Modern Mathematics: An International Movement? 3031111656, 9783031111655

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History of Mathematics Education

Dirk De Bock  Editor

Modern Mathematics An International Movement?

History of Mathematics Education Series Editors Nerida F. Ellerton, Illinois State University, Normal, IL, USA M. A. Ken Clements, Illinois State University, Normal, IL, USA

History of Mathematics Education aims to make available to scholars and interested persons throughout the world the fruits of outstanding research into the history of mathematics education; provide historical syntheses of comparative research on important themes in mathematics education; and establish greater interest in the history of mathematics education.

Dirk De Bock Editor

Modern Mathematics An International Movement?

Editor Dirk De Bock KU Leuven Leuven, Belgium

ISSN 2509-9736     ISSN 2509-9744 (electronic) History of Mathematics Education ISBN 978-3-031-11165-5    ISBN 978-3-031-11166-2 (eBook) https://doi.org/10.1007/978-3-031-11166-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Foreword

The curriculum of schools, universities, and other institutions has always been a matter of contestation. The history of the curriculum, and the areas and subjects within it, provide an important insight into the political and professional disputations that have been central to movements for change and reform. Teachers with the good fortune to spend a lifetime teaching their subject will know that curriculum development, implicitly and explicitly, is always on the agenda. There are some who argue, for example Hussain and Yunus (2020), that recent swings to more populist governments have increased the level of political engagement with schooling and curriculum. There is plenty of contemporary evidence to support this view. Donald Trump planned to set up a commission to promote “Patriotic Education” that would address, in his words, the “twisted web of lies” being taught in American schools (Wise 2020). The British politician, Michael Gove, when Secretary of State for Education, attempted to change the school history curriculum to reflect a more patriotic perspective on British history. He railed against the demonization of the British Empire (Watson 2021). Controversial views such as these go back a long way in curriculum history. In the late 1970s, I became interested in the way the reform of mathematics teaching in the 1960s, sometimes called the “New Maths” movement, had spread across Europe (Moon 1986). I went into the research with an open mind. What story would I find? What patterns emerged from those stories? I looked at reforms in England, France, and the then West Germany as well as Denmark and the Netherlands. In each context, there was a clear link between the heady, progressive social ideas of the 1960s and the reform of mathematics teaching. I pondered whether the impact of the Royaumont Seminar, well covered in this book, could have been achieved without the political climate of reform that characterized Western Europe in the 1960s. The French “Bourbakist”1 movement, which was so successful in reforming the understanding of mathematics research in the university, moved almost seamlessly into an advocacy for reforming the mathematics curriculum in schools. Looking at the rhetoric and proclamations of the day, there is a clear social as well as epistemological purpose in these activities. There were other indications that subject curriculum reform was inseparable from social and political trends. I found cartoons in German newspapers that parodied the hippy notion of Alef, the most well-known modern mathematics project. In England, the pushback against progressive ideas, represented by the series of Black Papers (Cox and Dyson 1969), made a great play of the mystique of modern mathematics. Curriculum and ideology were intertwined. And the momentum for reforming the mathematics curriculum was so strong that it bypassed traditional structures of curriculum control. Even in the formidably centralized French system, the schools were changing methods and textbooks well in advance of any reformulation of regulations.  Nicolas Bourbaki, a nineteenth-century French general, was the pseudonym used by a group of mathematicians who, from the mid-1930s on, advocated radical reform of the way mathematics was taught and researched within the university. In the 1950s, these “Bourbakist” ideas were seen by some as a model for mathematics in the school curriculum (see also Chap. 3 in this volume). 1

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Foreword

Schools, and to some extent universities, are buffeted by these ideological tremors. This works across the political spectrum. When I began teaching in London in the 1960s, I was involved in a notable British Curriculum initiative, The Humanities Curriculum Project, led by Lawrence Stenhouse. This acquired some notoriety, and in part foundered, because the treatment of race alienated all shades of political opinion (Elliott and Norris 2011). Today, in the wake of the Black Lives Matter movement, the controversies continue. Lawrence Stenhouse subsequently went on to do more than anyone to establish a rapprochement between competing perspectives on curriculum theory (Lawton 1983). One of the key aspects of his contribution to curriculum research was the accessibility of his work. He was confident with the historical and sociological literature, and this showed in the clarity of his arguments. He also saw the importance of comparative data and ideas that feature so strongly in this book. Curriculum historians have a vital role in describing and explaining the motives and rationale behind curriculum reform and change. Over the last half century, and more, significant progress has been made in appropriating the methodological and theoretical ideas of historians and social scientists to the study of curriculum history. These ideas are not static. Narrative history, the “longue durée” of Fernand Braudel and ongoing debates about how we do history, informs our task. As do the various schools of thought that have given sociology such an influence on the study of all aspects of education. The challenge to establish a strong cross-disciplinary base for curriculum history is ongoing. American authors such as Franklin (2006), Kliebard (2018), and Popkewitz (2013) have given greater rigor to these modes of enquiry. A few years ago, Bernadette Baker wrote a review of curriculum history research (Baker 1996). She built her arguments around the questions “What is the field? And who gets to play on it?” This book makes an important contribution to answering those questions, not only for mathematics education but for curriculum history more generally. The French historian Marc Bloch wrote that history was about finding, and following, the tracks left by previous generations. And to quote Bloch, “misunderstanding of the present is the inevitable consequence of ignorance of the past” (Bloch 1954, p. 43). Curriculum historians need publications such as this, as do future generations of curriculum reformers. Emeritus Professor of Education  Bob Moon The Open University, Milton Keynes, UK

References Baker, B.  (1996).  The history of curriculum or curriculum history? What  is  the field and who gets to play on it? Curriculum Studies, 4(1), 105–117. Bloch, M. (1954). The historians craft. Manchester, United Kingdom: Manchester University Press. Cox, C. B., & Dyson, A. E. (Eds.). (1969). Fight for education: A Black Paper. London, United Kingdom: The Critical Quarterly Society. Elliott, J., & Norris, N. (Eds.). (2011). Curriculum, pedagogy and educational research: The work of Lawrence Stenhouse. London, United Kingdom: Routledge. Franklin, B. M. (2008). Curriculum history and its revisionist legacy. In W. J. Reese & J. L. Rury (Eds.), Rethinking the history of American education (pp. 223–243). New York, NY: Palgrave Macmillan. Hussain, S., & Yunus, R. (2021). Right-wing populism and education: Introduction to the special section. British Educational Research Journal, 47(2), 247–263. Kliebard, H. M. (2018). Forging the American curriculum: Essays in curriculum history and theory. London, United Kingdom: Routledge. Lawton, D. (1983). Lawrence Stenhouse: His contribution to curriculum development. British Educational Research Journal, 9(1), 7–9.

Foreword

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Moon, B. (1986). The “New Maths” curriculum controversy: An international story. Barcombe, United Kingdom: Falmer Press. Popkewitz, T. S. (Ed.). (2013). Rethinking the history of education: Transnational perspectives on its questions, methods and knowledge. New York, NY: Springfield Link. Watson, M. (2019, July 7). Michael Gove’s war on historians: Extreme whig history and Conservative curriculum reform. Retrieved October 2, 2021, from https://blogs.lse.ac.uk/politicsandpolicy/michael-­goves-­war-­on-­historians/ Wise, A. (2020, September 17). Trump announces, “Patriotic Education” Commission, a largely political move. Retrieved October 2, 2021, from https://www.npr.org/2020/09/17/914127266/ trump-­announces-­patriotic-­education-­commission-­a-­largely-­political-­move?t=1633187691919

Contents

Foreword�����������������������������������������������������������������������������������������������������������������������������������     v Contents�������������������������������������������������������������������������������������������������������������������������������������     ix List of Figures���������������������������������������������������������������������������������������������������������������������������    xiii List of Tables�����������������������������������������������������������������������������������������������������������������������������    xix Abstracts.�����������������������������������������������������������������������������������������������������������������������������������    xxi Preface to the Series�����������������������������������������������������������������������������������������������������������������   xxix Preface to the Book�������������������������������������������������������������������������������������������������������������������   xxxi Author Biographies�������������������������������������������������������������������������������������������������������������������  xxxiii 1

Modern Mathematics: An International Movement Diversely Shaped in National Contexts���������������������������������������������������������������������������������������������������������     1 Dirk De Bock

Part I Preparing the Reform on Both Sides of the Atlantic 2

The Rise of the American New Math Movement: How National Security Anxiety and Mathematical Modernism Disrupted the School Curriculum �����������������������������    13 David Lindsay Roberts

3

The Early Roots of the European Modern Mathematics Movement: How a Model for the Science of Mathematics Became a Model for Mathematics Education���������������������������������������������������������������������������������������������������������������������������    37 Dirk De Bock

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The Royaumont Seminar as a Booster of Communication and Internationalization in the World of Mathematics Education�������������������������������������������������������������������������    55 Fulvia Furinghetti and Marta Menghini

Part II Implementation of the Reform Around the World 5

The Modern Mathematics Movement in France: Reforming to What Ends? The Contribution of a Cross-Over Approach to Modernity�����������������������������������������    81 Hélène Gispert

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West German Neue Mathematik and Some of Its Protagonists �������������������������������������  103 Ysette Weiss

7

New Mathematics in the United Kingdom: Projects and Textbooks as Driving Forces of Curriculum Reform�������������������������������������������������������������������������  127 Leo Rogers

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Modern Mathematics in Italy: A Difficult Challenge Between Rooted Tradition and Need for Innovation�����������������������������������������������������������������������������������������������������  147 Fulvia Furinghetti and Marta Menghini

9

 The Distinct Facets of Modern Mathematics in Portugal�����������������������������������������������  169 José Manuel Matos and Mária Cristina Almeida

10 Papy’s  Reform of Mathematics Education in Belgium: Development, Implementation, and Controversy�������������������������������������������������������������������������������������  199 Dirk De Bock and Geert Vanpaemel 11 A  Tale of Two Systems: A History of New Math in The Netherlands, 1945–1980����������������������������������������������������������������������������������������������������������������������������  217 Danny Beckers 12 Nordic  Cooperation on Modernization of School Mathematics, 1960–1967����������������������������������������������������������������������������������������������������������������������������  239 Kristín Bjarnadóttir 13 Reforms Inspired by Mathématique Moderne in Poland, 1967–1980�����������������������������  267 Zbigniew Semadeni 14 The  New Math in Hungary: Tamás Varga’s Complex Mathematics Education Reform���������������������������������������������������������������������������������������������������������������  285 Katalin Gosztonyi 15 New  Math and the South Slavs �����������������������������������������������������������������������������������������  303 Snezana Lawrence 16 The  Kolmogorov Reform of Mathematics Education in the USSR�������������������������������  319 Alexandre Borovik 17 The  Influence of Royaumont on Mathematics Education in the USA���������������������������  337 Jerry Becker and Bill Jacob 18 Aspects  of Canadian Versions of So-Called “Modern” Mathematics and Its Teaching: Another Visit to the Old “New” Math(s) �������������������������������������������  363 David Pimm and Nathalie Sinclair 19 New  Math in Latin America (and a Glimpse at Costa Rica) �����������������������������������������  383 Angel Ruiz 20 Modernizing  Mathematics Teaching: International Dialogues from Brazil�����������������  403 Elisabete Zardo Búrigo and Wagner Rodrigues Valente 21 Australian  School Mathematics and “Colonial Echo” Influences, 1901–1975�������������  423 Nerida F. Ellerton and M. A. (Ken) Clements 22 What  Did the “New Math Movement” Bring to Hong Kong in the 1960s and the 1970s (and Beyond)? �����������������������������������������������������������������������  453 Man-Keung Siu and Ngai-Ying Wong

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23 Modern  Mathematics: An International Movement, the Experience of Morocco���������������������������������������������������������������������������������������������������������������������������  471 Ezzaim Laabid 24 Modern  Mathematics Curriculum Reforms in Ghana: UK and USA Influences������������������������������������������������������������������������������������������������������������  489 Damian Kofi Mereku Epilogue: From Center to Periphery? The Global Dynamics of the New Math Movement�������������������������������������������������������������������������������������������������������  505 Composite Reference List�����������������������������������������������������������������������������������������������������������  511 Author Index�������������������������������������������������������������������������������������������������������������������������������  565 Subject Index�������������������������������������������������������������������������������������������������������������������������������  577

List of Figures

Figure 2.1 Max Beberman teaching a class of high school students, 1959. (Courtesy of the University of Illinois Archives)�������������������������������������������������������  23 Figure 2.2 Page from unpublished UICSM text material, 1954. (Originally owned  by Virginia Garrett Rovnyak, now in possession of David L. Roberts)���������������������  25 Figure 2.3 Howard Fehr, 1949. (Photograph by Alman Co. 590 Fifth Avenue, New York, NY 10019. University Archives, Rare Book & Manuscript Library, Columbia University Libraries) �������������������������������������������������������������������  25 Figure 2.4 Edward G. Begle, 1961. (By Mercado, School Mathematics Study Group Records, e_math_00582, The Dolph Briscoe Center for American History, University of Texas at Austin) �����������������������������������������������������������������������������������  28 Figure 2.5 SMSG writing group at Stanford University, summer 1960. (School Mathematics Study Group Records, e_math_00581, The Dolph Briscoe Center for American History, University of Texas at Austin)������������������������������������������������  29 Figure 3.1 Founding meeting of the CIEAEM in La Rochette par Melun, 1952, from left to right: Mrs. and Mr. F. Gonseth, J. Dieudonné, Lucien Delmotte, G. Choquet, L. Félix, F. Fiala, J. Piaget, unknown, C. Gattegno, unknown���������������  39 Figure 3.2 Covers of the two books from the 1950s with collective work by CIEAEM members. (Collection Guy Noël)�������������������������������������������������������������������������������  40 Figure 3.3 Working session in a park at the founding meeting of the CIEAEM in La Rochette par Melun, 1952���������������������������������������������������������������������������������  41 Figure 3.4 La Rochette par Melun, 1952 (from left to right, left: Mrs. and Mr. F. Gonseth, J. Dieudonné, and G. Choquet; middle: G. Choquet, F. Fiala, and C. Gattegno; right: F. Gonseth and W. Servais)�������������������������������������������������������������������������������  45 Figure 4.1 Choquet, partially hidden, Piaget, Gattegno, and Willy Servais in La Rochette par Melun in 1952. (Photo by Lucien Delmotte, collection Guy Noël)���������������������  57 Figure 4.2 Signatures in the Echternach treaty ���������������������������������������������������������������������������  68 Figure 4.3 Colloquium at Echternach in 1965. (Courtesy of Robert Kennes)�����������������������������  74 Figure 4.4 ICME-1, Lyon: Krygowska, Steiner, Papy-Lenger, Zoltán Dienes. (Courtesy of Gert Schubring)�������������������������������������������������������������������������������������  75 Figure 5.1 André Lichnerowicz (Sciences et Avenir, special issue No. 11 « La crise des mathématiques modernes » [“The crises of modern mathematics”], 1973, p. 87. (Photo by Robert Doisneau) �����������������������������������������  89 Figure 5.2 C’est un plan de dix ans pour l’expansion [It’s a ten-year plan for expansion]. (Newspaper excerpt from Paris-Normandie, November 5, 1956)�����������������������������  91 xiii

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List of Figures

Figure 5.3 La guerre des mathématiques [The mathematics war]. (Newspaper excerpt from l’Aurore, February 3, 1971)�������������������������������������������������������������������  94 Figure 5.4 La querelle des « mathématiques modernes » [The “modern mathematics” quarrel]. (Newspaper excerpt from Le Monde, January 20, 1972)�����������������������������  98 Figure 6.1 Heinrich Behnke (first row at the left) at the ICMI Colloquium Echternach in 1965. Next to Behnke in the front row (identification by Guy Noël): Frédérique Papy-Lenger, Gustave Choquet, unknown woman, André Lichnerowicz; second row: Robert Dieschbourg, two unknown men, Alfred Vermandel, Pierre Debbaut, Claudine Festraets-Hamoir (woman against the wall). (Photo collection P. Debbaut)������������������������������������������������������������������������������������� 107 Figure 6.2 Hans-­Georg Steiner, 1972. (Photo: Konrad Jacobs, Mathematisches Forschungsinstitut Oberwolfach Collection) ������������������������������������������������������������� 113 Figure 6.3 ICME-5 in Adelaide (1984), Bent Christiansen (with bag), Hans-Georg Steiner, and Claude Gaulin. (Photo collection H.-G. Steiner)������������������������������������������������� 121 Figure 7.1 Geoffrey Matthews (right) and Magda Jóboru (president of the Hungarian National Commission for UNESCO) at the international symposium on school mathematics teaching in Budapest, August 27–September 8, 1962. (Photo by Ferenc Vigovszki)�������������������������������������������������������������������������������������� 136 Figure 7.2 Sir Bryan Thwaites. (Thwaites 1972, p. ii).��������������������������������������������������������������� 140 Figure 7.3 Geoffrey Howson 1965����������������������������������������������������������������������������������������������� 140 Figure 7.4 Title figures of Rotation (p. 60) and Translation (p. 77) in Book T. (Howson 1964)����������������������������������������������������������������������������������������������������������� 141 Figure 8.1 Emma Castelnuovo (right) and Lina Mancini Proia in Pallanza, Italy, 1973������������� 151 Figure 8.2 Ugo Morin (left) and Bruno de Finetti at the UMI quadrennial Congress in Trieste, Italy, 1967. (Courtesy of Centro Ricerche Didattiche Ugo Morin)����������� 153 Figure 8.3 Michele Pellerey (right) with Tamás Varga in Bordeaux, France, 1973. (Courtesy of Raimondo Bolletta)������������������������������������������������������������������������������� 161 Figure 9.1 Timeline of modern mathematics in Portugal.����������������������������������������������������������� 170 Figure 9.2 Explaining the sum of more than two terms (“parcelas”). (d’Eça et al. 1974, p. 55)��������������������������������������������������������������������������������������������� 179 Figure 9.3 Solving equations using operators. (Biscaia et al. 1971, p. 116)������������������������������� 183 Figure 9.4 Bijection between the addition of vectors and the addition of integers. (Biscaia et al. 1971, p. 178) ��������������������������������������������������������������������������������������� 184 Figure 9.5 Projections of unitary vectors in two perpendicular directions. (Gomes and Pereira 1972, p. 114) ����������������������������������������������������������������������������� 184 Figure 9.6 Domain of an algebraic fraction. (Garcia et al. 1976, p. 51) ������������������������������������� 189 Figure 9.7 Lunch at the “First initiation course on the Cuisenaire method.” Gattegno is at the center, Nabais at his left. António Augusto Lopes is the fifth on the right. (Nabais 1965, p. 158)����������������������������������������������������������� 191 Figure 9.8 Exercise on sets requiring children to connect objects with geometric shapes. (Ensino Primário. Programas para o ano lectivo 1974–1975, 1974, p. 52) ��������������� 192 Figure 10.1 Papy experimenting with modern mathematics, early 1960s (left, at the blackboard a student proves the distributive law for union over intersection of sets; right, Christine Manet, one of his students, explains that the composition of relations is not commutative. (Hunebelle 1963)������������������� 202 Figure 10.2 Covers of Papy’s Mathématique Moderne����������������������������������������������������������������� 204 Figure 10.3 Children aged 6 at work with Papy’s minicomputer. (International Visual Aid Center, Brussels, late 1960s)��������������������������������������������������������������������� 211

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Figure 11.1 Figurendoos–handleiding [Figure box–manual], colorful brochure explaining the use of the logical blocks (1971)��������������������������������������������������������������������������� 224 Figure 11.2 New Math in Dutch primary schools. Ik doe–en ik begrijp [I do–and I understand], translated from the British Nuffield Project (1968)��������������������������������������������������� 224 Figure 11.3 Textbook author Paul Lepoeter adapted his work to the New Math era. Gids voor de nieuwe wiskunde voor de brugklas [Guide to the New Math for the first year] (1972)��������������������������������������������������������������������������������������������� 227 Figure 11.4 Moderne wiskunde voor ouders [Modern mathematics for parents] (1971), one of many initiatives to involve parents in New Math–notably not the one published by Malmberg��������������������������������������������������������������������������� 230 Figure 11.5 Bruno Ernst (left), advising Fred van der Blij (seated) during recordings of a television lesson on conic sections. (From Ernst (1968), Levende Wiskunde [Living Mathematics]) ����������������������������������������������������������������������������������������������� 231 Figure 11.6 Informatie-bulletin [Information brochure], introducing IOWO, and tentatively projecting plans outliving the scheduled time of funding of the institute (1972)������������������������������������������������������������������������������� 231 Figure 12.1 Bent Christiansen������������������������������������������������������������������������������������������������������� 243 Figure 12.2 Lennart Sandgren. (Blekinge museum, Sweden)������������������������������������������������������� 244 Figure 12.3 Teachers at the school on Niels Ebbesen Road, Frederiksberg, Denmark, in 1965; Agnete Bundgaard is the second from right in the middle row. (Frederiksberg Stadsarkiv)����������������������������������������������������������������������������������������� 247 Figure 12.4 The commutative law. The first introduction of addition and the “+” symbol. Age 7. (Bundgaard and Kyttä 1967, Vol. 1, p. 32) ��������������������������������������� 248 Figure 12.5 Subsets and ordering with one-to-one correspondence to the number line. Age 7. (Bundgaard and Kyttä 1967, Vol. 1, p. 29)����������������������������������������������������� 248 Figure 12.6 Associative law for addition. Age 7. (Bundgaard and Kyttä 1967, Vol. 1, p. 74)������� 249 Figure 12.7 Distributive law. Age 8. (Bundgaard and Kyttä 1968, Vol. 2b, p. 72)������������������������ 249 Figure 12.8 ICME-3, Karlsruhe 1976, Opening lecture. On the front row from the right: Heinz Kunle, Mrs. Kunle, Shokichi Iyanaga, Mrs. Heinrich Behnke, Bent Christiansen, Hanne Christiansen, Hans-Georg Steiner, Mrs. Steiner. (Courtesy of Gert Schubring and Livia Giacardi) ����������������������������������������������������� 251 Figure 12.9 Percentage of year cohort taking Bundgaard material up through grade 6 ��������������� 259 Figure 13.1 Stefan Straszewicz ����������������������������������������������������������������������������������������������������� 268 Figure 13.2 Zofia Krygowska��������������������������������������������������������������������������������������������������������� 268 Figure 13.3 Papy, presented as the “pope of modern mathematics.” (Caricature by Leon Jeśmanowicz, 1971)������������������������������������������������������������������� 271 Figure 13.4 Hans-Georg Steiner, Zofia Krygowska, and Hans Freudenthal at a meeting in Oberwolfach (Germany) preparing for ICME-3 on December 8, 1975. (Photo collection H.-G. Steiner)��������������������������������������������������������������������������������� 273 Figure 14.1 Varga (right) with János Surányi (left) and Ervin Fried (center) at Varga’s thesis defense in 1975. (Courtesy of Mária Halmos; published with the authorization of Varga’s family)������������������������������������������������������������������� 288 Figure 14.2 Varga among other eminent actors of the New Math era. (Caricatures by Leon Jeśmanowicz, CIEAEM meeting, Poland 1971) ������������������������������������������������������� 289 Figure 14.3 New topics for the elementary school math curriculum. (Varga 1982, p. 28) ����������� 295 Figure 14.4 Two of the three tables associated to the game with three disks. (Varga 1970, p. 426)��������������������������������������������������������������������������������������������������� 296 Figure 14.5 A probability problem from the fifth-grade textbook. (Eglesz et al. 1979, pp. 153–154)��������������������������������������������������������������������������������������������������������������� 298

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Figure 15.1 President of Ghana Kwame Nkrumah and President of Yugoslavia Josip Broz Tito arrive at the first non-­aligned conference in Belgrade, 1961. (Photographer unknown) ������������������������������������������������������������������������������������������� 306 Figure 15.2 Đuro Kurepa at Erlangen, 1976. (Photo: Konrad Jacobs, Mathematisches Forschungsinstitut Oberwolfach Collection) ������������������������������������������������������������� 307 Figure 15.3 At the Institute for Advanced Study in Princeton in 1959. From left to right: Edmund Hlawka (1916–2009), Marston Morse (1892–1977), and Đuro Kurepa. (Photographer unknown, Institute for Advanced Study, Princeton, NJ, USA)����������� 308 Figure 15.4 Miloš Radojčić. (Photographer unknown)����������������������������������������������������������������� 310 Figure 15.5 Judita Cofman in Ludwigsburg, 1991. (Photo: Erhard Anthes, Mathematisches Forschungsinstitut Oberwolfach Collection) ������������������������������������������������������������� 312 Figure 15.6 Problem 1.7 from Cofman 1990, p. 8. (Diagram by the author) ������������������������������� 313 Figure 17.1 1970 CSMP meeting organizers Burt Kaufman (left) and Hans-Georg Steiner. (Image from an appreciation collection of letters provided by Terry Kaufman)������� 341 Figure 17.2 Use of the representation of the context as a tool for thinking about the link between two forms of division ��������������������������������������������������������������������� 354 Figure 18.1 La mathématique: introduction à la notion de relation [Mathematics: introduction to the concept of relation] (left, Willy Servais; right, Lucien Delmotte), 1968. (Images from a film directed by Jacques Parent ©Office national du film du Canada)������������������������������������������������������������������������� 371 Figure 18.2 Experiences with numbers in color by Madeleine Goutard, 1974����������������������������� 372 Figure 18.3 Les mathématiques: un jeu d’enfants [Mathematics: A child’s game]. Caleb Gattegno and a pupil at work with the Cuisenaire rods, 1961 (Photo by John Howe ©Office national du film du Canada) ������������������������������������� 376 Figure 19.1 Cover of I CIAEM Proceedings��������������������������������������������������������������������������������� 388 Figure 19.2 Cover of II CIAEM Proceedings ������������������������������������������������������������������������������� 390 Figure 19.3 VI CIAEM, Guadalajara, Mexico, 1985, from left to right: E. Sebastiani (Brazil), A. Ruiz (Costa Rica), U. D’Ambrosio (Brazil), I. Harding (Chile), E. Lluis (Mexico), and G. Sánchez Vásquez (Spain) ������������������������������������������������� 393 Figure 19.4 Cover of 2012, 1–12 School Mathematics Curriculum approved by Costa Rican Education Authorities������������������������������������������������������������������������������������������������� 398 Figure 20.1 Euclides Roxo, professor of mathematics and director of Colégio Pedro II, in Rio de Janeiro. (Photo from Documentation Center of GHEMAT-SP)����������������� 405 Figure 20.2 Visit to Escola Vila Rica, in Santos, on October 7, 1971, by Zoltán Dienes, the first, from left to right. At his side, Osvaldo Sangiorgi, a key proponent of modern mathematics in Brazil. (Photo from Documentation Center of GHEMAT-SP)��������������������������������������������������������������������������������������������������������� 412 Figure 20.3 Maria Amábile Mansutti, teacher with extensive experience in the early school years, member of the team that prepared the National Mathematical Curriculum Parameters. (Photo from Documentation Center of GHEMAT-SP)������� 415 Figure 21.1 Illustration showing how Cuisenaire rods could be used to show that (g + b) + r equals g + (b + r) and that both “equal” 13��������������������������������������� 431 Figure 21.2 Zoltán Pál Dienes (c. 1970). (Image reproduced from Lawlor 2014) ����������������������� 433 Figure 21.3 A composite photograph of Dienes with Yarra Valley students in 1972. (Photographs by Ken Clements)��������������������������������������������������������������������������������� 438

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Figure 22.1 A page in Monoid’s Modern Mathematics for Secondary Schools (Book 1)������������� 457 Figure 22.2 Sample questions from the first New Math examination paper in 1969��������������������� 458 Figure 22.3 A page in Leung, Chen, and Chan’s Basic Mathematics (Book 4)����������������������������� 464 Figure 23.1 Cover of Aiouch et al. (1969), a textbook for the first year of secondary education (author’s collection)����������������������������������������������������������������������������������� 474 Figure 23.2 Cover of El Mossadeq & Peureux (1971), a textbook with exercises on internal laws of composition intended to prepare students for the second cycle of secondary education (author’s collection)���������������������������������������������������� 481 Figure 23.3 Cover of MEN (1976b), a textbook intended to 6th year of secondary education (experimental science section) (author’s collection)��������������������������������� 483

List of Tables

Table 9.1 Real rate of schooling in Portugal, by grade level, per school year���������������������������  172 Table 9.2 Number of students and certified teachers in liceus in Portugal per school year�������  172 Table 9.3 Number of provisional teachers in liceus in Portugal per year�����������������������������������  172 Table 23.1 Progression of the introduction of the elements of modern mathematics between the 3rd and the 4th year, reform of 1968�������������������������������������������������������  477 Table 23.2 Elements of the program of the 5th-year science, reform of 1968������������������������������  479

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Chapter 1 We reflect on the non-parallel origins and development of modern mathematics, as an educational movement, and its American counterpart New Math. The 1959 Royaumont Seminar played a decisive role in bringing together American and European reformers, acted as a catalyst, but did not lead to substantial reform cooperation on either side of the Atlantic. We pay attention to the pluriform nature of the movement(s), shaped by national traditions, existing educational systems, and societies at large. Moreover, we characterize the modern mathematics reform movement and list some of its main features. Adri Treffers’ and Hans Freudenthal’s model of classifying different approaches to mathematics education into four ideal types proved helpful. We conclude with some reflections on the rapid demise of modern mathematics, which in our view should not be regarded as a total failure, but was a breeding ground for a thorough reflection on mathematics education, nationally and internationally, and was the basis for the emergence of mathematics education as an autonomous scientific discipline. Chapter 2 In the 1940s, the teaching of mathematics in the secondary schools of the United States began to recover from a long period of disrespect. This augmented prestige was due in part to an increased demand for mathematically trained workers arising from World War II and the Cold War. At the same time, undergraduate mathematics instruction was undergoing revision, bringing it more into line with the “modern” viewpoint of research mathematicians, focused on unifying concepts and “structures.” There was a sentiment among a significant segment of mathematics educators that school mathematics had become too estranged from these exciting new developments. This environment encouraged, in the 1950s, the development of innovative secondary school curriculum programs, featuring higher levels of abstraction and precision of language. The University of Illinois Committee on School Mathematics (UICSM) was an early, and notably radical, exemplar, while the School Mathematics Study Group (SMSG) was the largest and best-funded program. By the end of the 1950s optimism that the “New Math” would fundamentally and permanently change the school curriculum for the better was widespread, although far from universal. Chapter 3 The cradle of modern mathematics in Europe is likely to be traced to the founding meeting of the International Commission for the Study and Improvement of Mathematics Teaching in 1952 in La Rochette par Melun (France). The organizer of that meeting, Caleb Gattegno, had chosen Mathematical and Mental Structures as a theme and succeeded in bringing together several “big names” from the fields of psychology, epistemology, and mathematics, including Jean Piaget, Ferdinand Gonseth, and the “Bourbakists” Jean Dieudonné, Gustave Choquet, and André Lichnerowicz. A few outstanding xxi

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secondary school teachers of mathematics participated too, among them Lucienne Félix and Willy Servais. Dieudonné explained the architecture of modern mathematical science, based on set theory and on the so-called “mother structures” of mathematics. Piaget linked these fundamental structures of mathematics with the stages of early mathematical thinking, as revealed by psychology. The claim of alignment between Bourbaki’s mother structures and Piagetian theory provided a strong argument for a substantial reform of mathematics education. Chapter 4 After a succinct description of the meeting opportunities for mathematics educators up to the 1950s, this chapter describes how, in the wake of the New Math/modern mathematics reform movement, meetings have become a fundamental tool for focusing on problems and potential of reform proposals. Bodies that have played the most relevant roles are ICMI, CIEAEM, OEEC/OECD, and UNESCO. In the conferences that followed the Royaumont Seminar, particular interest was turned to the search for new axioms for geometry, with many proposals and discussions. But modern mathematics was not just this; in other places, the attention was turned to more general questions of a methodological and social nature. This congress season has fostered the creation of new traditions such as the birth of journals specialized in mathematics education, and periodic conferences on mathematics education, as exemplified by the four-year ICMEs. Chapter 5 The reform of mathematics teaching in France in the 1960s and 1970s was one of several reforms which affected the disciplines of primary and secondary education at the time, while the school system structures were also profoundly modified. It was, however, in its course, scope, successes, and difficulties, a reform different from the others, which was emblematic of the period. Considering the reform of modern mathematics within the global dynamics of redefinition of the curricula, I will place it at the crossroads of different ambitions and requirements of modernity to better grasp its characteristics and aims. Firstly, I will deal with the new aims assigned to the French education system after World War II within the framework of a project of cultural, social, and economic modernity of the country. Then, I will examine the reform movement of the teaching of mathematics in relation to that of français and then to that of science. Finally, I will focus on a particular moment of the reform of mathematics itself and will return to its ambition of modernity and its contradictions in the socioeconomic context of France. Chapter 6 Today’s German perception of the New Maths movement in West Germany is strongly shaped by the view that the movement was a transfer of American ideas to Western European countries, that it was solely a reform of elementary school teaching, and that it failed. To understand this perception better without getting bogged down in numerous details of the eventful 1950s, 1960s, and 1970s, we concentrate on seven beliefs—presented in subchapters—about the West German New Math reform. It turns out that this reform of mathematics instruction was to a far greater extent aimed at mathematics teaching in the German Gymnasium. To understand the peculiarities of West German New Math, it will be also essential to see it in the context of earlier reforms of mathematics education and institutional reforms of the post-war period. It turns out that for a better comprehension of the failure of the reform one should go back even further in time and include considerations about the unfinished Meraner Reform.

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Chapter 7 The 1950s in the United Kingdom were marked by social and political reforms, which also led to new conceptions of mathematics teaching and learning. For mathematics, the Association for Teaching Aids in Mathematics, founded in 1952 by Caleb Gattegno and like-minded people, and internationally fostered by the International Commission for the Study and Improvement of Mathematics Teaching, played an important role in this. The 1959 Royaumont Seminar served as a booster for curriculum change in the UK, bringing in influences from the continent as well as from the United States of America. In its wake, several projects with accompanying textbooks and in-­service teacher training programs emerged in the early 1960s. Most influential were the School Mathematics Project for secondary education and the Nuffield Mathematics Project for primary education, projects that were also implemented, in part and/or adapted, in some countries outside the UK. From the 1970s onward, criticism of the reform reverberated more loudly and led to the fall of the new mathematics paradigm in the UK. Chapter 8 In this chapter, we first describe the Italian context when the proposals of reforms, generically indicated under the label of “new math” or “modern mathematics,” were developed all around the world. The current mathematics programs dated to 1945, the main topic was geometry, taught according to a rooted tradition based on Euclid. The conference of Bologna in 1961, which followed those of Royaumont, Aarhus, and Zagreb-Dubrovnik, stimulated the Italian mathematicians to consider also for their country reforms in the light of the proposals that emerged at the international level. With the collaboration of the Ministry of Education and under the aegis of OECD, they organized refresher courses on the new approaches suggested by modern mathematics, edited books, and supervised experiments in selected classes. The plan by the Ministry was not efficient; however, this ferment stimulated various meetings for developing new mathematics programs. These programs were never implemented and only a few notions of modern mathematics remained, but new ideas and new contacts began to circulate which slowly changed the Italian context. Chapter 9 The application of modern mathematics ideas in Portuguese schools took place from the 1960s to the end of the 1980s. A first experiment in the higher grades of secondary school that started in 1963 laid the ground for the ways in which the reform was developed later. The compartmentalized nature of the educational system led the several subsystems to develop distinct concretizations of the new ideas, from primary school to the higher grades of secondary school. We argue that modern mathematics in grades 5 and 6 became essentially a linguistic endeavor, contrasted with the reliance on logic as the backbone for the reform in grades 10 through 12. Reformers for the technical schools struggled to accommodate real-world applications into the abstract flavor of mathematics fostered by the new trends. In grades 7 through 9, curricular change meant the introduction of transformational geometry together with a rephrasing of old content into set theory. For each of these cases, program content, textbook implementation, and teacher formation are discussed. Chapter 10 The modern mathematics movement in Belgium is inextricably linked to Georges Papy, a flamboyant and uncompromising professor of algebra at the Free University of Brussels. From the late 1950s, Papy reshaped the content of secondary school mathematics by basing it upon the unifying themes of sets, relations, and algebraic structures. Meanwhile, he innovated the pedagogy of mathematics by functionally interweaving his rigorous discourse with multi-colored arrow graphs, filmstrips as nonverbal proofs, and playful drawings, as manifested in his revolutionary textbook series Mathématique

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Moderne. From 1961, his Belgian Centre for Mathematics Pedagogy coordinated various reform actions: Curriculum development, classroom experiments, and in-service teacher training. Although the Belgian mathematics education community was divided about Papy’s agenda, zeitgeist, media propaganda, and political support made it possible for Papy to realize his reform almost entirely. After the generalized and compulsory introduction of modern mathematics in Belgian secondary schools in 1968–1969, the primary schools followed in the 1970s. Chapter 11 New Math in the Netherlands will be described from the perspective of the ideals that were held within the institutes, instrumental in shaping Dutch education. In line with Bob Moon, and contrasting the analysis of some of the people who played a role in this history, we will describe the rise of realistic mathematics education as a realization of New Math, rather than as a breach with the past. Trust in mathematics and mathematicians played a role in the realization of a curriculum which accompanied the introduction of a modern school system, in 1968—supplanting the nineteenth-century system. Pillarization of Dutch society resulted in (1) a large number of stakeholders who all had their own ideas and power bases; and (2) the necessity to find common ground, which was found in a focus on individual learning processes and stressing the need for developing individuality. This gave Dutch New Math its distinct flavor. Chapter 12 After a seminar on new thinking in school mathematics held in Royaumont in 1959, four Nordic countries, Denmark, Finland, Norway, and Sweden, agreed to cooperate on school mathematics reform. A joint committee, the Nordic Committee for the Modernization of School Mathematics, declared a need for revising aims and content. Concepts from set theory, and the function concept, as well as greater precision in presentation, could promote interest, insight, and understanding of the subject. Working teams for three school levels: Grades 1–6, 7–9, and 10–12, wrote directives for joint experimental texts and teacher guides. A total of 1310 classes in the four countries took part in experimental instruction. More than 180,000 copies of experimental texts were produced. The Nordic cooperation on modernizing mathematics teaching was a remarkable experiment on the cooperation of independent nations. Gradually, each nation went its own way at grades 1–9, where comprehensive nine-year compulsory education was underway in each country. The experiments initiated a longneeded discussion about curriculum, stagnated in certain routines and topics, and had an impact on curriculum development in the redefined school systems. At grades 10–12, steps were taken to create coherence between the gymnasia and university level. Chapter 13 Efforts to modernize Polish mathematics education began in the first half of the twentieth century. After 1957, the moving spirit of the Polish reforms was Zofia Krygowska; a description of her role in Poland and in international fora is augmented with explicit quotations from her books and articles. In 1967, under a strong influence of the French and Belgian versions of New Math, a radical reform of Polish secondary mathematics education was introduced, followed by an equally radical reform of primary education. Unfortunately, the implementation of the latter was combined with a fundamental change of the whole 12-year schooling system to an unclear 10-year system. In 1980, when the reform reached grade 4, the Solidarity movement forced the government to abandon it; yet, some changes were irreversible. In 2008, the last remains of New Math disappeared from the Polish core curriculum. In the concluding part of this chapter, unique phenomena of New Math reforms and its ideology are discussed.

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Chapter 14 In Hungary, a reform movement called Complex Mathematics Education was led by Tamás Varga from 1963 to 1978. The Complex Mathematics Education reform is clearly inscribed in the context of the international reform movements of the New Math period but also bears specificities which can be traced back to a local, Hungarian tradition of mathematics and mathematics education: a heuristic epistemology of mathematics, and an approach to the teaching of mathematics which can be described with the terms “guided discovery.” With his work on this reform curriculum and the related experimentations, Tamás Varga himself contributed also to the development of international research on mathematics education, especially in the domains of logic, combinatorics, and probability. In this chapter, we present a brief chronology of the reform; an analysis of its historical context and the different influences which shaped it; the characteristics of the curriculum; and the related resources and expected teaching practices. We illustrate with some examples how the reform combined lessons from the New Math movement with local, Hungarian influence. Chapter 15 Yugoslavia was a young country when the Royaumont Seminar took place in 1959, a seminar that emerged from the New Thinking in School Mathematics initiative. This chapter, on Yugoslavia, seeks to illuminate that part of the mathematics education history of the country by looking at the three mathematicians who all contributed to a specific view of mathematics, mathematics education, and its practical manifestations. The contributions of Đuro Kurepa, Miloš Radojčić, and Judita Cofman are explored, and their lasting impact on mathematics education in the framework of the New Math in Yugoslavia and further away, on a global scale. Kurepa’s influence and prominence are explored through his role with the ICMI and his global networks; Cofman’s influence through her work in Europe: Germany and England—as much as her work in Yugoslavia; and Radojčić’s work reached Africa through the non-­aligned movement. Their contributions to mathematics education were diverse. Kurepa concentrated mostly on delineating the principles of teaching and learning and had leading roles in many institutions, nationally and internationally. Radojčić’s contribution to the philosophy of mathematics education was deeply colored by his epistemology based on anthroposophical principles. Finally, and perhaps the most enduring, was Cofman’s contribution to mathematics education in Serbia, England, and Germany, on structuring the learning of mathematics around problem solving. Chapter 16 In the Soviet Union, a reform movement in mathematics education was triggered by Andrey Kolmogorov in the 1970s, and it was followed by a counter-reform. This movement was rooted in the very different socioeconomic conditions of that time and place and followed a strategy with significant contrasts to similar programs in the United States, England, and France. This provides an interesting case study that may illuminate the way such movements arise and succeed or fail, and, at the social level, certain fundamental commonalities of constraints as well as significant differences according to local conditions. We shall show that the principal reasons for the failure of the Kolmogorov reform were political: (a) The reform ignored the reality of the socioeconomic conditions of the country; (b) The human factor was ignored, and very little attention was given to professional development and retraining of, and methodological help to, the whole army of teachers; and (c) An attempt to transfer mathematical content and methods from the highly successful advanced extension stream for mathematically strong and highly engaged children to mainstream education was an especially grievous error.

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Chapter 17 The influence of the 1959 Royaumont Seminar on US mathematics education is described. Five periods are discussed: The 1960s New Math, the post-­New Math of the 1970s and 1980s, the 1990s reform, the 2000s standards-based era, and the 2010s Common Core State Standards. The landmarks are initially described by tracing the teaching of geometry, but then other key Royaumont themes are developed: The use of set theory, logic, and mathematical structure; the use of problem solving and inquiry; and the role of research mathematicians. The views of three European mathematicians, Jean Dieudonné, René Thom, and Hans Freudenthal, whose writings in US publications stimulated important discussions, provide a second lens for examining the streams of thought that emerged. Examples from US mathematics education research, state policy documents, and instructional materials are given to illustrate the impact of the Royaumont meeting on the USA. Chapter 18 Not often cited as one of the countries making major contributions to new/modern math movements in school mathematics, the case(s) in Canada may seem of lesser interest to the international community. However, in this chapter, we show how the multiple forces acting on school mathematics curricula across the country led to a slow brew of quiet change. In addition to surveying existing writings on various Canadian contexts, we also examine some book and article publications written in the early 1960s that shed some light on the intricacies of this quiet change. We also highlight some of the significant reasons for Canada’s slow pace. Chapter 19 The “modern mathematics” reform (New Math) in Latin America is described, with an emphasis on its main international agents on the continent: The Inter-­American Committee of Mathematics Education (CIAEM) and the conferences that this organization nurtured. A distinction between the first four conferences that tried to propagate the reform and the fifth one is documented. In the latter, a separation from the New Math was evidenced, and it began a new stage in the evolution of these agents. The particular experience of the reform in Costa Rica is included not only to provide details of a special case but to highlight characteristics that to some extent were also present in other countries of the region. The reform in Costa Rica will be contrasted with a new and very ambitious mathematical reform that the country launched in the second decade of the twenty-first century. It can be seen as a “tale of two reforms.” The concluding remarks summarize some results that New Math (ideas and developments, or reactions toward them) provoked so far as the teaching of mathematics was concerned. Finally, comments on some elements of the current situation of CIAEM are offered. Chapter 20 This chapter discusses Brazil’s participation in international movements on the mathematics curriculum and how it shaped local proposals for mathematics teaching. We focus on three fundamental moments in the history of mathematics education in the twentieth century. The first was the creation of the International Commission on Mathematical Instruction in the early twentieth century and Felix Klein’s proposal of merging the different mathematical branches in the school curriculum. The second, known as the modern mathematics movement, resonated between the 1960s and 1980s. The third was the international mobilization launched by the World Conference on Education for All in 1990, which promoted the reorganization of the school curriculum around competencies development. By taking a look at different eras, we seek to understand how national debates and practices in mathematics teaching embraced international thinking before, during, and after the modern mathematics movement. The international becomes national, with different justifications. It appears that modernization

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is a recurring theme in curriculum reforms in Brazil. At different times, for changing reasons, different actors have advocated local reforms to bring mathematics teaching up to date with the most recent international trends. Chapter 21 Australia’s physical isolation from Europe and North America during the second half of the twentieth century meant that the main “New Mathematics” messages which emerged from the Royaumont Seminar of 1959 were slow to reach its shores. Nevertheless, in the 1950s and 1960s, primary school teachers in Australian schools were challenged to make greater use of structured aids such as Cuisenaire rods and Multibase arithmetic blocks, and the leaders of that movement were two European reformers—Caleb Gattegno and Zoltán Dienes—who both established groups of followers in some of the Australian states. At that same time, the ideas of Jean Piaget also became better known. But a lack of national organization and participation in international forums meant that any changes tended to be local, and a long-established “colonial echo” of British traditions continued to hold sway. The chapter closes with an outline of the work of the Schools Mathematics Research Foundation (SMRF) in Victoria, which, in keeping with the Royaumont themes, emphasized the importance of the language of sets, of structure, and of functions in upper-­secondary school mathematics. Another change was the move away from the canonical school mathematics curriculum, with its separate “subjects” of arithmetic, algebra, geometry, trigonometry, and calculus, toward more unified intended and implemented curricula. Chapter 22 Hong Kong joined the global trend of the New Math movement in the mid-­1960s, yet wrote its own story. Started first as a teaching experiment, the reform could not be contained on a small scale owing to the stakes involved. Yet Hong Kong managed to sail through, and eventually settle down with some harvest. It resulted in the localization of the mathematics curriculum and the growth of a professional community. In addition, Hong Kong strived to tackle teacher preparation, which is the heart of all educational reforms, through a long-term subtle change in its teaching culture. The notion of curriculum reform was conceived from an unconventional perspective. Teachers were gradually induced into curriculum initiatives in which a curriculum document was a summary of these changes rather than a blueprint for teachers to implement. Such an experience not only opened up a new horizon to future curriculum development but may also have shed light on curriculum as well as educational reform in general. Chapter 23 The implementation of modern mathematics in Morocco occurred in three main phases, from the early 1960s through the mid-1970s. In the first phase, the implementation mainly concerned the introduction of new vocabulary and symbols of set theory and algebraic structures into upper secondary education. In the second phase, which began in 1968, the latter notions were introduced in lower secondary education and were reinforced in upper secondary education by vector, affine, and analytical geometries. In the third phase, which began in 1971, there was a strengthening of modern mathematics by a restructuring of the content of the programs and by removing classical concepts of geometry from them. But, in 1975, the teaching of mathematics—which elicited a negative reaction from users of mathematics—was the subject of a national conference whose recommendations advocated changes in the mathematics programs taught in secondary schools so that they would be beneficial to the majority of students. Then, in 1978, the Minister of Education created a commission to discuss mathematics programs and made the necessary changes. Modern mathematics was abandoned gradually from 1983 to 1989. In this chapter, we describe the different stages and the peculiarities of implementation of modern mathematics in Morocco.

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Chapter 24 In this chapter, we examine the nature of the mathematics curriculum experienced in Ghanaian schools in the pre- and post-independence eras. In the early 1960s, when the school mathematics curriculum reforms reached Africa, there was a desire for change in all spheres of life, including education. There was also optimism that governments would initiate policies to change the educational systems to transform the nation’s youth into a completely literate working population for the rapid economic development of their country. The chapter discusses the nature of changes to school mathematics brought by the UK-led initiatives (grounded in the British School Mathematics Project tradition) and those that came with the USA-led initiatives (grounded in the American School Mathematics Study Group tradition). The chapter also examines the differences between the USA- and UK-led curriculum development approaches and their emphasis on content. Finally, factors which delayed the full implementation of the curriculum reforms, the criticisms of the reforms, particularities, and how the reforms have influenced the school mathematics curriculum in Ghana in the past five decades are discussed. Epilogue The New Math reform was a global phenomenon, which can be found in almost every country of the world. It may be seen as a manifestation of the universal character of mathematics. But equally, the uniformity of the New Math movement may be questioned. It could point to the dependency of non-­Western nations from ideas developed in some leading centers. The success of the New Math movement may even be linked to the longevity of colonial traditions. On the other hand, historians may look at the appropriation of the New Math ideology by local communities of scholars. The agency of local mathematicians and mathematics teachers may even be broadened to include a professed belief in the universality of mathematics, even if this does not fit local educational challenges.

Preface to the Series

Books in Springer’s series on the history of mathematics education comprise scholarly works on a wide variety of themes, prepared by authors from around the world. We expect that authors contributing to the series will go beyond top-down approaches to history, so that emphasis will be placed on the learning, teaching, assessment, and wider cultural and societal issues associated with schools (at all levels), with adults, and, more generally, with the roles of mathematics within various societies. In addition to generating texts on the history of mathematics education written by authors in various nations, an important aim of the series will be to develop and report syntheses of historical research that has already been carried out in different parts of the world with respect to important themes in mathematics education—like, for example, “Historical Perspectives on how Language Factors Influence Mathematics Teaching and Learning” and “Historically Important Theories Which Have Influenced the Learning and Teaching of Mathematics.” The mission for the series can be summarized as: • To make available to scholars and interested persons around the world the fruits of outstanding research into the history of mathematics education • To provide historical syntheses of comparative research on important themes in mathematics education • To establish greater interest in the history of mathematics education The present book is an important addition to the series. Chapter authors tell the story of worldwide developments before, during, and after what has become known as the period of the “New Math” or “modern mathematics.” Early chapters raise the question whether developments in the United States of America and Europe need to be seen as having different origins and aims. Many of the authors of chapters focus on how a seminar held in Royaumont (France) in 1959 brought together US and European mathematicians. Many of the leading participants at the seminar were wedded to Bourbakist views that school and college mathematics curricula needed to be brought into line with major mathematical developments of the past 100 years, especially those relating to algebraic structures, mathematical terminologies and notations, and geometry. Most of the chapters offer penetrating analyses regarding the spread around the world of ideas expressed at the Royaumont Seminar. Chapter authors in this book make reference to extensive supporting literatures. Readers are shown how thinking about modern mathematics in various countries differed, depending on the perceptions and involvement of m ­ athematicians, mathematics educators, mathematics teachers, education administrators, politicians, and captains of industry. Of special interest, however, are comments on betweencountry effects of events and interactions related to questions like: “Given Nation X’s unique cultural and historical background, what should be the intended mathematics curricula in its schools?” “Should the intended curricula be the same for all learners?” And, “Who should be responsible for bringing about changes to school mathematics curricula?” There is a common theme—how did different nations respond to the push for “modern mathematics,” especially during the period 1945–1980?

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We congratulate Dirk De Bock for his achievement in bringing this work to publication. A huge amount of very scholarly work was involved. Thank you, too, to all the contributing authors. We are proud to have this book as part of Springer’s History of Mathematics Education series. We hope that the series will continue to provide a multilayered canvas portraying rich details of mathematics education from the past, while at the same time presenting historical insights that can support the future. This is a canvas which can never be complete, for today’s mathematics education becomes history for tomorrow. A single snapshot of mathematics education today is, by contrast with this canvas, flat and unidimensional—a mere pixel in a detailed image. We encourage readers both to explore and to contribute to the detailed image which is beginning to take shape on the canvas for this series. Any scholar contemplating the preparation of a book for the series is invited to contact Nerida Ellerton ([email protected]), in the Department of Mathematics at Illinois State University, or Melissa James, at the Springer New York office. Normal, IL, USA 

Nerida F. Ellerton M. A. (Ken) Clements

Preface to the Book

Modern mathematics or New Math, as an educational phenomenon, originated in the 1950s in the United States of America and in Francophone Europe, in the euphoria of revival after World War II, and in the background of the Cold War. The American and European emerging movements, which evolved largely independently of each other in the 1950s, met at the legendary Royaumont Seminar (1959), the impetus for a global rollout, from the Americas to the whole European continent, from the Soviet Union to Ghana and Australia. It became a worldwide reform movement, perhaps the most radical that school mathematics had ever seen. Curricula converged from computational techniques and Euclidean geometry to a more abstract approach based on set theory, algebraic structures, and topology. Not only secondary education became involved; in quite a few countries modern mathematics also affected primary education and even kindergarten. It all occurred in no more than two decades. This could only be realized by a widely orchestrated and mediatized action engaging the diverse stakeholders: Mathematicians, mathematics teachers, educational officials, school boards, and publishing houses. Although in retrospect, many regarded modern mathematics as a fascinating yet failed experiment, no one will deny the strong influence it had exerted on the way school mathematics was taught and perceived. Despite that, the historiography of the movement is still in its infancy. Until the end of the twentieth century, only a limited number of authors had mapped the modern mathematics movement in their country, usually with only national objectives. A first attempt at an international overview, albeit on a European scale and focusing on primary education, was undertaken by Bob Moon in his oft-­quoted book The “New Maths” Curriculum Controversy: An International Story (1986). In the first decades of the twenty-first century, however, interest in the history of mathematics education in general, and in the phenomenon of modern mathematics in particular, increased considerably. Witness, among others, the various scientific conferences at which studies on the history of teaching mathematics can be presented—in particular, the specialized series of biennial International Conferences on the History of Mathematics Education, launched in Iceland in 2009. In these circles, scholarly interest in the historical phenomenon of modern mathematics became common, with several studies on different aspects of national reforms in the 1960s and 1970s, especially but not exclusively related to countries that had played a pioneering role in the movement (the United States of America, France, Belgium, etc.). At these specialized meetings, the idea also arose to unite forces and confront and integrate research findings on this key movement in the history of mathematics education in the second half of the last century, resulting in this edited volume. This volume basically consists of two parts. The first part documents the origins of modern mathematics on both sides of the Atlantic. In short, it can be concluded that the European debates were mainly related to a structural Bourbaki view on mathematics, while the American reform movement was stronger rooted in socioeconomic and political motives and from the start driven by the government. The European and American positions met briefly at the Royaumont Seminar (1959) and subsequent international meetings (organized by international bodies such as OECD, ICMI, and UNESCO), yet will largely go their separate ways afterward. The second and most extensive part of xxxi

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this volume analyzes how different countries or groups of countries incorporated the reform into their own educational systems and culture. It paints a picture of a movement with shared principles and beliefs, but which were implemented in very different ways. The authors of this volume have diverse scientific backgrounds; they are experts in the history of mathematics education, research mathematicians, or mathematics educators, and their relationship to the subject also varies (from “actor,” in some way or another, to neutral observer-researcher). Authors’ different backgrounds are also reflected in the nature of the different approaches to the subject. Some chapters offer a detailed historical account of events in their socioeconomic and political context; others are more reflective in nature, making comparisons with other historical reform movements, and/or providing insights that can support future developments. As the editor of this volume, I am indebted to several individuals and bodies: My colleagues and friends in the field of the history of mathematics education whom I regularly met at international conferences and who encouraged me to take the lead in this challenging publication project; the authors of the different chapters who responded enthusiastically to my call for collaboration (and were willing to revise their initial submissions); the editors of Springer’s History of Mathematics Education series who immediately favored the project; and the international scientific publisher Springer, who offered me this publication opportunity, and with care and flexibility molded this volume into its final form. Last but not least, I want to thank the board of the Faculty of Economics and Business of the KU Leuven, my employer, who provided me with the necessary time and resources to work on this project. Leuven, Belgium  Dirk De Bock December 2021

Author Biographies

Mária  Cristina  Almeida  is a teacher of mathematics in a secondary school in Portugal. She received her Ph.D. in mathematics education from Universidade Nova de Lisboa in 2013. She is the President of the Working Group on History and Memories of Mathematics Education of the Portuguese Association of Teachers of Mathematics. Currently, her scholarly interests span several areas, including the history of mathematics education, history of mathematics teacher education, and history of mathematical problem solving.

Jerry  Becker  received his doctorate under Professor E.  G. Begle at Stanford University. His research foci are problem solving in school mathematics, international mathematics education, mathematics teacher education, and mathematics curriculum development. He has published widely in various peer-reviewed journals and has organized and participated in numerous conferences and seminars that deal with problems and issues in mathematics education. He teaches courses in mathematics teacher education and advises both master’s and doctoral students. He has developed a number of email distribution lists and frequently posts notes on research, hot issues in mathematics education, announcements of national and international conferences and seminars, and special requests from mathematics education worldwide. Danny  Beckers  studied mathematics and cultural history at Nijmegen University. In 2003, he earned his PhD on a thesis that delved into the reasons for mathematics becoming a mandatory course in Dutch education, in the early nineteenth century. He published on various aspects of Dutch (mathematics) education, from late medieval until modern times. Currently, he teaches courses in history of mathematics, history of computer science, and history of AI to students at the Vrije Universiteit Amsterdam. His research projects are in history of education. He is writing a double biography of the Freudenthal couple and a history of mathematics education in the Netherlands, with a specific focus on educational practices and ideals.

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Kristín Bjarnadóttir  is a professor emerita at the University of Iceland, School of Education. She earned her BA degrees in physics and mathematics at the University of Iceland in 1968, MSc degree in mathematics at the University of Oregon in 1983, and PhD degree in mathematics education at Roskilde University, Denmark, in 2006. Her research concerns the history of mathematics education, in particular “Modern Mathematics.” Previously, she taught mathematics and physics at secondary schools in Iceland. She led the writing of Iceland’s national mathematics curricula for primary and secondary school levels, published in 1999. She has translated and been co-author of mathematics textbook series for lower secondary level, and she authored a textbook on discrete mathematics for upper secondary level. She wrote the chapter History of Arithmetic Teaching in The Handbook on the History of Mathematics Teaching (2014), and she has written numerous articles on the history of mathematics education in international journals and conference proceedings. She chaired the Nordic LMFK-congress Reykjavík in 1990, and she has been on the standing committee of biannual international conferences on the history of mathematics education, since the first one, held in Gardabaer, near Reykjavík, Iceland, in 2009. Dirk De Bock  is a professor of mathematics in the Faculty of Economics and Business of the University of Leuven (Belgium). His major research interests are history of mathematics education, psychological aspects of teaching and learning mathematics, the role of mathematics in economics and finance, and financial literacy. His recent research in the field of history of mathematics education focused on the role of Belgian mathematicians and mathematics teachers in the international modern mathematics reform movement of the 1960s, and has led to the 2019 monograph Rods, Sets and Arrows: The Rise and Fall of Modern Mathematics in Belgium, co-authored with Geert Vanpaemel, in Springer’s History of Mathematics Education Series. Ongoing research projects include the study of New Math or modern mathematics from an international perspective. Alexandre  Borovik  is a professor emeritus at The University of Manchester, UK, where he was a professor of Pure Mathematics since 1998 to his retirement in 2020. In 40+ years of his academic career he taught in universities in Russia, USA, UK, and Turkey—countries with very different education systems, and different socioeconomic and political environments for mathematics education. He is primarily a hardcore mathematician who published three monographs (as well as textbooks, etc.) and 95 papers, and who continues to pursue challenging research programs. He is also interested in methodology of mathematical practice in all its aspects, including teaching of mathematics and social, cultural, and political pressures on mathematics and mathematics education, and has published papers addressing these issues.

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Elisabete Zardo Búrigo  is a full professor at the Federal University of Rio Grande do Sul, Brazil. She has conducted research on teacher education, curriculum, and the history of mathematics education. Modern mathematics movement is one of her favorite research topics, addressed in national and international events and publications.

M. A. (Ken) Clements  has held academic appointments in six universities, located in three nations. In 2019 he retired after being professor within the Mathematics Department at Illinois State University (ISU) for 15 years. He is now an emeritus professor at ISU.  He has served as a consultant and as a researcher in many nations, and as co-editor of three International Handbooks of Mathematics Education—published by Springer in 1996, 2003, and 2013. He and his wife, Nerida Ellerton, have co-authored many books and refereed articles on mathematics education. He is honorary life member of both the Mathematical Association of Victoria (MAV) and the Mathematics Education Research Group of Australasia (MERGA). Nerida F. Ellerton  was a professor within the Mathematics Department at Illinois State University (ISU) between 2002 and 2018, and is now an emeritus professor at ISU.  She holds two doctoral degrees—one in Physical Chemistry and the other in Mathematics Education. Between 1997 and 2002 she was Dean of Education at the University of Southern Queensland, Australia. She has taught in schools and has held academic appointments in at four universities, and has also served as consultant in many countries. She has written or edited 18 books and has had numerous articles published in refereed journals or in edited collections. She and Ken Clements are joint editors of Springer’s History of Mathematics Education Series. Fulvia Furinghetti  is a retired full professor of Mathematics Education at the University of Genoa (Italy). Her research concerns mathematics education (computer science in mathematics teaching, proof, beliefs, teacher education, integration of history in mathematics education, students’ difficulties in algebra and problem solving, and public image of mathematics) and the history of mathematics education (journals on mathematics education, ICMI, modern mathematics, teacher 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 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 hundred years of ICMI. In 2001–2004 she chaired the International Study Group on History and Pedagogy of Mathematics (HPM) affiliated to ICMI.

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Hélène Gispert  is an emeritus professor of History of science at Paris-­ Saclay University since 2017, and is affiliated to the Research Unit Études sur les sciences et les techniques. She is an historian of 19th–20th centuries mathematics. Her main research topics, that she conducts in a social and cultural historical approach, include the circulation of mathematics via journals, the history of mathematical training, and the history of analysis. She also worked more broadly on the history of mathematics in France during the Third Republic (1870–1940), studying its productions, its actors, and its institutions.

Katalin Gosztonyi  is an assistant professor in mathematics education at the Eötvös Loránd University of Budapest, Hungary since 2016. She defended her PhD under joint supervision, in France and in Hungary in 2015, on the comparison of the French and Hungarian mathematics educational reforms of the New Math period. She works in mathematics teacher education and conducts research on Inquiry-Based Mathematics Education, especially the Hungarian “guided discovery” approach, on the teaching of discrete mathematics and on the role of history in mathematics education. She is scientific coordinator of the MTA-ELKH-ELTE Research Group in Mathematics Education founded by the Hungarian Academy of Sciences. She is member of the Board of the European Society for Research in Mathematics Education since 2021. Bill Jacob  is a professor emeritus of Mathematics at the University of California, Santa Barbara. He received his Ph.D. from Princeton University in 1979 under the direction of Simon Kochen. His research in pure mathematics has focused on the algebraic theory of quadratic forms and the structure of finite-dimensional division algebras. In 1987 he received an American Mathematical Society Centennial Fellowship. Throughout his career he has collaborated on numerous professional development projects for teachers and developed and taught courses for pre-service teachers. For two decades he worked with Mathematics in the City at City College New  York and is a collaborating author of the Contexts for Learning Mathematics K–5 instructional program. Ezzaim Laabid  is a professor of mathematics and history and epistemology of mathematics at the Cadi Ayyad University of Marrakesh. He is a member of the Interdisciplinary Research Laboratory in Didactics, Education and Training (LIRDEF) of the École Normale Supérieure. He is also a member of the African Mathematical Union Commission on the History of Mathematics in Africa (AMUCHMA) of which he was secretary from 2013 to 2016 and the president for the period 2017–2022. His PhD thesis, defended in 2006, focused on the relationships between mathematics and the science of inheritance in medieval times in Muslim West. His current research focuses on the history of Arab mathematics and the history of mathematics education. He has published around 20 chapters and papers on history of mathematics and mathematics education.

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Snezana  Lawrence  is a Senior Lecturer at the Department of Mathematics and Design Engineering, Middlesex University and is involved in various national and international initiatives to promote the use of the history of mathematics in mathematics education. She is the Chair of the History and Pedagogy of Mathematics International Study Group (an affiliate of the International Mathematics Union for 2020– 2024) and is a Diversity Champion of the Institute of Mathematics and Its Applications (UK). Snezana is the co-­editor (with Mark McCartney) of Mathematicians and Their Gods (2015, Oxford University Press) and her book A New Year’s Present from a Mathematician was published by Chapmann & Hall (2019, CRP Press). Snezana is Series Editor (History) for Recreational Mathematics, CRC Press/Routledge. José Manuel Matos  taught at a Normal School and for some years he was a high school mathematics teacher. For twenty years he taught at the New University of Lisbon. Recently he was a visiting professor at the Federal University of Juiz de Fora, Brazil. He completed his doctorate at the University of Georgia and held several positions several Portuguese research societies. He was editor of the first Portuguese research journal in Mathematics Education and coordinator of a research center in education. He has integrated research teams focused on mathematics learning, math class culture, school success, and historical studies and is the author and editor of several research books on these subjects. Marta  Menghini  is an associate professor in the Department of Mathematics of the University of Rome Sapienza. She is the author of numerous works in the fields of Mathematics Education, History of Mathematics, and History of Mathematics Education. She was the chief organizer of the international Symposium on the occasion of the centennial 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 on Elementary Mathematics from a Higher Standpoint (2016). Damian Kofi Mereku  has been a professor in Mathematics Education at the University of Education, Winneba (UEW), since 1995 when he completed his PhD research in Leeds University, UK. His research and teaching have been in the fields of curriculum studies in mathematics education, curriculum development, and continuing professional development (CPD) of teachers. He served as a Dean as well as the Head of Department in the Faculty of Science Education, UEW, for several years. He has authored books in curriculum studies and co-authored refereed articles in mathematics education, including Five Decades of School Mathematics in Ghana and The Ghana Country Reports for TIMSS 2003 and 2007. He has been on the national mathematics panel which revised the mathematics syllabuses for primary, junior, and senior high schools in Ghana since 1998.

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Bob  Moon  was a secondary school teacher and head teacher before moving into higher education. He was a professor of Education at the UK’s Open University from 1988 to 2010 where he was head of the Department for Curriculum and Teaching Studies. He founded The Curriculum Journal in 1990 and served as Chair of the Editorial Board, and then Editor, until 2012. He has been advisor to many national and international governments and organizations and from 2015 to 2020 he was Education Specialist Adviser on research to the UK’s Department for International Development (DFID). In 2018, in the Queen’s Birthday Honours, he was awarded a CBE for his services to education in developing countries.  David Pimm  was born in 1953 in London, England, and grew up there and started secondary school in September 1964. His year at Latymer Upper School was the last of “old math,” the following year of students started using Richard Skemp’s series of books entitled Understanding Mathematics. Across desks in school he subsequently saw scrawled “Skemp is hard,” “Skemp is mad,” and “Skemp is impossible.” David only briefly engaged with some sort of “new maths” (namely, set theory, group theory, two-by-two, and three-by-three matrices) in the fall of 1970, preparing for the Cambridge entrance examination in November that year. Attending the University of Warwick from 1971 to 1974 was the first occasion in which he actually engaged in depth with new(-ish) maths, such as group theory and linear algebra, including the analytic “creeping lemma,” a sort of continuous induction, and new and more accessible proofs in group theory (from the late 1950s). David Lindsay Roberts  has master’s degrees in mathematics and industrial engineering and a PhD in the history of science. He has taught mathematics and history at a variety of post-secondary institutions and is currently an adjunct professor of mathematics at Prince George’s Community College in Largo, Maryland, USA.  His historical research has focused on mathematics education in the United States from independence to the present. He is the author of Republic of Numbers: Unexpected Stories of Mathematical Americans through History, published by the Johns Hopkins University Press in 2019.

Leo Rogers  is a founding member of the British Society for the History of Mathematics and founder of the International Study Group on the History and Pedagogy of Mathematics (HPM). He has taught in primary and secondary schools in England, and as a trainer of teachers has worked with pupils and teachers on a number of European Community curriculum and research projects. His principal interests are the historical, philosophical, and cultural aspects of mathematics as they relate to the development of curricula, mathematical pedagogies, and individual learning.

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Angel  Ruiz  has been a professor and researcher at the University of Costa Rica since 1975. He is the (co-)author or editor of 36 books and more than 200 articles on the History and Philosophy of Mathematics, Mathematics Education, and Social and Political Philosophy. In the last decade, he co-produced over 200 on-line multimedia resources (courses and videos) for the Teaching of Mathematics for curriculum reform implementation. He was Vice President of the International Commission on Mathematical Instruction for two terms (2010–2016), member of the Commission for Developing Countries of the International Mathematical Union (2011–2018), Vice President (2003–2007), and President (since 2007) of the Inter-­ American Committee of Mathematics Education, Director of the Mathematics Education Network for Central America and the Caribbean (2012–2021). Angel led the team that designed the Costa Rican school mathematics national curriculum (grades 1–12) in force since 2012 and then he has nurtured its implementation as Director of the Mathematics Education Reform in Costa Rica Project of the Ministry of Public Education of Costa Rica. Zbigniew  Semadeni,  mathematician, worked in functional analysis (Banach spaces of continuous functions), theory of categories and functors, primary mathematics education, and in philosophy of mathematics; worked at the University of Poznań (Poland) in 1954–1961, at the Institute of Mathematics of the Polish Academy of Sciences (1962–1986, Deputy Director 1973–1985), and at the University of Warsaw (1986–2004; Director of the Institute of Mathematics 1991–1996); now a professor emeritus. Visiting professor at the University of Washington, Seattle (1961– 1962), York University, Toronto (1982–1983), University of Sydney (1984), University of California, Davis (1989–1990); is a member of the Executive Committee of the International Commission on Mathematical Instruction (ICMI) 1979–1982, vice president of ICMI 1983–1986; is author or coauthor of mathematics textbooks for grades 1–3 of Polish primary school (1990–2003) and books for teachers; and participated in all International Congresses on Mathematical Education (ICME) from 1969 (Lyon) to 2004 (Copenhagen). Nathalie  Sinclair  was born in 1970  in Grenoble, France, when her Canadian parents were there on sabbatical before returning to Calgary, Alberta, where her father taught. As a result, starting in 1976, she had no direct experience in elementary school in Calgary of what used to be a version of the new math. Her first direct encounter with “new maths” (apart from undertaking abstract algebra courses at McGill University) did not come until much later, when she used Lore Rasmussen’s 1960s Miquon math lab materials (see https://miquonmath.com) with her very young daughter, which included extensive involvement of Cuisenaire rods and strings used to explore Venn diagrams.

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Man-Keung Siu  obtained his BSc degree in mathematics/physics from the University of Hong Kong and earned his PhD degree in mathematics from Columbia University. Like the Oxford cleric in Chaucer’s The Canterbury Tales, “gladly would he learn, and gladly teach,” he still enjoys doing that after retirement in 2005 from more than three decades of teaching and learning at a university. He has published some research papers in mathematics and computer science, some more papers in history of mathematics and mathematics education, and several books in popularizing mathematics. In particular he is most interested in integrating history of mathematics with the teaching and learning of mathematics, actively participating in an international community of History and Pedagogy of Mathematics since the mid-1980s. He has offered a course on liberal studies titled Mathematics: A Cultural Heritage as well. Wagner Rodrigues Valente  is an associate professor at the Department of Education, Federal University of São Paulo, Brazil. He is also the president of the Associate Group for Studies and Research in History of Mathematics Education—GHEMAT Brazil (ghemat-brasil.com.br). He has written and co-authored several books and reference articles on the history of mathematics education. As a researcher he has coordinated international cooperation projects between Brazil and countries, such as Portugal, Spain, Switzerland, and France. He is currently developing research on the history of the professional knowledge of mathematics teachers. Geert  Vanpaemel  is a professor emeritus in history of science at the University of Leuven (Belgium). His research concerns the history of science in Belgium since 1500, the history of universities and education, and the popularization of science. He has published on the mathematical culture of the seventeenth century, in particular, with respect to the Jesuits, the history of statistics during the nineteenth century, and the introduction of modern mathematics in secondary schools. He recently finished a book on the history of nuclear science in postwar Belgium.

Author Biographies

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Ysette  Weiss  is a professor at the Mathematics Department of the University of Mainz (JGU) since 2010. She studied and held academic appointments in eight universities in five countries — in east and west. In her research activities, she is interested in history of mathematics education and the use of history in mathematics teaching. Since 2013 she is a speaker of the Arbeitskreis Mathematikgeschichte im Unterricht in the German Society of Didactics of Mathematics (GDM) and since 2020 a member of the Advisory Board of the International Study Group on the Relations Between the History and Pedagogy of Mathematics (HPM). In particular her focus lies on reforms in the mathematics teaching of the last two centuries. Moreover, she works on concept formation from the perspective of activity theory, problem didactics, elementary mathematics and learning by discovery, multidisciplinary teaching, schoolbook and text analysis, concept development, and the use of language in the context of mathematical learning videos. Ngai-Ying Wong  obtained his B.A., M.Phil., PGCE, and Ph.D. degrees from the University of Hong Kong, and M.A. (Ed.) degree from the Chinese University of Hong Kong. He is an honorary professor at the Department of Curriculum and Instruction, the Education University of Hong Kong and former professor at the Chinese University of Hong Kong serving as Chair, Board of Undergraduate Studies, as well as Directors of M.A. in Student Activities and MSc. in Mathematics Education. Possessing ten years of mathematics teaching experience in secondary schools, he was founding president of the Hong Kong Association for Mathematics Education and mathematics convenor at the Hong Kong Association for Science and Mathematics Education. He was also involved in the local mathematics curriculum development, including the Holistic Review of Mathematics Curriculum in Hong Kong.

Chapter 1

Modern Mathematics: An International Movement Diversely Shaped in National Contexts Dirk De Bock

Abstract  We reflect on the non-parallel origins and development of modern mathematics, as an educational movement, and its American counterpart New Math. The 1959 Royaumont Seminar played a decisive role in bringing together American and European reformers, acted as a catalyst, but did not lead to substantial reform cooperation on either side of the Atlantic. We pay attention to the pluriform nature of the movement(s), shaped by national traditions, existing educational systems, and societies at large. Moreover, we characterize the modern mathematics reform movement and list some of its main features. Adri Treffers’ and Hans Freudenthal’s model of classifying different approaches to mathematics education into four ideal types proved helpful. We conclude with some reflections on the rapid demise of modern mathematics, which in our view should not be regarded as a total failure, but was a breeding ground for thorough reflection on mathematics education, nationally and internationally, and was the basis for the emergence of mathematics education as an autonomous scientific discipline. Keywords American movement · CIEAEM · Educational reform · European movement · Dissemination of reform · Hans Freudenthal · Internationalization · Modern mathematics · National developments · New Math · OECD · Reform movement · Royaumont seminar · School mathematics · SMP · SMSG · Structuralist mathematics education

Introduction Modern mathematics (or New Math(s), or new mathematics, etc.) refers to a rather short but drastic change in the way mathematics was understood and taught in Europe, the United States of America, and in various other countries around the globe (Kilpatrick 2012). “‘New Maths’ perhaps more than any other curriculum reform caught the imagination of the world at large” (Moon 1986, p. 8). The roots of modern mathematics, as an educational movement, were in the early 1950s, the peak of its influence was in the 1960s, and the enthusiasm for it, declined from the mid-1970s onward. A main feature of the movement was the introduction of new teaching content, and new teaching materials and practices as a response to the perceived poor state of mathematics education after World War II.  Inspiration for new approaches to mathematics teaching came not only from developments in D. De Bock (*) KU Leuven, Leuven, Belgium e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_1

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pre-­War research in pure mathematics, especially from the actions of the French “Bourbaki” group, but also, particularly in the Anglo-Saxon world, from new applications that had emerged or were being further developed during the War (Zwaneveld and De Bock 2019). Modern mathematics had various national faces, but in general, the emphasis shifted from developing technical–computational skills to insights into mathematical structure, often but not exclusively pursued by the study of abstract concepts such as sets, relations, algebraic structures, number bases other than 10, etc. Other characteristics were the replacement of traditional synthetic Euclidean geometry by an algebraic, affine, or vector-based approach (or combinations thereof), with a special focus on geometric transformations as objects of study in their own right and as tools for proving. The teaching of calculus (or analysis) became theoretically more rigorous by building it on the concepts of limit and continuity, defined in a topological environment. The changes in mathematical content were accompanied by pedagogical innovations, moving away from lecturing and focusing on students’ self-activity. New “structural” materials, such as the Cuisenaire rods, Dienes’ logic blocks, and Papy’s minicomputer, were carefully designed to stimulate students toward guided discovery and enhanced conceptual understanding. Outstanding, charismatic reformers (such as Max Beberman, Caleb Gattegno, Zoltán Dienes, Frédérique Lenger, Georges Papy, etc.) gave the so-called “model lessons” in their countries and beyond, convincingly showing how modern mathematics could be successfully taught to all students. In many countries, the reform affected primary, secondary, and tertiary education, albeit to varying degrees and in top–down order (Moon 1986). Typically, curriculum reform first entered universities and then was advocated for the scientific strands of upper secondary education (or high school), with the argument of reducing the gap between school and university mathematics. Next, the “non-­ scientific” strands of general education, technical education, and middle school came into the reformers’ spotlight. Finally, within a space of three to five years, arithmetic-mathematics instruction in primary (or “elementary”) schools, and even kindergarten became involved in the reform.



An American Cradle and a European Cradle

Regarding the origins of modern mathematics (New Math) as an educational model, different views circulate. Nadimi Amiri (2017b), reporting her doctoral research on the modern mathematics reform in Luxembourg, has posited that the movement started in the United States of America during the 1950s, as “a series of new reform programs, known as the ‘New Math reform’,” and later “travelled to Europe through the support of the Organisation for Economic Cooperation and Development (OECD)” (p. 738), unambiguously identifying the 1959 Royaumont Seminar as “the first event that officially started the New Math reform movement in Europe” (Nadimi Amiri 2017a, p. 89).1 Barrantes and Ruiz (1998) seem to endorse the thesis that modern mathematics in Europe was an import from the American New Math, although they situated the “crossing of the Atlantic” one year earlier when they stated: Even though in Europe, in the 1950s, there was intellectual concern regarding the teaching of pre-university Mathematics, the initial drive towards reform was given in Edinburgh at the International Congress of Mathematicians in 1958. After a report by five American participants representing various groups in the United States, a wave of opinion gave voice to the need for a reform in the methods of teaching Mathematics in Europe. (p. 1)  In 1959, at the time of the Royaumont Seminar, the OECD was called the Organisation for European Economic Cooperation, or OEEC. It was formed in 1948 to administer American and Canadian aid under the Marshall Plan for the reconstruction of Europe after World War II. As an OEEC initiative, the Royaumont Seminar was partly funded by U.S. money (OEEC 1961a). Once the Marshall Plan was complete, Canada and the U.S. joined the OEEC nations, which created the OECD on December 14, 1960. The OECD entered into force on September 30, 1961 (OECD 2020). 1

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According to Moon (1986), however, the claim that “a ‘wave’ of development in the USA ‘crossed over’ to Europe, although it is oft repeated, may be too simplistic a picture” (p. 46). Instead, Moon argues that the preface of the Royaumont report (OEEC 1961a) shows that “a pattern of ‘parallel’ innovation would be a more appropriate characterization” (pp.  46–47). Kilpatrick (2012) and Vanpaemel and De Bock (2019) endorsed Moon’s thesis. Moreover, Vanpaemel and De Bock (2019) have argued that the American and European movements did not have quite the same background motives and developed largely independently during the 1950s. They met briefly at the 1959 Royaumont Seminar and at subsequent international meetings, but once again evolved largely independently thereafter. In the first part of this volume, pre-Royaumont developments in the United States of America (Chap. 2) and in Western Europe (Chap. 3) are described and analyzed. These analyses provide further support for the thesis of two parallel movements. According to many popular accounts, mathematics reform in the United States of America began in 1958 when, in the wake of the Sputnik scare which gripped the American public, a School Mathematics Study Group (SMSG) was established and massively funded with the goal of producing a new mathematics curriculum and accompanying study materials for American students. The real story, however, is more complicated and nuanced. David Lindsay Roberts (Chap. 2), retracing this story to the end of 1959, shows that there were in fact a variety of curricular experiments, commonly referred to by the unifying term New Math. Complaints about the sorry state of mathematics in American schools and calls for modernization appeared as early as 1950. The first response was given in 1952 when the University of Illinois Committee on School Mathematics (UICSM) drafted and implemented a curriculum that logically developed mathematical ideas and emphasized deeper levels of understanding. In the mid-1950s, the UICSM project was followed by several other efforts to reform school mathematics. The main aim of these early efforts was to ensure that secondary schools would be encouraged to offer high-quality mathematical education, which would lead to more students enrolling in university mathematics and science courses. Therefore, school mathematics had to be brought into line with twentieth-century mathematical thinking, both in content and style, i.e., with attention to axiomatic structure, logic, rigor, and precision of language. The launch of Sputnik in October 1957 created the political climate for a significant strengthening of efforts, leading to the creation of SMSG led by Edward G. Begle who organized a nationwide network of authors to write curricular materials for each grade of secondary school and later for elementary school as well. Many curriculum modernizers of the late 1950s and 1960s, including Begle, believed that, given the right framework conditions, the new (improved) mathematics was suitable for all students, not just those destined for university studies (Begle 1971; Kilpatrick 2012). Beginning in 1950, a similar reform movement emerged in Europe. Although it coincided with what was happening at the same time in the United States of America, mutual influence is unlikely, though both movements were influenced to some extent by the work of Bourbaki. The organization that initiated the European reform movement was the International Commission for the Study and Improvement of Mathematics Teaching (CIEAEM), formally established in 1952 after Caleb Gattegno, an Egyptian-born mathematician and psychologist, had paved the way for it in the previous two years. In Chap. 3, the editor of this volume reconstructs the debates at the first meetings of this group, particularly the meeting in 1952 where the Bourbakists met the Swiss psychologist Jean Piaget. Debates resulted in the assumption of an alignment of mathematical and mental structures, which became a main argument for the reform of mathematics education in Europe: The Bourbaki model for the science of mathematics became a model for mathematics education. In the mid-1950s, CIEAEM debates also moved to national levels, particularly in France and Belgium. In Europe, the first systematic experiment with modern mathematics in the classroom was not carried out until the 1958–1959 school year (De Bock and Vanpaemel 2018)—shortly before the Royaumont Seminar took place in the fall of 1959.

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Compared to the American SMSG, CIEAEM operated on a much smaller scale, was not funded and not affiliated with any official body or government, and had no interest in implementing a large-­ scale reform. Despite its impressive name, CIEAEM was and always remained a small informal group, an inside group with little reach. Some of its members would take a leading role in the subsequent modern mathematics reform, but before 1959 this was not visible to outside observers. It is unlikely that many American reformers were aware of the existence of the CIEAEM during the 1950s.



Dissemination of the Reform

The 1959 Royaumont Seminar was a crucial gathering for the modern mathematics (New Math) movement for two main reasons. First, for the first time in history, European mathematicians such as Jean Dieudonné, Gustave Choquet, and André Lichnerowicz, who were members of or had a strong link with Bourbaki, as well as American reformers such as Edward G. Begle, were actually brought together to engage in an in-depth discussion about future avenues for school mathematics. Second, for most OECD countries, Royaumont marked the launch of the movement; in others, such as France and Belgium, Royaumont accelerated a reform that had been emerging during the 1950s. In several non-­ OECD countries, efforts to reform the school mathematics curriculum resembling those taken by OECD countries were undertaken during the 1960s and early 1970s. In the second part of this volume (Chaps. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24), the post-Royaumont implementation of modern mathematics is documented, including case studies of countries in all continents. These studies show the different faces of the reform as shaped by national traditions in mathematics education, educational systems in place, and societies at large. The Royaumont Seminar also motivated communication and internationalization in the world of mathematics education. In Chap. 4, Fulvia Furinghetti and Marta Menghini retrace the reform debate at international meetings during the 1960s. In the first half of that decade, the position and approach to geometry in schools were central to this debate. At Royaumont, Jean Dieudonné had initiated this debate with a proposal to free geometry from the legacy of Euclid, but this led to controversy rather than agreement among the participants. In a follow-up meeting to Royaumont, in Zagreb-Dubrovnik (1960), a group of experts who had been appointed to draft a modern mathematics program for secondary education agreed on the introduction of set theory, algebra, analysis, probability theory, and statistics, but regarding geometry education, the outcome was an ambiguous compromise. For the final years (15- to 18-years-olds), an axiomatic and structural approach was recommended, while in the early years (11- to 15-year-olds), the emphasis would be put on a more intuitive approach (OEEC 1961b). From the mid-1960s, the first classroom experiments with modern mathematics received considerable attention in international forums. In particular, the audacious approaches of Caleb Gattegno, Zoltán Dienes, and the Belgian Georges Papy, combining mathematical rigor with innovative pedagogies, received considerable attention and appreciation. At that time, modern mathematics was in full preparation in most Western European countries, especially those which belonged to the OECD. During the second half of the 1960s, modern mathematics spread rapidly worldwide. The reform debate had already reached a number of countries in Eastern Europe, Latin America, Africa, and Asia in the early 1960s, countries that were not part of the original OECD.  In 1978, the International Commission on Mathematical Instruction published a report “Change in Mathematics Education since the late 1950s,” which included 16 countries around the globe, but no reference was made to changes in any South American nation (Freudenthal 1978). The report documented the reform efforts in each participating country in the preceding two decades, the different directions, the varying degrees of success, and the influences of educational systems. Many reform projects in African and Asian countries took inspiration from either the American SMSG or from the British counterpart of SMSG, the School Mathematics Project (SMP). Best known is an SMP offshoot, the African

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Mathematics Program, commonly known as the “Entebbe Project” because in 1962 it organized a workshop in Entebbe, Uganda, in which 11 English-speaking African countries participated (Swetz 1975). In Latin America, where reform was promoted from the First Inter-American Conference on Mathematics Education (1961), the movement was a concoction of ideas from the SMSG, of input from European reformers such as Dienes, Lucienne Félix, Gattegno, and Papy, and of work by mathematicians and mathematical educators from the home continent (D’Ambrosio 1991).



Characterization of the Reform

The country- and region-specific chapters in the second part of this volume (Chaps. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24), as well as other review publications (Kilpatrick 2012; Moon 1986; Phillips 2015; Servais 1975; Vanpaemel and De Bock 2019), illustrate the pluriform nature of the international modern mathematics/New Math movement. Anglo-Saxon interpretations, for example, differed quite essentially from continental European ones (see also Karp’s (2008) interview with Geoffrey Howson). And even within the same parts of the world, implementations differed, depending on national cultures and local educational systems. In some countries, the reform of mathematics education was restricted to a limited number of experimental classes driven by highly motivated individuals; in others, the curriculum was strictly determined by a central authority, which left little room for teachers’ own initiatives (Vanpaemel et al. 2012). In an extensive survey of the various reform movements of the 1960s and early 1970s in continental Europe, Willy Servais (1975) concluded that continental Europe seems more homogeneous than it actually is. In the evolution of their mathematics education some countries have been bold, even rash. Others are advancing more cautiously, more patiently, more deeply. (p. 55)

The question arises whether a unique and comprehensive characterization of the modern mathematics movement is possible at all. Was there something like a common core to which all of these national (and subnational) reform efforts adhered? At the risk of being one-sided and partly betraying a well-intentioned reform movement, we make an attempt anyway. Evidently, there was the naive theory of sets, a “new” framework for a unified presentation of mathematics, and a starting point for teaching, both in content and in method. Many people associated the new mathematics primarily with the language of sets and its iconic representations of Venn and arrow diagrams that began to dominate textbooks from the mid-1960s onward (De Bock and Vanpaemel 2019). More essential than “sets and arrows,” however, seems to us the determination to shape mathematics education from the standpoint of mathematical structure(s), starting from poor structures and gradually constructing more rich structures (see also Chap. 3 in this volume). The latter led, among other things, to a kind of global algebraization of mathematics education, especially at the secondary level, to the prioritization of the affine viewpoint in geometry, and to the deletion of most of the synthetic (Euclidean) geometry of figures (Rouche 1984). The concept of structure was central to Bourbaki’s attempts, beginning in the 1930s, to reorganize mathematical science, but after World War II, it became a key instrument and a privileged language of “modern” science in general, both in the natural and social sciences (see, e.g., Gispert 2010). Bourbakists often advocated the role of structures as tools for mathematical discovery (see, e.g., Bourbaki 1948). Although structures, as used by this prominent group of research mathematicians to organize and advance their science, could not have the same meaning for learners, structures were seen in the 1950s and 1960s as tools to organize and advance school mathematics, more specifically to achieve a better conceptual understanding of basic mathematical concepts and methods.

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Adri Treffers (1987), and later Hans Freudenthal (1991), proposed a model to characterize different types of mathematics education in terms of four ideal types—mechanistic, empiristic, structuralist, and realistic mathematics education—created with a view to a global orientation, although susceptible to nuances. The model was based on a double dichotomy (presence versus absence) of horizontal and vertical mathematization (roughly “mathematizing reality” versus “mathematizing mathematics,” respectively) in actual or intended learning processes. Modern mathematics approaches were labeled structuralist—the horizontal component was absent, but the vertical component was cultivated. In the nineteen sixties and seventies of our century, under the name of New Math, the structuralist view was advertised and propagated. … On behalf of the prestructured mathematics to be taught, a correspondingly structured world was invented of Venn diagrams, arrow schemes, “games” and so on, to be mathematised by the learner. This was, indeed, a kind of horizontally mathematising activity, yet it started from an ad hoc created world, which had nothing in common with the learner’s living world. It was mathematics taught in the ivory tower of the rational individual, far from world and society. (Freudenthal 1991, p. 135, italics in original)

The emphasis on mathematical structure (rather than the mastery of specific knowledge or technical– computational skills) implied the introduction of new content, materials, and practices in school mathematics. Although specific features varied across countries and regions, and obviously with the level of education, we attempt to list some main features of mathematics curricula in the 1960s and 1970s. • The so-called “fundamental” concepts like sets and relations became the starting points; “richer” concepts were described in terms of these concepts (e.g., geometrical objects were defined as “sets of points”). • Concept and problem representation were directed to the use of Venn diagrams and arrow graphs; students were discouraged from making their own problem visualizations. • Recommended curricula were oriented from abstract to concrete (e.g., in geometry: first points, then lines, and finally “rich” geometrical figures). • Precise formulations, exact definitions, and correct symbol use received much attention in verbal and written explanation and communication (e.g., a clear distinction was made between “numbers” and “numerals”). • There was a focus on conceptual mathematical understanding (e.g., by studying number systems with bases other than 10) and basic laws (commutativity, associativity, …) rather than on computational fluency and number facts. • Linear algebra became a “royal road” (Choquet 1964, p.  11) to affine and thereafter Euclidean geometry; much attention was given to geometric transformations and their underlying structures, less to geometric problem solving.



A Failed Reform?

The introduction of modern mathematics/New Math curricula was not without controversy, but in the 1960s critiques had little impact. A most outspoken “early” opponent was Morris Kline, who, along with 64 other renowned American and Canadian mathematicians, signed a memorandum denouncing the abstract tenor of modern mathematics and the neglect of practical applications (Ahlfors et al. 1962). The memorandum was well-noticed, also in Europe (see, e.g., De Bock and Vanpaemel 2019; Guitart 2020), but could not stop the modern mathematics movement, which was then in full swing. In the first half of the 1970s, the movement reached its peak internationally, but at the same time, clear signs of an upcoming decline appeared (Moon 1986). In 1973, Kline’s book Why Johnny Can’t Add: The Failure of the New Math (Kline 1973) was published, in which the author severely criticized the roles of mathematicians in propagating the New Math and associated recommended teaching practices.

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Kline’s (1973) book quickly acquired an iconic status. A major European critical voice of that period was that of the distinguished French mathematician René Thom. Thom wrote an oft-quoted opinion article in American Scientist (Thom 1971) with a response from Jean Dieudonné (1973), and was invited to deliver a plenary address at the Second International Congress on Mathematical Education (ICME-2) held in Exeter (UK) in 1972 (Thom 1973). Eight years after the Exeter Congress, at ICME-4 held in Berkeley (USA) in 1980, modern mathematics was no longer an issue at all (Zweng et al. 1983). In this international shortlist of famous New Math opponents, the figure Hans Freudenthal, President of the International Commission on Mathematical Instruction (ICMI) between 1967 and 1970, should not be left out. Freudenthal converted early to the critical camp (Freudenthal 1963) and helped ensure that a “different version” of modern mathematics could take root in the Netherlands (see also Chap. 11 in this volume). Critical voices with mainly national influence are discussed in Chaps. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, and 24 of this volume. In retrospect, modern mathematics/New Math occupied a rather short period in the history of mathematics education: In most countries, the innovation was stopped already one or two decades after its introduction. Even in countries that were mostly affected by modern mathematics, such as Belgium, France, and the United States, it had disappeared by the end of the previous century, being replaced by more constructivist or realistic approaches to mathematics education. In particular, in countries in which modern mathematics was also introduced at the primary level, it was typically a period of very short duration. Moreover, one may question whether, especially at the primary school level, daily classroom practice was affected by modern mathematics as much as reform documents and official curricula of that period suggest. It is apparent that computational and measurement techniques as well as word problem solving—key parts of the “pre-New Math curricula” for that level— were not dropped by primary school teachers during the period of modern mathematics (De Bock and Vanpaemel 2019). Although modern mathematics disappeared rather quickly from the international scene, in some countries isolated elements of it were kept (e.g., in Belgian primary education, Venn diagrams are still used to explain the relationships between the various regular plane figures). In countries in which modern mathematics was implemented in a less radical way, such as the United Kingdom, this was, even more, the case (“Not all of the ‘new’ mathematics found an established place in the school curriculum, but much did,” Howson 2013, p. 647). Moreover, regardless of the country considered, structural elements as well as forms of abstraction and formalization, began to play a greater role in mathematics education in one way or another. Today, however, these elements are no longer seen as starting points for teaching, but rather as final stages in students’ developing mathematical culture. The modern mathematics movement also generated national and international momentum in the community of mathematics teachers (and other people involved in mathematics education). In countries such as the United Kingdom and Belgium, (new) associations of mathematics teachers were founded in which the debate about an upcoming reform of mathematics was held. Committed teachers felt connected as collaborators in an ambitious and valuable educational project that transcended ideological boundaries (De Bock and Vanpaemel 2019). Internationally, at the end of the 1960s, partly from the ashes of the modern mathematics movement, the amount of international collaboration grew and mathematics education became recognized as an autonomous scientific discipline (see also Chaps. 4 and 18 in this volume). Under Freudenthal’s presidency, ICMI adopted an agenda that fostered a new and refreshing dynamic in thinking about and researching mathematics education. The anchor points were the establishment of the ICMEs (the first took place in Lyon, France, in 1969) and the creation of the journal Educational Studies in Mathematics (launched in 1968), both at Freudenthal’s initiative. It was followed by the Journal for Research in Mathematics Education (launched in 1970) and other research journals in mathematics education. Modern mathematics was by no means an overall success, but it was at the root of developments that probably would not have occurred without it.

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References Ahlfors, L. V., Bacon, H. M., Bell, C., Bellman, R. E., Bers, L., Birkhoff, G., …, Wittenberg, A. (1962). On the mathematics curriculum of the high school. American Mathematical Monthly, 69(3), 189–193. Barrantes, H., & Ruiz, A. (1998). The history of the Inter-American Committee on Mathematics Education. (Bilingual Spanish and English edition). Bogotá, Colombia: Academia Colombiana de Ciencias Exactas, Físicas y Naturales. Begle, E. G. (1971). Time devoted to instruction and student achievement. Educational Studies in Mathematics, 4(2), 220–224. Bourbaki, N. (1948). L’architecture des mathématiques [The architecture of mathematics]. In F. Le Lionnais (Ed.), Les grands courants de la pensée mathématique [Major trends in mathematical thinking] (pp. 35–47). Paris, France: Cahiers du Sud. Choquet, G. (1964). L’enseignement de la géométrie [The teaching of geometry]. Paris, France: Hermann. D’Ambrosio, B. S. (1991). The modern mathematics reform movement in Brazil and its consequences for Brazilian mathematics education. Educational Studies in Mathematics, 22(1), 69–85. De Bock, D., & Vanpaemel, G. (2018). Early experiments with modern mathematics in Belgium. Advanced mathematics taught from childhood? In F. Furinghetti & A. Karp (Eds.), Researching the history of mathematics education: An international overview (pp. 61–77). Cham, Switzerland: Springer. De Bock, D., & Vanpaemel, G. (2019). Rods, sets and arrows. The rise and fall of modern mathematics in Belgium. Cham, Switzerland: Springer. Dieudonné, J. A. (1973). Should we teach “modern” mathematics? American Scientist, 61(1), 16–19. Freudenthal, H. (1963). Enseignement des mathématiques modernes ou Enseignement moderne des mathématiques [Teaching modern mathematics or modern teaching of mathematics]. L’Enseignement Mathématique, s. 2, 9, 28–44. Freudenthal, H. (Ed.). (1978). Change in mathematics education since the late 1950s—Ideas and realization: An ICMI report. Educational Studies in Mathematics, 9(2–3). Freudenthal, H. (1991). Revisiting mathematics education. China Lectures. Dordrecht, The Netherlands: Kluwer. Gispert, H. (2010). Rénover l’enseignement des mathématiques, la dynamique internationale des années 1950 [Renewing mathematics education, the international dynamics of the 1950s]. In R. d’Enfert & P. Kahn (Eds.), En attendant la réforme. Disciplines scolaires et politiques éducatives sous la IVe République [Waiting for the reform. School disciplines and educational policies under the Fourth Republic] (pp. 131–143). Grenoble, France: Presses Universitaires de Grenoble. Guitart, R. (2020). Metamorphosis of geometrical teaching in France (1950–1969). In É. Barbin, K.  Bjarnadóttir, F. Furinghetti, A. Karp, G. Moussard, J. Prytz, & G. Schubring (Eds.), “Dig where you stand” 6. Proceedings of the Sixth International Conference on the History of Mathematics Education (pp. 73–86). Münster, Germany: WTM. Howson, G. (2013). The development of mathematics textbooks: Historical reflections from a personal perspective. ZDM—Mathematics Education, 45, 647–658. Karp, A. (2008). Interview with Geoffrey Howson. International Journal for the History of Mathematics Education, 3(1), 47–67. Kilpatrick, J. (2012). The new math as an international phenomenon. ZDM—Mathematics Education, 44, 563–571. Kline, M. (1973). Why Johnny can’t add: The failure of the New Math. New York, NY: St. Martin’s Press. Moon, B. (1986). The “New Maths” curriculum controversy: An international story. Barcombe, United Kingdom: Falmer Press. Nadimi Amiri, S. (2017a). Historicizing the journey of the New Math reform from the United States to Luxembourg in the 1960s and 1970s. (Unpublished doctoral dissertation). The Faculty of Language and Literature, Humanities, Arts and Education, University of Luxembourg, Luxembourg. Nadimi Amiri, S. (2017b). When and educational ideology travels: The experience of the New Math reform in Luxembourg. In A.  Chronaki (Ed.), Mathematics education and life at times of crisis. Proceedings of the Ninth International Mathematics Education and Society Conference (pp. 738–748). Volos, Greece: University of Thessaly Press. OECD. (2020). The OECD at 60. Paris, France: OECD. OEEC. (1961a). New thinking in school mathematics. Paris, France: OEEC. OEEC. (1961b). Synopses for modern secondary school mathematics. Paris, France: OEEC. Phillips, C. J. (2015). The new math: A political history. Chicago, IL: University of Chicago. Rouche, N. (1984). Bilan de la réforme et idées pour l’avenir [Review of the reform and ideas for the future]. Bulletin de la Société Mathématique de Belgique (Série A), 36(2), 133–145. Servais, W. (1975). Continental traditions and reforms. International Journal of Mathematical Education in Science and Technology, 6(1), 37–58. Swetz, F.  J. (1975). Mathematics curricular reform in less-developed nations: An issue of concern. The Journal of Developing Areas, 10(1), 3–14. Thom, R. (1971). “Modern” mathematics: An educational and philosophic error? American Scientist, 59(6), 695–699.

