In 1909, Beke became, as one of the three delegates from Hungary, a member of Internationale Mathematische Unterrichtskommission (IMUK). In the series of national reports on the state of mathematics instruction in Hungary, Beke wrote the report on the secondary schools (1910). In 1936, at the International Congress of Mathematicians in Oslo, he was awarded the title of Membre Honoraire de la Commission.1 It was Klein who had chosen Beke as the organiser and coordinator of the international survey on the results of the introduction of the elements of the differential and integral calculus into secondary schools. His report, which was presented to IMUK meeting 1914  in Paris, was about one of the most essential concerns in Klein’s reform programme. He emphasised this introduction, realised during the last 12 years, as an essential element of permeating mathematics teaching with the conception of functional thinking. Beke reports on the state of introducing the elements of the calculus, mainly in European countries (fifteen states), besides a few non-European ones (Australia, Brazil and the USA). He shows that elements of infinitesimal calculus are present in the official syllabuses in some German states (Bavaria, Wurtemberg, Baden, Hamburg), in Austria, Denmark, France, British

1  See Comptes rendus du Congrès International des Mathématiciens, Oslo 1936, Oslo: A. W. Brøggers Boktrykkeri. 1937. Vol. 2, p. 289.

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Isles, Italy, Romania, Russia, Sweden and Switzerland. Regarding Italy, where it had been proposed only for a new, modern school type, Beke addresses, in highly diplomatic terms, the refusal by the majority of the mathematicians. In extensive parts, he outlined various conceptions for the teaching of infinitesimal calculus, manners to present its applications (geometric, physical, etc.) and focussing in particular on the question of rigour. The interactions between calculus and other subjects and the reactions from secondary and university teachers are discussed. On this last point in particular, Beke points out that while secondary school teachers were enthusiastic, university professors expressed a certain coolness, if not hostility. The reason for this behaviour is to be attributed, according to Klein, to the lack of rigour of the textbooks of infinitesimal calculus and to the gulf which existed between teachers and academics. Beke concludes his paper by inviting university professors to support the reform movement by collaborating with secondary school teachers and writing textbooks.

Sources Beke, Emanuel. 1898. A homogén lineáris differenciálegyenletek rezolveseinek alapegyenleteiröl [in Hungarian]. Math. Termtud. Értesítö 16: 407–420. Beke, Emanuel. 1906. Bevezetés a differenciál és integrálszámításba. [Introduction to the Differential and Integral Calculus; known as the “little Beke”; in Hungarian, reeditions in 1920, 1965, 1967]. Budapest: Franklin. Beke, Emanuel. 1910, 1916. Differenciál és integrálszámítás, I–II, [The differential and Integral Calculus; known as the “big Beke”; in Hungarian]. Budapest: Franklin. Beke, Emanuel. 1925. Determinánsok. Budapest: Atheneum. [Determinants; in Hungarian]. Beke, Emanuel. 1926. Analitikus geometria. Budapest. [Analytic Geometry; in Hungarian]. Kántor, Tünde. 2006. Beke, Manó. In A panorama of Hungarian mathematics in the twentieth century I Biographies, ed. John Horváth, 567–568. Berlin-Heidelberg: Springer. Kántor, Tünde. 2014. Arcképek a 20. század magyar matematikusairól: Beke Manó [Portraits of the Hungarian Mathematicians in the 20th century: Emanuel Beke]. Polygon 22 (1–2): 3–20. Radnai, Gyula. 2019. Beke Manó levele Eötvös Lorándnak. Érintő [e-Journal of the Bolyai János Mathematics Society], March 11.

Publications Related to Mathematics Education Beke, Emanuel. 1896. Vezérkönyv a népiskolai számtani oktatáshoz [Handbook for arithmetic education in elementary schools; in Hungarian]. Budapest: Egyetemi nyomda. Re-editions in 1900, 1911, 1923. Beke, Emanuel. 1900. Tipikus hibák a matematika tanításában [Typical errors in mathematics instruction; in Hungarian], Magyar Paedagógia, I) 10(9), 520–530. Beke, Emanuel. 1909. Über den jetzigen Stand des mathematischen Unterrichtes und die Reformbestrebungen in Ungarn. In Atti del IV Congresso Internazionale dei Matematici 1908. Vol. III, ed. Guido Castelnuovo, 530–533. Roma: Accademia dei Lincei.

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Beke, Emanuel and Sándor Mikola. 1909. A középiskolai matematikatanítás reformja [The reform of mathematics teaching in secondary schools; in Hungarian]. Budapest: Franklin. Beke, Emanuel. 1914. Les résultats obtenus dans l’introduction du calcul différentiel et intégral dans les classes supérieures des établissements secondaires”. L’Enseignement Mathématique 15: 245–306.

Photo Courtesy of Sándorné Kántor Tünde Varga (Collection of Beke’s daughter Margit Beke).

654

Charles Bioche

12.2 Charles Bioche (Paris, 1859–Férrières- en-Brie, 1949): Appointed Honorary Member in 1936 Éric Barbazo

Biography Charles Bioche was born on 7 September 1859 in Paris, the youngest of four children. He studied first in Lyon and after in Paris at the Collège Stanislas. He was admitted at the École Normale Supérieure en 1879. There he met, among others, Henri Bergson and Gaston Milhaud who became his friends. In 1884 he got the Agrégation de mathématiques for becoming a mathematics teacher. He taught in Mâcon, Poitiers, and in Paris, firstly at Lycée Michelet, at his Collège Stanislas and, in 1897, at Lycée Louis le Grand, where he stayed until his retirement in 1925. The mathematical activity of Bioche concerns primarily geometry. His first article, published in the Bulletin de la Société Mathématique de France in 1886, deals

É. Barbazo (*) Établissement International Français, Houston, TX, USA

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with the theorems on the curvature of the surfaces of Euler and Monge mentioned in an article of Poisson in 1832. Afterward, the majority of his many works were dedicated to the properties of algebraic or ruled surfaces. He also studied the resolution of differential equations and properties of continuity of functions. Although he was not a university researcher, Bioche published his works in high-level journals such as Bulletin des Sciences Mathématiques, Intermédiaire des Recherches Mathématiques, Revue de Mathématiques Spéciales, Mathésis and Nouvelles Annales de Mathématiques. This production consisted of remarks or clarifications on the results of contemporary mathematicians. Bioche belongs to the category of those mathematicians who did not make fundamental discoveries but obtained just minor results, which nevertheless, as he claimed, gave him great pleasure. His name is linked to some rules for calculating integrals through the change of variables, commonly called “rules of Bioche”.

Contribution to Mathematics Education Bioche devoted the main part of his career to teaching mathematics at the secondary level. In 1898 Jean Griess, a former student at the École Normale Supérieure, and the publisher Henri Vuibert founded the journal L’Éducation Mathématique. This bimestrial journal concerned elementary mathematics teaching, that is, the mathematics in the programmes “of the literature classes of the classic and modern stream of Écoles Normales and first years undergraduate schools, of girls secondary schools, of schools of art and craft, of agriculture, of trade, and of the various administrations”,2 as stated in the debut issue. After Griess’ death, in 1899 Bioche became the editor of the journal. According to Vuibert, he worked at the journal with passion and curiosity and made it very interesting. As a matter of fact, Bioche knew well the issues treated in the journal because in 1895 he had published the Eléments de géométrie à l’usage des classes de lettres (Saint Cloud: Belin). Bioche contributed to many other journals. From 1911 to 1930, he was vice-­ president of the Association Générale des Membres de la Presse de l’Enseignement and handled its quarterly bulletin. From 1920 to 1930, he was one of the main editors of the Revue de l’École, a weekly educational journal that published articles of general pedagogy and of disciplinary didactic. From July 1930 to December 1931, he edited a publication entitled Les sciences à la première moitié du baccalauréat (publisher Hatier) whose main goal was to help the candidates in the baccalauréats de première to pass the very difficult science tests. The variety of journals to which Bioche contributed shows his commitment to the development of mathematics teaching in France in the first half of the twentieth  The original text is: “classes de lettres des enseignements classique et moderne, de l’enseignement secondaire des jeunes filles, des Ecoles Normales et primaires supérieures, des écoles d’arts et métiers, d’agriculture, de commerce et des diverses administrations”. 2

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century. Since the beginning, he was active in the Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP) founded in 1910 with the aim of supporting the reform of 1902, which deeply transformed the French scientific teaching in secondary school. He devoted a lot of time to the association and its bulletin, where he published many articles. He was president of APMEP from January 1921 until February 1923 and later on April 1924 until February 1925. The topics discussed in the association were in line with those treated in the journals where Bioche was contributing: the teaching to girls, the pedagogical problems linked to the reform of 1902 and the difficulties introduced in science teaching by the reform based on the principe d’égalité scientifique (scientific equality principle) launched in 1925. This reform inspired the publication Les sciences à la première moitié du baccalauréat mentioned before. In 1909 Bioche was appointed as a president of the Société Mathématique de France (S.M.F.) and assistant secretary of the French Subcommission of the Commission Internationale de l’Enseignement Mathématique (CIEM). In 1913, following the death of Carlo Bourlet and the resignation of Albert de Saint Germain and Charles-Ange Laisant, he became a delegate of the Commission. He was a member of the special subcommissions appointed with the task of preparing brief reports as a base for discussion in the congress organised by CIEM in Milan (1911). At this congress, he presented a report on the fusions of algebra with geometry, planimetry with trigonometry and stereometry with descriptive geometry. In 1912 he participated in the International Congress of Mathematicians in Cambridge and took an active part in the discussion about the mathematical training of the physicist at a university and in the discussion about intuition and experiment in mathematical teaching at secondary schools. In the Congress organised by CIEM in Paris (1914), he presented a report on the introduction of the integral and differential calculus in the programmes of the lycées. Thanks to his role in CIEM, he published some articles on the systems of teaching in various countries in the Bulletin de l’Association des Professeurs de Mathématiques. This provided the community of French teachers with a new acquaintance with foreign situations as for mathematics education. After his retirement in 1925, Bioche continued his editorial activity practically until his last days. Between 1930 and 1944, he published many articles on elementary geometry and arithmetical curiosities. In 1936, at the International Congress of Mathematicians in Oslo, he was honoured with the title of Membre Honoraire de la Commission International (Honorary Member of the International Commission).3 He passed away in the house of his son in Ferrières-en-Brie (France) on 19 August 1949.

 The original text is: “classes de lettres des enseignements classique et moderne, de l’enseignement secondaire des jeunes filles, des Écoles Normales et primaires supérieures, des écoles d’arts et métiers, d’agriculture, de commerce et des diverses administrations”. 3

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Sources D’Enfert, Renaud and Caroline Ehrhardt. 2019. The French Subcommission of the International Commission on Mathematical Instruction (1908–1914): Mathematicians committed to the renewal of school mathematics. In National Subcommissions of ICMI and their role in the reform of mathematics education, ed. Alexander Karp, 35–64. Cham, Switzerland: Springer. Brasseur, Rolland. 2020. Charles Bioche. Retrieved on 9 February 2020 from: https://sites.google. com/site/rolandbrasseur/6%2D%2D-­charles-­bioche Brasseur, Rolland. 2020. Charles Bioche. Dictionnaire des professeurs de mathématiques en classe de mathématiques de spéciales. Retrieved on 9 February 2020 from: https:// drive.google.com/file/d/0B71JfRYrV2lYUGNEWmFkNmVpY1U/view?resourcekey= 0-4-3th8e7WUg8363c62WMWA Maillard, Roland. 1950. Bioche (Charles Marie Paul). Association Amicale de Secours des Anciens Élèves de l’École Normale Supérieure: 23–24.

Publications Related to Mathematics Education Bioche, Charles. 1891. Introduction à l’étude de la géométrie moderne. Paris: C. Delagrave. Bioche, Charles. 1895. Éléments de géométrie à l’usage des classes de lettres. Saint Cloud: Belin. Bioche, Charles. 1911. Sur le congrès de Milan. Bulletin de la Société Mathématique de France 39: 522. Bioche, Charles. 1914a. Histoire des mathématiques. Paris: Belin Frères. Bioche, Charles. 1914b. La Conférence internationale de l’enseignement mathématique. Revue Générale des Sciences Pures et Appliquées 25(9): 466–469. Bioche, Charles. 1942. Petit guide pour étudier les mathématiques. Paris: Hatier.

Photo Source: Roland Brasseur’s website, courtesy of Francis Bioche.

658

Farid Boulad Bey

12.3 Farid Boulad Bey (Cairo, 1872–Cairo, 1947): Appointed Honorary Member in 1936 Gregg De Young

Biography The Boulad family traces its history back to the Melkite Catholic community of medieval Damascus, where they had developed a reputation for excellence in metalworking.4 Their workshops had a long-established reputation for producing fine swords, which discerning consumers regarded as a cut above the competition. Their products were widely sought after, even in Europe. The family also had a long history of both civic engagement and technical ingenuity. Following the massacre of Christians in Damascus in 1860, several members of the extended family emigrated

 The bulk of the personal information concerning Farid Boulad is drawn from obituaries published in French at the time of his death. Jean-Édouard Goby (1947) published his obituary essay in Bulletin de l’Institut d’Égypte. The choice of Goby, a French civil engineer and director of the Suez Canal Company from 1938 until its nationalisation in 1956, to prepare this death notice underlines Boulad’s international reputation and his career as a civil engineer. A second obituary notice appeared anonymously in Le Lien: Revue de Patriarcat (1947), the official journal of the Melkite Catholic Church. I thank Deacon Charbel Nassif from the Secretariat of the Melkite Patriarchate in Beirut for kindly providing me an electronic copy of this obituary notice. Unless otherwise noted, all biographical information derives from these two obituary notices. 4

G. De Young (*) The American University in Cairo, Cairo, Egypt e-mail: [email protected]

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in a search for more congenial circumstances. Members of the family migrated to Lebanon, to the USA and to Egypt. It was in Cairo that Farid Boulad Bey was born on 7 October 1872, the son of Youssef Boulad (1837–1910), who had come to Egypt as a young man and spent most of his career as an agricultural engineer in charge of managing the estates of Prince Ḥassan Pasha Ḥafez. Farid occupied the medial position in a family of nine children and was the eldest son (after four daughters). Following his birth, his parents were blessed with three more sons and another daughter (Aractingi 2019). Education Farid completed his secondary school education at the Collège des Frères (Khoronfish), receiving his baccalauréat in 1889. Even before he had completed his studies, he had begun to publish solutions to difficult mathematical problems in international journals, such as Revue des Sciences Mathématiques (Paris) and al-Muqṭaṭaf (Cairo). Farid then enrolled in the School of Agriculture in Giza, honouring the wishes of his father. He was not, however, particularly interested in agriculture. He much preferred the study of mathematics, both theoretical and applied, which he continued to study at every opportunity. His mathematical acumen was such that, when the Egyptian government announced a competition for a grant stipend to study abroad, Farid applied and won the first place. And so in 1892, he travelled to France to complete his higher education not in agriculture but in mathematics. Young Boulad did not waste his opportunity. Upon arrival in France, he immediately enrolled in several specialised mathematics courses at the École Duvignau de Lannau, a private preparatory school associated with the École Centrale des Arts et Manufactures, in order to strengthen his formal grounding in mathematics. He threw himself into his studies and, at the end of the year, completed his Cours préparatoires at the head of his class. The following year, based on his excellent performance, he was offered admission to both the École Nationale des Ponts et Chaussées and the École Centrale des Arts et Manufactures.5 Farid chose the former institution, and that choice proved to be a defining moment, shaping his professional career for the next four decades. His choice brought him into the elite halls of the old and prestigious engineering and applied science school in Europe. This was the age of metal constructions (perhaps most famously, the Eiffel Tower,6 which had

 Boulad would have been among the handful of “outside students” who were admitted each year to l’École Nationale des Ponts et Chaussées. Among these, the “foreign students” formed an important subunit. Although some were, for political reasons, exempted from the usual entrance examination (Connor 1913, pp. 16–17), this does not seem to have been the case with Boulad, who spent his first year in France studying mathematics in a preparatory school (Goby 1947, p. 22). 6  Even before his arrival in France, Boulad would have been familiar with major iron constructions, having witnessed the completion of the Imbaba railway bridge across the Nile in 1892 as well as the erection of the new terminal building on Ramsis Street in Cairo. This first Imbaba bridge was 5

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been inaugurated in Paris in 1889 for the World Exposition), and Boulad quickly chose this as his career direction. The future appeared very rosy. But only a few months later, he was forced to apply for a leave of absence because of his health. Despite these difficulties, he completed his diploma in 1898 and took up two short-term practicum positions with the French northern railways company, first with the Paris-Nord line and then on the Paris-Lyon line. The same year he married a French girl, Alice Bertrand (Aractingi 2019). Professional Engineer and Mathematician In 1899, after nearly 7 years in France, Boulad, with his young wife, returned to his homeland where he was appointed an engineer in the Zagazig district of the Egyptian state railway, a position he held for 2  years, before returning to Cairo. He was appointed Ingénieur et Vérificateur Principal in charge of bridges with the Egyptian National Railway, a position he held until 1924. The couple had one daughter, Marie, born in 1905. They divorced some time thereafter. Boulad never remarried – as a member in good standing in the Melkite Catholic Church, it would have been difficult to obtain religious sanction for a remarriage. Instead, Boulad apparently threw himself with new intensity into developing his career. Throughout his professional career, Boulad was engaged in important projects to develop and improve public works and transportation infrastructure. His work on the railway system of Egypt, for example, took place during a time of rapid development in railway technology that led locomotives to almost double in weight by the end of the century. A natural consequence was that the original bridges in the railway system needed either to be reinforced or rebuilt in order to sustain the effects of the rapid growth in the technology. This was precisely the kind of problem for which Boulad’s engineering training had prepared him. Boulad was personally involved in overseeing the project to reinforce the iron bridges at Kafr El-Zayat and Benha in the Nile Delta region for which he received the Décoration de l’Ordre du Medjidiyé in 1905. And he was responsible for the verification of all calculations involved in the planning of new iron bridges over the Nile at Zifta (1907), the Nile Barrages, Nag Hammadi, Mansoura (1913), and Imbaba (1924). One of the high points of his career was the planning and construction of the iron bridge spanning the Nile at Edfina in the Nile Delta. Several years after his return to Cairo in 1901, Boulad, as part of his official duties, began to cultivate ties with European-learned societies.7 Of course, he looked a “swing” bridge, a kind of bridge discussed by Boulad in the Congrès international de la construction métallique held in Liège (1930). 7  It is noteworthy that Boulad turned first to foreign societies, both mathematical and technical, in order to build his career. It was not until 1921 that he joined Egypt’s premiere intellectual society, the Institut d’Égypte, and only in 1924 did he join the Société Royale Égyptienne des Ingénieurs.

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first to France – his second homeland. In 1909 he joined the Société Mathématique de France and the Association Française pour l’Avancement des Sciences, and the next year he became a member of the Circolo Matematico di Palermo. He was also a member of the London Mathematical Society from 1912 to 1918 and was a member of the American Mathematical Society from 1913 to 1932. We can see that Boulad identified himself first and foremost as a mathematician who worked in the engineering field. As a mathematician, his primary contributions lay in nomography, the use of graphical techniques to solve equations having multiple variables. Within this subspecialty, Boulad’s contributions fell mainly in two areas. On the one hand, he pioneered the application of nomographic methods to solve problems in iron construction, calculating the stresses on iron beams in various constructions, for example. On the other hand, he was an active investigator studying fundamental problems in nomography, such as the disjunction of variables. Many of Boulad’s more theoretical papers were published in mathematical journals, but his contributions to the practical graphical solutions for equations such as those needed for design of bridges were published in a variety of technical and professional journals, especially Annales des Ponts et Chaussées. Nomography did not present a single unified solution for resolving these complex equations or for calculating the results of complex interacting forces. Each type of equation required a specific nomogram for its solution. Boulad had a remarkable ability to perceive, almost intuitively, the correct nomogram to solve a particular problem. Whenever he was presented with a question, he would almost invariably see at once the most appropriate approach to use in solving the question. Boulad’s contributions to nomography were highly regarded by his contemporaries. Indeed, Maurice d’Ocagne, the founder and systematiser of this field, singled out several of Boulad’s contributions for special mention in the second edition of his monograph (d’Ocagne 1921, pp. 456–468). His sentiments toward Boulad are most clearly evident in some unpublished notes that he allowed Goby to use in his obituary notice (1947, pp. 27–28): Among the researchers who, trained at my school, have endeavored to make their contribution to the development of nomography, Farid Boulad is, by far, the one who has particularly stood out. Moreover, more than any other, he has multiplied towards me the marks of the warmest gratitude for the teachings he owed me, and he was indignant, even with a vivacity of which I could only be deeply touched, about the attitude of certain authors who have forgotten me, voluntarily or not, which is quite inexplicable. Besides having a beautiful intelligence, Farid Boulad is a brave heart.8

 The original text is: “Parmi les chercheurs qui, formés à mon école, se sont efforcés d’apporter leur pierre au développement de la nomographie, Farid Boulad est, et de beaucoup, un de ceux qui se sont le plus particulièrement distingués. En outre, plus que tout autre, il a multiplié à mon endroit les marques de la plus chaude gratitude pour les enseignements qu’il m’avait dus, s’indignant même avec une vivacité dont je n’ai pu être que profondément touché, de l’attitude de certains auteurs qui ont fait montre à mon égard d’un oubli volontaire ou non, assez inexplicable. En même temps qu’une belle intelligence, Farid Boulad est un brave coeur”. 8

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For his part, Boulad had published an Arabic review of d’Ocagne’s monograph on nomography in al-Muqtaṭaf (1908) and another historical overview of nomography in Arabic in al-Handassa (1922). These, together with his regular research reports in French, helped spread the knowledge of nomography in the Middle East and North Africa. Moreover, he held d’Ocagne in such high esteem that he installed a portrait of him in a place of honour in his home at 10 rue Boustan al-Maqsi (Goby 1947, p. 28). But fame is fleeting and history has been less generous to Boulad. Despite d’Ocagne’s praise for his contributions, Boulad receives only one brief sentence from Harold Ainsley Evesham (1982) in his survey of the history of nomography. Hankins (1999) in his review of the origins of nomography made no mention of Boulad’s contributions. And Tournès (2016), in his most recent historical survey of nomography, also makes no mention of Boulad or his contributions. Perhaps this historical eclipse should not surprise us, though, since Boulad’s papers were not so much theoretical as practical applications of the basic principles to solve engineering problems. Boulad received a variety of academic and sociopolitical awards during his lifetime (Institut 1947, p.  248). In 1905, the Egyptian government awarded him the Décoration de l’Ordre du Medjidiyé in recognition of his services in reinforcing several Nile railway bridges. In 1922 the Académie des Sciences de Paris awarded him the Montyon Prize in mechanics for his work in nomography and geometry (American Mathematical Society 1923, p. 91). The following year, he was decorated by the French government with the Croix de Chevalier de la Légion d’Honneur. And in 1925, the Egyptian government elevated Boulad to the rank of Bey (second class). On his retirement in 1932, the Egyptian government further honoured his contributions to the state by awarding him the Ordre du Nil, Quatrième Classe. In addition, the French government awarded him the title Officier de la Légion d’Honneur in 1936.

Contribution to Mathematics Education Farid Boulad took special delight in attending scientific conferences in Europe – and especially in France, a country that remained dear to his heart following his university years in Paris. Beginning in 1909, he attended nearly every meeting of the Association Française pour l’Avancement des Sciences until the outbreak of World War I in 1914. In his presentations to these learned gatherings, he continued to develop aspects of nomographic theory and its applications to construction of bridges and similar large steel edifices. In 1912, Boulad attended his first International Congress of Mathematicians (ICM), which was held in Cambridge, where he presented a short communication on use of nomographic techniques to solve equations in four variables (Boulad 1913). His paper was published in the proceedings. But Boulad did not publish a detailed account of his activities at the conference. Following the end of the war, he

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continued to attend successive quadrennial meetings of the mathematicians  – in Strasbourg, Bologna, Zurich, and Oslo. The 1928 ICM in Bologna was the first in which Boulad was officially designated to represent the Institut d’Égypte.9 As representative of the society, he submitted a detailed report of his activities – at least his official activities – during the congress. In this report, he made no mention of the education section nor of his role in its activities. The earliest mention of the education section of the ICMs in Boulad’s reports appears to be that concerning the 1936 ICM in Oslo. By this time the education section – or at least communications on the teaching of mathematics – has generally been a regular feature of ICM meetings. Boulad seems to treat it as an established entity in his report. At this congress, he presented two papers in Section III (geometry), both based on traditional nomographic theory (Boulad 1937a, 1937b). In his report, Boulad noted these contributions with little fanfare, along with brief mention of the mathematical contributions of several colleagues. He devoted more space, however, to the fact that he had been requested to chair the second session in the education section10 – a request that suggests that by this time he had already developed some reputation among the members of this section. The first session had been devoted to the International Commission on the Teaching of Mathematics. It is clear from the tone of the remarks that Boulad was deeply interested in the mission of this Commission, although there is no record in his published work that he ever presented a paper to this August meeting. Boulad joined the Commission in 1913.11 Perhaps one of the stimuli leading Boulad toward the activities of the Commission may have been the repeated emphasis of d’Ocagne, the inventor of nomography, concerning the importance of mathematics in the education of engineers.12 Boulad remained a member of the Commission until his retirement in 1932. The precise nature of his contributions are obscure since he appears to have presented no papers or communications to the Commission during any of its meetings. Nevertheless, perhaps because he was one of the longest-­ serving members of the Commission, he was honoured at the 1936 ICM in Oslo with the title Membre honoraire de la Commission13 and was asked, as mentioned above, to chair the second session of Section VIII (Pédagogie). Boulad’s report

 See Atti del Congresso Internazionale dei matematici, Bologna 3–10 Settembre 1928, 6 Vols. Bologna: Zanichelli, 1929–1932. Vol. I, 1929, p.  27; L’Institut Égyptien d’Actuariat et de Statistique 1947, p. 249. 10  See Comptes rendus du Congrès International des Mathématiciens, Oslo 1936. Oslo: A. W. Brøggers Boktrykkeri. 1937. Vol I, p. 17. 11  See L’Enseignement Mathématique, 1914, p. 166; L’Institut Égyptien d’Actuariat et de Statistique 1947, p. 247. 12  See, for example, d’Ocagne’s concluding speech to the general assembly of the International Commission on the Teaching of Mathematics in Paris during April 1914 (Fehr 1914, pp. 211–222). Although it appears that Boulad did not attend this meeting, he may well have read the published summary and ensuing discussion. 13  See Comptes rendus du Congrès International des Mathématiciens, Oslo 1936. Oslo: A. W. Brøggers Boktrykkeri. 1937. Vol. 2, p. 289. 9

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(1936–1937, 240–241) on these education sessions is unusually complete, perhaps indicative of his personal interest in the aims of the Commission. What was it that piqued Boulad’s long-standing involvement with the International Commission on the Teaching of Mathematics? We can only speculate because Boulad has left us no public record of his motives. Perhaps it was his involvement in major engineering construction projects – projects requiring extensive and precise mathematical calculations – that first turned Boulad’s attention to the importance of mathematics in the education of professional engineers. Or it may have been his recognition that his own mathematical training had provided him the essential tools to carry out his professional responsibilities. Whatever the case may be, it is clear that Boulad did not approach mathematics education from the point of view of the traditional classroom instructor in mathematical sciences but rather as a professional consumer of mathematics who needed extensive mathematical skills in order to produce sound engineering solutions to the new problems that the rapidly developing iron construction techniques posed.

Sources Boulad, Farid. 1908. La nomographie et le livre du professeur d’Ocagne le fameux mathématicien français. Al-Muqtaṭaf (Feb 1908): 147–148. (In Arabic). Boulad, Farid. 1913. Extension de la notion des valeurs critiques aux équations à quatre variables d’ordre nomographique supérieur. In Proceedings of the Fifth International Congress of Mathematicians, eds. E.  W. Hobson and A.  E. H.  Love, Vol. 2, 295–299. Cambridge: Cambridge University Press. Boulad, Farid. 1930–1931. Compte rendu de la mission aux deux congrès internationaux de béton et béton armé et de la construction métallique tenus à Liège du 1 au 7 Septembre 1930. Bulletin de l’Institut d’Égypte 13: 37–45. Boulad, Farid. 1936–1937. Compte rendu de ma mission au 10e Congrès international des mathématiciens tenu à Oslo du 13 au 18 Juillet 1936. Bulletin de l’Institut d’Égypte 19: 239–241. Boulad, Farid. 1937a. Sur les formes des équations à 3 variables représentables par des abaques coniques à simple alignement. In Comptes rendus du Congrès International des Mathématiciens, Oslo 1936, Vol. 2, 168–169. Oslo: A. W. Brøggers Boktrykkeri. Boulad, Farid. 1937b. Sur la symétrie nomographique et les formes canoniques des équations à 4 variables représentables par des abaques à double alignement. In Comptes rendus du Congrès International des Mathématiciens, Oslo 1936, Vol. 2, 169. Oslo: A. W. Brøggers Boktrykkeri. American Mathematical Society. 1923. Notes. Bulletin of the American Mathematical Society 29: 91–92. Aractingi, Farid. 2019. Farid Boulad Bey. Retrieved 23 December 2019 from: https://gw.geneanet. org/aractingi?lang=en & n=boulad & oc=0 & p=farid Connor, William Durward. 1913. National school of bridges and highways. Paris, France. Washington [Govt. print. off.]. Evesham, Harold Ainsley. 1982. The history and development of nomography. PhD thesis, University of London. Evesham, Harold Ainsley. 1986. Origins and development of nomography. Annals of the History of Computing 8: 324–333. Fehr, Henri (ed.). 1914. Compte rendu de la Conférence international de l’ensignement mathématique Paris 1–4 Avril 1914. L’Enseignement Mathématique 16: 165–226.

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Hankins, Thomas L. 1999. Blood, dirt, and nomograms: A particular history of graphs. Isis 90: 50–80. Goby, Jean-Édouard. 1947. Farid Boulad Bey (1872–1947). Bulletin de l’Institut d’Égypte 30: 21–36. L’Institut Égyptien d’Actuariat et de Statistique. 1947. Farid Bey Boulad (1872–1947): Notice biographique communiqué par l’Institut Égyptien d’Actuariat et de Statistique. Le Lien 12 (number 5–6): 37–43. D’Ocagne, Maurice. 1921. Traité de nomographie. Deuxième edition. Paris: Gauthier-Villars. Rory, N. 1930. Akuella spörsmål vid järn-och stålkon-gressen i Liège 1930. In Teknisk Tidskrift: Sextionde Årgången – Väg- och Vattenbyggnadskonst, ed., W. Finné, 140–144, 161–163. Tournès, Dominique. 2016. Abaques et nomogrammes. Retrieved 26 December 2019 from: https:// hal.archives-­ouvertes.fr/hal-­01484563

Photo Source: https://industrielsbelgesenegypte.omeka.net/treinen (Made in Belgium. Industriels belges en Égypte (1830–1952) Musée royal de Mariemont, Morlanwelz). (Retrieved June 2021).

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12.4 Samuel Dickstein (Warsaw, 1851–Warsaw, 1939): Appointed Honorary Member in 1936 Ewa Lakoma

Biography Samuel Dickstein was born on 12 May 1851  in Warsaw. He was a great Polish scientist-­mathematician, pedagogue, historian of natural sciences, organiser of the Polish scientific life and propagator of mathematics. His childhood and the greater part of his adult life took place during a very difficult period in Poland, when the country was not independent and had been divided up between Austria, Prussia and Russia. At that time, Poles strongly aspired to reunite the country, fighting against the domination of invaders. Poland finally regained independence in November 1918. Dickstein was among the scientists who were the most active and did all in their power to restore the country, mostly by broadening Polish education, by developing scientific research and by popularising science. In Warsaw, occupied by Russia, during the period 1869–1874, all secondary schooling was in Russian, and there was no Polish university. Dickstein, after his

E. Lakoma (*) Military University of Technology, Warsaw, Poland e-mail: [email protected]

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secondary school education, entered the only post-secondary school in Warsaw, Szkoła Główna (the Main School), in 1866 and studied there until 1869. Then he entered the Imperial University of Warsaw, which was established in 1870 by Russian rulers in place of the Main School (Kitzwalter 2016). He studied mathematics until 1876. At the same time, he worked as a mathematics teacher in secondary school. In 1878 he began to implement his ideas of promoting Polish education and for 10 years directed his own private secondary school for mathematics and science. Moreover, he began to promote Polish mathematics, its history and the history of science in many ways: as a lecturer of mathematics and history of science, as a scientist, as an author of works in mathematics and history of science, as a translator into Polish of the significant foreign works concerning modern mathematics, as a publisher and editor of scientific journals and finally as an organiser of Polish and international scientific societies, especially those assembling outstanding young mathematicians, in order to create the best possible conditions for their scientific development. In 1888, together with Władysław Natanson (1864–1937) and Władysław Gosiewski (1844–1911), Dickstein created, edited and published the Polish journal Prace Matematyczno-Fizyczne (Mathematical and Physical Papers), issued in 1888–1952, which quickly gained international renown. In 1897 he founded the journal Wiadomości Matematyczne (Mathematical News), which he edited until 1939. The publication of both these journals was continued after World War II. Beginning in 1955 they were published as Annals of the Polish Mathematical Society: Series 1, “Mathematical Papers”, renamed Commentationes Mathematicae in 1967, and Series 2, “Mathematical News”. In 1881 Dickstein founded another periodical, Rocznik Pedagogiczny (Pedagogical Yearly),14 which he edited in 1881–1884 and 1921–1928. In addition to these activities as a publisher and editor, he wrote books and articles concerning number theory, vector algebra, set theory and history of mathematics. Thanks to his articles published in international journals, especially in Bibliotheca Mathematica, many Polish mathematicians became known to foreign readers and were included in the select group of co-founders of European mathematics. In the field of history of exact sciences, Dickstein collected many old documents, which became points of departure for further scientific research. The monograph dedicated to the outstanding Polish mathematician Józef Hoene-Wroński (1776–1853) and the preparation of documents for a monograph dedicated to Adam Kochański (1631–1700), especially the publication of the correspondence between Kochański and Gottfried Leibniz (1646–1713), were among his most important works. His historical research was appreciated not only in Poland: he was elected as vice-president of the International Academy of History of Exact Sciences. Dickstein was well known in the scientific world: he was a member of the Scientific Society in Liège and an honorary member of the Czech Mathematical Society. His versatile activities (author of some 260 works, co-organiser of a network of meteorological stations in Poland, co-founder of the Museum of Industry and Agriculture and

14

 See (III Zjazd Polskiego Towarzystwa Matematycznego 1938).

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of the Museum of the Tatra Mountains, donor – bequeathing his rich library of tens of thousands of volumes to the Warsaw Scientific Society) became an example to follow for many of his students and for the next generations of mathematicians. Dickstein was a founder of many Polish societies. The Society of Scientific Courses (Towarzystwo Kursów Naukowych) was the only university in Warsaw whose aim was to give an opportunity to gifted young people to study at university level – it existed from 1906 to 1918. Dickstein was its founder, chair-holder and lecturer. He was a founder in 1903 of the Warsaw Scientific Society (officially established in 1907), a member from 1890 of the Society of Friends of Science in Poznań, a member from 1893 of the Academy of Arts and Sciences in Cracow and a co-founder and a head of the Mathematical-Physical Circle (Koło Matematyczno-­Fizyczne), which was organised in 1905 in order to integrate mathematicians, physicists and natural scientists (ibidem). The main aims of all those societies were to develop scientific research, to make science more popular, to organise financial support for young scientists and gifted students and to publish scientific books and articles. In the domain of mathematics, those societies also fostered strong cooperation between their members, the sharing of mathematical experience and teamwork. That particular atmosphere was the main secret behind the future great successes of the Polish Mathematical School. In a few years, all activities in scientific societies appeared to bear fruit, when in 1915, during World War I, in Warsaw there were favourable political conditions for the founding of the Polish University of Warsaw (Kitzwalter 2016). The group of mathematicians was ready to work at that university. Dickstein obtained a position as professor of mathematics and gave the first year lectures on algebra. In 1919 he became honorary professor of mathematics and history of science. In 1921 he was awarded an honorary doctorate degree by that university. He gave lectures on higher algebra and the history of exact sciences until 1937. The ideals pursued by the Society of Scientific Courses, the Academy of Arts and Sciences, the Warsaw Scientific Society and others were continued in the context of the Polish Mathematical Society (Polskie Towarzystwo Matematyczne or PTM), which was established in 1919. There too Dickstein played a leading role as one of the most active members and as president of the society from 1923 to 1926. The high international reputation of Dickstein is proved by the fact that in the inaugural address at the International Congress of Mathematicians of 1932 in Zurich, he is mentioned among the five illustrious participants to the meeting. Dickstein died on 28 September 1939. His daughter and most of his family perished during German occupation of Poland in World War II.  His tomb is in the Jewish Cemetery in Warsaw.

Contribution to Mathematics Education Dickstein’s commitment to mathematics education is testified by his relentless involvement in founding societies, schools and journals, in writing papers and books addressed at all school levels. These activities were in line with his aim

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of promoting the Polish mathematical culture and promote international contacts. He translated into Polish many works of foreign mathematicians and modern foreign academic textbooks on geometry, history of mathematics and of astronomy, calculus and analysis, pedagogy, physics, higher mathematics, astronomy and algebra. Among the authors translated, there are Felix Klein, Charles-Ange Laisant, Ernest Lebon, Gino Loria, James Clerk Maxwell, Wilhelm Franz Meyer, Gösta Mittag-­Leffler, Simon Newcomb, Ernesto Pascal, Giuseppe Peano, Émile Picard, Paul Reis, Corrado Segre, Bernhard Riemann and Heinrich Weber. We see that not only mathematics works were considered but also works specifically dealing with pedagogy of mathematics, such as the translation of the famous text Initiation mathématique by Laisant (1908. Nauczanie początków matematyki. Warsaw: Księgarnia Polska). In his translations he introduced new Polish terminology for concepts of modern mathematics. Moreover, he was the author of textbooks of elementary mathematics and of mathematics textbooks for students in secondary and post-­secondary schools.15 He treated the development of the art of mathematics teaching as a very important activity, so it is no coincidence that he was a co-organiser and animator of the first Polish pedagogical centre  – the National Pedagogical Institute (Państwowy Instytut Pedagogiczny), bringing together pedagogues and teachers of mathematics and science. Dickstein participated in the ICM of Zurich (1932) as a delegate of the Ministry of Education and was thanked with four other mathematicians in the opening address. His name appears in the list of authors at the end of volume 32 (1933) of L’Enseignement Mathématique without reference to page numbers; it may be supposed that he was author with Stefan Straszewicz (named in the same list) of the report “POLOGNE.  La préparation des professeurs de mathématiques de l’enseignement secondaire” (pp. 365–374). In 1936, at the International Congress of Mathematicians in Oslo, he was honoured with the title of Membre Honoraire de la Commission International (Honorary Member of the International Commission).16 Dickstein’s work and activities are still a role model for many contemporary mathematicians in Poland. The Polish Mathematical Society, in 1979, established a prestigious prize named after Samuel Dickstein in order to honour Polish mathematicians for their outstanding achievements for the benefit of mathematical culture.

 Most books and works of Dickstein are available in Poland at the National Library in Warsaw, the University of Warsaw Library, the Jagiellonian Library in Cracow, the Library of the Nicolaus Copernicus University in Toruń, the Library of the Adam Mickiewicz University in Poznań, Central Mathematical Library of the Polish Academy of Sciences in Warsaw and the University of Gdańsk Library. 16  See Comptes rendus du Congrès International des Mathématiciens, Oslo 1936, Oslo: A. W. Brøggers Boktrykkeri. 1937. Vol. 2, p. 289. 15

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Sources III Zjazd Polskiego Towarzystwa Matematycznego [The third meeting of the Polish Mathematicsl Society]. 1938. Jubileusz 65-lecia działalności naukowej, pedagogicznej i społecznej profesora Samuela Dicksteina [The jubilee of the 65 years of Samuel Dickstein’s scientific, pedagogical and social activities, 3.10.1937]. In Wiadomości Matematyczne [Mathematical News]: memorial volume. Dickstein, Samuel. 1889. Note bibliographique sur les études historico-­mathématiques en Pologne. Bibliotheca Mathematica s. 2, 3:43–51. Dickstein, Samuel. 1896, Katalog dzieł i rękopisów Hoene-Wrońskiego [Catalogue of works and manuscripts of Hoene-Wronski]. Cracow: Akademia Umiejętności. Dickstein, Samuel. 1896. Hoene Wroński: jego życie i prace [Hoene Wroński (1776–1853): His life and work]. Cracow: Akademia Umiejętności. Dickstein, Samuel. 1922. Galileusz w Padwie [Galileo in Padua]. In Wiadomości Matematyczne [Mathematical News] 26: 91–102. Dickstein, Samuel. 1926. Algebra wyższa opracowana według wykładów Samuela Dicksteina [Higher Algebra – elaborated according to the lectures of Samuel Dickstein]. Warsaw: Komisja Wydawnicza Koła Matematyczno-Fizycznego słuchaczy Uniwersytetu Warszawskiego. Dickstein, Samuel and Aleksander Birkenmajer. 1933. Coup d’oeil sur l’histoire des sciences exactes en Pologne. In Histoire sommmaire des sciences en Pologne, 1–33. Cracow: Drukarnia Narodowa. Dickstein, Samuel and Eduard Bodemann. 1902. Korespondencya Kochańskiego i Leibniza [Correspondence between Kochański and Leibniz]. Warsaw: Druk J. Sikorskiego. Dickstein, Samuel and Adam Danielewicz. 1910. Zarys arytmetyki politycznej [Political arithmetic]. Warsaw: Wydawnictwo byłych Wychowańców Szkoły Handlowej im. L. Kronenberga. Domański, Czesław. 2013. Samuel Dickstein (1851–1939). Acta Universitatis Lodziensis Folia oeconomica 285: 11–13. Domoradzki, Stanisław, Zofia Pawlikowska-Brożek, and Danuta Węglowska (eds.). 2003. Słownik biograficzny matematyków polskich [Biographical Dictionary of Polish Mathematicians]. Tarnobrzeg: Państwowa Wyższa Szkoła Zawodowa. Duda, Roman. 2016. Samuel Dickstein, Monumenta Universitatis Varsoviensis, Warsaw: University of Warsaw Publishers. Fehr, Henri. 1936. Compte rendu de la réunion d’Oslo, 15 juillet 1936. L’ Enseignement Mathématique 35: 386–388. Kitzwalter, Tomasz (ed.). 2016. History of the University of Warsaw 1816–2016. Warsaw: University of Warsaw Publishers. Knaster, Bronisław. 1955. Wspomnienia o Dicksteinie [Remembrance on Dickstein]. Roczniki PTM, Prace Matematyczne [Annals of the Polish Mathematical Society, Mathematical Papers] s. 1, 1: 4–8. Mostowski, A. 1949. La vie et l’oeuvre de Samuel Dickstein. In Prace Matematyczno-­Fizyczne [Mathematical and Physical Papers] 47(1): VII–XII. Novy, Luboš. 1971. Dickstein, Samuel. In Dictionary of Scientific Biography, Vol. 4, 83–84. New York: Charles Scribner’s Sons. Rapport du Cercle mathematico-physique de Varsovie. 1911. L’Enseignement des mathématiques et de la physique dans les écoles privées de la Pologne. L’Enseignement Mathématique 13: 299–319.

Publications Related to Mathematics Education Dickstein, Samuel. 1891. Pojęcia i metody matematyki [Concepts and methods of mathematics]. Warsaw: Redakcja Prac Matematyczno-Fizycznych.

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Dickstein, Samuel. 1898. Poradnik dla samouków, Matematyka, Nauki przyrodnicze [Guidebook for self-instruction, Mathematics, Natural Sciences]. Warsaw: S. Michalski and A. Heflich. Dickstein, Samuel and Władysław Folkierski. 1904. Zasady rachunku różniczkowego i całkowego [Fundamentals of differential calculus and integral calculus]. Warsaw: Księgarnia E. Wende. Dickstein, Samuel. 1883a. Arytmetyka w zadaniach – Liczby całkowite [Arithmetic in exercises – Whole numbers]. Warsaw: Gebethner i Wolff. Dickstein, Samuel. 1883b. O muzeach pedagogicznych słów kilka [Short note on pedagogical museum]. Warsaw: Niwa. 202: 744. Dickstein, Samuel. 1883c. Początkowa nauka geometryi w zadaniach, książeczka I [Initial learning of geometry in tasks, little book I]. Warsaw: Gebethner i Wolff. Dickstein, Samuel. 1886. Arytmetyka w zadaniach – Ułamki – 1340 zadań [Arithmetic in exercises – Fractions – 1340 Tasks]. Warsaw: Gebethner i Wolff. Dickstein, Samuel. 1889. Geometrya elementarna [Elementary Geometry]. In Encyklopedia Wychowawcza [Educational Encyclopedia]. Warsaw: Gebethner i Wolff. Dickstein, Samuel. 1895. Arytmetyka w zadaniach  – Stosunki, proporcjonalność, kwadratury, sześciany – zadania różne [Arithmetic in exercises – Ratios, proportionality, squaring, cube – various tasks]. Warsaw: Gebethner i Wolff. Dickstein, Samuel. 1921. Arytmetyka w trzech częściach  – Ułamki [Arithmetic in three parts  – Fractions]. Warsaw, Cracow, New York: Gebethner i Wolff. Dickstein, Samuel and Stefan Straszewicz. 1933. POLOGNE. La préparation des professeurs de mathématiques de l’enseignement secondaire. L’Enseignement Mathématique 32: 365–374. The report is not signed, but attributed to Straszewicz in the list of authors at the end of the issue; Samuel Dickstein named in the same list, but without the page numbers referring to a text, is probably his co-author.

Photo Source: Mleczko, Paweł. 2012. Samuel Dickstein the Founder of Wiadomości Matematyczne. Wiadomości Matematyczne 48(2): 331–334.

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Federigo Enriques

12.5 Federigo Enriques (Livorno, 1870–Roma, 1946): Appointed Honorary Member in 1936 Livia Giacardi

Biography Federigo Enriques was born in Livorno on 5 January 1871 to Giacomo Enriques and Matilde Coriat. He attended secondary schools in Pisa, and already at that time he displayed his inclinations for mathematics. He completed his studies at the University of Pisa and then the Scuola Normale Superiore in that same city, where he had great mathematicians as masters, including Enrico Betti, Ulisse Dini, Luigi Bianchi, Vito Volterra and Riccardo De Paolis. He graduated brilliantly in 1891 and, after a year of specialisation in Pisa, he came to Rome in November 1892 to follow Luigi Cremona’s course, always as a specialisation student. Here the fruitful scientific partnership and the deep friendship with Guido Castelnuovo started, which was consolidated by the marriage of Castelnuovo with Elbina, sister of Enriques. L. Giacardi (*) University of Turin, Turin, Italy e-mail: [email protected]

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From November 1893 to January 1894, he spent a few months in Turin for further training with Corrado Segre. In 1894 he was appointed lecturer of projective and descriptive geometry at the University of Bologna, and 2 years later, following a competition, he obtained that professorship. In Bologna he married Luisa Miranda Coen. During his stay in Bologna, he was president of the Italian Philosophical Society from 1907 to 1913, and in this capacity, he organised and presided over the IV international congress of philosophy which was held in that city in 1911. From 1913 to 1915, he was president of the National Association of University Professors and formulated a project for reforming the Italian university. The Bolognese period ended in 1922 when Enriques moved to Rome, first to the chair of complementary mathematics and then of higher geometry. Those years in Bologna were the most fruitful of his life, because his most significant contributions in both the mathematical and philosophical fields date back to that period. Enriques left important traces of his multifaceted and tireless activity also in Rome; in particular he founded the National Institute for the History of Sciences at the University of Rome in 1923 and, connected with it, a University School of History of Sciences. In 1938 he was suspended from teaching and official positions because of the anti-Jewish racial laws; the university chair was returned to him only in 1944. His mathematical work mainly concerns algebraic geometry and, in particular, the theory of surfaces. Although he was attracted by the new directions in algebraic geometry promoted by Segre, it was in Castelnuovo that Enriques found the ideal guide and collaborator. Castelnuovo wrote: I immediately realized that … Federigo Enriques was a mediocre reader. On the page before his eyes he saw not what was written, but what his mind projected on it. So, I took another method: the conversation. … Then began those endless walks through the streets of Rome, during which algebraic geometry was the favourite theme of our speeches. … It is not an exaggeration to say that in those conversations the theory of algebraic surfaces was constructed according to the Italian line of research.17 (Castelnuovo 1947b, pp. 81–82)

Enriques’ research on surface theory developed into two directions: the first concerns the general theory of geometry on an algebraic surface and the determination of invariants for birational transformations; the second concerns the problems of classification of surfaces and in particular of the surfaces of the first genus. In the first line of research, which he started together with Castelnuovo, he obtained numerous, original and important results in more than 50 articles, which gave him international fame. The first period of collaboration with Castelnuovo allowed Enriques to formulate a systematic research project above all in the second direction

 The original text is: “mi accorsi subito che … Federigo Enriques era un mediocre lettore. Nella pagina che aveva sotto gli occhi egli non vedeva ciò che era scritto, ma quel che la sua mente vi proiettava. Adottai quindi un altro metodo: la conversazione. […] Cominciarono allora quelle interminabili passeggiate per le vie di Roma, durante le quali la geometria algebrica fu il tema preferito dei nostri discorsi. […] Non è esagerato affermare che in quelle conversazioni fu costruita la teoria delle superficie algebriche secondo l’indirizzo italiano”. 17

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of research mentioned above, i.e. the classification of surfaces in relation to the values of the genus, a theme that occupied him intensely throughout his life. Enriques’ method of work reveals the profoundly intuitive and almost empirical approach he adopted to study mathematical entities, which, in the words of Castelnuovo, “resembles that held in the experimental sciences” (Castelnuovo 1928, p. 194).18 Among the famous treatises that collect his geometric research, worthy of mention is the four-volume work written in collaboration with Oscar Chisini, Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche (Enriques and Chisini 1915–1934) which is characterised by the unmistakable methodological approach of Enriques. Alongside his important mathematical work, Enriques also wrote numerous significant papers on history and philosophy of science and on mathematics education and teacher training. Enriques earned many prizes and awards, including the mathematics prize of the Società Italiana delle Scienze detta dei XL (1896), the royal prize of the Accademia dei Lincei together with T. Levi-Civita (1907) and the Bordin prize of the Academie des Sciences of Paris for a work written in collaboration with Francesco Severi (1907). He was a member of numerous Italian and foreign academies and was awarded honorary degrees from various foreign universities. For his philosophical work, he was named, in 1937, a corresponding member of the Académie des sciences morales et politiques of the Institut de France. He died in Rome on 14 June 1946.

Contribution to Mathematics Education In addition to strictly scientific interests, Enriques and Castelnuovo also shared a profound attentiveness to education, although they were motivated by different reasons. While Castelnuovo’s engagement in educational issues sprang from social concerns, Enriques’ involvement was rooted in his interests in philosophical, historical and interdisciplinary issues and his studies on the foundations of geometry. In 1896, the teaching of projective geometry at the University of Bologna stimulated him to study the genesis of the postulates of geometry, taking as his starting point the psychological and physiological studies of H.  Helmholtz, E.  Hering, E. Mach and W. Wundt. From the correspondence, writings and documents conserved in the archives of the University of Bologna, it is possible to see how already in these early years Enriques had turned his attention to the needs of education, in particular the refusal to resort to artifices in the proofs, the importance of using intuition, the connections between elementary and higher mathematics, the use of  The original text is: “che assomiglia a quello tenuto nelle scienze sperimentali”. The entire quotation can be found in the biographical portrait of Castelnuovo by Livia Giacardi in the present volume. 18

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the history of mathematics as a tool for understanding the genesis of the concepts presented and a unified vision of science and culture (Giacardi 2012, pp. 216–222). In the evolution of Enriques’ cultural project and his vision of mathematics education, along with his experience in teaching, an important role was played by the influence of Klein (Giacardi 2012, pp. 223–229). During Klein’s second trip to Italy in 1899, the principal theme of conversation was the psychological genesis of postulates, and when Klein invited him to write a chapter on the foundations of geometry (Enriques 1907b) for the Encyklopädie der mathematischen Wissenschaften, Enriques reached him in Göttingen in 1903 to discuss this subject: In addition to talking about the foundations of geometry, we discussed didactic issues at length, and in just a few hours I learned a great deal from him about a lot of things I knew nothing about – specifically about the way in which mathematics teaching is developing in England and Germany.19 (F. Enriques to G. Castelnuovo, 24 October 1903, in Bottazzini, Conte and Gario 1996, p. 536)

It was thanks to Klein that a German translation of Enriques’ Lezioni di geometria proiettiva was published in 1903. In his introduction to this book, Klein expressed particular appreciation for Enriques’ intuitive but rigorous treatment of the subject, and underlined the impact of this kind of research on didactics (Klein 1903, p. III). Moreover, various Enriques’ initiatives addressed to teacher training were inspired by Klein’s work and activities as can be seen below.  nriques’ Epistemological Assumptions at the Basis of His Idea E of Mathematics Teaching From the first years of the twentieth century, Enriques had in mind a very precise cultural project, in which active research in the field of algebraic geometry and philosophical, psychological and historical reflections were all closely intertwined. Enriques’ aim was to communicate to his intended audience – mathematicians, scientists, philosophers and educators – his vision of a scientific humanitas in which the boundaries between disciplines were overcome and the abyss between science and philosophy was bridged. The most important epistemological assumptions which inspired his idea of mathematics education can be summarised as follows. Enriques held a dynamic and genetic view of the scientific process, describing it as a process of continuous development which establishes a generative relationship between theories and perceives in their succession only an approximation to truth (Enriques 1912a, p. 132). In such a vision of science, errors become valuable as well, because every error always contains a partial truth that must be kept, just as every truth contains a partial error to be corrected. As a consequence of this idea, Enriques criticised the tendency of teachers to present a mathematical theory in a

 The original text is: “… oltre che delle questioni sui principii abbiamo discorso molto di q­ uestioni didattiche e da lui solo in poche ore ho imparato tante cose interessanti, di cui non avevo mai avuto notizia, sullo sviluppo dell’istruzione matematica in Inghilterra e in Germania”. 19

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strictly deductive manner, because in this way it appears closed and perfect but leaves no room for further discovery. Instead, teachers should approach problems with a number of different methods, paying attention to the errors which have allowed science to move forward and indicating those questions which remain open as well as new fields of discovery (Enriques 1942). These views on science are connected to Enriques’ conception of the nature of mathematical research  – typical of the Italian school of algebraic geometry  – as something aiming above all at discovery and particularly emphasising the inductive aspects and intuition. This belief is naturally reflected in the style of teaching, which should, according to Enriques, take into account the inductive as well as the rational aspect of theories. Logic and intuition represent two inextricable aspects of the same process; teachers should therefore find the right balance between the two: the important thing is to distinguish clearly between empirical observation and intuition on the one hand and logic on the other hand. Teaching should above all emphasise “large-scale logic” (which considers the organic connections in science) over “small-scale logic” (refined, almost microscopic analysis of exact thought) and prepare young people gradually to develop a more refined and rigorous analysis of thought. Moreover, for Enriques science is a “conquest and activity of the spirit, … [which] merges in the unity of the spirit with the ideas, feelings and aspirations which find expression across the different aspects of culture” (Enriques 1938a, p. 130),20 so it is important to establish links between scientific disciplines (mathematics, physics, biology, etc.) and other intellectual activities (psychology, physiology, philosophy, history, etc.). Like Klein, he believed it was useful and necessary to maintain close ties between abstract science and applied sciences because pure sciences offer instruments that are needed for the purposes of applied science, and in their turn, applied sciences perform functions that are essential for stimulating the development of theoretical sciences, as history makes amply clear. As a consequence of this, according to Enriques, teachers should transmit a unified vision of knowledge to their pupils, because only by overcoming narrow specialisation can mathematics achieve its true humanistic and formative value. Another central aspect of Enriques’ epistemological vision is his belief that scientific developments can only be fully understood in their historical connections: thus, for Enriques history has a central educational role in both teacher training and in teaching practice: future teachers should study the origins of each theory, together with its developments, not some static formulation; young people too should be educated in the masterpieces of the masters (Enriques 1921b, p. 16). The history of science can also constitute an important auxiliary tool for education in making it possible to better understand certain concepts or properties.