1  Modern Mathematics: An International Movement

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Thom, R. (1973). Modern mathematics: Does it exist? In A. G. Howson (Ed.), Developments in mathematical education. Proceedings of the Second International Congress on Mathematical Education (pp.  194–209). Cambridge, United Kingdom: University Press. Treffers, A. (1987). Three dimensions. A model of goal and theory description in mathematics education. Dordrecht, The Netherlands: Kluwer. Vanpaemel, G., & De Bock, D. (2019). New Math, an international movement? In E.  Barbin, U.  T. Jankvist, T.  H. Kjeldsen, B. Smestad, & C. Tzanakis (Eds.), Proceedings of the Eighth European Summer University on History and Epistemology in Mathematics Education (pp. 801–812). Oslo, Norway: Oslo Metropolitan University. Vanpaemel, G., De Bock, D., & Verschaffel, L. (2012). Defining modern mathematics: Willy Servais (1913–1979) and mathematics curriculum reform in Belgium. In K. Bjarnadóttir, F. Furinghetti, J. Matos, & G. Schubring (Eds.),“Dig where you stand” 2. Proceedings of the Second International Conference on the History of Mathematics Education (pp. 485–505). Lisbon, Portugal: New University of Lisbon. Zwaneveld, B., & De Bock, D. (2019). Views on usefulness and applications during the sixties. In K.  Bjarnadóttir, F. Furinghetti, J. Krüger, J. Prytz, G. Schubring, & H. J. Smid (Eds.), “Dig where you stand” 5. Proceedings of the Fifth International Conference on the History of Mathematics Education (pp. 387–399). Utrecht, the Netherlands: Freudenthal Institute. Zweng, M., Green, T., Kilpatrick, J., Pollak, H., & Suydam, M. (Eds.). (1983). Proceedings of the 4th International Congress on Mathematical Education. Boston, MA: Birkhäuser.

Part I

Preparing the Reform on Both Sides of the Atlantic

Chapter 2

The Rise of the American New Math Movement: How National Security Anxiety and Mathematical Modernism Disrupted the School Curriculum David Lindsay Roberts

Abstract  In the 1940s, the teaching of mathematics in the secondary schools of the United States began to recover from a long period of disrespect. This augmented prestige was due in part to an increased demand for mathematically trained workers arising from World War II and the Cold War. At the same time, undergraduate mathematics instruction was undergoing revision, bringing it more into line with the “modern” viewpoint of research mathematicians, focused on unifying concepts and “structures.” There was a sentiment among a significant segment of mathematics educators that school mathematics had become too estranged from these exciting new developments. This environment encouraged, in the 1950s, the development of innovative secondary school curriculum programs, featuring higher levels of abstraction and precision of language. The University of Illinois Committee on School Mathematics (UICSM) was an early, and notably radical, exemplar, while the School Mathematics Study Group (SMSG) was the largest and best-funded program. By the end of the 1950s optimism that the “New Math” would fundamentally and permanently change the school curriculum for the better was widespread, although far from universal. Keywords  AMS · CEEB · CUPM · Curriculum reform · Educationists · Edward Begle · Howard Fehr · Life adjustment education · MAA · Mathematical structure · Max Beberman · Modern mathematics · NCTM · Nicolas Bourbaki · SMSG · Sputnik · Steelman report · UICSM · Uni High · Writing groups

Introduction The education reform activity which emerged in the United States after World War II, often called new mathematics, modern mathematics, or the New Math, arose from a confluence of factors. In brief, impetus from outside education, associated with the War and the subsequent Cold War with the Soviet Union, combined with the professional aspirations of a variety of educators to create an environment that encouraged experimenting with substantial changes to the mathematics curriculum, and to a lesser extent with the methods of teaching. It will be convenient to use “the New Math” as a designation for the period, but this should not be taken to mean that there was a single, unified phenomenon; there were a variety of curricular experiments. But it cannot be denied that there were certain commonalities among many of these experiments, and links among the persons and institutions involved. In this chapter, the story will be carried to the end of 1959, which will prove to be an appropriate place at which to pause for reflection.

D. L. Roberts (*) Prince George’s Community College, Largo, MD, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_2

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The reform activity in question is mainly remembered for its effects on the elementary and secondary schools, the institutions offering instruction from kindergarten to grade 12. But there were ­concurrent reform efforts at the post-secondary level shedding light on the school-level reforms. Most of the important actors at the school level were from colleges and universities. This chapter will use the term “mathematics educator” to refer to anyone, at any level, who teaches classes specifically devoted to mathematics. The word “mathematician” will be restricted to those with graduate training in mathematics to the level of a master’s degree or above, awarded by a department of mathematics. Such individuals will be distinguished from those with a master’s degree or above in education, or awarded by a department of mathematics education distinct from a department of mathematics. Such individuals will be termed “educationists.” Thus, a majority of college and university instructors of mathematics are mathematicians (and were so in the New Math period), though a substantial number are educationists, and some are both mathematicians and educationists. There are probably more educationists than mathematicians among secondary school mathematics teachers, while a non-negligible portion is neither (more so during the New Math than today). Many of the mathematics educators of the New Math period belonged to one or more professional organizations. The three organizations of most significance at the time were the American Mathematical Society (AMS), the Mathematical Association of America (MAA), and the National Council of Teachers of Mathematics (NCTM).1 There is some value in briefly summarizing the primary orientations of these organizations, but it should be noted that these summations do not necessarily explain the attitudes of all individual members. Furthermore, during the New Math period, and still today, there was considerable overlap among the memberships of the three organizations. The AMS, founded in the late nineteenth century, was and remains dominated by university mathematicians promoting mathematical research. The MAA branched off from the AMS in 1915 to focus on the role of mathematicians as university teachers. The NCTM was founded in 1920 by secondary school teachers of mathematics, but soon began to attract university educators, especially mathematics educationists (Roberts 2012). By the time of the New Math, these latter had become prominent in the leadership of the NCTM, although school teachers continued to make up a large part of the membership. Frequent reference will be made to two serial publications used prominently during the New Math period to publicize issues in mathematics education: the American Mathematical Monthly (often, hereafter, simply the Monthly), the official journal of the MAA, and the Mathematics Teacher, the official journal of the NCTM.



Emergence of Mathematical Workforce Demands in the 1940s

Through much of the nineteenth-century mathematics enjoyed a largely unchallenged place of prominence in American schools. But in the early twentieth century, burgeoning school populations were accompanied by greater concern about both fitting students for future employment and preparing them to be good citizens. The time allocated for mathematics in the curriculum was subjected to increased competition from other subjects. The result, according to one observer, was a “twenty-five year depression” in school mathematics over the period 1915–1940 (Duren 1967, p. 24). Many school systems reduced their mathematics requirements. A 1936 editorial in the Mathematics Teacher lamented that “St. Louis high school pupils will no longer be required to take elementary algebra, but may, if they wish, substitute a course in the history of Missouri” (Reeve 1936, p. 254). Many mathematics educators, in both the schools and in the universities, felt themselves to be on the defensive. Blame was often cast at the schools and colleges of education, at “progressive” educa The reader is hereby alerted that this chapter will employ these and other acronyms with regularity.

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tors, and in general at those outside of mathematics who failed to appreciate the subject. But mathematics educators were not unified in their diagnoses. Speaking in 1931, retiring MAA president John Wesley Young (1932), a PhD mathematician teaching at Dartmouth College, complained that the NCTM was being captured by nefarious forces: “Unfortunately, from some points of view at any rate, the National Council reflects the point of view of our schools of education rather than that of our departments of mathematics” (p. 7). An alternative view was offered by Eric Temple Bell (1935), a prolific research mathematician teaching at the California Institute of Technology, writing in 1935 as he was rising to prominence as a mathematics popularizer. Bell asserted that mathematics professionals should take much blame themselves, for failing to adequately explain the value of their subject: Are not mathematicians and teachers of mathematics in liberal America today facing the bitterest struggle for their continued existence in the history of our Republic? American mathematics is exactly where, by common social justice, it should be—in harassed retreat; fighting a desperate rear-guard action to ward off annihilation. Until something more substantial than has yet been exhibited, both practical and spiritual, is shown the non-­ mathematical public as a justification for its continued support of mathematics and mathematicians, both the subject and its cultivators will have only themselves to thank if our immediate successors exterminate both. (p. 559)

In this environment, beleaguered mathematics educators seized eagerly on world events to bolster their cause and present a more unified front. In early 1940, with Europe already engulfed in World War II and with many Americans anticipating US involvement, the AMS and the MAA jointly created a “War Preparedness Committee.” This committee reported in September 1940 that it had created a subcommittee on “Education for Service,” with the aim of “strengthening of undergraduate mathematical education in our colleges to the point where it affords adequate preparation in mathematics for military and naval service of any nature” (Morse 1940, p. 500). It was soon acknowledged that this goal required giving attention to the secondary schools as well. Presentations by the War Preparedness Committee were made at NCTM meetings in January and February 1941 (Morse and Hart 1941). The chairman of the Subcommittee on Education for Service, William Hart of the University of Minnesota, asserted that proper preparedness required increased mathematical training for both men and women. There was no suggestion that any new topics were needed in the secondary school curriculum, only that certain existing topics needed more emphasis and should be taught to a larger cohort of students. “It would be desirable if skilled industrial workers had substantial secondary mathematics, through the stage of computational trigonometry, with at least an intuitional knowledge of solid geometry” (p. 299). He believed that “without special effort on the part of the high schools” the military and supporting industrial base would not be properly prepared. He urged the NCTM to “advertise the utility of mathematics in industry and military service” (p. 301) and to create its own special committee to prepare for war. By the middle of 1941, the Subcommittee on Education for Service had created a “Report on Mathematical Education for Defense,” published in the June–July issue of the Monthly and in the November issue of the Mathematics Teacher. This went into further detail on the mathematical knowledge needed for military and industrial specialties. Here again, there was considerable mention of trigonometry and solid geometry, with some implied attention to algebra and plane geometry (Hart 1941a, b). An incident from October 1941 involving Admiral Chester Nimitz, at that time Chief of the Bureau of Navigation, proved eminently exploitable as a publicity tool by mathematics advocates. In correspondence with a University of Michigan faculty member, Nimitz had decried the inadequate mathematical preparation that had been displayed by college students seeking commissions in the Reserve Officer Training Corps. He firmly asserted that “a candidate for training for a commission in the Naval Reserve cannot be regarded as good material unless he has taken sufficient mathematics” (“The Letter of Admiral Nimitz” 1942, p. 214). The University of Michigan included Nimitz’s words in a widely circulated pamphlet. Although Nimitz appeared sincerely alarmed at the time, there is no evidence

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that he ever returned to this topic later in his career. Educators at all levels, however, would repeatedly invoke Nimitz’s concern as clear evidence of the need to protect and expand mathematics in the schools. Advantage was taken of the rise in Nimitz’s national prominence, as he was named Commander in Chief of the Pacific Fleet after the Japanese attack on Pearl Harbor in December 1941. In January 1942, the Nimitz correspondence was republished in full as part of an editorial in the North Central Association Quarterly, a journal for school teachers of science and mathematics in the north central states (“Would-Be Midshipmen” 1942). This was followed in February by similar editorial treatment in the Mathematics Teacher (Reeve 1942), as well as a partial reprint of the Nimitz letter in School and Society (“Deficiencies in Mathematics” 1942). Then in March, there appeared another full reprint, in the Monthly, in response to “the wish of many” (“The Letter of Admiral Nimitz” 1942, p. 212). Further Mathematics Teacher articles, in April (Betz 1942) and in October (Hedrick 1942), featured key paragraphs from Nimitz, the latter piece authored by Earle R. Hedrick, a former president of the MAA. In January 1943, the correspondence was once again reprinted in full in the Scientific Monthly (Morse 1943), a publication of the American Association for the Advancement of Science. This article was written by Marston Morse of the Institute for Advanced Study, a major research mathematician, and at that time both president of the AMS and chairman of the AMS-MAA War Preparedness Committee. In February 1944, with increasingly positive war news for the US and its allies, the NCTM launched a “commission to plan mathematics programs for secondary schools in the post-War period” (Commission on Post-War Plans 1944, p. 226). Its first report, issued in May of the same year, cited “the widely publicized letter by Admiral Nimitz” (p. 230), and drew several additional lessons from the war regarding the utility of mathematics education and the need for teachers to be more familiar with “a vast range of uses of mathematics” (p. 232). A second report of the Commission on Post-War Plans (1945) offered more detailed suggestions. None of its thirty-four “theses” proposed major curricular innovations. The emphasis was on trying to deal more effectively with long-existing issues. There was considerable emphasis on the importance for the student to grasp “meanings,” and to develop “power,” “maturity,” and “functional competence.” It was judged unwise to “differentiate” students until grade 9, but at that point, a “double track” was recommended, with “algebra for some and general mathematics for the rest” (p. 205). The problem of preventing the general mathematics student from developing feelings of inferiority or “insecurity” (p. 207) was worried over but judged solvable. The Commission also weighed in on the mathematics of the two-year “junior colleges,” and on teacher training, again without strikingly novel proposals. Many of these points were further publicized as part of a massive October 1947 federal government report on Science and public policy, issued by the President’s Scientific Research Board. This has often been called the “Steelman Report,” in reference to the chairman of the effort, John R. Steelman, a special assistant to President Truman. For the Manpower for research volume of the report, the Research Board turned to the Cooperative Committee on the Teaching of Science and Mathematics, which had been earlier created by the American Association for the Advancement of Science (AAAS), an organization that aimed to represent all the sciences, including mathematics. The Cooperative Committee included one representative each from both the MAA and the NCTM. When it came to mathematics in particular, this committee scanned the landscape and found ready to hand the 1945 report of the NCTM’s Commission on Post-War Plans, from which it borrowed liberally (Steelman 1947). The overall tenor of the Steelman Report was one of alarm about the nation’s ability to maintain the scientific research needed to support the military and economic challenges ahead. Cold War competition with the Soviet Union was not explicitly invoked, “but no responsible person can fail to recognize the uneasy character of the present peace” (Steelman 1947, p. 3), and it was noted that the Soviets were vastly increasing their research and development budgets.

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When it came to scientific “manpower” in particular, the Steelman Report’s repeated theme was “shortage” (Steelman 1947, p. 1): shortages of scientists now, and growing shortages in the future, owing to the shortage of qualified teachers at all levels. The importance of mathematics education for the overall health of American science was given considerable emphasis. “The battle for competence in science is probably won or lost by the mathematics instruction in the lower schools (grades 1 to 12)” (p. 62). Admiral Nimitz’s “drastic indictment” (p. 63) was once again trotted out. Other themes familiar from reports earlier in the 1940s appeared as well: the need for more attention to “meaning” (p. 65) in mathematics instruction; the value of teaching mathematics in conjunction with scientific applications; the desirability of separating students into two tracks in ninth grade (with acknowledgment of the difficulties of doing this). For the nurturing of future scientists, the Steelman Report tentatively suggested that more innovative changes to the standard mathematics curriculum might be desirable, and it was suggested that a comprehensive study be launched on this topic. It was noted that the kind of study envisioned would require substantial funds, and since the NCTM and the MAA were observed to be in no position to provide the underwriting, it was hoped that the recently proposed National Science Foundation (then under vigorous discussion within the Truman administration and Congress) might “make this investigation one of its first projects” (Steelman 1947, p. 82). The Steelman Report also advised, more explicitly than any of the mathematics-specific reports preceding it, that in addition to an increased output of scientific researchers and teachers, the nation needed to improve the general public’s “comprehension of the importance of the basic sciences,” including mathematics. It was asserted that “this crucial task … looms in the background throughout this study” (Steelman 1947, p. 57). The Steelman Report was widely publicized among mathematics educators. In 1948, there appeared two articles about it in the Monthly (Schorling 1948; Vance 1948), and it was lauded in the Mathematics Teacher as “that incisive document” (Betz 1948, p. 379). Highlights of the report’s “valuable guidance” (Schult 1949, p. 143) were quoted in the latter journal in 1949. At the same time as the Steelman Report, another arm of the federal government was helping to reinvigorate some of the anti-mathematical positions of the pre-War period. Life adjustment education for every youth (1948) issued by the US Office of Education, “recognized the need for a more realistic and practical program of education for those youth of secondary-school age for whom neither college preparatory offerings nor vocational training for the skilled occupations is appropriate” (p.  15). Mathematics was denigrated as a “tool” subject, and it was asserted that “the ability to achieve any real understanding of abstract mathematics is a gift accorded to the few” (p. 95). It was conceded that “engineering colleges can appropriately ask high schools to teach their prospective candidates considerable mathematics” (p. 107), but this was seen as a minor aspect of the high school mission. The alarms of the Steelman Report about insufficient mathematics instruction were nowhere in evidence.



The Promotion of “Modern” Mathematics for Undergraduates

Independently of the external pressures to expand the mathematically trained workforce in the 1940s, there was an increase of longstanding internal agitation to reform mathematics curricula at colleges and universities. Applied mathematics had received a considerable boost from European emigrees in the 1930s, but pure mathematics remained the dominant American research activity. Thus, when calls went out in the 1940s to “modernize” the undergraduate mathematics curriculum, although there might be mention of probability and statistics, it was more common to see logic, set theory, and even topology designated for increased attention. But the word “modern” was most explicitly invoked in regard to algebra, where it was wielded to denote the transition from an endeavor focused on solving equations to the more abstract study of axiomatically defined structures such as groups, rings, and

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fields. A key role was played by the German text, Moderne Algebra, written by Dutch mathematician Bartel L. van der Waerden (1930). This was hugely influential on American research mathematicians coming of age in the 1930s, in particular on Saunders Mac Lane, who would in 1941 co-author with Garrett Birkhoff A Survey of Modern Algebra (Birkhoff and Mac Lane 1941), a somewhat more elementary treatment than van der Waerden’s), but in the same spirit (Mac Lane 1997). It was also in the early 1940s that American mathematicians were introduced to a major European proponent of mathematical structure, the collective of brilliant young French mathematicians writing under the name Nicolas Bourbaki (see also Chap. 3 in this volume). Papers referencing Bourbaki began appearing in American journals in 1941, notably a paper in April of that year in the Annals of Mathematics by Jean Dieudonné (1941), one of the originators of the group. Another Bourbaki founder, André Weil, emigrated to the US in 1941, spending the academic year 1941–1942 at Haverford College, near Philadelphia. Weil and his Haverford colleague, Carl Allendoerfer, collaborated on a paper in differential geometry (Allendoerfer and Weil 1943). After intermediate stops at several other institutions, Weil in 1947 settled at the University of Chicago for a decade, before ascending to the Institute for Advanced Study in Princeton (Weil 1992). Weil rarely took more than a mild interest in students at any level of instruction, but his presence in the US as an exemplary figure ensured that some young American mathematics educators were exposed to the Bourbaki point of view as to the true nature of mathematics.2 Although the pedagogical influence of the approach to mathematics promoted by Bourbaki, and embodied in the books of van der Waerden and Birkhoff and Mac Lane, was at first confined to research mathematicians in training, effects were eventually felt elsewhere. One arena in which this occurred was the course required by many colleges for students with no aspiration to take any further mathematics, sometimes known as “liberal arts mathematics.” Such courses had been troublesome for many years. The simplest solution had been to offer a modest extension of high school algebra and geometry, what might today be called “pre-calculus.” This could also double as a remedial course for those aspiring to a technical major but with weak backgrounds. However, students often found this an uninspiring course, and the mathematics faculty, especially those who were research-oriented, were loath to teach it. Consequently, several leading universities began to experiment with alternatives. In 1940, members of the mathematics department at Cornell University, led by Burton W. Jones, prepared a set of course materials for “nonmathematicians,” including such innovative topics as probability, Lorentz geometries, and topology, while deliberately avoiding trigonometry and calculus. Emphasis was placed on “understanding,” “appreciation,” and “logical development.” Eventually, a commercial textbook emerged: Elementary Concepts of Mathematics, published in 1947 under the authorship of Jones. A second edition appeared in 1963, by which time Jones had moved to the University of Colorado (Jones 1963). In 1943 the College of the University of Chicago3 began offering a specially designed liberal arts mathematics course. This course underwent continual revision, publicized by Eugene P. Northrop of the College faculty in articles in the Monthly appearing in 1945 and 1948. Unlike at Cornell, the Chicago course was “designed for all students, regardless of the occupation or profession they may expect to enter” (Northrop 1945, p. 134). The solution adopted was to emphasize abstract mathematics. The argument was that on the one hand, this was the characteristic feature of mathematics to which all liberal arts students should be exposed, and on the other hand that mastery of the core abstractions would allow future specialization in any of a multitude of directions. In particular, Northrop (1945) touted the value of this approach for returning veterans. He surmised that said veterans, having been exposed in the service to “accelerated courses in mathematics, often at the hands of  The first reference to Bourbaki in the Monthly occurred in 1950 and will be noted later in this chapter. The first reference to Bourbaki in the Mathematics Teacher appears to have been a short note by Phillip S. Jones (1951). 3  At this time the department of mathematics in the College was administratively separate from the department inhabited by the University’s research mathematicians. 2

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inexperienced teachers” (p. 133), would now be ripe to learn what mathematics was truly about. To this end, the course would emphasize “logical structure” (p. 134), the “postulational development” (p. 135) of both geometry and algebra, and “the mastery of a conception of mathematical systems” (p. 137). By 1948, according to Northrop, the course had become more committed to “modern symbolic logic,” and now culminated in “a careful study of the commutative group as an example of a deductive system” (Northrop 1948, p. 3). Northrop prepared a text for the course, titled Fundamental Mathematics, revised twice in the late 1940s and available only at the University of Chicago Bookstore. A more comprehensive work, Concepts and Structure of Mathematics, was published by the University of Chicago Press in 1954, authored by eight members of the College Mathematics Staff, including Northrop. The authors candidly admitted that this book remained “a continuing experiment. In teaching from it we hope to learn to what extent a comprehensive system of fundamentals can be taught at an elementary level” (The College Mathematics Staff 1954, p. iii). Research mathematicians were also reluctant to burden themselves with courses specially designed for prospective teachers of school mathematics, while at the same time disparaging the offerings which departments and schools and colleges of education provided to these same students. Many mathematicians, by encouraging only their weaker students to go into school teaching, effectively implied that a desire to be a high school mathematics teacher was not a respectable aspiration for a strong student of mathematics (Langer 1952). Cletus Oakley of Haverford College, writing in the Mathematics Teacher (Oakley 1942), noted a related problem with graduate training programs for teachers. A master’s degree in mathematics, designed as a mere rite of passage to a research PhD, was of dubious value to a high school teacher. At the same time, the master’s degree in education, to which said teacher now turned as an alternative, was too often sadly lacking in mathematical substance, according to Oakley. He saw a need for “revolutionary change” (p. 308), reorienting the university mathematicians to take a greater role in training school teachers and in general taking a far greater interest in school mathematics. Carl Allendoerfer, then still a colleague of Oakley at Haverford, presented a paper on “Mathematics for Liberal Arts Students,” at a meeting of the MAA in September 1947. In this talk, he proposed ideas for a one-year terminal mathematics course for students not pursuing degrees in mathematics, science, or engineering. Tentatively titled “Methods of Quantitative Thinking,” this course would introduce “deductive logic, axioms, and abstract thinking,” as well as “a substantial treatment of probability and statistics” (Allendoerfer 1947, p. 576). Allendoerfer ended by noting that his ideas remained “in a formative stage” (p. 577). By the early 1950s, Allendoerfer and Oakley had combined their ideas for revitalizing undergraduate mathematics instruction and were trying out classroom experiments. Their collaboration continued even after Allendoerfer left Haverford for the University of Washington in 1951. A preliminary version of a textbook came out in 1953, available for use by other interested mathematics professors (Mann 1956). Publication of Principles of Mathematics, often referred to as “Allendoerfer and Oakley,” followed in 1955, from McGraw-Hill. The authors hoped to show that “some of the content and much of the spirit of modern mathematics can be incorporated in courses given to our beginning students” (Allendoerfer and Oakley 1963, p. v). The chapter headings revealed a mixture of old and new: Logic, The Number System, Groups, Fields, Sets and Boolean Algebra, Functions, Algebraic Functions, Trigonometric Functions, Exponential and Logarithmic Functions, Analytic Geometry, Limits, the Calculus, Statistics, and Probability. But most commentators felt that the balance was decisively toward innovation. One reviewer in 1956 asserted that the book “has a modern flavor, is in harmony with contemporary mathematical thought, and completely reoriented away from the traditional viewpoint” (Fehr 1956, p. 270). Another reviewer tellingly observed that the book was “much more interesting to the teacher than the usual text” (Mann 1956, p. 438) while noting that he had had disappointing results in trying to teach from it. This reviewer concluded that the Allendoerfer and Oakley book was simply too tough for beginning undergraduates. He suggested that “much of the

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contents of this text should be deferred till a junior-senior course based perhaps on the well-known text of Birkhoff and Mac Lane” (p. 439). The MAA as an organization became officially involved in the modernization movement in 1953. In that year, MAA president Edward J. McShane appointed a special committee to survey the state of undergraduate instruction in the United States. This committee, chaired by William Duren of Tulane, reported widespread dissatisfaction with the undergraduate program across the country, echoing many of the concerns expressed already in the 1940s. A central feature of the situation emphasized by the Duren committee was “the desertion of elementary teaching by the best mathematicians, old and young” (Duren et al. 1955, p. 512). The committee recommended “one, universal freshman course for all reasonably qualified students” (p. 512).4 It was to be composed of two parts: The first part would essentially consist of calculus, and the second part would begin with set concepts, move on to simple algebraic systems, and then to probability and statistics. Both the University of Chicago program and the Allendoerfer and Oakley book were mentioned as being in harmony with the committee’s efforts. The ad hoc Duren committee of 1953 was reconstituted in 1954 as a standing committee of the MAA, first designated as the Committee on the Undergraduate Program (CUP) and later, as other disciplines were forming their own similar committees, as the Committee on the Undergraduate Program in Mathematics (CUPM).5 One of its first activities was a writing session at the University of Kansas in the summer of 1954, supported by the Social Science Research Council (Duren 1967). This helped set the fashion for producing text material in such summer writing sessions, an approach which became central to later curricular reform programs. The resulting Universal Mathematics text proved to be markedly abstract, emphasizing sets, equivalence relations, functions, axiomatic systems, and other standard items from the toolkit of the twentieth-­century pure mathematician. The calculus portion ambitiously offered the unifying but recondite topic of Moore-Smith limits (Price 1955). The intellectual argument was once again that abstraction was the efficient way to promote future specialization. The implied professional argument was that research-oriented mathematicians would be much happier teaching material close in spirit and vocabulary to research mathematics. That is, introductory undergraduate mathematics would be redefined so that research mathematicians would continue to tolerate teaching it and would therefore continue to exercise some jurisdiction over teacher training and other portions of university mathematics often viewed by these mathematicians with distaste. The text produced in the summer of 1954, Universal Mathematics, Part I, was given a mass trial at Tulane in 1954–1955, using about 750 students. Duren’s assessment from 1956 was blunt: The book is not yet suitable as a textbook and caused considerable difficulty to students and instructors. The main trouble is that students cannot read it…. One thing is clear; and that is that Universal Mathematics is not adaptable to students whose high school background is scant. (Duren 1956, p. 200)

Nevertheless, Universal Mathematics would eventually exercise considerable long-term influence on undergraduate instruction, through its Part II on the elementary mathematics of sets, algebraic systems, and probability. As the CUPM worked on refining this text, it became aware that a similar effort was advancing at Dartmouth College, under the direction of John Kemeny. Kemeny was thereupon invited to join the CUPM (Kemeny et al. 1966). Thus, there are similarities between the book eventually produced by CUPM, titled Elementary Mathematics of Sets With Applications (Davis 1958), and the highly successful textbook of Kemeny, Snell, and Thompson, Introduction to Finite Mathematics, first published in 1957. This latter book was presented as an alternative to the traditional college mathematics sequence focused on calculus. Moreover, it was designed to include “applications to the biological and social sciences” (p. vii), and thus to demonstrate that physics was not the

 Italics in original.  With slight abuse of historical exactness, we will here use only CUPM.

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only science where mathematics was useful. Such finite mathematics courses would become popular at many colleges in subsequent years.



Emergence of Secondary School Reform

At the beginning of the 1950s, many mathematics educators were feeling more generally secure than before the War, and in possession of more ammunition against their still persistent critics. These mathematics educators felt justified in arguing that the nation needed more mathematically skilled citizens, and consequently needed more and better mathematics classes taught by more and better mathematics teachers at all levels. We have noted the ferment at the undergraduate level, as classrooms were filled to overflowing with returning veterans, and experiments were initiated with modernizing the curriculum taught in those classrooms. In 1950, Princeton University’s Solomon Lefschetz (1950), a senior research mathematician of great prominence, judged that the current state of mathematics made it inevitable that this modernization would eventually filter down to the schools: “Mathematical science is in the midst of a process of thoroughgoing unification and fusion. In due time, and with the usual pedagogical lag, it will deeply affect the teaching of our science even in the elements” (p. 111). That mathematical science was enjoying a “unification,” and that this should be encouraged and exploited, was also one of the main contentions of Bourbaki (1950) at this time, as exemplified in his widely noted Monthly article of 1950, “The Architecture of Mathematics.”6 But mathematician G. Baley Price of the University of Kansas, speaking at an NCTM meeting in 1951, pointed out a major hurdle to be overcome in bringing the good news of unification of mathematics to the schools: “Modern mathematics, including the modern applications, is almost unknown to high school teachers of mathematics” (Price 1951, p. 375). In fact, a significant incursion of modern mathematics at the school level would begin to take shape in the very year in which Price spoke. That this occurred at the University of Illinois was due to a combination of circumstances. The College of Engineering at the university was deeply concerned about the mathematical preparation of its students. Discussion of this issue was conducted at the same time as various university faculty members were contributing to wider debates about the entire American educational system. These debates implicated the secondary school curriculum, and since the university had an affiliated secondary school, University High School, innovations proposed by university faculty at the school level could be carried out expeditiously. Finally, and most important of all, there emerged a charismatic central figure, Max Beberman, audacious and adept both politically and mathematically, who was able to inspire students from elementary school to graduate school. Struggles with the tumultuous post-War educational environment inspired the College of Engineering at the University of Illinois in 1951 to specify the mathematical requirements for its entering students. A committee of six was created, consisting of two members each from the College of Engineering, the Department of Mathematics, and the College of Education. Interviews were conducted with students of engineering, with university instructors of engineering and mathematics, and with high school mathematics teachers. A document was produced: “Mathematical Needs of Prospective Students in the College of Engineering.” This listed 97 “indispensable” topics, ranging from elementary arithmetic to “addition and subtraction of vectors by components” (Henderson and Dickman 1952, p. 93). None of the topics would have been a departure from standard high school mathematics of the pre-War period. A supplementary list of thirteen “recommended” topics was appended, again with no surprises, ranging from facility with the slide rule to minutiae such as knowledge of “formulas for tangents of the half-angle” (p. 93).  This appears to be the first time Bourbaki was mentioned in any way in the Monthly. The article was a translation, by Arnold Dresden, of Bourbaki’s original French article of 1948. 6

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It was a tactful move to construct the committee from multiple sectors of the university. The University of Illinois was just then erupting in debate on educational issues, the role of colleges of education being especially subject to contention. Professor of Botany Harry J.  Fuller (1951) had attacked the colleges of education as “dreary intellectual sinks” (p. 32), in which “sound knowledge has been replaced by drivel” (p. 38). He argued for “the restoration of the humanities, the arts, and the sciences to their properly dominant position in our educational system” (p. 40). This was answered by members of the College of Education, first in a public address by the dean of the college, Willard Spalding, in which Fuller was labeled as “the bewildered botanist” (Cairns 1953, p. 237), and then in a somewhat more measured published paper by Edwin H.  Reeder (1951). Reeder argued that the growth of universal public education necessitated a move away from the kind of dominance advocated by subject matter specialists such as Fuller. These scholars were neglecting “the function of the school as a social institution” (p. 515), which should be concerned with “the whole child” (p. 519), not just the child as the acquirer of academic knowledge. Professor of History Arthur E. Bestor (1952) then answered Reeder. He called out Reeder’s “caricature” (p. 110) of academic scholars and the general “anti-intellectualism” (p. 113) of professors of education and the federal, state, and local bureaucrats abetting them. Bestor, who would go on to publish a widely cited book expressing his views in more detail (Bestor 1953), did admit that the scholars were “partly to blame” (Bestor 1952, p. 113) because of their abdication from school matters, especially the training of teachers. Stewart S. Cairns, chair of the Illinois mathematics department, chimed in with a talk delivered for the MAA in December 1952. Like Fuller and Bestor, Cairns (1953) was disdainful of the life adjustment educational movement, but he conceded that the increased influence of the colleges of education was a natural outcome of the “enormous task of coping with unprecedented hordes of students” (p. 234). He also diplomatically took care to note the “valuable cooperation of the College of Education at Illinois with mathematicians and engineers in certain forward-looking projects in secondary school mathematics” (p. 233). He was referring to the committee already mentioned, as well as to “a committee of similar composition” which had prepared “an experimental high school curriculum…put into at least partial operation at University High School” (p. 239). As will shortly be seen, this experimental curriculum mentioned by Cairns emphasized a deeper level of abstraction than one would have expected from a reading of “Mathematical Needs of Prospective Students in the College of Engineering.” No irrefutable explanation for this crucial development will be offered here, but a number of suggestive features of the actors involved, and their environment, will be noted. The committee that produced the experimental curriculum was designated as the University of Illinois Committee on School Mathematics (UICSM).7 This group was headed by Max Beberman (Figure 2.1), who had come to Illinois in 1950 after high school teaching experience in Alaska and New York (“Dr. Max Beberman is Dead at 45” 1971). He was initially hired as an instructor at the University High School (“News and Notices,” 1950), a secondary school under the jurisdiction of the university’s College of Education. “Uni High,” as it was known to its denizens, offered a five-year program, beginning with a combination of grades 7 and 8, referred to as the “subfreshman year,” followed by grades 9 through 12. Admission to the school was competitive, based on a test. The school attracted children of university faculty members and other highly educated parents in the area (Meeker 2000; Rovnyak 2007, 2021). Beberman, as he himself was always clear to note, was an educationist, not a mathematician (Beberman 1965). When he arrived at Illinois, age 25, he was still working on his doctorate, from Columbia University’s Teachers College, the great mecca of American educationists and an object of suspicion for many advocates of subject matter scholarship. Beberman’s doctorate, awarded in 1953,

 This group was for a time known as the University of Illinois Committee on Secondary School Mathematics (UICSSM), but it has been more generally known as UICSM, and this is the acronym we will employ. 7

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Figure 2.1  Max Beberman teaching a class of high school students, 1959. (Courtesy of the University of Illinois Archives)

would be an EdD (“Dr. Max Beberman is Dead at 45” 1971), the kind of certification scoffed at by Arthur Bestor as “a doctor’s degree with the teeth pulled” (Bestor 1952, p. 114). Beberman was initially assisted in creating UICSM curriculum material by Bruce Meserve of the University of Illinois mathematics department, a PhD mathematician eight years older than Beberman. Meserve joined UICSM just as he was finishing a textbook, Fundamental Concepts of Algebra (Meserve 1953), based on one half of a course called “Fundamental Concepts of Mathematics,” which he had been teaching for several years at the university. This course was designed for a broad student clientele, much like the audiences aimed at by other texts of the era, mentioned earlier. Like these texts, Meserve’s treatment claimed “a modern viewpoint” (p. v), with some attention given to groups, fields, and transfinite cardinals. On being told that there was little chance for promotion at Illinois if he insisted on focusing on pedagogical matters rather than research, Meserve left in 1954 to take a position at Montclair State College in New Jersey (Meserve 2003). There have been suggestions over the years that the high level of abstraction in the UICSM classes was due to Beberman acquiescing to the views of the Illinois mathematicians, especially Meserve’s successor on the committee, Herbert Vaughan (Raimi 2004). There is little evidence to support this view, and reports of Beberman’s personality tell against it. It is evident that he enjoyed abstract mathematics and absorbed it readily. It was Beberman, not Meserve or Vaughan, who was in the Uni High classrooms, actually teaching the material, and he taught it because he was sincerely convinced of its value, not just because it was suggested by someone with a PhD in mathematics. Late in life, Meserve estimated that he had written slightly more than 20% of the ninth-grade content for UICSM and that Beberman had written the rest (Meserve 2003). The early pages of the ninth-grade unit on “The Natural Numbers,” used in 1953, contain general remarks which epitomize the UICSM approach: This unit will be very different from anything you have done before in mathematics. The ideas of the unit are very simple, but they are used in ways that may seem strange to you at first…In many exercises the real problem is to understand how the exercise is answered rather than just to find the correct answer… You should bear in mind throughout this unit that its purpose is not to speed up, or make more accurate your arithmetic, but to

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David Lindsay Roberts increase your understanding of what you are doing when you do arithmetic and algebra with numbers. (UICSM 1953–1954, pp. 1–1 and 1–2)8

With some justice, such sentiments came to be associated in the public mind with the entire tendency of the New Math era. One need only recall the big laugh from the audience evoked by Tom Lehrer’s patter in the live rendition of his “New Math” song of 1965: “But in the new approach, as you know, the important thing is to understand what you’re doing, rather than to get the right answer” (Lehrer 1965). The ninth-grade UICSM course proceeded to introduce students to increasingly inclusive number systems: natural numbers, integers, rational numbers, real numbers, and complex numbers. This reflected an understanding of how best to provide a foundation for the numerical side of mathematics that had become standard among research mathematicians since the late nineteenth century. The ideal aimed at by these mathematicians was to carefully derive each expansion step based on the properties established at the earlier levels so that only at the bottom level could any mere intuitive notions be manifested. Edmund Landau’s Grundlagen der Analysis of 1930 (Landau 1960) was an exemplary instance, with all details spelled out. Bruce Meserve’s Fundamental Concepts of Algebra included an abbreviated treatment (Meserve 1953). Few American schools until UICSM had given this topic so much emphasis or covered it in such an early grade. UICSM did not attempt to match the meticulous rigor of Landau but was nevertheless strikingly abstract and in some ways more painstaking than Meserve. It began with a discussion of one-to-one correspondence of sets, leading to the definitions of the natural numbers, (1, 2, 3, …), as classes of sets that can be put into one-to-one correspondence with each other. “The number two is the class of all sets that match (∆ ∆)” (UICSM 1953–1954, p. 1–21).9 Once the basic properties of the natural numbers were established, the integers were defined as equivalence classes of ordered pairs of natural numbers (Figure 2.2). The positive integers were represented by pairs (a, b) with a > b, while the negative integers were represented by pairs where a < b. The class of all pairs (a, a) was defined as zero. The UICSM students were then guided through the rules for addition and multiplication of these pairs, and various properties were derived, culminating in the following observation, offered with evident satisfaction: “There are many properties of the integers that you have taken for granted in the past that can now be proved” (UICSM 1954, p. 2–36). A closer correspondence than Landau or Meserve to the UICSM development of the number systems is found in the work of Beberman’s primary advisor at Teachers College, Howard Fehr (Figure 2.3). Fehr was himself a 1940 EdD graduate from Teachers College. He had written a dissertation titled “A Study of the Number Concept of Secondary School Mathematics,” intended as “a complete logical development of the number system in the English language that is readily accessible to secondary school teachers of mathematics” (Fehr 1940, p. 7). Fehr here developed the integers as ordered pairs of natural numbers, precisely as in UICSM above, though not as in Landau or Meserve.10 While Beberman was his student at Teachers College, Fehr revisited the development of the integers from pairs of natural numbers in a Mathematics Teacher article in 1949. Here he advised that “To attempt any logical presentation, similar to the one here given, in the high school is highly impractical and out of order” (Fehr 1949, p. 175), an admonition that Beberman ignored with UICSM.

 Underlining in original.  Underlining in original. 10  Meserve’s algebra text developed the number systems in an essentially equivalent manner, but with a different order and with different symbolism. After first defining the natural numbers he proceeded to the positive rational numbers and only then defined negative numbers, by “considering pairs [a – b] of nonnegative rational numbers as symbols” (Meserve 1953, p. 20). Fehr (1940) considers this treatment, but rejects it because “an unknown meaningless form (a – b) has been set equal to an unknown meaningless form -c” (p. 70). The UICSM treatment was identical in order and symbolism to Fehr’s treatment. Landau’s treatment was entirely distinct from both Fehr and Meserve (Landau 1960). 8 9

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Figure 2.2  Page from unpublished UICSM text material, 1954. (Originally owned by Virginia Garrett Rovnyak, now in possession of David L. Roberts)

Figure 2.3  Howard Fehr, 1949. (Photograph by Alman Co. 590 Fifth Avenue, New York, NY 10019. University Archives, Rare Book & Manuscript Library, Columbia University Libraries)

While UICSM made a point of not separating mathematics into separate compartments, a student in the program in the 1950s received the equivalent of a traditional high school sequence of algebra, geometry, and trigonometry, plus some calculus. But the UICSM student’s mathematics experience was enriched with less common topics such as the number system development described above, as well as vectors, determinants, probability, logic, and set concepts (UICSM 1955, 1956b, c, d). In

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general, there was far more attention to precision of language than most high school students would have encountered. UICSM made frequent alterations to its material, with Beberman and his ­colleagues not hesitating to change directions.11 Former students report a lively class atmosphere: “They came in every day with freshly-run ditto sheets. They were trying to see what worked and we were the guinea pigs. They were VERY interested in our responses, which made it interactive. It was fun!” (Rovnyak 2021).12 In a document from October 1956, the UICSM summed up its approach and its accomplishments. It was noted that the first high school graduates experiencing a full four years of the program would be entering college in the fall of 1957, and it mused on the appropriate college mathematics into which these students should be placed. Certainly, they will be over-prepared for certain kinds of freshman courses… As far as can be determined at the present time, it appears that these students will feel most at home with courses of the Allendoerfer-Oakley type, the Universal Mathematics type, the University of Chicago Type, and the type of course developed by Professor May at Carleton College and by Professor Begle at Yale. (UICSM 1956a, p. 6)

Many observers would have considered the UICSM curriculum at variance with the traditional preparation of future engineers, but the project managed to sell its approach on the basis that delving into the roots of mathematical ideas was ultimately more efficient. It was concluded “very early in its work” that “there is little need to differentiate at the early adolescent level among prospective engineers, prospective physicists, prospective physicians, prospective accountants, etc.” (UICSM 1956a, p. 1). The flourishing of the UICSM even withstood the brief abdication of Beberman, who, as an assistant professor in the College of Education, departed Illinois for Florida State University in 1954 and returned as an associate professor in 1955 (“News and Notices” 1954, 1955; Raimi 2004). The success at Uni High led the project to offer the curriculum at a small number of other high schools in Illinois and nearby states. It was soon realized that teachers in these other schools would need supplemental training, both in their own classrooms and in special sessions at the university. The consequent expansion of the UICSM staff required additional resources (UICSM 1956a). In 1956, a grant was obtained from the Carnegie Corporation of New York to support this expansion, a development that gave the program national recognition. Articles mentioning the $277,000 grant appeared in Time (“Math & Ticktacktoe” 1956), Scientific American (“Modernizing High-School Math” 1956), the Christian Science Monitor (Tallmadge 1956), and the New York Times (“Study Revamped for Mathematics” 1956). Plans were made to expand further. Beberman was acutely aware that he was under special scrutiny as an educationist. In a 1965 letter, he described the 1950s as being the time “when the antieducationist movement was in full swing and the mathematicians took over.” He then explained how he had weathered this environment: I was probably the only educationist actively at work in the curriculum reform movement in the 1950s, and I was able to hobnob with the mathematicians because I was regarded as a “damn good” teacher of mathematics, not as an educationist; and mathematicians have deep respect for good teachers. (Beberman 1965)



Sputnik and Its Aftermath

In October 1957, the Soviet Union launched Sputnik, the first artificial satellite to orbit the earth. Concerns about deficiencies in the mathematically trained workforce of the United States, which had remained robust since World War II, now reached new heights. Federal funds rapidly became available for application to mathematics education. The 1947 Steelman Report had hoped for a time when  Other Uni High teachers of mathematics during the 1950s included David Page and Eugene Nichols (Rovnyak 2007).  Capitalization in original. Virginia Rovnyak received a PhD in mathematics from Yale in 1965.

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a National Science Foundation (NSF) would serve as a major underwriter of experiments in mathematics education. The NSF, finally born in 1950, had at first ventured modestly into this arena (Phillips 2015). Now it began to play a major role. It is important to note that by the time of Sputnik there were several efforts for reform of school mathematics already in existence, in addition to the UICSM. We describe some of these briefly here. The Boston College mathematics department in 1953, under its new chairman, recently minted physics PhD, Father Stanley Bezuszka, SJ, began experimenting with modernizing its mathematics offerings for freshmen and sophomores.13 Finding that these students had emerged from high school unprepared for the new terminology and concepts, Bezuszka decided on an effort for “the re-­education of high school teachers in the elements of contemporary mathematics” (NCTM 1963, p. 6). With this goal in mind, in the summer of 1957, he founded the Boston College Mathematics Institute. In consultation with the in-service teachers attending the Institute, Bezuszka produced a text, Sets, Operations, and Patterns, a Course in Basic Mathematics, appropriate for eighth or ninth grade. Related commercial texts were later published by Science Research Associates (Bezuszka 2002; NCTM 1963). In 1955, the Ball State Teachers College Experimental Program was launched by Charles Brumfiel, a recent mathematics PhD. Using the Ball State Laboratory School in Muncie, Indiana as a testing ground, this program developed a secondary school curriculum with major emphasis on axiomatic structure and logic. Textbooks on geometry and algebra were eventually published by Addison-Wesley (NCTM 1961, 1970). The University of Maryland Mathematics Project (UMMaP) began its activities in September 1957, under the leadership of John Mayor, a 1933 mathematics PhD who had devoted most of his career to mathematics education (“Candidates for N.C.T.M Offices” 1952). Members of the departments of mathematics, education, psychology, and engineering were involved as consultants. The aim was to produce curriculum material for grades seven and eight: “a bridge between arithmetic and high school mathematics” (NCTM 1961, p. 19). An emphasis was placed on number systems. Experimental trials were conducted in schools in the Washington, DC area. The project received funding from the Carnegie Corporation of New York and later from the NSF. Much of the UMMaP material was merged into the junior high school curriculum of a much bigger project, the School Mathematics Study Group, to be described presently (Garstens et al. 1960; NCTM 1970). In 1955, the College Entrance Examination Board (1959), observing the signs that the colleges were changing their expectations for incoming students of mathematics, appointed a Commission “to review the existing secondary school mathematics curriculum, and to make recommendations for its modernization, modification, and improvement” (p. xi). The 14-person Commission, a mix of college teachers of mathematics, high school teachers of mathematics, and trainers of mathematics teachers included Allendoerfer, Fehr, and Northrop, and was chaired by mathematician Albert W. Tucker of Princeton. Among those attending one or more of the Commission’s meetings were Beberman, Cairns, Mac Lane, Mayor, Price, and Vaughan. The resulting report, Program for College Preparatory Mathematics, made specific recommendations as to the mathematical topics appropriate for a college-­ bound student and the order in which these topics might best be taught. It was weighted toward the sort of modern, structural mathematics we have described earlier in this chapter, although more conservative than UICSM.  The final report did not appear until 1959, but preliminary versions had become available by the time of Sputnik (NCTM 1970). The launch of Sputnik created the conditions to produce the largest New Math program of all, the School Mathematics Study Group (SMSG). It was founded in early 1958, following a flurry of educational concern among prominent members of the research-oriented AMS, which thereafter had very little to do with it officially, although many of its members would continue to play active roles. The  Bezuszka’s highest degree in mathematics was a bachelor’s degree, but his education in mathematics was probably the equivalent of a master’s degree (Bezuszka 2002). 13

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announced aim was to prepare a model secondary school curriculum and to arrange publication of monographs to spark the interest of secondary school students in mathematics. Pedagogically minded physicists had organized a similar project a few years previous, and they provided inspiration and guidance to the mathematicians (Wooton 1965). But high school physics was a tiny enterprise compared to high school mathematics. The SMSG soon received funding from the NSF, funding which would eventually dwarf the dollars received by other projects. SMSG was dissolved in 1972 when this funding ceased (Phillips 2015). During SMSG’s lifetime, its one and only director was mathematician Edward G. Begle (Figure 2.4), a 1940 Princeton PhD under Lefschetz. For a few years, Begle had been an active research mathematician, but by the early 1950s, as a professor at Yale, his research activity had virtually ceased (Phillips 2015). Colleagues have suggested that he had lost confidence in his ability to make further contributions in this direction and therefore turned to other avenues in which to contribute to mathematics (Rickart 1999). In 1954, he published a calculus textbook, notable for its axiomatic approach to the subject (Begle 1954). He also became the secretary of the AMS, a position that gave him wide knowledge of the personalities and talents of university mathematicians throughout the country. Meanwhile, Begle’s observation of his school-age children had caused him great unease about the general state of school mathematics in the United States. When he was offered the opportunity to influence school mathematics by leading the newly founded SMSG, he volunteered (Begle 1999). Begle soon found himself running a huge enterprise to design new school curricula. This was not part-time consulting such as Meserve and Vaughan had been doing at Illinois. Yale was supportive, to a degree, but made it clear that this was not the kind of work that would be rewarded with a full professorship (Rickart 1999). He thereupon left Yale, in 1961, taking SMSG with him to Stanford University, where he became a professor in the School of Education and a supervisor of doctoral candidates in education. Thus, it was that Begle, a mathematician, became a significant trainer of educationists, one of his most lasting legacies (Romberg 1999; Wilson and Kilpatrick 1999). The SMSG involved hundreds of people from across the nation (Wooton 1965). Many of the mathematics educators mentioned earlier in this chapter participated in some way. The initial intent was to produce and rapidly test new curricular material for the secondary schools, although it later expanded to include material for the elementary schools as well. Begle’s method, beginning in June of 1958, was to produce most of this material during intense summer writing sessions (Figure 2.5), using teams composed of both university mathematicians and school teachers. This balance of mathematical expertise and practical classroom experience would yield, it was hoped, mathematically rigorous yet teachable text material.

Figure 2.4  Edward G. Begle, 1961. (By Mercado, School Mathematics Study Group Records, e_math_00582, The Dolph Briscoe Center for American History, University of Texas at Austin)

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Figure 2.5  SMSG writing group at Stanford University, summer 1960. (School Mathematics Study Group Records, e_math_00581, The Dolph Briscoe Center for American History, University of Texas at Austin)

Some SMSG material, in hastily produced paperbound form, was tested in classrooms as early as the fall of 1958. The number of participating schools nationwide rose rapidly thereafter. Paul Rosenbloom of the University of Minnesota volunteered his state as a major test bed for SMSG, setting up what he dubbed the Minnesota National Laboratory to evaluate how well students performed when using SMSG texts. Meanwhile, the monograph section of SMSG also went forward, eventually producing a book series called the New Mathematical Library (Wooton 1965). The level of abstraction in SMSG was markedly less than in UICSM, but the influence of mathematical modernizers could be detected. Set concepts and one-to-one correspondence were thoroughly embraced, and over time were pushed down further and further into the lower grade materials. High school students were led through many familiar topics but now with more attention to precision of language. The content for grades seven and eight emphasized the commutative, associative, and distributive properties far more than standard courses at that level and was also novel in its attention to probability. The SMSG geometry, with mathematician Edwin Moise as a lead author, attempted to give the subject “a sort of head start” (SMSG 1960, Preface) by assuming the students to have knowledge and intuition regarding the real number system, an approach suggested by mathematician George David Birkhoff in the 1930s. A topic initially offered in seventh grade which would create controversy was doing arithmetic in bases other than 10 (SMSG 1959). Even SMSG partisans conceded that some teachers became overly enthusiastic with this, especially when it began to be adopted in elementary schools (Rosenbloom 2000). In the 1960s, satirical wit Tom Lehrer would be unable to resist observing that base eight was “just like base ten really—if you’re missing two fingers” (Lehrer 1965).

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Concluding Remarks

The end of 1959 can be seen in retrospect as a high-water mark of optimism regarding the New Math in the United States. Most of the experimental curriculum projects that had begun earlier in the decade were still going strong, and new projects were being launched. The biggest project, SMSG, was on the upswing, its materials being eagerly adopted by an increasing number of schools all across the country. In the fall of 1959, it is estimated that 26,000 students were using SMSG texts (Wooton 1965). Educationist Howard Fehr, Max Beberman’s mentor, surveyed the scene in a Mathematics Teacher article in January 1959. “All over the country various groups, large and small, are embarking on projects to produce a better mathematics program. Will they succeed?” (Fehr 1959, p. 18). He had been active in mathematics education since the bad old days when the subject was fighting to survive in the schools. He now saw much yet undone, but he was hopeful. He admired the vibrant community of research mathematicians and like them lamented the disparity between their work and the school curriculum. He, therefore, embraced the modernization proposals and welcomed mathematicians seeking an active role in the schools: “The mathematicians desire to close the gap between 19th-century high school mathematics and 20th-century mathematical thinking at research frontiers” (p. 17). But Fehr warned that these mathematicians needed to work closely with teachers if they expected to have a long-term influence in the schools. At the same time, he advised that teachers, or at least some among them, needed to improve their scholarly knowledge of mathematics in order to work confidently with the mathematicians. He judged it essential for some teachers to aspire to such knowledge, though admitting that it would be no easy task for a high school teacher, weighed down with other responsibilities. Thus Fehr, neither school teacher nor mathematical researcher, was advising these two groups on how they should best proceed. True, he had once been a school teacher, but in 1959 he was speaking as a representative of the rapidly growing cadre of teacher educators and education researchers. Thus, Fehr was not only commenting on the jurisdictional complexity of the school education environment described in this chapter, but he was also illustrating it in his own person. How should the responsibility for school mathematics be parceled out among the variety of mathematics educators: school teachers, university teachers, mathematics researchers, education researchers, and teacher trainers? How take account of the assortments of firm credos and ambivalence found among the members of these groups? These were problems that would not be resolved in the 1960s or in subsequent decades. Alert observers in 1959 might have detected other signs that the future would not be smooth for school mathematics. Not everyone agreed that the general line taken by most of the New Math programs was a good thing. Mathematician Morris Kline of New York University, who would become the most well-known and persistent critic of the New Math, had already indicated the themes he would expand upon in the 1960s and 1970s. Since 1954, he had been attacking the undergraduate texts and programs we have described as “modern,” advocating instead what he called a “cultural approach” (Kline 1954, p. 299). He intensely disliked the trend to load the liberal arts mathematics course with abstract concepts such as groups, rings, and fields. The stress on “rigor” in Begle’s calculus book had for Kline reached “the heights of absurdity” (Kline 1955, p. 9). In April 1958, speaking at a meeting of the NCTM, he began to train his fire on the high school programs, based, he said, on conversations with New Math advocates and on articles he had read in the Mathematics Teacher. Here Kline branded modern topics as “peripheral” (Kline 1958, p. 419), and he scoffed at the claim that abstraction offered advantages of efficiency. Students should be led to appreciate mathematical applications to physical science, since “the social sciences are hardly sciences, let alone mathematical sciences” (p. 420). Teacher training was another substantial issue that was lurking. The UICSM had largely managed to insist that its materials be used only by qualified teachers who had undergone special training. The SMSG had similar aspirations, but the quickly ballooning size of the program would make this increasingly difficult to maintain. NSF-funded summer training institutes for teachers could not meet

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the demand. The problem would become an order of magnitude worse as the SMSG attempted to reach into the vast realm of the elementary schools, often staffed by teachers fearful of mathematics. There was also the politically loaded question of whether the new programs were to be for all students or for a select few. It will be recalled that the notion of distinguishing students on the basis of ability or of career plans was on the minds of the committee report writers of the 1940s. Adherents to the life adjustment movement had been maintaining that it was futile and even cruel to insist on challenging all students with abstract mathematics. Some New Math promoters would have essentially agreed. To the extent that the modernization of the school curriculum in the 1950s was an attempt to prepare students for a reoriented college curriculum, there were grounds to limit the new school programs to a minority and to evaluate its success only with this minority. UICSM, as originally confined to Uni High, was frankly elitist. One student from the class of 1957 has expressed his dismay that it failed to remain so: “I grew up in Champaign-Urbana on the Illinois prairie, and was in the first batch of students to pass through Max Beberman’s exciting ‘new math’ program for good students with good teachers, before it was eviscerated to serve a mass audience” (McCrimmon 2021).14 Beberman himself had more democratic aspirations, which he was able to satisfy at least momentarily with his teaching brilliance. When he visited schools, he would ask to be introduced to a class considered of low ability. He would then, in an hour, have that class doing mathematics with a facility not considered possible by their teacher (Romberg 1999). For anyone who took seriously the claim that attention to the abstract foundations of mathematics had pedagogical advantages, it was natural to attempt to teach it to all comers. It is clear that this was the desire of many curriculum modernizers, in addition to Beberman, but that they struggled to accomplish this in practice. Writing in 1960, Mayor, the director of UMMaP, proclaimed that “the new materials should make the teaching of mathematics and the learning of mathematics much more profitable and enjoyable for all pupils” (Mayor and Brown 1960, p. 377).15 Yet in 1959, the UMMaP had reported on “the desirability of an easier course for slow learners” (“Report to the Carnegie Corporation of New York” 1959, p. 5). One of the tenets of the 1958 Eleventh Grade Writing Group in 1958 was that “Individual differences in ability and motivation must be recognized even among college-capable students. Some material must be included for the student who has exceptional ability in mathematics” (Allen 1965, p. 41). Another writing group, reflecting on its 1959 experience preparing “Introduction to Secondary School Mathematics” for junior high students, noted the danger of premature classification of students wherein “there may well be undiscovered and undeveloped capacity for profitable study of more mathematics than heretofore had been supposed” (Keifer 1965, p. 67). Such issues, which today would be tagged with words such as “equity” and “access,” were discussed by the mathematics educators we have been surveying with remarkably little acknowledgment of the concurrent national struggle by African Americans for equity and access. It will be recalled that the Supreme Court decision of Brown v. Board of Education, a landmark event in the emerging civil rights movement in the United States, was issued in 1954. Whatever the reality of American schools in practice, their aspirations to democratic access were frequently invoked in comparing them to European schools. Such aspirations were firmly in the mind of Howard Fehr, who was making such comparisons in the 1950s, as he became increasingly interested in viewing mathematics education from a global perspective and in forging connections with European mathematics educators. He attended the International Congress of Mathematicians in Amsterdam in 1954 (Fehr 1955), and the following one in Edinburgh in 1958 (Fehr 1965), with most of his attention on the activities of these meetings relevant to school and college education. The  Kevin McCrimmon received a PhD in mathematics from Yale in 1965. The University of Illinois University High School graduated 37 students in 1957 (Uni Graduating Classes 1951–1960 2021). Running these names through the Mathematics Genealogy Project (2021) detected three PhDs in pure mathematics (including McCrimmon, and Rovnyak, noted earlier) and two PhDs in mathematical economics. 15  Italics in original. 14

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Edinburgh Congress included a United States sectional meeting on Mathematical Education, during which leading lights of the American curriculum modernization movement made presentations: Allendoerfer, Begle, Duren, Price, and Tucker. Fehr himself, the outgoing president of the NCTM, gave an address on “Instruction in Mathematics Around the World to Youth Age 6 to 15 Years” (Fehr 1965). He painted “a rather sorry picture of the type of study that was commonplace for most countries” (p. 37). The Americans, according to Fehr, inspired many other countries “to take action in a reform of mathematical education” (p. 37). Such European action based on American experience was forwarded especially by the Organisation for European Economic Cooperation (OEEC), which in late 1959 organized a two-week comprehensive seminar on mathematics education, with representatives from Europe, Canada, and the United States. Fehr was once again in attendance, in Royaumont, France, and indeed he would be the lead author of the report that came out of this seminar, not published until May 1961 (OEEC 1961). Prior to this, he published a short commentary in the Monthly based on a survey conducted in conjunction with the seminar. Here he made clear his enthusiasm for the direction being taken by the American New Math reforms. Of the mathematics content taught in European schools, he opined that in no case is it modern in the sense of the fine materials written by the School Mathematics Study Group and the University of Illinois School Mathematics Committee, or the program advocated by the Commission on Mathematics. (Fehr 1960, p. 799)

References Allen, F. (1965). The philosophy of the eleventh grade writing group. In Philosophies and procedures of SMSG writing teams (pp. 39–44). Palo Alto, CA: Stanford University. Allendoerfer, C. B. (1947). Mathematics for liberal arts students. American Mathematical Monthly, 54, 573–578. Allendoerfer, C. B., & Oakley, C. O. (1963). Principles of mathematics. New York, NY: McGraw-Hill. Allendoerfer, C.  B., & Weil, A. (1943). The Gauss-Bonnet theorem for Riemannian polyhedra. Transactions of the American Mathematical Society, 53, 101–129. Beberman, M. (1965, January 7). Letter from Beberman to unknown “school” person. SMSG records, CDL 4/2, Box 6, Folder B 64–65, Archives of American Mathematics, Center for American History, The University of Texas at Austin. Begle, E. G. (1954). Introductory calculus with analytic geometry. New York, NY: Henry Holt. Begle, E. (1999, April 19). Transcript of interview by David L.  Roberts. In R.  L. Moore Legacy Collection, 1890– 1900, 1920–2013. Box 4RM15. Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin. Bell. E. T. (1935). [Review of The poetry of mathematics and other essays, by D. E. Smith]. American Mathematical Monthly, 42, 558–562. Bestor, A. E. (1952). Aimlessness in education. Scientific Monthly, 75, 109–116. Bestor, A. (1953). Educational wastelands: The retreat from learning in our public schools. Urbana, IL: University of Illinois. Betz, W. (1942). The necessary redirection of mathematics, including its relation to national defense. Mathematics Teacher, 35, 147–160. Betz, W. (1948). Looking again at the mathematical situation. Mathematics Teacher, 41, 372–381. Bezuszka, S. J. (2002, June 6–7). Transcript of interview by David L. Roberts. In NCTM Oral History Project records, 1992–1993, 2002–2003. Box 4RM109. Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin. Birkhoff, G., & Mac Lane, S. (1941). A survey of modern algebra. New York, NY: Macmillan. Bourbaki, N. (1950). The architecture of mathematics. American Mathematical Monthly, 57, 221–232. Cairns, S. S. (1953). Mathematics and the educational octopus. Scientific Monthly, 76, 231–240. “Candidates for N.C.T.M offices––1952 ballot”. (1952). Mathematics Teacher, 45, 139–144. College Entrance Examination Board. (1959). Report of the Commission on Mathematics: Program for college preparatory mathematics. New York, NY: CEEB.

2  The Rise of the American New Math Movement: How National Security Anxiety and Mathematical…

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Commission on Postwar Plans. (1944). The first report of the Commission on Post-War Plans. Mathematics Teacher, 37, 226–232. Commission on Postwar Plans. (1945). The second report of the Commission on Post-War Plans. Mathematics Teacher, 38, 195–221. Davis, R. L. (1958). Elementary mathematics of sets with applications. Buffalo, NY: Committee on the Undergraduate Program, Mathematical Association of America. “Deficiencies in mathematics a serious handicap in the war crisis”. (1942). School and Society, 55, 178. Dieudonné, J. (1941). Sur le théorème de Lebesgue-Nikodym [On the Lebesgue-Nikodym theorem]. Annals of Mathematics, 42, 547–555. “Dr. Max Beberman is dead at 45; a creator of new mathematics”. (1971, January 26). New York Times, p. 36. Duren, W. L., Jr. (1956). Tulane experience with Universal Mathematics, Part I. American Mathematical Monthly, 63, 199–202. Duren, W. L., Jr. (1967). CUPM, the history of an idea. American Mathematical Monthly, 74-II, 23–37. Duren, W. L., Jr., Newsom, C. V., Prince, G. B., Putnam, A. L., & Tucker, A. W. (1955). Report of the Committee on the Undergraduate Mathematical Program. American Mathematical Monthly, 62, 511–520. Fehr, H. F. (1940). A study of the number concept of secondary school mathematics. New York, NY: Teachers College Columbia University. Fehr, H. F. (1949). Operations in the systems of positive and negative numbers and zero. Mathematics Teacher, 42, 171–175. Fehr, H.  F. (1955). The International Mathematics Union, the International Congress of Mathematicians, and the International Commission of Mathematical Instruction. Mathematics Teacher, 48, 267–269. Fehr, H. F. (1956). [Review of Principles of mathematics, by C. B. Allendoerfer & C. O. Oakley]. Scientific Monthly, 82, 270–271. Fehr, H. F. (1959). Breakthroughs in mathematical thought. Mathematics Teacher, 52, 15–19. Fehr, H. F. (1960). What mathematics is taught in European schools. Mathematics Teacher, 67, 797–802. Fehr, H. F. (1965). Reform of mathematics education around the world. Mathematics Teacher, 58, 37–44. Fuller, H. J. (1951). The emperor’s new clothes, or prius dementat. Scientific Monthly, 72, 32–41. Garstens, H. L., Keedy, M. L., & Mayor, J. R. (1960). University of Maryland Mathematics Project. Arithmetic Teacher, 7, 61–65. Hart, W. L. (1941a). On education for service. American Mathematical Monthly, 48, 353–362. Hart, W. L. (1941b). Progress report of the subcommittee on education for service of the war preparedness committee of the American Mathematical Society and the Mathematical Association of America. Mathematics Teacher, 34, 297–304. Hedrick, E. R. (1942). Mathematics in the national emergency. American Mathematical Monthly, 35, 253–259. Henderson, K. B., & Dickman, K. (1952). Minimum mathematical needs of prospective students in a college of engineering. Mathematics Teacher, 45, 89–93. Jones, B. W. (1963). Elementary concepts of mathematics. New York, NY: Macmillan. Jones. P.  S. (1951). Le mathématicien polycéphale [The polycephalous mathematician]. Mathematics Teacher, 44, 500–501. Keifer, M. (1965). Report of the preparation of Introduction to secondary school mathematics. In Philosophies and procedures of SMSG writing teams (pp. 67–72). Palo Alto, CA: Stanford University. Kemeny, J.  G., Snell, J.  L., & Thompson, G.  L. (1966). Introduction to finite mathematics. Englewood Cliffs, NJ: Prentice-Hall. Kline, M. (1954). Freshman mathematics as an integral part of western culture. American Mathematical Monthly, 61, 295–306. Kline, M. (1955). Pea soup, tripe and mathematics. Marco Learning Systems. www.marco-­learningsystems.com/pages/ kline/lecture.html. Kline, M. (1958). The ancients versus the moderns, a new battle of the books. Mathematics Teacher, 51, 418–427. Landau, E. (1960). Grundlagen der analysis [Foundations of analysis]. New York, NY: Chelsea. Langer, R. E. (1952). The things I should have done, I did not do. American Mathematical Monthly, 59, 443–448. Lefschetz, S. (1950). The structure of mathematics. American Scientist, 38, 105–111. Lehrer, T. (1965). Transcribed from “New Math,” on compact disc That was the year that was: TW3 songs and other songs of the year. Reprise. Life adjustment education for every youth. (1948). United States Office of Education, Division of Secondary Education. Mac Lane, S. (1997). Van der Waerden’s Modern algebra. Notices of the American Mathematical Society, 44, 321–322. Mann, W.  R. (1956). [Review of Principles of mathematics, by C.  B. Allendoerfer & C.  O. Oakley]. American Mathematical Monthly, 63, 437–439. “Math & Ticktacktoe”. (1956, July 23). Time, 68, pp. 78–80. Mathematics Genealogy Project. (2021). Welcome! – The Mathematics Genealogy Project (nodak.edu).

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Mayor, J. R., & Brown, J. A. (1960). Teaching the new mathematics. School and Society, 88, 376–377. McCrimmon, K. (2021). Kevin McCrimmon, Department of Mathematics, University of Virginia. Kevin McCrimmon Biographical Sketch (virginia.edu). Meeker, B. (2000, March 23). Personal communication. Meserve, B. E. (1953). Fundamental concepts of algebra. Cambridge, MA: Addison-Wesley. Meserve, B. E. (2003, April 3). Transcript of interview by David L. Roberts. In NCTM Oral History Project records, 1992–1993, 2002–2003. Box 4RM109. Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin. “Modernizing High-School Math”. (1956, August). Scientific American, 195, p. 50. Morse, M. (1940). Report of the war preparedness committee. American Mathematical Monthly, 47, 500–502. Morse, M. (1943). Mathematics and the maximum scientific effort in total war. Scientific Monthly, 56, 50–55. Morse, M., & Hart, W. L. (1941). Mathematics in the defense program. American Mathematical Monthly, 48, 293–302. NCTM. (1961). The revolution in school mathematics. Washington, DC: NCTM. NCTM. (1963). An analysis of new mathematics programs. Washington, DC: NCTM. NCTM. (1970). A history of mathematics education in the United States and Canada. Washington, DC: NCTM. “News and Notices”. (1950). American Mathematical Monthly, 57, 651–656. “News and Notices”. (1954). American Mathematical Monthly, 61, 726–732. “News and Notices”. (1955). American Mathematical Monthly, 62, 598–602. Northrop. E. P. (1945). Mathematics in a liberal education. American Mathematical Monthly, 52, 132–137. Northrop, E. P. (1948). The mathematics program in the College of the University of Chicago. American Mathematical Monthly, 55, 1–7. Oakley, C. O. (1942). The Coming Revolution—in Mathematics. Mathematics Teacher, 35, 307–309. OEEC. (1961). New thinking in school mathematics. Paris, France: OEEC. Phillips, C. J. (2015). The new math: A political history. Chicago, IL: University of Chicago. Price, G. B. (1951). A mathematics program for the able. Mathematics Teacher, 44, 369–376. Price, G.  B. (1955). A Universal Course in Mathematics. In Lectures on Experimental Programs in Collegiate Mathematics, Stillwater, United States: Department of Mathematics, the Oklahoma Agricultural and Mechanical College. Raimi, R. (2004). Chapter 1, Max. Work in progress, concerning the history of the so-called new math, of the period 1952–1975 approximately. https://people.math.rochester.edu/faculty/rarm/beberman.html. Reeder, E. H. (1951). The quarrel between professors of academic subjects and professors of education: An analysis. American Association of University Professors Bulletin, 37, 506–521. Reeve, W. D. (1936). Is mathematics losing ground? Mathematics Teacher, 29, 253–254. Reeve, W. D. (1942). The importance of mathematics in the war effort. Mathematics Teacher, 35, 88–90. “Report to the Carnegie Corporation of New York on the University of Maryland Mathematics Project (Junior High School) for the Year Ending October 31, 1959”. (1959). AAAS Archives, Education and Human Resources, 1956– 1970, D. Ost files and Science Teacher Improvement project (Box 5), 892/I-5-4. Rickart, C. E. (1999, June 14). Personal communication. Roberts, D.  L. (2012). American mathematicians as educators, 1893–1923: Historical roots of the “math wars”. Boston, MA: Docent. Romberg, T. (1999, Aug. 13). Transcript of interview by David L. Roberts. In R. L. Moore Legacy Collection, 1890– 1900, 1920–2013. Box 4RM108. Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin. Rosenbloom, P. C. (2000, December 6). Transcript of interview by David L. Roberts. In R. L. Moore Legacy Collection, 1890–1900, 1920–2013. Box 4RM19. Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin. Rovnyak, V. (2007, November 14). Personal communication. Rovnyak, V. (2021, March 5). Personal communication. Schorling, R. (1948). A program for improving the teaching of science and mathematics. American Mathematical Monthly, 55, 221–237. Schult, V. (1949). Are we giving our mathematics students a square deal? Mathematics Teacher, 42, 143–148. SMSG. (1959). Mathematics for junior high school, volume I (part 1) (revised edition). New Haven, CT: Yale University. SMSG. (1960). Mathematics for high school geometry (part I). New Haven, CT: Yale University. Steelman, J.  R. (1947). Science and public policy, vol. 4: Manpower for research. Washington, United States: Government Printing Office. “Study Revamped for Mathematics”. (1956, June 27). New York Times, p. 33. Tallmadge, E. (1956, November 10). Five pilot schools try new U. of Illinois method of teaching math. Christian Science Monitor, p. 15. The College Mathematics Staff. (1954). Concepts and structure of mathematics. Chicago, IL: University of Chicago.

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“The letter of Admiral Nimitz”. (1942). American Mathematical Monthly, 49, 212–214. UICSM. (1953–1954). Unit 1 THE NATURAL NUMBERS. Document in possession of David L. Roberts, gift from Virginia Garrett Rovnyak, Class of 1957, University High School, University of Illinois. Underlining in original. UICSM. (1954). Unit 2 (Optional): Integers and Rational Numbers. UICSSM-X1-54. Document in possession of David L. Roberts, a gift from Virginia Garrett Rovnyak, Class of 1957, University High School, University of Illinois. UICSM. (1955). Unit Two: Equations, functions, slope function. UICSM-1-55, Third Course. Document in possession of David L. Roberts, a gift from Virginia Garrett Rovnyak, Class of 1957, University High School, University of Illinois. UICSM. (1956a). The Work of the University of Illinois Committee on School Mathematics. Document found in AAAS Box 334/C-5-6, Office Files (John Mayor), 1961–1970, File: STIP mimeograph materials. UICSM. (1956b). Second Course. Document in possession of David L. Roberts, a gift from Virginia Garrett Rovnyak, Class of 1957, University High School, University of Illinois. UICSM. (1956c). Third Course. Document in possession of David L. Roberts, a gift from Virginia Garrett Rovnyak, Class of 1957, University High School, University of Illinois. UICSM. (1956d). Fourth Course. Document in possession of David L. Roberts, a gift from Virginia Garrett Rovnyak, Class of 1957, University High School, University of Illinois. Uni Graduating Classes 1951–1960. (2021). Uni Graduating Classes 1951–1960 | University of Illinois Laboratory High School Vance, E. P. (1948). The teaching of mathematics in colleges and universities. American Mathematical Monthly, 55, 57–64. Van der Waerden, B. L. (1930). Moderne algebra [Modern algebra]. Berlin, Germany: Springer. Weil, A. (1992). The apprenticeship of a mathematician. Basel, Switzerland: Birkhäuser. Wilson, J., & Kilpatrick, J. (1999, May 24). Transcript of interview by David L.  Roberts. In R.  L. Moore Legacy Collection, 1890–1900, 1920–2013. Box 4RM108. Archives of American Mathematics, Dolph Briscoe Center for American History, University of Texas at Austin. Wooton, W. (1965). SMSG: The making of a curriculum. New Haven, CT: Yale University. “Would-be midshipmen stumble over elementary mathematics”. (1942). North Central Association Quarterly, 16, 222. Young, J.  W. (1932). Functions of the Mathematical Association of America. American Mathematical Monthly, 39, 6–15.

Chapter 3

The Early Roots of the European Modern Mathematics Movement: How a Model for the Science of Mathematics Became a Model for Mathematics Education Dirk De Bock

Abstract  The cradle of modern mathematics in Europe is likely to be traced to the founding meeting of the International Commission for the Study and Improvement of Mathematics Teaching in 1952 in La Rochette par Melun (France). The organizer of that meeting, Caleb Gattegno, had chosen Mathematical and Mental Structures as a theme and succeeded in bringing together several “big names” from the fields of psychology, epistemology, and mathematics, including, Jean Piaget, Ferdinand Gonseth, and the “Bourbakists” Jean Dieudonné, Gustave Choquet, and André Lichnerowicz. A few outstanding secondary school teachers of mathematics participated too, among them Lucienne Félix and Willy Servais. Dieudonné explained the architecture of modern mathematical science, based on set theory and on the so-called “mother structures” of mathematics. Piaget linked these fundamental structures of mathematics with the stages of early mathematical thinking, as revealed by psychology. The claim of alignment between Bourbaki’s mother structures and Piagetian theory provided a strong argument for a substantial reform of mathematics education. Keywords  André Lichnerowicz · Caleb Gattegno · CIEAEM · Cognitive development · Éléments de mathématique · Ferdinand Gonseth · Genetic epistemology · Gustave Choquet · Hans Freudenthal · Jean Dieudonné · Jean Piaget · Lucienne Félix · Mathematical structure · Mental structure · Mother structures · Nicolas Bourbaki · Willy Servais

Introduction In the post–World War II euphoria of liberation and the momentum of reconstitution, debates about the renovation of education, structural and institutional, but also in terms of content and teaching methods, emerged in several West European countries. These post-War debates were often led or inspired by left-wing intellectuals from circles of the resistance, and outspoken opponents of fascism, such as Paul Langevin and Henri Wallon in France, and Frans Van den Dungen and Paul Libois in Belgium. These and other protagonists of national reform movements had ambitious objectives, D. De Bock (*) KU Leuven, Leuven, Belgium e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_3

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although the resulting action plans were not or only partially implemented during the 1950s. Central themes were the improvement of educational quality, the extension of compulsory schooling, the development of the student as a whole (intellectually, but also morally, artistically, and physically), and the democratization of education in the sense that students’ orientation should depend on their competences and not on their family, social, or ethnic background. Although educational reform initiatives of that period were initiated by national politics or academia, enthusiastic teachers also played an important role in the ongoing debates, either on an individual basis or through their professional associations and journals. After all, the position and orientation of various educational subjects, such as mathematics, science, and languages, also formed part of the reform discussions (De Bock and Vanpaemel 2019, Chapter 1; d’Enfert and Kahn 2010). The early post-War reform debates took place mainly at national levels. However, in the field of mathematics, there already existed a longstanding tradition of international cooperation (Furinghetti 2014). In 1908, the International Commission on Mathematical Instruction (ICMI) was founded, and in 1952, it acquired the official status of sub-commission of the International Mathematical Union. One of the purposes of ICMI was (and is) to provide a worldwide forum for the exchange and dissemination of national ideas and realizations in mathematics education at all levels. Although in the early 1950s, ICMI restarted its pre-War activity, mainly based on international inquiries and reports (see, e.g., Furinghetti and Giacardi 2008; Menghini et  al. 2008), a newly founded organization, 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 would come to play a more decisive role in the European reform debates of the 1950s. In the next section, we outline the origin and characteristics of this new organization in the field of mathematics education.



The International Commission for the Study and Improvement of Mathematics Teaching

In April 1950, Caleb Gattegno (1911–1988) brought together an international group of experts in mathematics, psychology, and education, including some experienced mathematics teachers, in Debden (United Kingdom). Although the number of participants was limited to about 15, the range of competencies had to allow for “a thorough reconsideration of the whole problem of the child and mathematics” (Gattegno 1947, p. 220). This meeting, followed by two similar meetings in 1951, one in Keerbergen (Belgium) and one in Herzberg (Switzerland), led to the official founding of the CIEAEM in La Rochette par Melun (France) in April 1952. The first executive committee consisted of the mathematician Gustave Choquet (of the University of Paris) as president, the cognitive psychologist Jean Piaget (Universities of Genève and Paris) as vice-president, and Gattegno (University of London) as secretary, a trio that would serve throughout the 1950s. Bernet and Jaquet (1998) mentioned another 20 “founding members” of the CIEAEM, among them Evert Willem Beth, Emma Castelnuovo, Jean Dieudonné, Lucienne Félix, Félix Fiala, Hans Freudenthal, Ferdinand Gonseth, André Lichnerowicz, and Willy Servais. According to these authors, this list of “founding members” rather reflected status but not necessarily a physical presence in La Rochette (Figure 3.1). But it was the then little-known mathematician, Gattegno, who was the founder and actual animator of the CIEAEM during its first decade, and one should ask why he was moved to create a new, albeit informal, organization in the field of mathematics education. Gattegno was born and grew up in Alexandria (Egypt) where he studied science and mathematics—mainly as an autodidact— and worked there for a number of years as a teacher and lecturer. In 1937, he obtained a doctorate in mathematics at the University of Basel (Switzerland). He would earn

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Figure 3.1  Founding meeting of the CIEAEM in La Rochette par Melun, 1952, from left to right: Mrs. and Mr. F. Gonseth, J. Dieudonné, Lucien Delmotte, G. Choquet, L. Félix, F. Fiala, J. Piaget, unknown, C. Gattegno, unknown

a second doctorate in 1952, this time in psychology, at the University of Lille (France). In 1950, he was appointed as a professor at the Institute of Education of the University of London. Gattegno was a charismatic figure, some would have called him a guru, but in any case, he was a very hard and passionate worker. Over the years, he organized numerous conferences in Europe, North and South America, and Japan, and published some 120 books and 500 articles. He was gifted in languages and could easily express himself in Arabic, English, French, German, and Spanish. His main research interests, at least in the early years of his career, involved the (cognitive) psychological aspects of the learning of mathematics. For the afore-mentioned meeting in Debden, he had accordingly chosen the theme “Relations between the curricula of mathematics in the secondary schools and the intellectual capacities development of the adolescent” (Brown et al. 2010; Noël 2018; Powell 2007). In his Preface to Félix (1986), Gattegno explained why he had founded the CIEAEM: Since 1946, I earned my living full time in the field of “mathematics education for the secondary school” by training teachers for that level in my seminars at the University of London. So I had personal reasons to study this teaching to improve the teaching for the future students of my students. So for me, the study should precede the improvement and it is in this order that I named the functions of the Commission. […] For the study we needed a group in which epistemologists, logicians, psychologists, mathematicians, and educators would learn from each other what they didn’t know in order to create a synthesis, which was new anyway, and needed by those who wanted to improve. (no pagination)

Gattegno and his newly founded organization focused markedly on the study of psycho-pedagogical and foundational issues in mathematics education; social, economic, or cultural dimensions received almost no attention (Gispert 2010). This exclusive position of the CIEAEM would be maintained throughout the 1950s, as is illustrated by the collectively endorsed statement that “differences resulting from cultures are less important than similarities resulting from the structure of science and mathematical thought” (Preface to Piaget et al. 1955, p. 6, signed “Le Bureau”). Although an in-depth historical analysis of the CIEAEM has not yet appeared, a few sources provide us with some insight into its early points of interest and practices. Because these documents were realized by people who were strongly committed to this organization, objectivity is not always fully guar-

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Figure 3.2  Covers of the two books from the 1950s with collective work by CIEAEM members. (Collection Guy Noël)

anteed. First, there exists a historical overview by Lucienne Félix (1901–1994), a French teacher and a loyal participant from the first meeting. Félix’s report, covering the period 1950–1984, was first published in 1985 (Félix 1985a) and revised and extended one year later (Félix 1986). This report not only dealt with some scientific aspects of the CIEAEM meetings, but also sketched the Petite Histoire—providing anecdotes and other details of daily-life and the “atmosphere” at the meetings. Second, two books with collective work were published during the 1950s: L’Enseignement des Mathématiques [The Teaching of Mathematics] (Piaget et al. 1955) and Le Matériel Pour l’Enseignement des Mathématiques [Materials for the Teaching of Mathematics] (Gattegno et al. 1958), both published by Delachaux et Niestlé, Switzerland (Figure 3.2). Third and last, on the occasion of the 50th meeting of the CIEAEM in Neuchâtel (1998), a special effort was made to collect and synthesize the available historical documents related to the history of the CIEAEM until then. This led to an informative brochure by Bernet and Jaquet (1998), including a list of the first 50 meetings, some internal documents from the private archives of Willy and Renée Servais, and excerpts from the aforementioned publications and Proceedings of CIEAEM meetings which appeared regularly from 1974 onward. Some information about CIEAEM’s early intentions can be found in an action plan, dated Easter 1952 and probably edited by Willy Servais (Bernet and Jaquet 1998). This action plan lists various short-term projects in which the distinct roles of university professors and of teachers are defined. The university professors were invited, through their contacts with their colleagues from other scientific disciplines, to determine the future direction of mathematics education at the secondary level. They were also expected to formulate requirements about the content and style of this education, and to take initiatives so that teachers would be able to realize these requirements. For teachers of the other educational levels, a number of more concrete objectives were identified: designing lessons as psychological experiments, analyzing a part of the curriculum or a specific technique, exploring new methods of expression, studying students’ reactions to an axiomatic approach in geometry. Moreover, four long-term projects were formulated (Bernet and Jaquet 1998, p. 23): • Analyzing the changes in the understanding of mathematics as an activity due to: –– The crisis in the foundations of mathematics –– The work of Bourbaki –– The progress of mathematical epistemology

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–– The teaching experiments at various levels –– The psychology • Studying the new requirements imposed on the mathematics program by: –– The growing industrialization –– The social transformations –– The global consciousness • Striving for a synthesis in the form of a functional program to be tested, first by the Commission members, and by more researchers along with the establishment of the facts and the discovery of the methods • Interesting the public through the Associations [for teachers] and gaining acceptance by UNESCO of an adequate method of dissemination at the international level. In particular, to take the initiative to organize national or international surveys, to set up specialized committees, to participate in the work of similar commissions initiated by other groups, etc. To realize its projects, meetings (in French: rencontres) were organized, in principle on an annual basis. These were to include both Anglo-Saxon and continental traditions in (mathematics) education. Two official languages, English and French, were used side by side during CIEAEM meetings, but French was clearly dominant in the 1950s. These meetings usually lasted for about 10 days and took place at a quiet and secluded location, sometimes even with sessions in the open air (Figure 3.3). They were not conceived as scientific conferences in today’s sense, with keynote lectures and paper presentations, but rather followed a seminar or workshop format. They were sometimes held in an informal and friendly atmosphere and were characterized by open exchanges of experience and constructive dialogue between academics from diverse disciplines, as well as experienced teachers of mathematics. Remarkably for the 1950s and early 1960s, several female teachers also played a significant role within the organization (Emma Castelnuovo, Lucienne Félix, and Frédérique Lenger, later also Anna Zofia Krygowska, etc.). Academics and teachers conversed and collaborated on an equal footing, there was no status-based hierarchy.

Figure 3.3  Working session in a park at the founding meeting of the CIEAEM in La Rochette par Melun, 1952

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Each meeting was organized around a specific theme, communicated prior to the meeting, and addressed some (psychological) aspect of the teaching and learning of mathematics of which the ­relevance was generally acknowledged by the CIEAEM members. The practical mode of operation during the meetings was highlighted by Renée Servais in the following way: The work of the Commission was based on discussions on the theme that had been chosen. The conceptions and experiences of the participants were confronted. Gattegno played a role of facilitator: Encouraging the work in the group, posing problems, making observations, reporting personal or collective research. … An important activity consisted in giving lessons to children who were not really selected (sometimes retarded children). These were not model lessons, but rather spontaneous teaching experiences. Errors, unexpected answers were examined, the aim was to improve the pedagogy, the psychological understanding of the child. (quoted in Félix 1986, as summarized by Noël 2018, p. 68).

These lessons, often given by Gattegno himself, were thus not intended to help students learn something from a teacher, but rather to help teachers learn something from the students, in an attempt to understand their thinking better (Lenger 1954–1955b).



Modern Mathematics as an Educational Project

Structures in Mathematics and Child Psychology From the very beginning, Gattegno had some clear ideas about the kind of studies that could lead to an improvement in mathematics education. More specifically, he recognized opportunities in aligning insights from modern psychology with contemporary developments in pure mathematics. The link between the two, as he saw it, was the concept of “structure,” incarnated in modern mathematics by the Bourbakists, and in psychology by Piaget. As a holder of doctorates in both mathematics and psychology, Gattegno must have known the work of Bourbaki at an early stage, and surely he was very familiar with Piaget’s work as he had been involved in translating it into English (Piaget 1952). We provide some basic information about both structuralist movements. Nicolas Bourbaki is a pseudonym adopted during the 1930s by a group of French mathematicians which undertook the monumental task of collectively writing a multi-volume treatise reorganizing contemporary mathematical knowledge in terms of modern mathematical structures. The treatise’s name Éléments de Mathématique [Elements of Mathematics] was a back-reference to Euclid’s Elements, a series of textbooks that exposed the mathematical knowledge of his time, about 300 years BC, and influenced the research and teaching of mathematics for millennia. A more direct source of inspiration for Bourbaki’s project was Bartel Leendert Van der Waerden’s epoch-making Moderne Algebra [Modern Algebra] (1930), providing a structural conception of algebra. Van der Waerden’s textbook represented and systematized a deep transformation of this mathematical discipline which had begun in the last third of the nineteenth century, promoting abstractly defined entities like groups, fields, rings, ideals, and others as the main focus of algebraic research (Corry 1996, 2009). From the mid-1930s, the founding members of Bourbaki, young but already among the best mathematicians of their generation, including Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné, and André Weil, started their ambitious project to rebuild and reorganize all—in their view—fundamental knowledge of mathematical science into a coherent whole, starting from set theory and the so-called “mother structures” of mathematics (i.e., algebraic structures, order structures, and topological structures) (Bourbaki 1948). The first volume, Théorie des Ensembles [Set Theory], its first chapter entitled Description de la Mathématique Formelle [Description of Formal Mathematics], appeared in 1939 (Bourbaki 1939); volumes on algebra and general topology followed in subsequent years. New chapters and re-editions have been published, but the influence of the group’s books has waned (Hartnett 2020). In L’Architecture des Mathématiques [The Architecture of Mathematics], the group’s manifesto, Bourbaki (1948) explained the key concept of a mathematical structure:

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It can now be made clear what is to be understood, in general, by a mathematical structure. The common character of the different concepts designated by this generic name, is that they can be applied to sets of elements whose nature has not been specified; to define a structure, one takes as given one or several relations, into which these elements enter …; then one postulates that the given relation, or relations, satisfy certain conditions (which are explicitly stated and which are the axioms of the structure under consideration). (Bourbaki 1948, pp. 40–41)

Bourbaki’s presentation was thoroughly axiomatic-deductive, always going from poor to rich structures, from the general to the particular, without ever generalizing a particular result. The language of exposition was very formal, the style uncompromisingly rigorous, deliberately excluding diagrams and external motivations (Corry 1992, 2009). In no way were the Éléments intended as a textbook for secondary education, they were not even intended as an introductory course at the university level, as is clearly stated in the treatise’s User Guide: The treatise takes up mathematics at the beginning, and gives complete proofs. In principle, it requires no particular knowledge of mathematics on the reader’s part, but only a certain familiarity with mathematical reasoning and a certain capacity for abstract thought. Nevertheless, it is directed especially to those who have a good knowledge of at least the content of the first year or two of a university mathematics course. (Bourbaki 1939, p. 4)

In psychology, structuralism is associated with the name of the Swiss scientist Jean Piaget (1896– 1980). Piaget is widely known for his genetic epistemology including a theory of cognitive development, concerning the nature and development of knowledge in the human mind (Piaget 1936). According to Piaget, children are born with very elementary mental structures (genetically inherited and evolved) on which later knowledge and learning are based. From extensive empirical research, Piaget concluded that the cognitive development of a child proceeds through different stages characterized by internally coherent structures. Four different stages were discerned: The sensorimotor stage (about 0–2 years old), the preoperational stage (about 2–7 years old), the concrete operational stage (about 7–11 years old), and the formal operational stage (about 12 years and older). In these successive stages, the child’s abilities for abstraction increase. The transition from one stage to the next occurs through the child’s active interaction with its environment and is facilitated by two types of processes: Assimilation, the incorporation of new information into an already existing cognitive structure, and accommodation, in which already existing structures are modified to incorporate new information. Piaget had always shown particular interest in the structure and genesis of scientific subject knowledge in individuals. From the 1940s, he published a series of monographs in which he dealt with the child’s gradual acquisition of fundamental logico-mathematical and physical concepts such as space, quantity, speed, time, and number (Piaget 1941, 1945, 1946a, b; Piaget and Inhelder 1948). The aim of each book was to trace in detail the stage-by-stage development: From the sensorimotor stage to the rational coordination of thought by formal logical operations (Strauss 1953). Piaget systematically presented and discussed experimental data in support of his findings. English translations of these later books, which appeared from the 1950s onward, contributed significantly to Piaget’s international renown, first in circles of child psychologists, later also among educationalists.

The Preparatory Meetings: Defining an Agenda Influenced by Piaget’s theory, and as evidenced by a series of articles in The Mathematical Gazette, Gattegno was already convinced in the late 1940s that the teaching of mathematics had to be better aligned with the child’s stages of cognitive development (Gattegno 1947, 1949, 1954). This strong conviction was at the root of his creation of the CIEAEM and would determine the agenda and direction of the three preparatory meetings held in 1950–1951. To reconstruct the debates during these three meetings, we had to rely, in the absence of other substantial sources, on the historical overview by Félix (1986), written about 35 years later and mainly based on notes and memories of herself and other early members of this group.

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For the first meeting in Debden (April, 1950), Gattegno had invited Piaget’s closest collaborator Bärbel Inhelder to give an introductory talk on the psychological foundations of the mathematical curricula for secondary schools. But Inhelder did not show up. The discussions within the small and heterogeneous group of participants—most of them had never met before—were initially rich and stimulating, but not always focused. At a certain point, Gattegno would have taken the lead and refocused the debates through an intervention in which he suggested a reconceptualization of algebra, usually conceived as calculating with letters that represent numbers, as an “activation of operational structures in relation to the logical creations of the human mind” (quoted from Félix 1986, p. 26). This was totally new for primary and secondary school teachers, a perspective to rethink their entire activity. Debden was a first attempt to create a forum for that purpose, but, referring to future meetings, Gattegno already announced: “I will have the Bourbakists, I will have Piaget, I will have Gonseth” (quoted from Félix 1986, p.  26), an ambition he would realize two years later in La Rochette par Melun, a crucial meeting for the later “modern mathematics” movement in Europe. The theme of the second meeting, held in Keerbergen in April 1951, was “The teaching of geometry in the first years of secondary schools.” Although there were some presentations outlining and comparing different approaches to the teaching of geometry, Gattegno had a different agenda for this and subsequent meetings. Building on what he had argued in Debden with regard to algebra, Gattegno explained what in his opinion should become the most important research focus of the group: Science has become the discovery of models and structures. The dynamic description of their evolution, which leads to all scientific notions, is the subject of teaching. Therefore, our problem is to study how the child’s consciousness evolves in its active dialogue with its universe of the moment, how it extracts notions from its perception of reality, how we can help it to master these structures, to associate them in order to establish rational and relational relationships with its evolving universe. (Quoted in Félix 1986, p. 27)

According to Gattegno, the meeting in Herzberg in August 1951, the third in the series, was aimed at “establishing the teaching of mathematics on a scientific basis, by making true use of all the insights offered by mathematics, psychology, anthropology, sociology, logic, and epistemology” (quoted from Félix 1986, p. 32). Gattegno had indeed succeeded this time in bringing together a number of first-rate scientists from these fields, including Beth, Fiala, Gonseth, and Piaget, along with some outstanding mathematics teachers from different levels of education. Ferdinand Gonseth (1890–1975), Swiss mathematician, philosopher of science, and epistemologist, endorsed a multidisciplinary-informed perspective on the teaching of mathematics in the following way: [The mathematics teacher] needs to be deeply informed about the structure of mathematics and its relations with other branches of knowledge, within a constantly renewed scientific methodology, a vision as free and as well as possible informed of the psychology, of which he has direct knowledge through the activity of the students and others, the needs of today’s society and the orientation of the world of tomorrow. (Quoted in Félix 1986, p. 34)

The meaning and role of structures became a central theme of the Herzberg meeting. Structures, both mental and other, had become more and more dominant in Gattegno’s description and understanding of different kinds of “learning” (Gattegno 1951). As quoted from Félix (1986), during this meeting Gattegno characterized in terms of structures both the content of mathematics (“a system of structures,” p. 36), research in mathematics (“studying the influence of one structure on another,” p. 36), and the teaching of mathematics (“translating existing mathematical structures, selected by the adopted functional curriculum, in terms of mental structures that the student already possesses,” p. 37). Participants were invited to trace mathematical structures as integrated into the various chapters of traditional curricula, and also to characterize mental structures as they manifested themselves when observing students who react freely while struggling with a mathematical problem. The alignment of mathematical and mental structures would be the official theme of the fourth meeting in La Rochette par Melun in April 1952.

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A Decisive Meeting Between the Bourbakists and Piaget The meeting in La Rochette par Melun, which could be deemed the founding meeting of CIEAEM, was again attended by Fiala, Gonseth, and Piaget, but now also by Jean Dieudonné (1906–1992), spokesman for the Bourbakists, and by Gustave Choquet (1915–2006) and André Lichnerowicz (1915–1998), two French research mathematicians whose work was strongly inspired by Bourbaki. As usual, a number of outstanding teachers of mathematics also participated (including Félix, Lenger, and Servais) (Figure 3.4). Gattegno had chosen the theme Mathematical and Mental Structures and the overall aim was to achieve a multidisciplinary synthesis to improve the teaching of mathematics. The collective work by Piaget et al. (1955) was largely based on the contributions to the meeting in La Rochette (although this is not mentioned in the book’s Preface). The debate was initiated by Dieudonné who outlined Bourbaki’s points of view, paying particular attention to the origin and essence of structures in modern mathematical science (Félix 1986). He argued that structures are by no means artificial constructs that appear out of nowhere; they are “explicitations” of ideas that were already present in the work of great mathematicians of the past, implicitly and under different guises, but which were not yet recognized as such. “A structure must be freed from the coating of history, but it is history that justifies its creation” (Félix 2005, p. 82, the italic part is Dieudonné’s quote). Their role in mathematical research was clarified by Lichnerowicz: “A structure is a tool that we search for in the arsenal we have at our disposal. It is not at this stage that it is created” (quoted from Félix 2005, p. 82). The Bourbaki manifesto elaborated on this as follows: “Structures” are tools for the mathematician; as soon as he has recognized among the elements, which he is studying, relations which satisfy the axioms of a known type, he has at his disposal immediately the entire arsenal of general theorems which belong to the structures of that type. Previously, on the other hand, he was obliged to forge for himself the means of attack on his problems; their power depended on his personal talents and they were often loaded down with restrictive hypotheses, resulting from the peculiarities of the problem that was being studied. (Bourbaki 1948, p. 42)

Choquet and Lichnerowicz also testified about how they actually used structures in their research (Félix 1985b, 1986, 2005). For the teachers in La Rochette, the vision of the Bourbakists and the way they practiced mathematics was nothing less than a revelation. After World War II, Bourbaki’s work was known to research mathematicians, but most secondary school teachers, even those who had graduated in mathematics, were completely ignorant of this “modern” evolution within mathematics (Noël 2022).

Figure 3.4  La Rochette par Melun, 1952 (from left to right, left: Mrs. and Mr. F. Gonseth, J. Dieudonné, and G. Choquet; middle: G. Choquet, F. Fiala, and C. Gattegno; right: F. Gonseth and W. Servais)

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Dirk De Bock Fascinating were the testimonies about how thinking in structures guided the research of scholars who wanted to entrust this to us: Dieudonné, Lichnerowicz, Choquet. … Bourbaki as I was waiting for. (Félix 2005, pp. 82–83)

During the discussion in which mainly the university professors participated, Dieudonné emphasized that, from Bourbaki’s point of view, the concept of structure presented itself in existing mathematics; the Bourbakists were not dealing with questions of a philosophical or metaphysical level, only common logic was used (Félix 1986). Associations of mathematical structures with extra-mathematical constructs were not suggested by Bourbaki, but they were established by Piaget, who explicitly related Bourbaki’s structures to the mental operations through which a child interacts with the world (Piaget 1955).1 More specifically, Piaget identified the fundamental structures and stages of early mathematical thinking, as revealed by (his) psychological research, with the mother structures in Bourbaki’s treatise: Now, it is of the highest interest to ascertain that, if we retrace to its roots the psychological development of the arithmetic and geometric operations of the child, and in particular the logical operations which constitute its necessary preconditions, we find, at every stage, a fundamental tendency to organize wholes or systems, outside of which the elements have no meaning or even existence, and then a partitioning of these general systems according to three kinds of properties which precisely correspond to those of algebraic structures, order structures, and topological structures. (Piaget 1955, pp. 14–15)

Piaget’s identification of Bourbaki’s mother structures with the basic structures of thinking, implying a harmony between the structures of “contemporary” mathematics and the way in which a child constructs mathematical knowledge, had a straightforward pedagogical implication: The learning of mathematics takes place through the mother structures of Bourbaki, the structures with which twentieth-­ century mathematicians had founded and built their science. Correspondingly, Piaget (1955) asserted that “if the building of mathematics is based on ‘structures,’ which moreover correspond to the structures of intelligence, then it is on the gradual organization of these operational structures that the didactics of mathematics must be based” (p. 32). In other words: A model for the science of mathematics was promoted as a model for mathematics education.



Dissemination and Early Developments at National Levels

The Ongoing Debate Within CIEAEM Some teachers, including Servais and Félix, immediately adapted their teaching practice to what they had learned in La Rochette: In the light of the discussions, Servais already foresaw the direction he had to bring into his teaching. On his return to Belgium, in his class, he made tests; they were conclusive and students told him: “Why were we not told earlier about the structures? We would have seen more clearly.” … I [Félix] decided to try the initiation at a very basic level with my students aged 13–14 and, as with Servais, it was received with relief thanks mainly to the simplicity of the language. (Félix 1986, pp. 35 and 46)

In the Journal of General Education, Gattegno dealt with the ideas which had been discussed in La Rochette and at the three preparatory meetings (Gattegno 1952). It had become clear to him that the problem of teaching mathematics should be seen as the creation of an actual synthesis of mind, mathematics, and the needs of the future. Referring to Bourbaki, he argued that in this synthesis, mathematics should be understood as a specialized activity dealing “with structures, rather than with so-called ‘mathematical objects’” (Gattegno 1952, p. 260). As far as the mind was concerned, psychologists “can investigate at what stage of mental development the child is capable of answering this or that question  Piaget’s chapter in Piaget et al. (1955) was the summary of his presentation in La Rochette (1952), as mentioned in a footnote to that chapter. From the book’s Preface, we know that Piaget wrote his chapter after reading the other ones. 1

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in the field of mathematics,” which “is particularly exemplified in Piaget’s work” (Gattegno 1952, p. 261). In Gattegno’s view, the first step in the realization of the intended synthesis consisted in: the organization of the child’s experience in such a way that he becomes aware that his mental activity yields mathematical structures having their own organization, which organization is called “mathematics.” (Gattegno 1952, p. 263)

Bourbaki’s reconstruction of mathematics from a limited number of basic structures, connecting different branches of this science and underlining its fundamental unity, “supported” by Piaget’s theory of cognitive development, would further stimulate the debate and ultimately lead to fundamental reform of mathematics education. In retrospect, the seeds for a structural approach to the teaching of mathematics, i.e., modern mathematics, were laid in La Rochette, where Bourbaki offered the mathematical rationale and Piaget provided the psychological justification. In the years following La Rochette, CIEAEM would continue to play an important role in disseminating, developing, and refining its modern view on mathematics education. From the mid-1950s onward, this view would also have an impact on national reform debates and related actions, particularly in France and Belgium. In 1953, at the CIEAEM meeting in Calw (Germany), the study of the connections between students’ thinking and the teaching of mathematics was on the agenda, in particular how teachers could contribute to the understanding of students’ mathematical thinking by transforming a regular lesson into an inquiry-oriented lesson (Lenger 1953–1954). “The modern mathematics at school” was the theme of the next meeting of the CIEAEM which was held in Oosterbeek (The Netherlands) in 1954. One of the questions concerned the adaptation of curricula “in the light of what we know about modern mathematics and the thinking of the child and the adolescent” (Lenger 1954–1955a, p. 58). Some positions were taken by participants who already partly focused their teaching on structures and relations. According to Lenger, however, it was primarily up to the teacher to introduce something of the spirit of modern mathematics into their teaching. Modern mathematics … is the result of an awareness of structures and the relationships between these structures. The thinking of the modern mathematician is relational. And it seems to me that relational mathematical thinking can be recreated by a child or adolescent if the teacher is aware of it and if he presents the appropriate situations. (Lenger 1954–1955a, p. 58)

From the mid-1950s, the CIEAEM meetings increasingly focused on a “new” topic, namely the role of concrete models, teaching materials, or teaching aids, such as cardboard models, light projections, meccano constructions, geoboards, mathematical films, electrical circuits, and the Cuisenaire rods (for a detailed overview, we refer to De Bock and Vanpaemel 2019, Chapter 2). This focus of interest would culminate in a meeting and exhibition in Madrid (April, 1957), and in the group’s second collective publication (Gattegno et al. 1958). Although the study of teaching materials was of more direct use to teachers, it did not contradict the prevailing structuralist trend. New teaching materials were typically stylized objects, suitable for active manipulation to guide students’ mental activities. In accordance with Piaget’s pedagogical principles, the aim was to promote student understanding, raise their level of abstraction, and enable them to “discover” mathematical relations and structures.

Early National Developments A systematic effort in the mid-1950s to map national developments in the teaching of mathematics at the secondary level was undertaken by UNESCO, in cooperation with the International Bureau of Education led by Piaget (UNESCO 1956). In an extensive survey of secondary school mathematics, in which 62 countries participated, Question 17 reflected a growing interest in modern mathematics: 17. (a) Are any modifications contemplated in the teaching of mathematics in secondary schools? If so, kindly indicate their nature. (b) To what extent does the evolution of modern mathematics affect secondary education? (p. 10)

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The review of the answers of the 62 countries stated: The question as to what precise extent modern developments in the mathematical field have affected the secondary teaching of mathematics was answered by some twenty of the countries. In some cases the reply states merely that those developments were taken account of in the formulation of the secondary mathematics syllabuses. In other cases details are given of definite modifications made or impending in those syllabuses, including the introduction of infinitesimal calculus, coordinate geometry, statistics, etc., and added stress on functions, vectors, the calculation of probability, differential and integral calculus, and applied mathematics. (pp. 26–27)

Mathematical structures were not included in this list. Willy Servais, who was a delegate for Belgium at the UNESCO conference in Geneva in July 1956, at which the results of the survey were discussed, did make explicit mention of modern mathematical structures in his report: To what extent can the more abstract … mathematical structures, discovered within classical mathematics and developed worldwide by today’s mathematicians, have a beneficial impact on secondary education? This is a very recent question to which pioneers in many countries are seeking an answer. The results obtained so far hold the promise of a pedagogical innovation. (Servais 1956–1957, p. 40)

In general, responses to the survey revealed an awareness of the need for changes in the mathematics curricula for secondary schools, but there was little or no mention of implemented reforms. In Europe, concrete reform initiatives first emerged in France and Belgium. Although the role of axiomatics had been discussed in limited circles in France since the early 1950s, it was only in the mid-1950s that actions to modernize the teaching of mathematics gained momentum (Barbazo and Pombourcq 2010). In 1955, at an international meeting in Sèvres, Choquet compared teachers of mathematics to guards of a museum, which show dusty objects, most of which are of no interest (Dupont and Heuchamps 1954–1955). In the preparatory notes for his lecture at this meeting, Choquet wrote: “The important thing now is to help our colleagues … those who were educated 20 years ago” (quoted from Félix 1986, p. 83). He added the deed to the word. In 1956–1957, at the Sorbonne University in Paris, Choquet organized, in collaboration with the Association des Professeurs de Mathématiques de l’Enseignement Public (APMEP) [Association of Teachers of Mathematics in State Schools] and the Société mathématique de France [French Mathematical Society], a series of 17 lectures by research mathematicians—Bourbakists or persons inspired by Bourbaki—to initiate secondary school teachers in the fundamental structures of modern mathematics. These lectures were edited by Félix, then published in the Bulletin of the APMEP, and finally collected in a monograph of the journal L’Enseignement Mathématique (Cartan et al. 1958). According to Félix (1955), in-service initiatives like this not only introduced teachers to a modern conception of mathematics, but also helped them to empathize with their students. Between the mathematics we teach in our classrooms and the books for researchers, such as the Éléments of Bourbaki, the distance is enormous. But, like modern mathematicians, by means of axiomatics, return to the roots and ask to be aware of structures, starting from the simplest ones, the teacher who, like a student, takes the effort to see the usual facts in a new light, will also see his teaching in a new light. Because his habits are challenged, he will better understand children who do not have our habits. (Félix 1955, p. 81)

From the beginning of the 1960s, mainly under the impetus of the APMEP, reform initiatives in France would follow each other in rapid succession (Barbazo and Pombourcq 2010; Barbin 2012; d’Enfert in press; d’Enfert and Gispert 2011). In Belgium, the Société Belge de Professeurs de Mathématiques [Belgian Society of Mathematics Teachers] was founded in 1953. Its creation was closely related to the action of the CIEAEM. Willy Servais (1913–1979), the Society’s first president, as well as other leading personalities of the organization, were regularly present at CIEAEM meetings and were well informed about “new” developments in the scientific field of mathematics. From the mid-1950s, a discussion about the quality of the official mathematics programs for the secondary level was launched in the Society’s journal Mathematica & Paedagogia (De Bock and Vanpaemel 2019, Chapter 3). Year after year, the trend toward modern mathematics became clearer. Ingredients were outlined (integrating some ideas of

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modern algebra, restructuring geometry education, initiating students in the fields of probability and statistics, …), but an overall picture was missing and ideas about a possible implementation diverged. An important next step was taken by Lenger and Servais who, in the margin of the 12th CIEAEM meeting at Saint Andrews (United Kingdom) in August 1958, drafted a concrete program for the teaching of modern mathematics (Le Programme B des Écoles Normales Gardiennes 1958–1959). Based on that program and with the permission of the Ministry of Education, the first Belgian experiments with modern mathematics in a school context were run during the school year 1958–1959 (De Bock and Vanpaemel 2018). At the famous Royaumont Seminar (November 23 to December 5, 1959), Servais, having been invited to present a detailed syllabus for a modernized approach to algebra for 12–18-year-olds, alluded to some experiments of the school year before. The syllabus proposed has not so far been tried out on any extensive scale, but isolated tests have shown that it is by no means utopian and that young people when faced with modern ideas show much greater aptitude than their reactions to traditional teaching might lead teachers to suppose. (Servais 1959, p. 4)

Discussion In the early 1950s, a new organization appeared on the international mathematics education scene—The International Commission for the Study and Improvement of Mathematics Teaching (CIEAEM). It was founded by Caleb Gattegno, at that time settled in England. The main aims included the study of the teaching of mathematics and the provision of a scientific base for that teaching so that improvements could follow. To reach the Commission’s goals, Gattegno believed in an interdisciplinary approach, in an alliance between modern developments in the fields of mathematics, epistemology, and psychology. Therefore, leading academics of these disciplines were brought together with receptive mathematics teachers in annual meetings during which each time one theme, related to the teaching of mathematics, was thoroughly discussed in a friendly and informal atmosphere. The relation between mathematical and mental structures was a main issue of inquiry during early CIEAEM meetings. The meeting in La Rochette (1952), between the French Bourbakist mathematicians on the one hand and the Swiss psychologist Piaget on the other, played a capital role in clarifying this relation and in establishing the organization’s commitment to further work along this line of thought. The mathematics teachers present, most of them rather ignorant about recent developments in mathematics, were impressed by Bourbaki’s structural approach to mathematics. Piaget, likely enthusiastic about his conversations with leading mathematicians of the time, identified their structures with the basic structures of cognitive development he had discovered. In 1972, at the Second International Congress on Mathematical Education (Exeter, United Kingdom), Piaget again voiced his belief in this fortunate harmonic link: [Cognitive development] involves a spontaneous and gradual construction of elementary logico-mathematical structures and that these “natural” (“natural” in the way one speaks of the “natural” numbers) structures are much closer to those being used in “modern” mathematics than to those being used in traditional mathematics. (Piaget 1973, p. 79)

The identification of the structures of contemporary mathematics with Piaget’s theory of cognitive development, together with the observation that there was a deep gap between mathematics as a living science and mathematics taught at school, had given a driving impetus to a modernization of mathematics education in Europe, to so-called “modern mathematics”, a reform movement both in terms of content and teaching methods. As stated by Charlot (1985), “modern mathematics appeared as the daughter of Bourbaki and Piaget” (p. 28), and she inherited from them—explicitly or more diffusely— structure, formalism, and active pedagogy. At the end of the 1950s under the influence of major

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international institutions, such as the Organisation for Economic Cooperation and Development (OECD), UNESCO, and ICMI, the psycho-mathematical debate that emerged within the CIEAEM community would be enriched by socio-economic and cultural arguments. Ultimately, it would lead to the introduction of modern mathematics in most European countries, first in secondary and later also in primary education. But the early roots of this movement reached as far back as the historic meeting in La Rochette par Melun, on which Servais later commented as follows: The moment was propitious for the renewal of the teaching of mathematics. There was, on the one hand, the object of a global and structured reconstruction under the impetus of the Bourbaki group and, on the other hand, its learning was studied from a psycho-genetical perspective by the school of Piaget. (quoted from Warbecq 2000, p. 9)

However, basing the teaching of mathematical structures on the development of intellectual structures which would be in harmony with them was not unproblematic. Charlot (1985) argued that the fact that both mathematicians and psychologists referred to “structures” did not prove anything in itself, and that the advocates of a reform never made a serious attempt to specify the nature of the links between the mathematical structures to be taught and the intellectual structures of the child. More fundamentally, Piaget’s position on mathematical education implies a hierarchical organization of mathematics from general structures to more or less specialized ones (as in Bourbaki), or, in Freudenthal’s terminology, “the precedence of poor over rich structures” (Freudenthal 1991, p. 29). Piaget explained the construction of geometrical knowledge from that perspective: Topological structures precede projective structures which, in turn, precede metric structures (Piaget 1955); an order which is “natural,” both from the viewpoint of Bourbaki and from a Piagetian view on knowledge acquisition, but opposite to the historical development of geometry (Bkouche 1997). Piaget tried to justify this type of inversion as follows: Historically, it seems that these elements were given before the discovery of the structure, and therefore the structure essentially plays the role of a reflexive instrument intended to show their most general character. We must not forget that, from a psychological point of view, the order of consciousness inverts that of the genesis: What comes first in the order of the construction appears last in the reflexive analysis, because the subject becomes aware of the results of the mental construction before reaching the intimate mechanisms of it. (Piaget 1955, p. 14)

This point of view was vigorously criticized, both from a mathematical and from an educational and psychological point of view. For example, Freudenthal (1983) argued: According to Piaget, topological concepts should precede Euclidean ones. We anticipate that this holds at most for such spatial relations as inclusion, exclusion, and overlapping, but these are relations which no mathematician would consider topological as psychologists do. The acceptance of truly topological properties—that is, stating equivalence by means of one-to-one continuous mappings—is certainly not an attitude that can be placed in early childhood; it is much too sophisticated to be expected of little children. Piaget and researchers who repeated his assertions or experiments were seriously confused. From the inability of little children to draw circles and squares so neatly that they reasonably differed from each other, they drew the conclusion of topological predominance. Yet at an early age children are able to distinguish clearly circles and squares, which is the only thing that matters. (p. 190)

Bourbaki’s hierarchy, “from poor to rich structures,” when taken as a guiding principle for the organization of mathematical teaching, was typified by Freudenthal (1973) as an “antididactic inversion” (p. 103), implying that an end product of mathematical activity, the most recently composed structure of mathematics, should be used as a starting point for mathematics teaching. Freudenthal’s viewpoint is shared by most contemporary scholars in the history and epistemology of mathematics education. For example, Barbin (2013) argued that the general, unifying structures of modern mathematics cannot be dictated a priori, but must be conquered. An antihistorical approach to the teaching of mathematics “ignores the time needed for these conquests from a superstition that the ‘last show’ of mathematics must be taught at all costs” (p. 129).

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References Barbazo, E., & Pombourcq, P. (2010). Cent ans d’APMEP (Brochure APMEP n° 192) [One hundred years of APMEP (APMEP booklet n° 192)]. Paris, France: APMEP. Barbin, É. (2012). The role of the French Association of Mathematics Teachers APMEP in the introduction of modern mathematics in France (1956–1972). In Proceedings of the ICME-12 Satellite Meeting of HPM (History and Pedagogy of Mathematics) July 16–20, 2012 (pp. 597–605). DCC, Daejeon, Korea. Barbin, É. (2013). Le genre « ouvrage d’initiation »: l’Exposé moderne des mathématiques élémentaires de Lucienne Félix (1959–1961) [The “book of initiation” genre: The modern exposition of elementary mathematics by Lucienne Félix (1959–1961). In É. Barbin & M. Moyon (Eds.), Les ouvrages de mathématiques dans l’histoire. Entre recherche, enseignement et culture [Mathematical books in history. Between research, teaching and culture] (pp. 117– 129). Limoges, France: Presses Universitaires de Limoges. Bernet, T., & Jaquet, F. (1998). La CIEAEM au travers de ses 50 premières rencontres [The CIEAEM through its first 50 meetings]. Neuchâtel, Switzerland: CIEAEM. Bkouche, R. (1997). Epistémologie, histoire et enseignement des mathématiques [Epistemology, history and teaching of mathematics]. For the Learning of Mathematics, 17(1), 34–42. Bourbaki, N. (1939). Éléments de mathématique: Théorie des ensembles [Elements of mathematics: Set theory]. Paris, France: Hermann. Bourbaki, N. (1948). L’architecture des mathématiques [The architecture of mathematics]. In F. Le Lionnais (Ed.), Les grands courants de la pensée mathématique [Major trends in mathematical thinking] (pp. 35–47). Paris, France: Cahiers du Sud. Brown, L., Hewitt, D., & Tahta, D. (Eds.). (2010). A Gattegno anthology: Selected articles by Caleb Gattegno reprinted from Mathematics Teaching. Derby, UK: ATM. Cartan, H., Choquet, G., Dixmier, J., Dubreil, P., Godement, R., Lelong, P., Lesieur, L., Lichnerowicz, A., Pisot, C., Revuz, A., Schwartz, L., & Serre, J.-P. (1958). Structures algébriques et structures topologiques (Monographies de l’Enseignement Mathématique, n° 7) [Algebraic and topological structures (Monographs of l’Enseignement Mathématique, n° 7)]. Genève, Switzerland: Institut de mathématiques, Université de Genève. Charlot, B. (1985). Histoire de la réforme des « maths modernes » ; idées directrices et contexte institutionnel et socio-­ économique [History of the “modern mathematics” reform; guiding ideas, and institutional and socio-economic context]. Bulletin de l’Association des Professeurs de Mathématiques de l’Enseignement Public, 352, 15–31. Corry, L. (1992). Nicolas Bourbaki and the concept of mathematical structure. Synthese, 92(3), 315–348. Corry, L. (1996). Modern algebra and the rise of mathematical structures. Basel, Switzerland-Boston, MA-Berlin, Germany: Birkhäuser. Corry, L. (2009). Writing the ultimate mathematical textbook: Nicolas Bourbaki’s Éléments de mathématique. In E. Robson & J. Stedall (Eds.), The Oxford handbook of the history of mathematics (pp. 565–588). Oxford, United Kingdom: Oxford University Press. De Bock, D., & Vanpaemel, G. (2018). Early experiments with modern mathematics in Belgium. Advanced mathematics taught from childhood? In F. Furinghetti & A. Karp (Eds.), Researching the history of mathematics education: An international overview (pp. 61–77). Cham, Switzerland: Springer. De Bock, D., & Vanpaemel, G. (2019). Rods, sets and arrows. The rise and fall of modern mathematics in Belgium. Cham, Switzerland: Springer. d’Enfert, R. (in press). La réforme des mathématiques modernes: union et désunion d’une communauté mathématique [The reform of modern mathematics: union and disunion of a mathematical community]. In P.-M. Menger & P. Verschueren (Eds.), Mathématiques: communautés et institutions [Mathematics: Communities and institutions]. Aubervilliers, France: Éditions de l’EHESS. d’Enfert, R., & Gispert, H. (2011). Une réforme à l’épreuve des réalités: Le cas des « mathématiques modernes » en France, au tournant des années 1960–1970 [A reality-tested reform: The case of “modern mathematics” in France at the turn of the 1960s and 1970s]. Histoire de l’Éducation, 131, 27–49. d’Enfert, R., & Kahn, P. (Eds.). (2010). En attendant la réforme. Disciplines scolaires et politiques éducatives sous la IVe République [Waiting the reform. School disciplines and educational policies under the Fourth Republic]. Grenoble, France: Presses Universitaires de Grenoble. Dupont, M, & Heuchamps, E. (1954–1955). Journées internationales d’information sur l’enseignement des mathématiques [International information days on mathematics education]. Mathematica & Paedagogia, 6, 68–83. Félix, L. (1955). L’axiomatique, nos élèves et nous [Axiomatics, our students and us]. Bulletin de l’Association des Professeurs de Mathématiques de l’Enseignement Public, 167, 81–85. Félix, L. (1985a). Aperçu historique (1950–1984) sur la Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques (CIEAEM). [Historical overview (1950–1984) on the International Commission for the Study and Improvement of Mathematics Teaching (CIEAEM)]. Bordeaux, France: l’IREM de Bordeaux.

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Félix, L. (1985b). Souvenirs d’une époque archaïque [Memories of an archaic period]. Bulletin de l’Association des Professeurs de Mathématiques de l’Enseignement Public, 352, 5–13. Félix, L. (1986). Aperçu historique (1950–1984) sur la Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques (CIEAEM). 2ième édition revue et augmentée [Historical overview (1950–1984) on the International Commission for the Study and Improvement of Mathematics Teaching (CIEAEM). 2nd revised and extended edition]. Bordeaux, France: l’IREM de Bordeaux. Retrieved October 11, 2020, from http://math. unipa.it/~grim/cieaem_files/CIEAEM_histoire_FLucienne_1985.pdf. Félix, L. (2005). Réflexions d’une agrégée de mathématiques au XXe siècle [Reflections of certified mathematics teacher in the 20th century]. Paris, France: l’Harmattan. Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht, The Netherlands: Reidel. Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Reidel. Freudenthal, H. (1991). Revisiting mathematics education. China Lectures. Dordrecht, The Netherlands: Kluwer. Furinghetti, F. (2014). History of international cooperation in mathematics education. In A.  Karp & G.  Schubring (Eds.), Handbook on the history of mathematics education (pp. 543–564). New York, NY: Springer. Furinghetti, F., & Giacardi, L. (2008). The first century of the International Commission on Mathematical Instruction (1908–2008). The history of ICMI. Torino, Italy: Dipartimento di Matematica dell’Università. Retrieved October 10, 2020, from http://www.icmihistory.unito.it/. Gattegno, C. (1947). Mathematics and the child. The Mathematical Gazette, 31(296), 219–223. Gattegno, C. (1949). Mathematics and the child, II. The Mathematical Gazette, 33(304), 108–112. Gattegno, C. (1951). Remarques sur les structures mentales [Notes on mental structures]. Enfance, 4(3), 239–250. Gattegno, C. (1952). A note on the teaching of mathematics. Journal of General Education, 6(4), 260–267. Gattegno, C. (1954). Mathematics and the child, III. The Mathematical Gazette, 38(323), 11–14. Gattegno, C., Servais, W., Castelnuovo, E., Nicolet, J.  L., Fletcher, T.  J., Motard, L., Campedelli, L., Biguenet, A., Peskett, J.  W., & Puig Adam, P. (1958). Le matériel pour l’enseignement des mathématiques [Materials for the teaching of mathematics]. Neuchâtel, Switzerland: Delachaux et Niestlé. Gispert, H. (2010). Rénover l’enseignement des mathématiques, la dynamique internationale des années 1950 [Renewing mathematics education, the international dynamics of the 1950s]. In R. d’Enfert & P. Kahn (Eds.), En attendant la réforme. Disciplines scolaires et politiques éducatives sous la IVe République [Waiting for the reform. School disciplines and educational policies under the Fourth Republic] (pp. 131–143). Grenoble, France: Presses Universitaires de Grenoble. Hartnett, K. (2020). Inside the secret math society known simply as Nicolas Bourbaki. Quanta Magazine. Simons Foundation. Retrieved November 12, 2020, from https://www.quantamagazine.org/inside-­the-­secret-mathsociety-known-­as-­nicolas-­bourbaki-­20201109/. Lenger, F. (1953–1954). Rencontre internationale de professeurs de mathématiques [International meeting of teachers of mathematics]. Mathematica & Paedagogia, 1, 55–56. Lenger, F. (1954–1955a). Rencontre internationale de professeurs de mathématiques [International meeting of teachers of mathematics]. Mathematica & Paedagogia, 4, 57–59. Lenger, F. (1954–1955b). VIIIe rencontre internationale de professeurs de mathématiques [8th international meeting of teachers of mathematics]. Mathematica & Paedagogia, 6, 86–88. Le Programme B des Écoles Normales Gardiennes [The future kindergarten teachers’ program B]. (1958–1959). Mathematica & Paedagogia, 16, 70–75. Menghini, M., Furinghetti, F., Giacardi, L., & Arzarello, F. (Eds.). (2008). The first century of the International Commission on Mathematical Instruction (1908–2008). Reflecting and shaping the world of mathematics education. Rome, Italy: Istituto della Enciclopedia Italiana. Noël, G. (2018). Regards sur Caleb Gattegno [Views on Caleb Gattegno]. Losanges, 41, 68–69. Noël, G. (2022). 1945–1960: Quinze années d’enseignement des mathématiques [1945–1960: Fifteen years of mathematics teaching] (Booklet). Mons, Belgium: SBPMef. Piaget, J. (1936). La naissance de l’intelligence chez l’enfant [The Origin of intelligence in the child]. Neuchâtel, Switzerland: Delachaux et Niestlé. Piaget, J. (1941). La genèse du nombre chez l’enfant [The child’s conception of number]. Neuchâtel, Switzerland: Delachaux et Niestlé. Piaget, J. (1945). La formation du symbole chez l’enfant: imitation, jeu et rêve, image et représentation [Play, dreams and imitation in childhood]. Neuchâtel, Switzerland: Delachaux et Niestlé. Piaget, J. (1946a). Le développement de la notion de temps chez l’enfant [The child’s conception of time]. Paris, France: Presses Universitaires de France. Piaget, J. (1946b). Les notions de mouvement et de vitesse chez l’enfant [The child’s conception of movement and speed]. Paris, France: Presses Universitaires de France. Piaget, J., & Inhelder, B. (1948). La représentation de l’espace chez l’enfant [The child’s representation of space]. Paris, France: Presses Universitaires de France.

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Piaget, J. (1952). The child’s conception of number (C. Gattegno & F. M. Hodgson, Trans.). London, UK: Routledge (Original work published 1941). Piaget, J. (1955). Les structures mathématiques et les structures opératoires de l’intelligence [The mathematical structures and the operational structures of intelligence]. In J. Piaget, E. W. Beth, J. Dieudonné, A. Lichnerowicz, G.  Choquet, & C.  Gattegno, L’enseignement des mathématiques [The teaching of mathematics] (pp.  11–33). Neuchâtel, Switzerland: Delachaux et Niestlé. Piaget, J. (1973). Comments on mathematical education. In A. G. Howson (Ed.), Developments in mathematical education. Proceedings of the Second International Congress on Mathematical Education (pp. 79–87). Cambridge, UK: University Press. Piaget, J., Beth, E.  W., Dieudonné, J., Lichnerowicz, A., Choquet, G., & Gattegno, C. (1955). L’enseignement des mathématiques [The teaching of mathematics]. Neuchâtel, Switzerland: Delachaux et Niestlé. Powell, A. B. (2007). Caleb Gattegno (1911–1988): A famous mathematics educator from Africa? Revista Brasileira de História da Matemática, Especial n° 1–Festschrift Ubiratan D’Ambrosio, 199–209. Servais, W. (1956–1957). L’enseignement des Mathématiques dans les écoles secondaires [The teaching of Mathematics in secondary schools]. Mathematica & Paedagogia, 10, 34–43. Servais, W. (1959). Fundamental concepts and their development in the school curriculum: Thoughts on the teaching of algebra in secondary schools. Paris, France: OEEC (OSTP) (typewritten text, not paginated). Personal Archives Willy Servais, Morlanwelz, Belgium. Strauss, A. (1953). [Review of The child’s conception of number by J. Piaget]. American Sociological Review, 18(6), 711–712. UNESCO. (1956). Teaching of mathematics in secondary schools (Publication No. 172 of the International Bureau of Education, Geneva). Paris, France: UNESCO Van der Waerden, B. L. (1930). Moderne algebra [Modern algebra]. Berlin, Germany: Springer. Warbecq, A. (2000). Hommage à Willy Servais: Willy Servais et la CIEAEM [Tribute to Willy Servais: Willy Servais and the CIEAEM]. Mathématique et Pédagogie, 126, 9–10.

Chapter 4

The Royaumont Seminar as a Booster of Communication and Internationalization in the World of Mathematics Education Fulvia Furinghetti and Marta Menghini

Abstract  After a succinct description of the meeting opportunities for mathematics educators up to the 1950s, this chapter describes how, in the wake of the New Math/modern mathematics reform movement, meetings have become a fundamental tool for focusing on problems and potential of reform proposals. Bodies that have played the most relevant roles are ICMI, CIEAEM, OEEC/OECD, and UNESCO.  In the conferences that followed the Royaumont Seminar, particular interest was turned to the search for new axioms for geometry, with many proposals and discussions. But modern mathematics was not just this; in other places, the attention was turned to more general questions of a methodological and social nature. This congress season has fostered the creation of new traditions such as the birth of journals specialized in mathematics education, and periodic conferences on mathematics education, as exemplified by the four-year ICMEs. Keywords  Aarhus · Arab region · Athens · Belgrade · Bogotá · Bologna · Bucharest · Budapest · Cambridge · CIEAEM · Communication · Dakar · Dalat · Dubrovnik · Echternach · Egypt · Entebbe · Frascati · ICMI · International Congress of Mathematicians · Internationalization · Modern mathematics · Moscow · New Math · OECD · OEEC · Reforms · Royaumont · Saigon · UNESCO · Utrecht · Zagreb

Introduction In the mid-twentieth century concomitant factors ranging from the increasing importance of mathematics and its applications for society to new insights on learning and teaching mathematics brought to light a need for “radical changes and improvements in the teaching of mathematics” (OEEC 1961a, p. 11) in many countries. As a result, a reform movement spread around the world, usually known as “New Math” in most English-speaking countries or “modern mathematics”—the label that we will use—in most European countries. The difference in labels is not just a question of language. Modern mathematics had its initial inspiration in the Bourbakist theories, while New Math is usually identified with the movement which sprung out of the School Mathematics Study Group (SMSG) in the

F. Furinghetti (*) University of Genoa, Genoa, Italy M. Menghini University of Rome Sapienza, Rome, Italy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_4

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USA. The two labels cover very different phenomena in the various countries involved in the movement of reform, nevertheless, these phenomena present commonalities in the period in which they developed, in some protagonists, and the intent to revising mathematics teaching and curricula. According to Kilpatrick (2012), the modern mathematics movement arose from a desire “to bring school mathematics closer to the academic mathematics of the twentieth century—to eliminate inane jargon, and make it better preparation for the mathematics being taught in the university” (p. 563). What is striking in the reform movement of modern mathematics is the acceleration in communication and the internationalization it promoted in the milieu of school mathematics. These aspects were rather new in this area, although they had already been advocated at the beginning of the twentieth century. The journal L’Enseignement Mathématique, founded in 1899, identified its mission and vision in internationalization, communication, and solidarity (Furinghetti 2003). In 1908 the foundation of the International Commission on the Teaching of Mathematics, aka CIEM or IMUK1, progenitor of the current International Commission on Mathematical Instruction (ICMI), was a further step in the path toward communication and internationalization. Despite these past events, for many years discussion on mathematics education was mainly confined to national contexts. On the contrary, the reform movement of modern mathematics, though it had different fallout in various countries, aroused the need to communicate and discuss issues of teaching and learning mathematics at the international level. The trigger of this new trend was the Seminar held in Royaumont in 1959, which stirred a series of conferences in the 1960s. The development of international bodies established with the purpose of providing institutional and financial support favored these events. In this chapter, we see how these conferences were carried out and how they contributed to rethinking mathematics education.



Meetings on Mathematics Education Before Royaumont

Nowadays attending international conferences, virtually or face-to-face, has become a common practice among scientists, but in the past, it was not so. On the occasion of the World’s Columbian Exposition held in Chicago (1893) in celebration of the four-hundredth anniversary of Christopher Columbus’s discovery of the “New World,” alongside the usual activities of this kind of celebration, a conference of mathematicians was organized by the University of Chicago’s Faculty of Mathematics with participants and reports from the United States and Europe. This was quite an exceptional event of international communication and in 1897 it was followed by the launch of the periodical International Congresses of Mathematicians (ICMs). In 1900, after the second of these congresses, the frequency became four years, with interruptions due to World Wars. For mathematicians, the ICMs provided one of the few opportunities to meet at the international level. For those interested in mathematics education, including teachers, there were even fewer opportunities. An opportunity was offered by the ICMs because, since 1900, their program included a section dedicated to the teaching of mathematics generally associated with other topics such as history, logic, or philosophy. In these sections, contributions to mathematics education were presented, usually in the form of short communications. The proceedings of the ICMs do not always contain the texts of these contributions, but rather regularly contain the titles and the names of authors. In a few ICMs there were also special contributions that were given more time, but never was a plenary talk dedicated to mathematics teaching. The only exception could be Felix Klein’s plenary in Zurich in 1897, which contains the word Unterricht (teaching) in the title. It was mainly concerned with university teaching of advanced mathematics. Already at the beginning of the twentieth century, the anomaly of this situation was felt and CIEM launched international meetings on mathematics education that marked its initial activity. The first was in Brussels (1910), the second in Milan (1911), and the third in Paris (1914). While taking into account the dif CIEM stands for Commission Internationale de l’Enseignement Mathématique, IMUK stands for Internationale Mathematische Unterrichtskommission (see, e.g., Howson 1984). 1

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ferences in context, we agree with Schubring (2008) who considers these congresses, in particular, that of Paris, to be the ancestors of the present ICMEs (International Congresses on Mathematical Education). The two World Wars and the CIEM’s lethargy in the 1920s prevented the continuation of initiatives such as the meetings of the 1910s. After World War II, the context was changed and the concern about mathematics education assumed new connotations. Since the early 1950s projects regarding mathematics education were developed in the United States of America and the launch of Sputnik in 1957 accentuated the interest in both science and mathematics education. Various initiatives were undertaken but their scope remained mainly national. Even the important conference called by the Education Committee of the National Academy of Sciences at Woods Hole in September 1959, which generated Jerome Bruner’s (1960) The Process of Education, was attended only by scientists from the United States of America. In Europe, things went differently thanks to 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], which created a tradition of international meetings on problems of mathematics education. The early years of this Commission are of extreme interest for the history of mathematics education, with particular reference to modern mathematics. The group was founded by Caleb Gattegno, who in 1950 invited some 30 scholars in Debden (UK) to set up an international commission for the study and the improvement of the teaching of mathematics from nursery to university (see Vanhamme 1991). He had some 13 positive answers and, after two further meetings in the next year, in 1952, CIEAEM was officially established in La Rochette par Melun (see Félix 1986). Gattegno, a mathematician and psychologist responsible for the training of prospective teachers at London University, was able to gather scholars from mathematics, philosophy, psychology, pedagogy, and teachers (Powell 2007). Figure 4.1 shows four important protagonists in the meeting of La Rochette par Melun. As it is claimed in the Préface of the book by Piaget et al. (1955), the Commission aimed to take all initiatives that, through study and action, could lead to a better understanding of the problems raised by the teaching of mathematics at all levels. In his foreword to the history of CIEAEM written by Lucienne Félix (1986), Gattegno stressed the importance of studying in-depth the problems related to mathematics teaching as a necessary premise for its improvement. Bernet and Jaquet (1998) have collected information for building an ex-post list of the founding members of CIEAEM, which includes scholars from European countries (Belgium, France, Germany, Italy, The Netherlands, Switzerland, UK) and one from Chicago. In the list of the active members of CIEAEM drawn up by Gattegno in 1957, we find, besides European members, Howard F. Fehr from

Figure 4.1  Choquet, partially hidden, Piaget, Gattegno, and Willy Servais in La Rochette par Melun in 1952. (Photo by Lucien Delmotte, collection Guy Noël)

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the USA and Celestino Galli from Uruguay (Bernet and Jaquet 1998). CIEAEM thus became a truly international group. In 1960 Gattegno resigned from the Commission after ten years as its secretary (Gattegno 1988). This was the beginning of CIEAEM’s gradual evolution towards new patterns, though it kept the practice of giving its annual meetings titles that should indicate the main topic. Until 1970, the number of participants in the meetings was around 40, but after that, this number increased, and as the years passed the Commission adopted better defined rules for its organization. CIEAEM still exists today and, despite having lost the innovative spirit of the early years, it has retained the initial characteristics of inclusiveness and the involvement of teachers. Some of the titles of the meetings held in the 1950s2 offer a kind of Manifesto of CIEAEM. In 1952, in La Rochette par Melun (France), the title “Mathematical and mental structures” referred to the lines of research developed thanks to the presence of the mathematicians Gustave Choquet, Jean Dieudonné, André Lichnerowicz, and the psychologist Jean Piaget, who are among the authors of the first book published by CIEAEM (Piaget et  al. 1955). In 1953  in Calw (Germany), the title was “Connections between pupils’ thinking and the teaching of mathematics,” and in 1955  in Bellano (Italy), it was “The pupil coped with mathematics—a releasing pedagogy”). In each case, the themes treated were related to psychology. Possibilities for updating mathematics programs were considered in 1954 in Oosterbeek (The Netherlands), where the title was “The modern mathematics at School”) and in 1955 in Ramsau (Austria), where the title was “The teaching of probability and statistics at the university and at school”. In the meeting at Madrid (Spain) in 1957, the title was “Teaching materials” and the focus was on classroom practice and the use of concrete materials. That theme was developed in the second book published by CIEAEM (Gattegno et al. 1958).

Royaumont Bodies Promoting Internationalization in Mathematics Education Acceleration in the phenomenon of the internationality of conferences was supported by two bodies—OEEC/OECD and UNESCO—which provided institutional and financial aid. OEEC became the acronym for the “Committee for European Economic Co-operation,” which was founded in 1947 and renamed the “Organisation for European Economic Cooperation” (OEEC) on April 16, 1948. It aimed to manage the substantial funds of the USA-financed Marshall Plan for implementing a European Recovery Program (ERP) addressed to the countries of Western Europe in the areas of industry, agriculture, infrastructures, energy, and technology. With this plan, the United States of America responded to the challenges that might be associated with education in socialist countries. The 18 members of OEEC were: Austria, Belgium, Denmark, France, Greece, Iceland, Ireland, Italy, Luxembourg, The Netherlands, Norway, Portugal, Sweden, Switzerland, Turkey, United Kingdom, and West Germany (originally represented by Bizonia and the French occupation zone) and the town of Trieste for a period before it returned to the Italian sovereignty. When the USA and Canada joined as members in the end of 1960, it was renamed “Organisation for Economic Cooperation and Development” (OECD). New statutes came into force on September 30, 1961. In the wake of the shock caused by the launch of Sputnik, OEEC was strongly committed to including education among its areas of intervention. In June 1958 OEEC opened an office in Paris—the Office for Scientific and Technical Personnel (OSTP)—the aim being to make science and mathematics education more effective and to promote a reform of the contents and the methods of mathematics instruction for 12- to 19-year-old students. UNESCO (the United Nations Educational, Scientific and Cultural Organization) is a specialized agency of the United Nations (UN) that sponsors international scientific projects with the purpose of  The complete list can be found in https://www.cieaem.org/index.php/en/meetings-en/previous-meetings

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promoting world peace and security through international cooperation in education, the sciences, and culture. UNESCO’s constitution came into force on November 4, 1946. In the 1950s, UNESCO ­supported initiatives in the field of mathematics education, including conferences organized by the International Mathematical Union (IMU), like for example, a conference that took place at the Tata Institute of Fundamental Research in Bombay from February 22 to 28, 1956, whose purpose was “to discuss, with special reference to South Asia, the problems of mathematical education at all levels, and to formulate plans for its sound development” (Report of the Executive Committee ..., 1956, p. 9). Members of ICMI and CIEAEM delivered talks at the Bombay conference. During the second half of the 1960s, and particularly during the period 1969–1974, the role of UNESCO in mathematics education increased, with special attention being given to developing countries. A series of UNESCO volumes entitled New Trends in Mathematics Teaching (Christiansen 1978; Hodgson 2009; Jacobsen 1996) was launched. Collaboration—both scientific and organizational—with ICMI, initiated by Marshall Stone, was continued by the next ICMI President, André Lichnerowicz. UNESCO financed the new journal Educational Studies in Mathematics and the creation of ICME (the International Congress on Mathematical Education), launched by Hans Freudenthal, and promoted advisory meetings with ICMI for discussing ICMI programs.

The Seminar of Royaumont: Not Only “Euclid Must Go!” According to Vanpaemel and De Bock (2019), in the 1950s, interactions between American and European actors in the field of school mathematics were limited. In 1952, the presence of Saunders Mac Lane and Stone in the Executive Committee of ICMI was the first concrete bridge between the two communities. Later on, some participants from the USA presented contributions on mathematics education to the ICMs of Amsterdam (1954) and Edinburgh (1958) (Gerretsen and de Groot 1954– 1957; Todd 1960). At Edinburgh, Fehr presented a report on ICMI’s first topic “Mathematical Instruction up to the Age of Fifteen Years.” Moreover, in the book Abstracts of Short Communications and Scientific Programme, issued to members during the Congress, five speakers were nominated by the United States Committee on Mathematical Instruction to deliver talks on experiments carried out in their country. Indeed, for the two sides of the Atlantic, the first real occasion for confrontation was offered by the Seminar of Royaumont entitled “New Thinking in School Mathematics,” held from November 23 to December 4, 1959, at the Cercle Culturel de Royaumont, Asnières-sur-Oise, France. It was organized by the OSTP and Marshall Stone, President of ICMI from 1959 to 1962, was the chair. The Royaumont Seminar can be seen as the beginning of a common reform movement to modernize school mathematics across the world, as evidenced by the OEEC recommendations to its member countries and Stone’s stress on the need for a “deep and urgent” reform in his introductory address (Stone 1961, p. 29). The vulgate has emphasized the folklore of Dieudonné’s claim “Euclid must go!” but, indeed, the event initiated long-distance effects which stretched well beyond geometry education. In short, the theme of the Seminar was not only the need for new thinking in both mathematics and mathematics education—including changes in curricula and teacher training—but also the development of appropriate follow-up action (OEEC 1961a). The basis for discussion should have been answers to a questionnaire sent in December 1959 to the countries participating in the program asking about current conditions of mathematics education in their countries (see OEEC 1961a, p. 7).3 The  The questionnaire (see Appendix B in OEEC (1961a), pp. 221–237) was sent to the OEEC member countries, and to Canada, the United States, and Yugoslavia. Only Spain did not answer, so the survey lists 20 countries. The results of the survey, divided by countries, with the exception of Canada. were also presented to OEEC (1961b). 3

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data were not available at the beginning of the Seminar, however, but answers to the questionnaire were nevertheless presented (partly summarized) and analyzed in the second part of the official report (OEEC 1961a, pp. 127–210), which was titled “Survey of Practices and Trends in School Mathematics.” The questions concerned: • • • • •

The number of hours of compulsory mathematics teaching in the different schools. Educational qualifications of those providing teacher training. Which authorities were competent to establish programs and select textbooks. Any ongoing reforms. The contents of the programs (more precisely, various topics were listed and it was asked, for each of them, in which school year was it carried out).

Answers to the survey provide an interesting source of data with respect to mathematics education in the countries participating in the program before the start of modern mathematics in Europe. Three delegates from every invited country—an outstanding mathematician; a mathematics educator or person in charge of mathematics in the Ministry of Education; and an outstanding secondary school teacher of mathematics—were invited to participate in the Seminar (OEEC 1961a). From the list of the participants (OEEC 1961a), we know that there were 33 delegates from 15 OEEC countries (Austria, Belgium, Denmark, France, Germany, Greece, Ireland, Italy, Luxembourg, The Netherlands, Norway, Sweden, Switzerland, Turkey, UK), one from Yugoslavia, three more from Canada and the United States, and three officers of OEEC. Only Norway sent three delegates. There were 13 guest speakers, of whom four were from France, three from the USA, two from the UK, and one from each of Belgium, Denmark, Germany, and The Netherlands. CIEAEM was represented by the guest speakers Choquet, Dieudonné, Félix, and Willy Servais and the Italian delegates Luigi Campedelli and Emma Castelnuovo. ICMI was represented by the guest speakers Begle (a future Executive Committee member) and Edwin A.  Maxwell (past Executive Committee member and future ICMI secretary) and by the delegates Otto Frostman (Executive Committee member), Kay W. Piene (past Executive Committee member), and the chair of the Seminar Stone (ICMI President). The 33 delegates included university professors and school teachers equally divided, and education-related officials. There were only two women, both of them secondary teachers—Castelnuovo and Félix. Among the 13 guest speakers, university professors were predominant. The studies by De Bock and Vanpaemel (2015) and Vanpaemel and De Bock (2019), which contain information taken from Servais’ archives, and by Schubring (2014a, b) based on OEEC/OECD’s archives, shed new light on the development of the Seminar, but its actual program remains difficult to reconstruct due to the many discrepancies found by Schubring (2014a) between the official report (OEEC 1961a) and the documents kept in the OEEC/OECD’s archives. As written in the preface, the official report of the OEEC (1961a) meeting was “drawn up” by Fehr with the assistance of Lucas N. H. Bunt. It had the same title as the Seminar—New Thinking in School Mathematics. The French version was titled Mathématiques Nouvelles [New Mathematics]. The report included the full texts of the interventions of Dieudonné and Stone, excerpts and summaries of various contributions, and reports on the discussions, comments, and conclusions, but in an order that was not necessarily that of the presentations at the Seminar. The conference was divided into three sections, dealing with (a) New Thinking in Mathematics, (b) New Thinking in Mathematical Education, and (c) Implementation of Reform. The report did not correspond to those three sections and is instead divided into Part I (Report) and Part II (Survey) each with five sub-chapters, with Roman numbering. According to Fehr (1961), the purpose of the Seminar was not “to formalize all mathematics instruction through an axiomatic and logical treatment,” but “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” (p. 209). The talks of Dieudonné and Stone represented the two poles of the way to tackle the reforms in teaching. The first focused on the subject matter, bearing in mind the

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developments of Bourbakist research (Dieudonné 1961). Dieudonné freed geometry from Euclid’s burdens and proposed a gradual program from the age of 14 upwards: After experimental work in geometry and algebra, pupils were to begin quite early to use the language of sets and to speak of axioms. Further topics would include matrices and determinants of orders 2 and 3, elementary calculus (functions of one variable), construction of the graph of a function and of a curve given in the parametric form (using derivatives), elementary properties of complex numbers, and polar coordinates. To carry out this program, the concepts of vector space, linear algebra, and quadratic forms were presented as being fundamental. In his opening address, Stone considered whether some new topics or subjects should be introduced and whether some traditional parts of mathematics which had lost their importance should be eliminated. He did not mention the objects of the changes. Nevertheless, different from Dieudonné’s talk, the heart of Stone’s address was its appeal for the development of specific didactic capacities, the creation of university research institutes, and professorships. Furthermore, he stressed the need to look at programs for elementary schools, and to develop efficiently the latent mathematical talents and interests of average children in order not to remove them permanently from the study of mathematics. This would bring on improvements in secondary schools. Dieudonné’s proposal drew expressions of approval and disagreement among participants, but the shared conclusion was not to remove Euclid entirely from the secondary school curriculum. Dieudonné’s proposal focused on the subject matter which overshadowed other important aspects. Texts associated with these latter aspects were discovered by Schubring among the documents in preserved files labeled “Background Paper for Section IV”4 (Schubring 2014b). Its title is “Psychological and Educational Researches into the Teaching of Arithmetic and Mathematics.” The authors were William Douglas Wall, an educator director of the National Foundation for Educational Research in England and Wales (NFER) and chairman of the international project for the “Evaluation and Educational Attainment,” and John B.  Biggs, mathematics educator, at that time Assistant Research Officer at NFER. In OEEC (1961a) the contribution is summarized in nine paragraphs from 344–352 (pp. 101–103) under the title “Extracts from Dr. Wall’s remarks.” Biggs is not on the list of the participants in the Seminar and his name does not appear in the proceedings. From the extracts published we know that the paper analyzed the state of empirical research on the teaching and learning of arithmetic and mathematics: According to the authors, current research was short-term, scattered, piecemeal, and mainly focused on one or two aspects of the problem at a time without considering the complex of interrelated variables involved. The paper called for large-scale, systematic, scientific research in actual classrooms, which should involve mathematicians, educators, teachers, and psychologists, and perhaps also logicians, philosophers, physicists, engineers, statisticians, and economists. Three types of research were called for: Preliminary fundamental studies, operational research to evaluate the results of the reforms, and action research which would provide feedback on the school systems. Wall and Biggs were able to identify most of the research topics which would later be developed in the field of mathematics education. The prevailing role of university professors had shifted the attention to the contents so that in the chapter “Summary and conclusions of the Seminar” out of 507 paragraphs only two (namely the numbers 417 and 418) were dedicated to “Research in mathematics education” (OEEC 1961a, p. 120) without there being any specific mention of the contribution made by Wall or Biggs. In the following, we will see that also in subsequent conferences of those years the approach to mathematics education only focused on contents.

 As explained above, the official sections of the Seminar are three. The Section IV found by Schubring (2014b) may have been suppressed or may correspond to sub-chapter IV of the report, which indeed contains the “Extracts from Dr. Wall’s remarks.” 4

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In the Aftermath of Royaumont

Royaumont had a significant effect on mathematics education from an international perspective, testified, among other things, by the impressive subsequent flourishing of international meetings. These meetings had an institutional character, being organized by national institutions often with the support of OECD or UNESCO. Participants were chosen by the organizers from among experts in the field and officials of the bodies supporting the meetings. There is one aspect, however, that represented a step back: In general, few or no teachers were invited to these meetings. Concerning teachers, the efforts of the reformers were mainly addressed to retraining courses, usually held at a national level: Only the Belgian training courses known as the Days of Arlon, organized annually (between 1959 and 1968) by the Belgian Society of Mathematics Teachers and the Ministry of Public Education, although primarily intended for the retraining of Belgian teachers, were over the years regularly attended by foreign guests. Of course, the themes treated in Royaumont also influenced activities already carried out in the mathematics education milieu such as at the International Congresses of Mathematicians and CIEAEM meetings.

Aarhus, Denmark In January 1960, the President of ICMI Stone appointed a committee consisting of Heinrich Behnke (president), Svend Bundgaard (chairman), John G.  Kemeny, Piene, Ole Rindung, Servais, Stefan Straszewicz, and Erik Kristensen (secretary) with the responsibility of organizing a seminar (to be held in Aarhus, Denmark) to advance the study of the following three topics which would be discussed at the 1962 International Congress of Mathematicians in Stockholm: (1) Which subjects in modern mathematics and which applications of modern mathematics can find a place in programs of secondary school instruction. (2) Education of teachers for the various levels of mathematical instruction. (3) Connections between arithmetic and algebra in the mathematical instruction of children up to the age of 15. Three sessions were devoted to the discussion of the three points mentioned above, but, as Bundgaard wrote in the preface, a decision was taken to concentrate most of the lectures on the discussion of “Modern teaching of geometry in secondary schools with particular emphasis on ways of treatment opened up by developments lately, in particular by the algebraization of mathematics” (Behnke et  al. 1960, p. ii). It was decided to publish only the lectures on geometry (by Behnke, Choquet, Dieudonné, Werner Fenchel, Hans Freudenthal, György Hajós, Günter Pickert) and the related discussions (by Bunt, Servais, János Surányi, Poul Thomsen, Tullio Viola). Kristensen was in charge of the report. Some of the authors (Bunt, Choquet, Dieudonné, and Servais) had attended the Royaumont Seminar. The meeting was held from May 30 until June 2, 1960. In the preface to the report, it is indicated that there were 30 participants from 10 countries. Of the seven talks, five proposed new axioms for geometry: Dieudonné introduced, as in Royaumont, the plane (or space) as a two- (or three-) dimensional real vector space completed by the dot product. He showed how the concept of angle could be defined by expressing its cosine using the dot product (but also argued that the angle is not a useful notion in mathematics). The long presentation by Choquet was a summary of his book (Choquet 1964). He presented axioms for the affine and metric plane, which—like those proposed by Dieudonné—were suitable for 17-year-old pupils and also “strong” axioms for Euclidean geometry based on the primitive concepts of distance and line

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reflection, which were suitable for younger pupils (age 15–16). Choquet defined an angle as a rotation that is the product of two line reflections. Pickert presented axioms based on the properties of vectors—seen as primitive concepts—and their linear combinations. Metric properties were introduced by axioms on congruence and perpendicularity. Hajós proposed that a large number of non-independent axioms concerning properties of Euclidean geometry should be introduced which would have to be reduced as the student became aware of links between them. Fenchel presented a list of axioms defining relations such as collinearity, coplanarity, congruence, and orthogonality. His exposition made wide use of logical symbols. Discussion of the five different proposals could not produce fruitful results, but there seemed to be some agreement on the fact that Euclidean geometry was too difficult, even in the form of transformation geometry. It was Behnke who distanced himself not only (implicitly) from Otto Botsch—who in Royaumont had spoken in favor of the German schools that experienced the geometry of transformations—but also explicitly from Felix Klein, who had always been against abstraction and in favor of intuition. Of course, Behnke recognized Klein’s work and merits, but also placed him in his historical period. Behnke was totally in favor of the modern approach and said he thought that Klein would have appreciated some of its aspects (but not all of them). There were attempts, in particular by Freudenthal, Servais, and Viola, to introduce a wider educational and pedagogical point of view. Freudenthal distinguished between a rigorous axiomatic approach and a deductive approach with local deductions in which Euclidean “facts” could be placed; he claimed that we must ask ourselves the question of why we want to teach axioms. And Servais stated that mathematization is as important as logical deduction. The distance between Freudenthal and Dieudonné was evident. When Freudenthal deemed it necessary to take into account the processes of the pupils and their psychological and pedagogical aspects, Dieudonné exclaimed “la psychologie, je m’en fiche” [Psychology, I don’t care] (Behnke et al. 1960, p. 104), and when Freudenthal claimed that with school mathematics it was necessary to see “the whole together,” Dieudonné answered that “we are here to discuss mathematics and nothing else” (p. 127).

Zagreb-Dubrovnik, Yugoslavia In 1960 OEEC organized a working session in Yugoslavia (August 21 until September 2 in Zagreb, from September 4 to 17 in Dubrovnik) in which a group of experts was charged with the study of a modern program for teaching mathematics in secondary schools. Two days later most members of this group attended the ICMI symposium on the “Coordination of the Teaching of Mathematics and Physics” held in Belgrade (September, 19–24). The experts—Emil Artin, Botsch, Choquet, Bozidar Derasimovic, Fehr, Cyril Hope, Kristensen, Ðuro Kurepa, Paul Libois, Laurent Pauli, Lennart Råde, Bruno Schoeneberg, Servais, Stone Pierre Théron, and Mario Villa—were from eight OEEC countries, Yugoslavia, and the United States of America. Nine of the participants had attended the Royaumont Seminar. The resulting proposals were collected in one volume OEEC (1961c), the preface of which explained that the programs for the first cycle were more flexible and adaptable, and those for the second cycle were aimed at students who were oriented toward scientific and technical studies. The text referred directly to school teaching. It was divided into five parts: Algebra (1st cycle); Geometry (11–15 years); Algebra (2nd cycle); Geometry (15–18 years); and Probability and Statistics. Each part contained objectives, prerequisites, programs, exercises, and comments. The program of the 1st cycle, Algebra (11–15 years), began with elementary notions on sets, relations between sets, and cardinal numbers, and continued with more classical themes such as ­numerical

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sets and their operations, the number line, linear functions and graphs, and simple equations. Then came structures such as rings, groups, and fields. The comments of this section contained almost 50 pages on sets, with exercises and applications connected to the other topics covered. A part with arithmetic exercises followed. The 1st cycle Geometry program was characterized by the use of algebra and by intuitive and descriptive definitions, with the use of concrete materials that could allow pupils to move on to abstraction. Geometric transformations were introduced as was a connection to algebra when treating graphs of simple equations. Simple logical deductions were proposed. In this section the programs were mixed up with the exercises: All the topics were presented through examples. The Algebra of the 2nd cycle was an abstraction of what was seen in the first cycle, including set theory, structures, group theory, the abstract notion of vector space, and much more. In the introduction, it was indicated that this was a “maximum” program. Exercises were not given, but a division by school years was proposed together with insights into structures, group theory, applications between structures, and linear algebra. Names of “official authors” for the individual parts of the programs were not given, but there is an “oral tradition” which attributes the geometry programs for the level 11–15 years to Libois, and to Choquet those for the level 15–18 years—actually, the second part of these programs presented axioms of the affine and the metric plane, which were very similar to those proposed by Choquet in Aarhus. The author of this second Geometry section asserted that the proposed programs represented a synthesis of the positions of the various countries. The study of geometry included vectors (and later their dot products), coordinates (with particular reference to the correspondence between the line and the set of real numbers), and properties of the Euclidean plane. The latter did not correspond to the study of geometry according to Euclid’s axioms; indeed, in the opinion of the author, the classical Euclidean properties had been studied for the most part in middle school experimentally, and they did not need a systematic study at this level. The pupils would be expected to understand that in addition to Euclidean geometry, there are other geometries (the reference was to affine geometry in particular). The study of these new geometries required reference to recent developments in mathematics (such as sets, groups, vector spaces, …). As for the final part of the text, the authors suggested ways for introducing probability intuitively, through games and statistics involving data collection. The main part was then dedicated to the contents for the second cycle, which coincided with those usually treated today. The comments to this part consisted of introductory paths to probability and descriptive statistics. It has to be noted that, since the program had been discussed by experts from 10 countries, it was expected that it should satisfy most of them and effectively represent a synthesis of the various positions. Indeed, it was presented as a common program. Nevertheless, discussion continued in subsequent conferences.

Bologna, Italy A conference entitled “A Discussion of the Aarhus and Dubrovnik Reports on the Teaching of Geometry at the Secondary Level” was held in Bologna from October 4 to 8, 1961 (BUMI 1962). It was sponsored by ICMI and the Commissione Italiana per l’Insegnamento Matematico (CIIM) [Italian commission for mathematical teaching]. The organizing committee consisted of the Italians Pietro Buzano, Villa, and Viola (chairman), the President of ICMI Stone, and ICMI Secretary Gilbert Walusinski. Fulvia Furinghetti and Livia Giacardi (to appear) have reported that initially Aleksandr Aleksandrov, a member of the ICMI Executive Committee, had proposed to ICMI President Stone that he organize a European meeting in the USSR in 1961 and Stone discussed this with Sergei Sobolev, but nothing came of it.

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The participants in the Bologna Conference (mathematicians, educators, ministerial officials, and observers of UNESCO and OECD) were from Belgium, France, Germany, Italy, Romania, Sweden, Switzerland, the USA, The Netherlands, and India (UNESCO observer). In his opening address, Stone pointed out that geometry had been the main object of discussion at Royaumont. Although there was a consensus on the need for action on geometry, the meetings of Aarhus and Zagreb-Dubrovnik had not reached an agreement on how to realize changes and this was the reason why the meeting of Bologna was organized. Stone (1963) presented a contribution on the choice of axioms for geometry in school. His conclusions went beyond the mere examination of mathematical arguments and underlined the need to carry out and examine the results of psychological studies—both analytical and experimental—and, from a practical point of view, to prepare students from primary school to face new concepts. In addition to those published in the journal L’Enseignement Mathématique (by Artin, Campedelli, Cartan, Freudenthal, Libois, Lucio Lombardo-Radice, Ugo Morin, Ruggero Roghi, Servais, Stone, Viola) there were the following contributions: –– Nyman Bertil, On a geometry text, based on Choquet’s axiomatics, in preparation in Sweden; –– Pauli Laurent, Pedagogical experiences in Switzerland on the teaching of geometry in secondary schools. The various presentations sparked lively discussions. Three contributions stood out and became a landmark of this conference. Firstly Artin (1963), who was among the experts in Zagreb-Dubrovnik, illustrated the work of mediation between three extreme positions attempted at that conference: To preserve the classical Euclidean method, to introduce Hilbert’s axiomatics, and to define Cartesian space as a vector space of dimension two. The final choice was the third way, introduced by axioms. Freudenthal (1963) focused on pedagogical aspects and advocated frank didactic research in mathematics education. According to him, the geometry programs proposed in Aarhus and Dubrovnik were based on what he called an “antididactic inversion” (p. 32) since they started from the finished mathematical product, which is the result of the activity of others, rather than promoting the student’s investigation and exploration for constructing the meaning of the concepts. In line with Freudenthal’s ideas were the interventions of Libois and the Italian Viola. On the opposite side, Henri Cartan (1963) was in favor of the new contents proposed in Aarhus and Dubrovnik and criticized those who, despite showing interest in reforms, in reality, did not want to make any changes. He advocated taking care not only of students who would stop their school careers after secondary school but also of those who would proceed to university. Moreover, he stressed the need to train teachers on the new mathematics content. He was convinced that in planning a radical reform, it was necessary to focus first on the mathematical content and then on educational objectives. On this point, he was not in agreement with Freudenthal (1963) who claimed that didactical problems had to be considered first. At the end of the Bologna conference, the participants were invited to make proposals for books inspired by the new trends which had emerged from the various discussions. In Chapter 8 of this volume, Furinghetti and Marta Menghini analyze the influence of this conference on the introduction of modern mathematics in Italy. Axiomatics had already been extensively treated by Choquet in his lecture at the Lausanne Seminar on “The teaching of analysis and relative manuals” sponsored by ICMI and the Swiss Mathematical Society and held from June 26 to 29, 1961 (see L’Enseignement Mathématique, 1960, s. 2, 6, 93–178).

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 tockholm, Sweden. Modern Mathematics at the International Congress S of Mathematicians in 1962 During the ICM held in Stockholm from August 15 to 22, 1962, reports on the three themes discussed in Aarhus were presented, based on materials submitted by different countries. Two reports referred also to modern mathematics. Kemeny (1964) presented a report “Which Subjects in Modern Mathematics and which Applications of Modern Mathematics can Find a Place in Programs of Secondary School Instruction?” based on submissions from 21 nations. He concluded that four areas of modern mathematics—elementary set theory, an introduction to logic, some topics from modern algebra, and an introduction to probability and statistics—were usually introduced, while there was no agreement on how far the axiomatic system should be applied to mathematics, and in particular to geometry. He further observed that the most frequent motivation for teaching modern mathematics was to prepare students for university and that the greatest difficulties were the shortage of qualified teachers and the lack of suitable textbooks. In his talk, based on 11 national reports, Straszewicz (1964) identified a general tendency to bring the school education of mathematics—even in the lower grades—closer to contemporary science and its new applications by gradually introducing a modern mathematical language. It was proposed, for example, to introduce fairly early the simplest notions of the algebra of sets and propositional logic, in order to highlight the structural properties of the different number systems. In an interview with Bernard Hodgson, Geoffrey Howson stated that in Stockholm (as well as in the previous ICM in 1958) he had been disappointed by the short amount of time dedicated to mathematics education and the lack of involvement of people such as Begle, Georges Papy, Servais, Félix … “who were doing things” (Hodgson 2008, part 1) and the too much attention paid to the words of university mathematicians not directly involved with schools. He complained about the lack of discussion between the two successive ICMs of 1958 and 1962, although he recognized that this criticism was partly answered by discussion at the Arlon meetings and CIEAEM.

Athens, Greece The wave of renewal that swept through many nations, mainly in Europe and North America, is witnessed by the congress held in Athens from November 17 to 23, 1963 (OECD 1964). This event was the first attempt to present the achievements and repercussions of modern mathematics in OECD countries. Forty mathematicians, mathematics teachers, and education officers participated. They were from the 20 OECD countries—Austria, Belgium, Canada, Denmark, France, Germany, Greece, Iceland, Ireland, Italy, Luxembourg, Norway, Portugal, Spain, Sweden, Switzerland, The Netherlands, Turkey, UK, and the USA. Fehr wrote the preface to the proceedings, and is referred to as “Rapporteur General.” The second cover says that during the conference “a forum of leaders in School Mathematics reform examined and discussed the new programs which O.E.C.D. has encouraged or actively sponsored for experimentation and adoption in Member Countries.” In the Appendices, information on some national projects was given. In the proceedings five major aspects of mathematics education were considered: (1) The subject matter content of a modern program in mathematics for the scientific line in secondary school (2) Methods of organizing and teaching a newly formed program (3) The use and purpose of pilot-experimental classes in modern mathematics (4) The role of applications in the teaching of newer branches of mathematics

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(5) The education and professionalization of teachers of secondary school mathematics As reported in the proceedings, the most significant outcome of the conference was a clear picture of what was meant by the “modernization of mathematics teaching” (p. 291). A genuine distinction appeared between (a) the updating of a traditional program, where the same sequence is maintained but modern concepts and treatments were injected into the curriculum; and (b) a new program completely modern, based on a total re-construction of the curriculum. The resolutions and recommendations took into consideration the main issues concerning mathematics teaching: The content of the courses (sets, relations, and functions, vector spaces, calculus, probability, statistics, the use of computers, mathematical logic, updating of notations and definitions), the need for research and information, experiments in classrooms, and their evaluation, teacher education, links with other disciplines and applications.

In the United States of America: The Conference in Cambridge (MA) Despite the contact established between the United States and Europe, Royaumont’s impact was different in the two communities. We cite as an example of these fact the conference sponsored by the National Science Foundation held in Cambridge (Massachusetts) from June 18 to July 12, 1963, to discuss the future of mathematics curricula. The steering committee consisted of Begle, Bruner, Andrew Gleason, Mark Kac, William Ted Martin, Edwin Moise, Mina Rees, Patrick Suppes, Stephen White, and Samuel S. Wilks. Some 25 mathematicians and users of mathematics from universities and industries attended the conference. Among them, there were some participants in the Woods Hole Conference of 1959—including Bruner—and members of the SMSG (School Mathematics Study Group). According to the report (Goals for School Mathematics, 1964; De Finetti 1965), the main purpose “was to reconsider the structure of mathematics education and to sketch a rough outline of a possible new framework for the primary and secondary school” (p. 196). Conference recommendations dealt more with methods of presentation than with specific mathematical content, as in the case of the “spiral” approach, in which every new topic would be introduced under low pressure and would then be reconsidered repeatedly each time with more sophistication, the importance of multiple motivation for the topics introduced, and the development and implementation of discovery procedures. Among the various themes discussed, we note the reference to the importance ascribed by Freudenthal to the early development of the child’s spatial intuition, and themes important in modern mathematics such as sets (without too much emphasis on theory and formal logic), elementary ideas of probability and statistical judgment. However, there were no explicit references to Royaumont nor to the follow­up meetings. Henry O. Pollack reported on this conference at the Athens meeting of 1963 (OECD 1964, pp. 90–94).

Frascati, Italy From October 8 to 10, 1964, 27 participants from six European countries and Argentina met in Frascati for discussing the topic “Mathematics at the coming to university. Real situation and desirable situation.” A few school teachers were present. The conference was organized by ICMI with the collaboration of CIIM and the Centro Europeo dell’Educazione (CEDE) [European Center of Education] in the villa Falconieri, the headquarters of CEDE. Lectures were delivered by Jean Bass, Behnke, Bruno de Finetti, René Deheuvels, Julien Desforge, Bo Kiellberg, Arnold Kirsch, Jacqueline Lelong, Carlo Felice Manara, Papy, Pickert, André Revuz, Hans-Georg Steiner, Walusinski. There were no proceedings. The short report (R. G. 1964) contained the titles of the lectures and the sum-

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maries of the talks by Behnke, Campedelli, and Papy. There were no final resolutions, but the participants expressed a wish to fill the gap between secondary and university mathematics education. Behnke pointed out that the means for reaching this goal was to remedy the shortage of teachers and to prepare them adequately. He also recommended avoiding excessive abstraction and axioms. Modern mathematics was extensively treated by Papy who reported on Belgian schools and de Finetti when discussing his proposal for the renewal of Italian programs.

From Milano Marittima, Italy, to Echternach, Luxembourg After 1960, CIEAEM continued with its yearly thematic meetings, many of which show a link to modern mathematics. As Félix (1986) phrased: It is the moment to become aware of possible post-Hilbert axioms for geometry, after the contributions of Choquet (1955–1965), Artin, H. Levi, Papy, Delessert, Revuz, Krygowska, Servais and many others who had understood the importance of the work of Dieudonné but also its limits at the educational level. This is the moment for an important reflection on the axiomatic approach, on the construction of the affine plane, on the alternative between a geometric or algorithmic solution. (p. 64)

In 1965, at the CIEAEM meeting entitled “The place of the geometry in a modern mathematical teaching” in Milano Marittima (Italy), Revuz (1965) proposed an approach to geometry that attempted to reconcile the positions expressed by Choquet and Dieudonné at the congress of Aarhus. Dieudonné had explicitly accused Choquet of being linked to naive realistic and synthetic methods (Dieudonné 1964). The solution by Revuz was based on proposals by Papy, which were quite similar to those of Choquet (1964) at the first stage, but which allowed a second stage in which the achievements of the first stage served as a base for new axioms for the further development of geometry, in agreement with Dieudonné’s positions (Vanpaemel & De Bock 2019). The proposal by Revuz was also discussed at the colloquium of Echternach in the same year (from May 30 to June 4), the link being established by Lichnerovicz, who was a member of CIEAEM and at that time President of ICMI.  The colloquium was organized by ICMI in collaboration with the Minister for Education and Cultural Affairs for the Grand Duchy of Luxembourg. The 92 participants were from eight European countries. The subject of the seminar was “Repercussions of Mathematical Research on Teaching.” The resulting book (Conference Echternach 1965) contains the texts of the lectures by Behnke, Camille Bréard, Bunt, Choquet, Jean de Siebenthal, Paul Debbaut, André Delessert, Dieudonné, Arthur Engel, Kirsch, Papy, Pickert, Charles Pisot, Revuz, Servais, and Steiner. Many lectures were about the axiomatic method, not only as regards geometry. Dieudonné spoke about the role of linear algebra in modern mathematics, Servais about axiomatization and elementary geometry (12–15 years), and Papy about the Euclidean vector plane in teaching (15 years) (Conference Echternach 1965). Revuz’s motion, known as “Convention de Ravenne et traité de Echternach” [Ravenna Convention and Echternach Treaty], was signed by Choquet and Dieudonné (Félix 1986) (Figure 4.2). At the end of the colloquium, in a meeting of ICMI with the representative of UNESCO, the publication of the already mentioned series of volumes entitled New Trends in Mathematics Teaching was planned, and

Figure 4.2  Signatures in the Echternach treaty

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Anna Zofia Krygowska was entrusted with the task of overseeing the publication (see Internationale Mathematische Nachrichten, 83, 1966, p. 3). In 1967 the first volume (UNESCO 1967) was issued.



Modern Mathematics Goes Beyond the Iron Curtain

We may say that the Echternach conference closed the first cycle of conferences on new thinking in school mathematics—a cycle that started with the suggestions of Royaumont and continued with the proposals of Aarhus, Zagreb-Dubrovnik, Bologna, and Athens. ICMI and OEEC/OECD were the main supporting bodies. The colloquium of Echternach was one of the last attempts to find a solution to the problem of teaching geometry, a problem which—after the intervention of Dieudonné in Royaumont—was perhaps given an excessive role. We will see that more general problems arose in other conferences.

Budapest, Hungary From August 27 to September 8, 1962, at the invitation of the Hungarian government, UNESCO organized an international symposium in Budapest aimed at dealing with the problems involved in reforming mathematics teaching in schools in the light of the results of the meetings already held in various countries (Hungarian National Commission for UNESCO 1963). The number of participants outside Hungary was limited to 18, chosen among mathematicians, teachers, psychologists, and educators; there were six Hungarians. For the first time in this kind of meeting, there were people from Australia, Japan, and the USSR. The main themes treated were: The mathematics curriculum; teaching methods and their psychological background; training and re-training of mathematics teachers. We note the explicit mention of psychology; indeed, the British scholar, Richard R.  Skemp, who participated in the conference, was the major pioneer in the psychology of mathematics education. ICMI was present through its President Stone and its officer Yasuo Akizuki. Summaries of each day’s discussions were prepared by rapporteurs and circulated on the following day (Hungarian National Commission for UNESCO 1963, p. 6). There were also “background papers” prepared before the symposium (p. 5). A chief rapporteur—Servais—prepared the “Conclusions and recommendations” (pp. 11–34). These recommendations included some nods to modern mathematics, which, nevertheless, due to the general approach to the problems adopted, was not a central issue of the symposium. In the “Introduction” (pp. 7–10)— which gave a survey of the activities—Stone claimed that “the results obtained in this Symposium are consistent with the conclusions reached by a number of recent conferences, both national and international, and represent a substantial advance beyond them” (p. 8). His conclusion was a call to establish a network of contacts and information.

 oscow, USSR. Modern Mathematics at the International Congress M of Mathematicians in 1966 The four-year International Congress of Mathematicians was held in Moscow from August 16 to 26, 1966. There were communications concerning mathematics education, but the short notes published in the booklet of abstracts and the proceedings make it difficult to evaluate the topics which were considered. In Volume 3 of the one-hour and half-hour addresses, there is the text of Papy’s lecture entitled “La géométrie dans l’enseignement moderne de la mathématique” [Geometry in the

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Modern Teaching of Mathematics] (pp. 82–89), which reported on the Belgian experiments at the secondary level (12- to 17-year-old students). This contribution was also published in L’Enseignement Mathématique (1966, s. 2, 12, 225–233). In the same volume, Jean Leray commented on Papy’s experiments in his article “L’initiation aux mathématiques” [The initiation to mathematics] (pp. 235– 241). Two general reports of ICMI, presented at the Congress of Moscow, were not included in the proceedings, but were published in L’Enseignement Mathématique: Pisot, Ch. 1966. Rapport sur l’enseignement des mathématiques pour les physiciens [Report on Mathematics Education for Physicists]. 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 [Development of Pupils’ Mathematical Activity and the Role of Problems in this Development]. L’Enseignement Mathématique s. 2, 12, 293–322.

Bucharest, Romania From September 23 to October 2, 1968, UNESCO conducted an international colloquium about “Modernization of Mathematics Teaching in European Countries” in Bucharest (Teodorescu 1968), with the cooperation of the Society of Mathematical Sciences of Romania and the Romanian National Commission for UNESCO. It was attended by 39 “experts” from 22 European countries and some 300 Romanian secondary school teachers. Among the experts were ICMI President Freudenthal and other members of the Executive Committee, Delessert, and Revuz. Problems of mathematics teaching were raised and discussed from the point of view of the scientific-­technical evolution in contemporary society. The term “modern mathematics” was used in the introduction by Nicolae Teodorescu (1968), but in a completely different sense, linked to electronic computers. “Representatives” of modern mathematics of the early 1960s, such as Revuz or Servais, were present, but their contributions related to more general questions on the teaching of mathematics. Even Papy’s talk was anything but uncompromising; he proposed a progressive approach to an increasingly abstract axiomatic system. In the discussion which followed, Krygowska questioned the legitimacy of some of Papy’s statements. For instance, Papy affirmed that in traditional Euclidean geometry, proofs of quite obvious facts were presented. Krygowska stated that this was not changed by the fact that we use modern axioms. In Papy’s books, students also proved “intuitive theorems.” The problem was a different one—it was a problem of methodology: Pupils must understand that they have to establish a logical relation between certain groups of theorems. They like to prove evident things if they know what they are proving. In turn, Sobolev criticized Papy’s alleged simplification of mathematics. It was mathematicians who needed to learn to simplify their way of presenting arguments. The final recommendations concerned: The impact of mathematics on contemporary schools; the teaching of modern mathematics; teacher education; international cooperation, including the contribution of ICMI.



Modern Mathematics in Other Hemispheres

Latin America In Latin America, contact with the New Math occurred initially through the textbooks of the School Mathematics Study Group (SMSG) (Barrantes and Ruiz 1998). A decisive impulse came from the First Inter-American Conference on Mathematics Education, held in Bogotá, Colombia, from

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December 4 to 9, 1961. Its organization was the result of the cooperation between ICMI and CIAEM (Comité Interamericano de Educación Matemática) [Inter-American Committee on Mathematics Education: IACME], which was founded in 1961 by a group of mathematicians and math educators from the three Americas. The conference was sponsored by the Organization of American States (OAS), USA’s National Science Foundation (NSF), the Rockefeller Foundation, the Ford Foundation, the Association of Colombian Universities, and UNESCO. Mathematicians and mathematics teachers attended as representatives or guests from American countries—23 according to Hugo Barrantes and Angel Ruiz (1998)—and there were a few European special guests from Denmark, France, and Switzerland. The keynotes addresses were delivered by Enrique Cansado, Laurent Schwartz, Guillermo Torres, and the participants in Royaumont Begle, Bundgaard, Choquet, Fehr, and Stone, who reported on what was going on in Europe and the USA and on the means to introduce innovations into Latin America: There was a need for suitable textbooks, curriculum changes, teacher training, and more. In response to the opening speech by the Education Minister, Stone, President of ICMI and the Royaumont Seminar, recalled the role of ICMI in mathematics education and encouraged collaboration with ICMI to promote regional cooperation. In fact, at the end of the conference, the first Executive Committee of CIAEM or IACME was established with Stone as pro tempore President (Stone remained in charge until 1972). After this event, the conferences of CIAEM were regularly held in different locations throughout Latin America. The main ideas put forward during the conference were: Teaching geometry in a new way with a focus on modern mathematics; pursuing the unity of mathematics through the study of fundamental structures relying on modern algebra; teacher education and in-service teacher retraining. The final resolutions concerned these ideas. Some talks, in particular those of Choquet and Fehr, aroused dissension both on the mathematical contents proposed and on their inadequacy to the context in which the proposals would be inserted. Among the many interventions, we note that Torres, of Mexico, claimed that it was not advisable to abandon entire topics from classical mathematics and then fall into formal definitions and concepts that would communicate absolutely nothing to students. He suggested that mathematics should be taught in accordance with its historical development, again in opposition to Choquet who considered this method outdated. Illustrative of the context in Latin America is the statement by Omar Catunda who claimed that the formula suitable to Brazil should be “al menos la Geometría de Euclides” [“at least Euclid’s geometry”] (Barrantes and Ruiz 1998, p. 13).

Africa and Asia Science education had become an important area of cooperation with newly independent and developing countries, many of which established their agencies for curriculum development. The momentum generated by the reform movements of the 1960s resulted in projects aimed at incorporating modern approaches, methods, and materials in the area of Africa and Asia. In 1961 the African Mathematics Program (AMP) was launched at a meeting in Dedham (Massachusetts) in which scholars in mathematics education, teachers, educators from Africa, the USA, and the UK met for planning educational innovation in African schools (Weaver 1965). Three seminars were then organized in Entebbe (Uganda) by the Entebbe Mathematical Centre, directed by Martin, to produce materials. Being linked to New Math the topics—introduced at all school levels— included sets and their use in teaching numbers, the related basic operations and their properties, intuitive geometry and measurement, statistics, probability, and motion geometry (Williams 1976). The third seminar in Entebbe, entitled “Mathematics” (Uganda, July 15–August 15), continued the work that had been started in the two preceding seminars. It was attended by 60 participants, including school teachers, from 12 countries (Ethiopia, Ghana, Kenya, Liberia, Malaya, Nigeria, Sierra Leone, Tanzania, Uganda, United Kingdom, United States, and Zambia).

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The Association of South-East Asian Institutes of Higher Education organized in Saigon and Dalat (Vietnam, November 9 to 13, 1964), a “Seminar on mathematics teaching in South-East Asia.” The 20 participants represented the universities of Malaya, Singapore, Thailand, and Vietnam (UNESCO 1967). The Seminar concerned the state of the art of mathematics teaching in the participating countries and the new trends in the training of teachers, engineers, economists, and researchers in mathematics. On these questions, final recommendations were made. The Commission Inter-Unions de l’Enseignement des Sciences (CIES) [Inter-Union Commission for the Teaching of Science, IUCST]—of the Scientific Council of Scientific Unions (ICSU) organized in Dakar (January 14 to 22, 1965) the “Congress on science teaching and its role in economic progress.” It was attended by 84 members (experts in different scientific disciplines) from nine African countries, four American countries, five Asian countries, and nine European countries. ICMI and UNESCO were represented at this conference. A parallel congress on mathematics teaching in conjunction with the teaching of other sciences was organized by ICMI in collaboration with the National Senegalese Commission of the Teaching of Mathematics (January 13 to 16, 1965). A similar attempt to import ideas and materials from other countries was carried out by UNESCO when in 1966 it was requested to assist the Arab Region [including Egypt] in improving mathematics education. Seminars, syllabus determination, textbook writing sessions, and training sessions for teachers were held, which resulted in a new mathematics course for secondary schools which was implemented in most schools in the Arab States. […] The Arab League Education, Cultural and Science Organization (ALECSO) extended the reform to intermediate level students, aged 13–15 and revised the UNESCO project books. (Jacobsen 1996, p. 1246)

As Malaty (1999) noted, 8 of the 22 authors of the textbooks for senior secondary school were from “outside the Arab world” (p. 238)—mostly the United States and the United Kingdom—and the books were written in English and then translated into Arabic. A lack of prior knowledge of the local context of some of the authors of the materials hindered the development of the project. Issues 2 and 3 of the journal Educational Studies in Mathematics (1978, Vol. 9) entitled “Change in Mathematics Education Since the Late 1950s—Ideas and Realization,” edited by Hans Freudenthal, provided an overview of initiatives carried out in the period of the reform movements in various countries.



Toward New Horizons

Utrecht 1964, Netherlands ICMI organized an international colloquium on “Modern curricula in secondary mathematical education” at Utrecht from December 19 to 225, 1964, under the presidency of Freudenthal. The organizing committee also included Behnke, Choquet, and Moise (Internationale Mathematische Nachrichten, 80, 1965, pp. 3, 8). There were 41 participants from 10 European countries, the United States of America, and Canada. Lectures were delivered by Max Beberman, Castelnuovo, Delessert, Jürgen Dzewas, Félix, Freudenthal, Theodorus Jacobus Korthagen, Krygowska, Servais, Steiner, Owen Storer, Straszewicz, Bryan Thwaites, Roelof Troelstra, Jacobus Van Lint, Leonardus Reinhard Joseph Westerman, and Alexander Wittenberg. There were no proceedings. UNESCO (1967) reported that the themes treated were grouped as follows: The general principles of the reform in the teaching of mathematics; reports of the work done in certain centers concerning the programs developed and

 In Steiner (1965) and UNESCO (1967) the final day is December 23. In the folder dedicated to Freudenthal (inventory number 1831) of the Noord-Hollands Archief the date is December 22. 5

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other activities; report on experiments already completed in modern teaching and projects concerning the setting up of certain fragments of mathematics in line with the new programs. Among other things, it emerged that the reforms of the teaching of mathematics were at different stages in the countries represented, and also that they were conceived and organized in different ways concerning such questions as, for example, the introduction of axioms and the role of geometry. Only the contributions by Castelnuovo (1966) and Wittenberg (1965) were published. The latter’s talk was remarkable for its insight into the teaching and learning of mathematics and for the proposals launched: The necessity of creating university chairs for mathematical education; the urgency of founding an international journal specifically dedicated to mathematics teaching.

Utrecht 1967, Netherlands At the beginning of his ICMI presidency, Freudenthal organized the colloquium “How to teach mathematics so as to be useful,” held in Utrecht from August 21 to 25, 1967. It was sponsored by the International Mathematical Union and the government of The Netherlands. The colloquium was attended by “34 active and 34 passive members” (Internationale Mathematische Nachrichten, 91, 1969, p.  36). Lectures were given by Behnke, Anne Brailly, Delessert, Engel, Trevor Fletcher, Freudenthal, Maurice Glaymann, Brian Griffiths, John M.  Hammersley, Matts Håstad, Murray Klamkin, Krygowska, Robert Cranston Lyness, Pisot, Pollak, Revuz, André Roumanet, Servais, Steiner, Jean Tavernier, Robert Walker, and Gail Young. The texts of the lectures were published in the new journal Educational Studies in Mathematics (1968, Vol. 1). The conference was centered on the “usefulness” of mathematics and its applications. In his intervention Hammersley (1968) openly declares his perplexities about modern mathematics. In the panel organized during the conference, besides applications of mathematics, modern mathematics was widely discussed and the panelists often referred to the concepts of structure, set theory, and logic (Behnke et al. 1968). This colloquium was followed by a meeting of the Executive Committee of ICMI, in which some important issues which shaped the future of ICMI and affected the community of mathematics educators were discussed (Delessert 1967). Freudenthal, President of ICMI, stressed the lack of ­effectiveness of the contributions in the sections dedicated to mathematics education at the ICMs. He proposed to organize an ICMI congress in 1969, with invited lectures and communications. The founding of a new journal more suitable to deal with the problems of mathematics education than L’Enseignement Mathématique, the current organ of ICMI, was also solicited. The two meetings of Utrecht were held under the influence of Freudenthal. More than a follow-up of Royaumont they could be seen as forerunners of the new trend in meetings and communication inaugurated by Freudenthal’s presidency.

Conclusions Dramatis Personae We have seen that the main actors in the story of modern mathematics have been the international bodies CIEAEM, ICMI, OEEC/OECD, UNESCO, and, with a marked national impact, the Study Groups in the United States of America. But this story is, first of all, made by scholars committed to mathematics education.  We refer here to the paper version of the first issue of 1969 of the Internationale Mathematische Nachrichten, which has not been digitalized. The number 91 of the volume was repeated for the first issue of 1970. 6

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While certain meetings were open to teachers and other agents in schools, the voice of mathematicians prevailed. Among the participants and speakers, there were some frequent names. The Bourbaki group was always well represented, in the first place by Choquet and Dieudonné, and sometimes by other Bourbakists (Cartan in Bologna and Schwartz in Bogotá). The Belgians Servais and Papy were frequent participants. Fehr was also very active and had institutional responsibilities in drawing up important reports. ICMI was present with its Presidents Stone and Lichnerowicz, and Behnke, vice-­ President, former Secretary, and President. The lists of participants in the meetings described above provide interesting data about the emergence of women on the international stage of mathematics education. Before the meetings under Freudenthal’s presidency, we identified only four women delivering talks—Castelnuovo, Félix, Krygowska, and Lelong. In the list of participants in the Utrecht meeting of 1964, there are Castelnuovo, Félix, Krygowska, and the French C. Tcherkawsky (Internationale Mathematische Nachrichten, 80, 1965, p. 8). She is probably the secondary school teacher Colette Tcherkawsky, author of textbooks (L’Archicube, 2017, February, 149–152). At the Frascati meeting of 1964, there was the Italian mathematics teacher Maria Albanese, a member of CIIM.  The list of participants in the colloquium of Bucharest (1968) included Castelnuovo, Krygowska, Lina Mancini Proia, Frédérique Papy-Lenger, and Marie-Antoinette Touyarot (teacher at the École Normale de Caen). In the available records of the conferences considered above, we have not found other mentions of women, except those participating as observers of UNESCO or secretaries. We know that there were no women officially participating in the conference of Bogotá (Barrantes and Ruiz 1998). We do not have enough information about feminine participants in the meetings linked to projects addressed to developing countries (Entebbe, Saigon, Dalat, Dakar, and Arab countries). The proceedings of the Echternach meeting in 1965 do not show a list of participants, but the photo published there (Figure 4.3) is an iconic representation of the poor rate of women to men. The scant presence of females is mainly due to the fact that the invited experts or simple participants were manly university professors, a category where women were poorly

Figure 4.3  Colloquium at Echternach in 1965. (Courtesy of Robert Kennes)

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represented. We note that the CIEAEM meetings, which have not been the object of our analysis, had a slightly different trend regarding the presence of women. It was at ICME-1 in 1969 that a clear opening to the “other side of the moon” started, as shown by the informal list of participants in ICME-1 sent to us by Erich Wittmann. This opening increased with each subsequent ICME and in other conferences such as those organized by the International Study Group on the Relations between the History and Pedagogy of Mathematics (HPM) and the International Group for the Psychology of Mathematics Education (PME).

Much Ado About Nothing? The wishes expressed in Utrecht 1967 were fulfilled: In May 1968, the first issue of the journal Educational Studies in Mathematics came out, and from August 24 to 30, 1969, the first ICME was held in Lyon, France (Figure 4.4). UNESCO contributed financially to both initiatives. Another story was beginning. The roots of the ferments in the milieu of mathematics education that marked the 1960s are to be sought in socio-political and cultural factors affecting the 1950s. About socio-political factors, it is illuminating what Howson (December 2021, personal communication to Furinghetti) claims about the School Mathematics Project, which started in 1961: In some ways it [the School Mathematics Project] was generated by the great part mathematics had played in the war, code-breaking, the beginnings of operational research, …. There was seen to be a need to update what was taught in schools to meet new demands for mathematical knowledge. […] significantly, the money to launch this [the School Mathematics Project] came almost entirely from industry and commerce where the need for people with advanced mathematics was increasing.

As to the cultural factors, there was a new approach to the problems of mathematics education which led to new projects and new working groups, for example, in the USA, the University of Illinois Committee on School Mathematics (UICSM) first and the School Mathematics Study Group (SMSG) later, and in Europe CIEAEM. The chapters in the second part of this book show the impact of modern mathematics in various countries. At the end of the 1960s, the decline of modern mathematics was beginning. Its definitive end was sanctioned in 1972 by René Thom’s talk at ICME-2 entitled “Modern Mathematics: Does it

Figure 4.4  ICME-1, Lyon: Krygowska, Steiner, Papy-Lenger, Zoltán Dienes. (Courtesy of Gert Schubring)

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exist?” (Thom 1973), and later by Morris Kline’s book Why Johnny Can’t Add: The Failure of the New Math (Kline 1973). However, not all was lost and there have been initiatives, which still exist, which owe their birth to modern mathematics. The most remarkable of these initiative are: The French Instituts de Recherche sur l’Enseignement des Mathématiques in 1969 (IREMs) [Mathematics Education Research Institutes], the German Zentrum für Didaktik der Mathematik (ZDM) [Centre for didactics of mathematics] in Karlsruhe in 1968, and Institut für Didaktik der Mathematik (IDM) [Institute for the Didactics of Mathematics] in Bielefeld in 1972. The establishment of IREMs sprang from the work of the Commission Lichnerowicz, a commission born to promote the reform of modern mathematics when it was realized that any discussion on mathematics teaching had to be based on a solid program of mathematics teacher education and research. As stressed by Maurice Glaymann, the IREMs were bodies independent from the Commission (Artigue 2008). This independence allowed them to survive the Commission Lichnerovicz, which ended its work in 1973. About the ZDM and the IDM, their creation was a kind of reaction to modern mathematics stimulated by the observation that renewal in mathematics teaching was above all a matter of methods of teaching and learning, rather than a matter of mathematical content (Steiner, personal communication to Menghini, 1989). Despite the failure, at least in its most radical form, of modern mathematics in various countries, we agree with Kilpatrick (2012) that “By the time the new math era ended […] everyone concerned with school mathematics had a much better sense of what was going on around the world” (p. 569). And indeed, it was the Royaumont Seminar that, by promoting communication in the discussion of mathematics education, generated the driving force behind most initiatives in the following decades.

References Artigue, M. (2008). Interview with Maurice Glaymann. Retrieved November 1, 2020, from https://www.icmihistory. unito.it/clips.php Artin, E. (1963). Les points de vue extrêmes sur l’enseignement de la géométrie [The extreme points of view on geometry teaching]. L’Enseignement Mathématique, s. 2, 9, 1–4. Barrantes, H., & Ruiz, A. (1998). The history of the Inter-American Committee on Mathematics Education (Bilingual Spanish and English edition). Bogotá, Colombia: Academia Colombiana de Ciencias Exactas, Físicas y Naturales. Behnke, H., Choquet, G., Dieudonné, J., Fenchel, W., Freudenthal, H., Hajós, G., & Pickert, G. (1960). Lectures on modern teaching of geometry and related topics. Aarhus, Denmark: Matematisk Institut (Aarhus Universitet), Elementaer Afdeling. Nr. 7. Behnke, H., Hammersley, J. M., Krygowska, A. Z., Pollak, H., Revuz, A., Servais, W., Sobolev, S., & Freudenthal, H. (1968). Panel discussion. Educational Studies in Mathematics, 1, 61–79. Bernet, T., & Jaquet, F. (1998). La CIEAEM au travers de ses 50 premières rencontres [The CIEAEM through its first 50 meetings]. Neuchâtel, Switzerland: CIEAEM. Bruner, J. (1960). The Process of Education. Cambridge, MA: Harvard University Press. BUMI. (1962). Il convegno di Bologna promosso dalla Commissione internazionale dell’insegnamento matematico [The Bologna conference promoted by the International Commission on Mathematical Instruction]. Bollettino della Unione Matematica Italiana, s. 3, 17, 199–214. Cartan, H. (1963). Réflexions sur les rapports d’Aarhus et Dubrovnik [Reflections on the Aarhus and Dubrovnik reports]. L’Enseignement Mathématique, s. 2, 9, 84–90. Castelnuovo, E. (1966). Un enseignement modern des mathématiques dans le 1er cycle secondaire [Modern teaching of mathematics in lower secondary education]. L’Enseignement Mathématique, s. 2, 12, 195–199. Choquet, G. (1964). L’enseignement de la géométrie [The teaching of geometry]. Paris, France: Hermann. Christiansen, B. (1978). The cooperation between ICMI and UNESCO. ICMI Bulletin, 10, 4–10. Conférence Echternach. (1965). Les répercussions de la recherche mathématique sur l’enseignement. Textes originaux des conférences faites au séminaire organisé par la C.I.E.M. à Echternach (G.-D. de Luxembourg) été 1965 [The impact of mathematical research on teaching. Original texts of the conferences given at the seminar organized by the C.I.E.M. in Echternach (G.-D. of Luxembourg) summer 1965]. Echternach, Luxembourg: Institut Grand-Ducal Section des Sciences Naturelles, Physiques et Mathématiques.

4  The Royaumont Seminar as a Booster of Communication and Internationalization in the World…

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De Bock, D., & Vanpaemel, G. (2015). Modern mathematics at the 1959 OEEC Seminar at Royaumont. In K. Bjarnadottir, F. Furinghetti, J. Prytz, & G. Schubring (Eds.), “Dig where you stand” 3. Proceedings of the third International Conference on the History of Mathematics Education (pp. 151–168). Uppsala, Sweden: Department of Education, Uppsala University. De Finetti, B. (1965). Programmi e criteri per l’insegnamento della matematica alla luce delle diverse esigenze [Programs and criteria for teaching mathematics in the light of different needs]. Periodico di Matematiche, s. 4, 43, 119–143. Delessert, A. (1967). Compte rendu de la séance de la C.I.E.M. tenu à Utrecht, le 26 Août 1967 [Minutes of the ICMI meeting held in Utrecht on August 26, 1967]. L’Enseignement Mathématique, s. 2, 13, 243–246. Dieudonné, J. (1961). New thinking in school mathematics. In OEEC, New thinking in school mathematics (pp. 31–46). Paris, France: OEEC. Dieudonné, J. (1964). Algèbre linéaire et géométrie élémentaire [Linear algebra and elementary geometry]. Paris, France: Hermann. Fehr, H. F. (1961). The significance of this report. In OEEC, New thinking in school mathematics (pp. 207–210). Paris: OEEC. Félix, L. (1986). Aperçu historique (1950–1984) sur la Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques (CIEAEM). 2ème édition revue et augmentée [Historical overview (1950–1984) on the International Commission for the Study and Improvement of Mathematics Teaching (CIEAEM). 2nd revised and expanded edition]. Bordeaux, France: IREM de Bordeaux. Retrieved December 31, 2017, from http://math. unipa.it/~grim/cieaem_files/CIEAEM_histoire_FLucienne_1985.pdf Freudenthal, H. (1963). Enseignement des mathématiques modernes ou Enseignement moderne des mathématiques [Teaching modern mathematics or modern teaching of mathematics]. L’Enseignement Mathématique, s. 2, 9, 28–44. Furinghetti, F. (2003). Mathematical instruction in an international perspective: The contribution of the journal L’Enseignement Mathématique. In D. Coray, F. Furinghetti, H. Gispert, B. R. Hodgson, & G. Schubring (Eds.), One hundred years of L’Enseignement Mathématique (Monographie 39 de L’Enseignement Mathématique), pp. 19–46. Genève, Switzerland: L’Enseignenement Mathématique. Furinghetti F., & Giacardi, L. (to appear). ICMI in 1950s and 1960s: Reconstruction, settlement, and “revisiting mathematics education”. In F. Furinghetti & L. Giacardi (Eds.), The International Commission on Mathematical Instruction 1908–2008: People, events, and challenges in mathematics education. Cham, Switzerland: Springer. Gattegno, C., Servais, W., Castelnuovo, E., Nicolet, J.  L., Fletcher, T.  J., Motard, L., Campedelli, L., Biguenet, A., Peskette, J. W., & Puig Adam, P. (1958). Le matériel pour l’enseignement des mathématiques [Materials for the teaching of mathematics]. Neuchâtel, Switzerland: Delachaux et Niestlé. Gattegno, C. (1988). Reflections on forty years of work on mathematics teaching. For the Learning of Mathematics, 8(3), 41–42. Gerretsen, J. C. H., & de Groot, J. (Eds.). (1954–1957). Proceedings of the International Congress of Mathematicians. Groningen, The Netherlands: E. P. Noordhoff N. V./ Amsterdam, The Netherlands: North-Holland. Goals for school mathematics. (1964). The American Mathematical Monthly, 71(2), 196–199. Hammersley, J. M. (1968). On the enfeeblement of mathematical skills by modern mathematics and by similar soft intellectual trash in schools and universities. Educational Studies in Mathematics, 1, 17. The full text is published in 1968, Bulletin of the Institute of Mathematics and its Applications, 4, 66–85. Hodgson, B. R. (2008). Interview with Geoffrey Howson. Retrieved November 1, 2020, from https://www.icmihistory. unito.it/clips.php Hodgson, B.  R. (2009). ICMI in the post-Freudenthal era: Moments in the history of mathematics education from an international perspective. In K.  Bjarnadóttir, F.  Furinghetti, & G.  Schubring (Eds.), “Dig where you stand”: Proceedings of the conference on “On-going research in the History of Mathematics Education” (pp.  79–96). Reykjavik, Iceland: The University of Iceland. Howson, A. G. (1984). Seventy-five years of the International Commission on Mathematical Instruction. Educational Studies in Mathematics, 15, 75–93. Hungarian National Commission for UNESCO. (1963). Report on the work of the international symposium on school mathematics teaching. Budapest, Hungary: Akadémiai Kiadó. Jacobsen, E. (1996). International co-operation in mathematics education. In A.  J. Bishop, K.  Clements, C.  Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 1235–1256). Dordrecht, The Netherlands/Boston, MA/London, United Kingdom: Kluwer. Kemeny, J. G. (1964). Which subjects in modern mathematics and which applications of modern mathematics can find a place in programmes of secondary school instruction?. L’Enseignement Mathématique, s. 2, 10, 152–176. Kilpatrick, J. (2012). The new math as an international phenomenon ZDM—Mathematics Education, 44, 563–571. Kline, M. (1973). Why Johnny can’t add: The failure of the new math. New  York, NY: St. Martin’s Press Random House. Malaty, G. (1999). The third world mathematics education is a hope for the world mathematics education development in the 21st century. In A. Rogerson (Ed.), Proceedings of the International Conference Mathematics Education into the 21st Century: Societal challenges, Issues and Approaches (pp. 231–240). Cairo, Egypt: The mathematics education for the future project.