 The original text is: “conquista e attività dello spirito [...] [che] si fonde nell’unità dello spirito colle idee, coi sentimenti, colle aspirazioni che si esprimono nei vari aspetti della cultura”. 20

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Institutional and Editorial Initiatives for Teacher Training Enriques used various strategies for carrying out his cultural project and spreading his vision of a scientific humanitas. The institutional and publishing initiatives, more strictly connected to mathematics education and teacher training, are really impressive.21 In 1900 Enriques published the Questioni riguardanti la geometria elementare, specifically designed for teacher training purposes and inspired by Klein’s Vorträge über Ausgewählte Fragen der Elementargeometrie (1895). The topics treated are congruence, equivalence, the parallel theory and non-Euclidean geometry, problems that can or cannot be solved with straightedge and compass and the constructability of regular polygons. In fact, like Klein, Enriques believed that the teaching of geometry could take advantage of the progress made by mathematics researchers and that secondary school teachers should possess a much broader knowledge of such progress so that their teaching is inspired by much larger perspective (Enriques 1900, pp. I–II). This work was enlarged in subsequent editions with the title Questioni riguardanti le matematiche elementari (Enriques 1924–1927) a collective work in which he involved his friends and colleagues and various secondary school teachers. The problem of teacher training was particularly important to Enriques and occupied him throughout his life. In 1906 on the occasion of the congress of the Federazione Nazionale Insegnanti Scuola Media held in Bologna, he explained his ideas regarding teacher training. He suggested the establishment of a “pedagogical degree” in addition to the “scientific degree”: the first 2 years of study would be dedicated to acquiring basic knowledge of the discipline, and by the end of that time, a distinction would be made between those who intended to dedicate themselves to research and those who wanted to teach. For the future teachers, the next 2 years would be aimed at providing professional training by means of: 1) courses on those parts of science that aim at a more profound understanding of the elements, 2) lectures on concrete questions of pedagogy that interest the various areas of teaching, particular in relation to the analysis of the textbooks, 3) exercises comprising practice teaching, partly in the university and partly in secondary schools, drawing, and experimental technique. (Enriques 1907a, p. 78)22

Closely connected with this proposal is the project to reform the Italian university presented by Enriques in 1906, during the First International Congress of Philosophy in Milan. His project had grown out of the ascertainment of the defects of the Italian university system which had, in his opinion, serious repercussions for research, teaching and the work world: the lack of interaction between the various faculties;  For further details see Giacardi (2012).  The original text is: “1) corsi su quelle parti della scienza che si riattaccano ad una più profonda visione degli elementi, 2) conferenze sulle questioni di pedagogia concreta che interessano i varii rami d’insegnamento, particolarmente in rapporto colla critica dei testi 3) esercitazioni comprendenti il tirocinio parte nell’università e parte in una scuola secondaria, il disegno e la tecnica sperimentale”. 21 22

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the excessive fragmentation of disciplines with programmes that were too heavy; and the tendency of each professor to defend his own discipline favouring the preeminence of already consolidated areas of research over those which were interdisciplinary or unexplored. The solution he proposed was that of conjoining in a single faculty of philosophy all of the theoretical disciplines – mathematics, physics, physiology, history, law, economy, etc. – so as to correspond to a unitary vision of knowledge, as opposed to the scientific-educational particularism. He also proposed the institution of “special schools of application” which were to group together professional teachings aimed at a specific career, the polytechnic schools for engineers and the polyclinical schools for physicians and the Scuole di Magistero for teacher training (Enriques 1908). In these years Enriques also started a long and successful series of textbooks (for geometry, algebra, trigonometry and infinitesimal calculus), in which he put his vision of mathematics teaching into practice, thanks also to the valuable collaboration of Ugo Amaldi. In 1903 the Elementi di Geometria, a textbook for secondary schools co-written with Amaldi, was published. It was characterised by a “rational inductive” method: beginning with a series of observations, the authors enunciate certain postulates from which the theorems that depend on them are developed by logical reasoning; from these theorems, they then continually return to observations or intuitive explanations. In this case as well, Enriques acknowledged Klein’s influence,23 and Klein mentioned this textbook in his essay on geometry teaching in Italy, “Der Unterricht in Italien”, praising the authors for having taken didactic requirements into consideration, thus reconciling logical rigour and intuition.24 Enriques’ relationships with Klein and his profound interest in mathematics education were certainly at the origin of his involvement in the activities of the International Commission on the Teaching of Mathematics from its founding. From 1908 to 1920, Enriques was an Italian delegate to the commission, together with Castelnuovo and Giovanni Vailati who, after his death in 1909, was replaced by Gaetano Scorza. In this capacity he participated in the conferences organised by the commission intervening in the discussions25 and preparing an important report on the changes in the teaching of mathematics in Italy since 1910. In this report, in particular, Enriques presents an account of the Giovanni Gentile reform of secondary schools (see below) which appears less critical than might be expected: he limits himself to pointing out the reduction in the number of the hours devoted to mathematics and the unsolved problem of teacher training. Instead, he gives ample space to the flourishing of new textbooks and presents his own many initiatives aimed at  See F. Enriques to F. Klein, 10 January 1905, in Giacardi 2012, p. 263.  Klein, Felix. 1925–1933. Elementarmathematik vom höheren Standpunkte aus, I Arithmetik, Algebra, Analysis, II Geometrie, III PräzisionsundApproximationsmathematik, II, pp.  245–250. Berlin: Springer (1st ed. 1908–1909). 25  He participated, for example, in the sessions devoted to the commission in ICM 1908 in Rome, in ICM 1912 in Cambridge (intervening on the mathematical training of the physicist at university, L’Enseignement Mathématique 14, 6, 1912: 503) and in the meeting in Milan in 1911 (intervening on rigour in secondary teaching, L’Enseignement Mathématique 13, 1911: 464–468). 23 24

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teacher training (Enriques 1929). For his various contributions to education, Enriques was appointed Honorary Member of the International Commission on the Teaching of Mathematics in the ICM held in Oslo in 1936. From 1919 to 1932, Enriques was president of the Mathesis Association, the national association of mathematics teachers. His desire to foster connections between mathematics and other sciences is evident from the new charter for the association, which, on 7 May 1922, welcomed teachers of physics into its ranks, and led the society to assume the new name: Società italiana di scienze fisiche e matematiche “Mathesis”. In 1921 he also took over direction of the Periodico di Matematiche which became the association’s publishing venue. During his direction of it (1921–1937, 1946),26 he gave it a strong imprint which is already evident from the title: Periodico di Matematiche. Storia—Didattica—Filosofia. In fact, according to Enriques’ project, the Periodico was intended to disseminate the idea of mathematics as an integral part of philosophical culture, as well as to fill the gap that existed in scientific education at that time in Italy. For this reason, he gave ample space to questions of methodology and philosophy, to elementary mathematics from an advanced standpoint, to physics and to history of mathematics and science, availing himself of the collaboration of mathematicians, physicists and historians of science (Ugo Cassina, Giulio Vivanti, Enrico Persico, Enrico Fermi, Ettore Bortolotti, Gino Loria, etc.). Enriques also encouraged an active collaboration on the part of teachers who wished to offer the contribution of their experience. In the first issue of the new series of the Periodico, Enriques published his famous article, “Insegnamento dinamico” (Enriques 1921b), which is practically a manifesto of his working programme and of his particular vision of mathematics education: active teaching, Socratic method, learning as discovery, the right balance between intuition and rigour, the importance of errors, the historic view of problems, the connections between mathematics and physics, elementary mathematics from an advanced standpoint and the educational value of mathematics. As mentioned above, in 1923 Enriques founded the National Institute for the History of Sciences and the following year the University School for the History of the Sciences connected to it with the threefold aim of giving an impulse to historical research, of achieving his ideal of a scientific humanitas and of training mathematics teachers. Analogous objectives inspired the book series Per la storia e la filosofia delle matematiche, launched in 1925 by Enriques, for which he also drew on the collaboration of secondary school teachers. The idea behind the series came, as he himself wrote, from teaching experience at the Scuola di Magistero (Enriques 1925, p. 7) and was addressed not only to teachers but also to secondary school students and in general to the educated public. Most of the volumes in the series are translations, with commentaries and historical notes, of those works by important authors of the past which might be of relevance to mathematics teaching, such as Euclid, Archimedes, Bombelli, Newton, Dedekind, etc. In fact, he believed that good  From 1921 to 1934, he was co-director with Giulio Lazzeri. From 1938 to 1943, Enriques’ name does not appear on the title page as a consequence of the racial laws passed in 1938. In 1946 he was co-director with Oscar Chisini. 26

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teachers should “study science not only in its static aspect, but also in its developing state” in order to “learn from history to reflect on the genesis of the ideas, and on the other hand, take an active interest in research” (Enriques 1938b, p. 190).27 Almost all of these last initiatives of Enriques took place after the Gentile reform. In 1923 the neo-idealist philosopher Giovanni Gentile, minister for Education in Italy’s fascist government, carried out a full and organic reform of the school system, according to which the secondary education was dominated by the classical-­ humanistic branch, which was designed for the ruling classes and considered absolutely superior to the technical-scientific branch. The principles of fascism and the neo-idealist ideology were opposed to the widespread diffusion of scientific culture and, above all, to its interaction with other sectors of culture. Humanistic disciplines were to form the main cultural axis of national life and, in particular, of education. This point of view was, of course, opposed to the scientific humanitas to which Enriques aspired. As president of the Mathesis Association, he engaged in intense negotiation with Gentile in the hope of avoiding the devaluation of science teaching. However, his pleas fell on deaf ears. Unlike Volterra and Castelnuovo, who were in absolute opposition to the Gentile reform, Enriques assumed and maintained a conciliatory position. In fact, he agreed with Gentile on several points: he conceived of knowledge as a personal conquest; he was in agreement with the need to fight encyclopaedism, and he considered education to be the free and unfettered development of inner energy. Furthermore, his ideal was to achieve a fusion between “scientific knowledge” and “humanistic idealism” in a “superior awareness of the universality of thought” (Enriques 1924, p. 4).28 Until his death in 1946, Enriques fought his battle for a scientific humanitas and was involved in teacher training, which he believed to be the crucial element for the formation of good schools and one of the channels for achieving his cultural project.

Sources Enriques, Federigo. 1956–1966. Memorie scelte di geometria. 3 Vols. Bologna: Zanichelli. Enriques, Federigo. Edizione Nazionale delle Opere: https://www.federigoenriques.org. This website includes also a rich secondary bibliography. (Retrieved June 2021). Enriques, Federigo and Oscar Chisini. 1915–1934. Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche, 4 Vols. Bologna: Zanichelli. Brigaglia, Aldo and Ciro Ciliberto. 1995. Italian Algebraic Geometry between the two World Wars. Queen’s Papers in Pure and Applied Mathematics, Vol. 100. Kingston, Ontario: Queen’s University.

 The original text is: “la scienza sia da loro appresa non soltanto nell’aspetto statico, ma anche nel suo divenire”, “apprenda dalla storia a riflettere sulla genesi delle idee, e d’altro lato partecipi all’interesse per la ricerca”. 28  The original text is: “superiore consapevolezza dell’attività universale del pensiero”. 27

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Bottazzini, Umberto, Alberto Conte and Paola Gario. 1996. Riposte armonie. Lettere di Federigo Enriques a Guido Castelnuovo. Torino: Boringhieri. Campedelli Luigi. 1973. Un cinquantenario: Federigo Enriques nell’insegnamento. In Atti del convegno internazionale sul tema: storia, pedagogia e filosofia della scienza, 75–90. Rome: Accademia Nazionale dei Lincei. Castelnuovo, Guido. 1928. La geometria algebrica e la scuola italiana. In Atti del Congresso Internazionale dei Matematici (Bologna 3–10 settembre 1928). Vol. I, 191–201. Bologna: Zanichelli. Castelnuovo, Guido. 1947a. Commemorazione del socio Federigo Enriques. Atti della Accademia Nazionale dei Lincei. Rendiconti Cl. Sci. Fis. Mat. Nat 8, 2: 3–21. Castelnuovo, Guido. 1947b. Commemorazione di Federigo Enriques. Periodico di Matematiche s. 4 25: 81–94. Ciliberto, Ciro and Paola Gario. 2012. Federigo Enriques: The First Years in Bologna. In Mathematicians in Bologna 1861–1960 ed. Salvatore Coen, 105–142. Basel: Birkhäuser. Giacardi, Livia. 2012. Federigo Enriques (1871–1946) and the training of mathematics teachers in Italy. In Mathematicians in Bologna 1861–1960 ed. Salvatore Coen, 209–275. Basel: Birkhäuser. Giacardi, Livia. 2019. The Italian Sub-commission of the International Commission on the Teaching of Mathematics (1908–1920). Organizational and Scientific contributions. In National Subcommissions of ICMI and their Role in the Reform of Mathematics Education, ed. Alexander Karp, 119–147. Springer Series: International Studies in the History of Mathematics and its Teaching. Israel, Giorgio. 1984. Le due vie della matematica italiana contemporanea. In La ristrutturazione delle scienze tra le due guerre mondiali. Vol. I: L’Europa, eds. Giovanni Battimelli, Michelangelo De Maria, and Arcangelo Rossi, 253–287. Roma: La Goliardica. Israel, G. 1992. F.  Enriques e il ruolo dell’intuizione nella geometria e nel suo insegnamento. In Enriques, Federigo and Ugo Amaldi. Elementi di geometria, pp. IX–XXI.  Pordenone: Studio Tesi. Israel Giorgio and Laura Nurzia. 1989. Fundamental trends and conflicts in Italian mathematics between the two world wars. Archives Internationales d’Histoire des Sciences, 39: 111–143. Klein, Felix. 1903 Zur Einführung. In Enriques, Federigo. Vorlesungen über projektive Geometrie (trans, by H. Fleisher), III. Leipzig: Teubner. Menghini, Marta. 1998. Klein, Enriques, Libois: variations on the concept of invariant. L’Educazione matematica 5, 3: 159–181. Nastasi, Pietro. 2004. Considerazioni tumultuarie su Federigo Enriques. In Intorno a Enriques. Cinque conferenze, ed. Luca Maria Scarantino, 79–204. La Spezia: Agorà. Nastasi, Tina. 2011. Federigo Enriques e la civetta di Atena. Centro Studi Enriques. Pisa: Edizioni Plus. Nurzia, Laura. 1979. Relazioni tra le concezioni geometriche di Federigo Enriques e la matematica intuizionista tedesca. Physis 21: 157–193. Pompeo Faracovi, Ornella ed. 1982. Federigo Enriques. Approssimazione e verità. Livorno: Belforte Editore, with the essays by Modesto Dedò and Tina Tomasi on Enriques and the educational issues. Pompeo Faracovi, Ornella. 2006. Enriques, Gentile e la matematica. In Da Casati a Gentile. Momenti di storia dell’insegnamento secondario della matematica in Italia, ed. Livia Giacardi, 305–321. Centro Studi Enriques. La Spezia: Agorà Edizioni.

Publications Related to Mathematics Education Enriques, Federigo. 1900. Questioni riguardanti la geometria elementare. Bologna: Zanichelli. German translation: Fragen der Elementargeometrie, 2 Vols. 1907–1910. Leipzig: Teubner. Enriques, Federigo. 1907a. Sulla preparazione degli insegnanti di Scienze. In Quinto Congresso nazionale degli insegnanti delle scuole medie. Bologna, 25-26-27-28 settembre 1906, Atti, 69–78. Pistoia: Tip. Sinibuldiana.

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Enriques, Federigo. 1907b. Prinzipien der Geometrie. In Encyklopädie der mathematischen Wissenschaften, Bd. 3-1-1, 1–129. Leipzig: Teubner. Enriques, Federigo. 1908. La riforma dell’Università italiana. Rivista di Scienza 3: 362–372. Enriques, Federigo. 1912a. Scienza e razionalismo. Bologna: Zanichelli. Enriques, Federigo. 1912b. Sull’insegnamento della geometria razionale. In Questioni riguardanti le matematiche elementari, 19–35. Bologna: Zanichelli. Enriques, Federigo. 1921a. Ai lettori. Periodico di Matematiche s. 4, 1: 1–5. Enriques, Federigo. 1921b. Insegnamento dinamico. Periodico di matematiche s. 4, 1: 6–16. Enriques, Federigo. 1924. Il significato umanistico della scienza nella cultura nazionale. Periodico di Matematiche s. 4 4: 1–6. Enriques, Federigo. 1924–1927. Questioni riguardanti le matematiche elementari. Raccolte e coordinate da Federigo Enriques. 3rd ed. Bologna: Zanichelli (2nd ed. 1912–1914; Anastatic rpt. Bologna: Zanichelli 1983). Enriques, Federigo. 1925. Gli Elementi d’Euclide e la critica antica e moderna (Libri I–IV). Rome: Alberto Stock. Enriques, Federigo. 1928. La riforma Gentile e l’insegnamento della Matematica e della Fisica nella Scuola media. Periodico di Matematiche s. 4, 8: 68–73. Enriques, Federigo. 1929. Italia. Les modifications essentielles de l’enseignement mathématique dans les principaux pays depuis 1910. L’Enseignement Mathématique 28: 13–18. Enriques, Federigo. 1938a. L’importanza della storia del pensiero scientifico nella cultura nazionale. Scientia 63: 125–134. Enriques, Federigo. 1938b. Le matematiche nella storia e nella cultura. Bologna: Zanichelli. Enriques, Federigo. [Adriano Giovannini, pseudonym]. 1942. L’errore nelle matematiche. Periodico di matematiche s. 4, 22: 57–65. Enriques, Federigo. 2003. Insegnamento dinamico. Centro Studi Enriques. La Spezia: Agorà, with essays by F. Ghione and M. Moretti. Enriques, Federigo and Ugo Amaldi. 1903. Elementi di geometria ad uso delle scuole secondarie superiori. Bologna: Zanichelli. Enriques, Federigo and Ugo Amaldi. 1914–1915. Nozioni di matematica ad uso dei licei moderni, 2 Vols. Bologna: Zanichelli. Enriques, Federigo, Francesco Severi and Alberto Conti. 1903. Estensione e limiti dell’insegnamento della matematica, in ciascuno dei due gradi, inferiore e superiore, delle Scuole Medie. Il Bollettino di Matematica, a. 2, n. 3–4: 50–56.

Photo Courtesy of the University of Turin.

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12.6 Charles-Ange Laisant (Basse-Indre, 1841– Asnières-sur-­Seine, 1920): Founder with Henri Fehr of the Journal L’Enseignement Mathématique Eduardo L. Ortiz

Biography Charles-Ange Laisant is remembered for his deep commitment to the advancement of mathematics teaching, for his role in the creation of valuable mathematical journals and for his contributions to the development of a truly international network of mathematicians. He was born on 1 November 1841 in the small village of Basse-Indre and studied at the École Polytechnique, Paris, and later at the École d’Application, Metz. In 1871 he received the Croix de la Legion d’honneur for his role in the defence of the Fort of Issy, at the time of the siege of Paris, in the final stages of the Franco-­Prussian War. At the end of that war, Laisant, and a group of his immediate friends, felt that an effort should be made to restore their country to the position it occupied in the

E. L. Ortiz (*) Imperial College, London, UK e-mail: [email protected]

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intellectual life of European nations at the beginnings of the century, that is, breaking with what they perceived as cultural isolation. These reformers shared the view that some basic changes in the structure of French society were necessary, particularly in its scientific community, which they perceived as antiquated and short in communication facilities, in particular between the capital city and the departments. In their view the situation was different in Germany, and some even suggested that these differences may have been an element in the Prussian victory. Such views had a substantial impact in the personal and scientific life of Laisant and on that of some of his friends. Later they made considerable efforts to renovate their country’s scientific community trying to set it on a more internationalist path. If some of these young men played a role in the 1870 war, on occasions of some significance as it was the case of Laisant, their actions after that war had a far larger and permanent impact on the scientific life of their country. It is not then surprising that the period immediately after the Franco-Prussian War was one of substantial institutional development in France. One of the directions in which a serious effort was then made was in attempting to bring about an avancement of science through a more organised and planned development of the community of scientists. In rapid succession, which contrasts with expansion in earlier periods, new societies for the promotion of science, either in general or in specific branches, began to appear. Such was the case of the new Association Française pour l’Avancement des Sciences, founded in 1872; it supported the development of science at a national level sponsoring, also, closer scientific interactions between Paris and the rest of the country. Other institutions created in that period tried to contribute to the development of specific branches of science at an advanced research level: among them the Société Mathématique de France, founded in 1872, and the Société de Physique, founded in 1873. In addition, these organisations attempted to increase the circulation of scientific information, and, for that, they quickly widened and deepened contacts with similar societies inside the country and abroad. The circle of scientists to which Laisant belonged promoted also a wider exchange of ideas and a better understanding between scientists across nations. In a later stage, in the last decades of the century, they embarked on new projects to promote wider channels of international scientific communication. They also began to consider, more dynamically than ever before, the need to focus on the improvement of the teaching of science and mathematics. Laisant’s life, as a mathematician and as a pioneer of a coherent international mathematical community, as well as his efforts in favour of the modernisation of mathematics education is paradigmatic of that important period in France’s intellectual history. Laisant Mathematician Laisant had a genuine concern with mathematical research and published contributions in several contemporaneous mathematics journals; his works covered topics of analysis, mechanics and geometry, the latter being his main and lasting interest. He

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could not be described, however, as a leading member of the brilliant French mathematics research elite of his time. Through the last third of the nineteenth century, Laisant and his friends, particularly Émile Lemoine (1840–1912), contributed significantly to the development of a branch of classical geometry concerned with a detailed study of special points of the triangle: the so-called Geometry of the Triangle. The development of this chapter of geometry is directly related to new ideas in the field of mathematics education that advocated for the eradication of a teaching of Euclidean geometry based, essentially, on memory. Supporters of the new views rightly claimed traditional teaching did not encourage creativity in the minds of students: the recitation of Euclid texts, almost by heart, was perceived as supporting the belief that nothing could be added to its already perfect formulation. By contrast, the profusion of geometrical properties brought about by studies on the Geometry of the Triangle allowed teachers to propose their students problems leading to the discovery of new geometrical results; clearly, the latter were not necessarily profound new results. However, reformist teachers believed that the experience would open the student’s mind to the idea that geometry, and more generally mathematics, is not a dead science, where further advance had become impossible. In the early 1870s, Capitaine de Génie Laisant’s main research interests were in the theory of curves and surfaces, where the possibility of using the methodology of vector calculus to simplify proofs was beginning to be considered. Through the work of Guillaume-Jules Hoüel (1823–1886), Laisant was attracted to the new ideas on équipollences introduced by Giusto Bellavitis (1803–1880) in Italy from the 1850s. Laisant perceived the power of that new tool, which he believed to be capable of giving plane geometry a calculus with operational rules close to those of ordinary algebra. In the early 1870s, he began to correspond with Bellavitis and later translated his main book into French, Exposition de la méthode des Équipollences, adding extensive historical and technical notes. The book was published by the new Parisian firm of Gauthier-Villars in 1874 and had a considerable impact in France and abroad. However, essential difficulties were soon encountered, when trying to extend equipollences to three-dimensional space. These complications made Laisant, as well as Hoüel, and Bellavitis himself, turn their attention to William Hamilton’s (1805–1865) new work on quaternions. Although Laisant was clearly not the first to consider quaternions in France, he was a pioneer in the process of the transmission of this new theory to his country. In 1877 he submitted a thesis on quaternions and their possible applications to mechanics to the Faculté des Sciences, Paris; he was awarded a Doctorat es sciences. In 1881 Laisant published an elementary treatise on Hamilton’s quaternions, far more readable than Hoüel’s earlier introduction of 1874. His work was based on the modern ideas of Peter Guthrie Tait (1831–1901). Most of Laisant’s mathematical work was done while he was either Examinateur d’admission or Repétiteur at the École Polytechnique, Paris. In November 1888 the French government created an official committee for the organisation of an international congress on mathematical bibliography; the congress met in Paris 1 year later. Following this initiative, an international permanent committee was later established with representatives from Europe and the USA;

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Henri Poincaré (1854–1912) was designated its president and Laisant its secretary. The commission compiled an index, called Répertoire Bibliographique des Sciences Mathématiques, for which Laisant designed a cross-reference system (système de renvois). Through his activity in the Répertoire, Laisant widened considerably the international circle of his acquaintances; in part through these connections, he was incorporated as a Foreign Member to several European Academies of Science. In 1903–1904 he was elected president of the Assoçiation Française pour l’Avancement des Sciences and vice-president of the Société Astronomique de France; he was also honoured by leading scientific societies across Europe. In 1904, in the Bulletin of the Société Astronomique, Laisant wrote an interesting piece on the social role of science, in which he anticipated some arguments reconsidered in the 1930s controversies on science and society. Involvement in French Politics In the same years Laisant was battling for the acceptance of equipollences and quaternions in France, he was fully committed to national politics. His political career started in Nantes, at a local level and around 1871; 3 years later he was reelected as a candidate of a left-republican party (Union Républicaine) founded by Léon-­ Michel Gambetta (1838–1882), future Premier of France. This is, precisely, the time when Laisant translated Bellavitis’ book. Two years on he broke with his old party, beating the official candidate in the national elections of February–March 1876. On that year he resigned his commission in the army and in 1877 joined the national chamber as a Député. This was also the year when, in parallel with his intense political activity, Laisant submitted his doctoral dissertation to the Faculté des Sciences, Paris. In 1879 Laisant became the director of the political journal Le Petit Parisien, and in the following year, he was sentenced for publishing articles attacking a leading official figure in the French army. In December 1880 he finished writing the textbook on quaternions mentioned before, published in 1881. One year later he became a member of Parliament, a Député for the Département de Loire-Inférieure, and from then on his political career continued in a rapid progression, but not without conflict. In 1883 an article of his, published in another fringe political journal, La République Radical, caused new serious conflicts; but in 1885 Laisant was reelected as a member of Parliament, now for the Seine constituency of Paris. Soon after, a new political figure, War Minister General Georges Boulanger (1837–1891), emerged in the French political scene. Laisant had met him before 1887, when he was active as a parliamentarian, in commissions working on a project on conscription (a topic on which Laisant battled for fairness and justice) and on the reorganisation of the army; these were, both, areas in which Laisant had a considerable professional experience. War Minister Boulanger proposed other reforms, some highly controversial, which finally forced him to resign his position. However, he began to emerge as a national political figure, promising to do real change. A

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political alliance of groups and personalities, holding very diverse views and objectives, was formed around Boulanger: Laisant joined that alliance. Contradictions inherent to such complex and heterogenous alliance would soon become apparent: the so-called affair Boulanger ended in 1889 after an electoral disaster. Deeply discredited and troubled by personal problems, Boulanger took his own life closing a unique period in French political life, which continues to attract the attention of historians and political scientists to this day. Although this political fiasco affected Laisant’s political credibility, he managed to win, the same year, a parliamentary seat in the 18th arrondissement of Paris, defeating the traditional candidate. Some years later, in 1893, after over 20 years in French national politics, he decided to quit. In his political life, Laisant acquired valuable organisational skills as well as multiple personal contacts, which he later used to support the advance of the national and international mathematical communities. Laisant was also a leader among those who wished to give the international mathematical community a more structured organisation and to promote closer personal interactions through regular international gatherings. This initiative led to the institution of regular meetings of mathematicians, which continue to this day through the efforts of the International Mathematical Union. After its first official Congress in 1900, in Paris, and following a suggestion made by Ferdinand Rudio (1856–1929) at the Zürich meeting of 1897, Laisant began compiling a world list of mathematicians. For this endeavour he counted with the assistance of the mathematician Adolphe Buhl (1878–1949); the list appeared in 1901–1902 under the title of Annuaire des Mathématiciens. This valuable early twentieth-century worldwide directory, which includes data on some 6500 mathematicians, was supposed to be reviewed annually, but, sadly, this could not happen.