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OECD. (1964). Mathematics to-day. A guide for teachers. Paris, France: OECD. OEEC. (1961a). New thinking in school mathematics. Paris, France: OEEC. French edition: Mathématiques nouvelles. Paris, France: OECE. OEEC. (1961b). School mathematics in OEEC countries—Summaries. Paris, France: OEEC. OEEC. (1961c). Synopses for modern secondary school mathematics. Paris, France: OEEC. Piaget, J., Beth, E.  W., Dieudonné, J., Lichnerowicz, A., Choquet, G., & Gattegno, C. (1955). L’enseignement des mathématiques [The teaching of mathematics]. Neuchâtel, Switzerland: Delachaux et Niestlé. Powell, A. B. (2007). Caleb Gattegno (1911–1988): A famous mathematics educator from Africa? Revista Brasileira de História da Matemática, Especial n° 1–Festschrift Ubiratan D’Ambrosio, 199–209. R. G. (1964). Seminario matematico internazionale Villa Falconieri – Frascati, 8–10 ottobre 1964) [International mathematical seminar (Villa Falconieri – Frascati, October 8–10, 1964)]. Archimede, 16, 314–330. Report of the Executive Committee to the national adhering organizations. Covering the period April 21, 1955–May 31, 1956. (1956). Internationale Mathematische Nachrichten, 47–48, 1–16. Revuz, A. (1965). Pour l’enseignement de la géométrie, la route est tracée [For the teaching of geometry, the road is drawn]. Mathematica & Paedagogia, 28, 74–77. Schubring, G. (2008). The origins and the early history of ICMI. International Journal for the History of Mathematics Education, 3(2), 3–33. Schubring, G. (2014a). The original conclusions of the Royaumont Seminar 1959, edited and commented by Gert Schubring. International Journal for the History of Mathematics Education, 9(1), 89–101. Schubring, G. (2014b). The road not taken—The failure of experimental pedagogy at the Royaumont Seminar 1959. Journal für Mathematik-Didaktik, 35(1), 159–171. Steiner, H.-G. (1965). Internationales Kolloquium in Utrecht ober modernen mathematischen Unterricht an der höheren Schule [International colloquium in Utrecht on modern mathematical education at high school]. Mathematisch-­ physikalische Semesterberichte, n. s., 12(1), 127–128. Stone, M. H. (1961). Reform in school mathematics. In OEEC, New thinking in school mathematics (pp. 14–29). Paris, France: OEEC. Stone, M. H. (1963). Le choix d’axiomes pour la géométrie à l’école [The choice of axioms for geometry at school]. L’Enseignement Mathématique, s. 2, 9, 45–55. Straszewicz, S. (1964). Relations entre l’arithmétique et l’algèbre dans l’enseignement des mathématiques pour les enfants jusqu’à l’âge de quinze ans [Connections between arithmetic and algebra in the mathematical instruction of children up to the age of 15]. L’Enseignement Mathématique, s. 2, 10, 271–293. Teodorescu, N. (Ed.). (1968). Colloque International UNESCO. Modernization of mathematics teaching in European countries. Bucharest, Romania: Editions didactiques et pédagogiques. Thom, R. (1973). Modern mathematics: Does it exist? In A. G. Howson (Ed.), Developments in mathematical education. Proceedings of the Second International Congress on Mathematical Education (pp.  194–209). Cambridge, United Kingdom: University Press. Todd, J. A. (Ed.). (1960). Proceedings of the International Congress of Mathematicians. Cambridge, United Kingdom: Cambridge University Press. UNESCO. (1967). New trends in mathematics teaching—Tendances nouvelles de L’enseignement des mathématiques. Vol. 1. Paris, France: UNESCO. Vanhamme, J. (1991). Un peu d’histoire/A short historical background. In A. Warbecq (Ed.), Proceedings of the 41st CIEAEM Meeting (Role and conception of mathematics curricula, Bruxelles, 1989) (pp. V–XV). Bruxelles: Le Comité organisateur. Vanpaemel, G., & De Bock, D. (2019). New Math, an international movement? In E.  Barbin, U.  T. Jankvist, T.  H. Kjeldsen, B. Smestad, & C. Tzanakis (Eds.), Proceedings of the Eighth European Summer University on History and Epistemology in Mathematics Education (pp. 801–812). Oslo, Norway: Oslo Metropolitan University. Weaver, J. F. (1965). African mathematics program The Arithmetic Teacher, 12, 472–480. Williams, G. A. A. (1976). The development of a modern mathematics curriculum in Africa. The Arithmetic Teacher, 23, 254–261. Wittenberg, A. (1965). Priorities and responsibilities in the reform of mathematical education: An essay in educational meta-theory. L’Enseignement Mathématique, s. 2, 11, 287–308.

Part II

Implementation of the Reform Around the World

Chapter 5

The Modern Mathematics Movement in France: Reforming to What Ends? The Contribution of a Cross-Over Approach to Modernity Hélène Gispert

Abstract  The reform of mathematics teaching in France in the 1960s and 1970s was one of several reforms which affected the disciplines of primary and secondary education at the time, while the school system structures were also profoundly modified. It was, however, in its course, scope, successes, and difficulties, a reform different from the others, which was emblematic of the period. Considering the reform of modern mathematics within the global dynamics of redefinition of the curricula, I will place it at the crossroads of different ambitions and requirements of modernity to better grasp its characteristics and aims. Firstly, I will deal with the new aims assigned to the French education system after World War II within the framework of a project of cultural, social, and economic modernity of the country. Then, I will examine the reform movement of the teaching of mathematics in relation to that of français and then to that of science. Finally, I will focus on a particular moment of the reform of mathematics itself and will return to its ambition of modernity and its contradictions in the socio-economic context of France. Keywords  1950–1970s · Cultural, social, and economic modernity · Democratization · Educational paradigm · Fouchet reform · France · Haby reform · La mathématique · Lichnerowicz Commission · Literary teaching · Modern mathematics · Orientation · REDISCOL · Reform · Science · Secondary education · Selection · Structural educational reforms

Introduction The reform of mathematics teaching in France in the 1960s and 1970s was one of several reforms that affected the disciplines of primary and secondary education at the time, while the school system structures were also profoundly modified. It was, however, in its course, scope, successes, and difficulties, a reform different from the others, which was emblematic of the period. In this chapter, I propose to draw on the work of the multidisciplinary research project REDISCOL–Réformer les

H. Gispert (*) UR Études sur les Sciences et les Techniques, Université Paris Saclay, Paris, France e-mail: [email protected]

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disciplines scolaires: Acteurs, contenus, enjeux, dynamiques (années 1950–1980)1—which consisted of a comparative and cross-over approach to the development and reform of several school disciplines in France. This work made it possible to consider the reform of modern mathematics within the global dynamics of redefinition of the curricula, to place it at the crossroads of different ambitions and requirements of modernity, and to better grasp its characteristics and aims. Thus, broadening the perspective in relation to previous studies,2 I will take up results and analyses concerning the reform of modern mathematics published by d’Enfert and Lebeaume (2015),3 which are too little known on the international scene because published in French. This will allow me to answer the request made to me for this chapter, to provide a comprehensive overview of the French modern mathematics movement in its socio-economic context.4 Firstly, I will deal with the new aims assigned to the French education system after World War II: To democratize, orientate, and select within the framework of a project of cultural, social, and economic modernity of the country. These aims have affected the roles and contents of the various disciplinary teachings, their respective positions and legitimacy, and the power relations between them. In this light, I will examine, in a second section, the reform movement of the teaching of mathematics in relation to that of français and then, in a third one, to that of science; finally, in a fourth section, I will focus on a particular moment of the reform of mathematics itself and will return to its ambition of modernity and its contradictions in the socio-economic context of France.5



New Goals Assigned to the French Education System: Democratize, Orientate, Select for a Cultural, Social, and Economic Modernity

The “reform of modern mathematics” was officially launched in October 1966 with the creation of a ministerial committee in charge of thinking about new mathematics programs inspired by so-called modern mathematics,6 programs which were to be implemented progressively from 1969. This reform can only be fully understood if it is placed in the context of the upheavals that the entire school system has undergone since the 1950s, particularly in post-elementary education. Secondary education (for 11–17-year-olds) faced a new situation. It had to move from educating a relatively narrow elite, giving primacy to classical studies—which had been its purpose since its creation at the beginning of the

 This project (Reforming school subjects: Actors, content, issues, dynamics, 1950–1980), led by Renaud d’Enfert, brought together between 2007 and 2013 more than 20 researchers specialized in the teaching of français (Throughout the text, I have chosen to keep the original French terms, written in italics, when I have not found an equivalent expression in English. Their meaning will be defined at their first occurrence. The discipline français here refers to the teaching of French language and literature), mathematics, technology, physical sciences, natural sciences, and physical education. I would like to thank Renaud d’Enfert who agreed to read a first version of this text and whose comments were so valuable. 2  The papers devoted to the reform of mathematics teaching in France that have appeared up to now have been mainly concerned with the study of mathematics teaching alone. Let us quote, for works in English: Barbin (2012), Gispert (2014), and Gispert and Schubring (2011). 3  This book is the third of those published in the framework of the REDISCOL research and on which I will rely here: d’Enfert and Kahn (2010), d’Enfert and Kahn (2011), and d’Enfert and Lebeaume (2015). 4  I will limit myself in this chapter to post-elementary education, i.e., for children aged 11–17. 5  I thank Renaud d’Enfert and Clémence Cardon-Quint for agreeing to let me take up elements of our joint chapter (Cardon-Quint et al. 2015), Muriel Guedj and Pierre Savaton for doing the same for our chapter (Gispert et al. 2015), and Philippe Alix and Renaud d’Enfert for our chapter (Alix et al. 2015). This is also the case for Renaud d’Enfert for our paper (d’Enfert and Gispert 2011). 6  That is, in France, inspired by the work and conceptions of Bourbaki with in particular the new central role of the notion of structure in mathematics, which was the core of what was called “modern mathematics.” 1

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nineteenth century7—to educating a growing proportion of the population, mainly in modern education without Latin; the latter became a rival to classical education, which had hitherto been the ­hallmark of the prestige attached to a secondary education reserved mainly for this urban social and learned elite. At the same time, it had to deal with new missions assigned by the State. It was supposed to ensure equality of students in terms of education, regardless of their social origin, and to meet the needs of the economy.

The 1950s—Changing Educational Paradigm: The Reversal of Priorities During the 1950s, after the war, several factors called into question the aims of secondary education. The baby boom caused an increase in the number of students to be educated, with an influx of students who, because they belonged to the working classes, were directed toward sections without Latin. In addition, new modern sections were created in the upper secondary school (students aged 15–18).8 In this early period, the discussions on modernization of secondary education, as well as the measures taken, focus on the distribution of the different disciplines and their place in the school curriculum. Which disciplines are best suited to educate the growing number of young people who pursue secondary education, to meet the new challenges of the time? What new balance should be struck between general knowledge and specialization? What general culture should be promoted? How to define modern humanities, scientific humanities, and technical humanities? Such a vocabulary inherited from the nineteenth century cannot, however, conceal the evolution of arguments and priorities.9 A vast movement of opinion is taking shape beyond the academic world, affecting the circles of administrators and government executives, and of industrialists in favor of strengthening scientific and technical education. Formative virtues of science are not the only ones promoted. So are the economic needs, which are better understood through state-commissioned planning studies and statistical data on the labor force, to achieve the modernization of the country which is seen as a necessity. This new priority found its first concrete expression in 1957 in changes in timetables, with Latin having to be given hours to mathematics in the premier cycle10 classes. Much more than a victory of the “moderns” over the “classics,” it was a change in the educational paradigm that took shape at the end of the 1950s: It was no longer a question of appreciating, as in the inter-war period, the respective cultural value of the various courses of secondary education; it was a question of evaluating the relevance of each discipline in the light of the needs of higher education and economy. A reversal of priorities is taking place.

 In 1950, 5% of an age group held the baccalauréat, which is the final exam at the end of the 7 years of secondary education; in 1960, this percentage reached 11%, and it was 20% in 1968. An overview of the evolution of the French school system in the 19th and 20th centuries and the place and role of mathematics education can be found in Gispert (2014). It should be noted that, in addition to the secondary education (provided by the lycées and collèges) discussed here, there was a prolonged schooling of the working classes provided by other types of establishments, the écoles primaires supérieures (EPS) until 1940 and the cours complémentaires until 1959. These two types of schooling were part of two distinct and parallel systems (in French, ordres scolaires): The primary system (with the EPS and cours complémentaires), and the secondary system (with lycées and collèges). Each of these ordres scolaires had specific teachers. 8  Secondary education is divided into lower and upper secondary school. The French expressions, which we will use, are: Premier cycle (for students aged 11–14) and second cycle (15–17) which ends with the baccalauréat. 9  For specific sources supporting the developments in this section and the next two, see the three chapters of d’Enfert and Lebeaume (2015) mentioned in Note 5. All these texts are online. 10  See Note 8. 7

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 he 1960s—Thinking Democratization and Modernization: Structural T Reforms and Disciplinary Issues The second period, which corresponds roughly to the 1960s, begins with two major structural reforms11 undertaken by the new regime then setting up in France, the Fifth Republic of General de Gaulle. They were designed above all to better adapt student flows to the needs of economy. They establish for all children from ages 12 to 16 compulsory education in a more or less complicated system of ­various écoles moyennes gathered—and this was new—in a unique premier cycle (the first 4 years) of the secondary system.12 Secondary education, mostly its four first years, had then a new public and must have new aims, providing education to children whose educational and social future would continue differently: By long and general studies on the one hand, and by practical studies or apprenticeship on the other hand.13 The diversity of these middle schools, largely adapted to the future planned for the students, was more a result of quantitative growth without any strong qualitative social change—more a factor of “massification”—than a factor of “democratization.” The situation is not without concern in relation to the labor force targets set by the State. At the beginning of the 1960s, the proportion of young people with a literary baccalauréat from the Philosophy series exceeded 40% and tended to increase, whereas the proportion of those with a scientific baccalauréat from the Mathematics series barely exceeded 25%, at the same level as those from the Experimental Sciences series. In 1965, the reform launched by Minister Fouchet brought about this major reversal of the priorities that, as we have seen, had begun to emerge in the 1950s. The traditional distinction between classical and modern sections at the entrance to the second cycle was replaced by a distinction between scientific, literary, and technical sections. Mathematics replaced Latin as the discriminating subject, decisive in the orientation of students. The reform established the new superiority of the scientific courses in the second cycle, the translation in the school of a claimed economic modernity. Once these structural reforms were underway, the debate on democratization and cultural and social modernity shifted to the content and methods of teaching: Did they have a role to play in the process of democratizing the school? Various studies, including those of Pierre Bourdieu, popularized by the publication of Les Héritiers (Bourdieu and Passeron 1964), have drawn an implacable conclusion: The social inequalities observed in schooling at the end of the 1950s were not only the result of the structure of the education system, but they also reflected the unequal success of students according to their social background. In the view of sociologists, not all school subjects play the same role in this process, particularly, as we will come back to in the next section: français and mathematics. During the 1960s, after the structural reforms were adopted, three ministerial commissions for disciplinary reform were created, for mathematics in 1966, for français in 1969, and for physical sciences and technology in 1970, all three having been claimed for more or less time by the corresponding associations of specialists.14 Institutional reforms—which modified the economy of the system of disciplines—as well as the theories of sociologists and psychologists, conditioned the reformers. The challenge for each discipline was to try to embody, through the renovation of its contents and methods, the ambitions of cultural, social, and economic modernity. The interests were sometimes contradictory, the ambitions openly competing, as we shall see in the following sections, especially in front  These were the Berthoin reform in 1959 and the Fouchet reform in 1963 followed, for higher education, by the creation in 1966 of new higher education structures, the university institutes of technology. 12  See below. Until then, compulsory education was available until the age of 14 and students were educated in the two separate primary and secondary school systems (see Note 7). 13  These latter were until then educated in the primary system. 14  These are the commissions known as Lichnerowicz for mathematics, Emmanuel for français, and Lagarrigue for physical sciences and technology, named after their chairman (see Alix et al. 2015). We return to these different commissions in the following sections. 11

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of the then uncontested domination of the mathematics reform movement. Indeed, the different spheres of mathematical, political, administrative, and economic actors, consensually promoted “modern mathematics,” as the universal language, socially neutral, of all the other fields of knowledge.

The 1970s—Disillusionment and Controversy The situation changed in the 1970s, when the reform euphoria was followed by disillusionment and controversy. Some disciplinary reforms got bogged down, such as those in français or the physical sciences and technology, while others, e.g., in mathematics, did not have the expected effects. Selection by mathematics throughout secondary education, although desired and promoted, became a societal problem. Modern, formal, and abstract mathematics caused difficulties for too many students, especially at the premier cycle level in the lower social classes. In addition, a political and cultural critique of science, though marginal, questioned the aims and applications of scientific research in society and the associated criteria of modernity. Finally, at the institutional level, the last stage of the structural reform of secondary schooling was put in place in 1975 under Minister Haby. The establishment of the collège unique15 without any specific courses of studies—which was supposed to replace the diversity of middle schools—completes the process of reform of the premier cycle of secondary schooling which had begun 15 years before. This collège unique aroused quasi-unanimous opposition, including that of unions and specialists’ associations. The fact that this reform did not have the means to bring about any real democratization of the educational system was one of the arguments put forward by some of them. In spite of the massification of access to secondary studies, they denounced this reform as something that would lead to exacerbating the social inequalities at school. In terms of teaching content, the question arose in each discipline to which extent it would be possible to meet the challenge of teaching all students the same thing during the first 4 years of secondary school, taking into account the heterogeneity of students and their curricula. The debates on the teaching programs were then inextricably linked to the controversies on what this collège unique could and should be. The Haby reform undertook new programs for all school subjects, but the new process to realize the reform programs had nothing to do with what had been done in the 1960s. Directly driven by the Ministry, the reform of each disciplinary content had to be submitted to general instructions that would fix priority goals. At the secondary level, the ministerial vision of the 1970s broke with that of the 1960s. The aim was now to reduce the weight of selection through mathematics and to enhance the value of non-­ scientific sections. In the 1970s, both politicians and experts no longer envisaged aligning student flows with the structure of the labor market. It is thus within the framework of the renewal of the cultural, social, and economic project that the school must carry, that the teaching of mathematics has been required to modernize, as well as the other disciplines. It is therefore both the overall logic and the logic specific to each discipline, including mathematics, that we must now grasp in order to better understand the stakes and pitfalls of the modernity claimed by the mathematics reform movement during these decades.

 The word collège meant in France, at this period, any school that trained students during the 4 years of the premier cycle of secondary studies. What was new with this last reform, is that all these schools which, hitherto, in the frame of the first reforms of Berthoin and Fouchet, had various status, got with the Haby reform a unique status; hence the expression collège unique. All the students of this 11–14-year-old level were supposed to be schooled in the unique and same way. With this Haby reform the status and role of lycées changed; they were to train only second cycle students (15–17-year-olds). 15

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 Mathematics and Lettres16: Competition and Emulation of Modernizing Ambitions From the mid-1960s onward, (modern) mathematics took over from Humanities as the dominant discipline in the school system. As a consequence, the task of orienting and selecting students throughout secondary education fell to mathematics, and mathematics alone, which accumulated in the 1960s all the symbolic advantages of democratization, modernity, and scientificity. Latin, which had played this role until then, was definitively disqualified for this purpose. In the period studied here, Humanities and mathematics form an elementary dynamic system at the heart of the more complex system formed by all the disciplines of secondary education; it affects and partly explains, the dynamics of each of these two.

Disciplinary Issues of Democratization As we have seen, the issues of democratization are a particularly sensitive point in the reform ambitions of the period, despite the limits of government structural reforms in this area. The modernization of the teaching of mathematics and the almost concomitant renovation of the teaching of français (Cardon Quint 2015) respond to radically different injunctions as to the role of their disciplinary content in the democratization process. Mathematics, and science in general, were considered more “democratic” than the literary disciplines17 without this belief being substantiated by any study at the time. The vision of a supposed social transparency of mathematics acquired all the more weight as the supporters of a reform of mathematics education presented “modern mathematics” as “mathematics for all.” Since the early 1950s, mathematicians and pedagogues, influenced by Jean Piaget, have promoted the idea of a correspondence between mathematical structures  and mental structures of the child (d’Enfert 2011; Gispert 2010). Provided it is properly taught, “modern mathematics” is therefore theoretically accessible to all. This idea became obvious as the mathematics curricula were gradually modernized. From the end of the 1950s, with the development of new pedagogical methods in the French school system, adapted to these “modern mathematics,” the discourse on the “maths bump” was more and more frequently challenged, both by the teachers of mathematics gathered within their association, l’Association des professeurs de mathématiques de l’enseignement public (APMEP) [Association of Mathematics Teachers of Public Education], and in official circulars: Mathematics, this mathematics, is for all students. We shall see below in the last section, that, contrary to Pierre Bourdieu’s analyses, this discourse is not primarily social in nature: It is above all psychological. The student is here an ideal figure disconnected from the concrete conditions that his or her schooling may take. However, the confrontation with the school realities of compulsory education, in the early 1970s, will in turn mark mathematics with the seal of social selection, as I have said. Favoring abstraction and a certain formalism, modern mathematics appeared to be more accessible to students from privileged backgrounds. The reform was thus accused of increasing achievement gaps between students according to their social origin (IREM de Lille 1971). But the fact remained that the reformers’ vision of modern mathematics  Lettres here is a general term for (1) French language and literature (lettres modernes); (2) French, Latin, and Greek languages and literatures (lettres classiques). We deal here only with general secondary education—which in the 1960s only enrolled two out of three children aged 11–14—excluding the field of vocational education, which has nevertheless been studied in the context of REDISCOL. See Lopez and Sido (2015) on the teaching of mathematics and français— the humanities were not taught there—in short, technical education. 17  A position defended among others by Pierre Bourdieu, see Cardon-Quint et al. (2015) and Gispert et al. (2015). 16

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c­ onferred on mathematics the aura of a universal language, presented and perceived as socially neutral, as opposed to the French language of the “educated bourgeoisie of the Paris region.” The new selection through mathematics, introduced by the Fouchet reform during the 1960s, therefore appeared socially legitimate. On the other hand, from the 1950s, (classical) literature was criticized for the socially marked nature of Latin, whose teaching was not offered to students in the Cours complémentaires (11–15-year-­ olds),18 institutions that played a central role in the prolonged schooling of working-class children in the 1950s and early 1960s. Maintaining classical education (with Latin) alongside modern education from the first year of secondary school would perpetuate a de facto segregation between bourgeois and working-class schoolings. But Latin is not the only one at issue. In the 1960s, following Pierre Bourdieu’s analyses popularized in the book Les Héritiers (Bourdieu and Passeron 1964), many people19 emphasized the specific link between linguistic abilities and social origin and criticized the school selection process centered on the mastery of language. It should be noted that the mechanisms denounced here in relation to literary teaching seemed to operate identically in the teaching of modern mathematics in the 1970s with the implementation of the reform (see below). But in the 1960s, it is the valorization of literary teaching, the study of the classics of French literature, already considered to be contrary to the economic interests of the nation, which were seen as an obstacle to the democratization of education because of its social effects.

General Education for All and Specialization The tension between general culture and specialization was present, as we have seen, throughout the period for all disciplines. However, it was argued and negotiated differently for lettres and mathematics, the benefit being again for mathematics. In the case of lettres, the reformers faced particular difficulties and were torn between orientations that are difficult to reconcile concerning the place of literary language (and literature) in the teaching of français.20 However, there are many common points between them, including the attractiveness of structuralism, inspired by the mathematical notion of structure promoted by modern mathematics; the reformers saw in structuralism the quest for a scientific rigor likely to renew and democratize literary studies. The success of the very young Association française des professeurs de français (AFPF) [French Association of Teachers of Français], created in 1967, which was the spokesperson and meeting place for the supporters of the renovation, gave them a semblance of unity as well as an institutional visibility. However, the debate was lively: Was it a question of working to improve the language skills of students, for the benefit of all other disciplines, or was it a question of modernizing literary teaching in specific content? The aspirations were not all concordant and it seemed difficult to reach a balance between general education—the only avowed and avowable ambition for teachers in français—and the temptation to specialize and technicalize the literary teaching, which was what the development of university research called for. The teaching of français could not be an initiation to a culture, whereas other disciplines would first of all provide access to knowledge. These two components, appearing to be difficult to reconcile in the teaching of français, were in contrast inseparable in the reform of modern mathematics. The orientation of the reform movement,

 See Note 7.  Such arguments were even used in 1968 by Edgar Faure, Minister of Education, to justify the abolition of Latin in the first 2 years of secondary education. 20  Specific references can be found in Cardon-Quint et al., (2015). 18 19

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built up since the beginning of the 1950s, in particular within the APMEP,21 was clear, ambitious, and shared: Mathematics is a universal language. The ministerial reform commission, the Lichnerowicz Commission, proclaimed this loud and clear in its Preliminary Report: Mathematics [La mathématique] plays a privileged role in the intelligence of what we call the real, both physical and social real. Our mathematics [Notre mathématique] secretes, by its very nature, the economy of thought and, thereby, alone allows us to classify, to dominate, to synthesize sometimes, in a few brief formulas, a knowledge which, without it, would end up resembling some annoying encyclopedic dictionary infinitely heavy.22 (Commission ministérielle 1967, p. 246)

It is indeed thanks to the effectiveness of the notion of structure, henceforth placed at the heart of mathematical activity itself, that mathematics is thought of as the common language and thinking tool not only for the physical sciences or the sciences for engineers but also for a large part of the biological and medical sciences, economics and human sciences. These were the principles that inspired the new programs, which were gradually introduced from the beginning of the 1969–1970 school year, both in the premier and second cycles of secondary education and in primary education (students aged 6–11 years). In mathematics, therefore, for the reformers, there was neither tension nor contradiction—until the early 1970s once more (see below, the last section)—between the content selected for its contribution to the general training of the mind and the knowledge which would be necessary for the future scientist, the future mathematician.

Dissimilar Dynamics of Reform In all respects, the reform of modern mathematics was a model for the advocates of the renovation of the teaching of français. The Association française des professeurs de français (AFPF), which spearheaded the renovation movement in français, was explicitly inspired by the approaches taken by the APMEP: The Charte de Chambéry, published by the APMEP in 1968 (APMEP 1968), was followed by the Manifeste de Charbonnières, published by the AFPF in February 1970 (AFPF 1970). Similarly, AFPF demanded that the Ministry set up a reform commission on the model of the Lichnerowicz Commission; it was created in July 1969 and set up in March 1970. But the dynamics of the reforms and the work of the respective commissions were far from being the same. The Lichnerowicz Commission took only 2 months to set out its guidelines in its “Preliminary Report,” and from its very first meetings, it began work on new curricula, which were officially implemented barely 3 years later, a year behind schedule. The Emmanuel Commission, charged by the Minister to reflect on the “crisis of the school” and the “crisis of culture,” and to develop curricula, only partly responded to this request. It refused to design programs without having developed a doctrine. It functioned primarily as a forum whose work was organized in the form of colloquia, discussing cultural as well as academic issues, and its work seemed to have little influence on government decisions. It took 2 years for it to produce its first policy text and it was elsewhere, within the General Education Inspectorate, that new curricula were drawn up. These divergences between the two reform movements were of course due to characteristics specific to each discipline. In mathematics, the reform dynamic is long-standing. The Lichnerowicz Commission inherited more than a decade of thinking from a mathematical community led by the APMEP and the Société mathématique de France on the major contemporary issues of mathematics and on a reform of its teaching. The dynamic was thus more coherent than for français, which resulted from the recent conjunction of various reform movements and critical reflections carried by a very  The APMEP has been working with academics in this direction since the early 1950s (Barbazo and Pombourcq 2010; d’Enfert 2010). 22  All translations were made by the author. 21

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young association of specialists, the AFPF. But even more, the creation of the AFPF, facing the powerful and long-established Société des professeurs de français et de langues anciennes, reflected a profound division among teachers of literature, classical and modern. This latter had chosen to back the defense of the classical humanities with that of secondary education. Beyond these different disciplinary histories, the context further widened the gap: In the press as well as in the political world, there was a strong consensus on the necessity of modern mathematics reform, even if this consensus was undermined at the beginning of the 1970s. The scientific aura of André Lichnerowicz (Figure 5.1), and thus of the commission, the dynamism of the APMEP, and the support of the Ministry of Education facilitated the progress of the reform. In français, the rejection of the past model did not lead—outside the sphere of the reformers—to an inveterate faith in the virtues and the necessity of the envisaged reforms, especially since the Minister placed the mission of the Emmanuel Commission in the perspective of “to reorganize the second cycle in order to reduce the quantitative importance of exclusively literary sections which in a modern society appear less and less justified.” Prospects were not as conquering as for mathematics. The ambitions of the reformers, both in mathematics and in lettres, were to stumble on the 1975 Haby reform as the two commissions had ceased their activity (for the Lichnerowicz Commission, see below). Haby chose, for the collège unique, to adapt teaching to the supposed “aptitudes” and the various educational and professional destinies of students, even if it meant reducing its “modern” features. The mathematics curricula, while retaining achievements of the modern mathematics reform, were partly rearranged and brought traditional basic knowledge and techniques to the forefront.23 In français, in defiance of the perspectives opened up by the renewal of literary studies, the programs re-established a list of “great” authors to be learned. The joint study of the two movements to reform the teaching of français and mathematics shows the extent to which the position of mathematics then became dominant in the public arena as well as in the school arena, dethroning the humanities and inheriting their function of selecting the elite. It also highlights how “modern mathematics”—and the associated renewal of mathematical teaching— was then adorned with all the intellectual, economic, and democratic virtues of modernity, at least until the early 1970s, for many reformers as for politicians. Within the scientific disciplines, this domination of modern mathematics became problematic.

Figure 5.1  André Lichnerowicz (Sciences et Avenir, special issue No. 11 « La crise des mathématiques modernes » [“The crises of modern mathematics”], 1973, p. 87. (Photo by Robert Doisneau)  On the debates about the mathematics curricula of the Haby reform, see Gispert and Schubring (2011).

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Mathematics: A Discipline in the Field of Science Teaching In 1955, a mathematics teacher of the APMEP pronounced, during the general assembly of the Association, the following formula reported in its Bulletin: “If we really need scientists, the Latinists will have to capitulate” (APMEP 1955, p. 152). This pithy formulation is a good illustration of the power struggles and the issues at stake in the 1950s. But above all, it shows that the members of the Association were at that time fighting to develop their discipline within a “science” block which, at school level, brought together mathematics, physical sciences, and natural sciences. Thus, in 1956, the three associations APMEP, Union des physiciens (UdP), which grouped together the teachers of physical sciences, and Union des naturalistes (UdN) signed a joint wish for a revaluation of scientific teaching and asked for the creation of three scientific sections of equal weight, each with a scientific focus (mathematics, physics, or natural sciences) and with substantial timetables. Still, in 1960, the problem of the shortage of teachers and the need to improve their recruitment and training provided a new opportunity for joint action by the associations of specialists in the field of science teaching. Ten years later, while the movement for modern mathematics was at its peak, divorce appeared consummated between mathematicians on the one hand and their fellow physicists and naturalists, on the other hand. Mathematicians had expanded their ambitions beyond the exact sciences as claimed by the Lichnerowicz Commission: Mathematics [La mathématique] has always been an auxiliary discipline of the physical sciences and the art of engineering. It has now become, on the same basis, an auxiliary discipline for a large part of the biological and medical sciences, as well as for economics and the humanities. (Commission ministérielle 1967, p. 246)

In the meantime physicists, faced with the persistent disaffection with scientific studies, pointed to the responsibility of mathematicians. They then wrote a declaration that was considered to be the foundation of their action of the commission for the reform of the teaching of physical sciences, the Lagarrigue Commission, created in 1971. They asserted the need to reform the teaching of the physical sciences and criticized the outdated nature of the physical science syllabus, its disappearance from certain courses of study or competitive examinations, and the “invasion of the most abstract mathematics.” The promotion of science in political and economic circles, the inversion of the hierarchy, and the balance of power between sciences and classical humanities in education do not seem to have benefited in the same way to the three school disciplines of the science block equally.

Science and Science Teaching: The Future of the Country In 1956, under the aegis of the new head of government, Pierre Mendès France, the Caen colloquium was held (Prost 1988), which defined, within the framework of a 10-year plan, measures that should make it possible to double the number of scientific students, triple the number of engineering students, and increase tenfold the number of research and scientific higher education personnel, even though the dominant school and university values were still those of the classical humanities and culture (Figure 5.2). Secondary education was not forgotten, since it would be affected by some of the proposed measures. Their aim was to increase the number of hours devoted to science in secondary education and to organize the awakening of scientific vocations so as to redirect toward science students already involved in other directions. The promoters of the colloquium also set priorities for research and higher education in each scientific discipline. In biology, the emphasis should be on genetics, embryology, and biochemistry; in chemistry, on structural, physical, and nuclear chemistry; in physics and mathematics, solid state, atomic, nuclear, and molecular physics and probability, statistics, information, and game theory should be given priority. The measures advocated had little or no concrete translation in the 1950s; in the mid-1960s, a survey conducted by Pierre Bourdieu (Revue de

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Figure 5.2  C’est un plan de dix ans pour l’expansion [It’s a ten-year plan for expansion]. (Newspaper excerpt from Paris-Normandie, November 5, 1956)

l’enseignement supérieur 1966) showed that students chose scientific preparatory classes and faculties by default. This survey by Bourdieu highlights a hierarchy of prestige between the different disciplines of the science block. Thus, the best students, most often from the classical sections of secondary education, or students from the highest social classes, preferentially chose university curricula devoted to general mathematics and physics, where mathematics dominates; students from more modest social backgrounds, coming more often from the modern sections of the second cycle of the secondary education, chose studies in mathematics, physics, and chemistry in which mathematics plays a lesser role; finally, students from the working classes, whose level of aspiration is low, would choose studies deemed easy and engage in more experimental university studies devoted to the physical, chemical, and natural sciences, without mathematics. If the development of these three disciplines was claimed as a whole in the discourses of the political and economic ruling circles of the 1950s and 1960s, mathematics clearly enjoyed a privileged success and audience. It has it be recalled that, in these times of Cold War, after the shock of the launch of the Soviet Sputnik in 1957, the particular promotion of mathematics and the renovation of its teaching was played out on the international scene. From the end of the 1950s onward, the international economic and political organizations, the OEEC and then the OECD, initiated specific actions aimed at teaching mathematics in order to improve the training of engineers and scientists (Gispert 2010).24 Their project was to introduce into scientific secondary education this so-called modern mathematics, which  See also De Bock and Vanpaemel (2019, Chapter 2) and the different chapters of the first part of this volume.

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was both an axiomatic and formal science centered on the notion of structure, and a science integrating new fields linked to the applications developed during World War II. This asymmetry between mathematics and the physical sciences, in particular, is all the more notable here as there have been projects to renovate physics teaching in the United States since the mid-1950s, strongly supported at the national level again after the launch of Sputnik, then at the international level by UNESCO. These first projects were mainly aimed at future physicists and it was not until the end of the 1960s that the whole of secondary education was taken into account. At the international level, these projects were supported by the OECD, among others, in a report entitled Enseignement actuel de la physique, whose arguments always refer to this tension between humanities and science, the latter being declared necessary for the future of society. For example: The traditional programs [of the physical sciences] are not able to ensure a balance between the literary and scientific disciplines in the face of the demands of a society in which science plays a more important role every day. (OCDE 1969, p. 5)25

In France, the national and international dynamics thus seemed to combine in the 1960s to make mathematics the key school discipline of the country’s modernity, the one whose teaching had to be modernized as a matter of priority; they thus relegated the other scientific disciplines and their respective reforms to the second place. It is illustrated, for example, by the history of the reform commissions. The Lagarrigue Commission was set up in 1971, more than 4  years after the Lichnerowicz Commission, and the Bergerard Commission, which should have been devoted to the natural sciences, has never seen the light of day.

“The False Quarrel” of Modern Mathematics26 The above-mentioned 1970 statement by the physicists of the UdP and the two learned societies of physics and chemistry detailed their criticism of the mathematics reform. They described it as: the school of dogmatism [which] has as its last concern, both to motivate its abstractions by initial reference to some concrete problem, and to ensure to provide the other disciplines with the mathematical (or, if one prefers, “computational”) tools they need. (Hulin 1992, p. 41)

This criticism was not only external to the mathematical community. It was shared within the Lichnerowicz Commission itself, where a split was taking shape. Emblematic was the resignation of one of its members, Charles Pisot, who left the Commission and participated in the creation of a new association, the Union of Teachers and Users of Mathematics, which campaigned to preserve the traditional training of scientists and engineers. A number theorist, he had presented the ICMI survey on mathematics education for physicists at the 1966 International Congress of Mathematicians in Moscow. The relationship between mathematics and physics, between their teaching, was in fact a subject of lively discussion within the Lichnerowicz Commission from its very first meetings.27 Considered as a universal language, a privileged instrument of thought for all science, including human and social

 See on this subject French (1996), Hulin (1996), Ogborn (1996).  Speech given by the Minister of Education, Olivier Guichard, at the opening of the Lagarrigue Commission’s work on May 21, 1971. 27  The minutes of the Commission’s meetings are kept in the archives of the Commission d’étude pour l’enseignement des mathématiques [known as the Lichnerowicz Commission], at the Centre des Archives contemporaines de Fontainebleau under the reference 19870205/1 to 6. 25 26

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science, as stated in the Preliminary Report of the Commission, mathematics28 would be closer to grammar than to experimental science, a position expressed by some in a session in April 1967. Others, in particular physicists, refuted the idea that “the physical world [is] only an illustration of a mathematical world” and wondered about the consequences of such programs for “the quarter of young people with concrete minds who need to touch matter and do experiments.” This expressed the concern of physicists who did not want young “concrete minds” who follow long studies to drop out scientific studies, put off by too abstract mathematics which would have nothing to do with the physical world. Against this background of tensions, sharpened by the controversies of the winter of 1970 over the mathematics programs for students who were 13- to 14-year old,29 Minister Olivier Guichard took a position about what he described as a “false quarrel” (Guichard 1977, p. 10) of modern mathematics. He did so symbolically before the Lagarrigue Commission, during its setup. Noting the opposition between modern mathematics and classical mathematics (useful for the teaching of physics), Guichard invited, in a “spirit of interdisciplinary collaboration” (p. 11) to ensure coherence between the mathematics and physical science programs. By affirming that “the complementary nature of the different sectors of science, the reciprocal support that the development of physics and that of mathematics lend to each other are ... banal evidence” (p. 10) the Minister wished to erase the disputes between teachers. While he addressed the relations between the scientific disciplines before the same commission on October 17, 1975, the speech of the new Minister of National Education, René Haby, differed significantly. He vigorously reaffirmed the eminently experimental character that the teaching of physics should favor, whereas “the most common examples are sacrificed to the pseudo-mathematical dressing of physics and chemistry” (Haby 1977, p. 23).30 These strongly dissimilar speeches contributed to a redefinition of the hierarchies between scientific disciplines. By emphasizing the complementarity of these disciplines, Guichard aimed to smooth out the hierarchies, while he valued the science block as a whole. In contrast, with Haby, there was a distinct and opposed specification of experimental sciences on the one hand and mathematics on the other. In his view, mathematics was considered as a language of its own, a vector of general culture and at the service of the other disciplines.31 It no longer was considered as one discipline within a block but rather as a major part of the overall structure, thus shattering the very notion of the scientific block, which had already been subjected to multiple tensions in the preceding decades.

 The French expression for modern mathematics is la mathématique (singular), which is a significant difference from the traditional les mathématiques (plural). 29  See the next section. 30  Let us note that this affirmation by the Minister of an essentially experimental physics which takes into account neither the work of modeling, and its limits, nor the status of the principles which link the fundamental concepts appears contradictory to the ambitions affirmed within the Lagarrigue Commission and the UdP. 31  This position was criticized particularly by the Société mathématique de France, which denounced the dissociation made by the Haby reform of mathematics and other sciences, as well as the opposition between modern and classical mathematics. 28

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 The Ambitions of Modernity of the Mathematics Reform: Confrontation with Realities32 In February 1971, on the occasion of the development of new mathematics programs for 13- and 14-year-olds, that is students in the 3rd and 4th years of premier cycle of secondary education, a major French daily newspaper, l’Aurore, titled a lead article “The mathematics war” (Figure 5.3). This title reflected, beyond the journalistic effect, the real crisis that the process of the modern mathematics reform was going through. However, as we have seen, this process had developed in a quasi-consensus of the various actors in the scholarly, educational, political, and economic spheres since the creation of the Lichnerowicz Commission. Faithful to its agenda, to its principles of action (Commission ministérielle 1967), the commission wanted the systematic action it envisaged not to provoke any intellectual or material disorder; it intended to proceed step by step and foresaw a progressive and planned transformation of the mathematics programs which would have been the object of prior experimentation. Before these programs for 13- and 14-year-olds, the programs for the first 2 years of the premier cycle and the first 2 years of the second cycle had been experimented with and came into effect at the beginning of the 1969–1970 and 1970–1971 school years without causing any particular controversy. This had not been the case in 1971 with the curricula for the 3rd and 4th years of the premier cycle. Here the commission came up against a major obstacle, the reform of the geometry curriculum, which marked the first strong break, from a mathematical point of view, with the traditional mathematics that had prevailed until then. Whereas the programs were planned to be effectively implemented from September 1971 and 1972, a first draft was only presented in December 1970 to the consultative commission, made up of representatives of the Ministry, unions, and parents, which had to give its opinion before final adoption by the Minister. It would be adopted, but it was with a massive abstention from the participants. It was a true disavowal of the Commission’s work. This caused a real crisis in the weeks that followed, both in the Commission, where dissident members, including Charles Pisot,

Figure 5.3  La guerre des mathématiques [The mathematics war]. (Newspaper excerpt from l’Aurore, February 3, 1971)  For the precise sources on which I rely in this last section, see d’Enfert and Gispert (2011).

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tried to push through an alternative draft, and in the Ministry, where compromise solutions were sought.

Aims and Content of the Premier Cycle Programs The new mathematics curricula for grades 3 and 4 of the premier cycle were part of the overall project of the Lichnerowicz Commission. Two principles guided their development. The first one was that mathematics is a deductive science, not an experimental science. A logical presentation of the different mathematical concepts must be favored in order to remove anything that could be based on intuition or a supposed evidence. As a result, there was no longer any need to have a “math bug” to understand and succeed in the discipline. We recognize here one of the ambitions of the reformers, to promote mathematics for all. The second principle was that mathematics forms a theory—“la mathématique”—which must bring together under a single structure the knowledge that has been presented so far in a scattered manner. Mathematical notions that did not lead to contemporary mathematical concepts or techniques were to be excluded. In this perspective, the ambition of these new programs was to teach in order to differentiate the physical world from its mathematical model. The draft program thus stated: “Whenever there is a risk of confusion, care will be taken to use distinct terminology for concrete objects and their mathematical models.” It was an important epistemological leap in relation to the programs of the first 2 years of the premier cycle, which only introduced “modern” notions to translate concrete situations. With these programs, students were expected to enter directly into a methodical learning of deductive reasoning. We will therefore start in the 3rd year with the affine geometry model. Mathematically, this is the simplest but, and this aspect has been a problem, it is also the furthest from the real world: It only speaks about aligning points, parallelism, and the intersection of lines, and does not know the notions of distance, angle, and orthogonality, which are left to the next year. In other words, this new geometry is a geometry without circles and right angles, i.e., a geometry without compasses and squares; it is only the year after that students have the mathematical tools to account for the physical world. These new geometry programs thus presented a double specificity that appeared to be emblematic of the current reform and its paradoxes. On the one hand, they were relatively sophisticated from a theoretical point of view; on the other, they proposed a geometry that seemed to be little usable, or at least less usable than traditional geometry. However, programs were made to be implemented: Teachers will have to teach them, and students were supposed to assimilate them. In the premier cycle, as the Berthoin and Fouchet reforms had just redefined it,33 these were two realities that the Commission’s modernizing ambitions would come up against.

The Problematic Reality of the Teaching Staff If there is a reality of which the Commission was aware since the beginning of its work, it was the reality of the teaching staff, of the different teaching staffs one should say, because many of the profiles of mathematics teachers were then so different. A vast majority of mathematics teachers were non-tenured and unqualified staff. But, there was another aspect of which the Commission was not fully aware: In this newly renovated premier cycle, the majority of permanent teachers came from the  That is to say, a renewed premier cycle combining, on the one hand, the former premier cycle of the lycées (reserved for the urban social and intellectual elite) and, on the other hand, the former EPS and cours complémentaires (for the schooling of the working classes and rural areas). See the first section, among others, Notes 7 and 11. 33

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old system of schooling of the working classes.34 These teachers were not university-educated mathematics teachers with the most prestigious teaching diplomas; moreover, unlike their colleagues, the lycées teachers of the old secondary system, they taught several subjects. Few of them had received any real mathematical training at baccalauréat level, and their mathematical culture was therefore not as advanced as that of lycées teachers. But that was not the only difference. These different professional origins also referred to different mathematical cultures, one inherited from the secondary school system reserved for the social elite, the other from the old primary school system. This old school duality instituted since the beginning of the nineteenth century, persisted in fact beyond the institutional disappearance of these two school systems that the Berthoin and Fouchet reforms achieved. The first of these two cultures was historically dominated by the ideal of an abstract, theoretical, deductive science, aimed at training the mind and providing a general culture. The second was based on a completely different conception of the mathematical discipline. Inheritor of the old primary system, this conception was more practical and emphasized the links with the other sciences and their applications. These two traditions thus carried different disciplinary—and even pedagogical—logics which, in the case of geometry, had a particular echo, the first insisting on geometry as a deductive theory, the second conceiving it above all as an experimental science. However, the members of the Lichnerowicz Commission were all, at that time, former students of lycées, all of them came from this schooling reserved for the elite; they were far from, and even ignorant of, the mathematical culture of popular teaching education, with which their draft program, particularly in geometry, collided head-on. But, whatever the Commission’s wishes, it was, for the vast majority, teachers from the world of primary education and non-tenured teachers who would have to apply these new programs. For lycée teachers who taught in the premier cycle, another difficulty arose: Most of them were not familiar with the “modern mathematics” which they had not learnt at university. Indeed, it had only been taught in faculties since 1958. This contradictory situation, which arose partly from the modernization of the school system and clashed with the reformers’ intentions of modernizing the contents of the mathematical programs, gave rise to two types of discourse. The first discourse, promulgated by Lichnerowicz and most of the Commission as well as by the APMEP, raised the flag of training. They considered that the training, both initial and in-service, of these teachers in these new mathematics, new programs, and new methods, was a primordial condition for the success of the reform. For them, there was therefore no reason to reduce the ambitions of the programs. By contrast, the general inspectors within the Commission as well as the central administration, advocated a certain moderation in the renovation, arguing that the great mass of teachers was not in a position to understand and assimilate the projected programs. Their view was shared by the unions. While they considered the reform to be a necessary undertaking, many aspects of which were generously positive, they denounced the lack of the necessary means to ensure, as a matter of urgency, the training of teachers and for this reason abstained when the programs were presented.

The Pitfall of the Different Goals of the Premier Cycle It was required that the mathematics programs which had been drawn up by the Commission had to be relevant to all students enrolled in the last 2 years of compulsory schooling. The aim was therefore to propose the same content, and teach it in the same spirit, to all the students in this age group, whose immediate educational destinies were radically different: Long studies leading to the baccalauréat for some, short studies and/or entry into working life for others.  See the first section, Note 7.

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This reality, which was new as we have seen, had first of all been perceived in a very idealized way by the Commission, the members of which had no experience of the diversity of schooling, apart from the long-term general courses of the old secondary system. For Lichnerowicz, in fact, “by its very nature the premier cycle is anti-segregationist, the reform must be democratic and it is up to the school to ensure that children, who are more or less the same, achieve the same performance by compensating for differences in social origin.” At a Commission meeting in 1967, when the question was raised as to the relevance of the draft curriculum for the third year of the premier cycle for students going on to short vocational education, one participant argued: “Should we teach outdated mathematics to less intelligent children?” In fact, smarter or not, only one-third of children in the premier cycle went on to long studies in the second cycle. This was the norm according to which the Commission worked, based on the forecasts of the Fifth Plan. The Commission was in fact blind to this reality: Social differences, far from being compensated for by the school, were perpetuated there; the Berthoin and Fouchet reforms had put in place the massification of the premier cycle, not its democratization. Nevertheless, in presenting its programs for the third and fourth years of the premier cycle, the Commission developed a rhetoric that enabled it to legitimize the axes of mathematical renovation “for all”: Since mathematics intervenes as a model construction for all concrete situations, its usefulness is therefore obvious, including for students in short-term courses. Actually, the Commission smoothed the question of school goals and relied on experiments carried out in classes where qualified teachers with long schooling taught students who would further their studies in higher education. Hence the problem of the democratic challenge of mathematics for all and the reduction of failure in mathematics was above all for the Commission that of the abstruse character of the old programs.

Save the Reform After the disavowal vote from the Advisory Commission, the Ministry of Education came under intense pressure, especially since Charles Pisot, who was firmly opposed to these programs, sent Minister Guichard a counter-project that reintroduced the notion of distance from the third grade of the premier cycle. As evidenced by the large number of letters the Minister received in January and February 1971, the pressure came mainly from mathematicians and physicists, both of whom were divided between those who approved the Commission’s options and those who supported the counter-­ project (Figure 5.4). The interpellation from the scientists caused embarrassment in the Ministry, in the central administration. Without a revision of the contested programs, the cabinet feared a serious crisis, provoking extreme reactions on all sides: The renovation of mathematics teaching could be interrupted or compromised if reasonable decisions were not taken quickly. Lichnerowicz was called upon to make concessions. He toned down the modern character of the programs, removed the words that make some people angry, such as the epithet “affine,” and minimized references to theoretical presuppositions: Geometry would no longer be presented a priori as a mathematical theory but would have to appear gradually to the students. Approved by the Commission, and supported by the APMEP, the new version of the programs was soon afterward ratified by the Minister who thus gave the go-ahead to its implementation in the classes from September 1971. At the same time, a Ministry circular of April 1971 on the teaching of numeracy in secondary education recalled “the interest and urgency of a methodical and persevering teaching of numeracy,” as well as the need to “contain purely speculative developments.” The acceptance of these programs did not mark the end of the polemic which entered the public debate for 2 years. The Ministry continued to defend the reform and modernization of mathematics teaching: Demanded by all, it was urgent, in his opinion, to carry it out, especially as it corresponded to pedagogical progress. However, the Ministry had to give up some ground with the effective

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Figure 5.4  La querelle des « mathématiques modernes » [The “modern mathematics” quarrel]. (Newspaper excerpt from Le Monde, January 20, 1972)

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i­ mplementation of programs that not only confused many teachers but also did not satisfy the APMEP reformers who considered the programs too heavy, too theoretical, and too restrictive. The APMEP made counter-proposals so that the initial ambitions—to make the same mathematics accessible to all students through a joint renovation of the contents and teaching methods—were really achieved. According to its leaders, this implied firstly an adaptation of the programs where the knowledge required from the students would be limited to a core of essential notions and know-how; secondly, teachers’ practices should be transformed (teamwork, work on cards, differentiation, etc.) in the direction of a real taking into account of the diversity of classes and the subsequent orientations of students. To the scientific logic of the academics of the Lichnerowicz Commission, who refused to see the mathematical edifice they have built distorted, the APMEP responded by putting forward a pedagogical logic, as close as possible to school realities—both students and teachers—in order to realize fully the first modernizing ambitions of the reform project. Whereas school curricula of the premier cycle were supposed to address almost the entire age group, and the establishment of curricular continuity between primary (6–10-year-olds) and secondary (from age 11) education was on the agenda, a new question arose. The capacity of the traditional disciplinary model of seven-year secondary education—its contents and methods, but also its aims— to contribute to the success of the democratization of education, was at the heart of the problem and was beginning to be questioned.



Concluding Remarks

Reforming to what ends is the question, I have tried to answer in this chapter about the modern mathematics movement in France. The answers appear multiple and are far from being strictly disciplinary, whether mathematical or pedagogical. This movement to reform the teaching of mathematics was part of the renewal of the cultural, social, and economic modernization project of France in the post-War decades known as the Trente Glorieuse. In a complete overhaul of the educational offer, the education system was asked to modernize its structures as well as its contents. This modernization involved the promotion of science, which was considered synonymous with progress. Mathematics, supported by political and economic circles, became the dominant discipline in the school system. The movement to reform the teaching of mathematics, prioritized by the political authorities, was a model for all the other disciplines, which also sought to promote a modernization of content and methods and to obtain a profound reform of their teaching. Two principles underpinned the modernization of the education system: The democratization of education and its adaptation to the needs of economy. But the objectives assigned to the education system were not without contradiction, which had repercussions on the reform of mathematics. One of the most obvious realities of these decades of reform was the substitution of mathematics for Latin as the pivot of elite education. This new function assigned to mathematics was at first socially legitimate and accepted. However, the response to the declared needs of the economy provoked an increased specialization, which led to a hardening of the selection process. A selection that soon characterized, and for some discredited, modern mathematics. Selection through mathematics was not limited to the training of elites, mathematics also inherited this role in the premier cycle, which had just been renovated to ensure its democratization. The reality of the massification, instead of a real democratization, of these first years of secondary education, hit hard one of the most cherished aims of the reformers involved in the modern mathematics movement. Their ambition that modern mathematics would be “mathematics for all,” proved to be illusory. The multidisciplinary history of the reform of the teaching of mathematics in France has thus highlighted the interweaving of the different registers of modernity carried by the different actors, individual or collective, involved in this reform. The modernity of the so-called modern mathematics, the modernity of la mathématique, and its teaching, have acquired meanings which, I believe, enrich the historiography of this reform movement with relevance.

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References AFPF. (1970). Propositions pour une rénovation de l’enseignement du français. Manifeste de Charbonnières [Proposals for a renovation of the teaching of français. Manifesto of Charbonnières]. Le Français aujourd’hui, 9, 1–32. Alix, P., d’Enfert, R., & Gispert, H. (2015). Des commissions pour réformer les disciplines (1945–1980) [Commissions to reform the disciplines (1945–1980)]. In R. d’Enfert & J. Lebeaume (Eds.), Réformer les disciplines. Les savoirs scolaires à l’épreuve de la modernité, 1945–1985 [Reforming the disciplines. School knowledge in the test of modernity, 1945–1985] (pp. 85–111). Rennes, France: Presses Universitaires de Rennes. APMEP. (1955). Assemblée générale du 29 mai 1955 [General Assembly of May 29, 1955] (Bulletin de l’APMEP No. 169). Paris, France: APMEP. APMEP. (1968). Charte de Chambéry, Étapes et perspectives d’une réforme de l’enseignement des mathématiques [Charter of Chambéry, Steps and perspectives of a reform of mathematical teaching] (Supplément au Bulletin de l’APMEP No. 263–264). Paris, France: APMEP. Barbazo, E., & Pombourcq, P. (2010). Cent ans d’APMEP [One hundred years of APMEP]. (Brochure APMEP No. 192). Paris, France: APMEP. Barbin, É. (2012). The role of the French Association of Mathematics Teachers APMEP in the introduction of modern mathematics in France (1956–1972). In Proceedings of the ICME‑12 Satellite Meeting of HPM (History and Pedagogy of Mathematics) July 16–20, 2012 (pp. 597–605). DCC, Daejeon, Korea. Bourdieu, P., & Passeron J.-C. (1964). Les héritiers [The inheritors]. Paris, France: Éditions de Minuut. Cardon Quint, C. (2015). Des lettres au français. Une discipline à l’heure de la démocratisation (1945–1981) [From letters to français. A discipline at the time of democratization (1945–1981). Rennes, France: Presses Universitaires de Rennes. Cardon-Quint, C., d’Enfert, R., & Gispert, H. (2015). Démocratiser, orienter, sélectionner: l’enseignement du français et des mathématiques dans le second degré (1945–années 1980) [Democratizing, orienting, selecting: The teaching of français and mathematics in secondary schools (1945–1980s)]. In R. d’Enfert & J. Lebeaume (Eds.), Réformer les disciplines. Les savoirs scolaires à l’épreuve de la modernité, 1945–1985 [Reforming the disciplines. School knowledge in the test of modernity, 1945–1985] (pp. 37–60). Rennes, France: Presses Universitaires de Rennes. Commission ministérielle. (1967). Rapport préliminaire de la commission ministérielle [Preliminary Report of the Ministerial Commission]. Bulletin de l’APMEP, 258, 246–271. De Bock, D., & Vanpaemel, G. (2019). Rods, sets and arrows. The rise and fall of modern mathematics in Belgium. Cham, Switzerland: Springer. d’Enfert, R. (2010). Mathématiques modernes et méthodes actives : les ambitions réformatrices des professeurs de mathématiques du secondaire sous la Quatrième République [Modern mathematics and active methods: The reformist ambitions of secondary school mathematics teachers under the Fourth Republic]. In R. d’Enfert & P. Kahn (Eds.), En attendant la réforme. Disciplines scolaires et politiques éducatives sous la IVe République [Waiting the reform. School disciplines and educational policies under the Fourth Republic] (pp. 115–129 ). Grenoble, France: Presses Universitaires de Grenoble. d’Enfert, R. (2011). Une réforme ambiguë : l’introduction des « mathématiques modernes » à l’école élémentaire (1960–1970) [An ambiguous reform: the introduction of “modern mathematics” in elementary school (1960– 1970)]. In R. d’Enfert & P. Kahn (Eds.), Le temps des réformes. Disciplines scolaires et politiques éducatives sous la Cinquième République: les années 1960 [The time of reforms. School disciplines and educational policies under the Fifth Republic: the 1960s] (pp. 53–73). Grenoble, France: Presses Universitaires de Grenoble. d’Enfert, R., & Gispert, H. (2011). Une réforme à l’épreuve des réalités : Le cas des « mathématiques modernes » en France, au tournant des années 1960–1970 [A reform tested by realities: The case of “modern mathematics” in France, at the turn of the 1960s–1970s]. Histoire de l’Éducation, 131, 27–49. d’Enfert, R., & Kahn, P. (Eds.). (2010). En attendant la réforme. Disciplines scolaires et politiques éducatives sous la IVe République [Waiting the reform. School disciplines and educational policies under the Fourth Republic]. Grenoble, France: Presses Universitaires de Grenoble. d’Enfert, R., & Kahn, P. (Eds.). (2011). Le temps des réformes. Disciplines scolaires et politiques éducatives sous la Cinquième République : les années 1960 [The time of reforms. School disciplines and educational policies under the Fifth Republic: the 1960s]. Grenoble, France: Presses Universitaires de Grenoble. d’Enfert, R., & Lebeaume, J. (Eds.). (2015). Réformer les disciplines. Les savoirs scolaires à l’épreuve de la modernité, 1945–1985 [Reforming the disciplines. School knowledge in the test of modernity, 1945–1985]. Rennes, France: Presses Universitaires de Rennes. French, A. P. (1996). De nouvelles orientations dans l’enseignement de la physique : le PSSC 30 ans après [New directions in physics education: the PSSC 30 years later]. In B. Belhoste, H. Gispert, & N. Hulin (Eds.), Les sciences au lycée. Un siècle de réformes des mathématiques et de la physique en France et à l’étranger [Science in the lycée. A century of mathematics and physics reforms in France and abroad] (pp. 287–295). Paris, France: Vuibert & INRP.

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Gispert, H. (2010). Rénover l’enseignement des mathématiques, la dynamique internationale des années 1950 [Renewing mathematics education, the international dynamics of the 1950s]. In R. d’Enfert & P. Kahn (Eds.), En attendant la réforme. Disciplines scolaires et politiques éducatives sous la IVe République [Waiting the reform. School disciplines and educational policies under the Fourth Republic] (pp. 131–143). Grenoble, France: Presses Universitaires de Grenoble. Gispert, H. (2014). Mathematics education in France (1800–1980). In A. Karp & G. Schubring (Eds), Handbook on the history of mathematics education (pp. 229–240). New York, NY: Springer. Gispert, H., Guedj, M., & Savaton, P. (2015). Les disciplines scientifiques dans le second cycle du secondaire : hiérarchie et rapports de force [Scientific disciplines in upper secondary education: hierarchy and power relations]. In R. d’Enfert & J.  Lebeaume (Eds.), Réformer les disciplines. Les savoirs scolaires à l’épreuve de la modernité, 1945–1985 [Reforming the disciplines. School knowledge in the test of modernity, 1945–1985] (pp.  181–198). Rennes, France: Presses Universitaires de Rennes. Gispert, H., & Schubring, G. (2011). Societal, structural and conceptual changes in mathematics teaching: reform processes in France and Germany over the twentieth century and the international dynamics. Science in Context, 24(1), 73–106. Guichard, O. (1977). Allocution du Ministre de l’Education Nationale [Speech given by the Minister of National Education]. Bulletin de l’Union des physiciens, 597(1), 9–12. Haby, R. (1977). Allocution du Ministre de l’Education devant la commission Lagarrigue, 17 octobre 1975 [Speech given by the Minister of Education to the Lagarrigue Commission, October 17, 1975]. Bulletin de l’Union des physiciens, 597(1), 19–26. Hulin, M. (1992). Le mirage et la nécessité [The mirage and the necessity]. Paris, France: Presses de l’École normale supérieure. Hulin, N. (1996). Constitution de la physique moderne et nouvelle conception de l’enseignement de la discipline [Constitution of modern physics and new conception of the teaching of the discipline]. In B. Belhoste, H. Gispert, & N. Hulin (Eds.), Les sciences au lycée. Un siècle de réformes des mathématiques et de la physique en France et à l’étranger [Science in the lycée. A century of mathematics and physics reforms in France and abroad] (pp. 55–68). Paris, France: Vuibert & INRP. IREM de Lille. (1971). Développer et moderniser l’enseignement des mathématiques? Pour quoi faire ? [Developing and modernizing mathematics education? For what purpose?]. Bulletin de l’APMEP, 277, 69–71. Lopez, M., & Sido, X. (2015). L’enseignement des mathématiques et du français dans l’enseignement technique court de 1945 à 1985. Identité singulière, dynamiques et temporalités spécifiques ? [The teaching of mathematics and français in short technical education from 1945 to 1985. Singular identity, dynamics and specific temporalities?]. In R. d’Enfert & J. Lebeaume (Eds.), Réformer les disciplines. Les savoirs scolaires à l’épreuve de la modernité, 1945–1985 [Reforming the disciplines. School knowledge in the test of modernity, 1945–1985] (pp.  137–154). Rennes, France: Presses Universitaires de Rennes. OCDE. (1969). Enseignement actuel de la physique [Current teaching of physics]. Paris, France: OCDE. Ogborn, J. (1996). Les Anglo-saxon sont-ils différents ? [Are Anglo-Saxons different?] In B.  Belhoste, H.  Gispert, & N. Hulin (Eds.), Les sciences au lycée. Un siècle de réformes des mathématiques et de la physique en France et à l’étranger (pp. 271–285) [Science in the lycée. A century of mathematics and physics reforms in France and abroad]. Paris, France: Vuibert & INRP. Prost, A. (1988). Les origines des politiques de la recherche en France (1939–1958) [The origins of research policies in France (1939–1958)]. In Reprint of the Cahiers pour l’histoire de la recherche scientifique, 1939–1989, 1988(1), 1–18. Revue de l’enseignement supérieur. (1966). Une étude sociologique d’actualité : les étudiants en sciences [A sociological study of present time: students in science]. Revue de l’enseignement supérieur, 4, 199–208.

Chapter 6

West German Neue Mathematik and Some of Its Protagonists Ysette Weiss

Abstract  Today’s German perception of the New Maths movement in West Germany is strongly shaped by the view that the movement was a transfer of American ideas to Western European countries, that it was solely a reform of elementary school teaching, and that it failed. To understand this perception better without getting bogged down in numerous details of the eventful 1950s, 1960s, and 1970s, we concentrate on seven beliefs—presented in subchapters—about the West German New Math reform. It turns out that this reform of mathematics instruction was to a far greater extent aimed at mathematics teaching in the German Gymnasium. To understand the peculiarities of West German New Math, it will be also essential to see it in the context of earlier reforms of mathematics education and institutional reforms of the post-War period. It turns out that for a better comprehension of the failure of the reform, one should go back even further in time and include considerations about the unfinished Meraner Reform. Keywords  Arnold Kirsch · Axiomatic method · Erlangen program · Georg Wolff · German curricular reform · Hans-Georg Steiner · Heinrich Behnke · Heinz Griesel · Hermann Athen · Meraner Reform · Motion geometry · Neue Mathematik · Nuremberg curriculum · Otto Botsch · Royaumont Seminar · Transformation geometry · Vector calculus · West Germany

Introduction The description of a reform of mathematics education in a period of time that is marked internationally and nationally by tensions, contradictions, and far-reaching social, cultural, economic, and technical changes cannot be portrayed through a single story. Many complexly interwoven stories give an impression of the driving forces, protagonists, and social practices that accompany reforms. In addition, these stories are told by narrators with different social practices and so, above all, are always stories from outside observers. What they pay attention to is also guided by the Zeitgeist of the

Y. Weiss (*) University of Mainz, Mainz, Germany e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_6

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observer. In his extensive and varied analysis of the international teaching reform of New Math,1 Moon (1986) writes regarding the special case of New Math in West Germany: New Math in West Germany represents the changing fortunes of a range of political and professional groups who were subject to a series of national and international pressure. (p. 182)

The perspective taken allows an interpretation of national, West German phenomena of an educational reform within the framework of an international reform movement (see also De Bock and Vanpaemel 2015; Howson et al. 1981; Kilpatrick 2012; Servais 1975). Moon’s presentation of an international reform from the perspective of the agents, and of local and international power and interest struggles, also opens up areas of tension between intended, implemented, and achieved curricula (see also Damerow 1977; Steiner 1980). In this chapter, the focus will be on some national characteristics of the teaching reform of New Math of the then Federal Republic of Germany, the so-called West Germany. The reform of mathematics education in the former German Democratic Republic (GDR) differs greatly from the West German. Its investigation would therefore go beyond the scope of this chapter.2 In the following, the main focus will be on the development of teaching methods in the German Gymnasium. German secondary school mathematics education is still shaped today by the major mathematics education reform “Newer Geometry” of the nineteenth century and by the Meraner Reform of “education for functional thinking” shaped by Klein’s ideas at the beginning of the last century. It is therefore obvious that the German peculiarities of these also international reforms in mathematics education should be taken into account when considering the West German reform movement New Math. Although a decade since World War II had passed, and therefore the end of the Nazi dictatorship, by the beginning of the reform, the West German New Math reform was also shaped by the Nazi past and the long process of coming to terms with that past in the post-War years. The ease with which mathematical didactic theories, existing teaching cultures of the Weimar Republic, and the mottos of the Meraner Reform could be used to spread racial theories in Nazi Germany called into question all educational theoretical traditions of German mathematics education (Segal 2003). The necessity of a deeper and new understanding of the former Meraner educational goals for all school types was generally recognized in the post-War years. However, the question of complicity in the crimes of German Nazism made the discussion in West Germany complicated for decades. Therefore, in the post-War period, both the classic concept of education, which is based on cultural assets, and the scientistic concept of education, which is based on scientific knowledge, were questioned (Klafki 1967; Wagenschein 1989). However, these educational concepts were also fundamental to the educational and curricular goals of Klein’s teaching reform. Against the background of the experience of a fascist nationalist dictatorship, education was now seen by many in the post-War period as the education of people to develop their personality (Lenné 1969; Wittenberg 1963). The goal of upbringing was not only to adapt to existing social conditions but also to critically question and change them. The dissolution of the German tripartite education system3 aimed at by the Allies in the post-War years led to an intensified discussion of the educational goals of Gymnasiums, in particular of mathematics education (Lenné 1969; Schuberth 1971; Wittenberg 1963). The training of mathematics teachers for the Gymnasium level, which took place exclusively at universities in West Germany and as specialist training, remained largely unaffected by these discourses (Huebener 1952). With a few exceptions (e.g. Freiburg, Karlsruhe, Münster), the educational goals of teacher training, which were mainly

 Although the European reform movement is usually referred to as modern mathematics, a translation of the French mathématique moderne, we opt to refer to the German movement as New Math, translated into German as Neue Mathematik. 2  One can find some aspects of the East German New Math reform in, e.g. Filler (2016). 3  While in East Germany the transition to a common mainstream school took place, the three-part school system remained in the western zones, contrary to the enacted school laws. 1

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scientific, were based exclusively on the canonically organized subject study of mathematics. To give the reader an impression of the specifics of the West German reform movement without getting bogged down in details, we apply the following approach: We use various beliefs to underpin or confront them with selected facts. To do this, we consider the biographies of various protagonists in context and analyse the reform proposals from different perspectives.



Modern University Mathematics and Backward School Mathematics

In retrospect, the teaching reform of New Math is primarily perceived as a demand coming from the mathematicians at universities to reduce the gap between outdated school mathematics at Gymnasiums and the state of modern mathematical research at universities and to promote the transition from high school to university. Focusing on these objectives, the great similarity of these demands with those of the Kleinian reform is striking. In the Meraner Reform, the overcoming of the gap between modern analysis and an algebraic school analysis without the concept of limit, differential, and integral calculus was up for debate: The teaching reform of New Math should now apparently overcome the break between the language of modern algebra at universities and upper secondary school mathematics without algebraic structures, set theory, and axiomatic methods. At the time of the Meraner Reform, differential and integral calculus, and the consideration of different geometries (e.g. projective and spherical geometry) were taught in lectures in mathematics for beginners, not only at universities but also at technical colleges. This meant that these topics were also part of the training for future Gymnasium teachers (Schubring 2016). But what about modern algebra and set theory at German universities? One might think that in Germany as a birthplace of set theory (consider Cantor, Dedekind, Fraenkel, Frege, Hausdorff, and Zermelo), the axiomatic method and modern algebra (Artin, Hilbert, Noether, Van der Waerden), Studienräte4 in the German Gymnasiums would have learned these languages as mother tongues during their university studies. Van der Waerden’s Modern Algebra was already in its 5th edition in 1955. Nevertheless, modern algebra in the sense of Bourbakism only began to gain acceptance at German universities from the early 1950s. Based on an analysis of various university courses on algebra from the beginning of the last century to the 1950s, Vollrath (1991) shows how inconsistent, depending on the lecturer and training location and, in the Bourbakist sense, outmoded the German university teaching of algebra was in many places. In the 1960s, a wave of new hires took place, favored by several factors, such as the expansion of education (Schubring 2016) and the age structure of the teaching staff in the Gymnasiums. In the case of these “newcomers,” it cannot be assumed that they had enjoyed a mathematics education saturated with morphisms and arrow sequences. Only a few, particularly mathematically gifted students, who then mostly did their doctorates, had worked scientifically with the tools of modern algebra. If one takes seriously the demand for the mathematical modernization of mathematics teaching at the Gymnasium level, then modernization of university education for teachers of the Gymnasium should have come before the school reform. However, this was not intended in the context of the initial planning of the West German reform of New Math, since it was assumed that modern university mathematics was already in place. One could now assume that the mathematical training of upper secondary school teachers, which was inadequate for the reform, was a problem of the time, but one which would have solved itself after a few years. However, it was also a problem of principle. Different positions prevailed at universities with regard to the approaches suitable for learning both higher mathematics and school mathematics. The deductive method and the intended role of “set theory” also attracted international criticism. As early as 1962, 65 well-known American mathematicians  Teachers at German Gymnasiums start their employment with this status as civil servants.

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adopted a well-considered memorandum for a curriculum reform beyond “New Math” (Ahlfors et al. 1962). In Germany, too, mathematicians at universities developed a critical attitude toward the use of the axiomatic method, which they saw overemphasized and did not reflect the real work of a mathematician5 (see also Athen 1966; Freudenthal 1971; Laugwitz 1965; Wigand 1965).

 University Mathematicians Who Wanted to Modernize School Mathematics Efforts to develop and improve university teaching of mathematics have a long tradition in Germany. Particularly noteworthy are the efforts of Otto Toeplitz (1881–1940) and Heinrich Behnke (1889–1979). These endeavours involved the development of various teaching formats as well as experimental, problem-oriented approaches as alternatives to the deductive teaching method. Including considerations of such endeavours to improve university teaching and to promote mathematical talents in the next generation is also essential if one is to understand the activities of mathematicians, who were of particular importance for the New Math education reform. We briefly introduce some of the reform protagonists from the mathematical community. Heinrich Behnke, professor at the University of Münster since 1927, was the founder and editor6 of the journal Semesterberichte zur Pflege des Zusammenhangs von Universität und Schule [Semester Reports on Maintaining the Connection between Universities and Schools] and the conferences of the same name.7 As the journal and conferences show, Behnke was particularly interested in improving the connection between university and school, and in the internationalization of these efforts. Within Germany, Behnke had succeeded in 1951 in Münster—the traditional home of high-quality mathematics teacher training—to set up the first “Seminar for Didactics of Mathematics” at a German university. He also initiated a new compendium (Behnke et al. 1958) of high school mathematics on a scientific basis. Heinrich Behnke was a member of the ICMI Executive Committee from 1952—the year ICMI was restored—until 1966 and served as president from 1955 to 1958 (Figure 6.1). From 1958, when Marshal Stone took over the presidency, Behnke assumed the role of vice-president.8 Behnke’s incredible productivity is also evident in 31 supervised doctorates.9 One of his doctoral students, Heinz Griesel (1931–2018), who did his doctorate on a mathematical subject, later played an essential role in the implementation of New Math through numerous scientific publications and textbooks. Mentioning Behnke in the context of New Math is also inevitable because of Hans-Georg Steiner (1928–2004), who studied in Münster and received early support from Behnke. Steiner is one of the theoretical minds of the New Math movement.10 By age, Steiner belonged to the “Flakhelfer”11 generation. In maintaining the connection between university and school, Behnke was not concerned with the development of practical school materials, but with the scientific and methodological support of mathematics teachers  Carl Ludwig Siegel even speaks of a disgrace of mathematics: “I’m afraid that mathematics will perish before the end of the century if the trend toward pointless abstraction—the theory of the empty set, as I call it—is not stopped” (Lang 1994, p. 19). 6  Behnke edited the Semesterberichte from 1929 (first with Otto Toeplitz) and after the War with Wilhelm Süss (1895– 1958) and Walther Lietzmann (1880–1959). In contrast to Süss and Lietzmann’s Nazi past, Behnke, whose first wife and son were Jewish, had no problems in the post-War years because of a possible systematic proximity to Nazi Germany (Heske 2021; Segal 2003). 7  One can find a detailed description of the development of the Semesterberichte in Klaus Volkert’s (2016) paper. 8  On the relationship between Behnke and Stone see, e.g. Furinghetti et al. (2020). 9  Detailed information about Behnke’s biography can be found in Uta Hartmann’s (2009) dissertation. 10  As early as 1959 Steiner presented concepts based on philosophical considerations for the implementation of modern mathematics in practical school journals (Steiner 1959). 11  Flakhelfer were 15- to 17-year-old students of upper secondary schools, who were born between 1926 and 1928 and deployed from February 1943 as part of the military service in the air War of Nazi Germany. 5

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Figure 6.1  Heinrich Behnke (first row at the left) at the ICMI Colloquium Echternach in 1965. Next to Behnke in the front row (identification by Guy Noël): Frédérique Papy-Lenger, Gustave Choquet, unknown woman, André Lichnerowicz; second row: Robert Dieschbourg, two unknown men, Alfred Vermandel, Pierre Debbaut, Claudine Festraets-Hamoir (woman against the wall). (Photo collection P. Debbaut)

working at Gymnasiums. This should be done through a closed connection between Gymnasiums and universities, which would allow for a better integration into the community of mathematicians (see Volkert 2016). Another mathematician of this generation who also endeavoured to develop the teaching of mathematics and network mathematicians was Martin Barner (1921–2020). From 1963 to 1994 he was the head of the famous German scientific centre Oberwolfach and from 1968 to 1977 chairman of the Deutsche Mathematiker-Vereinigung (DMV) [German Mathematical Society]. In the first year of his management of Oberwolfach, a conference dealing with questions related to the planned curricular reform of secondary school mathematics and mathematics education was organized in this centre of mathematical research. The organization of scientific conferences on mathematics education in Oberwolfach played an important role in the preparation of the curricular reform, as it directly motivated and influenced the novel educational policy measures. Another important university mathematician who played a role in the teaching reform, especially as an excellent science organizer, was Heinz Kunle (1928–2012). In 1970 he took over Behnke’s international function as chairman of the German subcommission of ICMI. Kunle took up Steiner’s idea to establish a European branch in Germany of the American reform project Comprehensive School Mathematics Program (CSMP), which was founded in 1966 (Griesel 2012). This took place in the form of the Zentrum für die Didaktik der Mathematik [Centre for the Didactics of Mathematics] at the University of Karlsruhe, with Steiner becoming director. The sponsor of the Centre was the Verein der Förderung der Didaktik der Mathematik [Association for the Promotion of Didactics of Mathematics], whose chairman was Kunle, deputy chairman Steiner, members Barner, Griesel, Arthur Engel, Vollrath, and the head of the CSMP Burt Kaufman. The influence of American New Math ideas on German reform was certainly also due to the close relationship between Steiner and his Centre’s staff and the CSMP. Steiner was also the European co-director of the CSMP. In the German tradition of content-related analysis (Hefendehl-Hebeker 2016), the Centre provided preparatory work for the

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CSMP curriculum projects. Initially aimed only at gifted students in grades 7 to 13, the curricula were redesigned for normal classes with a shift in age to include elementary school and even kindergarten. An important point for the internationalization of the activities of the Centre was the establishment of the Zentralblatt für Didaktik der Mathematik. The above-mentioned mathematicians supported the reform primarily in terms of scientific organization. Günther Pickert (1917–2015) in particular engaged in a substantive discussion. From 1960, Pickert was co-editor of the mathematical-physical semester reports and published numerous articles on university didactic and school mathematical problems (Steiner 1977). He was head of the mathematical institute at the University of Gießen, at which Arnold Kirsch (1922–2013) was a lecturer. Pickert too was a friend of Steiner, Griesel, and Kirsch (Behnke and Stowasser 1979). As an activist of the reform movement, Detlef Laugwitz (1932– 2000), a university mathematician, should also be mentioned. He was Steiner’s doctoral supervisor, which is somewhat surprising since he was almost the same age as Steiner and was one of the publishing critics of the Nuremberg curriculum, which was written in the spirit of Steiner’s views on New Math. The “combative” contributions to the discussion by Steiner and Laugwitz were printed in the journal Der Mathematikunterricht and commented on by Hermann Athen, an influential textbook author who helped prepare this curriculum and who will be discussed later. Laugwitz was particularly interested in the question of using the axiomatic method (Laugwitz 1966), as he was concerned with a completely constructive foundation of analysis. In Germany, he was considered a father of non-­ standard analysis. Another PhD student supervised by Laugwitz was Hans-Joachim Vollrath, who made numerous content-related contributions to the implementation of the new curriculum. The broad philosophical and mathematical-historical education of Detlef Laugwitz and his view of the development of concepts, which is not only based on a subject system, is later also mirrored by Steiner and Vollrath. In his doctoral thesis, Steiner studied the topic of mathematical theory of voting committees (Steiner 1969). This social-cultural contextualization of set theory Steiner used later repeatedly as a paradigmatic example for the application of the axiomatic method. The mathematicians presented appeared in leading roles in various networks in the 1950s and 1960s, whereby Behnke, Kunle, and Barner influenced the development of the reform New Math primarily in terms of scientific organization and their interests in higher education didactics. Pickert and Laugwitz also dealt with problems of school mathematics and elementarizations of higher mathematics for school mathematics lessons. Another important figure in the reform movement, who was both a university mathematician and a secondary school mathematics teacher, was Herbert Meschkowski (1909–1990). Meschkowski studied mathematics in Berlin. After the second state examination12 in 1936, since there were no positions, he did not find employment in a secondary school but taught difficult-to-educate boys in a Protestant institution. From 1939 he worked at a Gymnasium in Berlin. The time in the Protestant institution and the spiritual closeness to his wife, who worked as an educator in this monastery, had a lasting influence on his views on mathematics education. During the War, Meschkowski was active in engineering, just like Hermann Athen, to whom we will come later. He was open to economic justifications for reforms in mathematics education, like those given by the OEEC—later the Organisation for Economic Cooperation and Development, OECD (1961a): The advancing automation creates reductions in working hours for many professions. But the expansion of technology, the assertion in international economic competition imperatively demands the solid training of engineers, physicists, mathematicians, and chemists. (Meschkowski 1965, p. 183)13

Meschkowski received his doctorate in pure mathematics only in 1950—after several years of work in schools and industry. He was then appointed to a professorship at the Freie Universität Berlin [Free University of Berlin] and the Pädagogische Hochschule Berlin [Berlin College of Education]. During  German teacher certificate for Gymnasium.  All quotes in this chapter, as well as titles, have been translated by the author.

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the period of the theoretical foundation and preparation of the curricular reform, from 1962 to 1964, Meschkowski was rector of the Pädagogische Hochschule Berlin. Like Laugwitz, he worked in mathematics and was particularly interested in problems related to mathematics education as well as to the history of mathematics. Important mathematics educators of the pre- and post-War period were Kuno Fladt, Walter Lietzman, and Georg Wolff. They all held doctorates in mathematics and were important educational protagonists in Klein’s teaching reform. We will return to them when we look at the school practitioners who were instrumental in the New Math reform.



The Royaumont Seminar as a Theoretical Cradle or Practical Beginning of German New Math

The reference points for the beginning of the West German New Math reform are the Sputnik shock, following the launch of the first Sputnik by the Soviets in 1957, and the 1959 Royaumont Seminar organized by the OEEC. Both references point to economic and political rather than contentrelated causes of the New Math reform. In order to understand the specifics of the effects of the Royaumont Seminar on German education policy, we briefly discuss the invited participants and their backgrounds. Each participating country was invited to send three delegates, one outstanding mathematician, one official for mathematics in the Department of Education, and a third an outstanding secondary mathematics teacher. Although there were, as we have seen, German mathematicians interested in modernizing school mathematics, no German university mathematician participated in the Royaumont Seminar. The German delegation consisted of Heinrich Schoene, as a representative of the Ministry, and Hermann Athen, as an experienced school practitioner and textbook author. Lastly, Otto Botsch, also an experienced teacher and textbook author, gave a lecture on geometry teaching as an invited speaker. Heinrich Schoene (1910–1999) worked in the Rhineland-Palatinate (formerly a French zone) school service from 1950–1954 and in the Ministry of Education and Culture of the State of Rhineland-­ Palatinate from 1954 onward. He was also a delegate of the State Ministers of Education and the Arts in the Federal Republic of Germany (KMK) at the OECD. In this capacity, he was the editor of the Synopsis für moderne Schulmathematik (Schoene 1966), a translation of the report of an expert committee from the follow-up meeting of the Royaumont Seminar held in Dubrovnik in 1960. The original Synopses for Modern Secondary School Mathematics (OEEC 1961b) appeared as early as 1961 and formed the basis for the work on the curriculum reform for secondary schools and influenced the reform proposals of the various federal states. Between 1965–1967, Schoene was Secretary General of the German Education Council (1965–1975) in Bonn, the successor institution of the German Committee for Education (1953–1965). Based on the experience of the Gleichschaltung (enforced uniformity of society in Nazi Germany), the federal organizations in post-War Germany had been supported to strengthen pluralistic democratic structures. The primary responsibility for legislation and administration in the field of culture, with particular responsibility for language, school and university systems, education, radio, television, and the arts, lay with the German federal states. Like the KMK, the German Education Council was a body for national and transnational educational planning. Among other things, it had the task of coordinating and agreeing with the various laws, ordinances, and curricula applicable in the German federal states, due to the cultural sovereignty of the states in the education sector. With regard to national curricular and transnational teaching reforms, such as the New Math reform, the leadership of the Education Council was of strategic importance. Between 1967 and 1974, Schoene worked again in the Ministry of Education and Culture of the State of Rhineland-Palatinate and between 1975 and 1978 he was head of a regional pedagogical centre.

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Hermann Dietrich Athen (1911–1981) studied mathematics in Jena and Kiel and received his PhD in mathematics in Kiel in 1936 (Athen 1936). From 1936 onward, he was a scientific advisor for mathematical external ballistics in Berlin and published on topics in applied mathematics. In 1939 he received the prize of the Lilienthal Society for aviation research. During the War, he worked as a graduate engineer in Aachen, where he completed his habilitation14 in 1943. In 1947, he returned to his hometown Elmshorn, and from 1955 he worked as Oberstudienrat [senior teacher] and then Oberstudiendirektor [senior director]15 at the Bismarck-Gymnasium Elmshorn. Oberstudiendirektor was the fourth and highest promotion office in the civil service in the area of education and here also included the management of the secondary school as director. He had been a member of the Hamburg Mathematical Society since 1956 and a DMV member since 1965. Before and during the Royaumont Seminar, Hermann Athen was co-editor of the Elemente der Mathematik [Elements of Mathematics], a textbook series for Gymnasium that has existed since the nineteenth century. The other main editor, Georg Wolff (1886–1977), was the editor of this series during the Weimar Republic and an ardent advocate of Klein’s reform ideas, “Education for functional thinking, the training of perception and self-activity.” Athen was therefore familiar with Klein’s ideas for reform through his long-term work on the Elemente der Mathematik and his collaboration with Georg Wolff. Wolff was sceptical of the new approaches, in particular the axiomatic method, and founded the journal Praxis der Mathematik (Behnke and Stowasser 1979) in which he critically examined these ideas. By contrast, Hermann Athen embraced the new ideas. Even before the Royaumont Seminar and his textbook work, he had published on various topics in school practical journals.16 Athen played a leading role in the preparation of the Nuremberg curriculum (1965), which was the template for the development of the core curriculum (Rahmenlehrplan) of the KMK which was aimed at implementing the curricular reform of 1968. The core curriculum concerned both the content and methods of the upper secondary school level as well as the intermediate level. During the implementation phase of New Math, at the end of the 1960s and in the 1970s, Athen published, as author or co-author, numerous textbooks on the popularization and dissemination of set theory—in particular, an introductory course for parents on set theory.17 Together with the aforementioned Heinz Griesel, he published a new series of textbooks, Mathematik heute [Mathematics today]. This textbook series, which was based on the specifications of the KMK, differed greatly in regards to the mathematical concept development of the Elemente der Mathematik that remained in Klein’s spirit. Athen was also co-editor of the Lexicon of School Mathematics in five volumes (Athen and Bruhn 1976). Written in the spirit of Bourbaki, the collection was not a textbook, but pursued the encyclopaedic approach of a modern, uniform representation of existing school mathematics. It goes without saying that all subject areas of conventional and new school mathematics are covered, but presented under the uniform point of view of contemporary approaches in the style of New Math. The lexicon of mathematics is therefore not only aimed at the teacher, but also at the subject matter didactics, and the students of mathematics and educational sciences as well as at schoolchildren. (Athen and Bruhn 1976, p. V)

Athen is a particularly interesting protagonist in the New Math movement. He had an excellent and broad technical education in theoretical and applied mathematics. His practical work at school and his work as an author and editor of textbooks and journals led to his versatile approach to the reform of mathematical teaching. With Kunle, Athen published the proceedings of the Third International Congress on Mathematical Education (Athen and Kunle 1977), which took place in Karlsruhe in  A post-doctoral qualification that is a prerequisite for appointment as a full professor at a German university.  Teachers at the German Gymnasium were and are civil servants with growing status and had and have privileges like a permanent position. 16  See, e.g. Athen (1948a, b, 1955a, b, 1956, 1957). 17  In the 1970s, Athen increasingly became an author of advisory literature, reference books, and encyclopedias (Athen 1974; Athen and Ballier 1970; Athen and Bruhn 1976; Athen et al. 1972), as well as a pioneer for the use of calculators (Athen et al. 1974) and the introduction of statistics (Athen 1970). 14 15

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1976. However, perhaps due to his engineering activities during World War II and the technical orientation of the habilitation at the University of Aachen, he was open to the demands and justifications made by the OECD for the modernization of mathematics education based on economic necessities (OEEC 1961a). Although he had a doctorate and habilitation in mathematics, Athen has to be seen in the Royaumont Seminar as the representative of secondary school teachers and textbook authors. The third German participant in the Royaumont Seminar was Otto Botsch. As an invited speaker, Botsch gave a lecture on the modernization of geometry teaching (OEEC 1961a). His programmatic lecture, which stood at the beginning of the reform movement, will be analyzed later in this chapter, as it provides information about his intended reform goals. The analysis will also show the great scope for interpretation of his lecture, facilitated by the prevailing mathematical culture in Germany. Otto Botsch (1905–1990), like Athen, was a school practitioner and textbook author. As the headmaster of the Helmholtz Realgymnasium for boys in Heidelberg, he also played an important role in local education policy. He was co-author, especially for geometric aspects, of the aforementioned Lexicon of School Mathematics and author of the textbook for higher schools Geometry of Motion, Trigonometry of the Plane, Vector Geometry (Botsch 1956, new ed. 1958). The subject, which he also presented at the Royaumont Seminar, was dealt with in this textbook. His two-volume Matrices: Groups, Rings, Fields, and Vector Spaces (Botsch 1969) was also published several times. Playing with Number Squares: An Introduction to Higher-Dimensional Vector Spaces (Botsch 1967), Brain Twisters with and without Mathematics (Barnard and Botsch 1975), Plane Geometry. Size Ranges (Botsch and Eckhardt 1976), and A Geometry with 45 Points (Botsch 1977) are all textbooks that reveal not only his productivity but also the joy of an author displaying flexible, playful approaches to an axiomatic representation of objects of modern algebra. In summary, one can say that the German Royaumont participants, who were supposed to contribute to the development of a program to modernize school mathematics, had strong backgrounds in school practice. Through their teaching activities and their authorship of various approved textbooks (Stark 2011), they were well-acquainted with the implemented curricula in various federal states. All three had concrete experience with the school reforms of the post-War period and therefore had an idea of the modernizations that could be implemented in the teaching practice at that time, structured by the framework plan of the KMK from 1958 and the curricula that differed from state to state. School practitioners such as Hermann Athen, Otto Botsch, Georg Wolff, and Kuno Fladt, were among the mathematics educators who, through numerous publications, campaigned for the implementation of Kleinian ideas in the post-War period that had not been implemented in the Richertsche 1925 curriculum reform. Such contents were, for example, vector calculation, transformation geometry, and probability theory. Mathematics education journals which contained practical school implementations on such topics, but also the discussion of the axiomatic method that was already carried out in the 1950s, are Der Mathematikunterricht, Praxis der Mathematik  and Der Mathematische und Naturwissenschaftliche Unterricht (Schubring 2016).



The Networks

We have seen that the same agents appeared both in the networks of university mathematics, which were particularly interested in the development of university teaching of the subject and the teaching of mathematics in secondary schools, and in the networks of school practitioners, textbook authors, and mathematics educators. We are now trying to examine these networks from a social perspective, in addition to the perspective of personal acquaintances that we have already considered (see also Moon 2010). From a socio-historical perspective, one can differentiate the various protagonists of the New Math reform between different generations: The pre-War generation of mathematicians and upper second-

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ary school mathematics educators, the War generation whose career was interrupted by fascism and who in the post-War period formed the post-War secondary school as Studienräte, Oberstudienräte, Studiendirektoren and Oberstudiendirektoren, and the post-War generation, whose childhood and early youth fell into the Nazi period and the War years and who received their mathematical training as secondary school teachers at the universities in the post-War period. In his fundamental work for the West German history of mathematics education, Helge Lenné (1969) distinguishes two main developments of New Math, the strict and the moderate. This distinction is ideal and is made by responding to various questions. It is more difficult to establish the distinction in terms of specific people, as it is not uncommon for the same mathematics educators to have a rigorous reputation with respect to one issue and a moderate one for another. Lenné assigned Jean Dieudonné, Emil Artin, Georges Papy, Hans Hahn, Jürgen Dzewas, and also Meschkowski as the rigorous representatives. From his point of view, the moderate direction included Hans Freudenthal, Walter Jung, Hermann Athen, Richard Stender, and Franz Denk. Authors such as Botsch, Fladt, and Wolff, who wanted the final implementation of Klein’s reforms and vector calculus, Lenné would also assign to the moderate direction. Lenné mentioned Steiner as an example of a mathematics educator who published numerous characteristic works in both directions: In his case, a mediating point of view seems to become clear as well as a certain tendency to advance from originally strongly emphasized rigor to a position that is characterized by predominantly didactic argumentation. (Lenné 1969, p. 85)

Such an increasing role of pedagogy can also be observed among other New Math theorists of the post-War generation such as Griesel, Kirsch, and later Vollrath, Winter and Wittmann. Heinrich Winter and in particular Erich Wittmann took a critical position on the axiomatic method at an early stage and worked on an alternative methodology in Freudenthal’s spirit (Schubring 2018). By contrast, the university representatives engaged in New Math took rather rigorous positions. One explanation for the initially rigorous attitudes could be seen in the very small pedagogical components of university teacher education. The secondary school teacher training only took place at universities and consisted—almost exclusively—of higher mathematics. It is also worth noting that in the 1950s and 1960s, hiring opportunities for upper secondary school teachers were very good. The protagonists of the New Math movement presented here obtained doctorates in mathematics, so they were among the particularly gifted mathematics students; after completing their doctorates, they did not take the path of mathematical habilitation in mathematics, but instead passed the second state examination and went into the teaching profession, teaching mathematics at the Gymnasiums. Their relationship with the university was maintained through personal friendships, activities, teaching assignments, publications and editorial work in journals. In the mid-­ 1960s, as part of the educational expansion and the opening of Pädagogischen Hochschulen18 (PHs) [pedagogical universities], they were appointed to professorships at universities of education. Kirsch was a teacher for 10 years and held a professorship from 1966, Griesel worked in a school for 9 years, also as a head of department, had teaching assignments and was appointed to a professorship at a PH in 1967. Steiner’s career was different. He received his doctorate late (in 1969) and immediately after receiving his PhD was offered a professorship (Figure 6.2). All three theorists of the reform movement were not involved in the development of textbooks and teaching materials in the 1950s. In Germany before and immediately after World War II, the latter was based mainly on Lietzmann’s three-volume methodology of mathematics education and the content of Richter’s 1928 curriculum. Athen, on the other hand, had worked as a school practitioner and textbook author since 1951 on a traditional textbook, the educational goals and interpretation of which were determined by Klein’s ideas.

 For a more detailed description of this developments, see Schubring (2016).

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Figure 6.2  Hans-­Georg Steiner, 1972. (Photo: Konrad Jacobs, Mathematisches Forschungsinstitut Oberwolfach Collection)

Another interesting distinction between the various actors involved in the New Math was made by Behnke and Stowasser (1979). Griesel, Kirsch, Steiner, Pickert, and Barner were seen as the theoretical leaders: The content didactic backbone work for the immediate needs of practice is carried out by the teachers and didactics involved in the design and development of teaching materials: Athen, Bigalke, Engel, Freund, Holland, Hürten, Lauter, Röhrl, Schröder, H., Seebach, Linen, Vollrath, Winter, Zeitler… (Behnke and Stowasser 1979, p. 149)19

At the time of the publication of Behnke and Stowasser’s (1979) article, some of the mentioned “backbone workers” also held professorships at PHs and universities. The so-called backbreaking work seems to have meant a content didactical treatment of school mathematics, not only from the point of view of the mathematical subject systematics and the abundance of content20 but also from a school practical perspective. The mathematics educators who were appointed at a PH no longer dealt with upper secondary school mathematics, since teacher training for the Gymnasium remained at the universities. The PHs were responsible for teacher training for the Volksschule [elementary schools] and primary and lower secondary education. Another aspect that supports a differentiation between theoretical didactics of mathematics and pedagogical methodology was the change in the relationship between general educational science and didactics of mathematics. Didactics of mathematics was once the direct practical activity of imparting teacher knowledge and experience in textbooks, and methodologies, and teacher training and curriculum development were also part of its field. Didactics of mathematics was not any more the sum of unconnected activities, which institutionally and in terms of work areas were clearly assigned to two separate areas: The “Stoffdidaktik” [content didactics] for teaching

 Volkert (2016) comments the backbone work as “also later disparagingly noted” (p. 24).  The curricular problem of abundance of content (Stofffülle) is discussed in detail in Lenné (1969).

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Ysette Weiss mathematics at the Gymnasium, which had its institutional basis in Fachseminaren and Studienseminaren21 for Gymnasium teacher training, and the “Rechendidaktik” [arithmetic didactics] or “Rechenmethodik” [arithmetic methodology] for teaching at the Volksschulen which was studied at the PH. (Keitel 1980, p. 460)

With the introduction of the Hauptschule22 and the upgrading of arithmetic lessons at the Hauptschule in Fachunterricht [subject teaching], mathematics teaching was standardized and teacher training was made more scientific. This also softened the traditional separation of the branches: Rechenmethodik was for primary education at the PHs and Stoffdidaktik for upper secondary education. The attitude toward general didactics, which had played an overriding role with regard to general goals and principles of didactics of mathematics, became increasingly negative on the part of secondary school content didactics (Keitel 1983). Griesel (1975) defined the new scientific discipline of didactics of mathematics, which was supposed to replace content didactics, as coming “from the overall view of the existing didactic publications and discussion contributions” (p. 20) and described it as engineering, in which its most important research and development tasks would be the engineering production of practicable courses for the learning of mathematics. In the development of didactics of mathematics, both the methodological development of the courses anchored in the content didactics, in which psychological aspects played a role, as well as the student-oriented pedagogical view of the Rechenmethodik, came together in a universal overall view, struggling for a scientific identity, the development of didactics of mathematics in the broad sense. The New Math thus also created space for activities and projects which, through their novelty and otherness, differentiated from traditional Stoffdidaktik. Getting work scientifically published, in contrast to traditional Stoffdidaktik, opened doors to careers at PHs and universities. Since the goals of New Math were almost exclusively characterized by subject systematics and the modernization of subject content, this also led to narrow educational goals: In comparison with the general educational goals of traditional mathematics, it is noticeable that the New Mathematics did not develop any really new general goals, even more, any aspects of general, not directly related to the subject-related education, as well as the ethical references, with the exception of professional performance, appear largely eliminated. (Lenné 1969, p. 83)

At this point, we would also like to mention briefly Heinrich Bauersfeld (1926–2022), who, thanks to his textbook project “Alef,” is considered to be the theoretical head of New Math for elementary school. Bauersfeld studied to become a teacher at a Gymnasium, then earned a doctorate in mathematics and immediately switched to teaching at PHs and was appointed professor at a PH. In contrast to the theoretical leaders of the early New Math reform, which was initially intended primarily for upper secondary school mathematics, he had no practical school experience, in particular in the area of​ elementary school education. For a detailed description of his career and the Alef project, see Bauersfeld (1972), Hamann (2018), and Karp (2014). His work was grounded in the area of primary school education, a field traditionally related to diverse pedagogical and psychological perspectives. His approach to connect Stoffdidaktik with empirical (scientifically recognized) research avoided the teething problems described in the new scientific discipline of didactics of mathematics.



The New Math Reform vs. Meraner Reform

In the following, for a deeper understanding of the curricular reform, New Math will be studied in the context of the history of previous German curricular reforms of mathematics education at the Gymnasium level. It is striking that the great reforms, “Newer Geometry” of the nineteenth century, “Meraner Reform” of the first half of the twentieth century, and “New Math” have in common that, in  The practical teacher training for the Gymnasium was placed in Studien- and Fachseminaren, the duration was 2 years.  Detailed information about the type of school and the institutional reform can be found in Schubring (2016).