Contribution to Mathematics Education At a meeting of the Association Française pour l’Avancement des Sciences, in Caen, in August 1894, Laisant announced the foundation of a journal open to the international community and designed to share information among readers. Any subscriber could submit a question to the journal’s editors in the hope that some other reader may know the answer. Appropriately, the name of this journal was L’Intermédiaire des Mathématiciens: it lasted for over a quarter of a century. L’Intermédiaire was modelled on a successful British journal for the generally educated public founded in the 1840s, which later inspired journal’s editors in different countries of Europe and America. In 1899, 5 years after L’Intermédiaire first appeared, Laisant, together with Henri Fehr (1870–1954) from Geneva as co-editor, launched yet another mathematical journal: L’Enseignement Mathématique. It was designed to promote contacts between people engaged in mathematics teaching and also to serve as a forum to the discussion of new pedagogical ideas. After the Commission Internationale de

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l’Enseignement Mathématique was established, at the International Mathematics Congress of 1908, in Rome, L’Enseignement Mathématique became its official journal. In those years Laisant became also editor of the prestigious Nouvelles Annales de Mathématiques, founded in 1842. Laisant was also the author of several general books on mathematics and mathematics education, among them La Mathématique, Philosophie, Enseignement and L’Initiation Mathématique, published in Paris in 1898 and 1910, respectively. They were reprinted several times and had some impact on mathematics education worldwide. Through these books Laisant attempted to give the ideas of philosopher Auguste Comte (1798–1857) a mathematical context. However, his approach was not uncritical: Laisant remarked that the definition of mathematics as the science dealing with the indirect measurement of magnitudes, given by the master in his works, was insufficient and stressed the fact that mathematics deals also with the important notion of order; Laisant saw it as inherent to mathematics as it is measurement. At the international level, Laisant was national delegate to ICMI for France. In the ICM of Cambridge (1912), he gave a relevant contribution to the debate which followed David Eugene Smith’s talk. On a personal level, his contemporaries describe him as a warm, generous and independent man; many pages in his works, as well as actions in life, suggest this description may have been accurate. He was also a truly European character and a committed pacifist, who championed for closer international contacts, particularly between France and Germany. Laisant died in Paris, in 1920, after seeing the horror of the First World War and the division it created in an international mathematical community he had contributed so much to create and develop. However, even after such tragic events, many of his ideas had lived on and thrived. In his obituary Buhl referred to “the logical, idealistic and revolutionary spirit of Laisant”29 (p. 75). Laisant, as well as his friend Lemoine, had a serious interest in chamber music. With other polytechniciens he founded a music circle known as La Trompette, which met regularly in Paris; its members included Camille Saint-Saëns (1835–1921) and Henri Poincaré.

Sources Laisant, Charles-Ange. 1880. Giusto Bellavitis, Bulletin des Sciences Mathématiques s. 2, 4: 343–348. Laisant, Charles-Ange. 1881. Introduction à la méthode des quaternions. Paris: Gauthier-Villars. Laisant, Charles-Ange. 1887. Théorie et applications des équipollences. Paris: Gauthier-Villars. Laisant, Charles-Ange. 1887. Assemblé Nationale, Chambre des députés. Commission chargée d’examiner le projet de loi organique militaire, Rapport par le député M. Ch.-A. Laisant. Paris: Imprimerie de la Chambre des Députés. Altmann, Simon and Eduardo L. Ortiz (eds.). 2005, Mathematics and social utopias in France: Olinde Rodrigues. Providence: American Mathematical Society/ London Mathematical Society.

29

 The original text is “l’esprit logique, idéaliste et révolutionnaire de Laisant”.

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Auvinet, Jérôme. 2013. Charles-Ange Laisant. Itinéraires et engagements d’un mathématicien de la Troisième République. Paris: Hermann. Bellavitis, Giusto. 1874. Exposition de la méthode des Équipollences, traduit de l’Italien par M.  Laisant. Gauthier-Villars: Paris. (This work was first serialized in Nouvelles Annales de Mathématiques). Buhl, Adolphe. 1920. Charles-Ange Laisant. L’Enseignement Mathématique 21: 73–80. Freguglia, Paolo. 1991. I fondamenti dell’algebra degli immaginari secondo Giusto Bellavitis. In Giornate di storia della matematica, ed. Massimo Galuzzi, 143–168. Rende: Editel. Furinghetti, Fulvia. 2003. Mathematical instruction in an international perspective: the contribution of the journal L’Enseignement mathématique. In One hundred years of L’Enseignement Mathématique, eds. Daniel Coray, Fulvia Furinghetti, Hélène Gispert, Bernard R. Hodgson, Gert Schubring, 19–46. Monographie n. 39, L’Enseignement Mathématique. Gispert, Hélène (ed.). 1991. La France Mathématique: La Société mathématique de France (1872–1914). Cahiers d’Histoire et de Philosophie des Sciences Nouvelle Série, 34. Gispert, Hélène. 1991. 2002. Par la science, pour la patrie: L’Association Française pour l’Avancement des Sciences. Rennes: Les Presses Universitaires de Rennes. Ortiz, Eduardo L. 1996. The nineteenth-century international mathematical community and its connection with those on the Iberian periphery. In L’Europe mathématique, eds. Jeremy Gray, Catherine Goldstein and Jim Ritter. Paris: Edition de la Maison des Sciences, 321–344. Ortiz, Eduardo L. 1999. Quaternions abroad: some remarks on their impact in France. Acta Historiae Naturalium Necnon Technicarum 3, 295–302. Ortiz, Eduardo L. 2001. Proyectos de cambio científico y proyectos de cambio político en la tercera República: el caso de la teoría de los cuaterniones. Revista Brasileira de História da Matemática 1: 66–85. Also in Antonio Duran Guardeño and José Ferreiros Domínguez (eds.). 2015. El valor de las matemáticas, 141–164. Sevilla: Editorial Universida de Sevilla. Ortiz, Eduardo L. 2007. Babbage and French idéologie: functional equations, language and the analytical method. Chapter 2. In Episodes in the History of Modern Algebra (1800–1950), eds. Jeremy Gray and Karen Parshall, 22–57. Providence and London: American Mathematical Society/ London Mathematical Society. Zerner, Martin. 1991. Le règne de Joseph Bertrand (1874–1900). Cahiers d’Histoire et de Philosophie des Sciences. Nouvelle Série 34: 298–322.

Publications Related to Mathematics Education Laisant, Charles-Ange. 1893. Recueil de problèmes mathématiques classés par divisions scientifiques. Second edition in 1896. Paris: Gauthier-Villars. Laisant, Charles-Ange. 1898. La Mathématique. Philosophie. Enseignement. Paris: G.  Carré et C. Naud. Laisant, Charles-Ange. 1906. L’éducation de demain. Second edition in 1913. Paris: Les temps nouveaux. Laisant, Charles-Ange. 1906. L’Initiation mathématique. Paris: Hachette. (with many editions and translations). With Henri Delannoy and Émile Lemoine, Laisant edited: Lucas, Édouard. 1891–1896. Récréations mathématiques. Paris: Gauthier-­Villars et fils. Lucas, Édouard. 1895. L’arithmétique amusante. Paris: Gauthier-Villars et fils.

Photo Source: L’Enseignement Mathématique. 1920–1921. 21. (Photo at n.n.p.).

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12.7 Gino Loria (Mantua, 1862–Genoa, 1954): Appointed Honorary Member in 1936 Fulvia Furinghetti

Biography Gino Benedetto Loria was born in Mantua on 19 May 1862 to a well-off Jewish family. His father was Salomon Vita, also called Gerolamo or Girolamo, and his mother Anaide D’Italia. His brother Achille, a notable political economist and professor in various universities, was elected to the Italian Senate. In the years 1875–1879, he completed the secondary studies at the Istituto Tecnico Provinciale of his home town. From 1879 to 1883, he enrolled in the University of Turin, where he received the laurea in mathematics with a dissertation entitled “Intorno alla geometria su un complesso tetraedrale”, which was later elaborated into research on spherical geometry. His supervisor was Enrico D’Ovidio (Togliatti 1954, p. 115). In the years 1884–1886, while he was D’Ovidio’s assistant, he published 16 papers about applications of algebraic concepts to geometrical F. Furinghetti (*) University of Genoa, Genoa, Italy e-mail: [email protected]

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studies, the geometry of straight line and spheres, hyperspatial projective geometry, entities generated by algebraic correspondences between fundamental forms and Cremona spatial transformations (Terracini 1954, p. 403). With Corrado Segre, a major contributor to the early development of algebraic geometry in Italy, he wrote the paper “Sur les différentes espèces de complexes du 20 degré des droites qui coupent harmoniquement deux surfaces du second ordre”, published in the Mathematische Annalen (1884, 23: 2I3–234). At the end of 1883, he went to the University of Pavia for attending an advanced course. There he met Eugenio Beltrami who, some years later, encouraged his vocation as a historian.30 In 1886 Loria won the chair of higher geometry and 1891 became a full professor at the University of Genoa, where he lived the rest of his academic life. He also taught, on assignment, history of mathematics, analysis and descriptive geometry. In 1903 he married Ida Levi Gattinara and settled permanently in Genoa. He retired in 1935 but continued to be active in research. In Genoa, Loria contributed to establishing the Institute of Mathematics (now Department), which was initially born as a branch of the Naval School in the Engineering Faculty. In 1906 he launched the Scuola di Magistero (university courses to prepare elementary, middle and high school teachers), where he was the director and delivered courses and talks. In 1887 he created the Mathematical Library now in the Mathematics Department; see Togliatti (1973). He was dean of the Faculty of Sciences. WWII was a difficult time for the Jew Loria due to the racial laws promulgated in 1938 by the Italian government; see Varnier (2002). To escape the racial persecutions, in 1943 he left Genoa and was hosted in the mountains south of Turin by the community of Waldenses. He came back to Genoa in 1945 and continued to study and write papers. He died in Genoa on January 30, 1954. He bequeathed his private library to the Institute of Mathematics of the University of Genoa under the condition that a history of mathematics course should always be held; see Fenaroli, Furinghetti, Garibaldi and Somaglia (1989, p. 222). It contains a unique collection of books, mathematical journals, journals dedicated to mathematics teaching and 14,463 offprints sent to him by their authors from all over the world. Loria received many honours and awards, such as Prix Binoux of the Institut de France (twice) and the Silver Medal of the Association Française pour l’Avancement des Sciences. He was a member of the Accademia Virgiliana in his native Mantua, the Accademia delle Scienze di Torino, the Accademia Nazionale dei Lincei and other national and international academies. The book (Loria, 1937a) containing Loria’s selected papers mentions 278 titles. Raymond Clare Archibald (1939, pp. 17–30) lists 360 publications, 17 entries in the Enciclopedia Italiana, 4 edited works, 4 prefaces and several contributions to the journal L’Intermédiaire des Mathématiciens. The database edited by Aldo Brigaglia, Ciro Ciliberto e Edoardo Sernesi, lists 998 titles. This copious production mainly concerns three domains of research: geometry, history of mathematics

 (Loria 1914a, p. V) and Loria, Gino. 1926. Durante quarant’anni d’insegnamento: confessioni e ricordi. Il Bollettino di Matematica s. 2, 5: 65–77. 30

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and mathematics teaching. Loria’s research in geometry developed manly in the wake of the previous generation, instead of following the new trend of the geometrical studies carried out in Italy in the field of algebraic geometry (Archibald 1939, pp.  11–17; Terracini 1954, pp.  409–421; Pepe 2001). Meanwhile, Loria became more and more engaged in historical research. In the field of history of mathematics, he reached an international reputation so that Archibald (1939 p. 5) claimed he was “the dean of mathematical historians in Italy”. The first relevant work in history was Il passato ed il presente delle principali teorie geometriche published in 1887 (Regia Accademia delle Scienze di Torino, Memorie s. 2, 35: 327–376), which in 1896 became a book (Loria 1896). This work had many translations. In 1893 he published his famous treatise Le scienze esatte nell’antica Grecia (Regia Accademia delle Scienze di Modena, Memorie, Sezione di Scienze s. 2, 10: 3–I68 and 11: 3–237); this treatise had further editions, also as a book (Loria 1914a). In the guide book to the study of history of mathematics Guida allo studio della storia delle matematiche (1915. Milan: Hoepli), Loria recommended that mathematicians researching history use the rules followed in their research in mathematics (p. 7). In the article on cataloguing books entitled “Sui metodi di compilazione dei cataloghi bibliografici. Pensieri e desideri” (Bollettino di Bibliografia e Storia delle Scienze Matematiche. 1900. 3, July, August, September: 65–70), we find the following important advice: people who are in charge of the conservation of manuscripts shouldn’t forget that a library is a laboratory, not a museum (p. 70). In 1897 Loria published a 24-page supplement to the mathematics journal Giornale di Matematiche dealing with history of mathematics. In the following year, he succeeded in having an autonomous journal for historical studies entitled Bollettino di Bibliografia e Storia delle Scienze Matematiche, known as Bollettino di Loria. This journal was, with the Bibliotheca Mathematica of Gustav Eneström (founded in 1884), an international landmark for publications on the history of mathematics. The Bollettino was issued until 1919; in 1922 it appeared as a section of the journal for mathematics teachers Il Bollettino di Matematica, see (Furinghetti 2006). One of Loria’s aims was to contribute to creating the figure of the professional historian of mathematics. Giacardi (2013) illustrates Loria’s contribution to the development of historical studies in Italy. As told before, Loria had contacts with scholars all over the world. His books on history were reviewed in foreign journals. He wrote articles on the history of mathematics in various countries, see (Loria 1919a; 1919b; 1927). He was an active participant in the International Congresses of Mathematicians (ICMs) as a speaker, as chairman of sessions and as a member of the International Program Committee (in Rome). In the ICM of Zurich (1897), Heidelberg (1904), Rome (1908), Bologna (1928) and Zurich (1932), he presented contributions on history and in 1904 on mathematics teaching, see (Loria 1905a). In the ICM of 1932, he presented the report of the ICMI inquiry on teachers, see (Loria 1932b). Evidence of the large appreciation of Loria’s action at the national and international level is the book (Loria 1936) for the celebration of his retirement, which contains a selection of his articles. This book was sponsored by Loria’s Italian colleagues, teachers (mainly of Genoa and surroundings) and international colleagues.

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Among the latter, there are Archibald, Fernando de Vasconcellos, Henri Fehr, Lucien Godeaux, Jacques Hadamard, George Sarton and David E. Smith.

Contribution to Mathematics Education In the field of mathematics education, Loria was active at the international and national levels. He took part in the debate on the geometrical syllabus in Great Britain, which developed in the last decades of the nineteenth century (see Loria 1889; 1893a) and published articles in The Mathematical Gazette and other foreign journals. From 1904 he was a member of the editorial board of L’Enseignement Mathématique and was among its main contributors. He was one of the reactors to the article of Smith published in this journal in 1905, which advocated more international cooperation and the creation of a commission aimed at studying instructional problems in different countries. Giacardi (2019) reports that Loria contributed to the preparation of the didactic section in the ICM of Rome through contacts with David E. Smith. Loria’s main contribution to the activity of the Commission was the inquiry into the theoretical and practical education of mathematics teachers in different countries launched by the International Commission on the Teaching of Mathematics in 1914. The results were to be presented at the next ICM in 1916  in Stockholm. Because of WWI and the subsequent dissolution of the Commission, the work was suspended for over 14 years, and the report was presented at the ICM of Zurich in 1932; see (Loria 1932). Its full text was published in L’Enseignement Mathématique (Loria 1933). The same issue also contains the reports by the national delegations, some of which had already appeared in journals of their respective countries. In 1936 on the occasion of the ICM in Oslo, the International Commission on the Teaching of Mathematics gave the title of “honorary member” to Loria (together with eight other scholars) for his contribution to the activities of the Commission.31 One of the main interests of Loria was the education of mathematics teachers, which was a theme debated in Italy in his times. At the second national congress of Mathesis (Italian Society of Mathematics Teachers), he discussed the situation in Italy on this theme, see (Loria and Padoa 1909). He supported the institution of the Scuola di Magistero and announced, with regret, its suppression starting from the academic year 1920–1921 (Loria 1921b, p. 149). Some years later, he expressed the rightful worry of high school teachers that their abilities were not recognised (Loria 1924, p. 23). Loria was an active member of Mathesis; for a long time, he was the president of the Ligurian section of this association and organised regular local meetings with teachers. Most of these teachers were females, and in the meeting minutes, there is an echo of the discussions about women’s difficulties in having

 See Comptes rendus du Congrès International des Mathématiciens, Oslo 1936, Oslo: A. W. Brøggers Boktrykkeri. 1937. Vol. 2, p. 289. 31

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access to teaching in secondary school. Then Loria had the opportunity to reflect on a theme (the relationship of women with mathematics), which he had treated from a historical point of view at the beginning of the century in the talk “Donne matematiche” (Women mathematicians) delivered at the Academia Virgiliana of Mantua. The text of this talk was published in Atti e Memorie Della R. Accademia Virgiliana di Mantova (biennium 1901–1902. 1903. 21: 75–98). This text was translated into French (Les femmes mathématiciennes. 1903. Revue Scientifique, column “Histoire des sciences”, 4, 20: 385–392). Later on, Loria published his answer to the comment of Józefa Joteyko that appeared in the same journal (Encore les femmes mathématiciennes. 1904. Revue Scientifique, column “Variétés”, s. 5, 1: 338–340). Loria attentively followed the various events of the school world. He reviewed various textbooks for secondary schools in didactic journals and participated in the conferences on mathematics teaching. In particular, he took part in the debate on the renewing of mathematics programmes; see Loria (1906a; 1909). Thanks to his international connections, he was well informed about the programmes developed abroad; in particular, he was interested in the reform of secondary school in Germany (Loria 1906b). Of course, he devoted much thought to the teaching of geometry, which was one of the fields of his mathematical research. The teaching of geometry is the subject of the seminal paper (Loria 1893b), where he combined his knowledge of geometry with his knowledge of the history of its teaching. Though he was convinced of the importance of rigor, he was aware of the students’ low interest in geometry (ibidem, p. 98–99), and then he was favourable to any legitimate means to enlivening and keeping awake this interest. His project for teaching geometry was to present this subject not as a dead language but as a living language (ibidem, p.  110). In the paper (Loria 1900), he considered with interest the fusionism, a method of teaching geometry by blending plane and spatial geometry. Loria’s commitment to teaching mathematics was recognised internationally. About the situation of the teaching of geometry in Italy, Cajori (1910) wrote: This recent Italian emphasis upon extreme rigor has led to deplorable results with the less gifted pupils, and a reaction appears to be setting in. Under the leadership of Loria and Vailati, a movement is on foot favoring greater emphasis upon intuition, the introduction of some modern geometrical notions, the fusion of geometry with arithmetic, and the concession to the demands for practical applications made by this age of industrial development. Italy is entering upon a reform much like that of Germany and France. (p. 192)

A similar appreciation was expressed by Klein (Klein, Felix. 2016. Elementary Mathematics from a Higher Standpoint: Geometry, Vol. 2, Gert Schubring Trans. Berlin: Springer, pp. 249–250). Geometry was one of the poles for Loria’s action in approaching the problem of mathematics teaching. The other pole was the history of mathematics, which was his other main field of research. In the paper (Loria 1899), he outlines a programme addressed to mathematics students who will become school teachers. This programme focuses on topics they will cover in their teaching. In this way, he aims at overcoming the sense of daunting isolation felt by young persons just graduated in mathematics as they start teaching in secondary school. To this purpose, he suggested a 2-year course on the history of mathematics to be introduced in the Faculty

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of Sciences (ibidem, p. 20). In his project, the teaching of the history of mathematics should become as a coupling link between secondary and university education (ibidem, p. 22).

Sources Loria, Gino. 1936. Scritti, conferenze, discorsi sulla storia delle matematiche, raccolti per iniziativa e pubblicati sotto gli auspici della Sezione Ligure della Società ‘Mathesis’. Padova: Cedam. Data-base Brigaglia, Aldo, Ciro Ciliberto, and Edoardo Sernesi (Bibliografia Loria). Retrieved on 17 August 2020. http://www.mat.uniroma3.it/users/sernesi/BIBLIOGRAFIA/loria.htm Loria, Gino. 1896. Il passato ed il presente delle principali teorie geometriche. Torino: Clausen. Loria, Gino. 1899. L’Enseignement Mathématique, revue international. – Directuers: C. A. Laisant et H. Fehr. Periodico di Matematica 14: 224–225. Loria, Gino. 1914a. Le scienze esatte nell’antica Grecia. Milano: Hoepli. Loria, Gino. 1919a. Le matematiche in Ispagna ieri ed oggi. Scientia 25: 353–359; 441–449. Loria, Gino. 1919b. Le matematiche in Portogallo: ciò che furono, ciò che sono. Scientia 26: 1–9. Loria, Gino. 1921a. Storia della geometria descrittiva dalle origini ai giorni nostri. Milano: Hoepli. Cajori, Florian. 1910. Attempts made during the eighteenth and nineteenth centuries to reform the teaching of geometry. The American Mathematical Monthly 17, 181–201. Fehr, Henri. 1936. Commission Internationale de l’Enseignement Mathématique. Compte rendu de la réunion d’Oslo 15 juillet 1936. L’ Enseignement Mathématique 35: 386–388. Fenaroli, Giuseppina, Fulvia Furinghetti, Antonio C.  Garibaldi, and Anna M.  Somaglia. 1989. Collezioni speciali esistenti nella biblioteca matematica dell’Università di Genova. In Atti del convegno “Pietro Riccardi (1828–1898) e la storiografia delle matematiche in Italia” (Modena, 1987), eds. F. Barbieri and Franca Cattelani Degani, 219–230. Modena: Dipartimento di Matematica dell’Università. Furinghetti, Fulvia. 2006. Due giornali ponte tra ricerca e scuola: la Rivista di Peano e il Bollettino di Loria. In Da Casati a Gentile. Momenti di storia dell’insegnamento secondario della matematica in Italia, ed. Livia Giacardi, 181–237. Livorno: Agorà. Furinghetti, Fulvia and Annamaria Somaglia. 2005. Emergenza della didattica della matematica nei primi giornali matematici italiani. In História do ensino da Matemática em Portugal, eds. D. Moreira and J. M. Matos, 59–78. Beja: Sociedade Portuguesa de Ciências da Educaçao. Giacardi, Livia. 2006, Loria Gino. Dizionario biografico degli italiani. 66, 131–133. Roma: Treccani. Giacardi, Livia. 2013. Gino Loria Mantova 1862 – Genova 1954. In Il contributo italiano alla storia del pensiero. Appendice VIII della Enciclopedia Italiana di Scienze, Lettere ed Arti, vol. IV, Scienze, 10, eds. Antonio Clericuzio and Saverio Ricci, 632–636. Istituto della Enciclopedia Italiana: Roma. Janovitz, Alessandro and Fabio Mercanti. 2008. Gino Loria. Sull’apporto evolutivo dei matematici ebrei mantovani nella nascente nazione italiana. Monografie di EIRIS (Epistemologia dell’Informatica e Ricerca Sociale), rivista on line, 27–42. Retrieved 21 marzo 2013 from https:// diazilla.com/doc/811027/sull-­apporto-­evolutivo-­dei-­matematici-­ebrei-­mantovani-­nella.%20 Accessed%205%20January%202022 Mercanti, Fabio. 2008. Gino Loria. Retrieved January 30, 2020 from Fulvia Furinghetti and Livia Giacardi. 2008. The first century of the International Commission on Mathematical Instruction (1908–2008) https://www.icmihistory.unito.it/portrait/loria.php Pepe, Luigi. 2001. Gino Loria e i suoi ‘assidui studi’ di storia delle matematiche. In Contributi di scienziati mantovani allo sviluppo della matematica e della fisica, Atti del convegno nazionale

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della Mathesis (Mantova, 17–19 May 2001), eds. FabioMercanti and Luca Tallini, 227–234. Cremona: Tip. Cremonese. Togliatti, Eugenio G. 1974. La biblioteca di Matematica dell’Università di Genova: formazione e primi sviluppi. Atti dell’Accademia Ligure di Scienze e Lettere 30: 339–346. Varnier, Giovanni B. 2002. L’Accademia Ligure di Scienze e Lettere e le “leggi razziali” tra silenziose espulsioni e tarde reintegrazioni. Atti della Società Ligure di Storia Patria n. s., 41(2): 496–510. Archibald, Raymond Clare. 1939. Gino Loria. Osiris 7: 5–30. Natucci, Alpinolo. 1954. In memoria di G. L. Archimede 6: 81–84. Terracini, Alessandro. 1954. Commemorazione del Socio Gino Loria. Rendiconti Accademia Nazionale dei Lincei, Classe Scienze Fisiche, Matematiche e Naturali 17: 402–421. Togliatti, Eugenio G. 1954. Necrologio. Gino Loria. Bollettino della Unione Matematica Italiana 9: 115–118. Vollgraff, Johan Adriaan. 1955. Kurze Bemerkungen, und Zitate, über Vergangenheit und Zukunft der Mathematik, so wie Gino Loria (1862–1954) versucht hat sich dieselben zu denken. Synthese 9: 485–491.

Publications Related to Mathematics Education Loria, Gino. 1889. Rivista bibliografica: A.I.G.T. (a proposito di un libro recente). Periodico di Matematica 4: 125–127. Translated as: ‘Bibliographic review. A.I.G.T. (à propos of a recent book)’, Association for the Improvement of Geometrical Teaching, Nineteenth General Report January, 1893: 46–48. Loria, Gino. 1893a. A few remarks on the “Syllabus of modern plane geometry”, issued by the A.I.G.T.’. Association for the Improvement of Geometrical Teaching, Nineteenth General Report, January: 49–53. Loria, Gino. 1893b. Della varia fortuna d’Euclide in relazione con i problemi dell’insegnamento geometrico elementare. Periodico di Matematica 8: 81–113. Loria, Gino. 1897a. Euclide. Dizionario illustrato di pedagogia, Vol. 1, 575–580. Milano: Vallardi. Loria, Gino. 1897b. Matematica. Dizionario illustrato di pedagogia, Vol. 2, 631–638. Milano: Vallardi. Also in Il Pitagora. 1897, 3(1): 1–4; 3(2): 17–20; 3(3): 37–40; 3(4): 53–56; 3(5): 69–71. Loria, Gino. 1899. La storia della matematica come anello di congiunzione fra l’insegnamento secondario e l’insegnamento universitario. Periodico di Matematica 14: 19–33. Loria, Gino. 1900. La fusione della planimetria con la stereometria. (Una pagina di storia contemporanea). Periodico di Matematica 15: 1–7. Loria, Gino. 1905a. Sur l’enseignement des mathématiques en Italie. In Verhanlungen des dritten Internationalen Mathematiker-Kongresses, in Heildelberg vom 8. bis 13 August 1904, ed. A. Krazer, 594–602. Leipzig: B. G. Teubner. Loria, Gino. 1905b. Programmi del passato e Programmi per l’avvenire. Lettura fatta a Milano il 22 Aprile 1905 dal Prof. Gino Loria, in occasione della riunione regionale dei professori di matematica, promossa dall’Associazione ‘Mathesis’. Rivista Ligure di Scienze, Lettere ed Arti 27(3): 129–146. Loria, Gino. 1906a. Sulle riforme scolastiche da compiersi e in particolare su quelle relative all’insegnamento della matematica. Il Bollettino di Matematica 5(10-11-12): 187–193. Loria, Gino. 1906b. La riforma della scuola Media in Germania (Pagine stralciate da un Diario di viaggio). Rivista Ligure di Scienze, Lettere ed Arti 28(5): 323–333. Loria, Gino. 1909. La scuola media e la sua attuale crisi di sviluppo. Discorso inaugurale. In Atti del II Congresso della ‘Mathesis’ Società Italiana di Matematica, Padova, 20–23 Settembre, 12–29. Padova: Premiata Società Cooperativa Tipografica.

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Loria, Gino. 1913. Discorso. In Atti del III Congresso della ‘Mathesis’ Società Italiana di Matematica, Genova, 21–24 Ottobre 1912, 12–14. Roma: Cooperativa Tipografia Manuzio. Loria, Gino. 1914b. Les gymnases-lycées modernes en Italie. Zeitschrift für Mathematik und Naturwissenschaftlichen Unterricht 45: 188–193. Loria, Gino. 1918. L’insegnamento della Matematica nelle Scuole industriali secondarie dei principali paesi del mondo. Bollettino della Mathesis 10(2): 66–76. Loria, Gino. 1921b. La preparazione degli insegnanti medi di matematica (conferenza alla sezione ligure della Mathesis). Periodico di Matematiche s. 4, 1: 149–164. Loria, Gino. 1931. L’insegnamento della storia delle scienze in Italia. Archeion 13: 474–476. Loria, Gino. 1932a. La préparation théorique et pratique des professeurs de mathématiques de l’enseignement secondaire. In Verhandlungen des Internationalen Mathematiker-Kongresses Zürich 1932, ed. W. Saxer, Vol. 2, 363. Leipzig: Orell Füssli. Loria, Gino. 1932b. The training of mathematics teachers. The Mathematical Gazette 16(221): 331–336. Loria, Gino. 1933. La préparation théorique et pratique des professeurs de mathématiques de l’enseignement secondaire dans les divers pays. L’Enseignement Mathématique 32: 5–20. Loria, Gino. 1937. Teaching the history of mathematics, The Mathematical Gazette 21(245): 274–275. Loria, Gino and Alessandro Padoa. 1909. Preparazione degli insegnanti di matematica per le scuole medie. Relazione. In Atti del II Congresso della ‘Mathesis’ Società Italiana di Matematica, Padova, 20–23 settembre, Allegato A, 1–10. Padova: Premiata Società Cooperativa Tipografica.

Photo Courtesy of the University of Turin.

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12.8 Mihailo Petrović (Belgrade, 1868–Belgrade, 1943): Appointed Honorary Member in 1936 Snezana Lawrence

Biography Mihailo Petrović was born on 6 May 1868 in Belgrade (sometimes also cited as 24 April, due to the differences in calendars), the eldest child of Nikodim, a professor of theology, and Milica (née Lazarević). Of a middle class background, he completed his primary (1878) and secondary schooling (1885) and undergraduate studies in Belgrade, the latter at the Superior, or sometimes also called the Great School, in 1889. He then went on to study at the École Normale in Paris. This was not usual practice, at the time, in Serbia. Serbian mathematics education in the nineteenth century developed first under the Ottomans and after 1833 under the influence of the Austro-Hungarian Empire. The Belgrade University grew out of a succession of institutions; the most prominent (and still existing) was Matica Srpska, literally meaning “the Serbian

S. Lawrence (*) Middlesex University, London, UK e-mail: [email protected]

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Queenbee”, which was founded in 1826 in Pest to promote Serbian language, culture and science. This institution grew into the Lyceum, and the Lyceum into a Superior School. The first trained mathematician to teach at the Lyceum was Dimitrije Nešić, educated in Vienna and Karlsruhe Polytechnic. At the end of the nineteenth century, several mathematicians studied for doctorates at western universities: Dimitrije Danić at Jena (1885), Bogdan Gavrilović at Budapest (1887), Djordje Petković at Vienna (1893) and Petar Vukičević at Berlin (1894). Finally Mihailo Petrović went to Paris – it is not known why or how he chose this as his destination when all his contemporary countryman had studied in Germany or Austria. But as a consequence of his choice, Petrović established important links with the French academia and also government which he maintained for the rest of his life. Thus, although most of educational and cultural influences in the middle of the nineteenth century in Serbia were of Austro-Hungarian or German origin, the most prominent of Serbian mathematicians and the one who founded the national school of mathematics introduced the French mathematics and French mathematicians to his country. Studies in Paris and French Connections Petrović was awarded his doctorate in 1894 for the thesis entitled Sur les zéros et les infinis des integrals des équations différentielles algébriques. The examining commission consisted of Charles Hermite (1822–1901), Émile Picard (1856–1941) and Paul Painlevé (1863–1933). Petrović dedicated his thesis to Jules Tannery (1848–1910) and Paul Painlevé: he had met Tannery during the first year of his studies when he attended courses on differential calculus and differential equations. Painlevé came to Paris from Lille in 1891 to become a professor at the Sorbonne, having completed his doctorate in 1887. He and Petrović became good friends during Petrović’s studies in Paris. Both later gained friends from the political elites of their respective countries. In 1906 Painlevé became a deputy for the fifth arrondissement, the so-called Latin Quarter. He later became prime minister of France and twice – 1917 and later again in 1925. Petrović, on the other hand, became first the tutor and later a good friend of the Crown Prince George Blackgeorge (Djordje Karadjordjević, 1887–1972). Petrović and Painlevé continued their friendship after Petrović’s return to Belgrade. After Petrović’s insistence, Painlevé’s work on mechanics (Painlevé 1922) was translated by Ivan Arnovljević and published in 1828 in Belgrade as a textbook under the title Mehanika (Mechanics). Petrović also made friends with Charles Hermite, who already had another Serbian student, Mijalko Ćilić (1859–1912). Hermite taught Petrović higher algebra, and his son-in-law, Emil Picard, was another of Petrović’s examiners. Petrović and Picard became lifelong friends, and Picard drew on the work from Petrović’s thesis in his Traité d’Analyse (1901–1908) (Trifunović 1994, p. 27).

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Work in Belgrade Upon his return to Belgrade in 1894, Petrović was made a professor at the Superior School. At the beginning of 1905, this school was replaced by the newly founded University of Belgrade, and Petrović was appointed to the chair of mathematics. This position he held until his death in 1943. His main area of interest was classical analysis, and he wrote papers on the properties of real and complex functions defined by power series (Petrović 2004, p. 100). When he first returned to his city of birth, Petrović devoted some time to creating an analogue computer for solving a certain type of analytically non-integrable differential equations (Petrović 1908). He completed the machine, which he called Hydro Integrator, for the Universal Exhibition held in Paris in 1900 and was awarded the Bronze Medal for its invention (Petrović 2004). In the years prior to the First World War, Petrović worked on mathematical phenomenology, with a view of developing a mathematical apparatus which would be able to encompass “all facts” and link apparently unconnected phenomena in a mathematical fashion (Petrović 1911). His work on cryptography established the science in Serbia and, later, Yugoslavia. As an outcome of this work, during the First World War, he founded the theory of mathematical spectra, which has analogies with the spectral method in chemical analysis. This theory consists of dispersing unknown quantities from a problem into a numerical spectrum. The unknown quantities are dispersed, separated and determined in the same way as the spectral analysis is done in chemistry. This theory was to be applicable to arithmetic, algebra and infinitesimal calculus, and Petrović taught a course on it at the Faculté de Sciences at the University of Paris in 1928 (Petrović 1919, and 1928). During his tenure at the University of Belgrade, Petrović delivered some 16 different courses in mathematics, for half of which he produced notes, and published 3 textbooks based on his teaching. His opus includes some 257 published papers, 5 registered patents, 10 monographs and 31 articles and books on subjects ranging from fishing and the history of piracy (of the ancient, seaman type) to ethnography and travelogues. Petrović, the Author, Fisherman and a Musician Apart from being an extremely prolific mathematical author, Petrović was also one of the most universally liked individuals from the Serbian intellectual history. His love of mathematics and the ease with which he made friends with people in high places as well as the local people and the gypsies, his love of music and his travelogues and novels made him a role model for many young Serbian mathematicians. Petrović had his own band, called Suz (which recorded more than 2000 Gypsy music songs for Radio Belgrade between the two world wars). According to biographer of Petrović (Trifunović 1969), the repertoire of this band included over 700 folk dance melodies and hundreds of tunes from different parts of Yugoslavia. Unfortunately, none of the recordings of this band or the reported work of Petrović

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on collecting folk melodies survived; reportedly they were destroyed in the bombing of Belgrade in 1941 by the Third Reich (Trifunović 1969; Dumnić 2018). Petrović was a dedicated fisherman and even passed the examination of Master Fisherman in Serbia as soon as he came back from his studies in Paris. In later years, he drafted the national laws on fishery and helped define international treatises. His interest in fishing led him to participate in a number of French exhibitions to the Arctic and Antarctic, to islands in the south Indian Ocean, and to write travelogues and a novel based on these journeys. One of his articles on fishery was entitled Do fish sleep? (Petrović 1897), and his novel A novel about an eel (Petrović 1940), which describes the life of an eel and its travels, is still in print in Serbia.

Contribution to Mathematics Education Serbia changed relatively rapidly from having little or no mathematical culture at the beginning of the nineteenth century to having a thriving mathematical life under the leadership of Petrović. There were some advantages to this relatively short history. At the International Conference on Mathematics Teaching (Conférence International de l’Enseignement Mathématique) in Paris held in April 1914, the Serbian representative Bogdan Gavrilović reported that the introduction of infinitesimal calculus into schools was devoid of problems in their country. Modernisation did not pose a problem in a place where there was no tradition which could inhibit it: Among the nations that have barely overcome the first steps of civilization in their development, there is no tradition and an idea in general and especially a new idea, becomes very easily the very ideal of a generation. Consequently, in these circumstances the realization of this ideal is not prevented or delayed by matters of tradition.32 (Compte rendu … 1914, p. 332)

It was Petrović who established a new tradition of teaching and learning of mathematics in Belgrade and Serbia in general. His work in education led him to attend the international congresses as a representative of a small country which was only beginning to build its mathematical school. He not only established this school, but introduced a new spirit of learning at the University of Belgrade. Through his personal efforts, the library there grew steadily to include the main mathematical journals of the times, and he was said to have encouraged his students to learn from the sources they themselves found in this library – a new approach to learning in the times when learning by rote was still very much the modus operandi. His efforts to establish the national school (virtually all mathematical doctorates in Serbia between the two world wars were done under his supervision) established a far-reaching change in mathematics in the Balkans. Petrović’s doctoral students  The original text is: “Chez les nations qui ont à peine dans leur développement, passé les premiers seuils de la civilisation, il n’y a pas de tradition et une idée en générale et surtout une idée nouvelle, dévient très facilement l’idéal même d’une génération. Par conséquence, dans ces circonstances la réalisation de cet idéal n’est pas empêchée ou retardée par des questions de tradition”. 32

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were Sima Marković (who earned his PhD in 1904 and became a famous communist and as such disappeared and lost his life in Russia under Stalin), Mladen Berić (1912), Tadija Pejović (1923), Radivoj Kašanin (1924, who became professor at the University of Belgrade), Jovan Karamata (1926, who taught mathematics at the University of Belgrade and University of Khartoum), Dragoslav Mitrinović (1933, professor of mathematics and founder of mathematical institutes in a number of universities of former Yugoslavia), Danilo Mihnjević (1934), Konstantin Orlov (1934, professor of mathematics at the University of Belgrade) and Dragoljub Marković (1938). These mathematicians, the pupils of Petrović, jointly produced a further 361 doctoral students during their professional lives  – and from them the number of doctoral students who draw their genealogy to Petrović grows daily. Petrović’s work had a long-term effect on Serbian, and later Yugoslavian, study of mathematics in the first half of the twentieth century and beyond. In this way the influence of the French school was felt long after its main Serbian student became the founder of the national mathematical school. Petrović’s dedication to education extended to his belief that communication with the outside world must be part of the practice at the university. In 1914 he suggested to the Serbian Royal Academy that all scientific communications contain an abstract in French or German. From 1922 this became a standard practice, and it later included translations into English. Petrović was also one of the founders of the journal Publications Mathématiques de l’Université de Belgrade in 1932. In this perspective of internationalisation and communication, Petrović participated in many International Congresses of Mathematicians (ICMs). He presented papers in analysis at Rome (ICM of 1908), Cambridge (ICM of 1912), Strasbourg (ICM of 1920), Toronto (1CM of 1924), Bologna (ICM of 1928) and Zurich (ICM of 1932). In Cambridge he was the delegate of Serbia (associated country to ICMI) at the session of ICMI and presented the printed national report of his country. In the ICM of 1924, he was appointed delegate of the Serbian Royal Academy of Sciences and Arts, National Council of Research and University of Belgrade, in 1928 of the Serbian Royal Academy and University of Belgrade and in 1932 of the Serbian Royal Academy of Sciences. Petrović was appointed as one of the secretaries in the Interim Executive Committee of the International Mathematical Union in the years 1919–1920. In 1936, at the International Congress of Mathematicians in Oslo, he was honoured with the title of Membre Honoraire de la Commission International (Honorary Member of the International Commission).33

 See Comptes rendus du Congrès International des Mathématiciens, Oslo 1936, Oslo: A. W. Brøggers Boktrykkeri, 1937, II, p. 289. 33

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Sources Primary Bibliography Petrović, Mihailo. 1999. Collected works (in Serbian), Vols. 1–15. Belgrade: Institute for Textbook Publishing and Teaching Aids. Pilipović, Stevan, Gradimir V. Milovanović, and Žarko Mijajlović (eds.). 2019. Mihailo Petrović Alas Life Works Times. Belgrade: Serbian Academy of Sciences and Arts. Petrović, Mihailo. 1894. Sur les zéros et les infinis des integrals des équations différentielles algébriques. Thesis presented at the Faculty of Sciences, Paris, No. 823. Paris: Gauthier-Villars. Petrović, Mihailo. 1894. Sur les intégrals uniformes de équations différentielles du premier ordre et du genre zéro. Compte rendus hebdomadaires des séances de l’Académie des Sciences 118 (Janvier-Juin): 1190–1193. Petrović, Mihailo. 1895. O asimptotnim vrednostima integrala diferencijalnih jednačina. Belgrade: Serbian Royal Academy of Science. Petrović, Mihailo.1896. Sur l’equation différentielle de Riccati et ses applications chimiques [On Riccati’s differential equation and its application in chemistry]. Sitzungberichte der Königlich– Böhmischen gesellschaft der Wissenschaften [Meeting reports of the Royal Bohemian Society of Sciences], 1–25. Prague: Royal Bohemian Society. Petrović, Mihailo. 1896. Contribution à la théorie des solutions singulières des équations différentielles du premier ordre. Mathematische Annalen 52: 103–112. Petrović, Mihailo. 1899. Sure une manière d’étendre le théorème de la moyenne aux équations différentielles du premier ordre. Mathematische Annalen 55: 417–436. Petrović, Mihailo. 1908. Procédé élémentaire d’application des intégrals définies réelles aux équations algébriques et transcedants. Nouvelles Annales de Mathématiques s. 4, 8: 1–15. Petrović, Mihailo. 1911. Elementi matematičke fenomenologije, Belgrad: Serbian Royal Academy of Science. Petrović, Mihailo. 1919. Les spectres numériques. Paris: Gauthier-Villars. Petrović, Mihailo. 1928. Leçons sur les spectres mathématiques. Paris: Gauthier-Villars. Petrović, Mihailo. 1929. Prilog istoriji jednoga problema teorije funkcija [A contribution to the history of one problem in the theory of functions]. Glas 134: 87–90.

Secondary Bibliography Compte rendu de la Conférence international de l’enseignement mathématique. 1914. L’Enseignement Mathématique 16: 245–356. Dumnić, Marija. 2018. Mihailo Petrović Alas and Music. In Mihailo Petrović Alas, The Founding Father of the Serbian School of Mathematics. Serbian Academy of Sciences and Arts (SANU). Belgrade, Serbia. Lawrence, Snezana. 2008. A Balkan Trilogy – Mathematics in the Balkans before the First World War. Oxford Handbook in the History of Mathematics. Oxford University Press, 177–196. Painlevé, Paul. 1922. Les axioms de la Méchanique. Paris: Gauthier-Villars. Petrović, Aleksandar. 2004. Development of the first hydraulic analog computer. Archives Internationales d’Histoire des Sciences 54: 97–110. Picard, Emile. 1901–1908. Traité d’analyse. Paris: Gauthier-Villars et fils. Trifunović, Dragan. 1969. Letopis života i rada Mihaila Petrovića [The chronicles of the life and works of Mihailo Petrović]. Belgrade: Serbian Academy of Sciences and Arts (SANU). Trifunović, Dragan. 1994. Doktorska disertacija Mihaila Petrovića. Belgrade: Archimedes.

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Travelogues and a Novel Petrović, Mihailo. 1897. Da li ribe spavaju? [Do fish sleep?]. Belgrade: Serbian Literary Society. Petrović, Mihailo. 1932. Kroz polarnu oblast [Through the polar region]. Belgrade: Serbian Literary Society. Petrović, Mihailo. 1933. U carstvu pirata [In the realm of the pirates]. Belgrade: Serbian Literary Society. Petrović, Mihailo. 1935. Sa okeanskim ribolovcima [With ocean fishermen]. Belgrade: Serbian Literary Society. Petrović, Mihailo. 1936. Na dalekim ostrvima [On distant islands]. Belgrade: Serbian Literary Society. Petrović, Mihailo. 1940. Roman o jegulji [A novel about an eel]. Belgrade: Serbian Literary Society.

Publications Related to Mathematics Education Petrović, Mihailo. 1913. Medjunarodna komisija za matematičku nauku [International commission on mathematical instruction]. Prosvetni glasnik, Belgrade, 724–731. Petrović, Mihailo. 1926. Francuska matematika [French mathematics]. Letopis Matice srpske 307: 207–220. Petrović, Mihailo. 1928. Kriptografija, škola za učenje nauke u četrnaest knjiga [Cryptography, a school on teaching the science in fourteen books]. Official documents of the Kingdom of Yugoslavia, Secret Service, Cryptography section, Belgrade. Petrović, Mihailo. 1928. Jedno pitanje iz nastave o logaritmima [Issues about the ways logarithms are taught]. Glasnik profesorskog drustva 8: 368–370. Petrović, Mihailo, with B. Gavrilovic and I. Djaj. 1932. Uputsvo za pisanje matematičkih i naučnih radova na stranim jezicima [Guidance on publishing in foreign languages for mathematical and natural sciences]. In Srpska Kraljevska Akademija, Godisnjak, Belgrade, 280–283. Petrović, Mihailo. 1932. O zavisnosti medju veličinama u zadacima [On dependence between magnitudes in mathematical problems]. Matematicki list za srednju skolu, 273–276. Belgrade. Petrović, Mihailo. 1933. Greške matematičara [The mistakes mathematicians make]. In Glasnik Jugoslovenskog profesorskog drustva 13: 874–881.

Photo Source: Wikimedia Commons.

Wilhelm Wirtinger

705

12.9 Wilhelm Wirtinger (Ybbs, 1865–Ybbs, 1945): Appointed Honorary Member in 1936 Gert Schubring

Biography Wilhelm Wirtinger was born on 15 July 1865  in Ybbs, an Austrian town on the Danube River. Wirtinger studied at the University of Vienna; he passed the doctorate exam in mathematics there in 1887. Thanks to a travel grant, he continued studying abroad, in Germany, first in Berlin but mainly in Göttingen. There he was guided by Felix Klein doing research on synthetic geometry. Eventually, Klein became his friend. Wirtinger obtained the habilitation in mathematics in 1890, again in Vienna. After a short period in 1895 as a professor at the University of Vienna, Wirtinger accepted a call to a professorship at the University of Innsbruck. In 1896, Wirtinger returned to a full professorship at the University of Vienna. G. Schubring (*) University of Bielefeld, Bielefeld, Germany e-mail: [email protected]

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Wirtinger showed extraordinarily broad competences in mathematics. His first important publications were on theta functions. The solution of a problem in the theory of general theta functions earned him, in 1895, an award from the Göttingen Academy. He continued with work on algebraic functions and their integrals, then on Abelian functions, later on differential geometry and eventually on complex functions of several variables. Moreover, he did research in mathematical physics and in statistics. In the Encyklopaedie der Mathematischen Wissenschaften mit Einschluß ihrer Anwendungen, there are a number of articles by him (partial differential equations, calculus of variations, complex function theory). He was the editor of the journal Monatshefte für Mathematik und Physik. Mathematicians who studied with him in Vienna were, among others, Schreier, Gödel, Radon and Taussky-Todd. Wirtinger was a member of the academies of science in Vienna, Berlin, Göttingen and Munich and of the Papal Academy in Rome. He was awarded several distinctions, including honorary doctoral degrees from the universities in Hamburg and in Oslo. He died on 15 January 1945 in Ybbs.

Contribution to Mathematics Education When the Austrian subcommittee of IMUK was established in 1909, Wirtinger was nominated one of the three delegates. He continued in this function until the dissolution in 1920. In 1936, after the reestablishment of IMUK, Wirtinger was elected an honorary member of the Commission.34 For the IMUK meeting in Milan in 1911, he was a member of the committee preparing the reports on the mathematical training of students of the natural sciences and gave the report for Austria (1911. L’Enseignement Mathématique 13: 492–494). In 1933, he published, together with Hans Hahn and Erwin Kruppa, the report on the training of mathematics teachers in Austria.

Sources Wirtinger, Wilhelm. 1887. Über die Brennpunctscurve der räumlichen Parabel. Wien. Wirtinger, Wilhelm. 1895. Untersuchungen über Thetafunctionen. Leipzig: Teubner. Wirtinger, Wilhelm. 1926. Allgemeine Infinitesimalgeometrie und Erfahrung. Leipzig: Teubner. Wirtinger, Wilhelm. 1942. Zur Theorie der konformen Abbildung mehrfach zusammenhängender ebener Flächen. Berlin: Verlag der Akademie der Wissenschaften.

 See Comptes rendus du Congrès International des Mathématiciens, Oslo 1936, Oslo: A. W. Brøggers Boktrykkeri. 1937. Vol. 2, p. 289. 34

Wilhelm Wirtinger

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Dedekind, Richard and Heinrich, Weber (eds.). 1902. Bernhard Riemanns gesammelte mathematische Werke und wissenschaftlicher Nachlass [1892]. Nachträge, Max Noether and Wilhelm Wirtinger (eds.). 2. Aufl. [Leipzig, Teubner]. Wirtinger-Festband. 1936. Monatshefte für Mathematik und Physik 43.

Obituaries Caratheodory, Constantin. 1948. Wilhelm Wirtinger. In Jahrbuch der Bayerischen Akademie der Wissenschaften 1944–1948, Bayerischen Akademie der Wissenschaften, ed., 256–258. München: Beck. Hornich, Hans. 1948. Wilhelm Wirtinger. Monatshefte für Mathematik 52: 1–12 (with the list of publications).

Publications Related to Mathematics Education Wirtinger, Wilhelm, Hans Hahn, and Erwin Kruppa. 1933. Die Ausbildung der Mathematiklehrer an den Mittelschulen Oesterreichs. L’Enseignement Mathématique 32: 184–191.

Photo Source: Wikimedia Commons.

Author Index

A Abdeljaouad, J., 336 Abellanas, P., 203 Abramovitz, B., 323 Abrams, F., 336 Ackermans, S., 336 Acton, F.S., 308 Adams, F., 258 Adda, J., 314, 336, 452 Adler, J., 125, 127, 270, 273, 283, 284 Aguado, G., 336 Aguilera, N., 122, 268, 283 Ahlfors, L., 246, 566, 567 Ahmad, S., 336 Ai, S., 320 Aidi, S., 336 Aiton, É., 336 Aizpun Lopez, A., 336, 344 Akel, G., 336 Akizuki, Y., 53, 63, 69, 109, 190, 194, 199, 203, 218, 249, 251, 281, 336, 350–353 Akkerhuis, G., 336 Alcolea-Banegas, J., 323 Ale, S.O., 314 Aleksandrov/Alexandrov/Alexandroff, A.D., 62, 63, 69, 70, 189, 190, 192, 194, 249, 281, 354–357, 482 Alexanderson, G.L., 109 Alexandroff, P.S., 482 Algra, E., 336 Allard, F., 336 Allard, J.C., 336 Allen, D., 336

Allendoerfer, C.B., 307 Almeido Costo, A., 336 Alonso Juaneda, J., 336 Alsina, C., 121 Altschul, P., 336 Alvarez, J.M., 121 Alves, G., 318 Amato, V., 305 Amir-Moéz, A.R., 311 Amitsur, S.A., 204 Andersen, A.F., 50, 248, 280, 359–363 Anderson, J., 336 Andžāns, A., 317 Andrade, J., 299, 300 Andrew, G.E., 318 Andronov, I.K., 310 Anka, K., 336 Ansarov, R., 336 Apollodorus, P., 336 Appell, P.E., 22, 141, 142, 155, 203, 466 Archenhold, F.S., 6, 7, 9, 300 Archibald, R.C., 37, 151, 691–693 Archimedes, 679 Arellano, L.T., 336 Aristotle, 159 Armitage, J.V., 336 Arnold, H.-J., 318 Arnold, V.I., 557 Arnovljevi, I., 699 Artigue, M., 90, 97, 100, 111, 113, 115, 120, 122, 125, 127, 129, 131, 137, 268, 270–273, 283, 284, 318, 321, 581, 584, 586

© Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0

709

710 Artom, E., 303 Arzarello, F., 588 Arzumanyan, G.S., 310 Ascoli, G., 50, 53, 248, 249, 280, 305, 365–369 Ashkinuze, V.G., 311 Ashna, A., 320 Asiedu-Addo, S.K., 318 Askin, S., 318 Athen, H., 106, 306 Audibert, G., 336 Auger, P., 64, 160, 162 Augier, J.L., 336 Avital, S., 343 B Bacon, H.M., 549 Bacon, R., 147 Badesco, R., 336, 344 Badillo, E., 322, 323 Bagby, J., 336 Bagni, G.T., 387, 481, 556 Baidyk, T.N., 318 Bajpai, A.C., 336 Bajpaj, D., 312 Baker, F.W.G., 73, 206 Baker, H.F., 544, 545 Bakhvalov, N.S., 597, 598, 608, 609 Balakrishnan, A.V., 536 Balangangadharan, K., 204 Baley Price, G., 307 Ball/Ball Loewenberg, 127, 320 Ball, E., 336 Ball, J., 125, 268, 271, 283 Balle, E., 336 Banchoff, T.F., 312 Bandet, É., 336 Bandic, I., 170 Barbazo, É., 395, 438, 639, 654 Barbeau, E.J., 127 Barbey, G., 336 Bardou, G., 336 Barile, M., 432, 533 Barner, M., 85, 101, 108, 255 Barsky, D., 511 Bartolini Bussi, M.G., 127, 273, 284, 318 Barton, B., 127, 273, 274, 284 Bass, H., viii, 95–97, 100, 108, 109, 113, 115, 117–119, 122, 125, 127, 268, 270, 273, 283–285, 289, 322, 486 Bastad, M., 344 Batanero, C., 125, 130, 270, 283

Author Index Batdedat, A., 336 Batier, C., 336 Batier, P., 336 Bauersfeld, H., 77, 100, 203, 255, 614 Baugh, R., 336 Baur, A., 306 Bazin, S., 336 Bazzini, L., 419 Beardsley, L., 336 Beauchant, J., 336 Bebbe-Njoh, E., 314 Becker, J.P., 97, 335, 341, 344, 345 Becques, C., 336 Beddens, J., 344 Begle, E.G., 66, 82, 105, 109, 256, 259, 282, 336, 372–375 Begnaud, L., 337 Behari, R., 53, 59, 62, 109, 152, 249, 281, 377–380 Behboodian, J., 317 Behnke, H., 44, 48–55, 57–59, 61–63, 69, 70, 77, 79, 85, 88, 96, 97, 161, 165–167, 169–172, 174, 176, 180–183, 185, 186, 188–190, 192–194, 199, 203, 204, 247–249, 251, 253, 280, 281, 306, 308, 336, 382–385, 394, 412, 413, 440, 512, 526, 588, 612 Behrends, E., 319 Beke, E., 20, 35, 39, 247, 301, 504, 650–652 Bell, A.W., 337, 344, 419 Bellaiche, P., 337 Bellavitis, G., 685 Bellenger, E., 337 Beliot Rosado, F., 337 Belouze, B., 337 Beltrami, E., 691 Ben Jamaa, M., 337 Bénézet, L.P., 646 Bennett, R., 319 Benzaghou, B., 511 Bercioux, A., 337 Berezina, M., 323 Berger, M., 311 Bergson, H., 654 Berić, M., 702 Bernays, P., 540 Bernet, T., 54, 337 Bernhard, P., 537 Bernier, D., 337 Berri, M., 337 Bers, L., 554 Berthault, M., 337 Berzolari, L., 147, 148, 150

Author Index Beth, E.W., 50, 54, 171, 248, 280, 387 Betti, E., 672 Betz, W., 47, 247, 304 Bhatnagar, P.L., 105, 256, 337 Bianchi, L., 365, 591, 672 Bianconi, M.-L., 392 Bidger, M., 308 Bieberbach, L., 450, 451 Bied, J., 337 Biemel, R., 337 Bigalke, H., 337 Biggs, E.E., 310, 337, 344 Biggs, J.B., 67 Bilotskiyi, N.N., 319 Bioche, Ch., 19, 39, 40, 155, 247, 470, 654–656 Biratelle, H., 584 Birkhoff, G.D., 45, 541, 618, 619, 644 Bishop, A., 100, 264, 337, 344 Blackburn, W., 337 Blackgeorge, G., 699 Blanc, É., 77, 337 Blasco, F., 323 Blau, P., 337 Blonce, J., 337 Blum, W., 127 Blutel, E., 141, 142 Bodin, A., 337 Bogoslavov, V., 337 Bohr, H., 360, 362, 476 Boissier, M., 337 Bolaños, G., 322 Bollon, L., 337 Bolon, J., 337 Boltyansky, V.G., 311 Bombelli, R., 679 Bompiani, E., 48, 49, 52, 53, 55, 88, 161, 166–170, 287, 413, 518 Bonden, H., 337 Bondesen, A., 337 Bonnard, C., 337 Bonnetain, R., 337 Bonsu, O.M., 318 Borier, M., 337 Borisov, Y., 357 Bortolotti, E., 679 Bosland, P., 337 Bosveld D., 363 Botana, F., 317 Botella, L., 640 Bott, R., 436 Bottazzini, U., 400, 675 Boulad, F., 39, 40, 247, 658–664