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retrospect, each is described as having failed (Damerow 1977; Kitz 2015; Krüger 2000b). In order to understand the meaning of failure in the different contexts, examining keywords used in the different reforms (e.g. to differentiate and delimit the Meraner Reform from the New Math reform), will be beneficial. Such keywords are, for example, Abbildungsgeometrie [transformation geometry], Bewegungsgeometrie [motion geometry], and the unification of different mathematical areas. As earlier mentioned, in a very simplified way, Lenné (1967) distinguished between rigorous and moderate views of modern mathematics. Representatives of the rigorous perspectives intended to incorporate set theory (and thus new content) to convey concepts such as vector space, transformation, and relation axiomatically and to introduce linear algebra and algebraic structures such as groups, rings, and fields. For our considerations, we would like to differentiate the moderate perspective. Under the structural view, we understand the effort to convey structural mathematical terms such as vector space, transformation, and group in geometric contexts with an emphasis on the axiomatic method and a set-­theoretical foundation. A perspective shall be called “functional” when structural mathematical concepts such as vector space, transformation, and group are to be further developed in geometric contexts in the sense of the Meraner Reform meaning “education of functional thinking.” To gain a more complete understanding of the perspectives on the curricular goals of the New Math reform, it is also useful to name those for whom both the new topics of New Math and those of the Kleinian reform for secondary education went too far. We will refer to this as the conservative view. From the conservative point of view, the complexity of the topics and the abundance of content23 should be reduced; therefore the approaches and topics such as differential and integral calculus, multi-layered conic theory, vector calculus, and basic concepts of probability theory, all introduced by the Meraner Reform, should be simplified or eliminated. One can see that there are similarities between the groups divided in this way and that a closer look is required to identify which basic views are represented. For example, both the Rigorous advocates and the Conservatives strove to reduce the curricular content with the aim of covering (at first glance) fewer complex objects and subjects. As Lenné (1967) noted, the representation of different points of view can also depend on the mathematical field and, for example, unite different curriculum developers with regard to algebraic and geometric content. Understanding the position represented at the Royaumont Seminar by the German representatives is of particular interest when taking a closer look at the invited lecture on the modernization of Euclidean geometry given by the German school practitioner and textbook author Otto Botsch. In the guidelines “Modifying Euclidean Geometry” (OEEC 1961a, b, p. 114) of the Royaumont Seminar, the proposal presented by Botsch was recommended as a possible approach to the modification of Euclidean geometry. Other approaches also recommended were that of Dieudonné, that of the Commission on Mathematics of the United States (Beberman, 1958), and that of the School Mathematics Study Group, as well as those by Howard Levi and Gustave Choquet. With regard to the distinctions we have made, several Royaumont speakers would be assigned to the rigorous view, since they wanted to reduce the consideration of geometric objects as far as possible to allow for arithmetic and algebraic structures. In the following, we are interested in what references the program presented by Botsch called for modifying Euclidean geometry, and to the Meraner Reform, and to what extent the recommended program implemented the basic principles of New Math. The distinction we have made between structural and functional views relates primarily to ideas about modernizing geometry teaching. Both perspectives can be related to Felix Klein, but they concentrate on very different thoughts from Klein. The structural view relates to the Erlangen program, the functional view to Klein’s idea of f​ unctional thinking transferred to geometry. To understand this distinction, it is helpful to consider different meanings of the term “transformation geometry” in the context of German teaching culture. This also enables us to understand more completely the modernization of mathematics education proposed by Otto Botsch. The main focus of Botsch’s modernization was the dynamization of geometry (see Botsch 1955).  The problem of abundance of content has a long tradition in German mathematics education and is referred to as Stofffülleproblem (Lenné 1969). 23

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The study of transformations has a rich tradition in German geometry teaching, both in the context of the “Newer Geometry” movement and in the Meraner Reform. As early as the mid-nineteenth century, there were courses for upper secondary education in which equations for translations and rotations as well as properties of the transformations of the plane were derived (Bretschneider 1844). In the textbooks by Henrici and Treutlein (1897), the perspective of transformation geometry ran through all three volumes. The use of physical objects and paper folding mentioned in Botsch’s lecture can also be found in these textbooks. Here transformations were a means of establishing and analyzing relationships.24 The books by Henrici and Treutlein were published for the last time in the 1920s, but other textbooks written at the beginning of the previous century also focused on mappings, transformations, and their compositions rather than on geometric figures. Particular mention should be made of Kusserow’s (1928) spatial theory for work lessons.25 The problem of developing an axiomatic for transformation geometry, raised in the discussion of Botsch’s lecture at the Royaumont Seminar, also had a tradition in Germany (see Willers 1922, also Bachmann 1959; Schwan 1929). In the 1930s, the use of transformations for concept development of the notion of a group in middle school received new structural mathematical impulses (Fladt 1933a, b). Behrend and Morgenstern (1932) wrote in their foreword to the textbook Form and Abbildung [Form and mapping]: The systematic structure of the preliminary course follows the presentation of F. Klein in elementary mathematics from an advanced standpoint. So, it is not axiomatic, but group-theoretical. Since the existence of the transformation group is assumed, the congruence theorems and the axiom of parallels can be proven. (p. 5)

For the different variants of “transformation geometry,” the background theories, apart from Klein’s theory of conservation quantities and group effects, were projective geometry and axiomatization approaches in the sense of Euclid and Hilbert on the basis of the transformation concept. The Abbildungsgeometrie [mapping geometry] was in the tradition of the Erlangen program. Nonetheless, representatives of this approach also referred to Klein’s ideas about functional thinking (e.g. Niebel 1956). However, Klein did not raise the demand for transformation geometry and groups as an end in itself but instead saw transformations as an opportunity to think flexibly and functionally (see Bender 1982). The Bewegungsgeometrie [motion geometry] was thus in the spirit of the Meraner Reform. In contrast to axiomatic approaches through transformations, the Meraner Reform was about making figures dynamic, examining relationships through variation, and looking at constant changes in space and time. But these were the goals of the Bewegungsgeometrie. In the Bewegungsgeometrie, which was also the focus of Botsch’s lecture, transformations are not used to classify various geometries as was the case in Felix Klein’s Erlangen Program. In the Erlangen Program, the concepts of transformation and group served as principles of order. In contrast to this, in the Meraner Reform, the concept of transformation was understood as a heuristic-genetic method to connect areas isolated from each other by using the same method (Krüger 2000a). The unification is therefore methodical, from within. The Bewegungsgeometrie was not a further development of transformation geometry in the spirit of the Erlangen program but transferred the idea of the ​​ Meraner Reform from the functional concept in analysis to the movement concept in geometry. This should create an inner connection between arithmetic, mechanics, and geometry by considering the changeable, the movement.26 The concept of transformation in the sense of structural mathematics allowed an external unification, a uniform treatment and representation of different areas. While the transformation geometry in the Erlangen spirit was close to structural mathematics and axiomatization, this was not the case with the Bewegungsgeometrie. In the sense of our assignment, Botsch had a functional view of the modernization of geometry. It is therefore surprising that assuming that Botsch gave a lecture on  In German textbooks referred to as Verwandschaften.  Arbeitsschulpädagogik in the tradition of Georg Kerschensteiner was quite influential in German mathematics education over a long period of time. 26  This approach is taken in Botsch (1956). 24 25

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Bewegungsgeometrie at the Royaumont Seminar, it was recommended as a possible modernization in the sense of New Math. The ambiguity of the terms “unification,” “structure,” and “transformation” shown here also demonstrates the wide scope for action that mathematics teachers and educators had due to the different possibilities of interpretation. The problem of ambiguity was less pronounced with the concept of a set since it was a new topic that was also to be taught in a very uniform manner and according to clear guidelines. The introduction of set theory into schools, however, brought the subject of content abundance back to the forefront. The topic of content abundance had also been central during the Meraner Reform through the introduction of new topics. In contrast to other countries, such as the USA (Fey 1978), differential and integral calculus was already anchored in the curriculum and had been part of the mathematical school culture in Germany since Klein’s reform. The incorporation of basic statistical concepts was also ­carried out within the framework of the idea of the ​​ interdisciplinary application and visualization-­ oriented approach of the Meraner Reform in the various textbooks of the different federal states. The Meraner approach to solving the problem of content abundance was methodical. The arrangement of the topics according to the genetic principle meant adaptation to the psychological conditions of the child and heuristic-genetic approaches. The concentration principle implied a didactic formation of terms for the entire lesson around the main term function in algebraic, arithmetic, or geometric guises. The utilitarian principle could be understood, for example, as application orientation and context development (Krüger 2000b). The mastery of complex mathematical concepts should be reached through interconnectedness, accessible contexts, the choice of suitable examples such as conic sections, and the use of various methods at different levels. In the New Math reform, the problem of the abundance of content was hardly considered: … the New Mathematics has increased rather than decreased the number of topics in secondary school mathematics—possibly more pronounced than has ever happened before in the history of secondary school mathematics. (Lenné 1969, p. 95)

Comparing the goals of the New Math reform with those of the Klein reform, one can see many similarities: The narrowing of the gap between university mathematics and school mathematics, the easier access to scientific-technical courses, the penetration of school mathematics from the perspective of higher mathematics, unification of different areas of mathematics, the inclusion of psychological aspects in the choice of mathematical content, and the arrangement of the content according to didactic principles (Krüger 2000b). The methodological approaches and “contemporary” content differed, however.



The Radical Nuremberg Curriculum

When dealing with the delimitation of the Meraner Reform and the New Math reform, we have concentrated on the area of ​​geometry and conceptual questions. When examining the Nuremberg curriculum, which formed the basis for the New Mathematics Framework Curriculum for the Gymnasium (Nürnberger Lehrpläne 1965, 1966), we will focus on algebraic topics and application contexts. Here, too, it will be worthwhile to consider, albeit briefly, the initial situation in order to help understand more fully the curricular reform. The mathematical textbooks from the period of the Weimar Republic, in which the concept of a function with applications in natural science and technology was widely developed, were rewritten in Germany during the dictatorship of the Nazi regime. Applications were misused in the course of the ideologization of mathematics teaching in Nazi Germany to strengthen defence capability, support militaristic ideas, and indoctrination with racial theories. Eliminating the latter and replacing it with other applications, for example, in surveying and astronomy, was one of the first tasks that textbook authors faced in the post-War period. A further rewriting and reorganization of the curriculum in the textbooks of Gymnasiums was initiated by the guidelines for the conversion of

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the three-tiered school system into a uniform school system. The abolition of the tripartite school system had been decreed by the Allied powers. Since this meant the expansion of the Gymnasium line by two classes, a 5th and 6th class (see, e.g. Stark 2011), the curriculum had to be rethought. In this context, the publishers suggested new editions of the Gymnasium textbook series. In the 1950s, there were already various discussions in the German mathematics education community that involved changes in the secondary level curricula, as well as new editions and extensive revisions of textbooks. On the one hand, these discussions and content didactic suggestions were aimed at the implementation of ideas of the Meraner Reform that had been realized only rudimentarily so far, such as transformation geometry and a comprehensive implementation of functional thinking in each of the areas of algebra, geometry, arithmetic, and analysis. On the other hand, numerous suggestions for algebraization, structural mathematical penetration, and axiomatic presentation of topics were made in practical school journals, but also increasingly, from 1959, in the Semesterberichte. The first reform approaches could already be found at the end of the 1950s in the guidelines and 1958 framework plans of the KMK and in the secondary school curricula of the federal states (Keitel 1980). Characteristic formulations for the first reform efforts are: If possible, newer concepts (such as quantity, transformation, vector, group) should be used as long as they can be integrated organically into the lesson. (Damerow 1977, p. 182)

As early as the 1950s, Der Mathematikunterricht devoted three issues to vector calculation, with Hermann Athen’s contribution in the first issue providing an interesting overview of the current state in terms of vector calculus (Athen 1955a). Reidt-Wolff’s revision of the Elemente der Mathematik, published in the 1950s, contained numerous interesting suggestions for concept development of transformations. In the Semesterberichte, several works on these topics, including two by Athen (1955c, 1962), were published, some of which already dealt with vector spaces (see Volkert 2016). In 1966, Schoene’s commented German translation of the guidelines for the modern processing of mathematical curricula in secondary schools (OEEC 1961b) was published. This document was drawn up under the auspices of the OECD and expressed a perspective on education which primarily focused on the increase of economic power of the member states of the OECD. The underlying perspective on education as specialized professional training was not unconditionally in line with, and sometimes even contradictory to, German educational ideals in the 1950s and 1960s (Theodor W.  Adorno, Jürgen Habermas, Max Horkheimer, Wolfgang Klafki). The Oberwolfacher mathematics education conferences, the Münster conferences Tagungen zur Pflege des Zusammenhangs von Universität und höherer Schule [Münster Conferences to promote connection between university and upper secondary school], the work of the German subcommission of ICMI, as well as the publications in mathematics education journals, including lectures and proceedings, were of great importance in sharpening the ideas for reforming mathematics education. There were several proposals from the federal states for the development of a new nationwide curricular framework. From a curricular point of view, Lenné (1969) provides an analysis of 11 then-­ current proposed curricula from different German federal states. Schuberth (1971) subjected the various reform proposals to an analysis from the point of view of general educational goals. The 1962 Hamburg draft reform was the first and probably the most radical plan. Although it was claimed in the cover letter that the new plan did not deviate in content from the traditional curriculum (guidelines and framework plan for mathematics lessons of the KMK from 1958), and that only the perspectives and the linguistic representation had changed, in fact, this was the only draft in which the traditional concept development structure was not adopted (Damerow 1977). The 1965 Nuremberg framework curriculum was initially created by a commission made up of eight representatives from various federal states, including Athen, Botsch, and Steiner. At times, the commission was expanded to include almost 50 experts from schools and universities. The framework curriculum was drawn up exclusively by Gymnasium and university teachers, although it affected all types of schools. Several members of the commission were also authors of teaching materials for the Gymnasium. The framework curriculum was presented at the general meeting of the Hauptversammlung des Vereins zur Förderung des math-

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ematischen und naturwissenschaftlichen Unterrichts [Association for the Promotion of Mathematics and Science Education] and met with strong rejection (Athen 1966; Wigand 1965). The discussions were printed in the journals Der Mathematikunterricht and Praxis der Mathematik, which shows that at the time efforts were made to involve secondary school teachers on a broad basis. The intended methodological design of the curriculum proposals was also presented in Steiner’s (1965) paper about quantity, structure, and transformation as key concepts for modern mathematical teaching, as well as Griesel’s (1965) article about the concept “set” as a key idea, and Kirsch’s (1965) paper about the visualization of simple group homomorphisms. Lenné (1969) mildly expressed the reasons for the cautious reaction of the teaching staff in the following way: That here a didactically particularly active group of syllabus authors, who were at the same time particularly closely connected to higher mathematics, faced a majority of mathematics teachers who were essentially still committed to traditional mathematics. (p. 79)

Nevertheless, the rejection is astonishing, since the traditional canon was retained in the Nuremberg curricula, and set theory and structural mathematics were to be treated inductively. The usefulness of the concept of a set should be particularly evident in the formulation of the concept of variable, the theory of equations, relations and functions, especially in relation to definitions and magnitudes. The application of the axiomatic method was only intended as an example. Structure was seen as a guiding and ordering principle. Clarity was no longer understood in the Kleinian sense, but meant habituation and familiarity in dealing with terms and symbols (Athen 1966). A central problem with the proposals was the abundance of content. The Hamburg draft reform remained without practical consequences (Damerow 1977; Lenne 1969). This certainly contributed to making the Nuremberg framework curriculum much more traditional. As a result, however, new content was added to the existing extensive topics, structured in Kleinian sense, instead of organizing them according to other principles. An example that shows the interaction of traditional concept formation with the new ideas is the Berlin framework plan, where the concept of number field was used when dealing with number range extensions (see, e.g. Damerow 1977). In the new guidelines, it did not become a set-theoretical derivation of number range extensions, but served only to justify some properties of the number concepts, which were previously explained with the principle of permanence, now through the desired properties of a field. The cautious implementation of the reform plans, clinging to traditions, was probably also based on ideas of what could be implemented in saleable textbooks. As already mentioned, using the example of Athen and Botsch, the textbook authors were able to gain experience with the new editions of the post-War period, as well as with their saleability. The “Nuremberg curriculum” also concerned a reform of the natural sciences and thus the applications of mathematics. Both in the selection of the content and the scope of school lessons, the Nuremberg curricula went beyond the guidelines on which the curricula of the German Higher Schools Working Group (AGDHS) were based (see Kremer 1985). In physics lessons for the upper level, the plan was not only to modernize the traditional subjects of mechanics, such as models of light and electricity, but also to enlarge the focus on “atomic and nuclear physics.” Accordingly, the way of working in physics lessons should be based on the thinking and research methods of the natural sciences, a goal that contained a new quality insofar as it was intended to professionalize the students in physics lessons in accordance with scientific research practice. For example, in mechanics through the topics “Problems of space research” and “Basic facts of the theory of relativity,” as well as the expansion of the model concept of light through quantum mechanical considerations. (Nürnberger Lehrplan 1965, p. 10)

The new teaching content shows that the mathematization of the natural sciences had to be implemented in school science programs as well. The Volkswagenwerk Foundation, established in 1961 by the federal government and the state of Lower Saxony, influenced the steering of the science curriculum reform in a way that should not be underestimated. It was the first funding institution to enter into a systematically organized supra-regional support for educational research and training. The Volkswagenwerk Foundation was the largest science-promoting foundation in Europe. It focused primarily on reforming the content of mathematics and science education. In addition, large-scale

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experiments to redesign mathematics lessons in elementary schools were funded, and an extensive program was prepared to improve mathematics and science education at universities. It ranged from the development of new curricula for early classes to the establishment of model chairs for mathematics and natural science education and the establishment of the Institute for Didactics of Mathematics in Bielefeld (IDM).



The End in the Beginning

Despite the moderate goals of the Nuremberg framework curriculum, it was perceived as radical in many places. There are various content-related reasons for this, some of which have already been indicated, like a well-established German textbook culture with the principles of inductive mathematical concept development, internal unification of the topics by arrangement according to didactic and non-subject-systematic aspects, the fundamental importance of Anschauung [perception] and applications, as well as the concept of student self-activity. Furthermore, it can be assumed that the understanding of modern university algebra was limited among the teachers who were supposed to implement this reform. The reform of teacher training should have taken place beforehand. New Math activists did not teach future Gymnasium teachers, but teachers for elementary and primary education, which did not include upper secondary education. Mathematics education in PHs, which lasted at most six semesters, was not research-oriented and by no means modern. The problem of content abundance, which was exacerbated by the growing heterogeneity in classrooms due to educational expansion, was further enlarged by the additional New Math topics. Some radical theorists of New Math as professors at PHs changed their one-sided focus on subject-systematic problems; their perspectives became more moderate and pedagogically sound. The movement of the 1960s also brought back the discussion of the educational value of mathematics and its abuse during German Nazism. Discussions about institutional reforms of the German school system, which was perceived as socially unjust because of early selection, also intensified during the 1960s in the context of the West German 1968 movement. As Gerwin Schefer shows in his empirical study on the social image of secondary school teachers, a strong elitist consciousness was anchored in German Gymnasium teachers, coupled with fears of status loss and the resulting politically conservative views (Lenné 1967; Schefer 1969). On the one hand, the university-like use of deductive methods and uniform presentations seemed to support the self-image as a mathematician, but on the other hand, the appropriation of mathematics through familiarization and internalization of automated skills contradicted this elitist consciousness and traditional teaching. In the late 1960s and early 1970s, the reform movement was increasingly ideologized. The transfer of the reform, initially planned for the elitist Gymnasial mathematics, to elementary and intermediate levels led to the inclusion of a wider public in the discourse on educational policy. The relatively short period of 4 years given for implementing the specifications of the new framework plan of the KMK led to a forced production of handouts, teaching materials, and textbooks. For the upper school level, the incorporation of structural mathematical and axiomatic concept development was a high conceptual requirement, but the simplification of these ideas for the elementary and intermediate levels was an even higher one. Since most of the primary school teachers themselves had learned only the basics of mathematics and arithmetic methodology, not only the new methods but also the new contents were unfamiliar and strange to them. Instead of structuring simple paradigmatic examples, naming and dealing with abstract structures became more and more the subject of teaching. The teaching method suitable for this was provided by programmed lessons,27 which had been gaining popularity for all  The theories of teaching machines (Correll 1968) provided pedagogical approaches that, at first sight, fitted well with the axiomatic method and structural mathematics and seemed to offer solutions for both the problem of Stofffülle and that of the larger and more heterogeneous student population in Gymnasiums. 27

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school types since the late 1950s. At this point, however, a detailed discussion of the reform movement in elementary and secondary schools would go beyond the scope of this chapter. The reactions from the grassroots, i.e. the Gymnasium teachers who were supposed to implement the Nuremberg curriculum reform, to the reform itself showed that it was not sufficiently supported as a movement from below. Nevertheless, in an imposed reform from above, both new content and new methods were implemented, using textbooks and curricula as the actual instruments of the reform. Some of the associated problems were discussed in the context of transformation geometry. In 1972, the curricular reform of the upper secondary level was followed by a structural reform of the same level. The 1972 Bonn Agreement of the Conference of Ministers of Education and Cultural Affairs established a course and matriculation examination system that focused on minimum standards and learning goal orientation (Damerow 1977). Interconnected topics owed to the Meraner Reform, such as conic sections, disappeared from the compulsory curricula. As a result, and through the strengthening of the preparatory role of the Gymnasium for university studies, the analytic geometry in the Gymnasium was transformed into introductory linear algebra. A characteristic of German didactics of mathematics is its strong subject-matter didactic tradition (Hefendehl 2016). In particular, the Meraner Reform, with its focus on the independent activity of students, the important role of interdisciplinary perspectives and applications, as well as functional, interrelated thinking, led to a thorough content analysis of all traditional school mathematics with elementary means. As a result, at the end of the 1950s, there existed an extensive engineering production of practicable courses for the learning of Gymnasial mathematics. The discipline of mathematics education was concerned with the development of such courses and was thus a science of engineering. New Math gave space to develop alternative content didactical concepts, which were not based on the use of elementary methods but on those of modern algebra. The new pedagogical approaches to deductive concept development also created the possibility of a renewed demarcation from the classic subject-matter didactical “backbreaking work.” The discourse on mathematics education as an academic discipline, its promotion, international networking, and institutional development became the determining topic of German mathematics education in the 1970s (Figure 6.3). The election of three

Figure 6.3  ICME-5 in Adelaide (1984), Bent Christiansen (with bag), Hans-Georg Steiner, and Claude Gaulin. (Photo collection H.-G. Steiner)

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directors, Hans-Georg Steiner, Heinrich Bauersfeld, and Michael Otte, as well as the work areas and priorities of research at the IDM in Bielefeld, reflect these research interests (Biehler and Peter-Koop 2007; Hefendehl 2016; Schubring 2018). Perhaps it is therefore appropriate not to speak of the failure of the New Math reform but of the disappearance of Stoffdidaktik which was primarily determined by subject-systematic and subject-matter aspects.

References Ahlfors, L. V., Bacon, H. M., Bell, C., Bellman, R. E., Bers, L., Birkhoff, G., … Wittenberg, A. (1962). On the mathematics curriculum of the high school. American Mathematical Monthly, 69(3), 189–193. Athen, H. (1936). Über Flächen mit Bertrand-Kurven gleicher Familie unter besonderer Berücksichtigung der Razzabonischen Transformation [On surfaces with Bertrand curves of the same family with special reference to the Razzaboni transformation]. Würzburg, Germany: Dissertationsdruckerei und Verlag Konrad Triltsch. Athen, H. (1948a). Ebene und sphärische Trigonometrie (Bücher der Mathematik und Naturwissenschaften) [Plane and spherical trigonometry (Books in mathematics and natural sciences]. Hannover, Germany: Wolfenbütteler Verlagsanstalt. Athen, H. (1948b). Vektorrechnung (Bücher der Mathematik und Naturwissenschaften) [Vector calculus (Books in mathematics and natural sciences)]. Hannover, Germany: Wolfenbütteler Verlagsanstalt. Athen, H. (1955a). Die Vektorrechnung im Unterricht der deutschen höheren Schulen [Vector calculus in the teaching at German secondary schools]. Der Mathematikunterricht, 1(1), 5–21. Athen, H. (1955b). Einführung in die Statistik (einschl. Kombinatorik und Wahrscheinlichkeitsrechnung) [Introduction to statistics (incl. combinatorics and probability)]. Paderborn, Germany: Schroedel/Schöningh. Athen, H. (1955c). Vektorrechnung auf der Kugel [Vector calculus on the sphere]. Mathematisch-physikalische Semesterberichte, 4, 90–100. Athen, H. (1956). Nomographie [Nomography]. Frankfurt am Main, Germany: O. Salle. Athen, H. (1957). Einige zeichnerische Methoden in der Analysis [Some graphical methods in analysis]. Der Mathematikunterricht, 3(4), 65–81. Athen, H. (1962). Vektorielle Begründung der Trigonometrie [Vectorial foundation of trigonometry]. Mathematisch-­ physikalische Semesterberichte, 8, 83–94 Athen, H. (1966). Die Modernisierungstendenzen im Nürnberger Rahmenplan für Mathematik [The modernization tendencies in the Nuremberg framework for mathematics]. Der Mathematikunterricht, 12(3), 87–106. Athen, H. (1970). Wahrscheinlichkeitsrechnung und Statistik [Probability and statistics]. Paderborn, Germany: Schroedel/Schöningh. Athen, H. (Ed.). (1974). Mathematik zum Nachschlagen für Schule, Studium und Beruf: Ein Handbuch mit vielen Beispielen und Anwendungen [Mathematics for reference at school, university and at work: A handbook with many examples and applications]. Gütersloh, Germany: Bertelsmann. Athen, H., & Ballier, F. (1970). Rechnen und Mathematik für Schüler und Eltern: Nach den Empfehlungen und Richtlinien der Kultusministerkonferenz des Mathematikunterrichts [Arithmetic and mathematics for students and parents: According to the recommendations and guidelines for the teaching of mathematics of the Standing Conference of the State Ministers of Education and the Arts in the Federal Republic of Germany]. Gütersloh, Germany: Bertelsmann. Athen, H., & Bruhn, J. (Eds.). (1976). Lexikon der Schulmathematik und angrenzender Gebiete [Lexicon of school mathematics and related fields]. Köln, Germany: Aulis-Verlag Deubner. Athen, H., Bruhn, J., & Dietrich, G. (1972). Logik leicht gemacht: Für Schüler und Eltern [Logic made easy: For students and parents]. Gütersloh, Germany: Bertelsmann-Ratgeberverlag. Athen, H., Bruhn, J., & Pohlmann, D. (1974). Blitzrechnen mit dem Elektronik-Taschenrechner [Quick calculation with the electronic calculator]. München, Germany: Mosaik Verlag. Athen, H., & Kunle, H. (1977). Proceedings of the Third International Congress on Mathematical Education. Karlsruhe, Germany: Zentralblatt für Didaktik der Mathematik. Bachmann, F. (1959). Aufbau der Geometrie aus dem Spiegelungsbegriff [Constructing geometry from the concept of reflection]. Berlin-Heidelberg, Germany: Springer. Barnard, D., & Botsch, O. (1975). Hirnverzwirner mit und ohne Mathematik [Brain twisters with and without mathematics]. Köln, Germany: Aulis Verlag Deubner. Bauersfeld, H. (1972). Einige Bemerkungen zum “Frankfurter Projekt” und zum “alef”-Programm [Some remarks on the “Frankfurt Project” and the “alef” program]. In E. Schwartz (Ed.), Materialien zum Mathematikunterricht in der Grundschule [Resources for teaching mathematics in elementary school] (pp. 237–246). Frankfurt am Main, Germany: Arbeitskreís Grundschule e.V.

6  West German Neue Mathematik

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Beberman, M. (1958). An emerging program of secondary school mathematics. Cambridge, MA: Harvard University Press. Behnke, H., Fladt, K., & Süss, W. (Eds.). (1958). Grundzüge der Mathematik [Basic mathematics]. Göttingen, Germany: Vandenhoeck & Ruprecht. Behnke, H., & Stowasser, R.  J. K. (1979). Der gymnasiale Mathematikunterricht im Lichte der Wolffschen Unternehmungen [Teaching mathematics in Gymnasium in the light of Wolff’s undertakings]. Mathematisch-­ physikalische Semesterberichte, 26, 145–153. Behrend, F., & Morgenstern, A. (1932). Lehrbuch der Mathematik. Form und Abbildung [Textbook of mathematics. Shape and figure]. Breslau, Poland: Hirt. Bender, P. (1982) Abbildungsgeometrie in der didaktischen Diskussion [Mapping geometry in the didactical discussion]. Zentralblatt für Didaktik der Mathematik, 82(1), 9-24. Biehler, R., & Peter-Koop, A. (2007). Hans-Georg Steiner: A life dedicated to the development of didactics of mathematics as a scientific discipline. ZDM Mathematics Education, 39(1), 3–30. Botsch, O. (1955) Die Bewegung als methodisches Prinzip im Geometrie-Unterricht de Mittelstufe [Motion as a methological pinciple to teach geometry in lower secundary school]. Frankfurt, Germany: Diesterweg. Botsch, O. (1956, new ed. 1958). Bewegungsgeometrie [Motion geometry]. Frankfurt, Germany: Diesterweg. Botsch, O. (1967). Spiel mit Zahlenquadraten: eine Einführung in höherdimensionale Vektorräume [Playing with number squares: An introduction to higher-dimensional vector spaces]. Frankfurt am Main, Germany: O. Salle. Botsch, O. (1969). Die Matrix 1. Theorie und Anwendung in Algebra und Geometrie. Gruppen, Ringe, Körper und Vektorräume [Matrices 1. Theory and application in algebra and geometry. Groups, rings, fields, and vector spaces]. Frankfurt am Main, Germany: Diesterweg-Salle. Botsch, O. (1977). G 45, eine Geometrie mit 45 Punkten [A geometry with 45 points]. Frankfurt am Main, Germany: Diesterweg-Salle. Botsch, O., & Eckhardt, H. (1976). Ebene Geometrie. Grössenbereiche [Plane geometry. Size ranges]. Frankfurt am Main, Germany: Diesterweg-Salle. Bretschneider, C. A. (1844). Lehrgebäude der niederen Geometrie: Für den Unterricht an Gymnasien und höheren Realschulen [Teaching principles of basic geometry: For teaching at Gymnasiums and upper secondary schools]. Jena, Germany: F. Fromann. Correll, W. (1968). Programmiertes Lernen und Lehrmaschinen (Theorie und Praxis der Schule) [Programmed learning and teaching machines (Theory and practice of schools)]. Braunschweig, Germany: Westermann. Damerow, P. (1977). Die Reform des Mathematikunterrichts in der Sekundarstufe I. Band 1: Reformziele, Reform der Lehrpläne. Max-Planck-Institut für Bildungsforschung [The reform of mathematics education in lower secondary schools. Vol. 1: Reform goals, reform of curricula. Max Planck Institute for Human Development]. Stuttgart, Germany: Klett-Kotta. De Bock, D., & Vanpaemel, G. (2015). Modern mathematics at the 1959 OEEC Seminar at Royaumont. In K.  Bjarnadóttir, F.  Furinghetti, J.  Prytz, & G.  Schubring (Eds.), “Dig where you stand” 3. Proceedings of the Third Internatonal Conference on the History of Mathematics Education (pp. 151–168). Uppsala, Sweden: Uppsala University, Department of Education, Uppsala University. Fey, J. T. (1978). Change in Mathematics Education since the late 1950s—Ideas and realisation: USA. Educational Studies in Mathematics, 9(3), 339–353. Filler, A. (2016). Weg von Euklid und wieder zurück? [Away from Euclid and back again?]. Mathematische Semesterberichte, 63(1), 93–134. Fladt, K. (1933a). Die Auswirkung des Abbildungsgedankens in den heutigen Schulbüchern [The impact of the idea of mapping in today’s textbooks]. Unterrichtsblätter für Mathematik und Naturwissenschaften, 39, 155–158. Fladt, K. (1933b). Gruppenbegriff und Abbildung im mathematischen Schulunterricht [Group concept and mapping in school mathematics]. Zeitschrift für Mathematischen und Naturwissenschaftlichen Unterricht, 64, 204–205. Freudenthal, H. (1971). Die neuen Tendenzen im Mathematikunterricht [The new tendencies in mathematics teaching]. Neue Sammlung, 11(2), 146–153. Furinghetti, F., Giacardi, L., & Menghini, M. (2020). Actors in the changes of ICMI: Heinrich Behnke and Hans Freudenthal. In É Barbin, K. Bjarnadóttir, F. Furinghetti, A. Karp, G. Moussard, J. Prytz, & G. Schubring (Eds.), “Dig Where You Stand” 6: Proceedings of the Sixth International Conference on the History of Mathematics Education (pp. 247–260). Münster, Germany: WTM. Griesel, H. (1965). Die Leitlinie Menge—Struktur im gegenwärtigen Mathematikunterricht [Sets and structure as guiding principles in mathematics education ]. Der Mathematikunterricht, 11(1), 40–53. Griesel, H. (1975). Stand und Tendenzen der Fachdidaktik Mathematik in der Bundesrepublik Deutschland [State and trends of didactics of mathematics in the Federal Republic of Germany]. Zeitschrift für Pädagogik, 21, 19–31. Griesel, H. (2012). Zum Tode von Heinz Kunle [On the death of Heinz Kunle]. Mitteilungen der Gesellschaft für Didaktik der Mathematik, 38(93), 87–91. Hamann, T. (2018). Die “Mengenlehre” im Anfangsunterricht: Historische Darstellung einer gescheiterten Unterrichtsreform in der Bundesrepublik Deutschland [“Set Theory“ in early teaching: A historical account of a failed teaching reform in the Federal Republic of Germany]. Siegen, Germany: Universität Siegen.

124

Ysette Weiss

Hartmann, U. (2009). Heinrich Behnke (1898–1979): Zwischen Mathematik und deren Didaktik [Heinrich Behnke (1898–1979): Between mathematics and its didactics]. Frankfurt am Main, Germany: Peter Lang. Hefendehl-Hebeker, L. (2016). Subject-matter didactics in German traditions. Journal für Mathematik-Didaktik, 37(1), 11–31. Henrici, J., & Treutlein, P. (1897). Lehrbuch der Elementar-Geometrie: Gleichheit der Gebilde in einer Ebene (Band 1. 2. Auflage) [Textbook of elementary geometry: Equality of figures in a plane (Vol. 1, 2nd ed.)]. Leipzig, Germany: B. G. Teubner. Heske, H. (2021). Mathematikunterricht im Nationalsozialismus [Mathematics education under Nazism]. Mathematische Semesterberichte, 68(1), 119–142. Howson, G., Keitel, C., & Kilpatrick, J. (1981). Curriculum development in mathematics. Cambridge, United Kingdom: Cambridge University Press. Huebener, T. (1952). Have the German schools been democratized? The Modern Language Journal, 36(3), 115–119. Karp, A. (2014). Interview with Heinrich Bauersfeld. In A. Karp & D. L. Roberts (Eds.), Leaders in mathematics education: Experience and vision (pp. 31–55). Rotterdam, The Netherlands: Sense Publishers. Keitel, C. (1980) Entwicklungen im Mathematikunterricht. In Bildung in der Bundesrepublik Deutschland. Daten und Analysen. Band 1: Entwicklungen seit 1950. Max-Planck-Institut für Bildungsforschung, Projektgruppe Bildungsbericht [Education in the Federal Republic of Germany. Data and Analyses. Vol. 1: Developments since 1950. Max Planck Institute for Human Development, Project Group Education]. Stuttgart, Germany: Rowohlt, 447–449. Keitel, C. (1983). Zum Verhältnis der Mathematikdidaktik zur Allgemeinen Didaktik [On the relationship between didactics of mathematics and general didactics]. Zeitschrift für Pädagogik, 29(4), 595–603. Kilpatrick, J. (2012). The new math as an international phenomenon. ZDM Mathematics Education, 44(4), 563–571. Kirsch, A. (1965). Über die Veranschaulichung einfacher Gruppenhomomorphismen [About the visualization of simple group homorphisms]. Der Mathematikunterricht, 11(1), 54–68. Kitz, S. (2015). “Neuere Geometrie” als Unterrichtsgegenstand der höheren Lehranstalten. Ein Reformvorschlag und seine Umsetzung zwischen 1870 und 1920 [“Newer geometry” as a teaching subject in institutions for higher education. A reform proposal and its implementation between 1870 and 1920]. Unpublished doctoral dissertation, University of Wuppertal, Wuppertal, Germany. Klafki, W. (1967). Studien zur Bildungstheorie und Didaktik [Studies in educational theory and didactics]. Weinheim, Germany: Beltz. KMK. (1968). Empfehlungen und Richtlinien zur Modernisierung des Mathematikunterrichts an den allgemeinbildenden Schulen. Beschluß der Kultusministerkonferenz vom 3.10.1968 [Recommendations and guidelines for modernizing mathematics instruction in schools for general education. Resolution of the Conference of Ministers of Education and Cultural Affairs of October 3, 1968]. Sammlung der Beschlüsse der Ständigen Konferenz der Kultusminister der Länder der Bundesrepublik Deutschland, 611. Kremer, A. (1985). Naturwissenschaftlicher Unterricht und Standesinteresse zur Professionalisierungsgeschichte der Naturwissenschaftslehrer an höheren Schulen [Science teaching and professional interest in the history of professionalization of science teachers in institutions for higher education]. Marburg, Germany: Soznat. Krüger, K. (2000a). Kinematisch-funktionales Denken als Ziel des höheren Mathematikunterrichts—das Scheitern der Meraner Reform [Kinematic-functional thinking as a goal of higher mathematics education—the failure of the Meraner reform]. Mathematische Semesterberichte, 47, 221–241. Krüger, K. (2000b). Erziehung zum funktionalen Denken. Zur Begriffsgeschichte eines didaktischen Prinzips [Education for functional thinking. On the conceptual history of a didactic principle]. Berlin, Germany: Logos Verlag. Kusserow, W. (1928). Los von Euklid! Eine Raumlehre für den Arbeitsunterricht, durchgehend auf Bewegung gegründet [Away from Euclid! A spatial theory for work lessons, based on motion]. Leipzig, Germany: Dürr’scher Verlag. Lang, S. (1994). Mordell’s review, Siegel’s letter to Mordell, diophantine geometry, and 20th century mathematics. Mitteilungen der Deutschen Mathematiker-Vereinigung, 2(4), 20–31. Laugwitz, D. (1965). Zur sogenannten Modernisierung des mathematischen Schulunterrichts [On the so-called modernization of mathematics education]. Jahresberichte der Deutschen Mathematiker-Vereinigung, 67 (Beilage), 31–34. Laugwitz, D. (1966). Sinn und Grenzen der axiomatischen Methode [Sense and limits of the axiomatic method]. Der Mathematikunterricht, 12(3), 16–39. Lenné, H. (1967). Jugend zwischen Tradition und Demokratie: Struktur und politische Bedeutung der Einstellung von Oberschülern zu politischer Autorität [Youth between tradition and democracy: The structure and political meaning of secondary school students’ attitudes toward political authority]. Berlin, Germany: Duncker & Humblot. Lenné, H. (1969). Analyse der Mathematikdidaktik in Deutschland [Analysis of didactics of mathematics in Germany]. Stuttgart, Germany: Ernst Klett Verlag. Meschkowski, H. (1965). Mathematik als Bildungsgrundlage [Mathematics as the basis of education]. Braunschweig, Germany: Friedrich Vieweg. Moon, B. (1986). The “New Maths” curriculum controversy: An international story. Barcombe, United Kingdom: Falmer Press.

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Moon, B. (2010). The historical and social context of curriculum. In J. Arthur & I. Davies (Eds.), The Routledge education studies textbook (pp. 154–165). London, United Kingdom: Routledge. Niebel, W. (1956). Das Erlanger Programm und die Geometrie der Abbildungen [The Erlangen program and the geometry of mappings]. Der Mathematikunterricht, 2(2),7–19. Nürnberger Lehrpläne des Deutschen Vereins zur Förderung des mathematisch-naturwissenschaftlichen Unterrichts. (1965). Der Mathematische und Naturwissenschaftliche Unterricht, 18(1), 1–18. Nürnberger Lehrpläne. (1966). Kritiken und Erwiderungen [Criticisms and responses]. Der Mathematische und Naturwissenschaftliche Unterricht, 18(12), 433–443. OEEC. (1961a). New thinking in school mathematics. Paris, France: OEEC. OEEC. (1961b). Synopses for modern secondary school mathematics. Paris, France: OEEC. Schefer, G. (1969). Das Gesellschaftsbild des Gymnasiallehrers [The social image of secondary school teachers]. Frankfurt am Main, Germany: Suhrkamp. Schoene, H. (Ed.). (1966). Synopsis für moderne Schulmathematik [Synopses for modern secondary school mathematics]. Frankfurt, Germany: Diesterweg. Schuberth, E. (1971). Die Modernisierung des mathematischen Unterrichts. Ihre Geschichte und Probleme unter besonderer Berücksichtigung von Felix Klein, Martin Wagenschein und Alexander I. Wittenberg [The modernization of mathematics education. Its history and problems with special reference to Felix Klein, Martin Wagenschein, and Alexander I. Wittenberg]. Stuttgart, Germany: Verlag Freies Geistesleben. Schubring, G. (2016). Die Entwicklung der Mathematikdidaktik in Deutschland [The development of didactics of mathematics in Germany]. Mathematische Semesterberichte, 63(1), 3–18. Schubring, G. (2018). Die Geschichte des IDM Bielefeld als Lehrstück [The history of IDM Bielefeld as a lesson]. Aachen, Germany: Shaker Verlag. Schwan, W. (1929). Elementare Geometrie [Elementary geometry]. Leipzig, Germany: Akademische Verlagsgesellschaft. Segal, S. L. (2003). Mathematicians under the Nazis. Princeton, NJ: Princeton University Press. Servais, W. (1975). Continental traditions and reforms. International Journal of Mathematical Education in Science and Technology, 6(1), 37–58. Stark, J. (2011). 65 Jahre Lambacher Schweizer. Ein Klassiker—immer auf der Höhe der Zeit [65 years of Lambacher Schweizer. A classic—always up to date]. Stuttgart, Germany: Ernst Klett Verlag GmbH. Steiner, H.-G. (1959). Das moderne mathematische Denken und die Schulmathematik [Modern mathematical thinking and school mathematics]. Der Mathematikunterricht, 5(4), 5–79. Steiner, H. G. (1965). Menge, Struktur, Abbildung als Leitbegriffe für den mathematischen Unterricht [Sets, structure and functions as guiding principles in mathematics education]. Der Mathematikunterricht, 11(1), 5–19. Steiner, H.-G. (1969). Examples of exercises in mathematization: An extension of the theory of voting bodies. Educational Studies in Mathematics, 1(3), 289–299. Steiner, H.-G. (1977). Günter Pickerts Beiträge zur Didaktik der Mathematik. Aus Anlass seines 60. Geburtstages [Günter Pickert’s contributions to the didactics of mathematics. On the occasion of his 60th birthday]. Mathematisch-­ physikalische Semesterberichte, 24, 151–171. Steiner, H.-G. (1980). School curricula and the development of science and mathematics. International Journal of Mathematical Educational in Science and Technology, 11(1), 97–106. Volkert, K. (2016). Die “Semesterberichte” und die Entwicklung der Mathematikdidaktik in der Bundesrepublik Deutschland (1950–1980) [The “Semesterberichte” and the development of didactics of mathematics in the Federal Republic of Germany (1950–1980)]. Mathematische Semesterberichte, 63(1), 19–68. Vollrath, H. (1991). Betrachtungen zur Entwicklung der Algebra in der Lehre [Reflections on the development of algebra in education]. Mathematische Semesterberichte, 38, 58–98. Wagenschein, M. (1989). Erinnerungen für morgen. Eine pädagogische Autobiographie [Memories for tomorrow. An educational autobiography]. Weinheim: Germany: Beltz Verlag. Wigand, K. (1965). Foerdervereinstagung in Nuernberg. Die 56. Hauptversammlung des Deutschen Vereins zur Foerderung des mathematischen und naturwissenschaftlichen Unterrichts vom 11.-15. April in Nuernberg [Association for the Promotion of Mathematics and Science in Nuremberg. The 56th General Meeting of the German Association for the Promotion of Mathematics and Science Education from April 11 to 15 in Nuremberg]. Praxis der Mathematik, 7(5), 127–130. Willers, H. (1922). Die Spiegelung als primitiver Begriff im Unterricht [The reflection as a primitive concept in teaching]. Zeitschrift für Mathematischen und Naturwissenschaftlichen Unterricht, 53, 68–77, 109–119. Wittenberg, A.  I. (1963). Bildung und Mathematik: Mathematik als exemplarisches Gymnasialfach [Education and mathematics: mathematics as an exemplary secondary school subject]. Stuttgart, Germany: Klett.

Chapter 7

New Mathematics in the United Kingdom: Projects and Textbooks as Driving Forces of Curriculum Reform Leo Rogers

Abstract  The 1950s in the United Kingdom were marked by social and political reforms, which also led to new conceptions of mathematics teaching and learning. For mathematics, the Association for Teaching Aids in Mathematics, founded in 1952 by Caleb Gattegno and like-minded people, and internationally fostered by the International Commission for the Study and Improvement of Mathematics Teaching, played an important role in this. The 1959 Royaumont Seminar served as a booster for curriculum change in the UK, bringing in influences from the continent as well as from the United States of America. In its wake, several projects with accompanying textbooks and in-service teacher training programs emerged in the early 1960s. Most influential were the School Mathematics Project for secondary education and the Nuffield Mathematics Project for primary education, projects that were also implemented, in part and/or adapted, in some countries outside the UK. From the 1970s onward, criticism of the reform reverberated more loudly and led to the fall of the new mathematics paradigm in the UK. Keywords  Association for Teaching Aids in Mathematics · Association of Teachers of Mathematics · Bryan Thwaites · Caleb Gattegno · CIEAEM · Cockcroft Report · Contemporary School Mathematics Project · Cyril Hope · Edith Biggs · Geoffrey Howson · Geoffrey Matthews · Geoffrey Sillitto · Midlands Mathematics Experiment · Nuffield Mathematics Project · Royaumont Seminar · School Mathematics Project · Trevor Fletcher · United Kingdom

L. Rogers (*) Independent Researcher, Oxford, UK e-mail: [email protected]

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Socio-Political and Educational Context of the Reform

The period 1950–1965 saw great expansion in the Teacher Training Colleges, and the development of “training” into “education”1 with the inception of a new Bachelor of Education degree as a means of raising the academic status of teachers,2 and of achieving the political ideal of an all-graduate teaching profession. In 1951, the examination for school leavers was changed to the General Certificate of Education to cater for the older school leavers with a wider range of subjects, but there was little change in the mathematics curriculum. The Labour Government had proposed the changes to Comprehensive Schools3 in their first term of office, and began to fulfil their policy in 1965, thus bringing the grammar and secondary-modern pupils together to be taught (ideally) in mixed classes. In concert with this change, the Schools Council,4 the Nuffield Foundation,5 and Local Education Authorities began actively to encourage “grass roots” curriculum development. As far as the mathematics curriculum was concerned, the social changes exposed the difficulties and challenged many preconceptions about content, teaching methods, and expectations of pupils’ achievement (Swan 1950). At the same time, the “modern mathematics” movement was a significant force for curriculum change and was interpreted and developed by various groups in different European countries and English educational areas in quite different ways (Servais 1975). A considerable amount of material appeared in England from the United States, and the principal influence from Europe was the 1959 Royaumont Seminar which encouraged the modernization of the mathematics curriculum for both economic and pedagogic reasons (OEEC 1961a, b). In this and the next section, we will discuss some of the circumstances surrounding a significant paradigm shift in pedagogical practice, which concerns the foundation and early days of the Association of Teachers of Mathematics (ATM).

The Dominant Paradigm The dominant paradigm for educational practice was founded on the belief that some were born to lead, and others to follow. This was a result of the “ideological and pedagogical divide” described in Rogers (1999) and in the Fyfe Report (Fyfe 1947). Consequently, for mathematics, very different syllabuses, expectations, and pedagogical practices were to be found in the grammar and secondary modern schools. These traditions were transmitted and reinforced by the various reports published by the Mathematical Association (MA) and the Government (HMI 1958).6

 As a result of the Robbins report on Higher Education of 1963 (Committee on Higher Education 1963), teachertraining institutions were reformed as Colleges of Education, and “teacher education” became the terminology signifying both a wider range and a greater intellectual status of studies. 2  The Bachelor of Education degree was first established in 1964 after pressure from the professional associations and the teachers’ unions for an all-graduate teaching profession, and the first teachers graduated under this system in 1968. 3  The Labour (socialist) Government elected after World War II (1945–1951) enacted radical Keynesian policies and after a period in opposition returned for a further term (1964–1970). They began the conversion of state secondary schools to the comprehensive system in 1965. 4  The Schools Council for Curriculum and Examinations (1964–1984) was a quasi-autonomous organization funded by the government and set up to encourage and support curriculum development. 5  The Nuffield Foundation was an independent charity with a mission to advance educational opportunity and social well-being by funding research and innovation in education, justice, and welfare. 6  For a detailed account of the history of the Mathematical Association, its teaching reports, social contexts and its dealings with other organizations, see Price (1994). 1

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In 1956, the MA finally produced its long-awaited report, The Teaching of Mathematics in Primary Schools (MA 1956), on which work had started 17 years earlier. A number of internal committees had met to study the problem but failed to agree, and by the time the final committee was appointed in 1950, a new doctrine had emerged, which was then embodied in the 1956 report. Earlier differences of opinion are clear from the preamble to the report in which the committee members of the Primary School Sub-Committee declared: In fact, thinking and doing go so much hand-in-hand that it is almost as if the children were thinking in terms of action. It has also been suggested … that the fundamental patterns of mathematical thinking appear to be fundamental to all thinking. Hence, we have found it convenient to analyse our ideas about Mathematics in Primary Schools in a threefold way:

(i) The interaction between a child’s mind and the concrete situations of his environment—i.e., the substance of his experience; (ii) The growth of mind and mental powers as they are exhibited in the child’s growth towards mathematical thinking; (iii) The mathematical ideas themselves and their development and inter-relation. (MA 1956, p. 1)

The content of the final report clearly shows that the committee had adopted Piagetian principles and incorporated them into its message. The Introduction and first chapter set out the place of mathematics in primary school where not only we find the generally recognized reasons for teaching mathematics; utility and scientific value, personal enjoyment; and the child’s right to our heritage, but also the claim that the patterns of mathematical thinking are fundamental patterns of all thinking. In this report, we find a comprehensive development of ideas about number and space, about the organization of the classroom and different groupings of pupils, about different kinds of understandings and suitable kinds of assessments and records of achievement, and the different experiences by which pupils have progressed and the emphasis on understanding mental processes: “The true test of mastery of a mathematical principle or process is to be found in the ability to apply it to a new situation, and not in a repetition of an already standardized situation committed to memory” (MA 1956, p. 87). Of particular note is a section on “Material aids to teaching” (MA 1956, pp. 93–106) describing various apparatus, including graphs, diagrams, and displays with suggestions about where and how they may be employed to make a growing mathematical idea clearer through experiment, and to foster memorizing and skills (Gattegno 1963; Servais 1970). Finally, there is an annotated section on research on teaching, reinforcing the message about the nature of mathematical thinking. The willingness and ability to promote a study of research is another innovative idea not usually found in a manual intended for teachers at that time. In 1959, a report on Mathematics in Secondary Modern Schools was produced (MA 1959), intended to be read in conjunction with, and as a continuation of, the earlier Primary report. However, here the tone was quite different; in the statement of terms of reference, we find: The mathematical ideas and teaching methods here discussed are those which we consider suitable for the great bulk of secondary school population between 11 and 15 years of age, who have shown no early signs of readiness for Mathematics as an abstract study, or at least have not achieved the attainments in arithmetic traditionally associated with the ablest of their age group … Its use as a specialised tool may be needed by some later in life, but readiness for many mathematical techniques comes with an intellectual maturity only reached by most modern school pupils some time after entry, if at all. (MA 1959, pp. 1–2)

Clearly, expectations for these pupils were limited, “intellectual maturity” was the principal criterion, and the attitude was one of a vocational or manual rather than a professional future for the pupils— where few, if any, were expected to go on to any form of further education. At this time, many secondary modern schools were still operating, and there does not seem to be any hint that the writers of this report had learnt anything from the message in the Primary Report. A Second Report on the Teaching of Arithmetic in Schools was published in 1964 that discussed the teaching of arithmetic as part of a general mathematics course, from the end of primary school to age 16. This was intended

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… for teachers of boys and girls of good ability such as are found in grammar schools and in the grammar streams of comprehensive schools. There are many other pupils who are still not ready at 11-plus to begin such a course as is here envisaged and appropriate guidance for their teaching may be found in … Mathematics in secondary modern schools. (MA 1964, p. 1)7

By this time, the modern mathematics movement was already underway, and so arithmetic … should be taught in such a way that it leads on to the rest of mathematics, not only algebra and trigonometry, coordinate geometry and calculus, but also the new topics now being introduced into school syllabuses. (MA 1964, p. 3)

How arithmetic should be taught in such a way that it leads to subjects not normally taught to these pupils is not explained. Apart from statistics8 (which was not really a new topic), hardly any mention was made of any “new” mathematics. Toward the end of the report, we find some suggestions for classroom procedures: The teacher has … two lines of approach. The first is traditional and familiar: It is teacher-centred and relies on exposition and examples worked on the blackboard, but the pupil learns largely by listening. In the second the emphasis is on the pupil’s activity: The teacher’s task is to contrive the appropriate stimulus to learning, sometimes by posing a challenge or problems and guiding the pupil to discover a solution, at other times by suggesting and activity … leading to the discovery of a generalisation. (MA 1964, p. 84)

Apart from this brief hint at “guided discovery,” the suggestions only concerned classroom organization, the choice of routine examples, marking of pupils’ work, and so on, but provided no discussion of the actual pedagogical challenges produced by the new comprehensive educational system. Clearly, the problems of teaching pupils in mixed classes were being felt by members of the MA much earlier (Nunn 1951; Swan 1950) but even after the changes, it was difficult to break out of the long-­established pattern. Some teachers in comprehensive schools avoided the problems (of both pedagogy and new content) by sorting pupils into different “ability groups” according to standardized arithmetic tests.

Changes in the Primary and Secondary Schools Before 1952, the Mathematical Association (MA) was the principal authority whereby the beliefs and practices about teaching in secondary school were handed on. The members of the MA were largely university mathematics graduates who had little or no knowledge of psychology and often no pedagogical training, and so the broad consensus of expectation and action was driven by their training as mathematicians. This is not to deny that the members of the MA had high expectations for their pupils, but these pupils were mainly the privileged few who were found in grammar and private schools. The standards in primary schools, originally based on the “Revised Code” of 1862, gradually had slowly evolved, while educationists and official reports demanded more practical applications; classes became smaller and teachers better trained but despite the influence of Montessori and Froebel on the “nature of the child” and related practical activities, rote learning of arithmetic was hard to eliminate (Pinner 1981).

 Many comprehensive schools adopted the practice of “streaming” pupils into “able” and “less able” classes using standard arithmetic tests. 8  Statistics at this time consisted of some basic ideas of means, standard deviation and cumulative frequency, and this, together with a separate paper on Arithmetic, was available as an easy alternative to the full Mathematics examinations at age 16. 7

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By this time, the teachers in the Secondary Modern and Secondary Technical schools were mostly those who had attended a 2-year course in a teacher training college9 where mathematics was limited to basic arithmetic and mensuration, determined by the supposed mental capacities of their pupils. It became clear that teachers in the Secondary schools were very concerned with the problems of teaching their pupils, and as new entrants to the profession began to read accounts of the work of Piaget10 and realize that the problems could be seen from different points of view, the underlying epistemological and pedagogical beliefs which guided the established practices were seriously challenged. However, most Primary teachers were not members of the MA and consequently unaware of the 1956 Report, so apart from some individual efforts, there was little change in the classrooms.

The “New Mathematics” and the Creation of a New Association of Teachers Developments During the 1950s in Europe and in the USA After World War II, a number of mathematicians from European countries realized that their school and university mathematics curricula and pedagogical practices had to be revised, and so by 1950, a small group of mathematicians, educationalists, secondary school teachers, and psychologists concerned with the improvement of mathematics teaching met in Debden (United Kingdom) at the initiative of the mathematician and psychologist Caleb Gattegno, and in a follow-up meeting held in 1952 at La Rochette par Melun (France), they decided to create 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 with Gattegno as secretary11 (see also Chap. 3 in this volume). The first joint publication of the CIEAEM, L’Enseignement des Mathématiques appeared in 1955 initiated by a group of founding members—Jean Piaget, Evert Willem Beth, Jean Dieudonné, André Lichnerowicz, Gustave Choquet, and Caleb Gattegno (Piaget et al. 1955). The six chapters were unedited and represented independent statements of each writer’s beliefs. The preface (pages 5–9), was signed “Le Bureau” but most likely written by Gattegno, clearly showing the united intentions and clearly different interests of the authors: This book is the first joint publication of the CIEAEM, in which six of the founder members have participated— A psychologist, a mathematical logician, three professional mathematicians, and a pedagogue of mathematics. The texts have not been collated. Only Mr. Piaget has written his chapter after reading the others. The Bureau of the Commission, that is responsible for this book, decided to present a work which shows the variety of experience of the members, concentrated on the same problem, and they have believed that initially in presenting a mosaic, they will help the reader to become accustomed to the analysis and to see the contemporary preoccupations that comprise the true connection at the centre of the Commission. (Piaget et al. 1955, p. 5, translated from French by Leo Rogers)

The European input to the Royaumont and Dubrovnik meetings and corresponding proceedings (OEEC 1961a, b) was driven by the work of the CIEAEM (Vanpaemel et al. 2012), showing that the aim of the “New Mathematics” was to introduce new curriculum content and pedagogical practices to revitalize mathematics teaching at all levels. In his introductory address at Royaumont, Marshall

 The first government qualifications were established in 1846, initially for training Primary teachers, and training institutions were established as a consequence of the Education Act of 1870. From 1902, teacher training colleges and university departments trained women and men for Primary and Secondary schools. However, it was still possible to find teachers in Secondary schools without any training (Nunn 1951). 10  The National Froebel Foundation began publishing pamphlets on Piaget’s work in 1955. 11  Gattegno was “spiritus mentor” of CIEAEM.  See http://www.cieaem.org/?q=node/18 and http://www.icmihistory. unito.it/19371954.php. Lucienne Félix’s (1985) account of the period 1950–1985 can be downloaded. 9

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Stone emphasized the “growth of pure mathematics [and] the increasing dependence of scientific thought upon mathematical methods” (OEEC 1961a, p. 15). The need for the services of scientists of every description was becoming urgent. He referred to the growth in knowledge of psychological development and the need to “help the growing mind” (p. 23) with new teaching materials and methodologies including the use of films and television. He also stressed that universities should adopt a cooperative educational research agenda. Participants from the UK were Cyril Hope, principal lecturer at the City of Worcester Training College, and Frank William Land (1911–1990), senior lecturer in the Department of Education at the University of Liverpool; professor Edwin Arthur Maxwell (1907–1987), Queen’s College, Cambridge, was invited as a guest speaker. At this time, European mathematical curricula were quite varied, and early reforms in modernizing their mathematics programs were by no means uniform as it might have appeared from the outside (Servais 1975). Each country had its own mathematical and pedagogical traditions and modified their changes accordingly. Servais’ conclusion was “What we want is to update teaching of mathematics both in content and in method and to keep it alive as a permanent activity” (Servais 1975, p. 55), implying regular reflection and regeneration. Meanwhile, concerned for the improvement of mathematics in their own schools, in the USA, the School Mathematics Study Group (SMSG) was formed in 195812 leading to the publication of classroom materials that could be used in some English schools, and some well-designed introductions to various new mathematical topics. The Elementary Science Study (ESS), a program for unstructured open-ended discovery, was created in 1960. Concerned with developments in Soviet mathematical pedagogy, other organizations in the United States published a series of translations of pedagogical texts (Kilpatrick and Wirszup 1969–1972; Simon and Simon 1963). Much of this material was made available in England and enjoyed some success among teachers seeking innovative pedagogical approaches, but an English “home-grown” revolution took a rather different turn.

The Beginnings of the Association of Teachers of Mathematics During the early 1950s, some teachers of mathematics and those in the Training Colleges were influenced by the accounts of Piaget’s work13 and the news of the changes in mathematics teaching emanating from Europe. Caleb Gattegno was teaching at the Institute of Education in London where he informed his students and practicing teachers of the views of CIEAEM and other European pedagogical developments. Considering the effect of the social and political changes described above, and the confusion among students which seemed to exist in many mathematics classrooms, some teachers were eager to find a course of action that would alleviate the problems of teaching the “less able” children, in both Primary and Secondary Modern schools. There was a widely held belief that the current mathematics curriculum was poor, and a group of teachers, mathematics education students, and university lecturers decided to take radical action and form a new professional association for all teachers of mathematics. A clear division had developed between the established guardians of practice in the Mathematical Association and the radical ideas represented by Gattegno and his sympathizers. At the same time as he was becoming involved in the foundation of the CIEAEM and in translating some of the writings of Piaget, Gattegno had taken the time to write a series of papers expressing  The School Mathematics Study Group (SMSG), founded in 1958, was the largest and best financed of all the National Science Foundation projects of the era, by the combined efforts of the American Mathematical Society, Mathematical Association of America, and the National Council of Teachers of Mathematics (see also Chap. 2 in this volume). 13  Nathan and Susan Isaacs were some of the first to introduce Piagetian ideas to Primary Teachers in 1955 through the National Froebel Foundation. 12

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r­ adical views on curriculum and philosophical issues which would be published in the Mathematical Gazette, the journal of the Mathematical Association (Gattegno 1947, 1949, 1954a, b). The implications of emerging psychological theory, practical teaching experiments, and philosophical departures relating to education meant that many assumptions about teaching mathematics at all levels began to be seriously re-examined.14 Since it seemed impossible to persuade the MA to take steps to introduce these ideas into established practice, Gattegno, together with Roland Collins15 and other sympathizers, broke away from the MA and founded the Association for Teaching Aids in Mathematics (ATAM). There were twelve people—teachers from all over England—at the first meeting of a steering committee held on June 28th, 1952. These founding members had responded to a brief notice in the Mathematical Gazette, the journal of the Mathematical Association, inviting the formation of a teachers’ cooperative to produce teaching aids. (ATM 2004, no pagination)

These teachers were initially concerned with the plight of pupils in Primary and Secondary Modern schools whose mathematical diet was restricted to commercial arithmetic and mensuration, and who suffered from extremely limited views of pedagogical practice, so they placed great emphasis on the development and use of a range of innovative approaches to teaching and the use of “concrete” apparatus to assist pupils learning. These materials included the Cuisenaire rods and geoboards, where practical examples could bring new structural ideas to light, and new terms like visualization and imagery are brought into the language. In fact, it is when we are engaged in a dialogue with concrete material that the principal mathematical ideas emerge. Every perception or action derived from the concrete duplicates itself in mental imagery; this becomes structured and can then be recalled in its own right. The first objects for mathematical study are the relations between perceptions and actions made virtual in this way. (Servais 1970, p. 207)

The Mathematical Association had discussed the problem of aids to pupils’ understanding earlier (MA 1947), but even though Gattegno had published a paper on dynamic geometry showing his innovatively different approach16 (Gattegno 1954b), little action appears to have been taken until the Mathematical Association Primary Report of 1956 appeared (see above). In 1955, ATAM began to publish its regular journal Mathematics Teaching. Each of the earlier issues brought articles on different aspects of classroom activities, critiques of current textbooks, advertisements of meetings held in different parts of the country, together with news of a growing number of pamphlets for teachers, explaining new mathematical topics, reviewing books and apparatus, discussing psychological theory, and sharing experiences from the classroom. This variety of articles was a continuing feature of the Journal’s critical comment. Experiences were shared and many teachers wrote in to show interest to start up local activity groups. The first Annual General Meeting (AGM) of ATAM was held in February 1954, at the London Institute of Education. This opportunity provided a comprehensive exhibition that included mathematical teaching aids and models, the showing of a number of films; and demonstration lessons given by co-founders Roland Collins and Caleb Gattegno and other members to schoolchildren using the latest teaching methods. This event became a tradition, with the date of the AGM becoming the date of the Annual Conference. The Association grew in popularity, being the main provider of positive assistance to many primary and secondary teachers.

 Significantly, Gattegno (1954a) was the first person in the UK to highlight the importance of mathematical structures in the confrontation of mistakes in classroom mathematics. Also, the importance of these ideas could not reach the majority of Secondary Modern teachers, since they were not members of the MA. 15  Roland Collins was a teacher at Doncaster Training College and the author of Mathematical Pie, a periodical for school pupils. 16  In this paper he suggested the whole of geometry could be taught by this means. 14

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By the time of a conference held in Blackpool in 1958, the Association had 900 registered members and there were some 12 thriving local branches, and by 1959 membership had reached 1059 including a considerable number from educational institutions. Cooper (1985, p. 70), shows a table indicating the number of members in grammar schools, training colleges, and universities. At its AGM in 1962 so many secondary teachers and members of university teacher training departments had joined, that the decision was taken to change the name of the organization to the Association of Teachers of Mathematics (ATM), representing all teachers of mathematics, with this change taking effect from June 1962. The ATAM/ATM played a significant role in the development of new textbooks and syllabuses, particularly in the Midlands Mathematics Experiment (MME), co-founded in 1961 and run by Cyril Hope, which experienced rapid growth among “lower status” teachers who had been unprovided for by the MA, which was particularly pedagogically conservative and unwilling to adopt any creative views about “modern mathematics.” Hope argued: “We are not concerned with mere instruction designed to convince the child sufficiently for him to accept the teacher’s dogmas, but with the child’s creative activity by which he discovers and convinces himself and others of the truth and importance of his discoveries” (Hope 1958, p. 11). These views were recognized by Trevor J. Fletcher, Roland Collins, and Ian Harris, important ATM members whose original motivations lied in teaching aids and films and structural apparatus and an original enduring concern with pedagogy. Fletcher was a pioneer, maker of mathematical films, and the instigator of the first book by members of the Association, Some Lessons in Mathematics (Fletcher 1964). He was the founder of the ATM film group that was also led by Ian Harris with many contributions from other members, like Derrick Last, together with a creative photographer Lyndon Baker who helped produce designs for many covers of Mathematics Teaching. Baker was also responsible for the ATM Curves with pictures of children’s activities. The uses of new technologies were quite important. Bill Brookes created a series of films on transformation geometry on 8  mm film strips which were produced as film loops inside cassettes which could be viewed using a special projector. Trevor Fletcher, an early member of AT(A)M, and active member of CIEAEM in the late 1950s, was elected President of the Association in 1960 after Gattegno, and he was the first editor of Mathematics Teaching. He wrote a book on linear algebra in his ATM style (Fletcher 1972), and eventually became Staff Inspector for Mathematics i.e., the chief of all Her Majesty’s Inspectors—the system before the reforms of the Thatcher Government.



Some Curriculum Development Projects

The first large-scale experiments or projects with “new mathematics” in secondary schools in England and Wales were run almost simultaneously in the school year 1961–1962 (Howson 1978). In principle, schools or groups of schools had the right to ask to be examined at “Ordinary” or “Advanced” level based on their own syllabuses rather than their Board’s standard one. Several groups of teachers decided to make use of this right and began to devise “modernizing” schemes of work and propose new mathematics for the ordinary-level examinations, in line with the recommendations of the Royaumont and Dubrovnik meetings (OEEC 1961a, b). The most important were: The School Mathematics Project (SMP), started by Bryan Thwaites at Southampton University; the aforementioned Midlands Mathematics Experiment (MME), organized by the Royaumont participant Cyril Hope; and the Contemporary School Mathematics Project (CSM), run by Geoffrey Matthews at St. Dunstan’s College, a London secondary school (Rogers 2017). Mathematics in Education and Industry (MEI), a project focusing on advanced-level examinations (for students aged 16–18), was established in 1963 and was sponsored by British Petroleum, in conjunction with the MA. It was not only concentrated in some London schools but also attracted some public schools from elsewhere (Karp 2008).

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In addition to the four projects mentioned above, there were several smaller projects during the 1960s—for example, the Abbey Wood Mathematics Project, devised by David Fielker, which was a considerable success in terms of innovative pedagogy and as an exemplar of what teachers could achieve. The project was the most successful creative and independent school “modern mathematics” syllabus for upper primary and early secondary pupils of the time. Fielker was editor of Mathematics Teaching for many years. As editor, he was quick to comment on movements and projects. Another project was Richard Skemp’s Psychology Mathematics Project. Skemp published his early ideas on concepts and relations between aspects of mathematics before his well-known and often-cited paper on “Relational mathematics and instrumental mathematics” in Mathematics Teaching (Skemp 1976). His ideas were taken up by teachers and professional psychologists. Skemp’s teaching project failed—his principles were noteworthy, but the books did not go down well with teachers and were difficult to use in the classroom. Sales were poor. There was a lack of follow-up, and finally, the project closed. For a brief discussion of some of these smaller projects, we refer to Rogers (2017). Similar reform initiatives in Scotland, such as the Scottish Mathematics Group (SMG), promoted by Geoffrey Sillitto, are discussed in Rogers (2014). Sillitto produced a series of ideas in transformation geometry which were published in Mathematical Reflections (ATM 1970) after his death. Clearly, at the beginning of the 1960s, a consensus had grown among leading mathematics teachers about the need to reform secondary school curricula, or as Cooper (1982) stated: By the late 1961 … the “need” to reform school mathematics was seen as legitimate by actors in a number of arenas, including the industrial and the political. In this climate various groups of individuals, who had been involved in the conferences and activities of the preceding few years were to move onto … the drawing up of detailed proposals for curriculum development, and the search for funds to implement these. (p. 252)

Each of these projects developed texts that were first tried out in draft form in schools and then were prepared for commercial publication. Following the model of the SMSG in the US, these textbooks were written by teams of practicing teachers, but unlike the American model these teams had little or no substantial backing from university mathematicians (Thwaites 1966). No public (state) money was involved, and financial support for teachers to take time out for writing, was limited; the projects did not even receive assistance (or encouragement) from the Ministry of Education (Howson 1978). SMP and MME provided some training for teachers in the form of short courses (up to 1 week in duration) and the publication of Teachers Guides. Project writers also addressed teachers’ meetings, organized by the two teachers’ associations (MA and ATM) or by Local Education Authorities (LEAs). Although the basic approaches of the three projects, SMP, MME, and CSM, were remarkably similar—they all were to replace the standard Ordinary-level examinations and addressed new mathematical topics for the most able students—emphases and matters of detail differed. MME, for instance, set out to cover a wider range of abilities and designed its book to cater for the introduction of the (less-­ demanding) Certificate of Secondary Education (CSE) which was introduced in 1965. In terms of content, it laid more emphasis on vector geometry than the other projects. SMP would ultimately prove to be the most successful project, according to Howson (1978), because it had better financial backing (from industry) and it could also draw on a wider pool of experienced teachers–authors. All three projects respected the English tradition of applied mathematics and thus did not fall into the excesses of over-abstraction to be observed in some of the Continental and American versions of the modern mathematics/New Math reform. Thwaites (1966) explained: It is a common feature of English experiments that the fundamental mathematics is allowed to grow slowly (some would say rather too slowly) out of the everyday experiences of the children and out of all the varied concrete illustrations of mathematical thinking it is possible to present in the classroom. (p. 46)

Some “modern” pure mathematics was covered in the projects, such as groups, vectors, and transformations, but the English projects were, according to Geoffrey Howson, essentially practical in nature, taking examples from “real-world” situations. They were less set-driven and axiom-driven than, for

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Figure 7.1  Geoffrey Matthews (right) and Magda Jóboru (president of the Hungarian National Commission for UNESCO) at the international symposium on school mathematics teaching in Budapest, August 27–September 8, 1962. (Photo by Ferenc Vigovszki)

example, Georges Papy’s approach in Belgium (see Chap. 10 in this volume) or that of SMSG in the US (Karp 2008). There was additionally a focus on “modern” applied mathematics (data collection and representation, statistics, linear programming, binary systems and codes, flow diagrams, computer programming, …), topics that were also advocated by some lecturers at Royaumont (see, e.g., De Bock and Zwaneveld 2020), but which were often not addressed in Continental reform programs. In the early years of the “new mathematics” projects, the emphasis was on the mathematical content rather than on the teaching methods, which came to the forefront only gradually (Thwaites 1966). In September 1964, a “new mathematics” project for primary education was launched: The Nuffield Mathematics Project,17 named after the Nuffield Foundation and led by Geoffrey Matthews (1917– 2002) (Figure 7.1), at that time appointed to Chelsea College Centre for Science Education (Moon 1986; Rogers 2017). The age range 5–13 covered all primary years and transition into secondary school.18 The first Teachers Guides, published in 1967, used a conceptual progression based on Piagetian research. These explained the underlying mathematics with ideas for different activities, all illustrated with pupils’ work. Expressly written for teachers, they introduced the rationale for the project, explaining the need for change, with ideas for class organization and a new approach to evaluation. Modern structural ideas of mathematics were introduced: Sets, relations, number bases, and properties of operations, to help form a conceptual basis for calculation, together with properties of plane transformations and spatial relationships. There were also “checking up” guides that provided interviews using everyday objects to review children’s understanding of conservation, equivalence, and invariance. An important innovation of the Nuffield Project was the creation of mathematics centers where teachers could meet with advisers; these evolved into professional development centers to be found in most Local Authority areas. The National Foundation for Educational Research published a survey of the techniques of primary teachers (Williams 1971a, b), and the charismatic Edith Biggs19 published her Mathematics in Primary Schools (Biggs 1965), a popular practical guide for primary  More information about the Nuffield Primary Mathematics Project can also be found at http://www.nuffieldfoundation.org/nuffield-primary-mathematics-1964 18  At the time this project was set up, there were many “middle schools” in England, for pupils aged 9 to 13—this was a result of the freedom of the LEAs to organize their own provisions of education. 19  Edith Biggs (1911–2002) was appointed Her Majesty’s Inspector in 1950, and until the publication of her report, all documents issued by Her Majesty’s Inspectorate were anonymous. Obituary: Mathematics Teaching, 180, 2002, p. 33. 17

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teachers. The Nuffield project was adopted in many parts of the UK, inspiring many teachers, and even involving them in classroom research; this also provided the basis for the attainment tests found in the Assessment of Performance Units (APU) and the Concepts in Secondary Mathematics and Science (CSMS) investigations in the 1970s and 1980s. Another innovation that began in the 1970s was the use of television in the classroom. A number of programs were produced for the BBC, but they were rather static. Children were bored and teachers found them difficult to fit into the school program—teachers could do a better job by working on the same subject with the children. A group of ATM members came together, wanting to use the new media for teaching mathematics. Penguin Education had produced innovative publications called “Voices,” a mixture of pictures and pieces of paintings, poetry, or prose, which prompted the idea of a mathematics series of a similar kind for children. Out of this came the Leapfrogs Group, formed as a limited company called Leapfrogs Ltd.: This was led by Ray Hemmings, Derrick Last, Leo Rogers, David Sturgess, and Dick Tahta, all lecturers in Departments of Education from Universities or Training Colleges and Teachers’ Centers with wide experience of teaching children and students. The group had a collection of innovative publications of books for teachers and children. Independent Television offered to provide a series of programs on mathematics for school children from 7 to 9 years old. There was a considerable amount of discussion about children’s “span of attention” and how the programs could be structured. They were built on the principle that topics were created as “vertical” elements and themes could be “horizontal” running through a school term. “Presenters,” of whom some were already popular with children, could be used, as well as pieces of apparatus, “outside broadcasts” (visiting places that showed exemplars of the ideas), animations, and stories. A typical program of 20 min had four topics, each related to a different theme that would be running “horizontally” through the term. Missing a program did not matter too much because there was plenty of time to “catch up” later. A series of handbooks for teachers was published with suggestions on how to develop ideas in the film series and on how to develop the activities in the programs. Ending as “video maths,” the programs ran for about 12 years in different forms. The work on school TV that the Leapfrogs Group did for children incorporated the pioneering work that the ATM film group had produced, following Gattegno’s principles and the best of available contemporary technology. It did not attempt to “teach” mathematics to children, but produce “everyday” examples of situations that could lead to mathematics emerging in the hands of teachers who took advantage of the accompanying specially written literature. This was the real root of modern mathematics in the United Kingdom and the principles of ATM, the philosophy, and the ways of working have prevailed. Other projects in the 1970s, which began to use structural approaches to teaching, were sponsored by the Schools Council. That was the last attempt to produce a kind of unified curriculum before the Thatcher Government started to encourage the right wing of politics to publish works critical of new ideas and promoting “traditional” mathematics courses. By this time, SMP was big enough to be able to resist these cries for “traditional” mathematics where the principal actors were often regarded as part of the establishment.

Radical Mathematics in ATM Publications During the 1960s, ATM continued to produce publications. Most experimental texts of SMP, MME, and CSM owed much to these publications, in particular, to the examples of dynamic geometry and other “modern” topics to be found in Mathematics Teaching, and the book Some Lessons in Mathematics: A Handbook on the Teaching of “Modern” Mathematics (Fletcher 1964). This book was the result of a “group writing” project with 20 contributors. It was edited by Trevor Fletcher (Rogers 2019). Although Some Lessons in Mathematics was aimed at mathematics teachers, its con-

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tent and style were so uncompromisingly modern that most ordinary teachers found it very hard to accept either the recommended pedagogies or the mathematics, which they often found hard to understand. The core mathematical ideas were taken from a graduate course that exposed the structure and coherence of a series of fundamental ideas that run through the core of any modern secondary school program. Getting these ideas accepted was a challenge at any level because much of the content seemed to be out of place—yet Bryan Thwaites recognized its importance. In his Foreword to this book, he announced a challenge: Certainly at the secondary level, it has been left to a few individuals and small groups to begin to answer some of the classroom difficulties inherent in any reform of mathematical syllabuses. However, whatever the truth may have been in the past, suddenly, the situation is utterly altered. Some Lessons in Mathematics must, surely, mark the beginning of a new era in the history of English school mathematics; in an instant, a new standard is set by which other efforts are bound to be judged. It is a tremendous leap forward … The reader has only to open this book at random to recognise the authority with which it is written, the insight which it gives into the classroom situation and the comprehension which it displays of the rightful impact on school teaching of modern mathematics … Thus, though this book is neither a pupils’ text nor a teachers’ guide for some special syllabus, it is at once compulsory reading for everyone concerned in mathematical education: and one is left to congratulate the group of authors who have produced such a very remarkable contribution to the cause of the reform of school mathematical teaching. (Thwaites in Fletcher 1964, p. vii)

The first chapter began by examining what may be meant by mathematics and where it appeared in different forms in primary and secondary school, focusing on teaching new mathematics, new ways with old mathematics, contemporary applications of the subject, and the psychology of teaching it. It is a description of a new pedagogy. The structure of the “lessons” is not meant as a syllabus or course of study but as part of a larger context, as discussions of specific items as well as of possible ways that material could be taken into the classroom. The suggestions covered a wide range of abilities to suggest what might be possible in many different teaching and learning contexts. There is a description of what was meant by a “pedagogical situation.” A context that Willy Servais described in his visits to ATM from the CIEAEM group was summarized in the following way: Mathematics does not start with finished theorems in the textbook, it starts from situations. Before the first results are achieved, there is a period of discovery, creation, error, discarding and accepting results for appraisal and further discussion before progressing with further enquiry, in the situation pédagogique so at this stage, explanatory comments often place the ideas in a more advanced perspective by recognising elementary aspects of important ideas that might be introduced earlier. (Servais in Fletcher 1964, p. 2)

As Thwaites (above) remarked, it marks “the beginning of a new era in the history of English school mathematics”! The second important publication was Notes on Mathematics in Primary Schools (ATM 1967). Here 26 authors shared in the writing that was compiled in two meetings in January and April 1965, and here again, we had eight pages of pedagogical discussion and description. Importantly, the reader’s attention was directed toward the fact that the learning of mathematics begins before the child goes to school and continues through primary school and beyond. Mathematics was not just about number and space. This new emphasis was that in the classroom mathematics should become more varied and therefore more enjoyable when the child can explore situations that are within his experience, and which can yield quite profound mathematics. In this book teachers, lecturers, and student teachers had collected ideas for the newer methods of mathematics teaching in the classroom. There were eight pages of introduction, comprising a large number of activities, paper folding, various measurement activities, tessellations, commentaries from children and their teachers in classrooms, and ideas for teachers and in particular, there was a strong emphasis on the learner: Mathematics is the creation of human minds. A new piece of Mathematics can be fashioned to do a job in the same way, that, say a new building can be designed … mathematics cannot be an absolute, given a priori, or a science built entirely on observations of the real world … Because mathematics is made by men and exists only in their minds, it must be made or re-made in the mind of each person who learns it. In this sense, mathematics can only be learnt by being created. We do not believe that a clear distinction can be drawn between the

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­ athematician inventing new mathematics and the child learning mathematics that is new to him. The child has m different resources and different experiences, but both are involved in creative acts. We want to stress that the mathematics a child knows is, in a real sense, his possession, because by a personal act, he has created it. (ATM 1967, pp. 1–2)

Thus, the idea of teaching as a learner-centered activity should become a key aspect of the teacher’s work. Two further books appeared, Notes on Mathematics for Children (ATM 1977) and Mathematical Reflections (ATM 1970), which contained an assessment of Geoffrey Sillitto’s20 influence in reforming mathematics in Scottish schools and some of his previously unpublished work. The introduction of “modern mathematics” content, in its wider definition, into both the primary and secondary curricula, encouraged a substantial transition from the limited study of arithmetic and its applications for the majority of the school student population to an examination of a much wider range of topics in the subject, some of which were potentially exciting and intrinsically interesting. It is worth examining the impact of mathematics projects and other initiatives of the 1960s and to assess their contribution to the changes in mathematics teaching. Important aspects of the changes were the development of the “teaching method,” initially suggesting a single unique procedure into a wider aspect of pedagogy once an expression formerly used only with respect to the teaching of primary children. ATM also took up an aspect of the new teaching ideas that motivated elaboration of the idea of “the problem” after the publication of Polya’s How to Solve it was circulated in a cheap paperback version (Polya 1957). This prompted a series of publications developing many useful ideas about problem solving, and variations of its procedures with different groups of pupils, being popular in teacher training courses and student–teacher projects. A further significant and original outcome of these deliberations was the report prepared in 1966 for the Sub-Committee on Mathematical Instruction of the British National Committee for Mathematics entitled: The Development of Mathematical Activity in Children: The Place of the Problem in this Development (ATM 1966). Twenty-eight people contributed to this collection of writings that ranged from factual reports to “work in progress,” on the nature of mathematical activity, by exploring a wide range of “problems,” their relevance, contexts, aims, place, and function (in the widest sense) in the school curriculum and in learner-centered activity. It was presented to the Mathematical Instruction Section of the International Congress of Mathematicians in Moscow in 1966.

Zooming in on the School Mathematics Project The School Mathematics Project (SMP) proved to be more successful than any other project of the 1960s. It was adopted by far more schools than any other project and also outlived most of the others. Therefore, we take a closer look at it. In the early 1960s, SMP materials were developed for students of above-average ability (let us say, the top 25%), whether they were found in grammar (or public) schools or in the grammar streams of comprehensive schools. These children were taught by teachers who were thought to have the highest academic ability. Nowhere in the early materials was there any reference to teaching methods; “it was assumed that enlightened conventional teaching methods would still suffice” (Howson 1978, p. 191). From the mid-1960s onward, versions of the material were also produced for students of average ability in comprehensive and Secondary Modern schools. From then on, some attention was also paid to teaching methods (Breakell 2001).

 Geoffrey Sillitto, who died in 1966, was a lecturer at Jordanhill College of Education in Glasgow, a key member of ATM and a promoter of the Scottish Mathematics Group (see, e.g., Rogers 2014). 20

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The origins of SMP trace back to 1957 when a conference was convened at Oxford for the purpose of bringing together those who taught mathematics in school and those who used mathematics in real life (Cooper 1985). A number of changes in the secondary curriculum were discussed, both with respect to removing outdated topics and including elements of “modern” (pure and applied) mathematics, and although no consensus was reached, the conference signaled the beginning of the debate on modernizing secondary school mathematics in the UK (Breakell 2001). At a follow-up conference in Liverpool in 1959, two groups emerged: Representatives from industry, making a case for applied mathematics, and teachers of mathematics, members of ATM, who campaigned for the introduction of “modern” mathematical ideas into syllabuses, but also for pedagogical changes based on Piagetian ideas. In 1961, at a third conference held in Southampton and convened by Bryan Thwaites (born in 1923), both groups consolidated their positions, and an advisory committee was set up to consider reform. It directly led to the conception of SMP with Thwaites as a director (Figure 7.2); Geoffrey Howson (1931–2022) was appointed as a full-time editor/coordinator in 1962 (Figure 7.3). Soon, the drafting of school texts started, beginning with SMP Book T (“T” for “transition”) and SMP Book T4, commercially released as textbooks in, respectively, 1964 and 1965 (Howson 1964, 1965), and from September 1968, an experiment started in eight schools and then it moved on.

Figure 7.2  Sir Bryan Thwaites. (Thwaites 1972, p. ii).

Figure 7.3  Geoffrey Howson 1965

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Figure 7.4  Title figures of Rotation (p. 60) and Translation (p. 77) in Book T. (Howson 1964)

SMP Book T and Book T4 were intended for 13–15-year-old grammar school pupils who prepared for Ordinary-level examinations. Number, Sets, and Inequalities were the opening chapters. Sets were basically used as a language, not as a topic in its own right (Karp 2008). There was a great emphasis on geometry, not as a sequence of propositions but as motion geometry. Book T was devoted to simple transformations of geometrical figures, opening with reflections. Attention was paid to properties that are preserved under the transformation (and which ones are not). Other topics included the slide rule, area, and volume, displaying data, percentages and proportions, graphs and relationships, coordinates, trigonometry, binary arithmetic, geometry in three dimensions, and loci (Howson 1964). In Book T4, the geometric track was further developed, in particular the ways in which reflections, rotations, and translations can be combined. Examples of group tables appeared in this context, even though groups were not yet defined. Work on Cartesian coordinates was consolidated prior to the introduction of vectors and for the purpose of building a bridge between algebra and geometry. Matrices were introduced, highlighting the transformations from an algebraic perspective. It also provided a source for algebraic structures, finally leading to the definition of a group structure. There were also chapters on statistics and probability, linear programming, and practical arithmetic in everyday economics (Howson 1965). A particular characteristic of both books was their admirable presentation (Figure 7.4): One cannot leave the Books T and T4 without admiring their layout, illustrations and references to the literature, increasing the cultural value of the textbook series. Each introduction to a chapter is accompanied by a drawing and a reference to literature, such as by T. S. Eliot, Charles Dickens, William Shakespeare, Bertrand Russell, Alexander Pope, Christopher Marlowe, and Lewis Carroll. (Bjarnadóttir 2020, pp. 79–80)

By 1967, SMP Books T and T4, at that time regarded as obsolete, were replaced by SMP Books 1–5, starting at the age of 11 and preparing for the same ordinary-level examinations. Also in 1967, a new series, SMP Advanced Mathematics Books 1–4, was published to cover the syllabus for advanced-­ level examinations. This series focused not only on analysis and the algebraic framework but also on applied topics, such as statistics and mechanics. By 1965, experiments in comprehensive schools had made clear that the SMP materials were suitable for a wider ability range of pupils, but the presentation needed adaptation. This led to the SMP Books A–H,21 which could serve as a “modern” secondary mathematics course for comprehensive schools, starting at the age of 11. Books A–H were based on an adaptation of the material from Books 1–5, but the presentation and the complexity of the language and arguments of the original series were simplified so that the books “would be suitable at least for the top 75% of the intelligence range” (Thwaites 1972, p. 157). Therefore, this series was not considered appropriate for preparing students for official examinations (Howson 1967).  The first volume (Book A) was published in June 1968 and Book B in October 1968 with further books of the series planned for publication at six monthly intervals (Breakell 2001).

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SMP became immensely popular in the 1960s and project staff did everything they could to meet the demand. With the support of the Centre for Curriculum Renewal and Educational Development Overseas (CREDO), a body established in 1966 to provide help to developing countries on problems relating to curriculum development in schools, Thwaites and his colleagues made the SMP materials an export product, mainly in former British colonies (but not restricted to those). In Africa, two competing projects following the SMP model were established: The Joint Schools Project (JSP), which began in West Africa (see also Chap. 24 in this volume), and the East African School Mathematics Project (EASMP) in East Africa. For the latter project, the Advanced Mathematics Books 1–4 were re-written for high-attaining students in Kenya, Uganda, and Tanzania. However, as Schubring (2017) has argued, there was little adaptation to a proper cultural identity, instead the EASMP books were continuing more or less the cultural visions of the former colonial powers. Moreover, only an elite of students in these countries were reached, or as Geoffrey Howson put it in retrospect: In going round the countries, first of all you had to discount the fact that in whatever former colony you went to, you would find a handful of schools which were modeled on the English grammar school and which could match any English school. Often these schools were run by expatriates, used the English external examinations and sent their students on to English universities. But that was just the tip, the very tip of an enormous iceberg, and what was underneath was pretty disastrous in most countries. In fact, this is what increasingly worried me in that CREDO’s efforts were perhaps directed too much at this tip, at promoting new mathematics and science in these select schools, whereas one should be concentrating much more on what was below. (Howson in Karp 2008, p. 55)

Other versions of the SMP Books 1–5 were produced for the Southeast Asian market, specifically for use in English medium schools, in, for example, Malaysia and Hong Kong (see also Chap. 22 in this volume). In the mid-1970s, the era of Morris Kline’s Why Johnny Can’t Add: The Failure of the New Math (Kline 1973), the SMP textbooks, particularly Books A–H, were increasingly criticized in their country of origin. The writers were said to have grossly underestimated the difficulties students faced, particularly in number work, and with decimals and fractions (Ling 1987). The abstract and challenging content of the SMP Books A–H made them immensely popular with some teachers in comprehensive schools (and this reflected in their sales), but they “essentially failed to satisfy the overall needs of the range of students for which they were written” (Breakell 2001, p. 126). In response to the growing dissatisfaction with the materials in Books A–H, SMP convened a major conference in Bristol in July 1976 to discuss what it should do next for pupils aged 11–16. Subsequently, SMP embarked on a new pathway for this age group, with the aim of producing mathematical material for a wide range of learners, excluding the bottom 15% in terms of attainment (Ling 1987). It was the start of the differentiated SMP 11–16 Series, being used in the majority of comprehensive schools from the mid-1980s.



The End of Modern Mathematics in the UK

Through several projects, modern mathematics was introduced widely into British grammar schools in the early 1960s, and then into primary schools and the new comprehensive schools in the 1970s. Initially, the focus was on the modernization of mathematical content, but as a broader student population was targeted, more attention was also paid to innovation in teaching methods. From the mid-­1970s, criticism began to grow louder, but SMP, the leading secondary school mathematics project in the United Kingdom, developed a new 11–16 Series that did not just cater only to the most talented. Much new materials, such as coordinate geometry, probability, and statistics entered the

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11–16 curriculum and stayed there; other modern topics such as sets, relations, structures, and motion geometry disappeared almost completely. The late 1970s and 1980s marked a worldwide search for alternatives to modern mathematics. In the UK, in 1978, the Labour Government announced it would establish an inquiry into the teaching of mathematics in primary and secondary schools in England and Wales. It led to Mathematics Counts, better known by the name of its chairman as the Cockcroft Report (Cockcroft 1982). The report was very critical of modern mathematics and the way it was implemented in schools in the UK: The unexpectedly rapid expansion of modern mathematics courses meant that it was not long before many teachers were required to teach these courses without the benefit of introductory training. A further, and also very rapid, development was the extension of modern mathematics courses to pupils whose attainment was lower and the introduction of modern syllabuses in CSE mathematics examinations. Not all teachers possessed a sufficient mathematical background to enable them to appreciate the intentions underlying the new courses they were teaching. In consequence the material which was included in modern courses was often not presented as part of a unified structure but as a collection of disconnected topics whose relevance to the mathematics course as a whole did not become apparent to pupils. (Cockcroft 1982, pp. 81–82)

The report highlighted new themes, such as the usefulness of mathematics and its importance as a means of communication between different fields of application (Brown 2014). The acquisition of problem-solving and reasoning skills and attitudes, and the ability to apply them in everyday life, were promoted as important goals for “modern” mathematics education in primary and secondary schools.

References ATM. (1966). The development of mathematical activity in children: The place of the problem in this development. Nelson, United Kingdom: Author. ATM. (1967). Notes on mathematics in primary schools. Cambridge, United Kingdom: Cambridge University Press. ATM. (1970). Mathematical reflections. Cambridge, United Kingdom: Cambridge University Press. ATM. (1977). Notes on mathematics for children. Cambridge, United Kingdom: Cambridge University Press. ATM. (2004). An account of the first decade of AT(A)M. Derby, United Kingdom: Author. Retrieved October 18, 2021, from http://www.atm.org.uk/about/first-­decade.html. Biggs, E.  E. (1965). Mathematics in primary schools (Curriculum Bulletin No. 1). London, United Kingdom: The Schools Council. Bjarnadóttir, K. (2020). Royaumont’s aftermath in Iceland—Motion geometry, transformations and groups. In É. Barbin, K. Bjarnadóttir, F. Furinghetti, A. Karp, G. Moussard, J. Prytz, & G. Schubring (Eds.), “Dig where you stand” 6. Proceedings of the Sixth International Conference on the History of Mathematics Education (pp. 73–86). Münster, Germany: WTM. Breakell, J. (2001). The teaching of mathematics in schools in England and Wales during the early years of the Schools Council 1964 to 1975. Unpublished DPhil thesis, Institute of Education, University of London, United Kingdom. Brown, M. (2014). The Cockcroft Report: Time past, time present and time future. Mathematics Teaching, 243, 5–9. Cockcroft, W. H. (1982). Mathematics counts (Report of the Committee of Inquiry into the Teaching of Mathematics in Schools). London, United Kingdom: Her Majesty’s Stationery Office. Committee on Higher Education. (1963). Higher education: Report (Cmnd 2154). London, United Kingdom: Her Majesty’s Stationery Office. Cooper, B. (1982). Innovation in English secondary school mathematics: A sociological account with special reference to SMP and MME. Unpublished DPhil thesis, University of Sussex, United Kingdom. Cooper, B. (1985). Renegotiating secondary school mathematics. A study of curriculum change and stability. London, United Kingdom: Falmer Press. De Bock, D., & Zwaneveld, B. (2020). From Royaumont to Lyon: Applications and modelling during the sixties. In G. A. Stillman, G. Kaiser, & C. E. Lampen (Eds.), Mathematical modelling education and sense-making (pp. 407– 417). Cham, Switzerland: Springer. Félix, L. (1985). Aperçu historique sur la Commission Internationale pour l’Étude et l’Amélioration de l’Enseignement des Mathématiques (CIEAEM). [Historical overview (1950–1984) on the International Commission for the Study and Improvement of Mathematics Teaching (CIEAEM)]. Bordeaux, France: l’IREM de Bordeaux.

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Fyfe, H. W. (1947). Secondary education: A report of the advisory council on education in Scotland. Edinburgh, United Kingdom: Her Majesty’s Stationery Office. Fletcher, T.  J. (Ed.). (1964). Some lessons in mathematics: A handbook on the teaching of “modern” mathematics. Cambridge, United Kingdom: Cambridge University Press. Fletcher, T. J. (1972). Linear Algebra; through its applications. London, United Kingdom: Van Nostrand Reinhold. Gattegno, C. (1947). Mathematics and the child. The Mathematical Gazette, 31(296), 219–223. Gattegno, C. (1949). Mathematics and the child, II. The Mathematical Gazette, 33(304), 108–112. Gattegno, C. (1954a). Mathematics and the child, III. The Mathematical Gazette, 38(323), 11–14. Gattegno, C. (1954b). The idea of dynamic patterns in geometry. The Mathematical Gazette, 38(325), 207–209. Gattegno, C. (1963). Perception and action as bases of mathematical thought. In For the Teaching of Mathematics (Vol. 2, pp. 49–59). Reading, United Kingdom: Educational Explorers. HMI. (1958). Branch Reports. Mathematical Gazette, 42(341), i–vi. Hope, C. (1958). Filmstrips in mathematics. Mathematics Teaching, 6, 11–13. Howson, A.  G. (Ed.). (1964). Book T.  The School Mathematics Project. Cambridge, United Kingdom: Cambridge University Press. Howson, A.  G. (Ed.). (1965). Book T4. The School Mathematics Project. Cambridge, United Kingdom: Cambridge University Press. Howson, A. G. (Ed.). (1967). Advanced mathematics. Book 1. The School Mathematics Project. Cambridge, United Kingdom: Cambridge University Press. Howson, A. G. (1978). Change in mathematics education since the late 1950s—Ideas and realization: Great Britain. Educational Studies in Mathematics, 9(2), 183–223. Karp, A. (2008). Interview with Geoffrey Howson. International Journal for the History of Mathematics Education, 3(1), 47–67. Kilpatrick, J., & Wirszup, I. (Eds.). (1969–1972). Soviet studies in the psychology of learning and teaching mathematics (6 Vols.). Stanford, CA: School Mathematics Study Group. Kline, M. (1973). Why Johnny can’t add: The failure of the New Math. New York, NY: St. Martin’s Press. Ling, J. (1987). SMP activity in the 11–16 sector: 1961–86. In A. G. Howson (Ed.), Challenges and responses in mathematics (pp. 34–48). Cambridge, United Kingdom: Cambridge University Press. MA. (1947). The place of visual aids in the teaching of mathematics. The Mathematical Gazette, 31(296), 193–205. MA. (1956). The teaching of mathematics in primary schools. London, United Kingdom: Bell & Sons. MA. (1959). Mathematics in secondary modern schools. London, United Kingdom: Bell & Sons. MA. (1964). A second report on the teaching of arithmetic in schools. London, United Kingdom: Bell & Sons. Moon, B. (1986). The “New Maths” curriculum controversy: An international story. Barcombe, United Kingdom: Falmer Press. Nunn, T. P. (1951). The training of the teacher. The Mathematical Gazette, 35(311), 41–43. OEEC. (1961a). New thinking in school mathematics. Paris, France: OEEC. OEEC. (1961b). Synopses for modern secondary school mathematics. Paris, France: OEEC. Piaget, J., Beth, E.  W., Dieudonné, J., Lichnerowicz, A., Choquet, G., & Gattegno, C. (1955). L’enseignement des mathématiques [The teaching of mathematics]. Neuchâtel, Switzerland: Delachaux et Niestlé. Pinner, M. T. Sr. (1981). Mathematics: Its challenge to primary school teachers from 1930–1980. In A. Floyd (Ed.), Developing mathematical thinking (pp. 12–25). London, United Kingdom: Adison-Wesley in association with Open University Press. Polya, G. (1957). How to solve it. New York, NY: Doubleday Anchor Books. Price, M.  H. (1994). Mathematics for the multitude? A history of the Mathematical Association. Leicester, United Kingdom: Mathematical Association. Rogers, L. (1999). Conflict and compromise: The evolution of the mathematics curriculum in nineteenth century England. In P.  Radelet-De Grave (Ed.), Proceedings of the Third European Summer University on History and Epistemology in Mathematical Education (Vol. 1, pp. 309–319). Leuven/Louvain-la-Neuve, Belgium: Université Catholique de Louvain. Rogers, L. (2014). Mathematics education in the United Kingdom: Scotland. In A.  Karp & G.  Schubring (Eds.), Handbook on the history of mathematics education (pp.  269–282). New  York, NY: Springer Science+Business Media. Rogers, L. (2017). New conceptions of mathematics and research into learning and teaching: Curriculum projects for primary and secondary schools in the UK (1960–1979). In K. Bjarnadóttir, F. Furinghetti, M. Menghini, J. Prytz, & G. Schubring (Eds.), “Dig where you stand” 4. Proceedings of the Fourth International Conference on the History of Mathematics Education (pp. 325–347). Rome, Italy: Edizioni Nuova Cultura. Rogers, L. (2019). An appreciation of Trevor Fletcher: 1922–April 14, 2018. Mathematics Teaching, 266, 44–45.

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Schubring, G. (2017). Mathematics teaching in the process of decolonisation. In K.  Bjarnadóttir, F.  Furinghetti, M. Menghini, J. Prytz, & G. Schubring (Eds.), “Dig where you stand” 4. Proceedings of the Fourth International Conference on the History of Mathematics Education (pp. 349–367). Rome, Italy: Edizioni Nuova Cultura. Servais, W. (1970). The significance of concrete materials in the teaching of mathematics. In Association of Teachers of Mathematics, Mathematical reflections. Contributions to mathematical thought and teaching, written in the memory of A. G. Sillitto (pp. 203–208). Cambridge, United Kingdom: Cambridge University Press. Servais, W. (1975). Continental traditions and reforms. International Journal of Mathematical Education in Science and Technology, 6(1), 37–58. Simon, B., & Simon, J. (Eds.). (1963). Educational psychology in the USSR. Stanford, CA: Stanford University Press. Skemp, R. (1976). Relational understanding and instrumental understanding. Mathematics Teaching, 77, 20–26. Swan, F. J. (1950). Mathematics in the comprehensive school. The Mathematical Gazette, 34(309), 182–197. Thwaites, B. (1966). Mathematical reforms in English secondary schools. The Mathematics Teacher, 59(1), 42–52. Thwaites, B. (1972). The school mathematics project: The first ten years. Cambridge, United Kingdom: Cambridge University Press. Vanpaemel, G., De Bock, D., & Verschaffel, L. (2012). Defining modern mathematics: Willy Servais (1913–1979) and mathematics curriculum reform in Belgium. In K. Bjarnadóttir, F. Furinghetti, J. Matos, & G. Schubring (Eds.), “Dig where you stand” 2. Proceedings of the Second International Conference on the History of Mathematics Education (pp. 485–505). Lisbon, Portugal: New University of Lisbon. Williams, J. (1971a). Problems and possibilities in the assessment of mathematics learning. Educational Studies in Mathematics, 4(1), 135–149. Williams, J. (1971b). Teaching technique in primary maths. Slough, United Kingdom: NFER.

Chapter 8

Modern Mathematics in Italy: A Difficult Challenge Between Rooted Tradition and Need for Innovation Fulvia Furinghetti and Marta Menghini

Abstract  In this chapter, we first describe the Italian context when the proposals of reforms, generically indicated under the label of “new math” or “modern mathematics,” were developed all around the world. The current mathematics programs dated to 1945, the main topic was geometry, taught according to a rooted tradition based on Euclid. The conference of Bologna in 1961, which followed those of Royaumont, Aarhus, and Zagreb-Dubrovnik, stimulated the Italian mathematicians to consider also for their country’s reforms in the light of the proposals that emerged at the international level. With the collaboration of the Ministry of Education and under the aegis of OECD, they organized refresher courses on the new approaches suggested by modern mathematics, edited books, and supervised experiments in selected classes. The plan by the Ministry was not efficient; however, this ferment stimulated various meetings for developing new mathematics programs. These programs were never implemented and only a few notions of modern mathematics remained, but new ideas and new contacts began to circulate which slowly changed the Italian context. Keywords CIEAEM · CIIM (Commissione Italiana per l’Insegnamento Matematico) · Emma Castelnuovo · Experimental projects · Geometry · ICMI · Italian programs of mathematics · Luigi Campedelli · Modern mathematics · New Math · Reform movement · Ugo Morin · UMI (Unione Matematica Italiana)



Traditions in Italian Mathematics Education

In 1861, Italy became a unified country and 10  years later Rome became the capital, after its annexation in 1870. These events were the final step of a process of construction of the nation, which after the Vienna Congress in 1815 had been made up of some 10 little states. One of them, the Kingdom of Sardinia (including Savoy, Piedmont, Liguria, the county of Nice, and Sardinia), was the core of this process and the king of Sardinia became the king of Italy. One of the main tasks of the new state was the construction of a system of instruction, starting from the system inherited from the Kingdom of Sardinia enacted by the Minister of Education Gabrio Casati with the Law of November 13, 1859. F. Furinghetti (*) University of Genoa, Genoa, Italy M. Menghini University of Rome Sapienza, Rome, Italy © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_8

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An important step was the Coppino Reform of higher education launched in 1867, which introduced the Elements of Euclid as a textbook (Maraschini and Menghini 1992; Vita 1986). This introduction was intended to reinforce the educational value of the Italian High School: The teaching of geometry followed the Euclidean approach and was definitely seen as “gymnastics of thought.” Some paramount mathematicians, the geometer Luigi Cremona for one, were behind this choice. The Coppino Reform drastically reduced Casati’s programs of 1859, to eliminate “all those theories […] whose importance lies mainly in the applications they lead to, or in their subsequent development in higher mathematics”1 (Regio Decreto, October 10, 1867, No. 1492). In the following years, the Elements became non-­compulsory and were gradually replaced by textbooks that still had to follow the Euclidean method. The imprint had been given: In Italy, the “traditional route” became, without doubt, deductive and synthetic teaching of geometry via axioms and theorems, as done in Euclid’s Elements. According to Guido Castelnuovo, loyalty to Euclid’s Elements had overshadowed the theory of measure, just as it had happened with the Greeks, “who could not conceive of the ratio between two incommensurable measures as a number” and whose “sense of rigor rejected the use of approximations” (Castelnuovo 1919, p. 5). This position prevented teaching from penetrating “the heart of modern thought” (p. 5). For instance, Italy refused the curricular reform proposed by Felix Klein to the European countries at the beginning of the 1900s, which revolved around the concept of “functional thinking” (Schubring 1987). Indeed, the strongly “classic” setup that inspired the school programs of the newly unified Italy— and that remained a dominant feature in Italian schools—is evident when we refer to the question of the introduction of real numbers and the concept of function. Even the main structural reform of the Italian school system, known as the Gentile Reform, launched in 1923, which also introduced the Scientific Lycée, was characterized by a marked humanistic orientation and did not solve the question. From 1925 until the programs of the Allied commission of 1945,2 attempts to better the Gentile Reform by allowing an intuitive introduction of real numbers in the early years of High School were disregarded. Real numbers and analytic geometry continued to enter the curriculum very late, only in the third year, because as clarified above by Castelnuovo (1919), the construction of real numbers is geometric and they can only be introduced after the complete treatment of geometry (Marchi and Menghini 2013). This is the framework that explains some of the Italian resistance to change when the reform movements of the 1960s developed all around the world. In Italy, the reference movement was “modern mathematics” inspired by Bourbakism. In this chapter, we use the following labels to indicate the various levels of the Italian school system (see Ciarrapico and Berni 2017) for further information: • Primary school (age range: 6–11 years; grades: 1–5). • Secondary school: Middle school (age range: 11–14 years; grades: 6–8). High school (age range: 14–19 years).

 All the translations in this chapter are by the authors.  After 1945 the first reform of the Lycées occurred in 2010.

1 2

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The Reconstruction After World War II in Italy: National Initiatives and International Contacts in Mathematics Education

Like all nations that came out of the War, Italy had to deal with the problems of reconstruction. Although there were serious problems in the education system, chief amog them the illiteracy still present in the country, there was little attention paid to the importance of the school system, especially a good scientific education for a modern society. The first sign of an exit from stagnation was given in 1950 by the Ministry of Public Instruction, which entrusted the Centro Didattico Nazionale per la Scuola Secondaria Superiore [National Educational Center for High School], a body established following a law of 1942, with the task of organizing didactic specialization courses and experimental classes to inspire and encourage new teaching methods. In 1950, mathematicians such as Luigi Campedelli and Attilio Frajese, together with educators such as Aldo Agazzi and Gesualdo Nosengo, founded the Movimento Circoli della Didattica [Movement of Didactic Circles] and the journal Ricerche Didattiche [Didactical Studies] dedicated to the teaching of various disciplines. Soon after its foundation, the Movimento organized conferences and edited books, where the cultural background for renewing mathematics teaching was discussed (see Movimento Circoli della Didattica 1956). The previous initiatives had a general orientation toward all disciplines taught at school. As far as mathematics education was concerned, since 1895, an association of mathematics teachers (Mathesis) existed and the journal Periodico di Matematica (from 1921 Periodico di Matematiche), founded in 1886, became its official organ in 1898. Another journal for mathematics teachers, Archimede, founded by a secondary school teacher in 1902 with the title Il Bollettino di Matematica, was published regularly. As a general policy, the Periodico mainly dealt with Italian authors and themes. Archimede showed some openness to developments abroad. In the 1960s, this journal published articles not only with a Bourbakist imprint but also gave voice to the unease of teachers in relation to the proposals of modern mathematics. After World War II, the Italian community of mathematicians revived the activities of the Unione Matematica Italiana (UMI) [Italian Mathematical Union] and resumed relations with the international context so that in 1952 Italy hosted the first General Assembly of the reborn International Mathematical Union (IMU) (Lehto 1998). On this occasion, the old Commission on the Teaching of Mathematics founded in 1908 became a subcommission of IMU with the name “International Mathematical Instruction Commission” (IMIC). After 2 years, this name was changed into the current name “International Commission on Mathematical Instruction” (ICMI). The Italian Guido Ascoli was appointed as a Treasurer of the Executive Committee in the period 1952–1954 (L’Enseignement Mathématique, 1951–1954, 40, pp. 81–82). At the meeting of Geneva on July 2, 1955, under the presidency of Heinrich Behnke, the countries belonging to IMU were invited to appoint national subcommissions for ICMI and be represented by national delegates (L’Enseignement Mathématique, 1955, s. 2, 1, p. 198). In Italy, the Sottocommissione Italiana per l’Insegnamento della Matematica [Italian Subcommission for Mathematics Teaching] was established in 1954 (see BUMI 1954). The name Commissione Italiana per l’Insegnamento Matematico [Italian Commission for Mathematical Teaching] (from 1963 designated with the acronym CIIM) still exists, although CIIM is no longer a subcommission of ICMI: From 1975, it became a permanent subcommission of UMI. The successive presidents in the period 1954–1988 were Giovanni Sansone, Enrico Bompiani, Campedelli, Salvatore Ciampa, Vinicio Villani, Giovanni Prodi, Benedetto Scimemi, and Carmelo Mammana. They were mathematicians, but some of them showed a genuine interest in mathematics education. Usually, the president of CIIM was the ICMI representative. It was not until 1963 that school teachers were admitted to CIIM. In the period in question, ICMI was not the only agency aimed at discussing problems related to mathematics education: A prominent role was played by the Commission Internationale pour l’Étude

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et l’Amélioration de l’Enseignement des Mathématiques (CIEAEM) [International Commission for the Study and Improvement of Mathematics Teaching]. Among the Italian members of CIEAEM, two teachers—Emma Castelnuovo and Angelo Pescarini—were particularly active and contributed to establishing international contacts in the school milieu.



From Royaumont to Bologna

From November 23 to December 4, 1959, OEEC (Organisation for European Economic Co-operation, later on, OECD, Organisation for Economic Co-operation and Development)3 organized a Seminar on the reform of mathematics teaching at the Cercle Culturel de Royaumont in Asnières-sur-Oise (France). Delegates came from 15 OEEC countries (Austria, Belgium, Denmark, France, Germany, Greece, Ireland, Italy, Luxembourg, The Netherlands, Norway, Sweden, Switzerland, Turkey, UK), Yugoslavia, Canada, and the United States. As is well known, this event was a turning point in the history of modern mathematics (see De Bock and Vanpaemel 2015; Schubring 2014). The basis for discussion should have been the answers to a questionnaire sent in December 1959 to the countries participating in the program asking about the current conditions of mathematics education in their countries (see OEEC 1961a, p. 7),4 but data summarizing responses to the questions were not available at the beginning of the Seminar. Answers given to the questions were nevertheless presented and analyzed in the second part of the official report (OEEC 1961a, pp. 127–210), titled “Survey of Practices and Trends in School Mathematics.” The questions concerned: Number of hours of compulsory mathematics teaching in the different schools; educational qualification and teacher training; authorities competent to establish programs and select textbooks; any ongoing reforms; contents of the programs (more precisely, various topics were listed and it was asked, for each of them, in which school year it was carried out). This survey is an interesting source of information about mathematics education in the countries participating in the program before the start of modern mathematics in Europe. We report the following information concerning Italy, based on answers given by Campedelli to the questions of the questionnaire: • Italy was the only country, besides Ireland, in which no pedagogical studies were included in the programs for initial teacher training. • Italy was the only country where there seemed to be an oversupply of mathematics teachers.5 • The part of the questionnaire relating to the training of elementary teachers referred more precisely to “arithmetic teachers”; a few countries, including Italy, specified that the training of primary teachers concerned not only arithmetic but also geometry

 The “Organisation for European Economic Co-operation” (OEEC), born in 1947 as “Committee for European Economic Co-operation,” included 18 European countries. When the USA and Canada joined as members in the end of 1960, it was renamed “Organisation for Economic Co-operation and Development (OECD),” “Organisation de coopération et de développement économiques” in French. New statutes came into force on September 30, 1961. English and French are the official languages. 4  The questionnaire (see Appendix B in OEEC (1961a), pp. 221–237) was sent to the OEEC member countries, and to Canada, the United States, and Yugoslavia. Only Spain did not answer, so the survey lists 20 countries. The results of the survey, divided by countries, are also presented in (OEEC 1961b), with the exception of Canada. 5  This is in contradiction with various associations’ agendas around the years 1962–1964 (see, e.g., R. G. 1964a). 3

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The Italians at Royaumont The various countries adopted different selection processes for the delegates who were to represent them. In Italy, the CIIM stated that it would participate in the Seminar with two delegates (BUMI 1960), who were Campedelli and E. Castelnuovo, both key figures in the Italian milieu of mathematics education of those years. By a twist of fate, they both were linked to G. Castelnuovo, a distinguished mathematician, founder of the Italian school of algebraic geometry. He was also engaged in the development of mathematical instruction in Italy and abroad in the early decades of the 1900s. In 1908, he chaired the International Congress of Mathematicians in Rome, when the Commission from which ICMI sprung was founded. Later, he was a member of its Central Committee between 1913 and 1920 and vice-president between 1928 and 1932. Campedelli (1903–1978) started his academic career as an assistant professor of G. Castelnuovo in Rome. He researched in the field of algebraic geometry under the guidance of Federigo Enriques. At the end of the 1940s, when the impetus of the Italian school of algebraic geometry died out, Campedelli’s interests turned almost exclusively toward mathematics education and elementary mathematics, with particula reference to geometry, logic, and foundations of mathematics. He wrote textbooks for school and university, and many articles (see Furinghetti 2019). As mentioned before, he founded the Movimento Circoli della Didattica and animated the first national initiatives on mathematics teaching in the 1950s. In the list drawn up by Caleb Gattegno in 1957, he was among the “membres actifs” [active members] of CIEAEM (see Bernet and Jaquet 1998, p. 25), which were the members who constituted the national section and promoted the work of the Commission in their countries. Campedelli contributed to the second book published by CIEAEM (Gattegno et al. 1958) with a chapter oriented to classroom practices that included extensive use of concrete materials (Figure 8.1). In his papers, Campedelli did not take an explicit position about modern mathematics, which is surprising given that he had been one of the Italian delegates at the Royaumont Seminar. In particular, it is disappointing that such a prolific author did not report adequately in Italian journals for mathematics teachers about this event. As to E. Castelnuovo (1913–2014), daughter of Guido, the reader can find information in different papers (e.g., Furinghetti and Menghini 2014). Here we focus on her many international contacts: According to Théo Bernet and François Jaquet (1998), she was among the founding members of CIEAEM, and became the Italian representative at ICMI, starting with the 1975–1978 term (ICMI Bulletin, 1975, No. 5). During that term, she was also a member-at-large of ICMI. Through her text-

Figure 8.1  Emma Castelnuovo (right) and Lina Mancini Proia in Pallanza, Italy, 1973

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books, she contributed to the dissemination in Italy of some of the proposals born in the milieu of CIEAEM and the Belgian schools. For her, a fundamental reference and source of inspiration were Paul Libois and his work at the École Decroly in Brussels, which she often visited (Castelnuovo 1965). The attitude of E. Castelnuovo toward modern mathematics can be summed up by her exclamation at Royaumont, which she recalled in a private conversation during the forty-third meeting of CIEAEM in Locarno (1991): “But Professor [Jean Dieudonné], without the triangle, the table on which your speech sheets rest would collapse! [it was a plank placed on trestles]” (Équipe de Bordeaux 2009, p.  2). She recognized the unifying power of modern mathematics and introduced elements of set theory and transformations into her teaching. However, she did not give up the intuitive approach and her method of bringing students closer to the abstract, starting from their experiences in concrete situations. After an intensive teacher training course on mathematics teaching held in Arlon (Belgium) on July 2–4, 1960, organized by the Belgian Ministry of Public Education, she wrote that on the one hand, she felt all the charm of new studies which, thanks to their generality and simplicity, may unify the most disparate theories; on the other hand she did not consider it correct to introduce modern theories in their purest expression without making young people aware that they are the result of the work of centuries (see Castelnuovo 1960).

Bologna Conference The themes discussed in Royaumont were revisited and extended in follow-up meetings, in particular, in Aarhus (May 30–June 2, 1960), and in Zagreb-Dubrovnik (August 21–September 19, 1960) (see Notiziario. 1960; see also Chap. 4 in this volume). In those years, there were no Italians in the ICMI Executive Committee, but Tullio Viola participated in the ICMI meeting in Paris on December 7–8, 1959 (L’Enseignement Mathématique, s. 2, 5, 1959, p. 140), and Sansone and Mario Villa participated in that of Belgrade on September 21, 1960 (L’Enseignement Mathématique, 1959, s. 2, 5, p. 284). Viola represented Italy at the Aarhus meeting and contributed to the discussion (Behnke et al. 1960). Villa participated in the Zagreb-Dubrovnik meeting (BUMI 1960; OEEC 1961c), and Pescarini participated in the Budapest meeting of 1962 (Hungarian National Commission for UNESCO 1963). A good opportunity to come in direct contact with modern mathematics was given to Italians at the international conference held in Bologna (October 4–8, 1961) entitled “A discussion of the Aarhus and Dubrovnik reports on the teaching of geometry at the secondary level” (see Chap. 4 in this volume). ICMI and the Italian CIIM sponsored the meeting, and university professors, school inspectors, educators, and delegates from UNESCO and OECD were in attendance (BUMI 1962b). The organizing committee consisted of Pietro Buzano, Marshall Stone (ICMI president), Villa (chairman), Viola, and Gilbert Walusinski (ICMI secretary). The Italian delegation consisted of six ministerial inspectors, one representative of the Centri Didattici Nazionali (Ruggero Roghi), and university professors (Platone 1961; Stone and Walusinski 1963). Most contributions were published in L’Enseignement Mathématique (1963, s. 2, 9). The importance of the Bologna conference is twofold: At the international level, it offered a moment of synthesis of the various proposals and clarified the positions on modern mathematics of some famous mathematicians (Stone, Emil Artin, Henri Cartan, Hans Freudenthal); at the national level, it promoted the opening to modern mathematics in Italy. In his talk, the Italian Ugo Morin (1901–1968) proposed to introduce tools, such as real numbers, abstract algebra, set theory, and vectors, which could be used in teaching geometry. He noted that in Italy it would be difficult to implement the axiomatic structure proposed in Zagreb-Dubrovnik because

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Figure 8.2  Ugo Morin (left) and Bruno de Finetti at the UMI quadrennial Congress in Trieste, Italy, 1967. (Courtesy of Centro Ricerche Didattiche Ugo Morin)

of the different starting age of high school (14  years) in Italy compared to that of other countries (15 years) (Morin 1963). In the 1960s, Morin well understood what could be the important elements for renewing mathematics teaching and teacher training. For example, he advocated the introduction of psycho-­ pedagogical courses into the university curriculum for mathematics students who chose to become secondary school teachers (BUMI 1966). Morin had been Vice-President of UMI (Figure 8.2) and a member of CIIM.  His actions throughout his professional life demonstrated a strong political and social commitment (Tomasi 2018). He wrote textbooks and drafted programs for secondary schools that incorporated elements of modern mathematics. Morin’s intervention at the Bologna conference revealed his openness to new methods and contents, elements which were also present in his mathematical research. L’Enseignement Mathématique included the texts of the interventions of other Italians, who did not take an explicit position on modern mathematics. In particular, the conference opened with a report by Villa (1962), who on the one hand stressed the need to modernize mathematics teaching in Italian secondary schools, which, he said, had been “standing still” for almost half a century; on the other hand, he observed that any modernization should be carried out with appropriate caution. He argued that it was necessary first to institute refresher courses for teachers and then pilot classes in which some of the topics and methods that were to be introduced into mathematical teaching could go through a necessary, prudential experimental phase. Furthermore, he added that it was clear that the tests, the experiments, and the definitive programs required an adaptation to the traditions and needs of the various countries in which the modernization of mathematics programs would be undertaken. In Villa’s talk, caution and prudence were emphasized. Other talks by Italian representatives were those by Viola about didactics without Euclid and Euclidean pedagogy, and by Lucio Lombardo-Radice about geometry and culture in humanistic schools (Licei Classici). Campedelli and Roghi reported respectively on the refresher courses held in Italy and on the first experiments carried out in schools. The Italian contributions seemed to stay away from the nub of the problem, specifically modern mathematics. This suggests that a criticism expressed

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by Cartan (1963) toward those who wanted to innovate without changing anything referred to Italians. In particular, the aloofness of Campedelli in the discussion might be interpreted as a sign of his lack of interest in (or even dislike for) modern mathematics.



First Cautious Steps Toward Modern Mathematics in Italy

In the days immediately following the conference (October 9–10, 1961), a meeting of experts appointed by OECD took place to discuss which Italian classes should test the new proposals (BUMI 1962b). The meeting was attended by several Italian university professors. In a follow-up meeting on November 12, 1961, CIIM authorized Buzano, Campedelli, Morin, Villa, and Viola to plan an experiment in schools (BUMI 1962a). The Ministry of Education set up a commission (BUMI 1962b) to organize refresher courses and pilot classes in mathematics with a modern program; costs were to be borne half by the Ministry of Education and half by OECD. These refresher courses covered topics that originated from the Bologna conference such as set theory, geometric transformations, essential elements of abstract algebra and algebraic structures, the logical structure of geometry, and applied mathematics. Two volumes were published on the contents of the courses (Villa 1965, 1966). In the school year 1962–1963, the Ministry set up some 40 “pilot classes,” which concerned the last classes of the humanistic High School, the second-last classes of the scientific High School, and the High School for prospective teachers (see Notiziario. 1963). In these pilot classes, part of the mathematics program was traditional, while the other part covered topics chosen by the teachers from those developed in the refresher courses.6 At the above-mentioned meeting of CIIM in November 1961, prudence was recommended, and a need to listen to teachers’ opinions was stressed. The pilot classes were then extended to the istituti tecnici (technical institutes), i.e., high schools intended to train different types of professional workers, and to a few middle schools. In these experimental classes, essentially two texts were used: Per un Insegnamento Moderno della Matematica [For a Modern Teaching of Mathematics], published in 1963 (Bologna, Italy: Patron) by the Ministry of Education and OECD, and La Matematica Moderna nell’Insegnamento Secondario [Modern mathematics in Secondary teaching] published in 1965 (Rome, Italy: Eredi V. Veschi), written by the ministerial inspector Armando Chiellini. There was no lack of criticism of the new trends, such as that by Eugenio Togliatti expressed in a conference held in Rome in May 1962 on the theme “Scientific and humanistic teachings in the educational function of secondary school.” In his report we read: I do not see how the average of the students, even in the last high school class, can reach such a level of knowledge and logical education that they can appreciate the high value of the scientific synthesis that is contained in an abstract “structure.” […] I would not like this to end up in an arid formalism, and for the students, too difficult to appreciate. And again, how could current mathematics teachers, trained in another climate, change the direction of their teaching so radically? (Togliatti 1963, p. 44)

In 1961, a university curriculum for prospective teachers was established (Presidential Decree 26 July 1960, No. 1692). An algebra course replaced the chemistry course present in the previous curriculum of mathematics (Lombardo-Radice 1963). A new generation of mathematics teachers was emerging, for whom in the future, the problem of updating on new topics might be less pressing. A reform that came to fruition in those years was that of the middle school, approved in December 1962 (Law No. 1859, December 31, 1962). It led to the unification of the middle cycle (11–14 years), which before was divided into various types. Mathematics and “Elements of natural sciences” were to  The atmosphere of enthusiasm that animated some initiatives of that period is well expressed in the autobiography of Lina Mancini Proia (2003). It should be remembered that the teachers involved in the various experiments of those years acted mainly on a voluntarily basis. 6

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be taught by the same teacher (something which generated a lively discussion that continues today (see Ciarrapico and Berni 2017; Furinghetti 2015). The atmosphere of modernization of mathematics teaching in those years had only a timid influence on related programs. The contents were in line with the past, if not for the invitation to give examples of “correspondences and functions” and to consider issues “whose discussion involves operational and structural analogies.” The word “set” was never used. The methodological suggestions in this reform appear more modern. We read that “it is useful to refer to inductive procedures that start from observations, easy experiments, and empirical evidence” and that “constant care must be taken to harmonize arithmetic with geometry to give a unitary vision of mathematics” (Law No. 1859, December 31, 1962). Frequent use of graphs to help visualization was also recommended.

Proposals of New Programs for High School Besides the reform of the middle school mentioned above, which strictly speaking cannot be considered a modern reform, other proposals for new mathematics programs for high schools followed one another, also in view of a reform that should have taken place in October 1966, but was never implemented. In particular, we refer to the programs developed between 1963 and 1967 at the Italian conferences held in Gardone, Camaiore, and Frascati. Leading characters in this phase were again Morin, vice-­ president of UMI; Campedelli, president of CIIM; Viola, president of Mathesis; and Villa, director of the Centro Didattico Nazionale. The programs proposed at these conferences primarily focused on the first 2 years of high school, which, should be the same for all types of schools. In the following, we will present only the topics of geometry, which have been the most discussed. The first proposal was developed at the Gardone conference on May 27–29, 1963, as follows (Anonymous 1965): First Year of High School • The plane as a set of points and straight lines as its subsets. Properties of connection and parallelism. Direction of a straight line. • Order on a straight line and partition of the plane. Segments. Convex figures. Angles. Polygons. • The group of the congruences. Segments and angles as magnitudes. • Perpendicular lines. Reflections. • Euclidean properties of triangles and polygons. Second Year of High School • • • •

Circumference and circle. Area (extension) of a polygon seen as magnitude. Theorems of Pythagoras and Euclid.7 Geometric constructions. Overview of logic schemes and methods for proving theorems and solving problems.

According to Morin (1965), who participated in the drafting process, the commission that had developed the Gardone programs was still too tied to inaccuracies characteristic of the status of mathematics 50 years ago. We do not know what exactly Morin was referring to; we notice that the change  The theorems usually taught in Italy as theorems I and II of Euclid correspond to the two parts of proposition 8 of Book 6 of Euclid Elements. They are necessary to prove the Pythagoras theorem. All these theorems disappeared from the successive Camaiore Programs, but the Pythagoras theorem was re-introduced, along with the Thales theorem (i. e., the intercept theorem) in the Frascati Programs. 7

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seems to be more in language than in substance. The reference to the magnitudes should also be emphasized. The theory of magnitudes refers to the introduction of real numbers according to Euclid’s theory of proportions, that is, to a geometric introduction. In the Gardone programs, the term was used to avoid talking about measures, which correspond to an isomorphism between real numbers ­(introduced independently) and segments. The term magnitude was deleted in subsequent drafts and substituted with lengths. The second proposal was developed at a conference in Lido di Camaiore on October 3–5, 1964 (Morin 1965). First Year of High School (we only mention some of the listed points) 4. Affine properties of the plane: Incidence and parallelism, direction; translations, vectors, and point reflection. 6. Topological properties: Order on a straight line and partition of a plane. Segments, convex figures: Angles and polygons. Second Year of High School 9. Metric properties of the plane: Perpendicular lines, reflection about a line, group of congruences. Length of a segment and width of an angle (mentioning their measures). Applications to polygons. 12. Elements of analytic geometry: Cartesian coordinates on the line and on the plane. Equations of a translation. Representation of a straight line … There was also a first point on “elements of set theory,” a second point on “inner operations on sets of numbers representing groups, rings, and fields,” and an eighth point (in the second year) on “an intuitive approach to real numbers.” An interpretation of these programs was provided by Morin (1965). The aim was to introduce, in addition to the axioms of incidence, an axiom of existence of a collineation (point reflection) which fixes a point (No. 4), and an axiom on the properties of perpendicular lines, which allowed for the introduction of reflections about a line and therefore congruences (No. 9). So we observe that axioms asserting the existence of isometries appear (as also is in Gustave Choquet’s metric axioms), but geometric transformations are not yet explicit. They will appear later in the Frascati programs. The aim of all the programs was to “adhere to the new scientific and didactic needs, but also to update notations, expressions, definitions, to the current scientific level” (Morin 1965, p. 57). It was precisely the new expressions and definitions that made some proposals hermetic. Another moot point in these programs was that a large part, relating to set theory, order structures, and algebraic structures, was supposed to precede the teaching of geometry. At both the conferences of Gardone and Camaiore, a need was felt to renew contents and methods to promote scientific, cultural, and educational progress. Moreover, an urgency was stressed to organize refresher courses for all mathematics teachers and to make the necessary bibliographic tools available to them. With the Camaiore programs, the debate increased, with suggestions of prudence, but not of rejection being made. There was a general attitude in favor of the reform; what was most appreciated was the possibility of new “transversal” connections between the classical topics (the concepts of structure, set, vector, function, …) and, as E. Castelnuovo (1964) suggested with reference to geometry, a treatment by type of property rather than by type of figures. In addition to this, the concept of structure, which characterized the work of Bourbaki, was recognized as unifying different mathematical branches: “This is a far-reaching fact: Undoubtedly today there would not be such a widespread awareness of the unity of mathematics if Bourbakism had not existed” (Prodi 1995, p.  416). The Camaiore programs also reflected the idea that developments in mathematics (in particular abstract algebra and topology) could not fail to influence school teaching.

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The journal Archimede gave space to the curriculum debate. Indeed, the director of Archimede, Roberto Giannarelli,8 became personally involved in the activity of reform. Starting from 1964, ­comments and reports on the various proposals made at the conferences and working sessions followed one another in this journal. The journal Periodico di Matematiche also published some contributions on this subject from 1965 onward. The references that appear in the various papers related to the proposals always included the works of Choquet, Dieudonné, Georges Papy, etc. The Erlangen program was rarely mentioned. Francesco Tricomi (1965) complained, among other things, about the disappearance from the Camaiore programs, of the Pythagoras theorem, and theorems relating to the circumference. A much less cautious position was taken in the account of Bruno de Finetti (1965) at another international conference, held in October 1964 at the Centro Europeo per l’Educazione (CEDE) [European Center for Education] in Villa Falconieri (Frascati). The conference focused on an issue that was very important at the time, namely the transition from high school to university. Of particular importance was the teaching of geometry, which, as pointed out by Jacqueline Lelong (1965) formed the basis of secondary mathematics education but did not allow any extension to higher studies. According to de Finetti, “the main role for a radical simplification and revision of the mathematical tools and a unified vision of almost all topics, undoubtedly belongs to the notion of the linear (affine) system (and space). … Euclid’s geometry, so unnatural and heavy because of the lack of distinction between affine and metric properties, can therefore be left aside” (De Finetti 1965, pp. 123–124). He greatly appreciated the programs for the middle school proposed by Papy (1966), already being implemented in Belgium, and for high school by André Revuz (1965), which constituted a rather definitive model of the ideas to be realized. The Ministry’s intent to modernize school mathematics aroused mixed reactions from teachers. Those who had followed the experiments in the pilot classes were enthusiastic, but many others, the majority, were perplexed and assumed an attitude of refusal. Among these were the older teachers and the teachers of the middle school who seemed to be unable to deal with the proposed change because most middle-school teachers held degrees in the sciences and lacked specific mathematical preparation (see Temussi 1965). The problem of teachers’ lack of preparation had been mentioned by Behnke as an international problem at the aforementioned Frascati conference (see R. G. 1964a, b). The pilot classes, in which ministerial inspectors and teachers did their utmost, were not very successful for a variety of reasons including the fact that the project was experimented only in the two last school years (Ciarrapico and Berni 2017). Vincenzo Vita (1986) and Giro (1969) claimed that the results of the experiments in the pilot classes were never collected and publicized, nevertheless, their effects could be considered limited. In lycées and high schools for prospective primary  teachers—arguably the schools which most needed a change, no official reform was initiated, and—apart from the experiments—the mathematics programs remained those of 1945.



The (Blunt) Top of Modern Mathematics in Italy

During 1966 and 1967, a large commission gathered at Villa Falconieri in Frascati under the aegis of UMI and CIIM, and a long cycle of activities dedicated to programs was concluded with what became known as the Frascati programs (Giannarelli 1967; R. G. 1966). Participants in the two conferences of 1966 (February, 23–26) and 1967 (February, 16–18) were university professors—including Campedelli, de Finetti, Prodi, and Villani—secondary school teachers invited by UMI (BUMI 1966), the president of UMI and the members of its Scientific Commission,  Sometimes his contributions in the journal are signed R. G. or Giro.

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members of the Consiglio Superiore della Pubblica Istruzione [Higher Council of Public Education], officers of the Ministry of Education, and representatives of the National Educational Center for High School. At the end of the conferences, two proposals for new programs were formulated. The first, that of the 1966 conference, was published in the journal Archimede, and represents a unified proposal for the first two classes of high school (R. G. 1966); the second, formulated at the end of the work in 1967, is a minimum proposal for the final 3 years of classical or scientific lycées and appeared in the same journal (Giannarelli 1967). In these programs, the aims were to educate students in mathematical thinking, avoiding teaching conceived as training. It was recommended to proceed gradually from the intuitive to the rational aspects. UMI approved these programs (BUMI 1968), which were the subject of articles and conferences (we only mention the conference promoted by CIIM and UMI in Frascati from February 6 to 8, 1969, see Notiziario 1969). Although there has been no practical implementation, Frascati’s programs helped to raise awareness among the most attentive teachers of the need for renewal of mathematics teaching. De Finetti (1967), member of CIIM since 1964, who was one of the drafters of the programs, recalled the work that accompanied the discussion over the years and observed that: the most positively innovative aspect, if I am allowed to express myself in a paradoxical form, lies not so much in the introduction of some useful and interesting things, as in banning all the heavy and useless things with which children are tortured due to inveterate customs. (p. 80)

The program for the first 2 years presented two themes—algebra and geometry—both with aspects of modernity. It included topics such as the partition of a set and equivalence relations, structures such as ring, group, field, and—in case—lattice and metric space, presented through examples from different domains. As for geometry, in continuity with the middle school, geometric transformations (translations, rotations, and reflections) were treated with applications to segments, angles, triangles, and polygons. The program suggested an approach to geometry through several methods chosen by the teacher, including the analytical one. In the 3-year program for the final years, alongside traditional topics, new themes such as the geometric vector plane and the abstract vector space appeared. In the final year, probability and statistics appeared, timidly. It was de Finetti, a researcher in probability theory, who was opposed to introducing probability, believing that this topic could not be taught well at that level (De Finetti 1967). This was probably why the programs included only “Elements of probability and simple applications of statistics.” There is also a chapter entitled “afterthoughts and complements,” in which possible topics traditionally unrelated to mathematics in secondary school were listed—such as “non-Euclidean geometries,” “the projective extension of affine space,” “the introduction to mathematical logic and Boolean algebra,” “elements of topology,” “elements of game theory,” “equations of the third and fourth degree.” The section on geometry for the 5 years was structured as follows: First year • The plane as a set of points and straight lines as its subsets. Incidence, parallelism, direction. • Order on a straight line and partition of a plane. Segments, convex figures: Angles and polygons. Second year • Congruence (or isometries). Comparison of segments. Perpendicularity. Translations, rotations, and reflections. Applications to segments, angles, triangles, and polygons. Circumference and circle. Regular polygons. Intercept theorem and Pythagoras’ theorem. Third year • The geometric vector plane: Linear combinations, coordinates, translations. Systems of linear equations in two unknowns. Cartesian equation of a straight line, systems of two lines.

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• Complex numbers. Scalar product. • Group of congruences and similarities in the plane. Fourth year • Cartesian equation of the circumference, ellipse, hyperbola, and parabola. • Space as a set of points, lines, and planes as its subsets. Incidence and parallelism, half-spaces. • Geometric vector space. Scalar product in three dimensions. Perpendicularity. Distances. Angles between lines and planes. • Area of the plane figures: polygons, circle. Length of the circumference. Fifth year • Simple solids and their main properties. • Volumes of simple solids. Areas of rotation surfaces. • Abstract vector space. Its models and applications. To some extent, Gardone’s programs were adopted, but with an emphasis on geometric transformations. We still find, as in Camaiore’s programs, an intuitive introduction of real numbers and Cartesian coordinates in the second year. In the 3-year program among the aforementioned optional topics, there are “elementary transformations and their groups,” and also the “projective extension of the affine or Euclidean space,” but since the affine or Euclidean spaces were not explicitly named in the programs, it may be assumed that they fell within the study of the geometric vector plane in the third year, and in its extension to space with the scalar product in the fourth year. Analyzing these programs we see that, despite the stimuli given by the European reforms, the Italian proposals followed a different path, in particular different from the French one. The Italian mathematicians involved in the 1960s reform did not want to deviate too much from their tradition: There was a clear common desire to treat geometry in a synthetic way, adapted to the new trend. Anyway, they accepted to anticipate the Cartesian plane and the intuitive treatment of real numbers to the second year, and also accepted to give up the traditional axiomatic approach in favor of a different axiomatic approach. A controversial question still concerned when to introduce the “group of congruences,” and the role to be given to it. The problem, as clearly expressed by Morin already in the preface to the 1958 edition of his textbook (Morin and Busulini 1963), was how to establish the equality between figures, which is the basis of elementary geometry. In this respect, it was useful to “recover” the Erlangen program. Despite the contribution of Otto Botsch, the Erlangen program had been neglected at Royaumont (OEEC 1961a), but the study of transformation groups was present in various Italian debates on the reform of geometry programs (Menghini 2007). In the preface of the first edition of Morin and Busulini (1963), we read: If by suggestion we speak of movement and displacement, the geometric interest only concerns the consequent equality relation between figures, whose properties need to be postulated. This approach to the theory of equality is perfectly rational, indeed it characterizes elementary geometry following the famous Program of Klein, according to which in a geometry we study the properties of figures that are invariant for a specific group of transformations. (no pagination)

Thus, for instance, while Choquet postulated an isomorphism between the length of segments and the real numbers, Morin used the axioms of Euclidean geometry, postulating the existence of congruences that preserve segments and angles. We can conclude that the Frascati programs represented a point of balance between innovative choices and conservative positions. After the Frascati conferences of 1966 and 1967, more and more attention was given to learning problems. Research in the field of mathematics teaching intensified and numerous experimental initiatives were undertaken in which teachers participated enthusiasti-

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cally, supported by university professors. International developments in mathematics education contributed to this evolution by creating further opportunities for confronting ideas and experiments not only at the CIEAEM meetings but also in the newly established International Congress on Mathematical Education (ICME) and the meetings of the International Group for the Psychology of Mathematics Education (PME). At ICME-1 in Lyon, among the 655 active participants from 42 countries, there were 28 Italians (ICMI Bulletin, 1975, 5), most of them school teachers. But, as already mentioned, none of the reform proposals for high schools, went through (Ciarrapico and Berni 2017).



Initiatives in Grades 1–8

The Primary School Without ever appearing in official programs, modern mathematics found its place in primary schools and was even more diffused than in secondary schools. Unfortunately, lacking adequate teacher training, it was often reduced to the more superficial and folkloristic aspects of set theory and caused tensions between innovators and conservatives (Vita 1986). One of the first encounters with set theory happened even before the Royaumont Seminar. Indeed, in 1956, thanks to the initiative of Campedelli and E. Castelnuovo, the Italian translation of the volume Initiation au calcul [Introduction to calculation] (Piaget et al. 1950) was published. This volume included an important chapter by Jean Piaget on the genesis of the concepts of number and measure in the child. According to Piaget, the concept of number is based on two fundamental stages: First, equipotent sets are identified—through one-to-one correspondence—later they are ordered according to the number of elements (i.e., according to the inclusion of the smallest into the largest). This chapter gave rise to the first spontaneous and isolated experiments in school (Pellerey 1989). But the greatest dissemination occurred in the late 1960s, with the publication of several works on the teaching of mathematics in Primary School, including—between 1967 and 1969—the translation of the Nuffield Project (see Chap. 7 in this volume). This translation, published by Zanichelli (Bologna), was distributed by the Associazione per l’Incremento dell’Educazione Scientifica (AIES) [Association for the Increase of Scientific Education] chaired by Alba Rossi dell’Acqua and financed by Shell Italia. As Pellerey (1989) argued, Piaget never stated that the construction of the “logic of classes” (p. 30), or set theory, should precede the concept of number and arithmetic operations. However, for the introduction of the number concept, and in general of arithmetic, the set approach was almost universally accepted as a result of a “compromise” between Piaget and the Bourbakists. In school practice, it became usual to start by introducing logical concepts and operations on sets (including intersection, union, complement, …) and only later to introduce the concept of natural number “almost as if nothing had been built by the child” (Pellerey 1989, p. 30). This is, for example, the approach of Nicole Picard (1967), which served as a reference for the programs of the Lichnerovicz Commission (see Chap. 5 in this volume). Even the Nuffield Project, in which elementary school teachers collaborated and which cannot be seen as an emanation of academic choices, did not deviate from this approach, deriving, for example, the sum from the union of disjoint sets, and the product from the Cartesian product of two sets. According to Pellerey (1989), in those years Italian educators felt international pressure to renew the teaching contents, including those for primary schools. For this reason, the Centro Didattico Nazionale per la Scuola Elementare [National Educational Center for the Primary School] launched refresher courses, with contents that were inspired by the French proposals of the Lichnerovicz

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Figure 8.3  Michele Pellerey (right) with Tamás Varga in Bordeaux, France, 1973. (Courtesy of Raimondo Bolletta)

Commission for Primary School, by the British proposals of the Nuffield project, and by the books and materials proposed by Zoltán Dienes (in particular, his logic blocks and his multi-base arithmetic blocks) (see Pellerey 1989). A volume by Vittorio Duse (1969) that attempted integration between the various approaches, had a notable diffusion among primary school teachers. It was a text for teacher training, presenting a “higher point of view,” and included: Introduction to set theory, primitive concepts and tautologies, Venn diagrams, Dienes logic blocks, Cartesian product of two sets, and algebraic structures. In Italy, the influence of mathematicians led to exaggerated formalism and “rigor,” and Piagetian influence led to accentuating the use of manipulative materials, such as logic blocks. The first chapter of Duse’s book on set theory became the first chapter of all textbooks (not only for primary school) and this chapter—almost always detached from the rest of the book—persisted for a long time, often well beyond the end of the modern mathematics era. Under the influence of the new proposals, some structured projects about mathematics teaching in primary schools flourished. Worth mentioning is the project Ricerche per l’Innovazione del Curricolo Matematico nelle Elementari (RICME) [Studies for the renewal of mathematical curriculum in Primary School], led by Michele Pellerey and born from a collaboration with the Hungarian project at the Országos Pedagógiai Intézet (OPI) [National Pedagogical Institute], started by Tamás Varga in 1963 (Figure 8.3). The project began in some Roman schools and then spread to other Italian schools. Logic and set theory were still present here, but not predominant. The concept of number had an eclectic character, based on a constructive approach connected to the idea of recursive functions, rather than to sets (Pellerey 1989). A project coordinated by the University of Pavia (Ferrari et  al. 1982) also envisaged a varied approach, though obviously it still included activities linked to logic and set theory for the introduction of connectives, or activities on relations between sets. Unfortunately, these interesting projects did not have wide dissemination, but they made it possible to arrive, in 1985, at the formulation of proper programs for primary schools (Presidential Decree 104 of February 12, 1985), in which “logic and theory of sets are no longer the foundation of mathematics, but a means to analyze the mathematical discourse and guide its development” (Pellerey 1989, p. 176).

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The Middle School The reform of the middle school launched in 1963 had left some problems in mathematics teaching. To face these problems, in 1978, the Ministry of Education set up a commission of 60 members. E. Castelnuovo, Prodi, Francesco Speranza, and Villani were among the members for mathematics. The resulting programs were promulgated in 1979 (Ministerial Decree 268 of February 9, 1979). Those concerning mathematics were interesting from the point of view of contents and pedagogical suggestions. Modern mathematics was no longer in fashion (in Italy and elsewhere), but some traces remained, as the following passages illustrate: The language of sets can be used as an instrument of clarification, of unitary vision, and valid help for the formation of concepts. However, a separate theoretical discussion will be avoided, as it would be inappropriate at this level …. During the three years, whenever the opportunity arises, similarities and differences between different situations will be recognized, as an approach to the ideas of relationship and structure.

These programs incorporated some novelties shared with foreign programs: Elements of logic, probability, and statistics. The use of pocket calculators was recommended. In geometry, some types of transformations were introduced (isometries and similarities, in particular, homotheties). Through the experience based on the observation of shadows, the existence of other types of transformations was suggested. The 1979 programs were at the forefront of the international context but had little application in practice due to teacher resistance to change.



Experimental Projects for Mathematics

A New Student Population In the 1960s, there were changes in the school system, influenced by societal changes and student movements. The transition to mass schooling created a new school population. The final exam at the end of high school (Esame di Maturità), which was very demanding as it required control over all subjects in the program, was reformed (Law No. 119 of April 5, 1969). The main change was that the oral examination was reduced to an interview on two subjects assigned in the previous months, one of which was chosen by the candidate. Since in Italy the diploma given by the Esame di Maturità was a necessary condition to enter university, this examination reform facilitated access to university. Furthermore, a new law (Law 910 of December 11, 1969) allowed access to all university faculties to students with a high school diploma, not just the lycée diploma as had previously been the case. This has changed the university population both in their basic preparation and in number. Another change in student population at the end of the 1960s was generated by the marked attention for the technical institutes, that is to say, the types of schools for the age range 14–19 which were aimed at educating in applied disciplines (mechanics, electronics, chemistry, etc.). The renovation began with the programs of the Istituto Tecnico Commerciale (ITC) [Commercial Technical Institute] followed by those of the Istituto Tecnico Informatico (ITI) [Informatic Technical Institute]. In addition to some elements of computer science, the programs of these schools offered topics in statistics and operations research. These programs made just a few references to modern algebra, such as the concept of set and related operations, Venn diagrams, and laws of internal and external composition, but there was little integration with the rest of the program. On the other hand, there was no reference to synthetic geometry. In the mathematics programs of other types of technical institutes, there was nothing on modern algebra, as if the Bourbakist wave had never existed (Ciarrapico and Berni 2017).

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While the programs of the technical institutes did not require parliamentary approval and were therefore defined by a commission in charge, for the lycées the obligation to pass through parliament complicated reforms, so that many reform attempts were stalled due to political disagreements. This immobility led to the Decreti Delegati [delegated decrees] of 1974 (Presidential Decree 419 of May 31, 1974), which allowed the first experiments in schools. In fact, they authorized individual schools to make changes to their timetables and programs. The experiments could involve different school subjects and be carried out in entire sections of a school, therefore they needed to be more fully articulated than those involving the pilot classes for mathematics of the past. The opportunity offered by the delegated decrees was well exploited by some mathematics teachers, who—fed up with legislative immobility—started important innovations in teaching. But in most schools, nothing happened. Many of the teachers were quite unaware of the innovative issues that were emerging (Ciarrapico and Berni 2017). As far as mathematics is concerned, a strong impetus to start experiments certainly came from the Frascati programs. The opportunity provided by the delegated decrees favored the birth—around 1975—of the Nuclei di Ricerca in Didattica della Matematica [Units of research in didactics of mathematics] at numerous universities, in which university professors and teachers from primary and secondary schools collaborated. They promoted educational research projects, which became effective through classroom experiments. So, the Ministry began to approve proposals for experimental projects put forward by high schools and technical and professional institutes, to be implemented in one or two sections of these schools. Subsequently, in 1976, the Consiglio Nazionale delle Ricerche (CNR) [National Research Council] signed three funding contracts to connect the world of university research with the experiments of curricular innovations in schools. A first contract, entitled “Research aimed at teaching mathematics in High School,” was signed with UMI; it was coordinated by Villani, then president of CIIM. Research units headed by the Universities of Genoa, Naples, Parma, Pavia, Pisa, Rome, Turin, Trieste, and others participated in its realization. The experimental projects activated by this contract mainly followed the contents of the Frascati programs, with some changes, such as the anticipation of the teaching of probability and statistics, and the addition of contents related to computer science. The Nuclei reported on the results of their experiments at the annual UMI-CIIM conferences established in the 1970s. They became catalysts of theoretical and field research. The figure of the teacher–researcher began to emerge. Gradually, mathematics education began to take on the status of an academic discipline. A second contract, negotiated with Mathesis for primary schools, put into action the already mentioned RICME Project coordinated by Pellerey. The third contract, coordinated by Paolo Boero, concerned the middle school and was signed with the University of Genoa (“Development and testing of models for motivated teaching in middle schools”). Another project was promoted on a national scale by the CEDE (European Center for Education) in 1974, at the initiative of Mario Fierli. It experimented with the introduction of computer science in the first 2 years of some secondary schools. A few years later, the experimental programs of the various technical institutes were channeled into projects with a single denomination, practically uniform throughout the national territory. Later, the Minister of Education Franca Falcucci  started experimenting on a national scale. Of particular relevance for mathematics were the programs of the Piano Nazionale Informatica (PNI) [Informatics national plan] launched in 1985 and the Progetto Brocca (a project so called from the name of the parliamentary secretary Beniamino Brocca who developed it) launched in 1988. These programs, originating from the various experimental projects already carried out, were the result of a wide and articulate debate, which involved disciplinary associations (UMI, CIIM, Mathesis, …), as well as individual professors both at universities and at all school levels, engaged in experiments and educational research. The commissions that elaborated these programs were made up of experts in mathematics and education (teachers, researchers, inspectors) working inside and outside the Ministry of Education. We emphasize once again that all these experiments provided the only pathway to innovation, given the extent of immobility at the legislative level.

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From Projects to Textbooks In the period following the Royaumont conference, Italian mathematics education experienced important cultural and institutional openings, with the translation of foreign books addressing various forms of modern mathematics, meaning that there was now much greater access to the relevant international literature. Thus, for example, Choquet’s book L’Enseignement de la Géométrie (1964. Paris, France: Herman) was translated into Italian in 1967 (L’insegnamento della Geometria. Milan, Italy: Feltrinelli) with the preface of Pescarini. This book included a chapter presenting axioms for affine geometry, which could be extended to metric geometry with the introduction of the dot product, and an appendix offering a different set of axioms for metric (Euclidean) geometry. These latter axioms were well accepted by the Italian community of mathematics educators: They stated a 1–1 correspondence between the set of real numbers and the points on a line, and the existence of line reflection for every line, thereby allowing theorems to be proved using the length of a segment and isometries. Also Dieudonné’s book Algèbre Linéaire et Géométrie, Élémentaire (1964. Paris, France: Herman) was translated into Italian in 1970 as Algebra Elementare e Geometria Elementare (Milan, Italy: Feltrinelli) with the preface of Pescarini. After the translation in 1968 of Nuffield Project texts, in 1972, UMI published a translation of the first five volumes with accompanying guides of the School Mathematics Project. In 1979, two books were published in the Quaderni dell’UMI series: the first, edited by Candido Sitia, was a collection of significant interventions at the ICME conferences in Lyon (1969), Exeter (1972), and Karlsruhe (1976) and the second (Cenni di Didattica della Matematica) the Italian translation of Anna Zofia Krygowska’s treatise on didactics of mathematics, Zarys dydactyki matematyki (Vol. 1, second edition 1975). Papy too was made known in Italy through the Italian translation of his book Mathématique Moderne 6 (1972, Florence, Italy: Le Monnier; originally published by Labor-Didier in Brussels in 1967). Campedelli also took into account some experiences developed abroad and published by La Nuova Italia, a publishing house directed by him and E. Castelnuovo two books: A book by Dienes on Mathematics in the Primary School was published in 1977, and a translation of Trevor Fletcher’s Mathematics for Today’s School appeared in 1977–1978. In the series Strumenti per una Nuova Scuola [Tools for a New School] of the publisher Feltrinelli, alongside the Italian translation of the second of György Pólya’s works (Mathematical Discovery), texts by Dieudonné, Choquet, Papy, and Dienes appeared. Seminal Italian texts which Morin and Franca Busulini had published in 1958—thus before the Royaumont Seminar—were republished, for example, Elementi di geometria (Morin and Busulini 1963). This book, particularly the first part of it, was surely the most “Bourbakist” among the Italian textbooks that would appear later. It began with sets, correspondences, equivalent relations, and algebraic structures (but the authors explained that this part was not strictly necessary). Congruences were introduced through transformations, but the axioms were based on those of Hilbert, and affine geometry was treated after metric geometry. Campedelli (1970) published the book La Geometria dei Parallelogrammi [The Geometry of Parallelograms], which treated affine geometry using classical synthetic methods. He recognized the need to separate affine properties from metric properties, introducing—as an axiom—the affine case of Desargues’ theorem, to treat homotheties without having to resort to real numbers. The final part of the book included the classification of geometries according to Klein. From the second half of the 1970s, textbooks were published that resulted from the projects mentioned above. The experimental work carried out within the Prodi project led to the creation between 1975 and 1982 of the volumes Matematica come Scoperta [Mathematics as a Discovery]. Like the other texts mentioned below, these texts had some dissemination in teacher training and played an important role in introducing innovative contents and methods, but their spread in schools was small (Ciarrapico and Berni 2017). All these books were very different from the texts traditionally adopted

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by teachers, and—above all—included few exercises, which is the part that teachers liked to use the most. Other textbooks published in that period were: in 1977 Il Metodo Matematico [The Mathematical Method] by Lombardo Radice and Mancini Proia; in 1979 Il Linguaggio della Matematica [Language of Mathematics] by Speranza and Rossi Dell’Acqua; in 1981 Problemi e Modelli della Matematica, [Problems and models of mathematics] by Walter Maraschini and Mauro Palma; and in 1982 Matematica: Idee e metodi [Mathematics: Ideas and methods] by Villani and Bruno Spotorno. These books featured different approaches but were all equally committed to the cultural and methodological updating of mathematical teaching, with a common focus on teaching by problems. They included the introduction of geometric transformations and the basic concepts of probability and statistics. They also introduced sets, in different contexts ranging from probability to the solutions of equations, and to relations and functions. Similarly, groups were defined in all texts, based on the composition of geometric transformations or the properties of numerical sets. In geometry, the distinction between affine and metric properties was always underlined. To prove geometric properties, the texts by Prodi, Lombardo Radice and Mancini Proia, and Maraschini and Palma used Choquet’s metric-based axioms, with some variations. In the third volume of the text by Lombardo Radice and Mancini Proia, Choquet’s affine axioms were also presented. The experimental programs promoted by the Ministry of Education in 1985 and 1988 (PNI and Brocca) took off from the aforementioned texts and from the related projects. They left the choice to teachers about how to introduce geometry “using the geometry of transformations or following a more traditional path” (PNI, Ministerial Circular No. 24 of February 6, 1991). Sets remained, but groups disappeared due to a—perhaps excessive—reaction to abstract structures. Somehow, through the aforementioned books and the experimental programs, certain aspects of modern mathematics survived in Italy more than in many other countries. In particular, the book by Maraschini and Palma, the most widely adopted, fostered the spread of new and modern ideas, even among teachers less committed to renewal. Nevertheless, it can be said that these books did not open a new era, but rather gradually concluded what was—in the words of Paolo Linati—“the era of forgotten experiments and missed opportunities” (Linati 2012, p. 63).

Conclusions In 1965, the journal Archimede published a collection of notes about the introduction of modern mathematics written by a group of teachers of lycées and technical institutes. These teachers claimed to be aware of the importance of certain aspects of modern mathematics, but criticized the proposals made at the aforementioned meeting of Camaiore on the new contents of geometry. The main point of criticism was the absence of adequate teacher preparation. Furthermore, they stressed that there were very few textbooks dealing with the new approach to algebra. We add that the special books published for the pilot classes were not made available to all schools and that the results of the experiments were not publicized (Giro 1969). In Correale (1965) the criticisms were not limited to the institutions: We catch a veiled criticism of university professors who discussed modern mathematics in theory, but who failed to face the real problems of the school in practice. Despite these perplexities expressed by teachers, it must be recognized that the initiatives of the 1970s in which the mathematicians played an important role (UMI congresses on mathematics teaching, the projects launched by mathematicians, and the network of Nuclei with active involvement of teachers) were the trigger for going forward toward innovation. Times had changed, as had society, and perhaps the events of the 1970s would have happened anyway, but modern mathematics was an accelerator, something which provided suggestions on how to act to achieve results. We may say that the only remains of modern mathematics in Italian schools are the language of set theory and geometric transformations. Other innovations suggested by the foreign projects, such as

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the introduction of probability and statistics, have also been accepted but with difficulty. However, it should also be considered the other side of the coin: The substantial refusal to adopt modern mathematics in the Italian school has made it possible to preserve a tradition of geometric teaching that has disappeared in other countries. History has shown that the Italian school system is by its very nature conservative: From the first reform of Casati in 1859 to the years when modern mathematics appeared internationally, Italy has had only the Gentile Reform in 1923, with little variation brought in at the end of the Fascist period (Bottai Reform) and later on at the end of World War II (by the Allied military Government). As told at the beginning, there was a tradition that was difficult to eradicate. The point was, in William Sawyer’s (1955) words, that “The main difficulty in many modern developments of mathematics is not to learn new ideas but to forget old ones” (p. 65).

References Anonymous. (1965). Troppo «ermetici» i nuovi programmi [Too “hermetic” the new programs]. Archimede, 17, 102–103. Behnke, H., Choquet, G., Dieudonné, J., Fenchel, W., Freudenthal, H., Hajós, G., & Pickert, G. (1960). Lectures on modern teaching of geometry and related topics. Aarhus, Denmark: Matematisk Institut (Aarhus Universitet), Elementaer Afdeling. Nr. 7. (E. Kristensen was in charge of the report). Bernet, T., & Jaquet, F. (1998). La CIEAEM au travers de ses 50 premières rencontres [The CIEAEM through its first 50 meetings]. Neuchâtel, Switzerland: CIEAEM. BUMI. (1954). Notizie [News]. Bollettino della Unione Matematica Italiana s. 3, 9, 208–232. BUMI. (1960). Notizie [News]. Bollettino della Unione Matematica Italiana, s. 3, 15, 85–104; 305–355. BUMI. (1962a). Notizie [News]. Bollettino della Unione Matematica Italiana, s. 3, 17, 120–157. BUMI. (1962b). Il convegno di Bologna promosso dalla Commissione internazionale dell’insegnamento matematico [The Bologna conference promoted by the International Commission of Mathematical Teaching]. Bollettino della Unione Matematica Italiana, s. 3, 17, 199–214. BUMI. (1966). Notizie [News]. Bollettino della Unione Matematica Italiana, s. 3, 21, 102–118. BUMI. (1968). Notizie [News]. Bollettino della Unione Matematica Italiana, s. 4, 1, 154–171. Campedelli, L. (1970). La geometria dei parallelogrammi [The geometry of parallelograns]. Florence, Italy: Le Monnier. Cartan, H. (1963). Réflexions sur les rapports d’Aarhus et Dubrovnik [Reflections on the Aarhus and Dubrovnik reports]. L’Enseignement Mathématique, s. 2, 9, 84–90. Castelnuovo, E. (1960). Le matematiche moderne nell’insegnamento secondario al convegno di Arlon (Belgio) [Modern mathematics in secondary education at the Arlon conference (Belgium)]. Archimede, 12, 265–266. Castelnuovo, E. (1964). Didattica della Matematica [Mathematics Education]. Firenze, Italy: La Nuova Italia. Castelnuovo, E. (1965). Matematica d’avanguardia [Avant-garde mathematics]. Riforma della Scuola, 11, 3–19. Translated as “Mathématique d’avant-garde” in the volume La mathématique moderne [modern mathematics] (pp. 1–14) and as Belgique: Mathématique d’avant-garde [Belgium: Pioneering mathematics] (booklet). Castelnuovo, G. (1919). La riforma dell’insegnamento matematico secondario nei riguardi dell’Italia [The reform of secondary mathematics education in Italy]. Bollettino della Mathesis, 11(1–4), 1–5. Ciarrapico, L., & Berni, M. (2017). I curricoli di matematica, gli ordinamenti scolastici e le riforme dal 1940 al 2015 [The mathematics curricula, school systems and reforms from 1940 to 2015]. Bologna, Italy: Unione Matematica Italiana. Correale, G. (1965). «Fateci vedere come si fa» [“Let us see how it’s done”]. Archimede, 17, 104–107. De Bock, D., & Vanpaemel, G. (2015). Modern mathematics at the 1959 OEEC Seminar at Royaumont. In K. Bjarnadottir, F. Furinghetti, J. Prytz, & G. Schubring (Eds.), “Dig where you stand” 3. Proceedings of the third International Conference on the History of Mathematics Education (pp. 151–168). Uppsala, Sweden: Department of Education, Uppsala University. De Finetti, B. (1965). Programmi e criteri per l’insegnamento della matematica alla luce delle diverse esigenze [Programs and criteria for teaching mathematics in the light of different needs]. Periodico di Matematiche, s. 4, 43, 119–143. De Finetti, B. (1967). Le proposte per la matematica nei nuovi licei: informazioni, commenti critici, suggerimenti [The proposals for mathematics in the new lycées: information, critical comments, suggestions]. Periodico di Matematiche, s. 4, 45, 75–153.

8  Modern Mathematics in Italy: A Difficult Challenge Between Rooted Tradition and Need for Innovation

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Duse, V. (1969). Per un insegnamento moderno della matematica nella scuola elementare [For a modern teaching of mathematics in primary school]. Brescia, Italy: La Scuola. Équipe de Bordeaux  – AMPERES (2009). Pourquoi enseigner le triangle? [Why to teach the triangle]. Retrieved January 19, 2021, from http://educmath.ens-­lyon.fr Ferrari, M., Bazzini, L., Borghi, L., De Ambrosis, A., Isella Massara, C., Mosconi Bernardini, P., Trivelli Ricci, P., Vittadini Zorzoli, M., & Germani, L. (1982). Un progetto di educazione matematico–scientifica per il primo ciclo elementare [A mathematical–scientific education project for the first primary cycle]. L’Insegnamento della Matematica e delle Scienze Integrate, 5(1), 56–73. Furinghetti, F. (2015). Matematici e insegnamento della matematica a livello pre-universitario: Enrico Bompiani terzo presidente della CIIM [Mathematicians and the teaching of mathematics at pre-university level: Enrico Bompiani third president of CIIM]. La Matematica nella Società e nella Cultura, Rivista dell’Unione Matematica Italiana, s.1, 8, 75–109. Furinghetti, F. (2019). Riforme internazionali e innovazioni nazionali: Luigi Campedelli quarto presidente della CIIM dal 1964 al 1972 [International reforms and national innovations: Luigi Campedelli fourth president of CIIM from 1964 to 1972]. Matematica, Cultura e Società. Rivista dell’Unione Matematica Italiana, s. 1, 4, 269–288. Furinghetti, F., & Menghini, M. (2014). The role of concrete materials in Emma Castelnuovo’s view of mathematics teaching. Educational Studies in Mathematics, 87, 1–6. Gattegno, C., Servais, W., Castelnuovo, E., Nicolet, J.  L., Fletcher, T.  J., Motard, L., Campedelli, L., Biguenet, A., Peskett, J.  W., & Puig Adam, P. (1958). Le matériel pour l’enseignement des mathématiques [Materials for the teaching of mathematics]. Neuchâtel, Switzerland: Delachaux & Niestlé. Giannarelli, R. (1967). Progetto di programma minimo di matematica per il triennio liceale (Villa Falconieri, February 16–18, 1967) [Minimum math program project for the last three years of High School]. Archimede, 19, 60–63. Giro (1969). Perché nessun consuntivo sulle «classi pilota»? [Why no balance sheet on the “pilot classes”?]. Archimede, 21, 185–186. Hungarian National Commission for UNESCO. (1963). Report on the work of the international symposium on school mathematics teaching. Budapest, Hungary: Akadémiai Kiadó. Lehto, O. (1998). Mathematics without borders: A history of the International Mathematical Union. New York, NY: Springer Verlag. Lelong, J. (1965). Sur l’harmonisation des enseignements secondaire et supérieurs, et le rôle de la géométrie élémentaire [On the harmonization of secondary and higher education and the role of elementary geometry]. Periodico di Matematiche, s. 4, 43, 235–240. Linati, P. (2012). L’algoritmo delle occasioni perdute. La Matematica nella scuola della seconda metà del Novecento [The algorithm of lost opportunities. Mathematics in the school of the second half of the twentieth century]. Trento, Italy: Erickson Live. Lombardo-Radice, L. (1963). Il nuovo insegnamento di algebra al primo anno di matematica [The new course of algebra in the first year of mathematics]. Archimede, 15, 1–7. Mancini Proia, L. (2003). Autobiografia di un’insegnante di matematica [Autobiography of a mathematics teacher]. In M. Menghini & M. R. Trabalza (Eds.), Geometrie in cielo e in terra [Geometries in heaven and on earth] (pp. 19–47). Foligno, Italy: Edizioni dell’Arquata. Maraschini, W., & Menghini, M. (1992). Il metodo euclideo nell’insegnamento della geometria [The Euclidean method in the teaching of geometry]. L’Educazione Matematica, 13, 161–180. Marchi, M. V., & Menghini, M. (2013). Italian debates about a modern curriculum in the first half of the 20th century. The International Journal for the History of Mathematics Education, 8(2), 23–47. Menghini, M. (2007). La geometria nelle proposte di riforma tra il 1960 e il 1970 [Geometry in the reform proposals between 1960 and 1970]. L’Educazione Matematica, 28, 29–40. Morin, U. (1963). Geometria elementare classica e metodi moderni [Classical elementary geometry and modern methods]. L’Enseignement Mathématique, s. 2, 9, 76–83. Morin, U. (1965). Interpretazione di un progetto di matematica per il biennio liceale [Interpretation of a mathematics project for the first two years of the Lycée. Archimede, 17, 57–68. Morin, U., & Busulini, F. (1963). Elementi di geometria [Elements of geometry] (Third edition). Padova, Italy: CEDAM. Movimento Circoli della Didattica (Ed.) (1956). Didattica della matematica. [Didactics of mathematics]. Rome, Italy: Signorelli. Notiziario. (1960). Seminario della commissione internazionale per l’insegnamento matematico (I.C.M.I.) in Danimarca [Seminar of the International Commission for Mathematical teaching (I.C.M.I.) in Denmark]. Archimede, 12, 268. Notiziario. (1963). Classi pilota per l’insegnamento della matematica e della fisica [Pilot classes for the teaching of mathematics and physics]. Archimede, 15, 331. Notiziario. (1969). Riunione della C.I.I.M. a Villa Falconieri (Frascati) [Meeting of C.I.I.M. at Villa Falconieri (Frascati)]. Archimede, 21, 111–112. OEEC. (1961a). New thinking in school mathematics. Paris, France: OEEC.

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Fulvia Furinghetti and Marta Menghini

OEEC. (1961b). School mathematics in OEEC countries—Summaries. Paris, France: OEEC. OEEC. (1961c). Synopses for modern secondary school mathematics. Paris, France: OEEC. Papy, G. (1966). La geometria nell’insegnamento moderno della matematica [Geometry in the modern teaching of mathematics]. Archimede, 18, 81–91. Pellerey, M. (1989). Oltre gli insiemi [Beyond the sets]. Napoli, Italy: Tecnodid. Piaget, J., Bosher, B., & Châtelet, A. (1950). Initiation au calcul [Introduction to calculation]. Paris, France: Baurelier. (Italian translation 1956. Avviamento al calcolo [Introduction to calculation]. Florence, Italy: La Nuova Italia). Picard, N. (1967). À la conquête du nombre: Classe de C. P. Livre du maître [Conquering the number: Class of C. P. Teacher’s book]. Paris, France: OCDL. Platone, G. (1961). Convegno internazionale sull’insegnamento matematico [International conference on mathematical teaching]. Archimede, 13, 313–319. Prodi, G. (1995/1982). Tendenze attuali nell’insegnamento della matematica [Current trends in mathematics teaching]. L’Insegnamento della Matematica e delle Scienze Integrate, Numero speciale in onore del Prof. Giovanni Prodi [Special issue in honor of Prof. Giovanni Prodi], 18, 413–430 (First published in 1982. Rendiconti. Accademia Nazionale delle Scienze detta dei XL. Memorie di Matematica e di Scienze fisiche e Naturali, 7(17), 183–196. Revuz, A. (1965). La geometria in un insegnamento moderno della matematica [Geometry in a modern teaching of mathematics]. Periodico di Matematiche, 43, 242–245. R. G. (1964a). Riforme fatte e riforme in studio, con i resoconti dei due colloqui di Villa Falconieri [Reforms made and reforms being studied, with the reports of the two meetings at Villa Falconieri]. Archimede, 16, 9–42. R.  G. (1964b). Seminario matematico internazionale (Villa Falconieri  – Frascati, 8–10 ottobre 1964) [International mathematical seminar (Villa Falconieri – Frascati, October 8–10, 1964)]. Archimede, 16, 314–330. R. G. (1966). Nuovi programmi di matematica per i licei [New High School mathematics programs]. Archimede, 18, 124–129. Sawyer, W. W. (1955). Prelude to mathematics. Harmondsworth, United Kingdom: Penguin Books. Schubring, G. (1987). The cross-cultural ‘transmission’ of concepts—The first international mathematics curricular reform around 1900, with an Appendix on the biography of F. Klein. Occasional paper Nr. 92, IDM (Universität Bielefeld). Schubring, G. (2014). The road not taken—The failure of experimental pedagogy at the Royaumont Seminar 1959. Journal für Mathematik-Didaktik, 35(1), 159–171. Stone, M. H., & Walusinski, G. (1963). International Commission on Mathematical Instruction. Report of the period 1959–62. L’Enseignement Mathématique, s. 2, 9, 105–112. Temusssi, S. (1965). «Guardare in faccia la realtà e andare cauti» [Face reality and be cautious]. Archimede, 17, 112–114. Togliatti, E.  G. (1963). L’insegnamento della matematica [The teaching of mathematics]. In Atti del Convegno: Insegnamenti scientifici e insegnamenti umanistici nella funzione formativa della scuola secondaria. Roma, 8–10 maggio 1962 [Proceedings of the Conference: Scientific teachings and humanistic teachings in the formative function of secondary school] (pp. 31–46). Rome, Italy: Accademia Nazionale dei Lincei. Tomasi, L. (2018). Un ritratto di Ugo Morin nel cinquantesimo anniversario della sua scomparsa. L’Insegnamento della Matematica e delle Scienze Integrate, 41B, 403–431. Tricomi, F.  G. (1965). Accordare «progressisti» e «conservatori» [Matching “progressives” and “conservatives”]. Archimede, 17, 101. Villa, M. (1962). Il discorso del Prof. Villa [The speech of Prof. Villa]. Bollettino della Unione Matematica Italiana, s. 3, 17, 200–207. Villa, M. (Ed.). (1965). Matematica moderna nella scuola media [Modern mathematics in middle schools]. Bologna, Italy: Patron. Villa, M. (Ed.). (1966). Matematica moderna nelle scuole secondarie superiori [Modern mathematics in high schools]. Bologna, Italy: Patron. Vita, V. (1986). I programmi di matematica per le scuole secondarie dall’unità d’Italia al 1986. Rilettura storico-critica [Mathematics programs for secondary schools from the unification of Italy to 1986. Historical-critical reinterpretation]. Bologna, Italy: Pitagora.

Chapter 9

The Distinct Facets of Modern Mathematics in Portugal José Manuel Matos and Mária Cristina Almeida

Abstract  The application of modern mathematics ideas in Portuguese schools took place from the 1960s to the end of the 1980s. A first experiment in the higher grades of secondary school that started in 1963 laid the ground for the ways in which the reform was developed later. The compartmentalized nature of the educational system led the several subsystems to develop distinct concretizations of the new ideas, from primary school to the higher grades of secondary school. We argue that modern mathematics in grades 5 and 6 became essentially a linguistic endeavor, contrasted with the reliance on logic as the backbone for the reform in grades 10 through 12. Reformers for the technical schools struggled to accommodate real-world applications into the abstract flavor of mathematics fostered by the new trends. In grades 7 through 9, curricular change meant the introduction of transformational geometry together with a rephrasing of old content into set theory. For each of these cases, program content, textbook implementation, and teacher formation are discussed. Keywords António Augusto Lopes · Cuisenaire rods · Curricular reform · Curricular studies · Education in Portugal · History of mathematics education · João Nabais · José Sebastião e Silva · Leite Pinto · Mathematics education · Modern mathematics · OECD · Portugal · Teacher education · Teacher knowledge · Teaching experiment · Textbooks

Introduction We can date the first time that the “modern mathematics movement,” as it became known in Portugal, was publicly expressed in the country to the late 1950s. It occurred in 1957 at a meeting in a secondary school in Lisbon, Liceu Pedro Nunes, attended by Leite Pinto, the Minister of Education himself. The school was also a teacher education establishment and the event was the opening ceremony of the teacher education activities at the Liceu. It was reported in all daily newspapers and attended by many educators. The lecture, delivered by José Calado (1958), a teacher of mathematics, began by explaining the “spirit of modern mathematics” (p. 92) and outlined the actions to renew mathematics teaching conducted by the International Commission on Mathematical Instruction. He

J. M. Matos · M. C. Almeida (*) Centro Interdisciplinar de Ciências Sociais (CICS.NOVA), Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, Lisbon, Portugal e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_9

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Figure 9.1  Timeline of modern mathematics in Portugal

then proceeded with an account of the new Bourbakist proposals for school mathematics and concluded by asking the Minister to support the implementation of the new ideas in Portugal. The appropriation of the new ideas in Portugal was not a linear process, as it had interludes, changes of purposes, and variations in leadership. The purpose of this chapter is to give an account of how the new ideas were brought to schools. We reviewed studies dwelling on legislative pieces, textbooks, articles in specialized journals and daily newspapers, archival materials, and testimonies from participants. This chapter extends the work performed by our research group at the New University of Lisbon and intends to contribute to a cultural understanding of knowledge (Burke 2016), aiming for a description that reveals the underlying meanings and intentions of participants, and a frame that includes a broader cultural and social environment. The chapter begins with a review of the social and political context in which the reform was presented. After detailing the Sebastião e Silva experiment for grades 10 and 11 which began in 1963, we proceed by presenting in an almost chronological order how it was implemented into several subsystems. These subsystems show distinct but interconnected ways in which the new ideas were appropriated which reveal the distinct facets they took in Portugal. Figure 9.1 presents the dates at which the new ideas reached the different non-university types of schools and the dates at which we consider they ended, as new programs based on problem solving (grades 1–9) or centered on analysis (grades 10–12) were approved.



Context of the Modern Mathematics Reform

From 1930 to 1974, the New State (Estado Novo), a Portuguese regime ideologically similar to the Italian fascism, followed a corporativist constitution that structured a state ruled by one party that intended to oversee all branches of society. Dissention was repressed by the use of censorship and the political police. At the time, the country was also a colonial empire spreading into three continents. However, contrary to the Italian fascism, industrialization was seen as a menace to the “authentic” roots of the nation that were believed to lay on small rural communities. The end of World War II posed a serious challenge to that perspective and, in the late 1940s, the regime survived through internal repression (for example, many mathematicians were expelled) and by foreign complacency,

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p­ rofiting from a neutral but collaborative posture with the allied powers during the War. However, times were different from the 1930s, and the importance of economic development to maintain power in an increasing multilateral world was understood by part of the establishment. Moreover, the future of the empire was at stake, as all over the world, former colonies were becoming independent. Political change inside the regime was difficult to achieve, however. Some sought economic, technological, and scientific development, requiring better qualifications for the working population. A key change occurred in 1974 when a revolution overthrew the government, leading to a democratic system and the independence of the colonies. In education, post-War change essentially started to occur when Leite Pinto (1902–2000), an engineer and a former teacher of mathematics, became Minister of Education (from 1955 until 1961). During his tenure, collaboration with the Organisation for Economic Cooperation and Development (OECD) started and key development projects were enacted (Teodoro 1999). His dismissal, due to his sympathy for an attempt to change the regime, temporarily stalled changes, but a new Minister, Galvão Telles (1917–2010, Minister from 1962 until 1968), continued his work. Collaboration with OECD played an important role in educational change, as it stimulated circulation of knowledge by scientific meetings, practical courses, studies, and reports. The OECD involvement had the support from the industrial sectors which needed specialized technicians and from the liberal wing of the regime that understood the importance of international collaboration for furthering economic development. Although Galvão Telles oversaw major changes in the educational system, namely increasing mandatory schooling and creating a department for research and innovation within the Ministry of Education, a global overhaul of the system, under the banner “Democratization of school,” was undertaken in 1973 by another Minister, Veiga Simão (1929–2014), fostering mandatory schooling and teacher education, among other decisions. The democratic revolution of 1974 capitalized on most of the proposals of Simão and extended mandatory schooling again. The main changes, however, were not on the architecture of the educational system, but rather on the incorporation of democratic values in schools. It took some time to stabilize the system and was not until 1986 that a new global law for the educational system was enacted. From 1948 until 1968, the centralized Portuguese school system began with 4 years of compulsory primary education (for students aged 6–9). Those who wanted to pursue education had to choose between secondary schools (liceus) and technical schools. Programs at liceus were organized into three Cycles: 1st (10–11-years-olds), 2nd (12–14-years-olds), and 3rd (15–16-years-olds). In the third Cycle, students were preparing for university studies, targeting the liberal professions and senior technical staff. Technical schools’ curricula comprised a Preparatory Cycle (10–11-years-olds), followed by 3 years devoted to technical training, and did not give direct access to universities. In 1968, this arrangement was changed with the creation of the Preparatory Cycle for Secondary Education (CPES), which replaced the 1st Cycle of liceus and the Preparatory Cycle of technical schools. From 1976, a unification of the remaining secondary school years and the adding of an extra year led to the present configuration of mandatory 12 grades. This split into several sub-systems goes hand in hand with a split into several school cultures (or grammars of schooling, in the sense of Tyack and Cuban 1997), separated by teacher formation, school buildings, and governmental supervising bodies. The distinction between teachers for primary schools, for liceus, and for technical schools, and from 1968, for CPES schools, is reflected in the diverse characteristics adopted by the reform in Portugal. Modern mathematics reform took place in difficult school environments, as, from the 1950s until the end of the 1980s, the educational system faced important challenges. The first was the growth of the school population that began immediately after World War II when Portuguese families started to value post-primary education, which was seen as a path to social ascension. Table 9.1 shows the dramatic increase in 30 years in the number of students, especially after primary school.

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Table 9.1 Real rate of schoolinga in Portugal, by grade level, per school year 7–9 5–6 10–11, 12b (2nd Cycle Liceus, Courses (1st Cycle Liceus, Tech. Schools, Secondary Prep. Cycle Tech. Grade level 1–4 (Complementary School) School year Kindergarten (Primary) Schools, CPES) Cycle) 1960–1961 0.9 80.4 7.5 6.1 1.3 1970–1971 2.8 83.7 22.0 14.7 4.3 1980–1981 21.4 100.0 44.7 24.4 12.2 1990–1991 47.1 100.0 71.7 58.3 31.0 Source: Compiled from Estatística (2009, pp. 65–66) a Real rate of schooling: Percentage between the number of students enrolled in a given cycle of studies, at the normal age of frequency of that cycle, and the resident population of the same age levels b Eleven years until 1976–1977 and twelve from 1977–1978

A second challenge was created by the regime itself, as for many years, it limited the certification of teachers. Table 9.2 focuses on the liceus and compares the variation of the number of students with the variation in the number of certified teachers. The steady increase in the number of students enrolled in liceus (growing roughly 8% a year) was not matched by the number of certified teachers that remained practically unchanged. Table 9.2 Number of students and certified teachers in liceus in Portugal per school year School year 1948–1949 Number of students 18,858  Ratio of change Number of certified teachers 735  Ratio of change Source: Compiled from Lima (1963, p. 84)

1952–1953 24,386 1.29 816 1.11

1956–1957 33,535 1.38 797 0.98

1960–1961 46,618 1.39 822 1.03

From 1930 until 1969, the certification of secondary school teachers was only possible by a two-­ year training stage at specific liceus with a very short number of vacancies. Table 9.2 reflects an educational policy that deliberately intended to limit the growth of public secondary schools. Families, however, had a different perspective about the value of secondary education and, as school populations grew, there was a steady increase in the number of provisional teachers without certification. In the case of mathematics, these usually referred to individuals with an incomplete university education that included at least one discipline of mathematics. Table 9.3 shows how the number of provisional teachers changed in a 10-year span. Table 9.3 Number of provisional teachers in liceus in Portugal per year Year Number of provisional teachers  Rate of change Source: Compiled from Lima (1963, p. 85)

1948 14

1952 106 7.57

1956 340 3.21

1960 715 2.10

During the 1960s, this situation worsened, as only at the end of the decade opportunities for teachers’ certification increased and, in 1971, teacher education courses started in universities. This picture is even bleaker if we consider secondary school mathematics. In many schools, especially in

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d­ isadvantaged areas, there were virtually no certified mathematics teachers. The predominance of provisional teachers in secondary schools implied the need for yearly job appliances with competitive procedures for the appointment of a large percentage of school personnel, inhibiting school stability and delaying the start of the school year. Only in late 1980s, the prevalence of provisional teachers in secondary education faded away. Instability in the school system occurred also in other areas. From the late 1960s until the end of the 1970s, the shortage of schools was a problem and, even after the hasty construction of provisional facilities, schools remained crowded. Also, there was a shortage of textbooks, essentially due to the volatility of programs. Changes in school mathematics curricula are born into this background of teachers with low competencies, crowded schools, delays in the beginning of the school year, and a shortage of textbooks.

Beginnings The role of mediators was prominent in the beginning of the new ideas in Portugal, and the mathematician José Sebastião e Silva (1914–1972), who studied in Rome from 1943 to 1946, and had a close contact with Federigo Enriques, Guido Castelnuovo, and Emma Castelnuovo, among others, played a central role. As a university professor of mathematics, he maintained a regular presence at international meetings, including the 1952 meeting of the International Mathematical Union in Rome that reinstituted the International Commission on Mathematical Instruction (ICMI). In 1955, he was nominated Secretary of the Portuguese subcommission of ICMI, together with secondary school teachers José Calado (1903–1986) and Silva Paulo (1905–1976). In April 1957, a delegation appointed by the Portuguese government attended the 11th meeting of 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 Madrid. It was composed of Sebastião e Silva, and three secondary school teachers: José Calado, Jaime Leote (1902–1988) from a liceu, and Santos Heitor (1903–1993) from a technical school. The last two had responsibilities in initial teacher education. Calado, Heitor, and Silva were also authors of mathematics textbooks for secondary education. The Portuguese presence in Madrid remained in the memory of all participants and others as a key event in the diffusion of new ideas in Portugal. According to official rules, public servants had limitations when traveling abroad and, at least for more than 30  years, there was no participation of Portuguese teachers from non-university education in international events. After that, all Portuguese scholars who were able to attend the Madrid meeting were expected to write articles outlining innovative approaches to school mathematics presented at the Madrid meeting (Calado 1958; Heitor 1958; Leote 1958; Silva 1957). Later, articles by other teachers confirm that this trip was a turning point in Portuguese mathematics education. A few months after the CIEAEM meeting, the new ideas were publicly presented at the important meeting at the Liceu Pedro Nunes we referred to at the beginning of the chapter. This reunion brought together many influential stakeholders, including educators, directors, administrators, and the Minister of Education. José Calado, the speaker, presented an action plan: A curriculum revision with more class time for mathematics, and courses for teachers on modern algebra, algebra of logic, and foundations of mathematics. The meeting’s prominence was favorably reported in the newspapers of the time and even Rui Grácio (1921–1991), a teacher and a prominent member of the opposition to the regime, called it a “good omen” (Folha and Grácio 1958, p. 211). After this start, and until 1963, the new international trends on mathematics education continued to be discussed in professional magazines (Matos 2006). In the absence of courses for teacher education in the universities, and consequently, of a curricular research tradition, the new ideas were

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d­ iscussed in the three Liceus Normais, where pre-service teacher education was provided. In this period, trainees produced monographs re-writing Bourbakist mathematics for secondary education (Almeida 2013; Monteiro 2018). Also, foreign educators occasionally lectured in these schools. Martha Dantas, a Brazilian teacher, had an extended stay in 1958, during which she attended a university course on “classic”1 linear algebra. She frequently participated in pre-service seminars at the Liceu Pedro Nunes with Jaime Leote and gave a lecture detailing how the new ideas were spreading in Brazil (Dantas 1958). She gathered some materials, which, upon her return to Brazil, were used to support courses on logic (Garnica 2008). In 1962, Gustave Choquet also gave lectures at the Liceu Pedro Nunes, discussing a new axiomatics for geometry (Leote 1964), and Caleb Gattegno joined a course on Cuisenaire rods. Although there was no participation of Portuguese teachers of mathematics in international events, one young teacher, Manuel Sousa Ventura, stayed in Paris for almost a year and came into contact with Piaget’s theories (Ventura 1961). Publicly, however, nothing happened, except for a course on logic taught by Sebastião e Silva in 1958 at Liceu Pedro Nunes. The political context may have had some influence, as in 1961, when independence wars started in the colonies, the dictatorship experienced one of its main challenges. Internal dissention about changes in the regime led to dismissal of the energetic Minister, Leite Pinto, and, for some time, it was not “appropriate” to discuss changes of any kind. In 1962, Sebastião e Silva took two initiatives. The first was an article (Silva 1962), originally published in an Italian journal, outlining a plan for a curricular reform in secondary school mathematics. Choosing a “moderate” approach to change, he proposed almost no changes in the first 5 years of liceus. The last 2 years, however, would change considerably with the introduction of modern mathematical topics, such as logic, theory of sets, abstract algebra, probability, and statistics. However, these innovations must be carried out with extreme prudence and with the finest pedagogical touch if we do not want to create in students an invincible repulsion for mathematics or lead them to the acquisition of a completely sterilizing empty formalism.2 (Silva 1962, p. 25)

He believed that these changes were important because, on the one hand, they would improve students’ intellectual formation and, on the other hand, they would fill a growing gap between university and school mathematics. The second initiative was a mathematics course for teachers that he taught in Lisbon from November 1962 until June 1963. During the 20 sessions, logic was the main topic, but modern algebra was also addressed. There was a great interest in this course, which was attended by more than 100 teachers. Interest in the new ideas was also on the rise among primary school teachers. From the beginning of 1962, João Nabais (1915–1990), a priest who had graduated in psychology and pedagogy from the University of Louvain in Belgium, promoted many courses on the use of Cuisenaire rods and other materials in the teaching of arithmetic, as we shall see in another section. During this time, daily newspapers were reporting these events in small articles (Almeida et al. 2020) and, in March 1963, the newspaper Diário Popular published a series of four long articles with the common title “Revolution in teaching.” These were written by a journalist, Corregedor da Fonseca, who acknowledged the growing public interest in the subject. The first article described ideas that were circulating internationally (Fonseca 1963a). In the second article (Fonseca 1963b), Fonseca interviewed Sebastião e Silva who justified the need to “modernize” mathematics in secondary schools, referring the enormous scientific and technological developments that had taken place after World War II, and highlighting the role of mathematics in science, technique, industry, economy, and culture of the most developed countries. He sought a new mathematics in schools and advocated both the updating of the content being taught and the renewing of teaching methods.

 We use the term “classic” to refer to mathematical approaches preceding the new ideas.  All translations were performed by the authors.

1 2

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The third article by Fonseca (1963c) was based on an interview with Jaime Leote, the trainer of mathematics teachers at Liceu Pedro Nunes. Leote recalled his visit to Madrid 6 years ago and began by developing the idea that modern mathematics allowed students to perfect their thinking through abstractions from concrete situations. After listing the different materials and teaching methods used in educational practice at his school, he insisted on the need to update teachers’ knowledge, especially scientific knowledge, which he considered indispensable for the success of any introduction of modern mathematics in schools. The new approach presented mathematics in a unified way, relying on the language of the sets, giving special emphasis to algebraic structures and logic, but the scientific preparation that the degrees in mathematics provided to future teachers was out of date. The last article (Fonseca 1963d) addressed the experiments carried out by João Nabais in his primary school involving the use of manipulative materials, especially Cuisenaire rods.



Laying the Ground—The Sebastião e Silva Experiment

The first actual step to introduce the new ideas in schools started in July 1963 with the appointment of a Commission composed by Sebastião e Silva (president), Jaime Leote, Manuel Augusto da Silva (1910–unknown), António Augusto Lopes (1917–2015), teachers in charge of training secondary school teachers at the three Normal Liceus of Lisbon, Coimbra, and Porto, respectively. The Commission immediately started a curricular project in the school year 1963–1964 that became the best-known initiative associated with modern mathematics in Portugal. The project introduced changes in the two last years of liceus (currently 10th and 11th grades), as Sebastião e Silva outlined in 1962. He wrote a program and, in the school year 1963–1964, the three other members of the Commission taught three “pilot” classes to students with good grades in mathematics. Also, class time changed from 4 to 6 h per week and class size was reduced from about 40 to 20 students. In subsequent years, teachers from other schools would join the project, gradually increasing the number of classes involved. From December 1963, OECD agreed to co-fund the project and support it technically. The project also included the elaboration of a textbook by Sebastião e Silva (three volumes) that he complemented with a teachers’ book (two volumes) in which he included many digressions into philosophy, history, and didactics. In an interview published in a daily newspaper, Sebastião e Silva outlined the reasons for trusting the new curricular ideas: The first [reason is that the] introduction of new fields of mathematical knowledge in school programs does not require maintaining everything taught before; the second is that there is the possibility of integrating them into a new, unitary, perspective the essence of “modern” mathematics and “classical” mathematics—with the benefit of its intelligibility by the student; the third reason refers to the fact that mathematicians and pedagogues have been concerned with research methods, the preparation of didactic material, the experimentation of techniques that facilitate the understanding of mathematics by children and adolescents. (Silva, quoted in Innovation in mathematics teaching, 1964, p. 13)

Later, in an extensive interview (Fonseca 1965), he delineated the purpose of the experiment: School mathematics should urgently accompany advances in science, especially to fulfill the need for specialized technicians (physicists, mathematicians, etc.). For him, teaching logic was the starting point for change and, in fact, his first-year program started with symbolic logic and set theory, Cartesian products, binary relations, semigroups, and groups, followed by an axiomatic description of N—the set of natural numbers—rings and fields, isomorphisms, Boole algebras, calculation of approximate values. The second year began with the study of vectors and vector spaces, determinants, complex numbers, an intuitive study of Taylor series, a heuristic introduction to integral calculus, probability, and statistics (Almeida 2013).

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Silva emphasized on several occasions that the project aimed at changing both content and methods. The opening paragraph of his guide for teachers is clear: The modernization of the teaching of mathematics will have to be done not only in terms of programs, but also in terms of teaching methods. The teacher should abandon, as much as possible, the traditional expository method, in which the students’ role is almost one hundred percent passive, and try, on the contrary, to follow the active method, establishing dialogue with students and stimulating their imagination, in order to bring them, whenever possible, to the rediscovery [of mathematical topics]. (Silva 1975, p. 11)

A study of students’ workbooks (Silva and Valente 2008) showed that in pilot classes, “modern” topics were intertwined with “classic” ones, as the experimental textbook was used at the same time as textbooks from the regular program.

Teacher Formation Teacher formation was essential to the implementation of these innovations. From 1964 until 1969, during the summer holidays, members of the Commission and other teachers conducted the Oeiras courses, as they became known. These were 2-week teacher training courses at the Liceu de Oeiras near Lisbon. Teachers already participating in the experiment attended, and the Commission invited teachers that would be included in the following year. Occasionally other teachers were invited. Among the personal documents of António Augusto Lopes, there are some materials used in the Oeiras courses, all of them referring to modern mathematical topics: 36 exercises on binary relations, groupoids, groups, vector calculus, and vector spaces, linear algebra, geometric transformations, and numerical approximations (Almeida 2013). We did not find documents about teaching methods and António Augusto Lopes confirmed that this was not a priority: The teachers who accomplished a training course at a Liceu Normal, were aware of the new topics. But the other teachers, let’s say, most of them, weren’t familiar with the concepts of modern mathematics. They might have studied something, but they were not adequately prepared. … The concern was that teachers would overcome the initial difficulties, which are usually the hardest … and then deepen their study. … We did not teach didactics, it was integrated in our lessons. (Lopes, quoted in Almeida 2013, p. 230)

According to Lopes, the courses were primarily intended to provide teachers with knowledge of the content of modern mathematics in such a way that they acquired the capacity to communicate this content to students appropriately. As we have seen, the Commission overseeing the experiment included three teachers with responsibilities in pre-service teacher education in their liceus, and, naturally, modern mathematics was extensively discussed in these schools and all teacher candidates produced monographs on the theme. Some of them published in professional journals, contributing to the development of professional knowledge related to the new topics (Almeida 2013; Monteiro 2018). The experiment also fostered international contacts. Members of the Commission attended the Athens Seminar in 1963 (OECD 1964) and presented the new program and, following a trip to the first International Congress on Mathematical Education in 1969, they visited Emma Castelnuovo in Italy and observed some classes. In 1968, Frédérique Lenger also visited Portugal as an OECD expert and observed project classes, among other activities.

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Evaluating the Experiment Some data allow us to discuss the results achieved by the experiment (Matos and Almeida 2021). Firstly, the curriculum modeled by teachers in experimental classes produced a distinct class environment especially favoring students with an aptitude for mathematics and sciences. That was in line with expectations of results for the experiment, although the effects were not clear for the rest of the students. Also, by 1966, there was a significant number of teachers believing that the new program was taking an excessive amount of class time and that the new ideas should not be limited to a small number of students. As a project for curricular development, it followed a RDD paradigm (Howson et al. 1981) and was close to a theoretical approach (Pacheco 2001), as the experiment was based on materials developed by an authority (Sebastião e Silva) who supported a gradually developed practice conducted by expert teachers. By 1969, the experiment had lost much of its steam (Almeida and Matos 2021). Firstly, the legitimacy of the composition of the Commission, and therefore of the direction of the experiment, was compromised when, in 1969, the training of secondary school teachers was no longer limited to liceus normais, and the number of those responsible for this training was multiplied. Secondly, the center of the reform of modern mathematics was no longer concerned with developing an elite course for the preparation of specialized technicians. Now it was time to think about taking the new ideas to the majority of students. Although the relevance of the project diminished, some of its members (Sebastião e Silva and Augusto Lopes) played a central role in this new phase, as we.



Mathematics as a Language—The Program for CPES

As the Sebastião e Silva experiment unfolded, one member of the Commission, António Augusto Lopes, became involved in the application of the new ideas in Telescola, a project that preceded the CPES and was meant to bring schooling to a large number of children (grades 5 and 6) by the use of television (Almeida and Matos 2020). The project was a quick fix to cover for the lack of teachers and schools to support the overdue expansion of mandatory schooling for children from 4 to 6 years of age. Lopes, most likely influenced by his work at the Commission, decided to introduce modern mathematics in his televised lessons and in fact it was the first time that the ideas of modern mathematics had been displayed nationally. His first lesson was broadcasted in October 1965. The new perspective seemed to change the basis of several mathematical concepts: Addition was now the cardinal number of the union of two disjoint sets, subtraction became the cardinal of a complementary set, multiplication was defined by the cardinal of the Cartesian product of two sets, and division was obtained by the partition of a set into equipotent subsets. Fractions, however, were taught “classically.” He also tried to apply set operations to geometrical objects, but his results were not mathematically sound (Almeida and Matos 2020). The autonomous program of Telescola lasted for three school years until 1968. In October of that year, CPES the mandatory extension of schooling to 6 years began and Telescola, became its televised version.

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The New Programs Considerable enthusiasm surrounded the CPES program. This new cycle started at the same time as the long-time dictator was replaced and a small opening of the regime seemed to happen. Late 1968 and 1969 were times of hope that, however, did not last, as the dictatorial nature of the government was reaffirmed and only ended in 1974. CPES, however, brought together many of the aspirations of progressive educators, as its implementation favored global educational approaches (interdisciplinarity, for example), provided an emphasis on new teaching techniques (project and group work), and the use of technology (the overhead projector, for example). The new mathematics program, written by Sebastião e Silva, fitted into this general pattern of innovation, hoping to bring profound changes in the way in which mathematics was taught. Sebastião e Silva started by reasserting his belief that change should address both methods and content. Consistent with his concern for the need for “active” teaching methods, the first CPES program was dotted with methodological recommendations for grounding teaching on intuition and respecting children’s previous experiences. Constantly associating mathematical concepts with language, it lacked, however, any references to the applicability of mathematics. The new program was needed, he briefly stated, because of the increasingly fast technical and scientific developments, and the reorganization of the content facilitated mathematical structuring. Others, like for example, Joaquim Redinha, an inspector who oversaw the application of the program, also referred to the importance of Piaget’s research. The old program for the two grades approved in 1954 was centered on geometry and mathematical ideas were drawn from real-world ideas and situations. It occupied two and a half pages of the official diary. The new program was based on notions related to sets and their operations, from which arithmetic operations were developed. It also rooted the study of fractions in the language of sets. Geometry, on the other hand, occupied much less time and was approached in a much more “classic” manner, with occasional references to sets. The new program was very detailed, occupying 11 pages, twice the space taken up by the programs of the other disciplines.

Looking at the Textbooks We may better understand the new curriculum by comparing the last textbooks (Ribeiro 1964, 1965) of the old program with the first two textbooks approved in 1970 (d’Eça et al. 1971, 1974) by the ministerial authorities for grades 5 and 6.3 The book for grade 5 (d’Eça et  al. 1974) followed closely all topics in the program and started by introducing the basic concepts of set theory, which took up one-fourth of the pages. It was mostly a linguistic endeavor, and although the authors made a strong effort to use common words—essentially geographical names, names of everyday objects, or common social situations—talking about sets and their operations needed considerable new terminology like sets, elements, belongingness, disjoint, singleton and empty set, inclusion, identity, univocal (one-to-one) and biunivocal correspondences, and the cardinality of a set. Most of these words were not familiar to both students and teachers and, as the program required each new term to have a precise meaning, the book explained thoroughly each of them. For example, the book carefully clarified the difference between “is contained in” and “is an element of.” Related symbols were also introduced and explained. The same happened with arithmetical content. For example, as early as page 31 the book for the fifth grade (d’Eça et al. 1974) made a distinction between number (an abstract entity) and  Ribeiro’s books were “unique” textbooks because they were the only ones approved for use in these grades. They had their first edition in 1950 and underwent minor changes until 1966. D’Eça et al.’s books were approved from 1970 to 1974 and had no changes during that period. 3

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numeral (a sign referring to that abstract entity). Some words changed their meaning. For example, the term “equal,” which students had been using from primary school in arithmetic problems, was now literally forbidden,4 and had to be replaced by “identical.” Going through these initial chapters of the textbook for grade 5 (d’Eça et al. 1974), we cannot avoid the feeling that they essentially dealt with the learning of a new language and the same can be said about the book for grade 6 by the same authors (d’Eça et al. 1971). Tacitly, the books assumed that the language of sets and their relationships could be used to express qualities of common words as long as these were carefully chosen. This general tone of the books echoed suggestions included in the program, namely that the teacher should establish connections between normal language and the language of sets: The verb to be was to be used to indicate belongingness, inclusion, and equality; nouns as representations of elements of sets; and adjectives as ways to define sets using properties; and correct usage of and, or, and not, also became important. After introducing sets, d’Eça et  al. (1974) discussed the four arithmetic operations and related them to the language of sets. Comparing the new discourse about arithmetic operations with the old, we can observe striking differences. In Ribeiro (1965), addition was approached through a problem requiring students to add the distances between two cities. Addition involving more than two numbers, addition of segments, and perimeter were then discussed. All these topics were considered in relation to specific situations (travel or geometric lines). With d’Eça et al. (1974) the study of addition started by presenting two operations on sets, union and intersection, together with corresponding symbols and new terminology. Intersection was thoroughly discussed, by using the names of persons. After six pages, the union of two sets was presented as an operation and addition is another operation that transforms a pair of numbers into a new number. A sum is the cardinal number of the union of two disjoint sets. In contrast with the older book, this section resorted to names either of persons or places. Figure 9.2 exemplifies how these two operations were related. Firstly, three “mutually disjoint” sets of islands, A, B, and C were named. Secondly, successive operations between pairs of sets were performed yielding new sets. Finally, it was shown that the sum of the cardinals of the three sets equals the cardinal of their union.