711 Boulad, Y., 659 Boulanger, G., 686, 687 Bourbaki, N., 66, 105, 392, 393, 395, 396, 535, 566, 568 Bourguignon, J.P., 322 Bourret, M., 337 Bourtot, B., 337 Bouteille, M., 337 Bouvier, A., 337 Bovey, H.T., 144 Boyer, L.E., 305 Boys, G., 337 Bracewell, H., 337 Bradley, H., 337 Brasseur, R., 410 Bray, S., 337 Brecht, G., 319 Bressy, G., 337 Bridger, M., 306 Brigaglia, A., 691 Bright, D., 337, 344 Brischle, T., 310 Brkic, J., 337 Broadbent, T.A.A., 379, 573, 574 Broman, A., 312 Bron, A., 337 Brossard, R., 337, 344 Broué, M., 394 Brousseau, G., 126, 271, 337, 419 Brouwer, L.E.J., 451 Browder, E.F., 46 Brückner, M., 300 Bruhat, F., 583 Bruley, D., 337 Brumfied, E., 337 Brun, J., 337 Bruner, J.S., 66, 91, 103 Bruni, J., 337 Brusotti, L., 170 Bryan, G.H., 140 Brydegaard, M., 337 Buffon, L., 466 Buhl, A., 139, 687, 688 Buisson, P., 337 Bundgaard, S., 170, 189, 191, 193, 253, 305 Bunt, L.N.H., 77, 305–308, 337 Burckhardt, J., 568 Burgatti, P., 366 Burgin, M.S., 317 Burgues, C., 314 Burkhart, S., 337 Burkill, J.C., 218 Burlet, O., 432, 435, 436

712 Burrill, G., 130 Burscheid, H.-J., 8 Burton, L., 419 Bury, P., 337 Buzano, P., 63, 249, 367 Byers, V., 443 Byrne, C., 337 C Cabric, M., 337 Cajori, F., 694 Calame, F., 337 Calandra, A., 549 Callagy, J., 337 Calvet, R., 337 Campedelli, L., 204 Campogrande, P., 337 Cañas, J.J., 322 Cance, H., 337 Cantor, M., 602 Capitan, J., 337 Caratheodory, C., 383 Cardot, A., 170 Carlavilla, J.L., 322 Carleson, L., 108, 110, 258, 282 Carlyle, T., 137 Carrega, J.C., 337 Carroll, E., 337 Carrus, S., 303 Carson, G.E., 146, 301 Carss, M., 111 Cartan, É., 392, 433, 517, 639 Cartan, H.P., 77, 81–87, 97, 100, 102, 207, 208, 210, 214, 216, 218, 220–228, 233, 238, 253, 281, 383, 392–397, 435, 453, 485 Carter, B., 337 Carter, M., 37 Cartwright, M.L., 305 Carvalho e Silva, J., 127, 273, 284, 425 Casaravilla, A., 332 Cassina, U., 307, 679 Cassy, B., 320, 323 Castejón, A., 322 Castelnuovo, E., 54, 74, 105, 108, 126, 256, 337, 406, 632 Castelnuovo, G., vii, viii, 7, 19, 22, 29, 37, 39, 40, 51, 128, 147–150, 244, 246, 247, 279, 300, 347, 399–406, 592, 593, 672–675, 678, 680 Castrillón López, M., 322 Casulleras, J., 337 Cattaneo, C., 77

Author Index Cauchy, A.-L., 517 Causier, A., 337 Celanire, Y., 337 Chaillou, P., 337 Chalvron, J., 312 Chamma’a, S., 337 Chandrasekharan, K., 64, 72, 76, 85, 101, 105, 183–185, 198–202, 210, 225, 227, 228, 232, 255, 282, 379, 434, 494 Chapon, J., 337 Charnay, R., 337 Chasles, M., 475 Chatard, M., 337 Châtelet, A., 48–50, 52, 53, 55, 57, 58, 63, 79, 87, 88, 161, 164, 166, 167, 170, 172, 174, 178, 247, 280, 305, 385, 409–414, 512 Châtelet, F. (son of Châtelet, A.), 80, 85, 101, 199, 209, 255, 337, 409 Chatterji, S.D., 316 Chebyshev, P., 607 Cheng, S.-Y., 322 Cherkasov, R.S., 310 Chern, S.-S., 619 Chernick, S., 337 Cheroux, P., 337 Chesney, S., 337 Chevalley, C., 392, 393, 485 Chevrier, J.L.P., 306 Cheze, J., 337 Chick, H., 124, 270 Chini Artusi, L., 337 Chirac, J., 534 Chisholm Young, G., 28 Chisini, O., 674, 679 Chittenden, E.W., 552 Chiu, S.-Y., 319 Choquet, G., 54, 63, 66, 191, 198, 248, 249, 251, 356, 379, 512, 518, 582, 584, 642 Christiansen, B., 63, 76, 84, 101, 105–108, 110–114, 215, 256, 258–260, 282, 337, 417–419, 488 Churchhouse, R.F., 114, 262 Ciarlet, P.G., 535 Ciesielski, Z., 313 Cignetti, A., 337 Ćilić, M., 699 Clapier, C., 303 Clemenceau, G., 110 Clements, R.R., 114 Clifford, P., 337, 338 Cobb, P., 126, 274 Cochrane, M., 337 Coeckelberghs Bogaert, L., 337

Author Index Cohen, P., 511 Cointet, M., 338 Colmez, F., 338, 584, 588 Colmez, J., 338, 511 Colomb, J., 338, 344 Combe, M.-L., 338 Comte, A., 688 Conant, J.B., 548 Condette, J.-F., 409–412 Condevaux, G., 413, 414 Constantinescu, G., 320 Conte, A., 400, 675 Conti, A., 7, 300 Cooney, M.P., 314 Cooney, Th., J. (Tom), 258, 419 Cooper, D., 338 Coray, D., 124, 269 Córcoles, C., 323 Coriat, M., 672 Cornu, B., 114, 262 Cortial, M., 338 Cote, N., 338 Coulomb, J., 392 Courant, G., 557 Courant, R., 512 Courau, D., 338 Courseau, P., 338 Courvoisier, A., 338 Couty, R., 338 Cowley, E.B., 303 Coxeter, H.S.M., 256 Coyle, A., 338 Cramer, H., 306 Crawford, D., 338 Crawforth, D.G., 222, 419 Crelier, L.J., 302 Crepin, R., 338 Crespo Crespo, C., 319 Créspo Pereira, R., 305 Crighton, D.G., 530 Croteau, B., 338 Crouth, R., 338 Cuello Nebot, E., 322 Cuisance, S., 338 Curbera, G., 137 Curien, H., 536 Curien, N., 312 Cvrkušić, V., 338 Czuber, E., 22, 32, 40, 244, 279, 421–423 D Dalançon, A., 640 D’Alembert, J.-B., 535

713 Daltry, C.T., 56, 169, 305, 310 D’Ambrosio, U., 64, 108, 109, 112, 113, 125, 126, 258, 274, 282, 645 Damerow, P., 120, 264 Damian, B., 338 Danić, D., 699 Daqian, L., 320, 537 Darboux, J.G., 31, 466, 470, 572 Darmois, G., 306, 517 Daugherty, J.D., 338 Daumet, G., 338 Dautray, R., 535 Davalan, J.-P., 338 Davidson, L., 338 Davis, H.M., 310 Davis, Ph., 112 Davis, R.B., 442, 443, 549 De Amicis, E., 301 De Bary, C., 336 De Bock, D., 65, 68 De Castro, B., 536, 537 De Finetti, B., 518 De Galdeano, Z.G., 299, 301 De Groot, J., 305 De Guzmán, M., 118, 119, 121, 122, 124, 125, 131, 265, 266, 268, 283, 316, 318, 324, 425–429 De Hosson, C., 588 De Lagaye, F., 340 De Lange, J., 321 De Oliveira Bender, J., 341 De Paolis, R., 672 De Possel, R., 392, 393 De Puya, I., 313 De Rham, G., 64, 72, 80, 81, 138, 170, 198–202, 208–210, 214, 251, 281, 432–436 De Shalit, E., 321 De Siebenthal, M.J., 77 De Souza Dantas, M., 342, 345 De Turckheim, E., 114, 263 De Vasconcellos, F., 693 De Vos van Steenwijk, J.E., 303 De Young, G., 658 Deakin, M., 312 Deans, E., 338 Decelles, P., 338 Decombe, F., 338 Dedekind, R., 624, 679 Dedò, M., 338 Dehame, É., 338 Dehn, M., 476, 481 Delange, G., 312 Delavault, H., 338

714 Delcourt, P., 338 Delessert, A., 70, 72, 74, 77–84, 86, 97, 138, 199–201, 204, 205, 208, 210, 212–214, 218, 220–224, 251, 253, 281, 338 Delitala, G., 301 Delone, B.N., 354, 355 Delorme, M., 338 Delsarte, J., 392, 393 Delta, E., 313 Demana, F.D., 314 Demott, B., 620 D’Enfert, R., 16 Denjoy, A., 410, 511, 582 Denniss, J., 338 Derwidue, L., 338 Descartes, R., 324, 471 Desforge, J., 59, 170, 181, 249, 281, 338, 438–441 Desjardin, C., 338 Despotovic, R., 338 Desq, R., 338 Detry, N., 338 Deulofeu, J., 322, 323 Dev Sarma, B.K., 320 Dhuin, C., 338 Diansuy, M.A.A., 320 Dickstein, S., 39, 160, 247, 623, 625, 666–669 Dienes, Z.P., 97, 98, 338, 631, 641 Dieudonné, J., 54, 66–68, 250, 311, 392, 393, 396, 418, 485, 518 Dimitrić, R., 319 Dini, U., 591, 672 Dionne, J.J., 119 Dirac, P., 529, 530 Diserens, R., 338 Dobrovol’sky, V.A., 310 D’Ocagne, M., 40, 155, 661–663 Docev, K., 338 Doicinov, D., 338 Dokic, O., 338 Dolbilin, N., 125, 270, 283 Dolinsky, R., 306 Donoghue, E., 6–8, 10 Doraisany, L., 317 Dörfler, W., 258 Dorier, J.L., 320, 586 Douady, R., 586 Douglas, J., 246 D’Ovidio, E., 690 Downs, F., 549 Dravinac, N., 338 Drenckhahn, F., 306, 307 Dreyfus, A., 469 Druel, P., 338

Author Index Du Sautoy, M., 322 Dubecq, 302 Dubreil, P.J., 53, 249, 395, 411, 582 Dubrocard, J., 338 Duby, J.J., 338 Duceux, G., 338 Duceux, P., 338 Duchène, F., 338 Ducker, I., 338 Duclos, D., 338 Dulmage, A.L., 203, 338 Dumnić, M., 701 Dumont, M., 338 Dumousseau, G., 338 Dunkley, M., 338 Duponsky, E., 640 Dupont, E., 640 Dupont, M., 338 Dupont, P., 344 Duporcq, E., 299 Durapau, V.J., 314 Duren, W.L. Jr., 307 Durgan, M., 338 Duverger, Y., 338 Duvert, L., 338 Dykman, J., 338 E Eaton, P., 323 Eckmann, B., 64, 69, 70, 180–184, 187–189, 191, 192, 194–196, 198, 237, 482 Edwards, C., 338 Egéa, M., 338 Egsgard, J., 118, 338, 343, 344 Egyed, A., 338 Ehmann, E., 450 Ehresmann, C., 392, 393 Ehrhardt, C., 16 Eido, R., 338 Eilenberg, S., 393, 541 Eiller, R., 338 Einstein, A., 378, 401, 517, 623 Eixarch Ferrer, R., 324 Elbaz, R., 338 Elyoseph, N., 170 Emch, A., 301 Emery, E., 77 Emmer, M., 320 Engel, A., 78, 108, 338, 343 Engel, F., 478 Engquist, B., 322 Enriques, F., 7, 36, 39, 40, 147, 148, 247, 400, 401, 592, 593, 672–680

Author Index Epperson, J.A.M., 323 Erard, C., 338 Erdős, P., 630 Erlwanger, S.H., 108, 258, 282, 442–444 Errecalde, P., 338 Ershov, Y.L., 118, 265, 283 Escher, M.C., 174, 307 Eslinger, R., 312 Esser, A., 338 Establier, A., 303 Euclid, 68, 159, 250, 418, 550, 569, 679, 685 Even, R., 127 Evesham, H.A., 662 Exner, R.M., 338, 345 Eyles, J.W., 319 F Faddeev, L.D., 116, 263, 283 Fairthorne, R.A., 304 Falk de Losada, M., 125, 270, 283 Fang, J., 338, 344 Faragó, A., 303 Faragó, L., 308 Farkas, M., 312 Farmer, H., 338 Fasanelli, F., 107, 113 Fauvel, J.G., 107, 113, 121 Favard, J., 199 Fawcett, H.P., 305 Fayala, M., 338 Fehr, Henri., 4, 7–11, 13, 21–40, 47–50, 60, 124, 128, 139–147, 149–151, 153–156, 158–161, 175, 186, 242, 244–248, 279, 280, 285, 289, 299–302, 304, 422, 423, 446–448, 463, 483, 512, 525, 575, 604, 612, 663, 683, 687, 693 Fehr, Howard, 61, 63, 67, 100, 182–184, 193, 249, 306, 308, 338, 344, 483, 612 Feigl, G., 160 Feldman, L., 315 Félix, L., 54, 55, 66, 74, 338, 344, 395 Feluk, B., 313 Fenaroli, G., 691 Ferland, Y., 338 Ferlay, L., 338 Fermi, E., 679 Fernández, M., 322 Fernández Cara, E., 537 Ferrara, U., 338 Ferreira, J., 338 Ferrer, A., 323 Ferrieu, R., 413 Fessel, A., 338, 344

715 Feuerbach, L., 545 Fields, J.C., 245, 302 Fielker, D., 338 Fierro, M., 338 Figueiras, L., 322, 323 Finsterbusch, J., 300 Fiou, G., 338 Fischbein, E., 103, 107, 257, 338 Fischer, G., 318, 498 Fjelstad, P., 315, 319 Fladt, K., 59, 525 Flegg, G., 312 Fletcher, H.J., 344 Fletcher, T.J., 78, 103, 219, 258, 641 Fletcher, W.C., 16, 143 Fleury, F., 338 Fleury, P., 83, 213, 214 Flores, R.M., 323 Folsom, M., 338 Fonacier, J.C., 258 Fonteret, A., 339 Fontova, E.M., 314 Fonvieille, H., 339 Forde, J., 339 Forsyth, A.R., 143 Fort, J., 339 Fortin, J., 339 Fraenkel, A., 170 Francis, M., 339 Frank, B., 313 Fréchet, M., 511 Frege, G., 613 Freiman, V., 323 Freudenthal, H., ix, x, 1, 2, 43, 44, 54, 60–63, 68, 69, 73, 74, 76–90, 95–97, 100, 101, 103, 105, 107–109, 126, 170, 199, 202, 203, 205–214, 216, 218–228, 230, 232, 233, 238, 249, 251, 253, 255–257, 265, 271, 281, 282, 308, 313, 339, 356, 379, 396, 397, 418, 450–454, 481, 482, 588, 612, 635 Friant, J., 339 Fricke, R., 299 Fried, E., 339 Friis, J., 339 Fritsch, R., 319 Fritzlar, T., 319 Frostman, O., 63, 69, 70, 75, 79–83, 85–87, 100, 101, 170, 190, 193, 194, 201, 205–213, 216, 217, 220, 222–229, 231–233, 249, 251, 252, 255, 281, 282, 306, 457–459, 487, 512 Fuentes, M. Fueter, R., 246

716 Fujita, H., 116, 122, 123, 263, 282, 316, 494 Furinghetti, F., ix–xi, 1, 4, 43–90, 95–98, 100, 103, 104, 114, 129, 147, 164, 165, 169, 170, 172, 174, 176, 185, 196, 208, 211, 214, 216, 240–275, 279–286, 289, 297–324, 330, 335–345, 377, 385, 392, 397, 453, 485, 493, 588, 690–692 Furlani, G., 303 Fusaro, B.A., 315 Fyfe, D.J., 315 G Gabrovsek, L., 170 Gadner, M., 324 Gadon, P., 77 Galbraith, P.L., 124, 125, 127, 270, 283 Galdon, J.M., 315 Galo, J.R., 322 Gambetta, L.-M., 686 Garbe, J., 339 García Duarte, G. Jr., 322 García Monge, M., 426 García Pineda, M., 322 Garcia Pradillo, J., 339 García Valldecabres, M., 323 Garibaldi, A.C., 691 Gario, P., 400, 401, 675 Gattegno, C., 54, 220, 248 Gaulin, C., 100, 118, 119, 339 Gauthier, A., 170 Gauthier, R., 339 Gauthier, S., 49, 58, 409–411 Gavrilović, B., 699 Geary, A., 307 Gel’fand, I.M., 310 Geneva, L.M., 339 Gentile, G., 594 Gentile, M.L., 339 Gerard, M., 339 Gérardin, A., 301 Gerdes, P., 264 Gerretsen, J.C.H., 54, 173, 249, 305 Ghenciu, P.I., 323 Giacardi, L., ix–xi, 1, 2, 16, 43–90, 95–97, 100, 114, 129, 137–234, 236–238, 240–275, 285, 287–296, 325–334, 365, 385, 397, 399, 402, 403, 405, 453, 588, 591, 672, 674, 675, 677, 678, 692, 693 Gianati, C., 339 Giannoni, C., 339 Gibb, E.G., 339 Gibellato Valabrega, E., 339 Gil Clemente, E., 323

Author Index Gilbert, H., 339 Gilis, D., 339 Gilliespie, R.P., 339 Gillman, L., 204, 339 Gilman, B., 644 Gilsanz, M., 322 Giraud, G., 366 Gispert, H., 47, 66, 71, 73, 138, 269, 465, 516–519 Gjunter, N.M., 607 Glagoleva, E.G., 310 Glardon, M., 339 Glass, E., 339 Glaymann, M., 78, 79, 97, 101, 203, 211, 339 Gleason, A.M., 85, 97, 101, 255, 263, 314 Glocke, T., 312 Glorian, M.-J., 339 Glowinski, R., 537 Goby, J.-E., 658 Godeaux, L., 693 Gödel, K., 628, 706 Godfrey, C., 40, 143, 144, 300, 301 Goeringer, G., 339 Goffree, F., 339 Gol’dberg, Y.I., 310 Goldsmith, M., 219 Goldstein, C., 49, 58, 409 Goldziher, K., 301 Gonseth, F., 54, 77, 518, 519 González Manteiga, M.T., 322 González, M.O., 195 González Roldan, I., 339 Goos, M., 114 Gorner, F., 339 Gosiewski, W., 667 Gould, S.H., 356 Goullin, C., 339 Gouret, A., 339 Goursat, E., 466, 624 Grabiner, J.V., 316 Gradelet, M., 339 Gradelet, S., 339 Graf, J.H., 40 Graf, K.-D., 124, 319 Graf, U., 179 Gravel, H., 339 Graves, L.M., 304 Gréco, P., 585 Green, T., 109 Greenhill, A.G., 7–15, 21, 24, 30, 32–36, 39, 40, 140–147, 152, 158, 159, 236, 242, 244, 279, 299, 447, 461–464, 525, 604 Greig, M., 339 Grevholm, B., 120, 266

Author Index Griesel, H., 612 Griess, J., 655 Griffin, F.L., 305 Griffith, L.K., 318 Griffiths, H.B., 101, 254, 258, 311 Griffiths, Ph., 122, 125, 268, 271, 283 Grimm, K., 77 Grolier, Y., 339 Grootendorst, A.W., 339 Grossi Pillar, E., 339 Grötschel, M., 127, 273, 284 Guenoun, Y., 339 Guerrero-García, P., 322 Guggenbuhl, L., 306 Guilbaud, G.T., 584 Guillerault Astier, M., 339 Guirado-Granados, J.F., 322 Gundlach, B.H., 305 Gusev, V.A., 320 Gutierres Trobajo, R., 339 Gutzmer, A., 7, 9–11, 141, 142, 299, 300 Guy, R., 77 H Haas, R., 339 Hadamard, J., 22, 32, 37, 38, 40, 47, 50, 154, 155, 160, 244, 246, 279, 280, 451, 465–472, 575, 693 Haggmark, P.M., 339 Haimovici, A., 344 Hajós, G., 630 Halberstam, H., 108 Hale, W.T., 339 Halmos, P., 619 Hamdi, A., 339 Hamel, G., 38, 304 Hamilton, W., 685 Hammer, J., 315 Hanna, G., 79, 120, 266 Hansen, H.K., 306 Hansen, O.K., 339 Hansen, V., 343 Hansen, V.L., 320 Haralambie, J.P., 339, 344 Hardgrove, C., 339 Hardy, G.H., 463, 572 Harey, J.G, 344 Harkin, D.C., 308 Harley, R.M., 311 Härtig, K., 339, 632 Hartley, E.M., 339 Hartnett, W.E., 315 Harvey, J.G., 339

717 Hashagen, U., 500 Hashimoto, Y., 122 Hastad, M., 101 Hatzidakis, N., 301, 303 Haugazeau, D., 339 Hawking, S., 529 Hayman, M.R., 310 Hayter, R.J., 339 Hecke, E., 383 Heddens, J., 339 Heegaard, P., 38, 40, 160, 246, 280, 303, 474–479 Heffernan, M., 339 Heiede, T., 339 Heinke, C., 339 Helmholtz, H., 674 Henn, H.-W., 127 Hennequin, P.-L., 584 Henry, A., 339 Henry, M., 339 Hering, E., 674 Hermite, Ch., 410, 466, 699 Hernández, E., 324, 426 Hernández, S., 323 Hernando, B., 323 Herriot, S.T., 308 Herz, J.-C., 339 Hessenberg, G., 300 Heyting, A., 338, 451 Hight, D.V., 339 Hilbert, D., 299, 451, 500, 505, 518, 523, 524, 553, 613 Hill, M.J.M., 301 Hille, E., 165, 304 Hillel, J., 442 Hilton, P.J., 79, 108, 110, 258, 282, 310, 563 Hime, H.W.L., 147 Hirst, A., 116 Hirst, K., 116 Hirzebruch, F., 199 Hitler, A., 246 Hlavaty, J.H., 339 Hlawka, E., 203 Hobson, E.W., 40, 301 Hodge, W.V.D., 53, 61, 172, 249, 400 Hodgson, B.R., viii, 80, 90, 95–97, 99–104, 106, 108, 110, 113–115, 118, 119, 121–123, 125–128, 137, 259, 268–270, 273, 283–285, 287–289, 293, 316, 318, 486, 599 Hoene-Wroñski, J., 667 Hogbe-Nlend, H., 111, 260 Hohenberg, F., 306 Høigård, E., 578

718 Holcomb, G., 339 Hollings, C., 39, 138, 523, 526 Holme, A., 317 Holte, J.M., 320, 323 Holton, D., 114, 121, 268 Hood, R.T., 308 Hopf, H., 58, 59, 61, 69, 174, 181–184, 189–191, 198, 249, 281, 450, 451, 481–483 Hoüel, G.-J., 685 Hough, S.S., 40, 144 Houston, K., 124 Howe, J.A., 339 Howson, A.G., 44, 56, 69, 96, 101–104, 106, 108, 111, 113–117, 122, 131, 219, 241, 254, 260–264, 282, 325, 339, 344, 418, 419, 463, 487, 562 Hoyles, C., 126, 127, 271, 273, 284, 320 Hu, C.-L., 320 Huau, R., 339 Huertas, M.A., 323 Hug, C., 339, 344 Hughes, A.C., 317, 320 Hughes, D., 339 Hughes-Hallett, D., 317 Hukuhara, M., 170 Hurewicz, W., 451 Hurwitz, A., 623 Hussaini, M.Y., 529 Hutin, R., 339, 345 Hykšová, M., 422 Hynes, M., 339, 345 I Ibáñez, R., 324 Iglesias, M.T., 322 Iitaka, S., 494 Ilić, J.L., 313 Ilić-Dajović, M., 307, 308 Ilie, I., 345 Iliovici, G., 440 Inghilterra, C., 339 Isoda, M., 350 Ito, Y., 317, 321 Ivanov, P., 339 Iyanaga, S., 85, 101, 105, 106, 108, 109, 170, 204, 228, 230, 233, 234, 254, 256–258, 281, 282, 339, 485–488, 493–495 J Jackson, A., 393, 394 Jackson, C.S., 14, 16, 146

Author Index Jacobsen, E.C., 63, 80, 101, 120, 121, 259, 261 Jacquemier, Ph., 339 Jacquemier, Y., 339 Jahnke, H.N., 320 Jahnke, T., 314 James, R.D., 312 Janiszewski, Z., 624 Jaquet, F., 54 Jaramillo Quiceno, D., 323 Jaruzelski, W.W., 259 Jech, T., 511 Jeffery, F., 490 Jeffery, G.B., 339, 345 Jeffery, R.L., 48, 50, 161, 247–249, 251, 280, 490–491 Jeśmanowicz, L., 339 Jessen, B., 47, 48, 76, 201–202, 362 Jevremović, V.B., 339 Jin, G., 321 Johansson, I., 474, 476, 478 Jojon, C., 339 Joksimovič, Z., 339 Jones, Ph.S., 257 Jordan, C., 158, 467 Jørgensen, V.T., 474 Juan A.A., 323 Julia, G., 393, 451 Juve, Y., 203 K Kahane, J.-P., vii, 111–118, 120, 122, 260, 262–265, 282, 283, 316 Kaijser, S., 457, 635 Kaino, L.M., 321 Kaiser, G., 124, 320 Kajikawa, Y., 315 Kalmár, L., 629 Kamke, E., 47, 170, 306 Kampé de Feriet, J., 413 Kanning, E., 339 Kantor, J.M., 314 Kantowski, M.G., 258, 314 Kapelou, E., 322 Kapur, J.N., 84, 216 Karamata, J., 63, 199, 200, 209, 250, 251, 702 Karp, A., 13, 15, 16, 39, 504, 604, 622 Kašanin, R., 702 Katsap, A., 321, 323 Katsifli, D., 315 Katz, P., 339 Kaufman, B.A., 339, 345, 613

Author Index Kawada, Y., 105, 108, 109, 256, 258, 282, 486–488, 493–495 Kefuss, J., 340 Keitel, C., 117, 120, 264 Kekkonen, U., 569 Kelly, P.J., 263 Kemeny, J.G., 68, 192, 309 Kendal, M., 124 Kenderov, P.S., 119, 120, 125, 270, 283, 321 Kennedy, J., 115 Ketterer, H., 340 Khilchenko, L.Y, 319 Kidd, S., 323 Kieffer, L., 204, 340, 345 Kieran, C., 118 Kies, J.D., 340, 345 Kikoin, I.K., 598 Kilpatrick, J., 73, 96, 101, 109, 112, 114, 116, 118, 120, 121, 126, 129, 263, 265, 266, 274, 282, 283, 317, 372, 540, 548, 552, 618, 644 Kim, Y., 316 Kindel, M., 340 Kishore, M., 315 Kitzwalter, T., 622, 623, 667, 668 Klaasse, A., 340 Klahn, S., 340 Klamkin, M.S., 345 Klein, F., vii, viii, 3, 7–13, 15, 17, 18, 20, 21, 24–36, 38–40, 51, 52, 55, 87, 88, 126, 130, 138, 140–143, 145–149, 151, 152, 156, 186, 236, 242, 244, 245, 271, 279, 284, 299, 384, 385, 401–403, 405, 406, 422, 447, 448, 462, 463, 475, 478, 497–505, 525, 526, 593, 604, 650–652, 669, 675–678, 694, 705 Klein, M., 323 Kline, M., 105, 549, 550, 554 Knaster, B., 548, 624 Kneser, A., 481 Knoche, N., 612 Kobeisse, H., 340 Kochañski, A., 667 Koenig, G., 321 Koenigs/Königs, G., 153, 154, 302 Kokalidis, S., 322 Koksma, J.F., 59 Koldijk, A., 340 Kolmogorov/Kolmogoroff, A.N., 72, 199, 356, 557, 598 Komatsu, H., 264 König, F., 500 Kooi, O., 340 Kortesi, P., 319

719 Kovacevic, B., 340 Kovacs, G., 340 Kovda, V.A., 195, 196 Kozlic, F., 340 Kramer, M.S., 305 Krazer, A., 299 Krstic, S., 340 Krygowska/Krygovska, A.Z., 72, 73, 75–78, 84, 101, 104, 201, 215, 252, 254, 257, 258, 261, 311, 314, 340, 632 Kudrjavcev, L.D., 105, 108, 256, 282 Kuijk, W., 258 Kulnazarova, A., 167 Kumaresan, S., 127, 273, 284 Kunle, H., 77, 95, 100, 106, 613 Kuntzmann, J., 340 Kuragina, E., 319 Kuratowski, K., 64, 198, 624 Kureno, D., 310 Kureno, N.I., 310 Kurepa, Ð./G., 48–50, 54, 56, 59, 61–63, 69, 70, 85, 161, 169, 179, 182, 190, 193, 194, 247–249, 280, 281, 305, 307, 340, 345, 483, 510–514 Kürschák, J., 630 Kutzler, B., 317 Kyed, T., 340 Kyo, R.J., 340 L Laborde, C., 121, 266, 283 Lacko, M., 317 Laclavère, G., 80, 210 Lacondemine, P., 340 Lacourt, M.T., 340 Ladoux, F., 340 Lagrange, J.-B., 127 Lahoz-Beltra, R., 322 Laisant, Ch.-A., xi, 4, 124, 139, 347, 447, 656, 669, 683–688 Laitone, E.V., 310 Lakoma, E., 622, 666 Lamb, P.J., 340 Landau, E., 383 Lange-Nielsen, Fr., 304 Langford, W.J., 573 Lariccia, G., 340 Laub, J., 340 Lauginie, P., 114, 263 Laugwitz, D., 613 Laurain, H., 340 Laux, J., 340 Lavabre, S., 340

720 Lavigne, G., 340 Lavrent’ev/Lavrentieff, M.A., 199, 355, 356 Lawrence, S., 698 Lax, A., 645 Lax, P.D., 537, 557 Laymand, R., 340 Lazzeri, G., 679 Le Chatelier, H., 301 Le Guern, G., 340 Le Lionnais, F., 64 Le Minous, C., 340 Le Pezron, Y., 340 Lebesgue, H., 433, 438, 469, 470 Lebon, E., 669 Lebrija Trejos, A., 323 Leder, G., 121, 122, 266, 268, 283 Lee, P.-Y., 263, 265, 282, 283, 321 Leendert, B., 258 Lefebvre, H., 340 Lefebvre, J., 340 Lefebvre, P., 340 Legris, R., 340 Lehmann, D., 340 Lehmann, H.-L., 340 Lehto, O., 31, 37, 44, 46, 50–53, 70, 96, 100, 105, 106, 110, 111, 115–117, 150, 159, 165, 168, 206, 213, 232, 253, 258–261, 263, 282, 283, 312, 463, 483, 488 Leibniz, G.W., 640, 667 Lelong-Ferrand, J., 105, 256 Leman, G.M.J.G., 144 Lemire, L., 340 Lemoine, É., 685, 688 Leray, J., 254, 311, 393, 396 Lerman, S., 108, 122 Leung, F.K.-S., 124, 125, 127, 270, 273, 283, 284, 320 Levi, H., 310 Leviatan, T., 319 Levi-Civita, T., 674 Lévy, P., 467 Lewis, D.J., 319 Lewis, E., 340 Lewis, R., 340, 345 L’Hospitalier, Y., 340 Li, M., 321 Libois, P., 512 Lichnerowicz, A., 44, 54, 66, 71–76, 85, 88–90, 101, 199, 200, 203, 218, 232, 233, 251, 255, 281, 282, 516–519, 585 Lichtenberg, J., 77 Liddell, H.G., 159 Lie, S., 478

Author Index Lietzmann, W., 6–12, 14, 20, 37–39, 50, 51, 138, 160, 246, 280, 503, 522–526 Lievens, E.M., 340 Lighthill, M.J., 84–87, 101–105, 216–218, 221–229, 231–234, 254–256, 281, 282, 487, 528–531, 546, 612 Limoge, M., 340 Lim-Teo, S.K., 108, 495 Lindelöf, E., 566 Lindsay, R., 340 Linfoot, J.J., 315 Lingua, P., 340 Lions, J.-L., 105, 108, 110, 118, 256, 258, 265, 282, 283, 311, 533–538 Liouville, A.-M., 340 Lipschitz, R., 498, 602 Lipson, S., 340 Littlewood, J.E., 463 Lluis Riera, E., 116, 263, 282 Loase, J., 315 Lombardo Radice, L., 85, 101, 255 Lopata, G., 340 Lopes, A.A., 340 Lopez-Real, F.J., 124 Lorey, W., 498 Loria, G.B., 37–39, 145, 147–149, 246, 247, 299, 304, 406, 669, 679, 690–695 Loudot, J., 340 Louis, M., 210 Lovász, L., 127, 273, 284, 630 Love, A.E.H, 301, 461–463 Lovett, C.J., 340 Löwner, K., 450 Lubański, M., 314 Lubbe, A., 340 Lubuela, F., 340 Luciano, E., 147, 149 Lukankin, G.L., 258 Luna, E., 118, 265, 283 Lyapunov, A.M., 607 Lyness, R.C., 84, 216, 310, 340 Lysenko, T.D., 355 M Mach, E., 674 Mac Lane, S., 49, 50, 172, 183, 185, 248, 280, 306, 540–542, 550, 619 MacNab, D., 340 MacNarb, W., 340 Mafrica, D., 340 Mafrica Micotti, C., 340 Magarian, E., 340 Magenes, E., 537

Author Index Magers, D., 340 Malecamp, E., 340 Malgrange, B., 108 Malik, M.A., 313 Malliavin, P., 311 Malme(?)g, A., 340 Malpangotto, M., 421, 510, 560 Mammana, C., 120 Mancini Proia, L., 340 Mandelbrojt, S., 393 Manic, M., 340 Manolov, S., 340 Manotte, J., 340 Marasigan, J.A., 258 Marchuk, G., 536 Marcolongo, R., 303 Margulis, A., 310 Marinkovic, B., 340 Marjanovič, M.M., 510 Markov, A., 607 Markovič, S., 702 Markusevic, 340 Marle, C.-M., 518 Maroni, A., 303 Marotte, F., 639 Marques, R.M., 340 Martin, F., 340 Martin, W.T., 340 Martínez Calvo, M.C., 322 Mar(?)us, A., 340 Maschler, M., 309 Masferrer, C., 323 Maslova, G.G., 310, 340 Mason, J., 419, 443 Mateev, A., 340 Mather, J.N., 558 Matricon, J., 586 Matskin, M.S., 310 Matsumoto, S., 317 Mattedi Dias, A.L., 493 Matthews, G., 71, 219, 310, 312, 340, 345 Matthys, J.-C., 340 Maublanc de Chiseuil, P., 340 Mauduit, C., 317 Maurières, M., 340 Maxwell, E.A., 50, 54, 59, 63, 84, 85, 87, 101, 203, 216, 228, 231–232, 248, 249, 254, 280–282, 340, 544–546 Maxwell, J.C., 669 May, K.O., 305, 307, 309 Mayor, R., 340 Mazurkiewicz, S., 624 Maz’ya, V., 466 Mazzi, M.T., 340

721 McCarthy, M.D., 204 McCullough, M., 164 McKay, M., 339 McKean, H.P., 558 McNicol, S., 340 McVoy, K., 305 Meghannem, M., 340 Mehrtens, H., 258 Meier, F., 340 Mellin, E., 340 Melter, R.A., 313 Mendès-France, P., 412, 517 Menghini, M., x, 2, 50, 80, 82, 95–131, 212, 215, 254, 273, 385, 453, 588 Mercier, J., 340 Merigot, M., 340 Merlan, J., 341 Metenier, J., 341 Meyer, A., 422 Meyer, W.F., 300, 669 Meylan, R., 341 Michael, M., 315 Michalopoulos, K.P., 170 Michelot, G., 341 Michels, M., 341 Michtchenko, T.M., 321 Mijnlieff, A., 341 Miles, E.P. Jr., 313, 315 Milhaud, G., 654 Miloskovic, D., 341 Mimura, Y., 85, 101, 255 Mineur, P., 160 Mison, G., 341 Mitrinović, D., 702 Mittag-Leffler, G., 669 Miyazaki, K., 341 Mizar, K., 640 Möbius, A.F., 545 Mogensen, P., 362, 363 Mohr, T., 341 Mohrmann, H., 147, 148 Moise, E.E., 72, 77, 97, 199, 203, 204, 218, 251, 253, 281, 548–550 Mollerup, J., 361, 362 Momcilo, L., 341 Monod, G., 641 Montel, P., 393, 511 Montgomery, D., 105, 232, 256, 282, 552–554 Moore, C., 341 Moore, E.H., 540, 619 Moore, R.L., 372, 548 Morfin, M., 341 Mori, M., 319 Moris, M., 341

722 Morley, M., 319 Morse, M., 191, 193 Morvan, I., 341 Moser, J., 111, 260, 263, 282, 556–558 Moshnikova, J.M., 319 Motte, M., 341 Motto, M., 341 Mountebank, W., 641 Movshovitz-Hadar, N., 320 Mulder, S., 341 Mullins, C.W., 318 Mumford, D., 121, 266, 283 Munford, D., 341 Munisany Doraisany, S., 317 Munkholm, E.S., 474–476, 478 Munkholm, H.J., 474–476, 478 Murakami, H., 316, 317 Myx, A., 341 N Nachbin, L., 203 Nakashima, K., 317 Namikawa, Y., 122, 268, 283 Nano, A., 341 Narasimhan, R., 80, 209, 434 Narion, Y., 340 Nastasi, P., 405 Natanson, W., 667 Nebres, B.F., 111, 258, 260, 282, 322, 495 Nemetz, T., 263 Neovius, E.E., 565 Neovius, O., 565 Neshkov, K.I., 310 Ness, W., 307, 309 Neuberg, J., 14, 40 Neubert, E., 341 Neukomm, G., 630 Neumann, B.H., 105, 108, 109, 256, 258, 282, 341, 488, 560–563 Nevanlinna, F., 565, 566 Nevanlinna, R., 63, 64, 69, 189, 190, 198, 249, 251, 281, 565–569 Neville, E.H., 50, 51, 160, 246, 280, 571–575 Newcomb, S., 644, 669 Newman, M.F., 111, 260, 282, 562, 563 Newton, I., 463, 679 Niang, S., 204 Nichols, E., 341, 345 Nicholson, J., 464 Nicol, C., 108 Nicolas, A., 341 Nicollerat, M., 341 Nielsen, J., 474

Author Index Nietzsch, J., 319 Nijdam, A.D., 101 Nikolic, J., 341 Nikolic, M., 341, 345 Nimier, J., 516 Nirenberg, L., 558 Nishimura, K., 317 Niss, M., 90, 96, 97, 100, 104, 110, 113, 116–122, 125–128, 259, 263, 265, 266, 268, 282, 283, 318, 418 Noether, E., 482, 540 Noureddine, 341 Novak, J., 79, 85, 101, 203, 218, 219, 255, 341 Nunes, G., 324 Nunez Berro Maria, 341 Núñez del Prado, J.A., 322 Nunn, T.P., 301, 571, 574 O Oberman, J., 341 O’Brien, S., 341 O’Brien, T., 345 O’Donnell, J., 341, 345 O’Halloran, P., 119, 563 Oleary, S., 341 Olech, C., 260, 313 O’Mahonq, 341 Opial, Z., 101 Orloff, C., 309 Orlov, K., 341, 345 Ornstein, A., 341 Ortiz, E.L., 683 Osgood, W.F., 13, 15, 40, 604 Osorio Dos Anjos, A., 341 Osouf, R., 341 Osterwalder, K., 322, 558 Otte, M., 77, 100, 108, 203, 255, 419, 614 Oulevay, B., 341 Overn, O.E., 305 Owden, L., 312 Ozaki, Y., 319 P Pacheco Esteban J.P., 319 Paditz, L., 319 Padoa, A., 693 Painlevé, P., 699 Palazzo, E., 306 Palis, J. Jr., 118, 121, 122, 265, 266, 268, 283 Palmer, B.A.H., 341 Palmer, R., 341 Pampallona, U., 341

Author Index Pann, J., 341 Papazian, M.-J., 341 Papy, F., 71, 98, 341 Papy, G., 71, 85, 101, 203, 218, 309, 418, 641 Paquette, G., 341 Pareja-Heredia, D., 324 Parish, G.L.D., 341 Parshall, K., 46 Pascal, E., 669 Pascual Xufre, G., 341 Peano, G., 669 Pedley, T.J., 530 Pejović, T., 702 Pekonen, O., 565 Pélissier, G., 341 Penavin, V., 341 Pendlebury, C., 140 Penfold Alec, D., 341 Pennaneach, F., 341 Perdon, C., 341 Perea, C., 323 Pérès, J., 411 Pérez, A., 121 Pérez de Vargas Luque, A., 322 Perkus, R., 309 Perol, C., 341 Perole, Y., 341 Perret, J., 341 Perry, J., 6, 11, 14, 143 Persico, E., 679 Perucca, E., 368 Pervine, Y., 341 Pescarini, A., 79, 220, 341 Peters, N., 341 Petersen, J., 475 Petersen, P., 451 Petersen, R., 362 Petkantschin, B., 341 Petrie, P.A., 170 Petrovič/Petrovitch, M., 39, 40, 247, 698–702 Petrovsky, I.G., 252, 309 Piaget, J., 54, 58, 61, 66, 71, 101, 103, 104, 248, 389, 390, 413, 519 Picard, Ch.É., 31, 45, 153, 158, 245, 438, 451, 466, 669, 699 Picard, Ph.., 341 Pichaud, J., 584, 588 Pickert, G., 341, 612 Piene, K.W.K., 56, 59, 66, 68, 169, 192, 204, 249, 281, 305, 307, 309, 577–579 Pietzker, F., 141, 142 Pihl, M., 170 Pikaart, L., 341 Pincherle, S., 45, 149, 245

723 Pinchinat, J., 341 Pinchinat, R., 341 Pisot, Ch., 75, 77, 201, 252, 311 Pissavin, O., 341 Pizzo, A., 319 Platzker, O., 341 Pleijel, Å.V.C., 204, 341 Plimpton, G.A., 603 Plomp, T., 341 Plücker, J., 498, 499 Pluvinage, F., 114, 262 Pogorelov, A.V., 313 Poincaré, H., 293, 395, 411, 433, 467, 469, 471, 475, 500, 583, 686, 688 Poivey, M., 341 Pollak, H.O., 78, 85, 101, 108, 109, 111, 120, 215, 220, 230, 254, 260, 264, 282, 311, 318, 341 Polo-Blanco, I., 323 Pólya, G., 101, 103, 109, 110, 117, 304, 428, 554, 632 Pons Ballarin, R., 341 Ponteville, C., 319 Pontryagin, L., 536, 609 Pool, K., 341 Popov, K., 341 Porcel, N., 341 Portron, 341 Poske, F., 141, 142 Pototsky, M.V., 310 Pouget, C., 341 Pourkazemi, M.H., 317, 319, 321 Poussin, N., 341 Powell, M.J., 341 Pradine, A.-M., 341 Prave, M.-M., 341 Predescu, V., 341 Prevot, Y.J., 341 Prinits, O.I., 310 Przibram, H., 304 Pukades Duran, R., 341 Pustilnik, S.W., 315 Q Quadling, D.A., 117, 341, 345, 545, 546 Quardt, D., 319 Queysanne, M., 585 Quimpo, N.F., 258 R Ra, A.N., 310 Rabinowitz, P.H., 558

724 Rade, L., 341, 345 Radon, J., 706 Radovic, M., 341 Radtka, C., 413, 414 Rageul, L., 341 Raghunathan, M.S., 272 Rai, R., 342 Raizen, S., 342 Rakocevic, S., 342 Rakover, B.D., 316 Ralston, A., 114, 262, 321 Ramanujan, S., 572 Ramírez, C., 323 Ramírez-Uclés, R., 322 Ramskov, K., 360, 362, 476 Rapley, B., 342 Ravizza, P., 342 Ray, A.K., 314 Raziuddin Siddiqi, M., 204 Rehmann, U., 318 Reidel, D., 81, 208, 211 Reis, M.R., 315 Reis, P., 669 Rellich, F., 556 Remmert, R., 394 Rennie, B.C., 342 Rényi, A., 77, 631 Revuz, A., 76–79, 84, 87, 97, 100, 101, 203, 204, 215, 218, 220, 228, 253, 258, 281, 342, 581–588, 632, 642 Reyes, J.A., 323 Rice, A., 85, 528, 544, 571 Richard, F., 342 Richardson, L.F., 535 Richardson, M., 305 Rico, L., 121 Riemann, B., 467, 500, 669 Riesz, M., 457 Riley, P.E., 315 Rimer, D., 311 Rindung, O., 192 Ríos, M., 322 Riou, A., 342 Rivière Lanne, P., 342 Robert, A., 318, 581 Robert, F., 342 Robinet, J., 586 Robitaille, D., 118 Roczen, M., 323 Roderick, H., 194, 197 Roditi, É., 342 Rodrigues, A., 345 Rodríguez Del Río, R., 319 Rodríguez Trueba, M.I., 319

Author Index Roero, C.S., 147, 149 Rogalski, J., 586 Rogers, L.F., 313 Rogerson, A., 258 Roherberg, A., 477 Rollett, A.P., 170 Roman, T., 310 Romberg, H., 565 Romberg, M., 565 Romberg, T.A., 320, 344 Ronveaux, A., 310, 342, 345 Room, T.G., 342 Ros, R.M., 324 Rosenbloom, P.C., 342 Ross, W.D., 159 Rossing, E., 305 Rosskopf, M.F., 306 Rouget, J., 342 Roumanet, A., 342 Roumieu, C., 342 Rouquairol, M., 342 Rouquet, G., 342 Rousseau, R., 342 Roussel, Y., 342 Roveyaz, G., 342 Rowe, C.H., 377 Rowe, D.E., 499 Rowley, C., 313 Roy, M., 342 Rucker, I.P., 342 Rudio, F., 298, 687 Rueff, M., 72, 197, 204, 342 Runge, C., 301, 500 Rüping, H., 306 Rusiecki, A.M., 624 Russell, B., 572 S Sabaud, R., 342 Sabbatiello, E.E., 342 Sacco, M.P., 342 Safuanov, I.S., 321 Sahab, S., 319 Saint-Saëns, C., 688 Sakellariou, N., 303 Salas Palenzuela, I., 342 Salvano, J., 342 Samana, G., 342 Samson, J., 342 Sanatani, S., 315 Sandgren, L., 170 Sano, K., 319 Sansone, G., 512

Author Index Santonja-Gómez, F.J., 323 Santos-Palomo, Á., 322 Sanz-Solé, M., 321 Sarazin, M.L., 342 Sarrazin, E., 342 Sarton, G., 693 Satake, I., 316 Satoca, G.M., 342 Sauvy, S., 342 Savalle, A., 342 Sawada, T., 122 Sawyer, W.W., 163, 309 Saxer, W., 303 Scarpis, U., 19 Schaack, F., 342 Schieldrop, E.B., 304 Schiffer, M., 554 Schmidt, E., 450, 481 Schmidt, W., 321 Schneider, J., 317 Schotten, H.G., 141, 142, 300 Schouten, J.A., 248 Schramm, R., 342 Schubring, G., 3–40, 47, 51, 57, 58, 66–68, 73, 76, 95, 100, 106, 138, 146, 186, 245, 269, 382, 401, 403, 417, 446, 448, 450, 497–499, 501–504, 522, 525, 588, 601, 604, 611, 613, 614, 650, 694, 705 Schult, V., 342 Schur, I., 450, 481, 560 Schuster, M., 524 Schwamberger, K., 342 Schwartz, L., 394, 412, 433, 469, 530, 533, 534 Schwartzman, P.A., 342 Scimone, A., 405 Scorza, B.G., 38, 40, 160, 246, 280, 591–595, 678 Scorza-Dragoni, G., 592 Scoth, R., 149 Scott, H., 40 Scott, R., 159 Scroggie, G., 342 Sebastiao e Silvia, J., 204 Seeger, F., 419 Segre, C., 147, 148, 399, 402, 592, 669, 673, 691 Seiler, R., 322 Semadeni, Z., 105, 108, 111, 256, 258, 260, 282, 316, 342 Semenov, A.L., 127, 273, 284 Semushkin, A.D., 310 Sengenhorst, P., 306 Seppälä, M., 322

725 Serrat, C., 323 Serre, J.-P., 311, 393, 394 Servais, W., 54, 61, 76–78, 84, 101, 182, 192, 215, 253, 342, 343, 418, 483, 632 Severi, F., 674 Sfard, A., 121, 126, 274 Shahvarani-Semnani, A., 317, 319, 321 Shakespeare, W., 158 Shanks, M.E., 309 Shapira, M., 342 Shapiro, B., 345 Shaposhnikova, T., 466–468 Sharygin, I.F., 122, 268, 283, 597–599 Shelley, N., 115 Shikhanovich, Y.A., 311 Shuard, H.B., 342 Shulman, L.S., 342 Shumaker, M., 258 Shvartsburd, S.I., 311 Shvartsman, L., 323 Sibagaki, W., 342 Sibille, J.L., 342 Sidoli, N., 79 Siegel, A., 321 Siegel, C.L., 556, 557 Siegmund-Schultze, R., 577 Sierpinska, A., 118, 120, 121, 265, 266, 283, 314, 317, 419 Signetto, F., 342 Signoret, C., 129 Sime, M., 103 Simola, I., 170 Simon, M., 6–8, 300 Simons, F.H., 263 Sinisa, J., 342 Sitia, C., 342 Sittignani, M.G., 303 Siu, M.K., 317, 354, 597, 606 Slavka, M., 342 Slomska, A., 314 Smid, H.J., 76, 442, 490 Smirnov, V.I., 607 Smith, 342 Smith, D.A., 318 Smith, D.E., x, 4, 6–13, 15–17, 21, 22, 24–26, 28–40, 90, 127, 138–151, 153, 155–160, 236, 244, 246, 273, 279, 280, 284, 300, 301, 304, 326, 440, 503, 601–604, 639, 688, 693 Smith, E.M.R., 309, 313 Smith, P.A., 304 Smithies, F., 163 Smolec, I., 77, 342, 345 Soares, R., 313

726 Sobel, M., 109 Sobolev, S.I., 319 Sobolev, S.L., 70, 77, 85, 97, 101, 203, 204, 218, 220, 230, 253, 254, 281, 282, 311, 342, 606–609 Solberg, N., 304 Soler, A.J., 342 Solovay, R.M., 511 Somaglia, A.M., 691 Soos, G., 342 Soós, P., 630 Sooy, J.M., 342 Sørensen, H.K., 359, 474 Sorger, P., 342 Sossinsky, A., 322 Šourek, A., 40, 300 Sousa Ventura, M.J., 342 Stacey, K., 124, 270 Stäckel, P., 20, 141, 142, 147, 148, 299 Steegmann, C., 323 Steen, C.M., 342 Steffes, P., 342 Steinbring, H., 419 Steiner, H.G., 75, 77–79, 95, 100, 101, 104–106, 201, 203, 218, 220, 253, 255–258, 261, 282, 314, 318, 342, 343, 345, 384, 419, 488, 611–615 Steiner, M., 342 Steinitz, E., 481 Stenström, V., 308 Stéphanos, C., 7, 40 Stephens, J., 219, 342 Stergiou, G., 310 Stewart, I., 321 Stewart, S.A., 342 Stigler, J., 318 Stillwell, J., 317 Stolyar, A.A., 310 Stone, M.H., 44, 46, 47, 49, 50, 52–55, 57, 59, 62–71, 73, 88, 89, 162–174, 179, 187–198, 237, 248–251, 280, 281, 287, 356, 379, 385, 412, 413, 512, 618–620 Storer, W.O., 307, 308, 342 Størmer, C., 246 Stosic, R., 342 Straszewicz, S., 68, 72, 77, 192, 199, 204, 251, 281, 309, 342, 622–626 Streefland, L., 69 Stroth, G., 319 Studzinsky, H., 342 Sturgess, D.A., 342 Sturm, R., 481 Sugiyama, K.-I., 122 Sulzer, B., 342

Author Index Suppantschitsch, R., 40, 300 Suppes, P., 309 Surányi, J., 85, 101, 105, 204, 228, 230, 254, 256, 257, 281, 342, 343, 628–632 Surányi, L., 628 Suslin, M., 511 Süss, W., 170, 383 Suter, A.K., 342 Suydam, M., 109 Swain, R.L., 305 Syer, E.H.C.H., 170 Szabó, L., 628 Szendrei, J., 116, 342 T Tagg, D., 342 Tailleu, G., 342 Taimina, D., 317 Tait, P.G., 685 Takagi, T., 485, 493 Takizawa, T., 317 Tall, D., 318 Tanner, R.C.H., 308 Tannery, J., 143, 410, 466, 699 Tarski, A., 388 Tassy, J., 342 Taussky-Todd, O., 706 Taylor, C.A., 263 Taylor, P.J., 127 Tazzioli, R., 45 Tellechea Armenta, E., 324 Temam, R.M., 536, 537 Temple, G., 530 Tennenbaum, S., 511 Terracini, A., 691, 692 Thalberg, O.M., 478 Therond, C., 342 Thiele, T.N., 475 Thieme, H., 300 Thille, P.M., 342 Thoft-Christensen, P., 342 Thom, R., 104, 393 Thompson, T.M., 315 Thullen, P., 383 Thwaites, B., 77, 79, 84, 97–100, 103, 203, 204, 216, 218, 219, 253, 281, 309, 342, 530, 531 Tikhomirov, V., 599 Timerding, H., 20 Tisseyre F., 322 Tobies, R., 28 Todd, J.A., 307 Todorčević, S., 510

Author Index Todorovic, L., 342 Togliatti, E.G., 690, 691 Tondeur, F., 322 Tong, Z., 321 Torrent, J.A., 323 Torres Hernández, R., 324 Tosi, A., 342 Touré, S., 258 Tournès, D., 662 Touyarot, M.A., 343 Towers, D.A., 315 Tradif, M., 342 Trejos Alvarado, M., 323 Treutlein, P., 141, 142 Tricomi, F.G., 308, 366 Trier, V.V., 477 Trifunovic, D., 699–701 Trost, E., 170 Tucker, A.W., 307, 308 Tumura, Y., 343 Turner, N.D., 310, 311, 313, 314 Tzitzéica, G., 40, 160 U Urrea-Henao, J., 324 Upton, C., 11 V Vailati, G., 300, 401, 402, 525, 593, 678, 694 Väisälä, K., 170 Valenza, A., 343 Van Arsdel, J., 343 Van Dantzig, D., 61, 182, 483 Van Den Briel, J.K., 343 Van Der Krogt, B., 343 Van Dormolen, J., 343 Van Lint, J.H., 114, 116, 118, 262, 265, 283, 316, 635–637 Van Maanen, J., 121 Van Speybroeck, J.O., 343, 345 Van Yzeren, J., 310 Vance, I.E., 343 Vandenberghe, R., 343 Vanhamme, J., 343 Vanpaemel, G., 65, 68 Varenne, G., 343 Varga, T., 116, 117, 343, 631 Varona, J.L., 321 Vasco, C.E., 121, 256, 283 Vasquez-Martinez, C.R., 321 Vassiliev, V.A., 271 Vaughan, H.E., 307, 308

727 Veblen, O., 45, 247, 482 Veit, B., 343 Ventadoux, M., 343 Verdera, J., 321 Verhoef, W., 343 Veronese, G., 299, 399 Verstappen, P., 419 Verset, A., 343 Verset, C., 343 Vervoot, G., 343 Vetulani, Z., 314 Vidal, C., 322 Villa, M., 170, 306 Villani, V., 120, 267, 318 Villat, H., 302 Villella, J., 319 Villeneuve, J.A., 343 Vincent, Jill, 270 Vincent, John, 270 Visscher, M.B., 164 Vitali, L., 343 Vivanti, G., 679 Vogeli, B., 343, 345, 622 Voisin, P., 343 Volk, W., 323 Volterra, V., 35, 158, 302, 672, 680 Von Linde, C., 500 Von Mises, R., 450 Von Neumann, J., 450, 535, 553, 557 Von Ungern-Sternberg, J., 28, 504 Von Wutheanau, S., 323 Vonk, G.A., 343 Vopni, S., 343, 345 Vredenduin, P.G.J., 204 Vuibert, H., 655 Vuilleumier, F., 343 Vukièević, P., 699 Vysin, J., 343 Vyvyan, R., 343 W Waits, B.K., 315 Wall, W.D., 67 Wallbank, S.A., 343 Walter, M.I., 343 Walusinski, G., 63, 66, 69, 190, 193, 249, 281, 584, 639–642 Wang, G.-H., 162 Wang, J.-P., 122, 268 Wansink, J.H., 170, 343 Wäsche, H., 212, 343 Watanabe, R., 343 Watson, F.R., 343