Figure 9.2  Explaining the sum of more than two terms (“parcelas”). (d’Eça et al. 1974, p. 55)

 It was actually forbidden as testifies the school experience of one of the authors of this chapter.

4

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Both books, the old (Ribeiro 1964) and the new (d’Eça et al. 1974), proceeded by discussing the properties of addition: Commutativity, associativity, and neutral element. In the new book, these properties were explained with numerical examples supplemented with operations on sets, whereas the old book only resorted to numerical examples. This pattern was followed for the other three arithmetic operations. In the new book, subtraction was the cardinal of the complement of a set, multiplication was repeated addition, and division was the decomposition of a set into disjoint subsets with the same cardinal, usually requiring the introduction of new terminology (“partitive” and “partitive-­ multiplicative” numerals, for example). Arithmetic operations and their properties take another fourth of the new book. The idea of multiplication as the cardinal of the Cartesian product of two sets—as it had been presented in Telescola—was not retained. For the remaining topics, the two books presented much less differences. Both discussed fractions and geometry similarly, although the new book usually framed the content in the language of sets. The new book for grade 6 (d’Eça et al. 1971) started with a revision of the language of sets and arithmetic operations from the previous year and the rest was similar to the old book (Ribeiro 1964). Although the new books we mentioned were the only ones approved, other titles circulated, commonly being used as “working books” or “books of exercises.” An analysis of some of these (Matos 2009) shows similar patterns. There was an initial large discussion of sets and their relationships, and arithmetic operations were approached as operations on sets. From 1974, the viewpoint that associated mathematics with the language of sets gradually took a firmer ground. The discipline of mathematics, which belonged to the area of scientific initiation together with the natural sciences, was moved to the area of communication joining Portuguese and foreign languages. The emphasis was now clearly on the linguistic dimensions, and the need for connecting mathematics to real-world experiences diminished. This also corresponded to the introduction of new terminology, especially in the study of fractions. Fractions were now extensively discussed as operators. This option remained until the end of the 1980s.

Teacher Formation Introducing the new programs to teachers proved to be a challenging task. Teachers of mathematics in CPES could have a degree in a wide range of areas, from mathematics, biology, pharmacy, economy, etc., but in the initial years, a significant number of them came from the preparatory technical schools and had a background as primary school teachers, supplemented by a limited course in secondary school mathematics. Several months before the beginning of CPES, a course for teachers featuring António Augusto Lopes was broadcasted. This course was intended for provisional teachers and about 7000 of them enrolled in it. It also drew the attention of many other teachers for the new mathematical ideas and television sets that had been installed in some secondary schools to satisfy demand. Lopes’ course comprised thirty-five 15-min lessons with topics on the “algebra” of sets; binary relations, equivalence relations, and order relations; mappings from one set to another set (functions defined in one set with values in another), composition of mappings, isomorphisms; binary operations (composition laws) (Almeida 2013), far exceeding the requirements of the program. It was, however, the first time many teachers had the opportunity to become acquainted with the new ideas. The introduction of the new program in October 1968 was also accompanied by seminars for teachers on the new topics (Wielewski and Matos 2009). The first, organized at the end of November 1968, focused on sets and their operations. More advanced themes, such as groups, equivalent classes, and functions, were also discussed. The conclusions of this seminar, written by Inspector Joaquim Redinha, were sent in January 1969 to the schools and we can see that teaching the new program was not an easy task for teachers.

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One of the problems that is emerging, with some seriousness, is the extension of the Program in relation to the three weekly times. Teachers complain about this. It was clarified that it is convenient to dose time, not to develop certain chapters too much; to insist on the essentials; to avoid the presentation of complicated examples, particularly if they were divorced from reality and inappropriate to the child’s intellectual development. In particular, the case of books and teachers exaggerating examples about sets was addressed; these notions were integrated in the program, mainly as support of some concepts (specifically of number theory). (Ofício-­circular No. 191, DSCPES, January 14, 1969, Informação, 1, p. 1).

Two months after the introduction of the new program, teachers were complaining about its extensiveness, and the Inspectorate was concerned that teachers were spending too much time on sets. Maybe Joaquim Redinha’s view that the teachers believed that the program was too extensive was confirmed. However, official recommendations were limited to avoiding complex examples of sets. In July 1969, at the end of this first year, in a private letter, Sebastião e Silva called the delay in the program “scandalous” (Silva 1969), and vehemently maintained that the program was neither too difficult nor too extensive. The two problems plaguing the new curriculum—its extensiveness and how a basic teaching of sets and their operations could be delivered—were the focus of much discussion in 1969–1970, during the second year of the new program. Several letters from the Inspectorate, all written by Redinha, were sent to schools in October 1969 with instructions addressing both problems in relation to the operation of the program at the grade 5 level. Firstly, several items were removed from the program: The instructions related to the association of mathematics with language, and to the application of the properties of the arithmetic operations for the justification of algorithms. Secondly, a stern recommendation was given that the introduction to set theory should take at most 2 weeks. Most of the content of these official letters were related to language issues and to the kinds of examples of sets that should and should not be used in class. The problem endured and two investigations conducted in 1972 and 1986 confirmed that only the most basic terminology (sets, their basic operations, and associated terminology) was retained and remained the only successful topic of the modern mathematics reform in CPES (Almeida and Matos 2021). The investigations also showed that most of class time of the first year was devoted to this topic, which naturally lowered the amount of time available for other topics, namely geometry and sharpening calculation skills. Both studies also showed that neither students nor teachers had coped well with other notions of modern mathematics. In the 1972 examination, for example, students’ performance on a very simple arithmetic task was hindered by the use of modern language, and a 1986 survey showed that teachers avoided teaching the most complex modern approaches to subtraction and fractions. A teacher certification program for the new CPES started in 1969. It reproduced the secondary school model of a one-year professional internship (before 1969 it had consisted of 2  years) in a school supervised by a teacher from that school. Another teacher oversaw the process in several schools of the same region. This program gradually promoted the differentiation of specialists in teacher education for CPES, centered in schools and in the Ministry of Education. Frequent regular meetings and seminars on broad educational topics helped to establish a common professional knowledge among them. Guy Brousseau lectured seminars in 1977 and 1978 and, by 1978, international connections were established and Leonor Filipe and Natália Vaz, became members of the CIEAEM, initiating the regular presence of several CPES teachers at international meetings that culminated with the 35th CIEAEM meeting being held in Lisbon in 1983. Essentially influenced by francophone authors, this dynamic group would play a central role in the mid-1980s in a movement that led to the creation of the Portuguese Association of Teachers of Mathematics.

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 The Struggle to Apply Modern Mathematics to Real-World Situations—The Technical School Experiment After World War II, one of the attempts to improve the Portuguese educational system was to systematize and reinforce professional schools, and to engage in an ambitious attempt to build new ones. And, in fact, during the 1950s, several schools had sufficient funds to acquire technological gear adequate for educational purposes. These were proud schools with proud teachers who had a clear mission to produce skilled workers and excellent intermediate technicians. The culture of these schools, valuing functionality and geared to the application of knowledge, was very different from the school culture in liceus which focused on preparing students for higher studies in the universities, and this difference had an impact on the ways in which the new ideas were appropriated.

Purpose and Methods for the Experiment The innovative character of the mathematics program of CPES required major transformations in the programs of the cycles which followed. In the case of the technical schools, these changes began to be addressed at the end of 1966. Following the first colloquium for teachers on modern mathematics in December 1966, a journal that supported the experiment, Folha Informativa dos Professores do 1.° Grupo E.T.P, was published from January 1967 to 1972. Although it had the consent of the authorities, it was not an “official” journal. It usually had about 16 stapled A4 pages, and was printed in a technical school near Lisbon where Aires Biscaia, a mathematics teacher, was the Director. Contrary to the implicit hierarchy of the Sebastião e Silva experiment, this was a grass roots movement and the journal was meant for teachers and written by teachers. It published many articles concerning the new proposals, detailed accounts of teachers’ seminars, including the different opinions about the reform, there was a section on problems, and much information about the international movement was provided. In times of strict control of the media, especially looking to suppress all kinds of dissent, this was indeed a very different type of journal. In January 1968, a Commission was appointed to oversee the reform in technical schools for grades 7 through 9. Directed by Santos Heitor, a teacher with responsibilities in teacher education, it also included Aires Biscaia, Francelino Gomes, Jorge Monteiro, and Vítor Pereira, all frequent contributors to Folha. This Commission coordinated the experiment on modern mathematics that began in the school year 1968–1969. Like the experiment in liceus, the plan started with 10 classes of Industrial Courses, and the number of classes involved was gradually increased. The experiment was accompanied by the production of textbooks. By 1970–1971, the program for the first year was applied to all students, thus ending the experimental phase for that year. By 1973–1974, the experiment was concluded. From 1970, a new course of 3 years (grades 7 through 9) replaced the former technical courses. It was thought that it was premature to start a professional formation in these grades and the intention was to design broad-spectrum curricula similar to those of liceus, foreseeing the future integration of the two. The mathematics program, however, had first been designed at the beginning of 1968, when the experiment started, at a time when these intentions had not been clearly formulated. The program of 1968 (and a revised version in 1970), even though acknowledging the need to include more general topics, intended to integrate modern mathematics into a professional formation by keeping the applied characteristics of the program as much as possible. Santos Heitor, the future coordinator of the experiment, explained his views on the objectives of the new programs for technical schools. He called for:

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1. The promotion, for more demanding professions, of mathematical skills and knowledge (masters, methods or work planning agents, technical agents, engineers, etc.). 2. An increasingly accentuated participation in a technological and economic society, a participation that even our labor legislation suggested. 3. A “mathematical mentalization” capable of allowing the individual to apprehend the complexity of structures in the current world. [This should be] at the top of the objectives, until it became possible to produce “professional mathematicians.” In summary: The more strictly utilitarian purpose of the mathematical training of the future workers must be interpenetrated [by] an authentic phase, gradual and progressive, of formative education of “mathematicians.” By “mathematicians” we understand, here, individuals capable of apprehending mathematical relationships and, hence, more general logical relationships. This bivalent, formative, and informative conception, would have to admit a gradual development, according to the apprehension capacities verified in the students. It will not be an exaggeration to admit that this gradual [development] will condition the individual’s position according to his economic and cultural promotion. (Heitor 1967, pp. 4–5, quotation marks in the original).

The Experimental Textbooks For the first year, the new program had two foci. The first was the study of relations and mappings. Looking into the textbook (Biscaia et al. 1971), we can see that this option allowed for the study of first-degree equations using the composition of bijections. The process started with a game in which for each ball with a number that a student would put inside an area encircled by a rope, another student would put a ball with twice that number into an area encircled by another rope. This was shown to be the operator 2x. The game was then extended to other operators, x + 1, etc., and, using the composition of applications, this was used to solve equations (Figure 9.3). The texts then proceeded to the study of direct and inverse proportionality as functions. “Real-life” problems were included in the book.

Figure 9.3  Solving equations using operators. (Biscaia et al. 1971, p. 116)

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The second focus was the study of Z, the set of integers, and its operations. Addition and its properties were taught considering a bijection between integers and vectors on a line (Figure 9.4). This method was not followed for the other arithmetic operations on integers that were studied by using numbers only The program also involved an applied nature and included the performance of operations with the slide rule. Several pages in the experimental textbook were devoted to calculations to determine squares and cubes. Moreover, “real” problems were included as well—for example, proportionality problems involving dimensions of iron sheets, traveling situations, relative density of several materials, spring experiments, etc., and operations with vectors included problems featuring soccer teams. The program for the second year first included the study of Q—the set of rational numbers—the solution of first- and second-degree equations, and systems of equations, and later transformation geometry and trigonometry. The study of the properties of Q was accomplished by occasional references to vectors and included graphical representations of mappings. As was the case in the first year, more than half of the examples were related to real-life situations (a shaft in a mine, physics examples, etc.). Solving systems of equations was first performed graphically and later algebraically. The second part of the program focused on transformation geometry. Parallel projection and operations with vectors were followed by rotations, symmetries, and homotheties. The program ended with trigonometry and sine and cosine were presented as projection functions that transform angles between unitary vectors and a horizontal direction into the module of their projections on perpendicular directions. Figure 9.5 shows an exercise that asks for the completion of modules of these projections. Previously the book informed students about some of these values for specific angles.

Figure 9.4  Bijection between the addition of vectors and the addition of integers. (Biscaia et al. 1971, p. 178)

Figure 9.5  Projections of unitary vectors in two perpendicular directions. (Gomes and Pereira 1972, p. 114)

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The third year was a recapitulation and consolidation of topics from previous years, occasionally introducing new themes: “Approximate” calculus, equations of a straight line, and sequences. There were few differences between the initial program and the final version adopted in 1970. The main difference was that the initial program included transformations of the plane as a starting point for trigonometry. These were scrapped in 1970, and in 1974, the program was again revised. In the first year, the main change was that addition in Z was no longer introduced through operations with vectors. All sections relating to applications were moved to the end. There are no well-documented records showing the ways in which the experiment was run. Through occasional comments in Folha and in the programs, we believe that beginning students lacked adequate knowledge of topics from CPES. Consequently, the 1974 programs included extensive reviews of those topics (fractions, for example). Moreover, the last topics in the programs were not being taught and were systematically included at the beginning of the following year.

Teacher Formation The experiment involved many seminars with teachers. In October 1967, immediately before the beginning of the experiment, which started in January 1968, 40 teachers gathered in a “Perfecting course for teachers.” The course was preceded by a questionnaire sent to all teachers of mathematics in technical schools asking for their suggestions about the organization of the course and opinions on modern mathematics. During the 4  days, participants were lectured about sets, vectors, mappings (functions), trigonometry, proportionality, equations, and symbology. Other courses followed, covering many topics of modern mathematics: Logic and sets, binary relations, groups, geometric transformations, etc. In 1968, the Belgian teacher Roger Holvoet participated in one of them. The new ideas were not readily accepted, especially at the first seminars (Rodrigues et al. 2016). On the one hand, there were those who proposed that mathematics for workers had to go beyond the mere repetitive application of procedures in order to give them intellectual tools to deal with problems requiring higher-order thinking skills. On the other hand, some argued that mathematics for workers had to be applied, and abstractions, as beautiful as they might have looked, were not easily applied in real-life contexts. Looking through the lens of time, we are witnessing the fading of technical schools which disappeared in the mid-1970s, as the importance of general education for grades higher than 6 became more widely recognized. Consequently, a tradition of applied mathematics that had been established in the late 1940s was disappearing. Although based on the RDD model, it is clear from the study of Folha that this experiment involved a large pool of decision makers. Although there was a clear leadership, shared by five or six participants, the hierarchical environment of the Sebastião e Silva experiment, centered around a university professor who developed both the program and the textbook, did not occur here. Now we had a reform “imagined by teachers and made by teachers” much akin to the practical experiences of curricular development (Pacheco 2001). We find it remarkable, that at this time, when all teacher associations were forbidden, democratic discussions about curricular options were occurring freely, both in the pages of Folha and at teachers’ meetings. Following the trend for extending the number of years of schooling, a new course for grades 10 and 11 started in 1973. The new mathematics program included approximate calculus, analysis up to integration, combinatorial analysis and probability, analytic geometry, matrices, and solid geometry. In 1977, the content sequence was rearranged and probability, approximate calculus, integration, and solid geometry—coincidentally, all areas of content enhancing mathematical knowledge adapted for future technicians—were removed. Perhaps these topics were not needed by prospective students, or teachers were not prepared to teach them.

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Structuring Geometry—Curso Geral in Liceus

The introduction of CPES in 1968 demanded curricular changes in the succeeding courses of technical schools and in grades 7–9 of liceus. As we have seen in the previous section, the former began discussing the new ideas late in 1966, and an experiment was started in 1967. In liceus, however, there was no preliminary discussion of the new curriculum and almost no teacher preparation and, when the classes of the new curso geral [general course] started in October 1970, the new textbook was not available. Although officially these were experimental classes, in fact, all students of grade 7 (and later 8 and 9) took the new course.

The New Curricula, as Expressed in the Unique Textbooks So far, we could not trace either a program or any document introducing the new curricula and we conjecture that, as there was a textbook that had been especially prepared, people in charge thought that there was no need for a curriculum overview. Therefore, we will discuss the “program” as it was expressed in the official textbooks (Costa and Anjos 1971, 1974; Costa et al. 1973). After a brief review of the arithmetic of natural numbers using sets and fractions as operators, as in CPES, the book for grade 7 (Costa and Anjos 1971) presented the set Q of rational numbers as magnitudes that could vary in opposite directions. Performing addition and subtraction with these new numbers was introduced as deposits or withdrawals, and their properties were established as a means to extend previous addition and subtraction tables. Multiplication and division followed a similar path. The study of equations in Q was followed by first-degree equations, and systems of equations. To that point, the “new” curriculum was very close to the “classic” one. Modern mathematics treatment was only apparent in the use of symbols and the systematization of arithmetic using the inclusion of the sets N, Z, and Q. The next topic, binary relations, however, clearly showed the new ideas. Venn diagrams and truth tables were extensively used, leading to the Cartesian product of sets. Properties of binary relations were discussed. Mappings, as a special case of binary relations, were presented and the terms injective, surjective, bijective, domain, codomain, range, inverse, and composition were introduced using Venn diagrams. These concepts were then applied to numerical functions, including graphical representations of direct and inverse proportionality. Costa and Anjos (1971) then dealt with geometry, and after a brief revision, vectors and translations were presented. Vectors were defined as mathematical entities characterized by a “direção” [direction] understood as property of a set of parallel lines, a “sentido” (also translated as direction) understood as the property common to a set of parallel rays pointing in the same direction, and a length. The operation of addition of a point to a vector was then used to define translations whose properties were presented and used to discuss angles in intersecting and parallel lines. The composition of translations was then associated with the addition of vectors. Rotations and symmetries followed, discussed as mappings. The set of plane isometries was then studied. The book ended with “Euclidean” construction problems, such as equality of triangles, perpendicular bisectors, and the circumcenter, incenter, orthocenter, and barycenter of a triangle. The book for grade 8 started with a revision of rational numbers, first-degree equations, and problems to be solved by systems of equations (Costa and Anjos 1974). Then powers with integer exponents were addressed, followed by multiplication of polynomials, square roots and Pythagoras’ theorem, irrational and real numbers. Up to that point, neither topics nor sequence differed from previous “classic” programs for grade 8. The new approach occurred with a discussion of the elements of R+—the set of positive real numbers—as operators, echoing how elements of Z+ were treated in CPES. Operations on real numbers were then presented as extensions of similar operations on rational

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numbers. Cartesian graphs of functions of the form y = ax2 were then introduced, followed by operations with quadratic radicals. Costa and Anjos (1974) then turned to the product of a vector by a real number and, using real numbers as operators, its properties were discussed. Homotheties, defined as mappings which transform points, were presented along with their properties, and these were classified as reductions, isometries, and enlargements. Similarities were then analyzed as transformations on line segments. The book ended with an introduction to trigonometry. The grade 9 book began with a linguistic discussion, distinguishing designations and propositions (Costa et al. 1973), echoing the way similar topics had been dealt with in CPES. Terminology included designations, propositional expressions, and equivalence of conditions—which was used to formalize equivalence principles for equations. The conjunction of conditions was applied to systems of equations. Properties of R followed, and led to intervals and the solution of inequalities. The solution of seconddegree equations followed, together with graphs and radicals. The book continued with logarithms, starting in base 2 including the exponential function and finishing with base-10 logarithms. In geometry, angles, arcs, and circumferences led to trigonometric ratios followed by a chapter on Euclidean geometry and deduction. Space isometries and homotheties, common solids, and areas and volumes concluded the book. In summary, these books essentially addressed “classic” arithmetic and algebraic topics, slightly changing usual terminology, and embedding the discussion in the structure of the sets Z, Q, and R. The classic program was framed within a hierarchy of sets and their operations. Geometry, however, was approached in a new way. Although some “classic” topics remained, there was a clear intent to structure it using a hierarchy of transformations—isometries and homotheties—based on the use of vectors and their operations.

Evaluating the New Curricula We have some basis for evaluating curricular success for these grade levels as, from 1977 until 1979, a major project involving the collaboration of Swedish educators assessed the teaching and learning in grades 7–9, and published a total of 16 reports (Matos and Almeida 2021). Five of these reports referred specifically to the teaching and learning of mathematics between 1975 and 1979. These reports were very critical of the curricula. For example, one of them stated: By analyzing the curricula for the 7th, 8th, and 9th grades it appears that they are curricula typical of the first generation of modern mathematics; they show, moreover, a keener interest in teaching pure mathematics that will make the students good mathematicians. (Catela and Kilborn 1979, p. 41)

Textbooks were also assessed by the project. At the time, there was a dominant collection of textbooks in use in schools and alternative manuals were residual. Shortages in the use of and access to textbooks were reported and their content drew strong criticism from the evaluators (Catela and Kilborn 1979). Firstly, they argued that books were almost replicas of the programs, which could pose problems for students and teachers. They indicated that the programs used concepts and sequences that were very different from regular teaching so that the “pure” logical structure of the books did not facilitate their understanding by teachers who were not comfortable with the meanings and assumptions on which these concepts and sequences were based. Secondly, the structure of the books was questioned. Students were often required to go through five or more pages of text before having the opportunity to tackle problems or become engaged with tasks. The content of the books contrasted with coeval experimental books for the same grades in technical schools. Examples and exercises in the latter are frequently related to real-world situations or applied content from other disciplines. In liceus, however, there were very few exercises, and almost all situations were abstract.

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The quality and extent of student learning were also studied in the project. Performances were much lower than expected (the project expected average ratings of 50%), even for 7th-grade content tested in grade 8. On average, 7th-grade students achieved a rating of 13%, and 8th graders achieved 24% on the content of grade 7 and 25% on the content of grade 8. Students in grade 9 scored 29% (Catela 1980). As these tests were developed in cooperation with the authors of the program, the evaluators argued that the authors were not in touch with the realities of classrooms, as most of them had only had teaching experience in “privileged” schools. Referring to the initial tests that were used, Catela (1980) stated: These trial tests were constructed by the authors of the programs, which are also teachers. The level of the tests M[-]I and M-II (1st [trial] version) shows, therefore, the notion that [these] teachers have of the level of their own classes, and it was ultimately proved that this notion contains higher expectations than the actual knowledge of students. (p. 24)

Answers given to the test questions involving modern mathematics were particularly disappointing. For example, performance on a question about binary relations was very weak: Sixteen 7th-grade classes (64% of the total of 7th-grade classes) and twelve 8th-grade classes (55% of the total) had an average percentage of correct answers less than 10%. According to the expectations of the authors of the test, the average percentages should have been 50%. It was, however, 8.8% in grade 7, and 9.7% in grade 8, which at the same time corresponded to the lowest percentage of correct responses of the test. The study of the percentages of correct answers only told part of the story. Looking at more detailed data, we detected many classes in which few or no students answered correctly. Although a few classes had slightly better results, these were still very far from expected results. In the Northeast region, in particular, data showed a high number of null responses (33 out of 57 classes, whereas in the region of Lisbon the corresponding result was 28 out of 84. The best results (more than 30% correct answers on average, but still below the 50% threshold) were obtained in three groups of schools in the city of Lisbon which had teachers well acquainted with the reform. The poor performance of students on questions concerned with binary relations seemed to be due to two factors—shortcomings in the teaching process, and intrinsic difficulty. In other words, it is very likely that, not having learned binary relations during their initial scientific formation, teachers tended either to teach it inappropriately, or not to teach it at all, as we have seen happening with other “modern” topics in CPES. But even classes taught by teachers conversant with the new ideas, like many in Lisbon, performed poorly. Even those teachers seemed not to be able to teach binary relations in such a way that learning would remain stable as students moved from grade 7 to grade 8, contrary to what happened in other “classical” items. Faced with this situation, the Ministry of Education in 1980 published “Minimum programs.” Binary relations and vectors and their operations were dropped, and consequently, transformation geometry lost most of its formal approach. Some “Euclidean” content (e.g., circumcenter, incenter, orthocenter, and barycenter) was also removed.



Mathematics as Logic—The Program for the Last Years of Liceus

In 1973, modern mathematics reached the last years of secondary schools (grades 10 and 11). By now, the Sebastião e Silva experiment was fairly consolidated and hundreds of teachers had attended the Oeiras courses. The new programs (and the associated textbooks) for these grades followed closely Silva’ experimental curriculum. Grade 10 started with mathematical logic and related it to operations on sets. Algebra of propositions, quantifiers, and Boole laws followed. Logical operations were then extensively applied to the discussion of fractional equations and inequalities. Figure 9.6 shows how logic was applied to the determination of the domain of an algebraic fraction. Several logical operations were performed and the result was expressed as unions of intervals. The authors twice asked students to say “why?” (“porquê?”). The expectation was that students would answer “because the denominator cannot be zero” and “because of the law of the cancellation of the product.” One of De Morgan’s laws was also applied.

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Figure 9.6  Domain of an algebraic fraction. (Garcia et al. 1976, p. 51)

The program continued by revising binary relations. A fairly “classical” approach to analytic geometry followed, including the study of conics, and logic was again used to discuss sets of points defined by the conjunction or disjunction of conditions. Geometry ended with linear programming. Functions in R as special cases of mappings were then studied. Terms like domain, codomain, range, inverse, surjective, and composition were discussed. Algebraic and logical procedures were preferred. Graphical representations were used to illustrate functions with several branches defined by the conjunction of conditions. Quadratic and biquadratic functions and trigonometric functions followed. The program ended with groupoids and combinatorial calculus. Grade 11 essentially consisted of the study of analysis: Sequences, limits, continuity, derivatives, and primitives. Then, rings, fields, an axiomatic study of N, and the set of complex numbers, followed by trigonometry. The program ended with vectors and geometric transformations. Some changes occurred by the end of the 1980s. The study of the straight line was deepened, and conics and linear programing were removed in 1977, as were binary relations in 1978. Also in 1978, analytical geometry was revised with the integration of vectors and the scalar product. Three-­ dimensional analytical geometry was briefly added in 1979 and removed in 1980. Some focal points of modern mathematics, such as groupoids, rings, and fields, were removed in 1979. However, other aspects such as the reliance on logic, especially in grade 10, as the basis for the discussion of algebraic expressions, remained until the end of the 1980s. From the beginning in 1973, there were complaints that the program was too extensive. And in fact, in 1980, a minimal program for grade 11 repeated almost half the content of grade 10, although maintaining all the other topics. The authors of the program never acknowledged there was a problem with the length of the program and essentially responded by specifying the number of classes attributed to each topic, implicitly suggesting that teachers were at fault by paying too much attention to logic in grade 10.

The Reliance on Logic A common trait of the programs for these grade levels was their reliance on logic. Matos and Monteiro (2020) argued that framing mathematical content in logic can be detected from the early years of the Sebastião e Silva experiment, and guided much of the effort to develop new pedagogical

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content knowledge in liceus in the 1960s. They argued that Sebastião e Silva’s preference for logic can be traced back to his early papers and guided most of his research in Italy in the early 1940s. Many years later, reflecting on the experiment and its later influence on the curricula of grades 10–12, António Augusto Lopes questioned the role of logic at the beginning of grade 10: The introduction to Logic should have been a support [for teaching], not an end. … The [10th-grade] teachers later exaggerated the transmission to the students. … The initial part of the Algebra of sets and Logic could be taught in a month, but the teachers who did not have adequate information, stalled, and some spent much more time than necessary. (Lopes, quoted in Almeida 2013, p. 233)

Logic was placed at the beginning of the program for grade 10 and from our personal experience, many teachers spent the first 3 months on the subject.

The Absence of Teacher Formation Compared to CPES, the social dynamics of teachers in grades 7 through 12 was low. We did not detect teacher-centered activities fostering the development of new professional knowledge other than occasional short-term educational courses for teachers. Investment in teacher formation for these grades was considerably lower than the large investment made with teacher courses for CPES or the technical schools. The lack of time given to checking with teachers regarding the application of the new curricula reflects the hierarchical relationship established with schools by the ministerial department in charge of these grade levels. Results on the tests for grades 7, 8, and 9, as discussed above (Catela 1980), can be extrapolated to understand how those in charge of this department related to common teachers: “We know very well what mathematics should be taught at these grade levels and, if there is a problem, it is your fault,” seemed to be the message. These persons had been deeply involved in the Sebastião e Silva experiment, and their stance fits well with the kind of curricular development procedures implicitly used in the experiment. Sebastião e Silva became ill in 1970 and died in 1972 cutting international connections. Again, contrary to what happened in CPES, there was no attempt to reestablish connections with international bodies. For example, in 1976, in the Proceedings of the third International Congress on Mathematical Education, he is still indicated as the Portuguese representative to ICMI (Matos 1989).



The Primary Schools

In primary education, the first initiatives to introduce new dynamics in mathematics teaching were initiated by João Nabais, who was a very energetic person, graduated in psychology in Louvain, Belgium, and in 1959 founder of the Colégio Vasco da Gama, near Lisbon—which was a school for primary and secondary students, from which he promoted many activities related to teacher education and educational research, delving into educational psychology and the didactics of several disciplines (Candeias 2007). As a promoter of the use of materials in primary school mathematics, he organized many courses on Cuisenaire rods. The first took place at his Colégio in April 1962 and was taught by Caleb Gattegno (Figure 9.7). In total, 135 primary and secondary school teachers from all over the country participated. In October 1962, the second Cuisenaire course was held in Oporto, taught by António Augusto Lopes and Nabais. Until 1967, almost 20 courses on the Cuisenaire method were carried out in several parts of the country with a large number of teachers attending. In one of these courses, in 1965, participated Madeleine Goutard (Candeias 2007) and in 1968, Roger Holvoet gave a course in another of his initiatives. All this work was complemented by the publication of several books on mathematics teaching using the Cuisenaire rods (Candeias 2007).

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Figure 9.7  Lunch at the “First initiation course on the Cuisenaire method.” Gattegno is at the center, Nabais at his left. António Augusto Lopes is the fifth on the right. (Nabais 1965, p. 158)

Nabais’ activities were not limited to primary schools. In 1969, he organized the seventh Seminar on Psychology and Pedagogy, attended by teachers of several grade levels. In this Seminar, Georges Papy presented, over 10  days, a course on modern mathematics and pedagogy of mathematics, in which he addressed topics such as sets and their operations, relations and their properties, equivalences, the binary system, cardinal numbers, addition, multiplication, real numbers, and so on. During the Seminar, Papy also guided some round-table discussions on those topics and gave two public lectures. In this Seminar, in addition to a course on educational technology and school guidance, Nabais organized several sessions about mathematics teaching (Candeias 2010). Others, among them António Lopes, Baptista Martins, Fátima Dias, and Silva Paulo, also conducted courses on the use of materials in primary schools, and the official journal Escola Portuguesa also addressed the new ideas, even though the first programs influenced by the modern mathematics movement would only be introduced after the change of political regime in 1974. An experiment with 6–7-years-olds was also developed in the late 1960s by the Gulbenkian Foundation (Candeias 2007). These programs, in line with the CPES programs, assumed that the child’s maturity must be considered for the purposes of progression in learning. The intention was to adapt teaching to the modern currents of pedagogy and psychology, taking the principles of Escola Nova [Active School] as a fundamental reference.

The New Programs With the programs instituted in 1968, mathematics content was divided into arithmetic and geometry. The use of the term “mathematics” as a unified set of knowledge only occurred in 1974. These programs focused on relationships between arithmetic, geometry, and reality and highlighted the importance of solving problems which children at the level of development experienced difficulty in solving (Almeida and Candeias 2014). The new program of 1974 introduced considerable changes in the first grade but did not change the other three grades. Two optional mathematics programs were presented: Program A, resulting from an adaptation of the 1968 program, and Program B, developed

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in the path of modern mathematics. The teacher could decide to follow either of the two options; however, if the choice was Program B, he or she had to inform the institutional department in charge of primary education to receive adequate support. Both options for the first grade presented mathematics as a discipline, and “objectives” were defined for teaching prior to the presentation of the content. Program A was essentially a list of topics with some “suggestions” for the teacher. Predicting that Program B would require a more careful preparation by the teachers, this program included detailed suggestions. Program B differed from Program A with respect to the topic of sets. Program A avoided the term “set” and only referred to “collections,” observation of collections, elements of a collection, having more objects than, having fewer objects than, and having as many objects as. And it did not suggest activities or exercises for the classroom. However, in Program B, such matters were assumed to have great relevance, occupying the first 3 months. This program provided a series of notes, objectives, and 33 exercises, accompanied by drawings of sets with different approaches for the first three themes: (1) Introduction of sets, (2) sets, partition of a set, subsets, and (3) the idea of correspondence. The use of Cuisenaire rods and Logic Blocks was suggested. The introduction of sets aimed at the acquisition of basic vocabulary that would precipitate the use of correct mathematical expressions. As activities suitable for the introduction of sets, the program recommended those that consist of observing, manipulating, comparing, and classifying objects (with attributes such as shape, color, and size being especially considered). Collective exercises in which children compared their opinions with those of their peers, and individual exercises using improvised or structured material were recommended. Geometry was also addressed under the heading “Observation of the shape of solid bodies.” It was suggested that after observing, manipulating, comparing, classifying, and grouping objects with common characteristics, students should observe solid bodies and describe them (Figure 9.8). Program B had no topic related to geometry.

Figure 9.8  Exercise on sets requiring children to connect objects with geometric shapes. (Ensino Primário. Programas para o ano lectivo 1974–1975, 1974, p. 52)

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In 1975, a new program for primary education came into effect. The intention was to modernize the teaching of mathematics more through activities than by changing the language used. Topics covered were numbers, addition and subtraction, multiplication and division, measure, money, and geometry. Although the program did not stress the word “set,” there was a note saying that “collection” and “set” should be used in their usual sense, as synonyms. The topic also attempted to integrate numbers and their operations. The 1975 program was amended in 1978, taking into account the opinions of primary school teachers. Content was now organized into five major themes: Sets, geometry, whole numbers, fractional numbers, and fundamental quantities. “Sets” comprised the definition and representation of sets, subsets, and operations with sets. Geometry was centered on the organization of space and geometric transformations. There were also some changes in terminology, as was the case with binary operations with whole numbers. Fractional numbers integrated three subthemes: Problematic situations, fractions, and decimal numbers. In 1980, programs changed again but this time they remained in force throughout the 1980s, therefore introducing stability in primary education. In mathematics, the topics were: sets, structuring of space and fundamental elements of geometry, numbers and counting, lengths, surfaces, volume and capacity, time and order, weight, area and money. These themes spanned the 4 years with increasing complexity. In the first year, sets, listing of properties, subsets, singleton and empty sets, union of two sets, and complement of a set were addressed, and these were expanded, in the third year, by the partition of a set into subsets with the same number of elements. Throughout the program, problematic situations were presented as suggested activities. Some content, that had been abandoned in the two previous programs, namely the study of angles and geometric solids, returned.



The End of Modern Mathematics

Modern mathematics prevailed on curricular perspectives in Portugal for about 30 years and, for the first 20, there were very few negative public reactions. Among them, we found one early article by Francisco Gonçalves, a prominent teacher and textbook author, who questioned the new trends claiming that the real problem was the 50% of students who fail examinations in the 2nd and 3rd cycles of liceus, despite the “successive simplifications that we present in our courses” (Gonçalves 1961, pp. 546–547). Directly questioning the bases of a potential reform, he stated that: In the field of education, it is necessary to establish an economy of effort that leads us to put aside subjects and methods that were already old 50 years ago and that, at the same time, should not be willing to implement all the novelties that come from Paris, even if they come with the warranty of the fascinating name of Nicolas Bourbaki. (Gonçalves 1961, p. 548)

And he added that, while recognizing the importance of scientific preparation for a few, general social indicators and international comparisons suggested that the country needed, primarily, to invest in “mathematics for crowds and not for elites” (p. 548). By the end of the 1960s, two articles published in the daily press reported some discomfort with the new approaches. The first, by Luís de Albuquerque, a mathematician at the University of Coimbra and author of numerous books on the history of mathematics and education, disagreed with the optimistic view reported in an account of an experience at a technical school (Albuquerque 1968). He argued that considering the way the experience was organized (a smaller class was formed with selected students), the conclusions obtained could hardly be generalized. The second, by João Ilharco (1969), questioned the content of one of the books used in CPES. The first year of the Preparatory Cycle is, in general, attended by ten-year-old students who, within what is natural and normal, will only very exceptionally be able to understand the series of abstractions with which they are put in contact in the first weeks of class. (Ilharco 1969, p. 1)

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The article included examples of what, according to him, were mathematical terms too complicated for children. He added a list that included most of the new terminology about sets proposed by the program. In the late 1970s, using the possibility to exercise freedom of speech and association made possible by the 1974 revolution, strong criticism started to mount about the programs, especially those for grades 7–12. From April to July 1981, several meetings of the newly founded Portuguese Society of Mathematics (SPM) were devoted to this topic and, in the last one, a document about the programs was approved (Os programas em debate 1982). Reacting against what was perceived to be “a critical situation,” it stated that modern programs were thought to render mathematics hermetic, formalized, and unnecessarily complicated by a great emphasis on symbols and linguistic games, and foreign to reality and applications. Curricula should change and integrate: –– A strong component making use of problems, i.e., great relevance to the role of problems as a means to develop an investigative and discovery spirit …; –– A strong focus on the practical side [of mathematics], by using calculators … and computers. –– A special attention to the applications of mathematics and its relationships to other disciplines, [adopting] a marked interdisciplinary sense. In sum, an increased relevance to the formative aspect (Os programas em debate 1982, p. 20) was needed. These three dimensions (problems, technology, and applications), which departed from the official curricular options at the time should integrate the backbone of the idealized mathematics curriculum proposed by the document. The document also discussed curriculum development. Criticizing previous practice of writing programs in closed commissions appointed by the Ministry of Education, new curricula, should be developed by a commission composed of persons indicated by the universities, the secondary schools, the Portuguese Society of Mathematics, and lastly the Ministry of Education. The 1981 meetings of the SPM brought together mathematicians, teachers in recently created teacher education courses, and young teachers of mathematics of grades 7 through 12. Some of the latter were still following courses at universities, and showed discomfort with respect to the state of mathematics teaching and learning in the nation. This new generation of teachers was very actively discussing curricular alternatives, mostly based on proposals of the National Council of Teachers of Mathematics in the USA, and rejecting the dominant francophone perspectives (Matos 2011). Later, in 1986, the Association of Teachers of Mathematics (APM) was created, joining several groups of innovative teachers from all non-university grades. The dominant perspective in the documents published by APM valued problem solving, the use of manipulative materials, and technology, and rejected what was perceived as a formalist approach to mathematics, based on logic and removed from the real world and the applications of mathematics. The new programs for grades 1–9, published in 1989, reflected this perspective and marked the end of the dominance of modern mathematics in these grades. Later, in 1995, the new program for grades 10–12 would be based on analysis and would downgrade the role of logic. For example, no longer would the study of fractional equations be performed as a logic problem. Instead, they would be treated immediately as functions, and their domains, Cartesian representations, etc., would be studied as such.



Concluding Remarks

In this chapter, we have tried to show the multiple facets that modern mathematics took in Portugal as it was applied in the different educational subsystems. We emphasized distinct options, namely, the use of manipulatives in primary school, the valorization of language in CPES, the focus on structures

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in grades 7–9 of liceus, and the study of advanced mathematical concepts framed in logic as in the Sebastião e Silva experiment and in the final grades of liceus. We also discussed the dilemmas faced by the technical schools as they struggled to find applications for abstract notions. Despite this diversity, it would be wrong to imagine that there were no common traits underlying the reform. On the contrary, in all these variations, we find a concern for the unity of mathematics and an effort to bring all school mathematical knowledge under a common umbrella—specifically, the language of sets and associated structures. Another common trait was the enthusiasm of hundreds of reformers and teachers in their struggle to bring modernity to what was perceived as a stalled curriculum, that until 1974 possibly corresponded to their image of a stalled country. Acknowledgements  This text was developed within the research group in History of Mathematics Education of the Unidade de Investigação e Desenvolvimento of Universidade Nova de Lisboa. We would like to thank, in particular, Alexandra Rodrigues, Ana Santiago, Cecília Monteiro, Rui Candeias, and Teresa Monteiro, with whom we discussed some of its draft versions. Funding from the Portuguese Fundação para a Ciência e a Tecnologia supported our work through the projects PTDC/CED-EDG/32422/2017 and UIDB/04647/2020 of CICS.NOVA - Interdisciplinary Center of Social Sciences of Universidade Nova de Lisboa.

References Albuquerque, L. (1968, October 18). Reflexões sobre o ensino da matemática «moderna» [Reflections on the teaching of “modern” mathematics]. Diário de Lisboa, pp. 3–4. Almeida, M. (2013). Um olhar sobre o ensino da matemática, guiado por António Augusto Lopes [A perspective on mathematics teaching guided by António Augusto Lopes] (Unpublished doctoral dissertation). Universidade Nova de Lisboa, Lisbon, Portugal. Almeida, M.  C., & Candeias, R. (2014). Os programas de matemática do ensino primário, da Telescola e do Ciclo Preparatório do Ensino Secundário [Mathematics programs for primary education, Telescola and the Preparatory Cycle of Secondary Education]. In A. Almeida & J. M. Matos (Eds.), A matemática nos programas do ensino não-­ superior (1835–1974) [Mathematics in non-university programs in Portugal (1835–1974)] (pp. 39–68). Caparica, Portugal: UIED and APM. Almeida, M., & Matos, J.  M. (2020). O conhecimento do professor em tempos de mudança curricular. O caso da Telescola portuguesa (1965–1967) [Teacher knowledge in times of curriculum change. The case of the Portuguese Telescola (1965–1967)]. Historia de la Educación, 30, 91–110. Almeida, M., & Matos, J. M. (2021). A avaliação da experiência de matemática moderna nos liceus portugueses [The evaluation of the experience of modern mathematics in Portuguese liceus]. REMATEC, Revista de Matemática, ensino e Cultura, 16, 43–58. Almeida, A. J., Matos, J. M., Almeida, M. C., & Candeias, R. (2020). A matemática moderna nos jornais diários de Lisboa [Modern mathematics in daily newspapers in Lisbon]. São Paulo, Brasil: Livraria da Física. Biscaia, A., Gomes, F., Monteiro, J., Pinto, M. H. P., Heitor, S., & Pereira, V. (1971). Matemática. Edição experimental para o 1° ano [Mathematics. Experimental edition for the 1st year]. Lisbon, Portugal: Amílcar de Matos Marques. Burke, P. (2016). What is the history of knowledge. Malden, MA: Polity Press. Calado, J. (1958). Sobre o ensino das matemáticas elementares [About teaching elementary mathematics]. Palestra, Revista de Pedagogia e Cultura, 1, 89–105. Candeias, R. (2007). Contributo para a história das inovações no ensino da matemática no Primário: João António Nabais e o ensino da matemática no Colégio Vasco da Gama [Contribution to the history of innovations in the teaching of mathematics in primary school: João António Nabais and the teaching of mathematics at Colégio Vasco da Gama]. (Unpublished master’s thesis). Universidade de Lisboa, Lisbon, Portugal. Candeias, R. (2010). Contributos para a história das inovações no ensino da matemática no primário: João António Nabais e o desenvolvimento e divulgação de materiais didácticos [Contributions to the history of innovations in mathematics teaching at primary level: João António Nabais and the development and dissemination of teaching materials]. Quadrante, 17(1), 47–77. Catela, M. E. (1980). Ensino Secundário Unificado. A aprendizagem da matemática em perspectiva: 9° ano de 1978–79 e sua relação com os 7° e 8° anos de 1977–78 [Unified Secondary Education. Learning mathematics in perspective: 9th year of 1978–79 and its relationship with the 7th and 8th years of 1977–78]. Lisbon, Portugal: GEP. Catela, M. E., & Kilborn, W. (1979). Ensino Secundário Unificado. A aprendizagem da matemática em 1977–78. 7° e 8° anos [Unified Secondary Education. Learning mathematics in 1977–78. 7th and 8th years]. Lisbon, Portugal: GEP.

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José Manuel Matos and Mária Cristina Almeida

Costa, A. A., & Anjos, A. O. ([1971]). Compêndio de matemática. 1° ano do ensino liceal (antigo 3° ano) [Compendium of mathematics. 1st year of high school education (formerly 3rd year)] (1st edition). Porto, Portugal: Porto Editora. Costa, A. A., & Anjos, A. O. (1974). Compêndio de matemática. 2° ano do ensino liceal (antigo 4° ano) [Compendium of mathematics. 2nd year of high school (formerly 4th year)] (1st edition). Porto, Portugal: Porto Editora. Costa, A. A., Anjos, A. O., & Lopes, A. A. (1973). Compêndio de matemática. 3° ano do ensino liceal (antigo 5° ano) [Compendium of mathematics. 3rd year of High School (formerly 5th year)] (1st edition). Porto, Portugal: Porto Editora. Dantas, M. (1958). Os congressos do ensino da matemática no Brasil [Congresses on the teaching of mathematics in Brazil]. Palestra, Revista de Pedagogia e Cultura, 3, 97–103. d’Eça, M. N., Cássio, M. G., Costa, M. A., & Gomes, M. V. (1971). Vamos estudar matemática. Ciclo Preparatório do Ensino Secundário. 2.° ano [Let’s study mathematics. Preparatory Cycle of Secondary School. 2nd year] (6th edition). Porto, Portugal: Livraria Avis. d’Eça, M. N., Cássio, M. G., Costa, M. A., & Gomes, M. V. (1974). Vamos estudar matemática. Ciclo Preparatório do Ensino Secundário. 1.° ano [Let’s study mathematics. Preparatory Cycle of Secondary School. 1st year] (9th edition). Porto, Portugal: Livraria Avis. Ensino Primário. Programas para o ano lectivo 1974–1975 [Primary school. Programs for the academic year 1974– 1975]. (1974). Lisbon, Portugal: Ministério da Educação e Cultura. Estatística, I. N. (2009). 50 Anos de estatísticas da educação. Vol. 1 [50 Years of education statistics. Vol. 1]. Lisbon, Portugal: Instituto Nacional de Estatística. Folha, R., & Grácio, R. (1958). Bom augúrio [Good omen]. Labor, Revista de Ensino Liceal, 22, 211–218. Fonseca, C. (1963a, March 6). Revolução no ensino—(1) Uma nova concepção da matemática inteiramente diferente da tradicional principia a ser conhecida no nosso país [Revolution in Education—(1) A new conception of mathematics entirely different from the traditional one begins to be known in our country]. Diário Popular, p. 6. Fonseca, C. (1963b, March 7). Revolução no ensino—(2) A introdução das matemáticas modernas no ensino secundário e a sua necessidade [Education revolution—(2) The introduction of modern mathematics in secondary education and its need]. Diário Popular, p. 7. Fonseca, C. (1963c, March 8). Revolução no ensino—(3) A formação do professor de liceu (mais do que a elaboração de novos programas) é indispensável para o rejuvenescimento do ensino secundário—afirma o dr. Jaime Leote, metodólogo de Matemática [Education revolution—(3) Secondary school teacher education (more than the development of new programs) is essential for the rejuvenation of secondary education—says dr. Jaime Leote, Mathematics Methodologist]. Diário Popular, p. 13. Fonseca, C. (1963d, March 9). Revolução no ensino—(conclusão) É preciso que atinjam a escola primária os novos métodos didáticos da matemática [Revolution in Education—(Conclusion) The new teaching methods of mathematics must reach primary school]. Diário Popular, p. 7. Fonseca, C. (1965, July 23). É urgente remodelar os programas de Matemática [It is urgent to change the Mathematics programs]. Diário Popular, pp. 1, 19, 29. Garcia, M., Anjos, A., & Ruivo, A. (1976). Compêndio de matemática 1° ano Curso Complementar 1° volume [Mathematics Compendium 1st year Complementary Course 1st volume]. Porto, Portugal: Empresa Literária Fluminense. Garnica, A. (2008). Resgatando oralidades para a história da matemática e da educação matemática brasileiras: o movimento matemática moderna [Recovering oralities for the history of Brazilian mathematics and mathematics education: the modern mathematics movement]. Zetetike, 16(30), 163–215. Gomes, F., & Pereira, V. (1972). Matemática. Edição experimental para 1972/73. 2ª parte — 2.° ano. Cursos Gerais de Índole Industrial [Mathematics. Experimental edition for 1972/73. 2nd part—2nd year. General Industrial Courses]. Lisbon, Portugal: Amílcar Matos Marques. Gonçalves, F. M. (1961). O ensino das matemáticas no momento presente [The teaching of mathematics at the present time]. Labor, Revista de Ensino Liceal, 25(202), 546–554. Heitor, A.  S. (1958). Comentário sobre a XI reunião da Comissão Internacional para o Estudo e Aperfeiçoamento do Ensino da Matemática [Commentary on the XI meeting of the International Commission for the Study and Improvement of Mathematics Teaching]. Escolas Técnicas, Boletim de Acção Educativa, VI(23), 269–284. Heitor, A.  S. (1967). Artigo preparatório do 2° Curso de Aperfeiçoamento dos Professores de Matemática (E.T.P.) [Preparatory article for the 2nd Improvement Course for Mathematics Teachers (E.T.P.)]. Folha Informativa dos Professores do 1° Grupo (E. T. P.), 9(July), 1–5. Howson, G., Keitel, C., & Kilpatrick, J. (1981). Curriculum development in mathematics. Cambridge, United Kingdom: Cambridge University Press. Ilharco, J. (1969, January 18). Ácerca do ensino da matemática no Ciclo Preparatório [About the teaching of mathematics in the Preparatory Cycle]. República, pp. 1, 8. Leote, J. (1958). Tendências actuais do ensino da geometria [Current trends in the teaching of geometry]. Palestra, Revista de Pedagogia e Cultura, 1, 37–48.

9  The Distinct Facets of Modern Mathematics in Portugal

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Leote, J. (1964). A actualização do ensino da matemática no nível secundário, vista através das reuniões da O. C. D. E. [Updating the teaching of mathematics at secondary level, seen through the meetings of the O.C.D.E.] Palestra, Revista de Pedagogia e Cultura, 21, 110–123. Lima, I. (1963). Sobre o recrutamento e formação dos professores de matemática dos liceus [On the recruitment and training of high school mathematics teachers]. Palestra, Revista de Pedagogia e Cultura, 18, 83–96. Matos, J.  M. (1989). Cronologia recente do ensino da matemática [Recent chronology of mathematics teaching]. Lisbon, Portugal: Associação de Professores de Matemática. Matos, J. M. (2006). A penetração da matemática moderna em Portugal na revista Labor [The penetration of Modern Mathematics in Portugal in the Labor journal]. Unión, Revista Iberoamericana de Educación Matemática, 5, 91–110. Matos, J.  M. (2009). Changing representations and practices in school mathematics: The case of modern math in Portugal. In K.  Bjarnadóttir, F.  Furinghetti, & G.  Schubring (Eds.), “Dig where you stand” Proceedings of a Conference on On-going Research in the History of Mathematics Education, Garđabær, Iceland, June 20–24 2009 (pp. 123–137). Reikyavik, Iceland: University of Iceland. Matos, J. M. (2011). Identity of mathematics educators. The Portuguese case (1981–1990). In M. Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education (pp. 1740–1749). Rzeszów, Poland: University of Rzeszów. Matos, J. M., & Almeida, M. (2021). Evaluating modern mathematics curricula. Matemáticas, Educación y Sociedad, 4(1), 57–72. Matos, J. M., & Monteiro, T. M. (2020). Construindo o conhecimento pedagógico do conteúdo em tempos da matemática moderna: as múltiplas facetas da lógica [Building pedagogical content knowledge in times of modern mathematics: the multiple facets of logic]. HISTEMAT—Revista de História da Educação Matemática, 6(2), 8–25. Monteiro, T. (2018). Formação de professores de matemática no Liceu Normal de Pedro Nunes (1956–1969) [Formation of teachers of mathematics at the Normal School of Pedro Nunes (1956–1969)] (Unpublished doctoral dissertation). Universidade Nova de Lisboa, Lisbon, Portugal. Nabais, J. A. (Ed.) (1965). Cadernos de Psicologia e de Pedagogia: Revista de Ciências da Educação, 1(3–4). Lisbon, Portugal: Centro de Psicologia Aplicada à Educação. OECD. (1964). Mathematics to-day: A guide for teachers. Paris, France: OECD. Ofício-circular No. 191, DSCPES, January 14, Informação, 1. (1969). Lisbon, Portugal: Archives of the General Secretariat of the Ministry of Education. Os programas em debate [The programs under debate]. (1982). Boletim da Sociedade Portuguesa de Matemática, 5, 18–22. Pacheco, J. A. (2001). Currículo: Teoria e práxis [Curriculum: Theory and praxis]. Porto, Portugal: Porto Editora. Ribeiro, Á. S. (1964). Compêndio de matemática para o 2.° ano do curso liceal [Textbook of mathematics 2nd grade of secondary course]. Lisbon, Portugal: Livraria Popular Francisco Franco. Ribeiro, Á. S. (1965). Compêndio de matemática para o 1.° ano do curso liceal [Textbook of mathematics 1st grade of secondary course]. Lisbon, Portugal: Livraria Popular Francisco Franco. Rodrigues, A., Novaes, B. W. D., & Matos, J. M. (2016). A cultura escolar em conflito: ensino técnico e matemática moderna em Portugal [School culture in conflict: technical education and modern mathematics in Portugal]. Revista Diálogo Educacional, 16(48), 381–402. Silva, J. S. (1957). XI reunião da Comissão Internacional para o Estudo e o Melhoramento do Ensino da Matemática [XI meeting of the International Commission for the Study and Improvement of Mathematics Teaching]. Gazeta de Matemática, 66–67, 31–32. Silva, J. S. (1962). Sur l’introduction des mathématiques modernes dans l’enseignement secondaire [On the introduction of modern mathematics in secondary education]. Gazeta de Matemática, 88–89, 25–29. Silva, J. S. (1969). Unpublished letter to António Augusto Lopes dated October 6. Archive AAL (1,4). Silva, J. S. (1975). Guia para a utilização do Compêndio de Matemática (1° volume) [Guide for using the Mathematics Compendium (1st volume)]. Lisbon, Portugal: GEP. Silva, M., & Valente, W. (2008). A matemática moderna em Portugal: o que dizem os cadernos dos alunos? [Modern mathematics in Portugal: what do students’ notebooks say?] Quadrante, XVII(1), 77–92. Teodoro, A. D. (1999). A construção social das políticas educativas. Estado, educação e mudança social no Portugal contemporâneo [The social construction of educational policies. State, education and social change in contemporary Portugal]. (Unpublished doctoral dissertation), Universidade Nova de Lisboa, Lisbon, Portugal. Tyack, D., & Cuban, L. (1997). Tinkering toward utopia: A century of public school reform. Harvard, MA: Harvard University Press. Ventura, M. (1961). O ensino das matemáticas nas escolas secundárias [The teaching of mathematics in secondary schools]. Labor, Revista de Ensino Liceal, 25(199), 263–361. Wielewski, G.  D., & Matos, J.  M. (2009). O currículo de Matemática prescrito no início no Ciclo Preparatório do Ensino Secundário português [The Mathematics curriculum prescribed at the beginning of the Preparatory Cycle of Portuguese Secondary Education]. In J.  A. Fernandes, M.  H. Martinho, & F.  Viseu (Eds.), XX Seminário de Investigação em Educação Matemática [XX Seminar on Research in Mathematics Education] (pp. 239–248). Viana do Castelo, Portugal: Associação de Professores de Matemática.

Chapter 10

Papy’s Reform of Mathematics Education in Belgium: Development, Implementation, and Controversy Dirk De Bock and Geert Vanpaemel

Abstract  The modern mathematics movement in Belgium is inextricably linked to Georges Papy, a flamboyant and uncompromising professor of algebra at the Free University of Brussels. From the late 1950s, Papy reshaped the content of secondary school mathematics by basing it upon the unifying themes of sets, relations, and algebraic structures. Meanwhile, he innovated the pedagogy of mathematics by functionally interweaving his rigorous discourse with multicolored arrow graphs, filmstrips as non-verbal proofs, and playful drawings, as manifested in his revolutionary textbook series Mathématique Moderne. From 1961, his Belgian Centre for Mathematics Pedagogy coordinated the various reform actions: Curriculum development, classroom experiments, and in-service teacher training. Although the Belgian mathematics education community was divided about Papy’s agenda, zeitgeist, media propaganda, and political support made it possible for Papy to realize his reform almost entirely. After the generalized and compulsory introduction of modern mathematics in Belgian secondary schools in 1968–1969, the primary schools followed in the 1970s. Keywords  Belgian Centre for Mathematics Pedagogy · Belgian experiments · CIEAEM · Days of Arlon · Frédérique Lenger · Georges Papy · Implementation of reform · Kindergarten teachers · Léon Derwidué · Mathématique Moderne · Minicomputer · Modern mathematics · Structuralist approach · Teacher recycling · Teacher re-education · Teacher training · Willy Servais

Introduction As a small country, Belgium played a pioneering role in the worldwide modern mathematics movement of the 1960s and early 1970s. Compared to most other countries, the Belgian reform movement started early (before the Royaumont Seminar, which was held in 1959), was quite radical, and survived for a long time (until the early 1980s). The movement was driven by the passionate reformer Georges Papy, president of 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 during the 1960s, an often invited expert at international conferences, and the author of a groundbreaking textbook series Mathématique Moderne. Papy inspired reformers all over the world, but this in no way meant that his ideas were welcomed uncritically in D. De Bock (*) · G. Vanpaemel KU Leuven, Leuven, Belgium e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_10

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other countries. Reformers outside Belgium—as well as some within Belgium—considered Papy’s project unfeasible for most secondary school students and teachers. Nevertheless, because of their refreshing approach, his textbooks in particular were often recommended as “compulsory literature” for pre-service and in-service teachers. Although Papy is recognized as the leading, if not uncontested, architect of the modern mathematics reform movement in Belgium, the path toward the reform was paved by others. As shown in Chap. 3 in this volume, discussions on the direction of a modernization process of mathematics education were held within the CIEAEM from the early 1950s onward. Several Belgians, including Lucien Delmotte, Louis Jeronnez, Frédérique Lenger, Paul Libois, and Willy Servais, actively participated in these discussions. When, in 1953, the Société belge de Professeurs de Mathématiques (SBPM) [Belgian Society of Mathematics Teachers] was founded by active CIEAEM members, the quality and possible improvement of secondary school curricula became an important concern, in particular at the Society’s annual conferences and in its journal Mathematica & Paedagogia (De Bock and Vanpaemel 2019, Chap. 3). The debates within the Society culminated in the development in August 1958 of an experimental program for the teaching of modern mathematics by Frédérique Lenger and Willy Servais (Le programme B des écoles normales gardiennes 1958–1959), and in an experiment based on that program.



Toward Modern Mathematics at the Secondary Level

The First Experiment with Future Kindergarten Teachers The first Belgian experiment with modern mathematics, based on the Lenger-Servais program, was run during the school year 1958–1959 in two schools for future kindergarten teachers in the French-­ speaking part of Belgium (one in Arlon and the other in Liège). The experiment was led by Frédérique Lenger and Madeleine Lepropre, and participants were 15–16-year old students who certainly did not belong to the top streams of education for mathematics (“Some of them did not hide their fundamental hostility toward mathematics … and the persons who teach it,” Papy 1968, p. 27). The experimental course (3 h of mathematics per week) started with fundamental notions from set theory, related to the genesis of natural numbers, and from topology as a basis for the study of geometry (Lenger and Lepropre 1959). After introducing the notion of set, the course continued with set-theoretical topics such as relations of inclusion and equality of sets, the main operations on sets (intersection, union, and Cartesian product), and correspondences between sets (e.g., one-to-one correspondences). Examples were taken from students’ experiences and from school life. The second part of the course included an arithmetic and a geometric track (treated in parallel in, respectively, one and 2 h per week). In arithmetic, operations with natural numbers and properties of these operations were discussed from a set-theoretical perspective, as well as their application in the decimal system. Geometry started with some intuitive topological notions (e.g., open and closed figures, interior and exterior of a closed curve) and culminated in a study of the basic plane figures. This study was primarily oriented to geometrical transformations and led to the concepts of symmetry, congruence, and similarity. According to Lenger and Lepropre, the experiment was a success. As their main evidence, they referred to the encouraging lively and active response from students to the new material. Education was provided in these classes in an atmosphere of happiness. The hostility of the students toward mathematics had completely disappeared. We saw vividly that today’s children are in resonance with the mathematics that is currently in use. (Papy 1968, p. 29)

If the experiment did prove one thing, Lenger and Lepropre maintained, it was that modern mathematics did not put off students, but on the contrary inspired them with a “taste for mathematics.” The

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experimenters invited other teachers to share their experiences and to find still better ways to make the teaching methods more active. After the first year of experimentation with modern mathematics, Lenger and Servais realized that they needed the assistance of an academic mathematician to help them with the mathematical problems that would arise with the design of new teaching programs for follow-up experiments. Georges Papy (1920–2011), a professor of algebra at the Université libre de Bruxelles [Free University of Brussels], was contacted (Papy 1968). Papy was a promising research mathematician who had not yet shown a strong interest in educational problems. However, he was not completely unknown among mathematics teachers. Papy had published in Mathematica & Paedagogia (a primarily mathematical article on the scalar product in which he argued that it would be advantageous to introduce some of such “unifying concepts” into secondary school mathematics; Papy 1954–1955), and he had also intervened in debates at the 1956 SBPM annual conference (with a position in favor of teaching ­concepts of modern algebra at the secondary level; Boigelot et al. 1956–1957). It is also worth mentioning that about that time Papy was assigned as secretary of a newly formed committee at his university that was to deal with the teaching of basic mathematics (Le Soir, April 27, 1958, pp. 1–2). Papy responded positively to Lenger and Servais’ request for mathematical help in future experiments. However, Papy did not confine himself to providing technical advice, but immediately took charge of the project. In September 1959, he started his own experiment in the École “Berkendael,” a school for kindergarten teachers in Brussels, and expanded his experimental actions year after year. In this, “Berkendael” period, Lenger became Papy’s partner, both professionally and personally (they married on October 1, 1960). From then on, Georges and Frédérique Papy (or Frédérique PapyLenger) would form a complementary team “driven by a shared vision and commitment that would guide the movement for more than a decade” (Noël 1993, p. 56). The experiments of the Papys would eventually lead to a generalized introduction of modern mathematics in Belgian secondary schools (starting in September 1968), and about a decade later, in Belgian primary schools.

A Ten-Year Experimental Trajectory at the Secondary Level Papy’s first classroom experiments were built on Lenger and Lepropre’s work. During the 1959– 1960 and 1960–1961 school years, Papy himself taught two experimental classes to future kindergarten teachers, 15–16-year-olds, in the École “Berkendael.” It was Papy’s first attempt at teaching of mathematics at the secondary level (to students who were not particularly gifted in mathematics). For a research mathematician, this must undoubtedly have been a culture shock. At first sight, Papy’s “Berkendael” course (Papy 1960) looked like a tough university course for future mathematicians, rather than a textbook for future kindergarten teachers. Papy built up his discourse from sets and relations, concepts he illustrated with simple and varied examples from elementary mathematics and from daily life (some likely generated by the students themselves). However, the emphasis soon shifted from these “concrete” examples to the basic definitions and principles, the precise terminology, and the symbolic language of set theory which served as a thinking tool and unifying element throughout the whole course, in particular, for an introduction to geometry, arithmetic, and topology. Structures of order and equivalence were revealed and emphasized, and from the very beginning, Papy promoted rigor and abstraction. Also, logical-deductive reasoning and proving were essential ingredients of Papy’s structuralist approach, activities for which the students could rely on the representational tools of Venn and arrow diagrams. Papy deliberately left little room for intuition (in its common sense) which evidently raised the course difficulty level. However, Papy’s structural and abstract view on mathematics and his tendency to detach mathematical entities from concrete, intuitive objects were embedded in a pedagogical approach that proved to be very effective (Randour 2003). He was a master in interacting with the students, bringing them

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Figure 10.1  Papy experimenting with modern mathematics, early 1960s (left, at the blackboard a student proves the distributive law for union over intersection of sets; right, Christine Manet, one of his students, explains that the composition of relations is not commutative. (Hunebelle 1963)

step by step and without effort closer to correct mathematical conclusions. Furthermore, in his “Berkendael” experiments, Papy introduced a pedagogical innovation that would become his trademark, namely multicolored graphs and various other enlightening visualizations, all strongly appealing to the aesthetic and affective side of learning mathematics (Figure 10.1). Along with Papy’s (and Lenger’s) talents as teachers, this attractive and refreshing pedagogical approach undoubtedly contributed to the success of the experiment. Papy’s merit is not so much in the content […] but in the teaching methods. Professor Papy and his wife show genuine talent. (Robert Baillieu, professor of mathematics at the Catholic University of Leuven, quoted by Stievenart 1968, pp. 18–19)

Apparently, if packaged in an appealing pedagogical approach, the abstract tenor of the mathematics in the spirit of Bourbaki (1939) was no obstacle for the future kindergarten teachers of “Berkendael.” On the contrary, according to the teacher–mathematician in charge, they proved to be very receptive to this kind of advanced mathematics. “Papy judges the results fully satisfactory. […] The supreme logic of higher mathematics is directly assimilated by any moderately gifted mind” (De Latil 1960, p. 543). Although Papy evaluated his experiment with 15–16-year-old future kindergarten teachers as successful, he considered it necessary to start the reform efforts from an earlier age. In May 1961, he published his Suggestions pour un nouveau programme de mathématique dans la classe de sixième [Suggestions for a new mathematics curriculum in the first year of secondary schools] (Papy 1961), based on his experiences in “Berkendael.” In this curriculum, Papy proposed the theory of sets as the starting point for the teaching of mathematics from the age of 12. While the space of Euclid could for a long time serve as the framework for a unified presentation of basic mathematics, it can no longer today, but its role can now be fulfilled by the universe of sets. Moreover, as it has been proved by experiments carried out in America, England, Russia, Poland and in our country, the teaching of the basic notions of set theory fascinates young students. It therefore seems sensible to propose that topic as starting point in secondary education. (Papy 1961, p. 21)

In addition to the language of sets and relations, Papy’s curriculum proposal for the first year of secondary schools included the ring of integers, the binary and decimal numeration system, and an initiation to affine plane geometry.

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Thanks to a positive recommendation by Henri Levarlet, general director of secondary education at the Ministry of Education, Papy’s curriculum proposal was approved with an experimental status and permission was granted to continue and expand the experimental trajectory (Papy 1968). Based on his experimental curriculum, Papy started a new experiment from September 1961 in the first years of 12 schools (representing some 30 classes) for general secondary education (12–13-year-olds) (and from then on gradually in the two subsequent years). For the extension of the experiment to the second and the third years (13–15-year-olds), Papy developed a new experimental curriculum (Papy 1962), including, for the second year, the ordered field of real numbers and the real vector plane and, for the third year, the Euclidean vector plane and some elements of “classical” algebra (the equation of a straight line, the square root, functions of one real variable, polynomials, and the solution of systems of linear equations). Soon the experiment was expanded to several dozens of schools all over Belgium. Papy’s experiments at the secondary school level resulted in his revolutionary textbook series entitled Mathématique Moderne (1963–1967), which we will discuss in the next section. In order to coordinate the experimental trajectory and related initiatives, Papy had founded on May 24, 1961, the Centre Belge de Pédagogie de la Mathématique (CBPM) [Belgian Centre for Mathematics Pedagogy] of which he also became the chairman. It brought together a number of reform enthusiasts, both from universities and from secondary education. The Centre’s goal was formulated in its Articles as “the study, the improvement and the reform of mathematics teaching. In particular, it will contribute to the promotion, the development and the diffusion of the teaching of modern mathematics” (Papy and Holvoet 1968, p. 133). This goal was realized by the development of experimental curricula, new textbooks, and teachers’ courses, the organization of large-scale actions of teacher re-education (which will be discussed in a separate section), and by continuing the experimental trajectory. From 1968 onward, the CBPM also published its own journal Nico, a clear reference to Nicolas Bourbaki. During the 1964–1965 school year, when the students who started the experiment in 1961 arrived in their fourth year of secondary school, the experiments were extended to the upper grades (15–18-­year-olds), first in the scientific study streams and in 1967 also in the “non-scientific” streams. Core themes for the upper secondary level were linear algebra, mathematical analysis (founded on topology), and higher arithmetic (Papy 1968). According to Holvoet (1971), statistics and probability theory were also included (but we have not found any trace of that in the documents of the CBPM from the 1960s). Unfortunately, a detailed experimental curriculum was published only for the fourth year in the scientific streams, (CBPM 1966). This experimental curriculum was a mixture of new elements (combinatorial analysis, whole-number arithmetic), traditional elements (such as, for example, approximate calculations and quadratic equations), and repetitions and extensions of subjects that already had been taught in previous phases of the experiment (real vector spaces and the Euclidean vector plane). As the generalized introduction of modern mathematics in the first years of secondary schools approached, and the pressure was mounting to start with the first year, the completion of the experimental efforts for the upper classes seemed to have lost its direction and vigour. While the programs for the lower grades (12–15-year-olds) had been extensively tested in experimental classes, it has, unfortunately, not been the case for those for the upper grades. Only one of the classes involved in the experiment in 1961 received, under Frédérique Papy’s direction, an experimental program throughout the six years of secondary school. That is not much to draw conclusions from. (Noël 1993, pp. 59–60)

Mathématique Moderne In 1963, Papy started with the publication of Mathématique Moderne (in collaboration with his wife Frédérique), a textbook series revolutionary in content and layout, based on his previous classroom experimentation and intended for the teaching of modern mathematics to students from 12 to 18

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Figure 10.2  Covers of Papy’s Mathématique Moderne

(Figure 10.2). The series clearly shows how Papy reshaped mathematics education, both in terms of content and didactics. In the first volume of the series (Papy 1963), Papy introduced the language of sets and relations, represented by Venn and arrow diagrams, respectively. These diagrams were intensively used for concept development, reasoning, and proof. The algebra of sets receives ample attention, not only because of its intrinsic value and interesting “new” applications but also because this algebra differed in several aspects from the usual “algebra of numbers,” and thus could contribute to a better understanding of the latter. The symmetric difference of sets provided the first example of a group structure. In a geometric track, the plane was introduced as “an infinite set of points” and straight lines were introduced as subsets of the plane, whose mutual positions were explained with Venn diagrams. Papy paid considerable attention to proof and logical-deductive reasoning. A few initial propositions were selected as axioms, from which some simple and intuitively clear properties of parallelism and perpendicularity were proved. According to Papy, self-evident properties are particularly suited for learning to reason correctly and for understanding the essence of proof. Papy also included some basic topological notions—he differentiated between an open and a closed “disk” and a circle (which only includes the “perimeter”). To visualize these notions, the red-­ green “traffic-light” convention (for parts that were ex/included) was introduced. New concepts, such as relations of order and equivalence, were commonly introduced with simple and familiar situations that encouraged the student “to take an active part in building the mathematical edifice” (p. vi). The geometric track also included an introduction to transformation and vector geometry. Translations or vectors were defined as classes of equipollent couples of points, the set of which forms a group under composition. In this section, Papy introduced the didactical tool of proof by film fragments: A sequence of suggestive images, from which a line of thought could be seen, was presented and students were asked to add justifications. The assimilation of a proof involves several stages which we should try to keep separate. The first step is for the student to understand the film so that he can explain it in informal language. Next he must be able to reconstruct the argument himself. After this comes the stage where more formal justifications are required. Only after all this do we turn our attention to the proper setting out of the proof. (p. viii)

Regarding algebra, Papy first anchored students’ pre-knowledge about numbers and their operations in a set-theoretical framework. Natural numbers were defined as cardinal numbers of finite sets and the addition and multiplication of such numbers were related to, respectively, the union and Cartesian product of sets. The positional notation of numbers was revisited by studying the binary system. To introduce integers and their addition, Papy proposed a combat game with red and blue counters, representing oppositely signed numbers that “kill” each other when coming in the same compartment.

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Properties of the operations with integers were strongly emphasized and led to the discovery of a group and ring structure. The first volume of Mathématique Moderne concluded with a chapter on (abstract) groups, bringing together and systematizing several “concrete” examples from the previous chapters. In the second volume of the series (Papy 1965), the field of real numbers was constructed, in a mathematically rigorous way. Papy’s starting point was a process of binary graduation of a straight line. By inserting the axioms of Archimedes and continuity, he established a one-to-one correspondence between the points on a line and the set of numbers, represented by terminating or non-­ terminating binaries, at a certain moment called “real numbers.” Then, the order and additive structure of the points (vectors) on that line were transferred to the set of real numbers. For the multiplicative structure, Papy first defined the multiplication of real numbers by means of homotheties (= homothetic mappings) and then deduced the basic properties of multiplication from the corresponding properties of the composition of homotheties. The ordered field of real numbers showed up as the ultimate reward. The rational numbers were defined after the real numbers and their structure appeared to be an ordered subfield of that of the real numbers. The real vector plane served as an example of the general concept of vector space and as a basis for affine analytic plane geometry. In Euclid Now (Papy 1967a)—the third volume in the series—the axiomatic-deductive building up of plane geometry was continued and finally resulted in a contemporary vector-based exposition of Euclidean (metric) geometry for 14–15-year-old students. Euclid’s Elements exposed the basic mathematics of his time, about 300 years before J.-C. The monumental work of Nicolas Bourbaki presents, at the highest level, the basic mathematics of today. The “MMs” [= Papy’s textbook series] want to expose the Elements of today’s basic mathematics for adolescents … and people of any age and schooling who wish to initiate themselves in the mathematics of our time. (p. vii)

Transformations and groups which were generated by these transformations played a key role in Papy’s construction of (Euclidean) geometry. Isometries were defined via the composition of a finite number of (perpendicular) line reflections. The different types—translations, rotations, reflections, and glide reflections—and their possible compositions received considerable attention. Colorful classification schemes based on Venn diagrams were presented and group structures were highlighted. Each time a group was discovered, it provoked an Aha-Erlebnis: When a student recognized a known abstract structure in a new setting, it was hoped that he or she might be able to apply all previously learned knowledge and skills about this structure to that setting. Over the past half century, mathematics has switched from the artisanal stage to the industrial stage. The machine tools of our factories made it possible to save human muscular effort. The great structures of contemporary mathematics allow to save the human mind. (p. vii)

Transformation approaches were also promoted as an alternative to traditional methods in school geometry as it was claimed that such approaches were much more intuitive and universal. The outdated artisanal technique based on congruence of triangles must be abandoned in favor of translations, rotations, and reflections, which are much more intuitive and whose scope goes far beyond the framework of elementary geometry alone. (p. ix)

Once the group of isometries was established, the fundamental concepts of Euclidean (metric) geometry could be introduced. The distance between a pair of points and the length of a line segment were defined by means of isometries (and from then on, isometries gained their etymological meaning of “length preserving transformations”). Definitions of the norm of a vector and the scalar product of two vectors followed. The “natural structure” for Euclidean geometry—a vector space equipped with a scalar product—was thereby created. Classical results, such as the Pythagorean theorem could be proved easily within that structure. Certain statements, once fundamental, are reduced to the rank of simple corollaries. That they now stop cluttering up the memory of our students. If necessary, they would be able to retrieve these results by routine use of one of the machine tools of modern mathematics. (p. ix)

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The fourth volume in the series remained unpublished. The fifth volume (Papy 1966) presented an introduction to combinatorial analysis and higher arithmetic, based on the theory and language of sets and relations. In the sixth volume (Papy 1967b), Papy first retraced, in brief, the laborious path from the original “intuitive” (synthetic) axioms of geometry to the establishment of a Euclidean vector plane structure, the path that the students had followed from the age of 12 to 15. This summary was intended to prepare these students for the second step which Papy described as a psychological reversal: The structure of a Euclidean vector plane was taken as a new and “unique” starting axiom for the further development of plane geometry. This approach opened the way for the future study of higher-­ dimensional Euclidean spaces, in particular for building up solid geometry. At the end of the book complex numbers were introduced as direct similarities. By relying on the structure of the latter and isomorphism, it was proved that complex numbers form a field, extending the field of real numbers. Volumes of Mathématique Moderne were translated into several languages, including Danish, Dutch, English, German, Italian, Japanese, Romanian, and Spanish. To the best of our knowledge, these translations served mainly as a source of information for teachers and all those who participated in the reform debate in other countries (“You can accept or reject Papy’s choices for mathematics education, but in any case you can’t ignore them,” Campedelli and Giannarelli 1972, p. v). However, it is difficult to overestimate the impact of Papy and his Mathématique Moderne on the international mathematics education scene during the 1960s (De Bock and Vanpaemel 2019, Chap. 6). Papy acted as an uncompromising modern mathematics ambassador at major international conferences of that period and defended, with verve and authority, his views on the modernization of mathematics teaching. Already at the 1963 OECD conference in Athens, Papy presented an extended sneak preview of the mathematical content and methodological approach of the first two volumes of Mathématique Moderne (Papy 1964). Papy’s design of teaching modern mathematics was well received by the other OECD experts: The example given by […] Mr. Papy were stimulating as to what can be accomplished by a proper blend of modern mathematical ideas with very conscious psychological methods of presentation. When students are directed toward the discovery of mathematical patterns and the self-construction of mathematical entities (such as the real numbers), motivation and permanency of learning are greatly enhanced. (OECD 1964, p. 296)

Large-Scale Recycling of Teachers: The Days of Arlon The content, approach, and results of the experiments of Papy and his team were largely disseminated among Belgian mathematics teachers. Several initiatives were taken by different actors, but the Journées d’Arlon [Days of Arlon] undoubtedly had the greatest impact. In July 1959, instigated by Lenger, the SBPM organized, in the Belgian city of Arlon, an intensive teacher training course on set theory and the principles of topology (Papy 1959). The course, which lasted 3 days and was attended by about 150 participants, was the first edition of the Arlon days, a series of annual in-service teacher training courses—at that time usually called “recycling courses”—aimed at introducing Belgian teachers to modern mathematics, both in terms of content and didactics. The second edition of the Arlon days (1960), organized by the Belgian Ministry of National Education and Culture, was devoted to the study of relations and graphs—two core ingredients of Papy’s “Berkendael” course which was distributed among the participants. From 1961 to 1968, the CBPM took charge of the practical organization of the Arlon days, in collaboration with the Ministry which supported the initiative financially and morally. Already in a 1962 circular, Victor Larock, socialist Minister of Education of Belgium at that time, drew teachers’ attention to the Arlon days and to other in-service training courses. Their purpose, he said, was to expand their knowledge of modern mathematics, to reflect on the educational problems raised by the teaching of contemporary mathematics, to convince themselves of their extreme importance and to collaborate for finding solutions. (Larock 1962, p. 8)

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Henri Janne, the successor of Larock and also a French-speaking socialist and a former rector of the Université Libre de Bruxelles, pledged his full support for the reform activities of his political “friend” Papy.1 In his opening address to the Arlon days of 1964, of which he had accepted the presidency, Janne formulated his endorsements in the following way: I would like to congratulate the promoters, and especially pay tribute to the effort that has been done for many years by Professor Papy who, in addition to his activity as a scientific creator, has made himself an apostle—the word will not shock him—of the current reform. His effort has an incontestable international influence, and I believe, is currently undisputed. … The CBPM … did an extraordinary effort to spread the new mathematics and a pedagogy of its teaching. (Janne 1964, pp. 9–10)

For this and other more intrinsic reasons, the Arlon days, held each year at the beginning of the summer holydays, became more and more successful, with about 600 participants attending at its peak. Most of those who attended were Belgian mathematics teachers, but there were also some university professors, inspectors, political officials, and foreign guests. From the third to the tenth edition the following themes were programmed: Groups (1961), vector spaces (1962), exterior algebra and determinants (1963), new paths in the teaching of analysis (1964), the Euclidean vector plane (1965), the teaching of mathematical analysis in the second year of the scientific stream (1966), the teaching of integral calculus in the first year of the scientific stream (1967), and the position of calculation in a modern teaching of mathematics (1968). The insights and materials of the Arlon days were further disseminated by working groups which were coordinated by the CBPM and locally led by benevolent instructors. These working groups, more than 20 in total, were active in all main Belgian cities and reached yearly up to 3000 teachers (Holvoet 1968). In relative terms, however, this was still a very small minority of the Belgian teaching staff for mathematics (Adé 1973–1974). The days of Arlon had great impact, even outside Belgium. In an interview from the 1980s, Piet Vredenduin, a prominent mathematics educator from the Netherlands who participated to the courses as a foreign guest, looked back: I have learned a lot in Belgium. They were ahead of us. Every year Papy organized a weeklong course on modern subjects in Arlon. These were excellent. (Goffree 1985, p. 163)

For Belgian mathematics teachers the Arlon days were not just one of the many in-service training courses focusing on new mathematics and its didactics. The days have been described as an exciting experience, connecting many people who felt themselves being part of a big and ambitious project, across the linguistic and ideological boundaries which were still strongly present in Belgian society. At that time (and still now) most schools in Belgium belong to one of two mighty educational networks, one representing the “free” (usually Catholic) schools, and the other, uniting the publicly-run schools. Of course, mobilizing actors of these two educational networks and of the two main linguistic Belgian communities (the French- and the Dutch-speaking) had a strategic-political dimension—Papy needed all these actors’ and their organizations’ support for his reform to succeed—but as Vanhamme (1991) testified, the unifying power of mathematics was also one of Papy’s profound convictions.



Implementation and Controversy

During the mid-1960s the reform movement was in a winning mood. In the school year 1963– 1964, a working group of the CBPM had developed an improved version of the experimental programs from 1961 (and 1962) for the lower secondary level (CBPM 1964), the structure of which was of course very similar to that of Papy’s Mathématique Moderne. At that time, the original programs for respectively the first, second, and third years were already run in about 100, 20, and 5 secondary school classes, respectively. In his opening lecture to the sixth edition of the Arlon days (1964),  Between 1963 and 1964, Papy himself had been a member of the Belgian Senate for the Socialist Party.

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Minister Janne had officially allowed the improved modern mathematics program as an alternative for the traditional mathematics curriculum for 12–15-year-olds: This program has the great advantage of being fully taught and of taking into account the experiments that have already been realized … In view of the quality of this working group and of the evidence provided by the previous experiments, I decided to authorize this program, of course on an experimental and optional basis, as early as the next school year [1964–1965]. (Janne 1964, p. 11)

The decision to make the optional program compulsory in the first years of secondary education from September 1, 1968, was made by Janne as Minister of Education in an outgoing government and announced in a circular of May 14, 1965, which also urged teachers to prepare (Janne 1965). However, the success story of Papy and his CBPM during the 1960s did not mean that all members of the Belgian mathematics education community were in favor of the ongoing reform. Opposition came, for example, from Mathématique et Technique (MATEC) [Mathematics and Technique], an organization of mathematics teachers in technical schools, who deplored the loss of geometrical representations and the emphasis on logic and abstract concepts. Papy’s curriculum proposals isolated mathematics from other courses such as technical drawing, for which understanding of spatial forms and representations was required. At the academic level, the opposition to Papy’s reform was led by Léon Derwidué (1914–1971), a professor at the Faculty of Engineering of the University of Mons. Derwidué rejected what he believed to be a one-sided emphasis on the axiomatic, logical, and structural aspects of mathematics, and pleaded for a renewal that took into account the needs of the engineers and other users of mathematics, whose advice had been disregarded in the reforms (Derwidué 1962). Moreover, Derwidué did not see any good argument for teaching set theory to young children, a theory that “serves first and foremost to provide a logical, solid, and precise basis for reasoning, a role that can only be appreciated by sufficiently advanced minds” (Derwidué 1962, p. 6). He doubted whether this theory—as well as other modern mathematics content—could be used for the exposition of many classical topics whose knowledge he still considered essential (from the point of view of the users of mathematics). And even for students who would later devote themselves to pure mathematics, Derwidué was not convinced that learning mathematics on the basis of a Bourbaki-style presentation provided a good starting point: Moreover, is it not appropriate to start the mathematical education of every teenager, even those predestined to the purest mathematics, with the useful and concrete aspect, which, in its further development, will naturally reveal problems for which rigorous treatment appears necessary? (Derwidué 1962, p. 10)

Opponents of the reform, however, had little chance in the mid-1960s. Papy, backed by a loyal and powerful base in politics and academia, succeeded in vigorously defending “his” modern mathematics, both in academic and more popular forums, and did not hesitate to ridicule his opponents’ views. Smet and Vannecke (2002) described how Papy, at a symposium under the slogan “Ahead with the reform,” organized on December 1, 1966, in the Brussels Palace of Congresses, in front of 1700 participants, vociferously denounced traditional mathematics as obsolete and worthless. Some compared Papy’s discourse with that expected at a political meeting, but he did not convince everyone, even though his experiments and other actions received ample attention in newspapers and popular magazines of the time. By labelling his approach as “mathématique moderne,” he already condemned any opponent as being outdated or even reactionary. It also reduced the debate to a dilemma, as formulated by Papy in one of his famous one-liners: “la mathématique de Papy ou les mathématiques de papa?” [Papy’s mathematics or daddy’s mathematics?]2 (Mawhin 2004). Papy divided the Belgian (and parts of the international) mathematics education community: He left no one neutral—he created dedicated followers as well as fierce opponents—“Papy’ists” and “anti-Papy’ists” (Colot 1969).  In French “la mathématique,” singular, refers to the unified (modern) mathematics, in contrast to “les mathématiques,” plural, referring to (traditional) mathematics as an umbrella term for several subdisciplines with little or no interrelationships. 2

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When the due date of the reform approached, a drastic change in the political climate seemed to reverse Papy’s chances. In March 1966, a new government was formed and the liberal Frans Grootjans became Minister of National Education. Papy’s (socialist) political family, to which also the aforementioned Ministers Larock and Janne belonged, did not participate in this new government. Grootjans then had to take the ultimate decision about the mandatory introduction of modern mathematics, as announced by Janne in 1965. Soon, some doubts about the wisdom of going ahead with the decision arose. In a short communication “Modern mathematics for a time not in education” (De Standaard, December 16, 1966, p. 8), Grootjans took a reserved position. The Minister was probably influenced by a group of representatives from the Faculties of Science and Engineering. The Faculties of Science, which organized the studies in (higher) mathematics, were divided on the issue of modern mathematics in secondary schools—Brussels and Leuven were in favor, Ghent and Liège were against—but the Faculties of Engineering were also involved, as “users of mathematics,” and they were unanimously against. Not surprisingly, the CBPM immediately expressed concern about a possible suspension of Janne’s earlier commitment. On January 4, 1967, Grootjans clarified his position: You can be assured that I will not take any decision without being informed by all the authorities responsible for the teaching and application of mathematics. Only after I will be in possession of all advice on this matter, will I announce my position without prejudice. (Mathematica & Paedagogia, 31, 1967, p. 76)

To obtain the desired advice, Grootjans installed two national study commissions: A University Commission and a Commission for Secondary Education. The University Commission, installed on March 20, 1967, firstly had to advise the Minister about the necessity to change the current mathematics curriculum of the scientific streams to meet the needs of the university, higher education, and trade and industry (and was thus composed of representatives of these sectors,3 but complemented with two CBPM members including Frédérique Papy). Secondly, this commission had to enumerate in detail the crucial mathematical knowledge that should be provided by the secondary level, without developing a specific curriculum. Despite strong disagreements and tensions—the delegation from the University of Ghent withdrew its cooperation after the first meeting—the University Commission reached a compromise in favor of reform on September 12, 1967 (Feusels 1979). Although the Commission’s recommendations and proposals had a marked modern mathematics signature, essentially reflecting the views of Papy and his team and not those of the University of Liège, some positions were more moderate (Commission Universitaire 1967). For example, a number of topics that Papy had considered outdated or useless, such as common plane and solid figures and their properties, relationships between sides and angles in a right-angled triangle, trigonometric formulas, spherical triangles, combinatorial analysis with and without repetition, estimation of numerical expressions, and error propagation, were nevertheless recognized as essential. The Commission for Secondary Education, consisting of inspectors and informed teachers at the secondary level, was then asked to elaborate in detail the final curriculum, taking into account not only the University Commission’s advice, but also the traditional curriculum, and the results of the experiment with the optional (experimental) curriculum. The Commission for Secondary Education soon agreed on a new curriculum for the first year of the secondary school. Although Georges Papy was not personally involved in its preparation, the Commission generally followed his view—both in terms of content and method—and thus, the new curriculum strongly resembled the CBPM’s experimental curriculum (CBPM 1964). It consisted of six main sections: Sets, relations, natural numbers, integers, geometry, and “acquisitions to maintain.” Only in the last section, in which some basic arithmetical and geometrical knowledge and skills from the primary school were repeated, applied, and expanded in a more or less intuitive way, was there any concession to the proponents of the classical approach. On April 11, 1968, a ministerial decision con The universities appointed their own delegation, each consisting of one representative of the Faculty of Science and one of the Faculty of Engineering Science. 3

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firmed the generalization of this modern curriculum, from September, 1968, in the first year of the general divisions of all secondary schools run by the state (Ministerie van Nationale Opvoeding/ Ministère de l’Éducation Nationale 1968). The Catholic network implemented the reform in general secondary education at the same moment, but with a slightly different program (Nationaal Verbond van het Katholiek Middelbaar Onderwijs 1968). Technical schools, which had not previously experimented with modern mathematics, started 1  year later in a similar vein (Ministère de l’Éducation Nationale 1969).



Modern Mathematics in Belgian Primary Schools

A Reform Prepared in Various Experiments In September 1967 the CBPM started the experiment Frédérique, which was aimed at preparing the reform at the primary level (Papy 1970, 1971). The experiment started with a class of 6–7-yearolds in the primary section of the École “Berkendael.” Frédérique set out the general objective as follows: In the attempt to renew the teaching of mathematics at the primary level that I have undertaken since September 1967, one of my main objectives is to build, with the help of children, a house of mathematics. … For the student, the unitary structure offers security and comfort, essential elements of a climate that favors the development of intelligence and knowledge in mathematics. (Papy 1971, p. 160)

More specifically, the experiment aimed at introducing children to the relational world of modern mathematics, as well as initiating them, progressively, to calculation techniques in “sets of ever richer types of numbers” (Papy 1970, p. 95). Tools to realize these goals were, in addition to Venn diagrams and arrow graphs, Cuisenaire rods, Dienes logiblocs, and the minicomputer. The minicomputer was not a computer or calculator, but a new teaching aid developed by Papy. It was a two-dimensional abacus with plates that were subdivided into four square sections, each ­colored according to the coding system of the Cuisenaire rods (Figure 10.3). In these plates, numbers were represented in a binary way by counters that could be played up and down, corresponding to the operations of doubling and halving (Papy 1969). Although the minicomputer was primarily based on the binary number representation, it also could be used for base 10 representation of numbers (by putting different plates, each representing a digit, next to each other). The method we used to introduce the 6-year old child to mechanical or mental numerical computation uses the decisive advantages of the binary over any other positional numeral system, while taking into account the decimal context in which we are housed. (Papy 1969, p. 333)

In the early 1970s, Frédérique published annotated accounts of her experimental classes in a four-­ volume book series entitled Les Enfants et la Mathématique [Children and Mathematics] (Frédérique 1970–1976). The series’ style—with many colorful figures intended to elicit a mathematical idea or line of thought—resembled that of Papy’s Mathématique Moderne, but the impact was much smaller partly because Frédérique’s approach was almost not reproducible by “ordinary” teachers (“Frédérique’s great didactic talents enable her to achieve results with young children that will be unattainable for many,” Vredenduin 1975–1976, p.  165). Nevertheless, some modern mathematics enthusiast circles outside Belgium showed interest—such as, for example, Burt Kaufman’s team in the USA which developed the Comprehensive School Mathematics Program and appointed Frédérique as director of research, a role that she fulfilled between 1973 and 1978 (Braunfeld 1973). In September 1968, a second and larger-scale modern mathematics experiment started— namely the Waterloo experiment, conducted in the preparatory section of the Athénée Royal de Waterloo and led by Louis Jeronnez (Jeronnez and Lejeune 1972a, b).

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Figure 10.3  Children aged 6 at work with Papy’s minicomputer. (International Visual Aid Center, Brussels, late 1960s) The fundamental goal of our Waterloo experiment is to promote education that can better shape students’ thinking, that promotes the spirit of personal research, that encourages children to think rather than to master techniques. (Jeronnez and Lejeune 1972a, p. 69)

Although the experiment was mainly focused on mathematical reflection, a lot of attention was paid to numbers, operations, and arithmetical skills that were deliberately trained. In this respect the Waterloo experiment distinguished itself from Frédérique’s approach. Students’ arithmetical skills were developed and individually supported through the manipulation of the Cuisenaire rods, which played a central role in the Waterloo experiment. In both Frédérique’s and in Jeronnez’s experiments, attention was given to the discovery of mathematical structures at an early stage. The Waterloo experiment was highly regarded in the French-speaking Community of Belgium: To Papy’s dismay, the entire in-service training of primary school teachers in state-run schools was entrusted to the Waterloo group (Papy 1979). The experimental efforts in the French-speaking Community of Belgium resulted in new curricula in the early 1970s. The main components were logic, sets and operations on sets, relations, structures, numbers, operations on numbers and their properties, measurement, an introduction to geometry, and word problems. The general aim was to find a balance between mathematical reasoning and the development of arithmetical skills in a renewed framework of sets and relations (Jeronnez and Lejeune 1972a). In Flanders, the Dutch-speaking Community of Belgium, a series of comparable experiments led to a reform of the mathematics curricula in the second half of the 1970s. Criticism of modern mathematics in primary education, and of its implementation, did not surface until the 1980s (e.g. Feys 1982).

Modern Mathematics in Daily Primary School Practice The compulsory introduction of modern mathematics in primary schools in the 1970s thoroughly reshaped both the content and the didactics of the discipline at that level. This impressive operation was accompanied by the publication of new textbooks and the organization of various kinds of in-­ service training courses for teachers and even for parents. We discuss some major changes in primary school practice resulting from the reform. Sets and relations became the main ingredients of the new approach to mathematics education, not only as learning objectives in their own right but especially

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also as a means to frame most of the “traditional” mathematical contents (which the children were still expected to learn). Knowledge of numbers and arithmetic preserved their importance, but attention shifted from being able to calculate quickly and accurately, and to perform standard procedures (such as the “rule of three”) to insight into number systems, operations, and structures. To promote thinking and understanding, numerical situations and operations were often represented with different tools (Venn diagrams, arrow graphs, Cuisenaire rods). To gain a better understanding of the decimal system, some addition and subtraction problems in systems with number bases other than 10 were proposed. Word problems received less attention (but were still included in the curricula). Most reformers looked down on “applications of shop, garden, and kitchen mathematics” (Barbry 1974, p. 121). To visualize problems from everyday life, children had to use the appropriate tools of modern mathematics (such as schemes based on arrow or Venn diagrams). Children were not encouraged to make their own informal visualizations of a problem situation. Probably the most radical change took place in the teaching of geometry. The plane, represented by the symbol Π, became an “infinite set of points” and lines and geometrical figures became “subsets of Π.” In particular, the hierarchical order of the different plane figures was considered to be essential. Relations, such as “all rectangles are parallelograms,” were highlighted and visualized in the language of sets. In the “exploration of space”—a curriculum component for 10–12-year-olds—this trend was continued with the classification of polyhedra according to diverse criteria. Solving applied problems about geometrical figures was considered less important. Besides, the correct use of an unequivocal terminology and symbol use was considered to be of utmost importance. Therefore, inaccuracies from the pre-modern mathematics programs were eliminated. For instance, a clear distinction was made between a “circle” and a “disk.” A circle only referred to the border of the plane figure, and thus its area was no longer π r2 but 0. The course in geometry also provided an introduction to transformation geometry. New topics, such as “reflection through an axis” and “axes of symmetry,” had to prepare children to an extensive study of transformation geometry at the secondary level. From the age of 10 onward, children were introduced to what was called logical thinking. In this special part of the mathematics course, they were expected to learn to use correctly the connectives “and” and “or” (“and/or”) and their negation by the logical operator “not.” In a next phase, children were also trained in the correct use of expressions such as “at least,” “at most,” “not all,” “only if,” “if and only if” and so on. Dienes logiblocs (a set of objects with restricted and well-defined features: rectangle, triangle, or disk; yellow, blue, or red; small or large, and thick or thin), with which all kinds of sorting and classification activities (“logical games”) were devised, were a popular teaching material for promoting logical thinking: These blocks are used systematically all year round. This material is just fantastic. The little children can play with them as much as they want and structure little by little. The child gains a lot of experiences because he or she constantly discovers new aspects. It is exactly that self-discovery aspect that I have learned to appreciate in this new approach. (Mogensen 1970–1971, p. 241)

Although the Belgian curricula for primary mathematics were seriously affected by modern mathematics, it is unclear how drastically day-to-day teaching practices for mathematics were actually affected by it. It is apparent that computational and measurement techniques as well as word problem solving—key parts of the “old curriculum”—were not dropped by primary school teachers during the period of modern mathematics (especially not in Flanders and in the Catholic network where the influence of Papy and his collaborators tended to be less strong) (see, e.g. Verschaffel 2004). These skills were still considered important, although it was less evident that they needed to be integrated into the philosophy of modern mathematics.

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Discussion In the late 1950s, the Belgian modern mathematics movement found its leader in the strong personality of Georges Papy, professor of algebra at the Brussels University. Papy designed and carried out audacious experiments, developed new curricula and teaching materials, and engaged teachers through large-scale in-service education programs. Papy’s actions were coordinated by the Belgian Centre for Mathematics Pedagogy, which had been founded in 1961, and received ample attention in the international mathematics education community. With the founding of his Centre, Papy was about 10 years ahead of similar institutes that were created in other Western European countries, such as France (IREMs), Germany (IDM), and the Netherlands (IOWO). However, although “study” of mathematics teaching was explicitly mentioned in the Centre’s Articles, fundamental psychologically oriented research was not Papy’s trademark: Papy wanted to move quickly to improve the teaching of (modern) mathematics. In 1963 Papy published the first volume of the groundbreaking textbook series Mathématique Moderne, based on his experimental trajectory and intended for the teaching of modern mathematics to 12–18-year-olds. Inspired by the work of Bourbaki, Papy reshaped the content of secondary school mathematics by basing it on the unifying themes of sets, relations, and algebraic structures. Meanwhile, he proposed an innovative pedagogy using multicolored arrow graphs, playful drawings, and non-­ verbal proofs by means of film strips. In contrast to other influential textbooks of the time, such as those produced by the School Mathematics Study Group in the USA or the School Mathematics Project in England, Papy’s textbooks were only used for teaching in experimental classes. When from 1968 to 1969 onward modern mathematics was made compulsory in Belgian secondary schools, the official programs were different and less ambitious than those developed by Papy and his team. Likely, this is a main reason why the series remained incomplete; in particular, it did not provide the necessary material for teaching modern mathematics at the upper secondary level. The series produced by Papy served as a major source of inspiration, both in terms of content and style, for mathematics educators and textbook developers during the 1960s and early 1970s, the period in which the modern mathematics reform was prepared and implemented in several countries. The implementation of modern mathematics at the secondary level was preceded by a process of about 10 years of experimentation. The experiments, which basically examined whether a new subject, a particular curriculum, or a specific approach was feasible at a certain age with certain students, were always considered successful by those in charge. There have never been thorough, comparative evaluations of the extent to which specific educational goals were met. The final introduction of the reform at the secondary level did not happen in a serene atmosphere; in primary education, the controversy was less strong or even non-existent. For more than 20 years, modern mathematics was the dominant paradigm for the teaching and learning of mathematics in Belgium. Proper notations and symbols, the use of the right jargon, and theory development received increased attention, barriers between mathematical subdomains were largely eliminated, and geometry education was redirected toward transformation and vector geometry. During the 1960s and 1970s, the vast majority of teachers and educators in Belgium expressed little or no criticism of modern mathematics. Although not necessarily inclined to reform, most remained silent; critics could not count on much support anyway. In the 1980s, partly influenced by international developments, criticism swelled, both with respect to the Bourbaki ideology for teaching mathematics and with respect to the way modern mathematics was implemented in Belgian schools. The criticisms of the early 1980s sounded loudest and sharpest at the primary level (e.g. Feys 1982), where modern mathematics was introduced last and probably least thoughtfully. It paved the way for the collapse of modern mathematics in Belgium, both in primary and secondary education, for a “reform of the reform,” and for the emergence of new visions on teaching and learning mathematics in the 1990s.