728 Watson, J.R., 343 Weber, H., 669 Weidig, I., 343 Weil, A., 105, 392, 393, 485, 511, 566, 619 Wells, P.J., 343 Wendt, G., 63, 162, 164 Wertheimer, M., 56 Westerthof, B.J., 343 Weyl, H., 540 Wheeler, D.H., 118, 119, 258, 313 White, P., 343 Whitney, H., 104, 108, 110, 111, 114, 258, 260, 282, 343, 442, 444, 644–646 Whitney, W.D., 644 Whittington, B., 343 Wigand, K., 306, 309 Wiggins, K.L., 315 Wilhelm, A., 343 Williams, E.M., 219, 343 Wilson, B., 114, 262 Wilson, J., 343 Wilson, J.H., 315 Wiredu, E., 318 Wirszup, I., 220, 343 Wirtinger, W., 39, 40, 160, 247, 705–706 Wittenberg, A., 73, 74, 252, 550 Wittmann, E., 343, 345, 419 Wolff, K.G., 306, 343 Wolff, L., 343 Wooten, W., 549 Wootten, A., 343 Wu, H.-H., 322 Wundt, W., 674 Wussing, H., 258, 313 Wydeveld, E.J., 343 X Xambó, S., 322 Xiao, S., 320 Xu, Z.-L., 319

Author Index Y Yaglom, I.M., 311 Yamabe, H., 553 Yamaguti, M., 114, 262 Yamashita, H., 317 Yang, D., 321 Yanosko, B.J., 315 Yaschenko, I., 320 Yaseen, A.A.K., 314 Ydesen, C., 167 Yeshurun, S., 343, 345 Yoshida, K., 317 Young, J.W.A., 13, 40, 145, 604 Z Zaleski, J., 623 Zaller, M.A., 343 Zammit, C., 343 Zammit, J., 343 Zaouli, M., 343 Zappa, G., 592 Zariski, O., 304 Zawdowski, W., 313 Zehnder, E., 557 Zehren, C., 584 Zermelo, E., 623 Zervos, M., 303 Zervos, P., 302 Zeuthen, H.G., 475, 476 Zhang, D., 121, 266, 283 Ziegenbalg, J., 321 Zimmermann, B., 319 Zimorya, D., 343 Zippin, L., 553 Zitarelli, D.E., 620 Zoll, E., 343, 344 Zund, J.D., 645, 646 Zúñiga Becerra, B., 324 Zweng, M.J., 109, 343 Zygmund, A., 619 Zykov, A.A., 310

Subject Index

A All-Russian Conference on Mathematical Education in Dubna (2000), 124, 269 American Association for the Advancement of Science (AAAS), 373, 613 American Mathematical Society (AMS), 46, 373, 541, 549, 553, 557, 607, 619, 661 Apartheid policy, 115, 262 Applied mathematics, 5, 20, 85, 128, 130, 271, 318, 402, 461, 500–502, 513, 528–531, 542, 554 Assessment in Mathematics Education, 117, 261, 265, 440 Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP/ APM), 66, 71, 395, 439, 519, 583–586, 639, 640, 642, 656 Association Française pour l’Avancement des Sciences, 661, 662, 684, 686, 687, 691 Association Générale des Membres de la Presse de l’Enseignement, 655 Associazione Mathesis/Mathesis, 366, 369, 401, 402, 405, 593, 679, 680, 693 Aufruf an die Kulturwelt, 27, 30, 403, 504 B Basic Components of Mathematics Education (BACOMET), 108, 419 Bollettino della Mathesis, 405 Breslauer Unterrichtskommission, 525 British Association for the Advancement of Science, 462, 573

Bulletin de l'Association des Professeurs de Mathématiques de l’Enseignement Public / Bulletin de l'Association des Professeurs de Mathématiques, 395, 519, 584, 640, 642, 656 C Centenary/centennial of ICMI (2008), 96, 127, 128, 130, 273, 285, 588 Centennial of L’Enseignement Mathématique (2000), 269 Central Committees of CIEM/IMUK (1908-1936), 242, 244, 246, 279–280 Central Midwestern Regional Educational Laboratory (CEMREL), 213, 613 Central Powers, viii, 31, 32, 35, 45, 150, 153, 245 Centre Belge de Pédagogie de la Mathématique, 418 CIEM/IMUK/ICMI inquiries, 55, 56, 76, 90, 114, 179, 242–244, 284, 305, 423, 440, 483, 692, 693 Circolo Matematico di Palermo, 592, 661 Classes Nouvelles de la Libération, 641 Cold War, 43, 45, 58, 70, 264, 411, 434, 469 Colloquium in Bucharest (1968), 79, 80, 212, 632 Colloquium in Lausanne (1967), 77, 79, 253 Colloquium in Utrecht (1964), 74, 252 Colloquium in Utrecht (1967), 78, 79, 81, 207, 209, 216, 224, 253, 588, 632

© Springer Nature Switzerland AG 2022 F. Furinghetti, L. Giacardi (eds.), The International Commission on Mathematical Instruction, 1908-2008: People, Events, and Challenges in Mathematics Education, International Studies in the History of Mathematics and its Teaching, https://doi.org/10.1007/978-3-031-04313-0

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730 Comité Interamericano de Educación Matemática/Inter-American Committee on Mathematics Education (CIAEM/ IACME), 64, 70, 105, 120, 121, 195, 197, 198, 256, 620 Commissie Modernisering Leerplan Wiskunde (CMLW), 454 Commission for the system of marks (Karaktersystemutvalget), 578 Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques (CIEAEM), 52, 54, 55, 60, 66, 71, 74, 76, 90, 98, 192, 220, 248, 389, 395, 418, 518, 587, 629 Commission Lichnerowicz, 73, 518, 585, 642 Commission on Mathematical Education of the African Mathematical Union (AMUCME), 267 Committee on the Teaching of Science (CTS), 85, 101, 105, 108, 111, 116, 118, 123, 206, 232, 254–256, 258, 260, 263, 265, 281–284, 291, 292, 419, 517, 636 Comprehensive School Mathematics Program (CSMP), 345, 613 Computer in mathematics education, 98, 99, 103, 113, 114, 117, 203, 219, 230, 261, 262, 313–318, 321, 427, 428, 442, 569, 614 Congress in Dakar (1965), 74, 252, 352 Congress of CIEM/IMUK in Cambridge (1912), 16, 19–23, 243–244, 302 Congress of CIEM/IMUK in Milan (1911), 16, 19, 57, 186, 243, 256, 406, 656, 678, 706 Congress of CIEM/IMUK in Paris (1914), 16, 19, 20, 23, 57, 244, 327, 403, 406, 422, 423, 470, 526, 651, 656 Congress of the European Society for Research in Mathematics Education (CERME), 268, 270 Connections between the teaching of mathematics and physics, 5, 19, 64, 68, 77, 79, 195, 196, 250, 253, 404, 405, 458, 512, 531, 594, 632, 642, 676 Constitutive Assembly of IRC in Brussels (1919), 31, 150, 245 D Deutsche Mathematiker Vereinigung (DMV), 11, 142, 171, 177, 500, 503, 523 Deutscher Ausschuß für mathematischen und naturwissenschaftlichen Unterricht (DAMNU), 523, 525, 526

Subject Index Deutscher Verein zur Förderung des mathematischen und naturwissenschaftlichen Unterrichts, 501, 523 Developing countries, 44, 45, 97, 99, 101, 104, 105, 131, 231, 262, 271, 272, 312, 419, 427, 536 Dissolution/liquidation of CIEM/IMUK, ix, 1, 18, 31–37, 151, 154–156, 245, 328, 401, 448, 504, 526, 693 E East Asian Regional Conference on Mathematics Education (EARCOME), 124, 127, 267 Educational Studies in Mathematics, 77–79, 81, 97, 99, 109, 207, 208, 211, 213, 253, 344, 397, 453, 588, 632 Erlanger Program, 499 Exclusion of the Central Powers from the international scientific cooperation, viii, 31, 32, 45, 150, 153, 245 Executive Committees of ICMI (1952–2009), 248, 249, 251, 253, 254, 256, 258, 260, 262, 263, 265, 266, 268, 270, 273, 280–284 F Fall of the Berlin Wall (1989), 264 Financial issues/funding of CIEM/IMUK/ ICMI, 11, 13, 14, 16, 33, 38, 44, 51, 52, 55, 63, 71, 87, 89, 96, 106, 113, 122, 141, 167, 173, 187, 198, 224, 231, 251, 261, 267, 294, 296, 453, 463 Finnish Union of Teachers of Mathematical Sciences, 568 First Africa Regional Congress of ICMI in Johannesburg (2005), 272 First Asian Technology Conference in Mathematics (ATCM), 267 For the Learning of Mathematics (FLM), 110, 259 Friction between ICMI and IMU, ix, 1, 44, 49, 52–54, 57–59, 69, 80–87, 89, 90, 169, 214, 222–228, 287, 397, 453 Fusionism, 5, 19, 51, 243, 656, 694 G Gender and mathematics education, 120, 263, 266, 311, 322 General Assembly of ICMI

Subject Index Adelaide (1984), 261 Berkeley (1980), 110, 259 Budapest (1988), 264 Copenhagen (2004), 271 Exeter (1972), 87, 228, 229 Karlsruhe (1976), 256 Laval (1992), 265 Makuhari/Tokyo (2000), 269 Monterrey (2008), 274, 286, 288 Sevilla (1996), 267 I Ibero-American Conference on Mathematics Education (CIBEM), 265 ICMI as a permanent subcommission of IMU, ix, 1, 44, 138, 247, 280, 329, 446, 465, 474, 522, 571, 591 ICMI Award Emma Castelnuovo Medal, 126 Klein Medal, 126, 271, 274, 505 Freudenthal Medal, 126, 271, 274, 454 Honorary Membership (1936), xi, 39, 155, 247, 347, 399, 401, 446, 650, 651, 654, 656, 658, 663, 666, 669, 672, 679, 690, 693, 698, 702, 705, 706 ICMI Bulletin, 99, 104, 110, 118, 124, 241, 294, 296, 494 ICMI-China Regional Conference on Mathematics Education, 265 ICMI News (Electronic Newsletter), 96, 273 ICMI Renaissance, 76–87, 90, 97–101 ICMI Seminar on “Calculators in school teaching” in Luxembourg (1978), 257 ICMI Study ICMI Study-1 Conference in Strasbourg (1985), 114, 115, 262 ICMI Study-2 Conference in Kuwait (1986), 114, 115, 262 ICMI Study-3 Conference in Udine (1987), 114, 115, 263 ICMI Study-4, 114, 115, 263 ICMI Study-5 Conference in Leeds 1989), 117, 264 ICMI Study-6 Conference in Calonge (1991), 17, 265 ICMI Study-7 Conference in Höör (1993), 120, 266 ICMI Study-8 Conference in Washington (1994), 266, 317, 320 ICMI Study-9 Conference in Catania (1995), 120, 266, 267

731 ICMI Study-10 Conference in Luminy (1998), 121, 267 ICMI Study-11 Conference in Singapore (1998), 121, 268 ICMI Study-12 Conference in Melbourne (2001), 124, 269, 270 ICMI Study-13 in Hong Kong (2002), 127, 270 ICMI Study-14 Conference in Dortmund (2004), 127, 271 ICMI Study-15 Conference in Aguas de Lindoìa (2005), 127, 272 ICMI Study-16 Conference in Trondheim (2006), 127, 272 ICMI Study-17 Conference in Hanoi (2006), 273 ICMI Study-18 Conference in Monterrey (2008), 274 IMU Bulletin, 232, 255, 458 Institut de Recherche sur l’Enseignement des Mathématiques (IREM), 77, 97, 100, 107, 519, 583, 586–588 Institut für Didaktik der Mathematik (IDM), 77, 100, 203, 255, 258, 313, 487, 488, 588, 614, 615 Instituut voor de Ontwikkeling van het Wiskunde Onderwijs (IOWO), 77, 100, 107, 255, 452, 453 International Association for Statistical Education (IASE), 130, 260, 265, 274 International Conference on Teaching Statistics (ICOTS), 260 International Congress of Mathematicians (ICM) ICM in Rome (1908), 4–9, 143, 144, 156, 160, 175, 242, 298, 300, 301, 401, 402, 447, 463, 525, 604, 678, 692, 693, 702 ICM in Cambridge (1912), 15, 16, 19–23, 156, 160, 170, 176, 243, 244, 298, 301–302, 504, 526, 604, 656, 662, 678, 688, 702 ICM in Strasbourg (1920), 32–36, 45, 138, 153, 154, 158, 159, 245, 298, 302, 448, 702 ICM in Toronto (1924), 36, 245, 298, 302, 702 ICM in Bologna (1928), ix, 37, 45, 160, 245, 246, 298, 302–303, 328, 401, 470, 663, 692, 702 ICM in Zurich (1932), 38, 145, 160, 246, 298, 303–304, 328, 360, 440, 470, 485, 625, 663, 668, 669, 692, 693, 702

732 International Congress of Mathematicians (ICM) (cont.) ICM in Oslo (1936), xi, 39, 47, 138, 155, 160, 175, 246, 280, 298, 304, 347, 401, 446, 465, 470, 474, 522, 523, 526, 571, 575, 579, 591, 625, 651, 656, 663, 669, 679, 693, 702 ICM in Cambridge (1950), 43, 45, 46, 247, 298, 304–305, 378 ICM in Amsterdam (1954), 55, 57, 60, 63, 88, 168–171, 174, 176, 179, 248, 298, 305–307, 367, 385, 388, 389, 412, 458, 483, 512, 579 ICM in Edinburgh (1958), 57, 60–62, 183, 184, 249, 298, 307, 308, 378, 379, 512, 579, 612 ICM in Stockholm (1962), 68, 251, 298, 308, 309, 458, 567, 579, 612, 626 ICM in Moscow (1966), 71, 72, 75, 201, 202, 252, 298, 309–311, 567, 612 ICM in Nice (1970), 85, 219, 254, 298, 311, 394, 609 ICM in Vancouver (1974), 106, 256, 298, 312 ICM in Helsinki (1978), 258, 260, 298, 312, 313, 487, 488, 494, 563, 567 ICM in Warsaw (1983), 259–261, 298, 313, 314 ICM in Berkeley (1986), 263, 298, 314–316 ICM in Kyoto (1990), 264, 298, 316, 486, 494 ICM in Zurich (1994), 266, 298, 316–318 ICM in Berlin (1998), 268, 298, 318–320, 558 ICM in Beijing (2002), 270, 298, 320, 321 ICM in Madrid (2006), 273, 298, 321–324 International Congress on Mathematical Education (ICME) ICME-1 in Lyon (1969), ix, 81, 82, 87, 90, 95–101, 207, 212, 213, 254, 325, 330, 335–345, 352, 385, 487 ICME-2 in Exeter (1972), 84, 86, 95, 101–105, 117, 215, 218, 219, 221, 224–230, 255, 487, 531, 609 ICME-3 in Karlsruhe (1976), 16, 105–108, 115, 233, 256, 258, 385, 419, 454, 487, 488, 494, 562, 614, 636 ICME-4 in Berkeley (1980), 108–111, 259, 419 ICME-5 in Adelaide (1984), 111–116, 119, 261, 444, 495, 562 ICME-6 in Budapest (1988), 116–118, 264, 428, 562

Subject Index ICME-7 in Quebec (1992), 118–121, 265, 427 ICME-8 in Sevilla (1996), 121–122, 267, 428 ICME-9 in Tokyo (2000), 122–125, 269 ICME-10 in Copenhagen (2004), 125–127, 271, 272, 615 ICME-11 in Monterrey (2008), 127–130, 274, 286, 288 International Council of Scientific Unions (ICSU), 63, 73, 83, 85, 89, 101, 105, 108, 111, 115, 116, 118, 123, 167, 173, 200, 206, 210, 213, 214, 232, 254–256, 258, 260, 262, 263, 265, 281–283, 291, 292, 413, 419, 517, 636 International Group for the Psychology of Mathematics Education (PME), 103, 107, 108, 114, 257, 263, 454 International Mathematical Olympiads (IMO), 110, 250, 314, 563, 597, 599, 629 International Mathematical Union (IMU), passim International Organization of Women and Mathematics Education (IOWME), 115, 263 International Research Council (IRC), 31, 45, 150, 153, 167, 245 International Study Group for Mathematical Modelling and Applications (ICTMA), 124, 271 International Study Group on the Relations between the History and Pedagogy of Mathematics (HPM), 107, 113, 257, 261 Inter-Union Commission on Science Teaching (IUCST/CIES Commission Interunions de l’Enseignement des Sciences), 73, 74, 82, 83, 206, 209, 211, 213, 214 J Japan Society of Mathematical Education (JSME), 256, 261, 351, 352, 488, 563, 609 Journal for Research in Mathematics Education, 99, 255 K Klein Project, 130, 131, 273 KöMaL, 630 Kvant, 598

Subject Index L Laboratory of Mathematics, 6, 58, 143, 163, 306 Launch of Sputnik (1957), 65, 249 L’Éducation Mathématique, 655 L’Enseignement Mathématique, xi, 4, 9, 13, 24, 26, 39, 48, 55, 60, 63, 79–81, 114, 115, 138, 139, 151, 153, 155, 157, 168, 186, 199, 207–209, 241, 242, 244, 247, 249, 250, 269, 270, 325, 327, 413, 422, 434, 447, 448, 453, 463, 503, 588, 602, 683, 687, 688, 693 L’Intermédiaire des Mathématiciens, 687, 691 M Martial law in Poland (1981), 259 Matematisk Tidsskrift, 360, 361, 477 Mathematical Association, 143, 219, 463, 530, 531, 546, 562, 573–575 Mathematical Association of America (MAA), 153, 215, 310, 373, 541, 549, 603 Mathematics curricula, 55, 56, 65–68, 104, 105, 124, 194, 256, 309, 310, 312, 316, 319, 321, 344, 345, 351, 373, 374, 396, 405, 441, 454, 478, 494, 498, 502, 503, 542, 554, 613, 615, 620, 625 Mathematics Curriculum Committee (CMC), 491 Meeting of CIEM/IMUK in Brussels (1910), 16, 186, 243, 526 Meeting of CIEM/IMUK in Cologne (1908), 10–14, 140, 141, 143, 242 Modern Mathematics/New Math, ix, 1, 54, 65–68, 71, 73, 74, 76, 88, 89, 100, 103, 104, 109, 207, 250–253, 309, 310, 374, 396, 487, 494, 519, 554, 568, 579, 584, 585, 614, 632, 640, 642 N National Council of Teachers of Mathematics (NCTM), 107, 373, 443 National delegate to CIEM/IMUK/ICMI, 8, 9, 11–16, 38, 39, 47, 53, 58, 70, 111, 141, 170–173, 177, 217, 223, 227, 242, 243, 287–292, 323, 325–329, 331–333, 405, 413, 422, 435, 458, 477, 487, 512, 526, 593, 604, 625, 640, 642, 651, 656, 678, 688, 702–706 National Subcommissions of CIEM/IMUK/ ICMI, viii, 3, 11–16, 18, 36, 38, 39, 49, 50, 52, 53, 59, 70, 72, 76, 78, 82, 83, 86, 87, 90, 95, 96, 186, 188, 189, 197, 202, 243, 244, 385, 405, 422, 477, 504, 604, 626, 656

733 New Trends in Mathematics Teaching/ Tendances nouvelles de l’enseignement des Mathématiques, 72, 73, 80, 89, 90, 100, 101, 106, 120, 252, 253, 256, 258, 632, 636 Nordisk Matematisk Tidskrift, 579 Norsk Matematisk Tidsskrift, 477, 579 Nuremberg Laws, 246 Nyt Tidsskrift for Matematik, 360, 477 O Organisation for European Economic Co-operation/Organization for Economic Co-operation and Development (OEEC/OECD), ix, 1, 46, 65–68, 89, 192, 193, 196, 247, 250, 271, 374, 396, 453, 546, 579, 619 P Pan-African Congress of Mathematicians in Rabat (1976), 256 Periodico di Matematiche, 679 Permanent Secretariat of ICMI, 75, 79, 205, 206, 252, 518 Pfingsttagung, 180, 384 Pipeline Project, 271 Polish Commission for Mathematical Curricula and Textbooks for Elementary and Secondary Schools, 625 Pop Maths Roadshow, 264 Popularization of mathematics, 117, 264, 268, 270, 316–321, 323, 324, 630 Pre-school education/Kindergarten, 74, 99, 100, 103, 374, 518 Primary education, 63, 67, 71, 84, 99, 103, 104, 108, 165, 219, 230, 344, 519 Q Quantum, 598 R Racial laws, 366, 673, 679, 691 Recherches en Didactique des Mathématiques, 110, 259 Reconstitution of CIEM/IMUK (1928), 1, 3, 36–37, 246, 328, 401 Reconstitution of the International Commission on Mathematical Instruction (1952), 1, 44–50, 247, 329, 384

734 Reconstitution of IMU (1952), 44–50, 247 Regional Conference, 105, 120, 122, 124, 127, 230, 231, 255, 256, 261, 265–267, 352, 488, 495, 609 Research in Mathematics Education, 267 Revue de l’École, 655 Rigour and intuition in mathematics teaching, 19, 51, 103, 128, 244, 301, 361, 369, 478, 526, 542, 545, 652, 656, 676, 678, 679, 694 Rocznik Pedagogiczny, 667 Role of mathematics and mathematician in society, 56, 60, 79, 80, 169, 179, 248, 367 Role of problems in mathematics teaching/ Problem solving, 56, 72, 75, 110, 201, 252, 310, 311, 314, 319, 323, 427, 428, 458, 598, 685 Role of psychology in mathematics education, 54, 56, 57, 60, 61, 67, 76, 103, 107, 173, 257, 389, 443, 454, 504 Royaumont seminar (1959), 65–68, 89, 196, 250, 374, 396, 453, 458, 546, 579, 619, 857 S Sängerkriege, 384 School Mathematics Study Group (SMSG), 66, 105, 109, 305, 306, 351, 368, 373, 374, 512, 549, 550 Scuola di Magistero, 678, 679, 691, 693 Secondary education, 4–7, 9, 17–20, 51, 60, 61, 63–68, 71, 72, 74, 75, 104, 108, 130, 162, 163, 165, 186, 189, 193, 230, 242, 244, 249–252, 300, 301, 303, 307–311, 317–321, 344, 345, 360–363, 365, 366, 368, 369, 385, 396, 401–403, 406, 411, 413, 418, 439, 459, 471, 487, 488, 494, 499, 502–504, 518, 519, 524, 542, 578, 594, 599, 612–614, 625, 631, 642, 651, 652, 656, 679, 680, 694 Secondary School Mathematics Curriculum Improvement Study (SSMCIS), 344, 613 Semesterberichte zur Pflege des Zusammenhangs von Universität und Schule aus den mathematischen Seminaren, 384 Seminar für Didaktik der Mathematik, 58, 384, 612 Seminar in Bologna (1961), 65, 68, 71, 193, 197, 250, 396, 642 Seminar in Lausanne (1961), 197, 198, 250, 435

Subject Index Seminar in Zagreb–Dubrovnik (1960), 65, 68, 250 Société Mathématique de France (SMF), 66, 394, 395, 440, 517, 584, 585, 640, 642, 656, 661, 684 Solidarity Program, 118–121, 265, 427 Southeast Asian Conference on Mathematics Education (SEACME), 108, 122, 257, 488, 495 Southeast Asian Mathematical Society (SEAMS), 108, 120, 495 Soviet-British Seminar on Mathematical Education in Oxford (1981), 259 Special Committee on the Teaching of Science (SCOTS), 64, 65, 195–197 Swiss Association of Mathematics Teachers, 448 Symposium in Aarhus (1960), 51, 189, 191, 193, 250, 418 Symposium in Belgrade (1960), 70, 191–194, 250, 458, 512, 632 Symposium in Echternach (1965), 74, 75, 201, 252 Symposium in Frascati (1964), 73, 252, 518, 642 Systematic Cooperation between Theory and Practice in Mathematics Education (SCTP), 419 T Tagung zur Pflege des Zusammenhangs zwischen Höherer Schule und Universität, 384, 612 Teacher training, 17, 25, 30, 38, 60, 82, 84, 230, 251, 254, 256, 257, 272, 284, 320, 324, 344, 368, 369, 401, 402, 418, 419, 440, 501, 519, 575, 585, 586, 598, 603, 631, 674, 676–680 Teaching methods, viii, 6, 9, 14, 19, 47, 60, 61, 68, 90, 102, 103, 149, 201, 305, 315, 344, 384, 405, 414, 513, 524, 595, 631, 641, 678, 679, 694 Teaching of differential and integral calculus, 5, 8, 19–21, 244, 301, 310, 314, 315, 321, 324, 406, 423, 470, 503, 504, 593, 652, 656 Teaching of geometry, 5, 11, 14, 19, 51, 60, 61, 67–69, 103, 120, 142, 189, 191, 193, 230, 249, 250, 261, 267, 307, 308, 313, 319, 396, 488, 525, 574, 593, 599, 613, 615, 642, 677, 678, 685, 694 Teaching of statistics, 130, 260, 261, 274, 307, 308, 323

Subject Index Terms of Reference, x, 44, 52–54, 58, 59, 69–71, 85, 88–90, 96, 97, 101, 111, 123, 127, 169, 172, 174, 188, 190, 191, 204, 206, 217, 228, 248, 250, 260, 270, 271, 280, 284, 285, 287–296, 325, 334, 385, 488 Tertiary education/university instruction, 19, 63–65, 72, 73, 75, 89, 100, 104, 108, 194, 195, 198, 201, 244, 252, 257, 265, 268, 269, 301, 302, 310, 312, 318–321, 360, 384, 435, 476, 501, 502, 517, 518, 562, 567, 584–586, 593, 594, 636, 652, 656, 677, 678, 691, 695 Textbooks, 163, 250, 251, 257, 309, 313, 321, 353, 356, 360–363, 365, 368, 369, 373, 374, 413, 414, 418, 422, 428, 433, 462, 470, 478, 487, 494, 501, 513, 524, 525, 541, 545, 549, 550, 585, 593–595, 598, 599, 603, 604, 608, 625, 636, 651, 652, 669, 678, 686, 694, 699 The Mathematical Gazette, 284, 463, 546, 573, 574, 693 The Mathematics Student, 62, 379 Twin Towers attack, 269 U Unione Matematica Italiana (UMI), 45, 193, 245, 396 United Nations Educational, Scientific and Cultural Organization (UNESCO), ix, 1, 46, 57, 63–65, 71–73, 77, 79, 80, 82, 87, 89, 90, 100, 101, 105–108, 114, 117, 120, 126, 130, 160, 162–165, 167, 173, 193–198, 210–212, 224–228, 233, 247, 250–253, 255, 256, 258–261, 264, 265, 269, 272, 313, 329, 352, 379, 396, 411, 417, 419, 444, 487, 517, 518, 632

735 W World Federation of National Mathematics Competitions (WFNMC), 112, 119, 266 World Mathematical Year 2000 (WMY 2000), 267, 269, 534, 537 World War I/The First World War/WWI, viii, ix, 1, 15, 16, 23, 24, 31, 44, 47, 57, 138, 144, 145, 158, 244, 328, 365, 382, 393, 401, 403, 410, 438, 440, 448, 468–470, 481, 501, 504, 526, 572, 604, 623, 651, 662, 668, 688, 693, 700 World War II/The Second World War/WWII, viii–x, 1, 3, 43, 44, 66, 87–89, 175, 179, 220, 247, 280, 284, 328, 351, 355, 383, 384, 434, 441, 446, 448, 451, 452, 465, 468, 470, 474, 486, 518, 522–524, 528, 548, 560, 571, 582, 591, 597, 607, 611, 618, 619, 623–625, 629, 635, 640, 641, 645, 667, 668, 691 Y Young European Researchers in Mathematics Education (YERME), 270 Z ZDM–Zentralblatt für Didaktik der Mathematik/ZDM–The International Journal on Mathematics Education/ ZDM–Mathematics Education, 99, 212, 253, 613 Zentrum für Didaktik der Mathematik, 77, 81, 95, 100, 212, 253, 613