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References Adé, J. (1973–1974). Vijf jaar moderne wiskunde … en dan …? [Five years of modern mathematics … and then …?]. Nova et Vetera, 51(2), 90–96. Barbry, R. (1974). Doelstellingen voor wiskundeonderwijs op de basisschool [Objectives for the teaching of mathematics in primary schools]. In Doelstellingen, evaluatie, structuren van hedendaags kleuter- en lager onderwijs II (Achtentwintigste Pedagogische Week ingericht in de Rijksnormaalscholen van Brussel II, Brugge en Hasselt voor het onderwijzend personeel van het kleuter- en lager onderwijs) [Objectives, evaluation, structures of contemporary kindergarten and primary education II (28th Educational Week organized in the Teaching Colleges of the State of Brussels II, Bruges and Hasselt for kindergarten and primary school teachers)] (pp. 111–135). Brussels, Belgium: Ministerie van Nationale Opvoeding en Nederlandse Cultuur, Directie van het Kleuter- en het Basisonderwijs. Boigelot, G., Dupont, M., & Vanhamme, W. (1956–1957). Le congrès de Mons—17 et 18 novembre 1956 [The conference of Mons—November 17 and 18, 1956]. Mathematica & Paedagogia, 11, 29–33. Bourbaki, N. (1939). Éléments de mathématique: Théorie des ensembles [Elements of mathematics: Set theory]. Paris, France: Hermann. Braunfeld, P. (1973). The role of algebra in a modern K-12 curriculum (including a description of the CSMP algebra program). International Journal of Mathematical Education in Science and Technology, 4(2), 175–185. Campedelli, L., & Giannarelli, R. (1972). Presentazione [Preface]. In G. Papy, La geometria piana nella matematica moderna [Plane geometry in modern mathematics] (pp. v–vii). Firenze, Italy: Le Monnier. CBPM. (1964). Programme de mathématique pour les écoles normales moyennes (section mathématique) proposé par le Centre Belge de Pédagogie de la Mathématique et adopté à titre expérimental par le Ministère de l’Éducation Nationale et de la Culture depuis l’année scolaire 1963–1964 [Mathematics curriculum for teaching colleges (mathematics section) proposed by the Belgian Centre for Mathematics Pedagogy and adopted on an experimental basis by the Ministry of National Education and Culture]. In Arlon 6 (pp. 25–42). Brussels, Belgium: Author. CBPM. (1966). Projet de programme pour la classe expérimentale de troisième scientifique [Draft curriculum for the fourth year of secondary school in the scientific streams]. Mathematica & Paedagogia, 30, 27–31. Colot, L. (1969). Lettre d’un professeur de mathématique à un collègue [Letter of a mathematics teacher to a colleague]. Mathematica & Paedagogia, 38, 34–40. Commission Universitaire. (1967). Liste des matières et désidérata du programme proposé par la Commission Universitaire de Mathématique installée par Mr. le Ministre F. Grootjans [List of contents and desiderata of the program proposed by the University Commission for Mathematics installed by the Minister F. Grootjans] (published in Mathematica & Paedagogia, 33, 1968, pp. 85–93). De Bock, D., & Vanpaemel, G. (2019). Rods, sets and arrows. The rise and fall of modern mathematics in Belgium. Cham, Switzerland: Springer. De Latil, P. (1960). Sensationnelles expériences en Belgique: Les “Mathématiques modernes” enseignées dès l’enfance [Sensational experiments in Belgium: “Modern Mathematics” taught from childhood]. Sciences et Avenir, 164, 539–544. 553. Derwidué, L. (1962). L’enseignement de la mathématique moderne [The teaching of modern mathematics]. Bulletin d’Informations de l’Association des Ingénieurs de la Faculté Polytechnique de Mons, 9(3), 5–9. Feusels, D. (1979). Vernieuwing in het wiskundeonderwijs [Innovation in mathematics education]. Unpublished master’s thesis, Department of Mathematics, Katholieke Universiteit Leuven, Belgium. Feys, R. (1982). Moderne wiskunde: Een vlag op een modderschuit [Modern mathematics: a flag on a mud barge]. Onderwijskrant, 24, 3–37. Frédérique. (1970–1976). Les enfants et la mathématique [Children and mathematics] (Vols. 1, 2, 3, 4). Brussels, Belgium-Montréal, Canada-Paris, France: Marcel Didier. Goffree, F. (1985). Ik was wiskundeleraar [I was a mathematics teacher]. Enschede, The Netherlands: Stichting voor de Leerplanontwikkeling. Holvoet, R. (1968). De werkgroepen van het Centrum [The working groups of the Center]. In G. Papy (in collaboration with P. R. Burgraeve, R. Holvoet, F. Papy, & A. Terfve) (Eds.). Arlon 10 (pp. 90–111). Brussels, Belgium: CBPM. Holvoet, R. (1971). Over de opbouw van de methode-Papy [About the structure of the method Papy]. In J. H. Wansink (Ed.), Didactische oriëntatie voor wiskundeleraren II. Tweede druk. [Didactical orientation for mathematics teachers II. 2nd edition] (pp. 405–449). Groningen, The Netherlands: Wolters-Noordhoff. Hunebelle, D. (1963). Ils nous ont ouvert les portes de la nouvelle mathématique [They opened the gates of the new mathematics to us]. Réalités, 211, 16–23. Janne, H. (1964). Une journée décisive [A decisive day]. Mathematica & Paedagogia, 26, 7–12. Janne, H. (1965). Programme optionnel de mathématique [Optional program of mathematics] (ministerial circular). Brussels, Belgium: Ministère de l’Éducation Nationale et de la Culture. Jeronnez, L., & Lejeune, I. (1972a). L’expérience de Waterloo d’un enseignement moderne de la mathématique à l’école primaire [The Waterloo experiment on a modern teaching of mathematics at the primary school]. Mathematica & Paedagogia, 53–54, 69–80.

10  Papy’s Reform of Mathematics Education in Belgium

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Jeronnez, L., & Lejeune, I. (1972b). L’expérience de Waterloo d’un enseignement moderne de la mathématique à l’école primaire (II) [The Waterloo experiment on a modern teaching of mathematics at the primary school (II)]. Mathematica & Paedagogia, 55, 211–221. Larock, V. (1962). Vernieuwing van het onderwijs in de wiskunde [Renewal of the teaching of mathematics] (ministerial circular). Brussels, Belgium: Ministerie van Nationale Opvoeding en Cultuur. Lenger, F., & Lepropre, M. (1959). Initiation aux mathématiques fondamentales dans une classe de première normale gardienne [Initiation to fundamental mathematics in a first year class of training of kindergarten teachers]. In Journées d’études 8: L’initiation aux mathématiques—Problèmes psycho-pédagogiques [Pedagogical days 8: Initiation in mathematics—Psycho-pedagogical problems] (pp. 17–24). Brussels, Belgium: Ministère de l’Enseignement Public, Secrétariat Général de la Réforme de L’enseignement Moyen et Normal. Le programme B des écoles normales gardiennes [The future kindergarten teachers’ program B]. (1958–1959). Mathematica & Paedagogia, 16, 70–75. Mawhin, J. (2004). Cinquante ans de mathématiques en Belgique: un survol [Fifty years of mathematics in Belgium: an overview]. Mathématique et Pédagogie, 145, 3–22. Ministère de l’Éducation Nationale. (1969). Programme de mathématique pour la première année d’études de l’enseignement technique secondaire inférieur dans les écoles de l’État [Mathematics program for the first year of the technical secondary schools of the state of the lower grade]. Brussels, Belgium: Author (published in Mathematica & Paedagogia, 35, 1969, pp. 5–13). Ministerie van Nationale Opvoeding/Ministère de l’Éducation Nationale. (1968). Leerplan wiskunde voor het eerste leerjaar van het rijksmiddelbaar onderwijs van de lagere graad/Programme de mathématique pour la première année d’études de l’enseignement moyen du degré inférieur [Mathematics program for the first year of the secondary schools of the state of the lower grade]. Brussels, Belgium: Author (published in Mathematica & Paedagogia, 34, 1968, pp. 8–21). Noël, G. (1993). La réforme des maths moderne en Belgique [The reform of modern mathematics in Belgium]. Mathématique et Pédagogie, 91, 55–73. Mogensen, O. (1970–1971). Wiskunde in de aanvangsjaren van het lager onderwijs [Mathematics in the early years of primary education]. Persoon en Gemeenschap, 23(5), 235–244. Nationaal Verbond van het Katholiek Middelbaar Onderwijs. (1968). Leerplan wiskunde voor de zesde van de humaniora [Mathematics program for the first year of general secondary education]. Brussels, Belgium: Author (published in Mathematica & Paedagogia, 34, 1968, pp. 22–34). OECD. (1964). Mathematics to-day. A guide for teachers. Paris, France: OECD. Papy, F. (1968). Une expérience pédagogique de dix ans en Belgique [A pedagogical experiment of ten years in Belgium]. In G. Papy (in collaboration with P. R. Burgraeve, R. Holvoet, F. Papy, & A. Terfve) (Ed.), Arlon 10 (pp. 25–89). Brussels, Belgium: CBPM. Papy, F. (1969). Minicomputer. Educational Studies in Mathematics, 2(2–3), 333–345. Papy, F. (1970). L’expérience Belge à l’école primaire [The Belgian experiment in primary school]. In A.  Revuz (Ed.), New trends in mathematics teaching—Tendances nouvelles de l’enseignement des mathématiques, Vol. II (pp. 95–111). Paris, France: UNESCO. Papy, F. (1971). Initiation à la notion de groupe [Initiation to the notion of group]. Mathematica & Paedagogia, 48–49, 160–179. Papy, G. (1954–1955). Le produit en géométrie [The product in geometry]. Mathematica & Paedagogia, 6, 8–17. Papy, G. (1959). Aperçu général de la théorie des ensembles. Quelques notions essentielles de topologie [General outline of set theory. Some basic notions of topology] (Report of Arlon 1, July 4–6, 1959). Brussels, Belgium: Presses Universitaires de Bruxelles. Papy, G. (1960). Premiers éléments de mathématique moderne [First elements of modern mathematics]. Brussels, Belgium: Author. Papy, G. (1961). Suggestions pour un nouveau programme de mathématique dans la classe de sixième [Suggestions for a new mathematics program in the sixth class]. Mathematica & Paedagogia, 20, 20–29. Papy, G. (1962). L’enseignement de la mathématique dans le tronc commun [The teaching of mathematics in the common core]. Mathematica & Paedagogia, 23, 47–60. Papy, G. (1963). Mathématique moderne 1 [Modern mathematics 1]. Brussels, Belgium-Paris, France: Didier. Papy, G. (1964). Methods and techniques of explaining new mathematical concepts in the lower forms of secondary schools. In H. F. Fehr (Ed.), Mathematics to-day—A guide for teachers (pp. 99–147). Paris, France: OECD. Papy, G. (1965). Mathématique moderne 2. Nombres réels et vectoriel plan [Modern mathematics 2. Real numbers and the vector plane]. Brussels, Belgium-Montréal, Canada-Paris, France: Didier. Papy, G. (1966). Mathématique moderne 5. Arithmétique [Modern mathematics 5. Arithmetic]. Brussels, Belgium-­ Montréal, Canada-Paris, France: Didier. Papy, G. (1967a). Mathématique moderne 3. Voici Euclide [Modern mathematics 3. Euclid now]. Brussels, Belgium-­ Montréal, Canada-Paris, France: Didier. Papy, G. (1967b). Mathématique moderne 6. Géométrie plane [Modern mathematics 6. Plane geometry]. Brussels, Belgium-Montréal, Canada-Paris, France: Didier/Brussels, Belgium: Labor.

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Papy, G. (1979). Unpublished letter dated October 30, 1979 to J. Dumortier, general director of the Organization of Studies Division of the Belgian Ministry of Education. Papy, G., & Holvoet, R. (1968). Centre Belge de Pédagogie de la Mathématique à Bruxelles: Statuts [Belgian Centre for Mathematics Pedagogy in Brussels: Articles]. In G. Papy (in collaboration with P. R. Burgraeve, R. Holvoet, F. Papy, & A. Terfve) (Ed.), Arlon 10 (pp. 133–135) (Reprint from Act 2472 published in the annexes of the Moniteur Belge of June 8, 1961). Brussels, Belgium: CBPM. Randour, C. (2003). Frédérique et Papy, créateurs d’images mathématiques [Frédérique et Papy, creators of mathematical images]. Lecture given at the 29th annual congress of the Société belge des Professeurs de Mathématique d’expression française, Forest, Belgium. Retrieved November 14, 2020, from http://www.rkennes.be/papy.ppt.htm. Smet, R., & Vannecke, A. (2002). Historiek van het technisch en beroepsonderwijs 1830–1990 [History of technical and vocational education 1830–1990]. Antwerp, Belgium-Apeldoorn, The Netherlands: Garant. Stievenart, P. (1968). La guerre des math aura-t-elle lieu? [Will the math war take place?] Special, 153, 16–19. Vanhamme, J. (1991). Papy, un esprit ouvert [Papy, an open mind]. In Un présent du passé, un passé du présent. Fascicule à Georges Papy à l’occasion de ses 70 ans [A presence of the past, a past of the present. Fascicle to Georges Papy on the occasion of his 70 years] (pp. 72–73). Arlon, Belgium: SBPMef. Verschaffel, L. (2004). All you wanted to know about mathematics education in Flanders, but were afraid to ask. In R. Keijzer & E. De Goeij (Eds.), Rekenen-wiskunde als rijke bron [Arithmetic-mathematics as a rich source] (pp. 65–86). Utrecht, The Netherlands: Freudenthal Instituut. Vredenduin, P.  G. J. (1975–1976). [Review of Les enfants et la mathématique 3 [Children and mathematics 3] by Frédérique]. Euclides, 51(4), 163–165.

Chapter 11

A Tale of Two Systems: A History of New Math in The Netherlands, 1945–1980 Danny Beckers

Abstract  New Math in the Netherlands will be described from the perspective of the ideals that were held within the institutes, instrumental in shaping Dutch education. In line with Bob Moon, and contrasting the analysis of some of the people who played a role in this history, we will describe the rise of realistic mathematics education as a realization of New Math, rather than as a breach with the past. Trust in mathematics and mathematicians played a role in the realization of a curriculum that accompanied the introduction of a modern school system, in 1968—supplanting the nineteenth-century system. Pillarization of Dutch society resulted in (1) a large number of stakeholders who all had their own ideas and power bases and (2) the necessity to find common ground, which was found in a focus on individual learning processes and stressing the need for developing individuality. This gave Dutch New Math its distinct flavor. Keywords  1950s · 1980s · Adri Treffers · Educational ideals · Educational systems · Edu Wijdeveld · Fred Goffree · Hans Freudenthal · History of arithmetic education · History of mathematics education · Leon van Gelder · Lucas Bunt · Modern mathematics · New Math · Susan Freudenthal · The Netherlands

Introduction In 1972, the Groningen professor Willem Jan Brandenburg (1921–2009) introduced, in a public lecture, a mathematical model of how pupils learn (Brandenburg 1972).1 His presentation illustrated both the overwhelming trust in mathematics, as well as what he considered to be the most pressing problem: How to teach mathematics to a substantial number of pupils? What he left out, was the goal of mathematics education. This could be the (experimental) result of optimization within such a model. Mathematicians, mathematics educators, and teachers agreed that pupils had to understand their mathematics. But what exactly it was that pupils had to understand, and how this could be observed, was quite another matter.  The model introduced by Brandenburg was similar to the model that had been introduced by the German cybernetics enthusiast Wolfram Menzel, to whom he referred. 1

D. Beckers (*) Vrije Universiteit Amsterdam, Amsterdam, The Netherlands e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_11

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At the same time, Brandenburg had a very clear opinion about this matter. As a teacher trainer and with 10 years of teaching experience, he had published his PhD thesis only a few years earlier. In this, he presented modern mathematics as a necessity: An obligation to pupils and society, a society that needed more graduating pupils and better, continuous teacher training (Brandenburg 1968). His thesis marked an important moment in Dutch mathematics education, 1945–1980: 1968 was the year that a new school system replaced the nineteenth-century system. A modern school was now in place, but one, according to Brandenburg, in dire need of modern mathematics. Brandenburg was a marginal figure within Dutch secondary schools. Leon van Gelder (1913– 1981), his boss at Groningen university, however, was a central character in Dutch primary education. He favored the introduction of New Math2 in primary schools (Van Gelder 1959). Although among school teachers mocked as “the wizard from the North” (Vos and Van der Linden 2004, p. 104), Van Gelder was a teacher who had worked his way up to the position of a full professorship in pedagogy at Groningen university teacher training college. One of his goals was the introduction of the “middle school.” Originating from his socialist convictions, Van Gelder wanted students to stay together in mixed ability groups until the age of 16, thereby preventing career decisions at an early age. His biographer called him “the spider in the web” of educational politics in The Netherlands (Amsing 2016). Although that is definitely an acute observation, his middle school plans failed. This illustrates the sharp distinction between primary and secondary schools—culturally, socially, and budgetary. Even though the crevice was stepped over by thousands of pupils annually, the difference in personnel, pedagogics, teacher training (see Van Essen 2006), and textbook authors was huge—effectively making the “middle school” unattainable. It is fair to say that the two systems mentioned in the title of this chapter reflect just as much the systems in place before and after 1968, as the systems serving pupils aged 4–12 and 12–18. The stories of Van Gelder and Brandenburg hint at various ideals playing a role in shaping Dutch education between 1945 and 1980. Socialist convictions, mathematical principles, and educational ideals were all involved. To the Dutch government in the decades after World War II, education was considered of the utmost importance. Hence, time and effort went into reshaping the educational system. The archive of the ministry (Archive Ministry of Education) bears witness to the awe-inspiring amount of work done by a group of mathematics educators. The work by Brandenburg and Van Gelder was inspired by that work. Dutch New Math was at the core of educational modernization, as it was at the core of two school systems—in both meanings of the phrase. In this overview of the history of New Math in The Netherlands, we will take the educational ideals as the leading principle behind the modern mathematics curriculum. In the first section, while describing the Dutch school system, we will introduce the various stakeholders. In the next three sections, chronologically, the 1950s, 1960s, and 1970s will be discussed. These may be considered as three stages in an unfolding story, where respectively the Mathematics Working Group, the Commissie Modernisering Leerplan Wiskunde (CMLW) [Committee for Modernization of the Mathematics Curriculum], and the Instituut voor de Ontwikkeling van het Wiskunde Onderwijs (IOWO) [Institute for the Development of Mathematics Education] established the mood for the realization of New Math ideals.

 In this chapter, the phrase “New Math” will be used as synonymous to “modern mathematics” or “modernized mathematics,” since the Dutch educators participating in the discussions during the 1950s and 1960s did not distinguish between those expressions. As will be made clear, “New Math,” to the Dutch, had a broader meaning than has been outlined in Chap. 1 of this volume. 2

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The Dutch School System(s)

The post-War Dutch school system bore the remnants of a nineteenth-century social class system. Originally, lower education offered elementary arithmetic, reading, and writing to the lower social classes. Middle-class citizens would send their children to separate schools. Higher education (that is, for the higher social classes) consisted of gymnasium and university (Blom 2004). During the nineteenth century, several new opportunities emerged and state funding expanded to include more schools. In 1863, the middle classes received further educational opportunities at the so-called HBS,3 where pupils would receive more general knowledge of sciences, French, German, or English, and a polytechnic institute, where they could receive training as an engineer. The lower classes were given more educational opportunities in schools with extended curricula (ULO, MULO4) and vocational training institutes (Smid 2000). Lower, middle, and higher education were kept separate, and although during the early twentieth century, the class distinctions in education became somewhat blurred, switching between lower, middle, or higher education was neither envisaged nor encouraged (Beckers 2017a). One controversy made state-funded education typical. That was the matter of which religion should be the basis of education. Strong Protestant presence and rising Catholic powers made it impossible to strive for the French laïcité, but the issue would regularly return. A compromise was reached in 1917, when a law was issued guaranteeing no state interference with curricula, and a funding of every school that had sufficient backing from a substantial group of people. Non-state-funded education became marginal, and pillarization of Dutch society became a fact (Kennedy 2017). Not just confessional schools profited from this law. Within socialist circles, new ideas about education were popular as well. In 1936, the radical pedagogue Kees Boeke (1884–1966) founded a Dutch branch of the New Education Fellowship (NEF) (Hooghiemstra 2013). The NEF aimed for “improving education,” which implied more attention to the individual pupil. According to the most radical ideals within NEF, social-class differences should not play a role at all, and a meritocratic educational system should come in its place. The less radical voices favored schools on the basis of Montessori, Freinet, or Dalton principles, where pupils could develop their personalities.5 After the War, the Dutch NEF branch came into the spotlight. The post-War governments were in favor of renewing education. Government intervention in the curriculum or didactics was out of the question, but having teachers and teacher educators thinking about that, thereby improving education, was encouraged. Most notably, the Mathematics Working Group, part of the NEF, came into the picture. This working group would become important in Dutch mathematics education. In 1951, the (Catholic) Minister of Education, Theo Rutten, announced a concept law that would introduce a modern school system, abandoning the social class-based system. In 1963, after a number of debates and compromises, a law was passed, effective in 1968—because of its size, tongue-in-­ cheek named Mammoth Law (Mellink 2014). Under this new law, distinctions between (M)ULO,

 HBS is an acronym of “Hogere BurgerSchool” [Higher Civic School], a school type destined for pupils aged 12–18, from the middle classes. The curriculum of HBS would typically not include Classic languages. 4  ULO and MULO are acronyms for, respectively, “Uitgebreid Lager Onderwijs” [Extensive Lower Education] and “Meer Uitgebreid Lager Onderwijs” [More Extensive Lower Education]. In fact, they offered to the lower social classes forms of what we would today call secondary education. 5  The idea of developing personality in conjunction with intellect was not generally shared, but meritocratic ideas were widely accepted. In papers on education, Voorbereidend Hoger en Middelbaar Onderwijs (VHMO) [Preparatory Higher and Middle Class Education] became fashionable in the 1920s (and more ubiquitous in the 1950s), deriving unity from the age group, instead of the social class structure. The combination of gymnasia and HBS schools in so-called lycea was a precursor to that idea (see Beckers 2017a). Schools offering “extended education” to pupils from the lower classes were excluded from this definition, although in practice some of these school functioned as a form of secondary education (Smid 2000). 3

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HBS, and gymnasium, now called MAVO, HAVO, and VWO6 (gymnasium as a category remained, as part of the compromise), became gradual: These were all part of secondary education, the follow-up of primary school, which was no longer connected to social class distinction. Moving (upward) from MAVO to HAVO, or HAVO to VWO, became a normal option, entrance examinations were abandoned. This implied the necessity of curriculum reforms. Dutch lower education, until 1968 including schools for extended lower education that would become MAVO, was serviced by three Pedagogical Centers (PCs): A Catholic, a (Protestant) Christian, and a socialist. All three offered pedagogical advice to schools and organized conferences—sometimes in allegiance with the NEF. When the government in the 1950s urged these centers to bundle their efforts, they did so in the 3PC group. In practice, the 3PC group acted as a Hydra-esque fourth center, which in turn also asked for special funding from the government, and acted in various allegiances with other stakeholders. Social class differences as well as pillarization were cultural imperatives in the early twentieth century. Superimposed upon the various Protestant, Catholic, socialist, and liberal organizations that represented teachers, the 1920s witnessed the rise of two associations for mathematics teachers: One for the gymnasia, Liwenagel (in 1921) and one for middle-class schools, Wimecos (in 1925). After the War, these organizations worked jointly in various ways— for example, in some modest curriculum reforms during the 1950s (Smid 2012, 2017). Both used the journal Euclides as their publication channel (Maassen 2000). In 1968, Wimecos, in line with the new law, decided to modernize its organization to include gymnasium, VWO, HAVO, and MAVO teachers, under the new name Nederlandse Vereniging van Wiskundeleraren [Dutch Association of Mathematics Teachers]. It took a couple of years before also Liwenagel ceased to exist.



Episode 1: 1945–1960, New Math Rising

This section has two purposes. First, it will illustrate that from the end of World War II, several stakeholders in Dutch mathematics education were striving for renewal. Second, it serves as a setting of the stage of the developments after 1960. Instead of striving for completeness, we want to show the diversity of ideals and the raw reality of various interests. Dutch mathematicians were involved in secondary mathematics education (see Beckers 2016a). This involvement was visible in three ways. First, they taught many of the students who would become mathematics teachers at HBS and gymnasia (see Blom 2000). Second, they contributed to the various forums for mathematics teachers, most notably to Euclides. Last but not least, they met teachers in their capacity of examiner, during the annual oral HBS and gymnasium examinations—a habit that would be discontinued in 1968. When mathematics became a productive force after the War, mathematicians also took a more active stance (Alberts 1998). To teachers, they emphasized the usefulness of their subject, but the main idea behind modern mathematics to them was the unity that they saw in geometry, trigonometry, and algebra. For them, the language of set theory and structures embodied this unity, and would make the school subject more in line with current academic practice. Mathematicians acted through two organizations: the Mathematical Centre and the Mathematical Society—which, in practice, acted as one. The Mathematical Centre offered teachers courses in modern mathematics. The Dutch Mathematical Society had an education committee. Its approach was both top-down, in that it told exactly what good  MAVO, HAVO, and VWO are acronyms for, respectively, “Middelbaar Algemeen Voortgezet Onderwijs” [Lower General Secondary Education], “Hoger Algemeen Voortgezet Onderwijs” [Higher General Secondary Education], and “Voorbereidend Wetenschappelijk Onderwijs” [Pre-university Education]. 6

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mathematics was about, and bottom-up, in that it never (openly) interfered with how and what was taught. The committee guaranteed connection to educators by having a teacher among its members. This teacher, Johan Wansink (1894–1985), would publish news related to the committee in Euclides. Wansink also kept Euclides readers updated about foreign news. He had a keen eye for the more beefy quotations.7 In 1956, for example, he quoted Morris Kline from the Mathematical Gazette, who in a tirade had written about a certain type of college mathematics teacher who “know the mathematical theory of ideals, but they certainly are not familiar with the ideals of teaching” (Wansink 1956, p. 65). Thus, through the lens of Wansink, Dutch mathematics teachers were made aware of the international New Math scene. With the notable exception of Lucas Bunt (1905–1984), the other members of the education committee were full professors at a Dutch university. Bunt (1949) actively approached teachers with questionnaires about their ideas on the curriculum. Apart from his activities on behalf of the education committee, and as a teacher at the teacher training institute of Utrecht University, after reading Piaget, in the 1950s he became an advocate for didactical experiments. Although dismissive of the conclusions that Piaget drew from his experiments on the acquisition of number concepts, he was intrigued by the observation and interviewing methods that Piaget used. His criticism of Piaget’s theories (Bunt 1950) was often copied by others. Bunt also favored curriculum renewal. He published both a history of mathematics textbook (Bunt 1954) and an introduction to statistics for Dutch secondary education (Bunt 1956). Both were intended for use by students of the A-groups, the linguistic-oriented strands in HBS and gymnasium. Wansink and Bunt were also among those who, directly after World War II, joined the Mathematics Working Group of the Dutch NEF branch. This was a growing group of teachers and teacher trainers, receptive to new ideas about teaching mathematics. Begun in the 1930s, as a social-democrat group of teachers who wanted to introduce pupils intuitively to the field of geometry, after 1945 the group secured itself of some scientific conscience and political influence by involving the Utrecht professors Minnaert and Freudenthal. By 1950, the intuitive introduction to geometry was generally accepted— although varying in timespan from a few weeks to the first years of secondary education. The new focus of the working group was on “learning to think,” or “understanding mathematics.” Rote learning would stand in the way of the individual in modern society, who would have to learn to make up his own mind. How to measure understanding or how to teach it, however, were not clear. The Mathematics Working Group experimented with new approaches and reported on its experiments. Piet van Albada (1905–1997), a teacher at the Amsterdam secondary Montessori school, presented material he had developed for his pupils, including experiments in which the pupils were involved in tasks involving tilings and positioning before they started a more formal course in axiomatized geometry (De Moor 2001). Rudolf Troelstra (1917–2012) reported on experiments with transformation geometry—which he performed with a couple of his colleagues (De Moor and Groen 2012). Wim Bos (1916–2004), a teacher who, together with his colleague Paul Lepoeter (1916–1978), conducted experiments in class management, had pupils working at their own pace, was also a regular contributor. In their opinion, mathematics was learned by doing, and they structured their lessons accordingly, by giving pupils ample opportunity to do mathematics from textbooks with many questions inviting practice, and with only brief instructions (Bos and Lepoeter 1954; Smid 2004). By far the most active member of the working group was Pierre van Hiele (1909–2010) who, together with his wife Dina van Hiele-Geldof (1911–1958), authored a new series of textbooks on algebra (Van Hiele and Van Hiele-Geldof 1948) and geometry (Van Hiele and Van Hiele-Geldof 1950). His ideas about level stages of learning geometry, matured during discussion at the Mathematics Working Group (Broekman 2005). The “levels theory” developed by Van Hiele was inspired by  In my opinion, Smid (2017) is far too modest in his assessment of Wansink’s ideas and influence. Of all the people, who claimed an overview over didactical issues in mathematics in the 1950s and 1960s, Wansink was the most knowledgeable (see Wansink 1966). 7

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Piaget, but like Bunt he rejected the conclusions of the psychologist. His theory appealed to many people within the Mathematics Working Group, because he connected his stages to a specific and easily recognizable use of language that the teacher could observe with his pupils. Although its members were growing in numbers and influence, the 1950s Mathematics Working Group did not represent mainstream Dutch mathematics teachers, many of whom adhered to teaching Euclidean geometry and algebra by presenting theorems and proofs in class.8 Nor did the Group represent core business within the NEF. The majority of NEF were pedagogues, concerned mainly with primary education—the other Dutch educational system. Through this connection, as well as through the connection within the academic teacher training institutes, the ideas of New Math sipped through to primary education. These two were not naturally compatible. Christiaan Boermeester, as a ULO teacher and NEF participant, was fed by the pedagogical debates, but also had to answer to his mathematical conscience: I attempt to make the first steps on the dangerous road of didactics of geometry education. A dangerous road it is, because never should didactics result in a violation of the most fundamental aspects of a subject. And where is this danger more ominous than with the teaching of mathematics in general, geometry in particular?9 (Boermeester 1955, p. 4)

Leon van Gelder was a regular speaker at the conferences of the Christian Pedagogical Centre. There, he would stress the importance of using materials to visualize numbers during the phase of early arithmetic. Van Gelder mentioned materials by Montessori and Decroly, noting that these all had pros and cons and also recognizing that the materials could help pupils abstract the mathematical sense of number (Van Gelder 1958). The text was soon expanded to a book format (Van Gelder 1959), which would be used as a textbook for future primary school teachers. Van Gelder took the introduction of New Math for granted. To him, it was self-evident that more pupils had to be better trained. Of the New Math ideas, he favored experimenting (children should try things themselves), the playing, the new ideas about class organization, but he did not necessarily opt for the language of set theory. On that aspect of the New Math, he would change his mind in the mid-1960s. New Math pur sang was the basis of projects that turned these teaching efforts into measurable units, which were tested in various circumstances, in attempts to establish which techniques offered the best results. The Mathematics Working Group was active in that way, by consistently inquiring into the nature of what exactly it was, that students had learned, and how it could be observed. They regarded language as fundamentally important. Pupils who were able to express themselves were not only easier to observe but also had an easier job acquiring new knowledge. Therefore, the members of the working group tried to find ways to eliminate the language barrier (see De Miranda 1956). This implied more individualized education. Those who liked set-theoretical language emphasized that it was so different from ordinary language that it did not favor higher social circles; or they emphasized that the change of language between the mathematics class and the other classes, made the pupil realize mathematics was something different. Using contexts that the pupil could relate to was one way to stimulate good use of language. This could range from giving the pupil an incentive in real-life examples, to an intuitive introduction to mathematical subjects, to staging class conversations, having the pupils work more for themselves (learning by doing), even writing essays. In all cases, the teacher was guiding students to mathematical knowledge and the correct use of language therein. The psychological research institute in Amsterdam run by Adriaan de Groot (1914–2006) was involved in “New Math” research from a different perspective. De Groot, like the Mathematics  Publishers also played their part in curriculum reforms (see Pingel 2010). New series of textbooks that were published were indicative of teachers embracing modern teaching. 9  Literally: [In dit werkje] heb ik getracht om enige stappen te zetten op de zo gevaarlijke weg van de didactiek van het meetkunde-onderwijs. Gevaarlijk, want nimmer mag de didactiek leiden tot aanranding van de wezenlijke kenmerken van het betreffende vak en waar is dit gevaar groter dan juist bij het wiskunde-onderwijs in het algemeen en het meetkunde-onderwijs in het bijzonder? 8

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Working Group, was concerned about the large number of dropouts from secondary school. He regarded this as a problem inherent to the school system—not something to be solved by didactics. In his view, the 1950s Dutch school system was merely copying social class structure, because teachers favored “their own kind.” His research was therefore directed at finding ways to eliminate this result of teaching practice (Bussato 2014). In his didactical experiments, De Groot was in league with the 3PC group, and in the mid-1950s started an experiment that aimed at developing an instrument to test geometrical understanding. The launch of the project showed his awareness of the different didactical approaches. Leaving that as a choice to the teacher, he wanted to establish the greatest common divisor in what teachers themselves regarded “understanding” of specific geometrical theorems (De Groot 1957). Troelstra cooperated, but several members of the Working Group opposed. They resented the statistical approach, which in their view could never yield anything sensible with respect to solid education. It was considered biased in its premise that results from large groups of pupils could reveal anything about an individual’s learning progress (De Miranda 1957). In general, mathematicians and mathematics teachers were in favor of teaching statistics, but mostly because they wanted to promote carefulness in interpreting statistical results. Judging from the quires of Euclides and the NEF archive, the number of ideas was incredible. Many more probably left no paper trace. New Math ideas were vibrant within the schools and school services of all the pillars. The Mathematics Working Group succeeded in joining people from across pillars and encouraging and enabling them to listen to each other.



Episode 2: The 1960s, a New Curriculum Committee

After the 1959 Royaumont Seminar, Lucas Bunt gave the first impetus to more structural funding for research into curriculum reforms. Active within 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, he was in touch with the Belgian and French reformers (see De Bock and Vanpaemel 2019), and already half-won over to New Math. Perhaps it was his ambition rather than his didactical conviction that made him grasp the opportunity (Zwaneveld and De Bock 2019), but he did it anyway. In an effort to secure funding, Bunt rallied partners and on January 12, 1961, presented his ideas, including a 10-year estimate of costs. The proposal met with a lukewarm reception. Within the Ministry, it was decided not to take up Bunt’s ideas. Moreover, Ministry personnel restricted the task assigned to the Committee to considering the curricula put forward by HBS and gymnasium— whereas Bunt had included the last year of primary school. In April 1961, the CMLW was created as a committee to advise on a modern mathematics curriculum. The Amsterdam mathematician Leeman, like Bunt a participant in Royaumont and at the time chairman of the examination committee, acted as its chair (Archive Ministry of Education, inv. nr. 99). In his installation speech, in July 1961, the Secretary of State emphasized that a New Math curriculum was necessary since mathematics had changed. For physics and chemistry, it was quite clear that the science of the nineteenth century was no longer in the textbooks for secondary schools. Mathematics had to follow suit. Moreover, the important role of mathematics in society warranted the work of the committee. He explicitly referred to the Mathematics Working Group (Stubenrouch 1961). In his response to the speech, Leeman gracefully accepted the task bestowed upon him, and immediately showed his diplomatic skills by noting that the Secretary of State was, of course, right: A new curriculum should take as many pupils as possible to the higher spheres of modern, abstract mathematics. At the same time, the tradition of teaching Euclid was perhaps justifiable for psychological reasons, if an introduction to abstract mathematics could best be achieved via Euclidean geometry (Leeman 1961). Thus, he kept the peace among all interested parties.

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Figure 11.1  Figurendoos–handleiding [Figure box–manual], colorful brochure explaining the use of the logical blocks (1971)

Figure 11.2  New Math in Dutch primary schools. Ik doe–en ik begrijp [I do–and I understand], translated from the British Nuffield Project (1968)

Controversies flared up. In this section, first, the new work outside the CMLW will be discussed, to continue with that of the Committee. This way, the reader may see that the CMLW, from the start, succeeded in incorporating and bringing forward many initiatives. Bunt, disappointed because he had been given a minor position in the CMLW, continued to be involved with his research. In 1968, out of the blue even to his co-workers, he accepted a position in the United States of America and disappeared from the Dutch scene. Until then, however, his interest

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in New Math was not as much in set theory, as it was in the introduction of new subjects and the innovative approach of programmed instruction—which he saw as allowing for more individualized instruction. With a couple of schools, he started testing a translation (by Harrie Broekman) of a U.S. series of textbooks developed by the School Mathematics Study Group (SMSG). The tests ran over several years, and many adaptations were made before the textbooks were finally published (Bunt 1968).10 New textbooks cashed in on the ideas discussed in the Mathematics Working Group. The Group maintained that students should do mathematics, geometry started in space and descended to two dimensions, the use of translations, rotations, and mirroring. Van Hiele revised the textbooks he had written together with his wife (Van Hiele and Van Hiele-Geldof 1963). The Mathematics Working Group stayed close to its 1950s line of working—experimenting with new ways of having their pupils understand, either the algorithms or the logic behind them—but refraining from using set theory. A notable exception was a series of geometry textbooks (Maassen and Oosten 1961), delivered with a sachet of plastic sticks to be used with the first chapters of the first book to help the pupils develop intuition. In this series, set-theoretic language was used and introduced in the first chapter of the second volume. It was motivated by the observation that sets played a fundamental role in algebra but could be more intuitively introduced in geometry. The authors were in favor of set-theoretic language in schools, since, the authors argued, it furthered (mathematical) simplicity and it brought unity in mathematics (Maassen and Oosten 1962). The idea of using set theory as a linguistic tool was enthusiastically received within the CMLW. All the university professors, as well as textbook author Piet Vredenduin (1909–1996), considered it easy to learn and concise (Vredenduin 1966); they pointed to the advantages which were noted by the Mathematics Working Group in the 1950s. Following the Belgian example—some were active in continuous teacher training there—CMLW first opted to create more enthusiasm among teachers by offering introductory courses in modern mathematics (Beckers 2015b). The Committee also wanted to introduce new subjects to the curriculum. They brought forward their ideas on that in several conferences—for example, with mathematics teachers who attended a Conference at the Christian Pedagogical Centre (CPS 1964). Most notably, it was hoped that a teacher course on computer mathematics held in 1967 would spark interest in this subject among the participants (CMLW 1967a). CMLW produced experimental textbooks focused on the HBS and gymnasium curricula. A very formal algebra textbook (Kindt and Maassen 1967), entirely in set-theoretic language, was prepared, as was a geometry textbook offering an intuitive introduction, starting with folding cubes and such activities, and then introducing set theory later (Troelstra and Kuipers 1965). The latter was tested in Troelstra’s school and based on the earlier experiments by Troelstra. Both textbooks were produced under the guidance of Hans Freudenthal (1905–1990), a renowned mathematician who specialized in algebraic topology. He was acquainted with Troelstra via the Mathematics Working Group and would have been pleased to have Troelstra’s loyalties taken away from De Groot—whose work he resented. Freudenthal was a member of the CMLW from the beginning. Although inactive within the CMLW until 1965, he started to become more enthusiastic and influential in the mid-1960s (Beckers 2016b; Beckers 2019; La Bastide van Gemert 2015). A third textbook was produced at Groningen University. Leo Westerman produced a vector geometry textbook, introducing vectors in two- and three-dimensional space and linear transformations, in  Programmed instruction, mostly in the guise of textbooks, was popular. With funding from the Christian Foundation for Research into Education, the working group programmed instruction had already started experimenting in the early 1960s. One result was a series of textbooks on algebra (Bouman et al. 1965). As didactical innovations, these experiments were also embraced by the Dutch NEF branch, who organized huge annual conferences on the subject since 1965. Van Gelder, who also liked to experiment with closed-circuit television and other technologies to further education, was one of the driving forces behind these conferences (Archive WVO, inv. nr. 37). The CMLW experimented with programmed instruction with a textbook on probability (Karman n.d.). See Beckers (2015a) for more details. 10

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a formal way, making students aware of group structures (Westerman 1966). These textbooks would form the basis of the curriculum plans that the Committee finally offered to the government (CMLW 1967b, 1969). In those plans, the texts supervised by Freudenthal, and two more texts covering ­statistics and some analysis, were supposed to be the basis for the Mathematics I course. Westerman’s textbook could serve as a basis for the Mathematics II course, for gifted students.11 Through Susan Freudenthal-Lutter (1908–1986), whose pedagogical career had molded her husband’s interests in mathematics education, NEF and New Math ideals mingled. In 1965, she cooperated with Martinus Jan Langeveld (1905–1989), director of the teacher training institute at Utrecht (Bos 2011; Levering 2015). Susan Freudenthal wanted to develop Jenaplan12 arithmetic education. In cooperation with Langeveld and two of the pedagogical centers, she made plans to design and test textbooks. She suggested to have Pierre van Hiele involved, whom she knew well, but Langeveld wanted to keep him out: Van Hiele would draw the project into the direction of what was happening in secondary schools, whereas for many pupils, primary school would finalize their educational career (Jenaplan archive, box 42, minutes October 1, 1965). The argument may have been genuine, but apart from illustrating the crevice between Dutch primary and secondary education, Langeveld kept the influence of mathematicians from the pedagogical realm of primary education. In the late 1960s, Susan Freudenthal would assist in the realization of an arithmetic series. The title of both the series of textbooks and the accompanying journal, were intended to keep teachers using the method theoretically up to date. The title was Denken en Rekenen [Thinking and arithmetic], and the publishing house Malmberg aimed at their traditional customers, Catholic school teachers. Valeer van Achter, of Belgian origin, became the mastermind behind this series, but in it, Susan Freudenthal also explained the use of logical blocks in the classroom—without going too quickly or deeply into abstract reasoning (Figure 11.1). She advised to observe the play and conversations of the pupils and help in recognizing the important remarks (Freudenthal-Lutter 1969). The series of textbooks referred explicitly to the work of Nicole Picard, Zoltán Dienes, and Tamás Varga as sources of foreign inspiration and hailed set-theoretic language—and mathematics in general—as a way to a more democratic and inclusive society (Van Achter 1969). Susan Freudenthal was in touch with what was going on in secondary education, and although she generally agreed with Langeveld that primary education should set its own goals, she was not blind to the ever-growing number of pupils attending secondary schools. In 1966, she helped the Dutch NEF organize a conference on arithmetic education. Van Gelder, in his Groningen institute, meanwhile focused on change in schools, because schools were not only there to provide education, but now also had to instill norms and values in their pupils (De Jong 2020; Van Gelder 1968b). With Gerrit Krooshof (1909–1980), whom he knew from Dutch NEF circles, and two enthusiastic young teacher trainers, Fred Goffree (1934–2020) and Edu Wijdeveld (born in 1932), he published on New Math (Krooshof 1968; Van Gelder 1968a; Wijdeveld and Goffree 1968) and managed to persuade the publishing company Wolters-Noordhoff to visit England to become informed with respect to the British Nuffield project (Goffree et al. 1968) (Figure 11.2). Wijdeveld was also involved in the CMLW, and after defining the Wiskobas project (see below), he proposed to have it financed and brought to fruition under the wings of CMLW. Within CMLW, people were aware that it was imperative to have support from various parties. The education committee of the Dutch Mathematical Society showed its commitment to CMLW in various ways (Loonstra and Vredenduin 1962; Van der Sluis 1969; Van Hoorn and Guichelaar 2018; Vredenduin 1967; Wessels 1969). While creating alliances, CMLW succeeded in expanding its assignment to cover mathematics education at all levels: Including MAVO, primary education, and  Distancing themselves from the old A and B curricula in HBS and gymnasium, the course names “wiskunde I” [mathematics I] and “wiskunde II” [mathematics II] could not have been chosen better. 12  Jenaplan refers to a school system based on an educational philosophy conceived and founded by the German pedagogue Peter Petersen (1884–1952).

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Figure 11.3  Textbook author Paul Lepoeter adapted his work to the New Math era. Gids voor de nieuwe wiskunde voor de brugklas [Guide to the New Math for the first year] (1972)

both lower and higher vocational training sectors. By the end of the 1960s, Edu Wijdeveld published a New Math introduction on behalf of teachers and teacher trainers, which started by stating that the modernization of mathematics could have no other consequence than introducing New Math (Wijdeveld 1969a, b). The unity of mathematics never shone brighter! But there was a downside to all the success. Progress was made only slowly, and the constant pleas for more funding made people within the Ministry nervous. Both in 1964 and 1967, CLMW asked for funding to start a more permanent institute. Mathematics was not only recognized as an important productive force but as a cultural force also contributing to management theories and psychology. De Groot, during a trip to the United States of America, had been impressed by the Educational Testing Service (see De Groot 1966). Among teachers, multiple-choice tests—the design of which was a core business of this Testing Service—were not popular, so De Groot was careful in maneuvering. In 1968, the Ministry decided to fund a permanent institute to assess the level of primary school pupils to determine their best fitting place in the new secondary education system: Centraal Instituut voor Toetsontwikkeling (CITO) [Central Institute for the Development of Tests] (Haas 1995). The mathematicians were not amused. To the government, this was efficiency: Virtually all Dutch primary school pupils would be subjected to the tests, so CITO also held the promise of offering insight into the effectiveness of schools. More importantly, CITO was also given an advisory role on the committee overseeing the construction of state exams for secondary education. It was a construction that had been suggested by the 3PC group, which was supportive of De Groot’s ideas (Archive Ministry of Education, inv. nr. 96). CMLW distanced itself from CITO. In fact, CITO gnawed at the very foundations of their ideals: Didactics and assessment of mathematics education should be in the hands of those who understood what the final goals were (see Smid 2015). The funds would be much better spent on a didactical institute, as proposed by the CMLW. This was more than political tactics (which it was too): Mathematicians were convinced that psychologists were particularly bad in collecting and assessing statistical data (Freudenthal 1975a, b; Monna and Van der Put 2004).

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With the new curriculum drawing near in 1968, the need to publish textbooks that were adapted to the new curriculum became urgent (Figure 11.3). Van Hiele, with the help of Boermeester and others, published a whole new series of textbooks (Boermeester et al. 1967–1971).13 These textbooks explicitly combined algebra and geometry as one subject and gave set theory a fundamental place. Starting from intuitive introductions and many constructions and other exercises that pupils could do themselves and were intended as “starters” for classroom discussion, the first volumes introduced the ideas of set and vector. The authors introduced the idea of commutativity with a reference to several arithmetical procedures that pupils should have been able to perform, having them note that 5 times 12 is the same as 12 times 5, but a similar observation could not be made with respect to division. Sets and Venn diagrams were introduced, making the pupil aware that commutativity also held with respect to, for example, the union of sets. Their most notable competitors were Mathematics Working Group members Krooshof and Jacobs who, with a group of authors (Troelstra would also join them), started translating and rewriting a Scottish series of textbooks. Their first publications appeared in 1967 under the name Moderne Wiskunde [Modern Mathematics]. Pupils were introduced to set-theoretic language, which was used both in chapters on algebra and geometry; for example, calculating the solution set of an equation— while Van Hiele would simply have them calculate (a) solution(s), refraining from forcing students to use set-theoretic language at this point. The first volume of the series started with observations of a matchbox. On the second page, the student was told that the mathematical abstraction was called a cuboid and, a few pages later, nets were introduced and questions requiring some insight into folding were asked. The second chapter started with the introduction of sets and from then on set-theoretic language was used throughout the book. Although this New Math series had a rather slow start, teachers were enthusiastic about the many exercises in the textbooks, offering material for both practice and class discussion, which made the textbooks particularly useful, also for those teachers who were reluctant to present proofs to pupils. Proofs were tucked away in a couple of examples. The focus was not so much on proof, as on the precise use of language with the help of sets. The teacher who wanted proof could generalize the examples in class (Jacobs et al. 1967). Moderne Wiskunde would become the market leader in the late 1970s. The CMLW had the support of government officials in the late 1960s. At the same time, CITO was the proverbial plan B. In 1970, the secretary of state decided to grant the mathematicians their institute, be it temporarily. According to Wijdeveld (2003), the backing of Hans Freudenthal was imperative. Whether that was really crucial, or that, as we suspect, the Ministry was convinced that they needed the cooperation of the mathematicians and whether it was the personal charm of Wijdeveld that provided the final push, one cannot know for sure, but in 1971 IOWO was founded.



Episode 3: The 1970s, IOWO

In the March-April 1972 issue of Euclides, the IOWO presented itself. Editor-in-chief Krooshof, also one of the driving forces behind the Mathematics Working Group, secured the whole double issue to a (completely anonymous!) presentation of the various plans and ideas of the new Institute. In this double issue, IOWO showed that it was well aware that many opinions about mathematics education existed, and it presented itself as an inclusive organization, aimed at bringing harmony to the various organizations contributing to research in didactics of mathematics. A number of initiatives were mentioned explicitly. A consensus was still being sought on the age level of the pupil to be brought to an understanding of modern mathematics (Krooshof 1972).  Also people active in the CMLW published textbooks: Van Dormolen (1968), Van Dormolen (1969), Kuipers et al. (1969), Westerman (1969–1970), Kindt et al. (1969). 13

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One of the organizations that was mentioned explicitly was the didactics committee of the Dutch Association of Mathematics Teachers. Not coincidentally, it was launched just before the IOWO. It was clearly an attempt by the teachers’ association to claim didactics as a domain of mathematics teachers. The careful phrasing about the relation of the committee to the CMLW already voiced that. They strove for keeping in touch and listening to each other—emphasis on the last two words (Broekman 1970). In practice, the didactics committee would become a link between the teachers and the IOWO, not least because the people in the committee got along so well. Among them were Harrie Broekman (born in 1938) and Fred Goffree. Within IOWO, work was divided into two teams: Wiskobas and Wiskivon14; the former focusing on primary education, the latter on secondary education. Hans Freudenthal was the scientific director. Behind the scenes were the general director Edu Wijdeveld who was very active on behalf of the primary school plans, and the CMLW, presided by Van der Blij, which remained as a board and ran projects in higher education (Figure 11.5). Look on Luck beautifully illustrated IOWO goals. This was a project in cooperation with a couple of schools and Dutch television, aimed at students of the last grade of primary and the first grade of secondary school (11–13-year-olds). The project had New Math written all over it: The subject of statistics, television lessons, involvement of parents (see also Figure 11.4), integration of primary and secondary school, and the meticulous use of a precise language. It allowed pupils to be actively engaged in doing mathematics and used situations the children would know—such as games and scenes from the television broadcast. In class conversations, the pupils were guided to more general conclusions. For example, in these class conversations, a distinction was made between a probability you could calculate and a probability you would find by statistical means—in a 1970s humoresque way denoted as, respectively, “weetkans” [“know” probability] and “zweetkans” [“sweat” ­probability]. Precise use of natural language was favored over the use of formal language—which was fitting since many primary schools were still in the process of incorporating New Math. The project was met with a lot of enthusiasm from teachers and educational researchers (Beckers 2017b). From the start, IOWO projected to outlive its funding by creating project plans and schedules, reaching way beyond 1975 (Figure 11.6). Developmental research was central. With this phrase, IOWO emphasized its eclectic approach to developing education (see De Moor 1999). To the IOWO team, developmental research was an almost communal effort. Many of their publications do not mention names. In a special issue of Educational Studies in Mathematics (Freudenthal et  al. 1976), IOWO celebrated the retirement of Hans Freudenthal—a special celebration since the Minister had given assurances that IOWO could continue for another 5 years. The focus was on what IOWO tried to achieve with developmental research. The emphasis was on both the connection between teachers and educational researchers and between education and testing of educational results. The educational researcher would be in the school, observing activities and developing materials in cooperation with teachers, testing these materials, adjusting them, and starting again by doing observations. Contrasting CITO, micro-didactics, observation of children’s reasoning, was considered key to test the results of education. Children should preferably discover by themselves and thereby learn mathematics better. One theme that was also addressed in this special issue, was “problem-oriented mathematics instruction in rich contexts” (Freudenthal et al. 1976, p. 327). Problem orientation was a clear concept: Educational researchers wanted teaching to start from a problem that students could relate to. Pupils had to be aware of a goal that was meaningful to them. What exactly made a context “rich,” was not entirely clear. Sometimes the context was rich because it could be solved by various mathematical strategies; sometimes it was rich because it offered educational opportunities, not necessarily mathematical ones. The ideas surrounding “rich contexts” were not discussed in the special issue, but during  Wiskobas and Wiskivon are acronyms for, respectively, “Wiskunde in het basisonderwijs” [Mathematics in primary education] and “Wiskunde in het voortgezet onderwijs” [Mathematics in secondary education]. 14

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conferences they were (Archive Freudenthal, inv. nr. 408, 411). One of the successful contexts was that of Gulliver, which allowed reasoning about fractions and proportions. Freudenthal was, and would remain, skeptical about using fairy tales as a context (“midget mathematics,” he would call it), but recognized that children liked it and that it worked (Archive Freudenthal, inv. nr. 412). IOWO was not the only place where developments were taking place. In 1971, the publishing house Malmberg published the translation of some French New Math textbooks for teacher training colleges and parents (Figure 11.4) who wanted to acquaint themselves with the new mathematics. The introduction to sets (Duvert et al. 1971a) and relations (Duvert et al. 1971b) were accompanied by a more practical set of punched cards designed for classroom use, and a booklet explaining the ideas behind and practical ideas with these cards explained entirely in the language of set theory (Colomb and Glaymann 1971). Malmberg also issued Dienes’ logical blocks. Dienes’ idea that the earlier children were submerged in arithmetic and mathematics, the more natural their learning process would become, was voiced in the manual (Meijers n.d.). These activities of the publishers were natural reactions to the changing demands of the customers. IOWO would embrace many of these initiatives, and in testing and trying new things, in talking and working with the teachers, they would look for what worked best—and try to find out what it was that made it work. In that way, IOWO was inclusive. It was also inclusive in the sense that it tried to unify didactics across all levels of mathematics education. Wiskobas made itself visible in Euclides, attempting to bridge the gap between primary and secondary school teachers (Goffree et al. 1971a, b). In 1978, Treffers described a new model for describing and assessing goal-oriented mathematics education. Not only did he implicitly reject the possibility of making a mathematical model out of didactics (see Wansink 1979) but he also completely neglected the unity of mathematics. It was not a bad thing to apply local reasoning if it helped the pupil to talk about the subject. In his conception of arithmetic teaching, calculating in base 8 had the purpose of helping pupils to reflect on what they were doing when calculating in base 10. If it did not serve that purpose, it had no place in the curriculum. Developing “thinking strategies” had gained a central place in (elementary) mathematics educa-

Figure 11.4  Moderne wiskunde voor ouders [Modern mathematics for parents] (1971), one of many initiatives to involve parents in New Math–notably not the one published by Malmberg

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Figure 11.5  Bruno Ernst (left), advising Fred van der Blij (seated) during recordings of a television lesson on conic sections. (From Ernst (1968), Levende Wiskunde [Living Mathematics])

Figure 11.6  Informatie-bulletin [Information brochure], introducing IOWO, and tentatively projecting plans outliving the scheduled time of funding of the institute (1972)

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tion: Individual strategies had to be checked by the teacher but were welcomed. These thinking strategies could be stimulated by using “rich contexts”: A phrase that had acquired a more precise meaning as a context that would allow pupils to develop their own strategies, enabling them to shape their own learning process and to continue developing that process later in life (Treffers 1978). In turn, the Wiskivon team was inspired by Wiskobas. In a project to develop mathematics education for the lower vocational training institutes, both teams cooperated. New Math themes such as statistics and computer algorithms were present, but the attempts to connect them to the ideas of these pupils led to abandoning the use of set-theoretical language (IOWO 1973; Smid 2011). Wiskivon itself was successful in creating texts on a variety of subjects, from computer mathematics to complex numbers. Where Wiskobas used the journal Panama Post to reach out to its users, Wiskivon published Wiskrant, which was a newspaper-like medley, allowing the readers a glimpse at projects and offering ideas for lessons on various subjects.



Aftermath: The End of a Didactical Institute

Within IOWO, there was not a soul who believed the Ministry would discontinue their project. So, the end came as a blow to many. In an outcry toward the entire community, in various newspapers and journals, IOWO staff tried to appeal to the fact that years of expertise and an institute of international renown were about to disappear (see IOWO 1979). But all was to no avail. The end of IOWO was not the end of the projects it had launched, but the unifying charisma it had possessed as an institute, was lost. Ideas lived on in several strands of educational research and practice. In a publication by Fred Goffree on didactics of mathematics (not arithmetic!) for primary school teachers, for example, he used base 8 arithmetic as an introduction to prospective teachers to help them remind how difficult it was to learn arithmetic (Goffree 1982–1985). Most notably, the Wiskobas project remained vibrant, because Ed de Moor (1933–2016) founded the Nederlandse Vereniging voor de Ontwikkeling van het Reken-Wiskunde Onderwijs (NVORWO) [Dutch Association for the Development of Primary School Mathematics]. It was an idea that was casually brought to De Moor by Freudenthal but executed by him and many of his former Wiskobas co-workers (De Moor 2005). By using the NVORWO as a platform, they succeeded in keeping the Wiskobas spirit alive. The association continued organizing conferences for primary school teachers and managed to keep continuing education as a national activity by coordinating efforts. The mathematics department of Utrecht University showed its loyalty by keeping a small part of the Wiskivon staff. One of their first acts was the founding of the journal Nieuwe Wiskrant, as a successor of the IOWO publication Wiskrant. In hindsight, one may conclude that this was definitely a brilliant move, since the journal succeeded in rallying secondary school mathematics teachers. The government even provided some funding by asking for advice on the first restructuring of the mathematics curriculum: A clear sign that the Ministry took the new tasks in education—deciding on the curriculum—very seriously. The first report on this restructuring, a rather modest scrapping of some of the most exuberant use of set-theoretic language in use at secondary school mathematics, met with opposition (see Vredenduin 1980). Nevertheless, this opposition proved to be a minority opinion among mathematics teachers. The so-called HEWET15 project would run as developmental research in the early 1980s, under the guidance of Martin Kindt (born in 1937) and Jan de Lange (born in 1943).

 HEWET is an acronym for “Herverkaveling Wiskunde I en II” [Reshuffling Mathematics I and II].

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Final Remarks

It is time to revisit the stories of Brandenburg and Van Gelder from the introduction. Within the Dutch educational field, fragmented by pillarization and an old division of social classes, developing new ideas in education implied making coalitions and compromises. The success of the Mathematics Working Group, CMLW, and IOWO, was in their ability to gather many of the stakeholders, rallying them to the cause of improving mathematics education. It was only within CMLW and IOWO that the primary and secondary school systems really joined forces. What was specific about the Dutch approach to New Math? At first glance: Not very much. The focus on acquiring a better understanding of mathematics was part of the post-War educational renewal, inspired by new applications and a new role of mathematics in society. As everywhere, mathematicians were anxious to present the unity and relevance of their subject. This was most obviously visible in the curricula: Statistics, automatic computers, discrete mathematics, linear algebra, multi-­ base arithmetic, and other subjects were included. Teaching mathematics was approached as a mathematical task. Focus on the language used in the mathematics lessons was one of the important strands, in which many believed. However, the focus of both CMLW and IOWO was also, and more importantly, on the teachers and teacher trainers, who had to do the actual work. They were looking for ways to have their pupils best served, and that increasingly was recognized as happening when the pupil could grab hold of his own learning process. It was the focus on individuality in the Dutch educational context that led the Wiskobas project to turn to natural language. At first, natural language was used to make very clear and precise statements, but in the mid-1970s, the ideas started to shift toward making observations of the language used by pupils. The micro didactics of individual learning processes was as mathematically sound as good statistics, and it was considered infinitely better than CITO testing groups of pupils. It was faith in mathematics that opened up the possibility of founding both CMLW and IOWO. It was that very faith that made politicians embrace the more pragmatic view of CITO psychometrics, to decide on the capabilities of pupils by comparison on the basis of statistics. Instead of unifying primary and secondary school mathematics by using the language of set theory, mathematics education evolved into a more loose series of techniques and ideas, focused on local reasoning. The gain was clear: More students were able to grasp the mathematics taught in this way. What remained was a linguistic peculiarity: The Dutch still speak of primary school mathematics (reken-wiskunde), or literally “arithmetic-mathematics.” This still reminds of a unity that was perceived mathematically by mathematicians and had psychological and aesthetic connotations to the other interested parties. The history of New Math is a story of ideals that were ultimately not achieved. But it is also a story about these ideals, and how they were embedded in a culture that lost its high-strung expectations and faith in mathematicians. What made these people start in the 1950s, what kept them going in the 1960s and 1970s, and what made them continue even after IOWO was dissolved, were their ideals. Beautiful ideals that turned out to be incompatible with the changing socio-political climate.

Sources Archive Hans Freudenthal, Regionaal Archief Noord-Holland, Haarlem. Archive Jenaplan, based at the Stichting Jenaplan, Zutphen. Archive Ministry of Education, Archief Ministerie, afdeling VHMO, entry 2.14.43, block O27580, Nationaal Archief, Den Haag. Archive WVO, Archief IISG Amsterdam.

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References Alberts, G. (1998). Jaren van berekening: Toepassingsgerichte initiatieven in de Nederlandse wiskunde-beoefening 1945–1960 [Years of calculation: Application-oriented initiatives in Dutch mathematical practice 1945–1960]. Amsterdam, The Netherlands: AUP. Amsing, H. T. A. (2016). Vader van de middenschool: Leon van Gelder (1913–1981) [Father of the middle school: Leon van Gelder (1913–1981)]. In V. Busato, M. van Essen, & W. Koops (Eds.), Zeven grondleggers van de onderwijskunde [Seven founding fathers of educational science] (pp. 242–282). Amsterdam, The Netherlands: Bert Bakker. Beckers, D. (2015a). Geprogrammeerde cursussen [Programmed courses]. Euclides, 91(2), 8–9. Beckers, D. (2015b). Perfectioneringscursussen [Perfection courses]. Euclides, 90(5), 22–23. Beckers, D. (2016a). Marvellous mathematics. How mathematicians wanted to improve the quality of life in post war Europe, 1945–1975. DVT: Dějiny věd a techniky, 49(4), 227–248. Beckers, D. (2016b). Wiskundige voor een betere wereld: Hans Freudenthal (1905–1990) [Mathematician for a better world: Hans Freudenthal (1905–1990)]. In V. Busato, M. van Essen, & W. Koops (Eds.), Zeven grondleggers van de onderwijskunde [Seven founding fathers of educational science] (pp. 113–158). Amsterdam, The Netherlands: Bert Bakker. Beckers, D. (2017a). HBS: Van modern instituut tot fossiel in 100 jaar [HBS: From modern institution to fossil in 100 years]. Studium, 10, 149–159. Beckers, D. (2017b). Kijk op kans [Look on luck]. Euclides, 92(4), 23–25. Beckers, D. (2019). Why to publish on mathematics education so as to be useful? Educational studies in Mathematics and its founder Hans Freudenthal. Educational Studies in Mathematics, 101, 1–11. Blom, K. (2000). Van de acten van bekwaamheid. De akten K1, K5 en lo wiskunde en hun geschiedenis [About the acts of aptitude. The acts K1, K5 and lo mathematics and their history]. In F. Goffree, M. van Hoorn, & B. Zwaneveld (Eds.), Honderd jaar wiskundeonderwijs [One hundred years of mathematics education] (pp. 89–104). Leusden, The Netherlands: NvWL. Blom, S. (2004). Education. In D.  Fokkema & F.  Grijzenhout (Eds.), Dutch Culture in a European perspective: Accounting for the past, 1650–2000 (pp. 197–228). Assen, The Netherlands: Van Gorcum. Boermeester, C. (1955). Over meetkunde-onderwijs en psychologie [About geometry education and psychology]. Groningen, The Netherlands-Djakarta, Indonesia: J. B. Wolters. Boermeester, C., Burger, B., & Van Hiele, P. M. (1967–1971). Van A tot Z [From A to Z] (Textbook series, 15 volumes; later volumes by Van Hiele, Kok, and Schuring). Purmerend, The Netherlands: Muuses. Bos, J. (2011). M. J. Langeveld. Pedagoog aan de hand van het kind [M. J. Langeveld. Pedagogue on the basis of the child]. Amsterdam, The Netherlands: Boom. Bos, W., & Lepoeter, P. E. (1954). Wegwijzer in de meetkunde I [Guide to geometry I]. Amsterdam, The Netherlands: Meulenhoff. Bouman, F., Geerts, W., & Lock, D. J. (1965). Algebra 1. Een geprogrammeerde cursus voor het VHMO [Algebra 1. A programmed course for VHMO]. Den Haag, The Netherlands: Meulenhoff. Brandenburg, W.  J. (1968). Modernisering van het wiskunde-onderwijs [Modernization of mathematics education]. Groningen, The Netherlands: Wolters-Noordhoff. Brandenburg, W. J. (1972). Mathematische modellen in de didaktiek [Mathematical models in didactics]. Groningen, The Netherlands: Wolters-Noordhoff. Broekman, H. (1970). Didactiekcommissie van de Nederlandse Vereniging van Wiskundeleraren [Didactics committee of the Dutch Association of Mathematics Teachers]. Euclides, 46, 8–11. Broekman, H. (2005). Helpen met leren helpt! Een hommage aan Pierre van Hiele [Helping to learn helps! A tribute to Pierre van Hiele]. Euclides, 80, 266–270. Bunt, L. N. H. (1949). De leerstof van ons wiskunde-onderwijs. Een onderzoek naar opvattingen en gebruiken dienaangaande [The subject matter of our mathematics education. An examination of views and practices in this regard]. Groningen, The Netherlands-Batavia, Indonesia: J. B. Wolters. Bunt, L. N. H. (1950). The development of the ideas of number and quantity according to Piaget. A synthesis and a criticism. Groningen, The Netherlands: J. B. Wolters. Bunt, L. N. H. (1954). Van Ahmes tot Euclides. Hoofdstukken uit de geschiedenis van de wiskunde [From Ahmes to Euclid. Chapters in the history of mathematics]. Groningen, The Netherlands: J. B. Wolters. Bunt, L.  N. H. (1956). Statistiek voor het voorbereidend hoger en middelbaar onderwijs [Statistics for preparatory higher and secondary education]. Groningen, The Netherlands-Djakarta, Indonesia: J. B. Wolters. Bunt, L.  N. H. (1968). Algebra voor de brugklas. Een geprogrammeerde cursus [Algebra for the first year. A programmed course]. Groningen, The Netherlands: Wolters. Bussato, V. (2014). A. D. De Groot (1914–2006): Meester in de psychologie [A. D. De Groot (1914–2006): Master of psychology]. In V.  Bussato, M. van Essen, & W.  Koops (Eds.), Van fenomenologie naar empirisch-analyt-

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ische ­psychologie [From phenomenology to empirical-analytical psychology] (pp.  249–314). Amsterdam, The Netherlands: Bert Bakker. CMLW. (1967a). Heroriënteringscursus “Computer Wiskunde.” Algorithmen en rekenmachine [Reorientation course “Computer Mathematics.” Algorithms and calculator]. Utrecht, The Netherlands: CMLW. CMLW. (1967b). Interimrapport met toegevoegde discussienota’s [Interim report with added discussion notes]. Utrecht, The Netherlands: CMLW. CMLW. (1969). Toelichting op het leerplan wiskunde van de mavo, onderbouw havo en onderbouw vwo [Explanatory notes to the curriculum for mathematics in mavo, junior havo and junior vwo]. Utrecht, The Netherlands: CMLW. Colomb, J., & Glaymann, M. (1971). Verzamelingen, logica en ponskaarten [Sets, logic and punch cards]. ‘s Hertogenbosch, The Netherlands: L. C. G. Malmberg. CPS. (1964). Stemmen uit de praktijk. Vernieuwing van het wiskundeonderwijs [Voices from the field. Renewing mathematics education]. Den Haag, The Netherlands: CPS. De Bock, D., & Vanpaemel, G. (2019). Rods, sets and arrows. The rise and fall of modern mathematics in Belgium. Cham, Switzerland: Springer. De Groot, A.  D. (1957). Een inzicht-test voor meetkunde [A test for understanding in geometry]. Euclides, 32(6), 218–224. De Groot, A.  D. (1966). Vijven en zessen. Cijfers en beslissingen: Het selectieproces in ons onderwijs [Fives and sixes. Grades and decisions: The selection process in our education system]. Groningen, The Netherlands: Wolters-Noordhoff. De Jong, W. (2020). Civic education and contested democracy. Towards a pedagogic state in the Netherlands post 1945. Cham, Switzerland: Springer. De Miranda, J. (1956). Het afbeelden van getallenrijen en het werken met grafieken in de wiskundeleergang van de eerste klas M.O. [Mapping sequences of numbers and working with graphs in the first grade of secondary education]. Euclides, 32(1), 15–27. De Miranda, J. (1957). Schoolvorderingentest of onderzoek naar verkregen inzicht [School progress test or examination of acquired understanding]. Mededelingenblad van de Wiskunde Werkgroep der WVO, 5(5), 4–6. De Moor, E. (1999). Van vormleer naar realistische meetkunde [From primary school geometry to realistic geometry]. Utrecht, The Netherlands: CD-B. De Moor, E. (2001). Het kistje van Van Albada [Van Albada’s box]. De Nieuwe Wiskrant, 21(2), 32–43. De Moor, E. (2005). HF en de NVORWO [HF and the NVORWO]. In H. ter Heege, T. Goris, R. Keijzer, & L. Wesker (Eds.), Freudenthal 100. Speciale ter gelegenheid van de honderdste geboortedag van Professor Hans Freudenthal [Freudenthal 100. Special edition [of Panama Post and Nieuwe Wiskrant] on the occasion of the 100th birthday of Professor Hans Freudenthal] (pp. 132–134). Utrecht, The Netherlands: Freudenthal Institute. De Moor, E., & Groen, W. (2012). Kijkmeetkunde, een ander uitgangspunt (1970–1980) [Vision geometry, a new perspective (1970–1980)]. Nieuw Archief voor Wiskunde, 5(13), 248–253. Duvert, L., Gauthier, R., & Glaymann, M. (1971a). Moderne wiskunde-opdrachten serie I.  Verzamelingen [Modern mathematics tasks series I. Sets]. ‘s Hertogenbosch, The Netherlands: L. C. G. Malmberg. Duvert, L., Gauthier, R., & Glaymann, M. (1971b). Moderne wiskunde-opdrachten serie II. Relaties [Modern mathematics tasks series I. Relations]. ‘s Hertogenbosch, The Netherlands: L. C. G. Malmberg. Freudenthal, H. (1975a). Een internationaal vergelijkend onderzoek over wiskundige studieprestaties [An international comparative survey of mathematical study performance]. Pedagogische Studiën, 52, 43–55. Freudenthal, H. (1975b). Pupils’ achievements internationally compared—the IEA. Educational Studies in Mathematics, 6, 127–186. Freudenthal, H., Janssen, G., Sweers, W., Van Barneveld, G., Bosman, J., Van den Brink, F., … Wijdeveld, E. (1976). Five years IOWO. Educational Studies in Mathematics, 7(3), 185–367. Freudenthal-Lutter, S. J. C. (1969). De logiblokken [The logical blocks]. Denken en Rekenen, 1, 10–18. Goffree, F. (1982–1985). Wiskunde & didactiek voor aanstaande leraren basisonderwijs [Mathematics & didactics for prospective primary school teachers] (3 volumes). Groningen, The Netherlands: Wolters-Noordhoff. Goffree, F., Sinnema, J.. & Wijdeveld, E. (1968). Modern math in de basisschool. Indrukken van een experiment in Edinburgh [Modern math in elementary school. Impressions of an Edinburgh experiment]. Groningen, The Netherlands: Wolters-Noordhoff. Goffree, F., Treffers, A., & Wijdeveld, E. (1971a). Wiskunde op de basisschool? (I) [Mathematics in primary school? (I)]. Euclides, 46(8), 309–314. Goffree, F., Treffers, A., & Wijdeveld, E. (1971b). Wiskunde op de basisschool? (II) [Mathematics in primary school? (I)]. Euclides, 47(2), 45–53. Haas, E. (1995). Op de juiste plaats. De opkomst van de bedrijfs- en schoolpsychologische beroepspraktijk in Nederland [In the right place. The emergence of the professional practice of business and school psychology in the Netherlands]. Hilversum, The Netherlands: Verloren.

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Hooghiemstra, D. (2013). De geest in dit huis is liefderijk. Het leven en De Werkplaats van Kees Boeke (1884–1966) [The spirit in this house is lovely. The life and school of Kees Boeke (1884–1966)]. Amsterdam, The Netherlands: Arbeiderspers. IOWO. (1973). Wiskunde lbo. Startpunt leerplanontwikkeling [Mathematics lbo. Starting point curriculum development]. Utrecht, The Netherlands: IOWO. IOWO. (1979). Positie IOWO in discussie [Position of IOWO in discussion]. Euclides, 54(10), 404. Jacobs, H.  J., Kniep, W.  J., & Krooshof, G. (1967). Moderne wiskunde voor voortgezet onderwijs. Deel 1 voor de brugklas [Modern mathematics for secondary education. Part 1 for the first year]. Groningen, The Netherlands: Wolters-Noordhoff. Karman, D. (n.d.). Wat is mijn kans? Een geprogrammeerde instructie [What are my chances? A programmed instruction]. Utrecht, The Netherlands: CMLW. Kennedy, J. C. (2017). A concise history of The Netherlands. Cambridge, United Kingdom: CUP. Kindt, M., & Maassen, A. J. T. (1967). Proeve van een gemoderniseerde algebra [First draft of modernized algebra] (2 volumes). Utrecht, The Netherlands: Pressa Trajectina. Kindt, M., Maassen, A.  J. T., & Van Oosten, C.  P. S. (1969). Moderne algebracursus [Modern algebra course]. ‘s Hertogenbosch, The Netherlands: Malmberg. Krooshof, G. (1968). Wiskunde is het ontdekken van relaties [Mathematics is the discovery of relations]. In L. van Gelder, E. Wijdeveld, F. Goffree, & G. Krooshof (Eds.), Moderne wiskunde in het basisonderwijs [Modern mathematics in primary education] (pp. 61–80). Groningen, The Netherlands: Wolters-Noordhoff. Krooshof, G. (Ed.). (1972). I.O.W.O. ter gelegenheid van de officiële opening [I.O.W.O. on the occasion of its official opening]. Euclides, 47(7–8), 237–333. Kuipers, M. G., Siepelinga, J., Troelstra, R., & Tromp, G. (1969). Gemoderniseerde meetkunde. Op basis van afbeeldingen [Modernized geometry. On the basis of mappings]. Groningen, The Netherlands: Wolters-Noordhoff. La Bastide van Gemert, S. (2015). All positive action starts with criticism. Hans Freudenthal and the didactics of mathematics. Dordrecht, The Netherlands: Springer. Leeman, H. T. M. (1961). Antwoord op de installatiereden [van Stubenrouch], uitgesproken door de voorzitter [Response to the installation speech [by Stubenrouch], delivered by the president]. Den Haag, The Netherlands: CMLW. Levering, B. (2015). Praktische wetenschap als levenslange ambitie. Martinus Jan Langeveld (1905–1989) [Practical science as a lifelong ambition. Martinus Jan Langeveld (1905–1989)]. In V. Bussato, M. van Essen, & W. Koops (Eds.), Vier grondleggers van de pedagogiek [Four founding fathers of pedagogy] (pp. 97–166). Amsterdam, The Netherlands: Bert Bakker. Loonstra, F., & Vredenduin, P.  G. J. (Eds.). (1962). Modernization of mathematical teaching in the Netherlands. Groningen, The Netherlands: J. B. Wolters. Maassen, A.  J. T., & Oosten, C.  P. S. (1961). Planimetrie I [Plane geometry]. ‘s Hertogenbosch, The Netherlands: Malmberg. Maassen, A. J. T., & Oosten, C. P. S. (1962). Mappings, relations, functions. In F. Loonstra & P. G. J. Vredenduin (Eds.), Modernization of mathematical teaching in the Netherlands (pp. 43–47). Groningen, The Netherlands: J. B. Wolters. Maassen, J. (2000). De vereniging en het tijdschrift [The association and the journal]. In F. Goffree, M. van Hoorn, & B. Zwaneveld (Eds.), Honderd jaar wiskundeonderwijs [One hundred years of mathematics education] (pp. 43–56). Leusden, The Netherlands: NvWL. Meijers, J. (n.d.). 35 kleuterspelletjes met Malmbergs Logiblokken [35 preschool games with Malmberg’s logical blocks]. ‘s Hertogenbosch, The Netherlands: Malmberg. Mellink, A.  G. M. (2014). Worden zoals wij: Onderwijs en de opkomst van de geïndividualiseerde samenleving sinds 1945 [Becoming like us: Education and the rise of the individualized society since 1945]. Amsterdam, The Netherlands: UvA. Monna, A., & Van der Put, M. (2004). Antonie Frans Monna. Ambtenaar en wiskundige [Antonie Frans Monna. Civil servant and mathematician]. Nieuw Archief voor Wiskunde, 5(2), 136–146. Pingel, F. (2010). UNESCO Guidebook on textbook research and textbook revision. 2nd edition. Paris, France-­ Braunschweig, Germany: UNESCO. Smid, H. J. (2000). Wiskundeonderwijs op bijna vergeten scholen. Wiskunde op de (m)ulo. [Mathematics education in nearly forgotten schools. Mathematics at the (m)ulo]. In F. Goffree, M. van Hoorn, & B. Zwaneveld (Eds.), Honderd jaar wiskundeonderwijs [One hundred years of mathematics education] (pp. 121–138). Leusden, The Netherlands: NvWL. Smid, H. J. (2004). Bos en Lepoeter of: de terugkeer van de zelfwerkzaamheid [Bos and Lepoeter or the return of self-­ effort]. Euclides, 79, 254–259. Smid, H. J. (2011). Het geheugen [Memory]. Euclides, 87, 317–320.

11  A History of New Math in The Netherlands

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Smid, H.  J. (2012). The first international reform movement and its failure in the Netherlands. In K.  Bjarnadóttir, F. Furinghetti, J. M. Matos, & G. Schubring (Eds.), “Dig where you stand” 2. Proceedings of the Second International Conference on the History of Mathematics Education (pp. 463–475). Lisbon, Portugal: New University of Lisbon. Smid, H. J. (2015). Zestig jaar hart voor wiskundeonderwijs [Sixty years of heart for mathematics education]. Utrecht, The Netherlands: NVvW. Smid, H. J. (2017). Johan Wansink and his role in Dutch mathematics education. In K. Bjarnadóttir, F. Furinghetti, M. Menghini, J. Prytz, & G. Schubring (Eds.), “Dig where you stand” 4. Proceedings of the Fourth International Conference on the History of Mathematics Education (pp. 369–381). Rome, Italy: Edizioni Nuova Cultura. Stubenrouch, G.  C. (1961). Installatierede, uitgesproken door de staatssecretaris van Onderwijs, Kunsten en Wetenschappen [Installation speech delivered by the State Secretary of Education, Arts and Sciences]. Den Haag, The Netherlands: CMLW. Treffers, A. (1978). Wiskobas, doelgericht [Wiskobas, goal-oriented]. Utrecht, The Netherlands: IOWO. Troelstra, R., & Kuipers, A. M. (1965). Proeve van een gemoderniseerde meetkunde [First draft of a modernized geometry]. Utrecht, The Netherlands: CMLW Van Achter, V. (1969). Nicole Picard. Denken en Rekenen, 1, 33–37. Van der Sluis, A. (1969). Computers en algoritmen [Computers and algorithms]. Groningen, The Netherlands: Wolters-Noordhoff. Van Dormolen, J. (1968). Algebra voor havo en vwo [Algebra for havo en vwo]. Den Haag, The Netherlands-Brussels, Belgium: Van Goor. Van Dormolen, J. (1969). Analyse volgens het leerplan wiskunde I voor het vwo [Analysis according to the curriculum mathematics I for vwo]. Den Haag, The Netherlands-Brussels, Belgium: Van Goor. Van Essen, M. (2006). Kwekeling tussen akte en ideaal [Prospective primary school teacher between act and ideal]. Nijmegen, The Netherlands: SUN. Van Gelder, L. (1958). Het leerproces van het rekenen [The learning process of arithmetic]. In De didactiek van het rekenen. Verslag van de werkweek Leraren wiskunde aan Prot. Christelijke kweekscholen [The didactics of arithmetic. Report of the work week Teachers of mathematics at Protestant Christian teacher training schools for primary education] (pp. 28–57). n.p.: CPS. Van Gelder, L. (1959). Grondslagen van de rekendidactiek [Foundations of didactics of arithmetic]. Groningen, The Netherlands: J. B. Wolters. Van Gelder, L. (1968a). Modern rekenen. New Mathematics in de basisschool [Modern arithmetic. New Mathematics in primary school]. In L. van Gelder, E.  Wijdeveld, F.  Goffree, & G.  Krooshof (Eds.), Moderne wiskunde in het basisonderwijs [Modern mathematics in primary education] (pp.  5–16). Groningen, The Netherlands: Wolters-Noordhoff. Van Gelder, L. (1968b). Veranderingen in het onderwijs [Changes in education]. Groningen, The Netherlands: Wolters-Noordhoff. Van Hiele, P. M., & Van Hiele-Geldof, D. (1948). Werkboek der meetkunde [Workbook of geometry]. Purmerend, The Netherlands: Muuses. Van Hiele, P.  M., & Van Hiele-Geldof, D. (1950). Werkboek der algebra [Workbook of algebra]. Purmerend, The Netherlands: Muuses. Van Hiele, P. M., & Van Hiele-Geldof, D. (1963). Van figuren naar begrippen [From figures to concepts]. Purmerend, The Netherlands: Muuses. Van Hoorn, M., & Guichelaar, J. (2018). De geschiedenis van het tijdschrift Pythagoras deel 1: Oprichting en succes [The history of the journal Pythagoras part 1: Founding and success]. Nieuw Archief voor Wiskunde, 5(19), 9–18. Vos, J., & Van der Linden, J. (2004). Waarvan akte. Geschiedenis van de MO-opleidingen, 1912–1987 [Noted. History of secondary education programs, 1912–1987]. Assen, The Netherlands: Van Gorcum. Vredenduin, P. G. J. (1966). De bewijskracht van de diagrammen van Venn en de implicatie [The power of proof of Venn diagrams and the implication]. Euclides, 42, 33–41. Vredenduin, P. G. J. (1967). Verzamelingen [Sets]. Groningen, The Netherlands: J.B. Wolters. Vredenduin, P. G. J. (1980). Verzamelingen, functies, relaties en het HEWET-rapport [Sets, functions, relations and the HEWET report]. Euclides, 56, 49–53. Wansink, J. H. (1956). Didactische revue [Didactic review]. Euclides, 32, 54–69. Wansink, J. H. (1966). Didactische oriëntatie voor wiskundeleraren I [Didactic orientation for mathematics teachers I]. Groningen, The Netherlands: J. B. Wolters. Wansink, J. H. (1979). Wat beoogt Wiskobas? [What does Wiskobas aim at?]. Euclides, 55, 2–6. Wessels, J. (1969). Rekenen met kansen [Calculating with probabilities]. Groningen, The Netherlands: Wolters-Noordhoff. Westerman, L. R. J. (1966). Meetkunde met vectoren [Geometry with vectors]. Utrecht, The Netherlands: CMLW. Westerman, L.  R. J. (1969–1970), Meetkunde met vectoren I, II [Geometry with vectors I, II]. Groningen, The Netherlands: Wolters-Noordhoff.

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Wijdeveld, E. J. (1969a). Nieuwe Wiskunde I: Taal en logica [New Mathematics I: Language and logic]. Groningen, The Netherlands: Wolters-Noordhoff. Wijdeveld, E.  J. (1969b). Nieuwe Wiskunde II: Structuren [New Mathematics II: Structures]. Groningen, The Netherlands: Wolters-Noordhoff. Wijdeveld, E. (2003). Omzien in verwondering [Looking back in wonder]. Euclides, 78(5), 218–225. Wijdeveld, E. J., & Goffree, F. (1968). Experimenten en onderzoekingen [Experiments and investigations]. In L. van Gelder, E. Wijdeveld, F. Goffree, & G. Krooshof (Eds.), Moderne wiskunde in het basisonderwijs [Modern mathematics in primary education] (pp. 17–60). Groningen, The Netherlands: Wolters-Noordhoff. Zwaneveld, B., & De Bock, D. (2019). Lucas Bunt and the rise of statistics education in the Netherlands. In U. T. Jankvist, M. van den Heuvel-Panhuizen, & M. Veldhuis (Eds.), Proceedings of the Eleventh Congress of the European Society for Research in Mathematics Education (pp. 2196–2203). Utrecht, The Netherlands: Utrecht University.

Chapter 12

Nordic Cooperation on Modernization of School Mathematics, 1960–1967 Kristín Bjarnadóttir

Abstract  After a seminar on new thinking in school mathematics held in Royaumont in 1959, four Nordic countries, Denmark, Finland, Norway, and Sweden, agreed to cooperate on school mathematics reform. A joint committee, the Nordic Committee for the Modernization of School Mathematics, declared a need for revising aims and content. Concepts from set theory, and the function concept, as well as greater precision in presentation, could promote interest, insight, and understanding of the subject. Working teams for three school levels: grades 1–6, 7–9, and 10–12, wrote directives for joint experimental texts and teacher guides. A total of 1310 classes in the four countries took part in experimental instruction. More than 180 000 copies of experimental texts were produced. The Nordic cooperation on modernizing mathematics teaching was a remarkable experiment on the cooperation of independent nations. Gradually, each nation went its own way in grades 1–9, where comprehensive 9-year compulsory education was underway in each country. The experiments initiated a long-needed discussion about curriculum, stagnated in certain routines and topics, and had an impact on curriculum development in the redefined school systems. In grades 10–12, steps were taken to create coherence between the gymnasia and university level. Keywords  Agnete Bundgaard · Axioms of the number field · Bent Christiansen · Comprehensive 9-year compulsory education · Experimental instruction · Experimental texts · Function concept · Lennart Sandgren · Matts Håstad · Modern mathematics · Nordic Committee for the Modernization of School Mathematics · Nordic cooperation · Place value notation systems · Probability · School mathematics reform · Set concept · Statistics · Vectors

Introduction An influential seminar on new thinking in school mathematics was held in Royaumont, France in 1959 by the OEEC, the Organization for European Economic Cooperation, later OECD. One of the seminar’s final recommendations was that each country would have its own unique way of making a reform—of introducing new material, and of experimenting with possible programs. Channels were to be provided for communicating the results of these programs and of experiments between countries to utilize the best thinking of all countries in stimulating new ideas (OEEC 1961, p. 125). K. Bjarnadóttir (*) School of Education, University of Iceland, Reykjavík, Iceland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_12

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International school mathematics reform movements were parts of reconsiderations of diverse humanitarian values following World War II, such as access to education for all, the democratization of mathematics, the relevance of psychology in mathematics education, and the need to take all school levels from primary to university level into consideration and ensure coherence. Economic concerns were the main driving force of OEEC in its engagement in the Royaumont Seminar. The premises of the Royaumont Seminar were that changes in cultural, industrial, and ­economic patterns called for a basic change in educational patterns. More people must be better trained in scientific knowledge (OEEC 1961, p. 107). Education, in particular technical education, would contribute substantially to economic and social progress. This argument was well-received in a world that was recovering from the calamities of war. The Nordic participants in the Royaumont Seminar agreed to cooperate on reforming their school mathematics. Three Nordic countries were represented at the seminar: Denmark, Norway, and Sweden. Nordic cooperation between the three countries and Finland operated during 1960–1967. An official report was published by the end of the project, written in Nordic languages (NR 1967b), while some sections were also published in English (NR 1967a). The final report gave an account of the results, but not necessarily of the operating process. Documents on the activities of the joint Nordic reform movement, used in this account, are preserved in the Swedish National Archives in Stockholm. This chapter contains an examination of the Nordic cooperation in modernizing school mathematics and the driving forces behind it. The research method draws on archived documents about the cooperation, the final report, and scholars’ accounts of the introduction and implementation of modern mathematics in the Nordic countries. The question is to which degree four independent states with some heritage in common, but different situations in other respects could create a common policy in the field of mathematics education, in particular at the compulsory school level.

The Nordic Countries The Nordic countries are a well-defined group. They cooperate at the Nordic Council, an official body for formal inter-parliamentary cooperation. The Nordic Council was founded in 1952 by Denmark, Iceland, Norway, and Sweden whose languages are of North-Germanic origin. The four countries were also members of OEEC, later OECD. Finland, not a member of OEEC, joined the Nordic Council in 1955. The Nordic countries have a long history in common. Iceland was settled from Norway around the year 900 and became its tributary in 1262. Denmark, Norway, and Sweden joined in the Kalmar Union from 1397 until 1523 when Sweden separated from the union, and Norway and Iceland became under Danish rule. Finland was part of Sweden from around 1150 to 1809. From the twelfth century until the 1350s, large-scale migration from Sweden to Finland resulted in Swedish settlements in southern and western coastal areas of Finland. Swedish became an official language in Finland, presently spoken by 5% of the population, while Finnish, spoken by the majority, is a Uralic language, related to Estonian and Hungarian. In the sixteenth century, Lutheranism, the evangelic Lutheran protestant religion, became formally established in various principalities, including the kingdoms of Sweden and Denmark. The Protestant Reform, according to its general approach of assuming the population was literate, set out to issue school ordinances in subsequent decades to establish gymnasia in larger towns that would prepare its students for university studies. The gymnasium school system was introduced in Sweden in 1626. In Denmark, Norway, and Iceland, the former catholic cathedral schools were gradually converted to gymnasia. In the 1960s, gymnasia in all the Nordic countries began at grade 10 (i.e., at the age of 16). All the Nordic countries had their share of World War II. Denmark and Norway were occupied by the Germans with considerable resistance. Finland was in war with the Soviet Union during 1939–

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1940, followed by conflicts, both against and alongside the Allies. Iceland was first occupied by the British and later protected by the United States. Sweden was neutral but received refugees from its neighboring countries. There was a need to restore the societies after the war. Denmark, Norway, and Finland needed economic restoration while Sweden was least affected due to its metal resources and steel industry. When the Nordic countries took up formal cooperation by establishing the Nordic Council in 1952, the premises were that each nation could use its own North-Germanic language. This was a hindrance to the Finns of whom 95% had a language of different origin as a mother tongue, even if Swedish was also an official language taught at schools. It also hindered Icelanders whose language had developed differently from the others in its isolation. To counteract this, Danish was the first foreign language at school in Iceland until 1999. By 1960, when a Nordic cooperation on reforming school mathematics was formed, the total population in the five countries was around 20  million inhabitants, where Sweden was the most populous, with 7.5 million, Denmark, Norway, and Finland with around 4 million each, and Iceland with 0.2 million having only 1% of the total population. In the 1960s, the Nordic countries were reorganizing their education systems and extending their compulsory education to a uniform 9-year program. Grade 1 of compulsory education began in the year when a pupil became 7 years old. Denmark had new school legislative acts by which compulsory education became stepwise a 9-year homogeneous school: The 1958 Act, followed by curriculum guidelines—popularly called Blå betænkning, the Blue Memorandum—and the 1975 Act. Earlier, a decision had to be made at the age of 13 if the pupil would prepare for technical education or for gymnasium, leading to university entrance (Skolelovgivningen i Danmark, efter 1521). Similar developments, which aimed at the democratization of the school systems, took place in Norway, Sweden, Finland, and Iceland. In all the countries, the debates extended over many years, coinciding with the period of the implementation of modern mathematics. The grounds for reforms were fertile.

Modern Mathematics In the mid-1960s, when modern mathematics was spreading around the world, the arguments were that school mathematics was considered fallen into a rigid system and needed new thinking. The basic premise of modern mathematics in schools was to enhance understanding and place less emphasis on training skills and rote learning. A part of the ideology was that new concepts could unite the many topics of mathematics, providing increased clarity and exactness, and reducing the gap between university mathematics and school mathematics. A central factor of the modern mathematics curriculum was the set concept. Another important concept was function, which had been introduced at the gymnasium level in the early 1900s with the Meraner reform movement, led by Felix Klein (Krüger 2019). Some elements of logic were often included. Attached to the new concepts in school mathematics was a new symbolic language. There was an emphasis on structure, such as basing the arithmetic operations on the axioms of the rational and real numbers. Geometry was built up more as an axiomatic system than in terms of mensuration. New topics were also introduced. Elementary statistics, elementary probability, and the Cartesian coordinate system were introduced at the compulsory school level, as were elementary combinatorics, modular arithmetic, and place-value notation systems with bases different from ten, such as five or seven, and the binary notation system. This could lead to the group concept, another important structure.

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Soon after the Royaumont Seminar, Lennart Sandgren, representing the Swedish Ministry of Education at the seminar, wrote letters to a Danish guest speaker at Royaumont, Svend Bundgaard (U 2)1, and the Norwegian delegate Ingebrigt Johansson (U 3). In the letters, dated December 31, 1959, Sandgren suggested that it would be most suitable to organize the cooperation under the Cultural Commission of the Nordic Council. The letters were accompanied by a memorandum (U 1) with five points about a prospective Nordic cooperation: 1. At the Royaumont Seminar, it was assumed that modernizing mathematics teaching demanded new textbooks and support to teachers, in addition to groups of experts who would prepare syllabuses, etc. It was suggested that OEEC would support groups of regional cooperation. 2. The participants at the seminar, among them Svend Bundgaard and Ole Rindung from Denmark, Ingebrigt Johansson and Kay Piene from Norway, and Lennart Sandgren from Sweden, agreed that there were good reasons for cooperation within Scandinavia, for example the school systems and syllabuses were relatively similar. The benefits were many: More possibilities to engage experts, any duplication of work could be avoided, cooperation was economically preferable, and more experimental results could be obtained. 3. A regional committee would be established with members from Denmark, Iceland, Norway, and Sweden, and, if finances would not be a hindrance [from OEEC], Finland should be added. The regional committee should be established as soon as possible. Its tasks would be to work out syllabuses, create textbooks for the pupils and teachers’ guides, evaluate the need for reforms in teacher education and for in-service courses for teachers, and initiate experiments with new textbooks. Furthermore, based on achieved experience, to propose desirable changes in the school mathematics programs could be formulated and proposed to governmental bodies in the various countries. The committee’s concern would be all school levels, while the first steps would be at grades 7–9 and the gymnasium level, grades 10–12. The committee could make use of material from other bodies, such as the School Mathematics Study Group (SMSG) at Yale University in the United States. The material produced by the committee would be provisional and be made available to whoever would use it as a basis for textbooks produced by textbook publishers. 4. The composition of the committee was specified. It should include university mathematicians, mathematics teachers, inspectors, and users of mathematics, such as engineers. There were also instructions on the committee’s working teams for each school level, and their experts; the frequency of their meetings; and secretarial staff to begin with. 5. A provisional budget was presented. Denmark, Finland, Norway, and Sweden were to share costs in the proportion 4 : 4 : 4 : 7 as Sweden had the largest population. Iceland was not mentioned. The share of Iceland with its tiny population would be small, while transport to and from Iceland was expensive at that time. No documentation has been found at the National Archives of Iceland about the matter.

 References marked U# denote numbered outgoing correspondence, while those marked I# denote incoming correspondence from and to the central office of the NKMM committee in Stockholm. The documents are preserved at the Swedish national archives (SE/RA/2717). 1

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Nordic Committee for the Modernization of School Mathematics A meeting of the new committee, appointed by the Nordic Cultural Commission, was held on October 3–4, 1960, in Stockholm, chaired by Sandgren, with four representatives from each country, Denmark, Finland, Norway, and Sweden (U 8; U 9). The 16 representatives comprised the committee, which became called, in Swedish, Nordiska kommittén för modernisering av matematikundervisningen (NKMM) [Nordic Committee for the Modernization of School Mathematics]. The members of the committee were as follows: • From Denmark (D): Erik Kristensen (1922–2006), Bent Christiansen (1921–1996) (see Figure  12.1), Ole Rindung (1921–1984), and Agnete Bundgaard (1909–1995) (sister of Svend Bundgaard (1912–1984)). • From Finland (F): Matti Koskenniemi (replaced by Paavo Malinen in 1962), Yrjö Juve (1919– 1967), Harkko Helvelahti, and Inkeri Simola (1930–2012). • From Norway (N): Ingebrigt Johansson (1904–1987), Kay Piene (1904–1968), Henrik Halvorsen (replaced in 1965 by Ragnar Solvang (1930–2018)), and Torgeir Bue (1912–1995) (replaced by Karsten Kjelberg in 1963). • From Sweden (S): Lennart Sandgren (1926–2009) (see Figure  12.2), Sixten Thörnquist, Göran Holmström, and Thure Öberg (resigned in 1963). The Swede Matts Håstad (1931–2019) was appointed as secretary, to be understood as executive officer, and the committee’s address was decided within the Swedish Ministry of Education in Stockholm. The costs were to be shared in the proportions 1 : 1 : 1 : 2, Sweden carrying the largest share. The committee decided to set up three working teams, each working on a specified school level. They were to write preliminary drafts of syllabuses. The teams consisted of the following members: • Grades 1–6: Torgeir Bue (N), Agnete Bundgaard (D), Veikko Heinonen (F), and Charles Hultman (S). Erik Kristensen (D) was appointed as an expert in clear mathematical questions. • Grades 7–9: Bent Christiansen (D), Kay Piene (N), and Inkeri Simola (F). • Grades 10–12: Ingebrigt Johansson (N), Ole Rindung (D), and Sixten Thörnquist (S). Plans for writing proposals for course syllabuses and teacher guides were discussed. The teams for grades 7–9 and grades 10–12 were expected to present their proposals in January 1961. More time

Figure 12.1  Bent Christiansen

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Figure 12.2  Lennart Sandgren. (Blekinge museum, Sweden)

was needed for the team for grades 1–6, as it was necessary to start by analyzing the present material with respect to pedagogical, psychological, and mathematical aspects, so proposals on course syllabuses for that level were to be expected in the autumn 1961. Small-scale teaching experiments could be performed in the academic years following the presentations of the proposals. On November 23, 1960, Sandgren wrote to Rindung (U 23), reporting that the possibilities of grants from the OEEC were more than expected.



Experimental Texts and Experimental Teaching

The initial working premises of the committee were that there was a need for revising aims and developing a new content, and secondly that better results might be achieved by changing the way mathematics was presented in the schools. Simple concepts from set theory, and the general concept of function, as well as greater precision in presentation, could promote interest, insight, and understanding of the subject alike (NR 1967a, p. 45). The working teams for the different levels laid down general principles in their directives to be followed in writing experimental texts. The texts were to be given a relatively broad structure so that pupils—except at the lowest level—could study them on their own. Brief comments to teachers were to be provided with hints on methods. Teams of writers (2–3 persons each team) to work out and prepare the texts were appointed. They received a set fee, but the copyright for the texts reverted to the writers after a given period, usually 4 years. Some texts were ready as early as the summer of 1961, others were successively completed through the beginning of 1966. According to the final report, 90 classes took part in experimental instruction at grades 1–6, 450 classes at grades 7–9, and 770 classes at grades 10–12. In most classes, more than one experimental text was used, and more than 180 000 copies of each experimental texts were produced. It was assumed that activities at grades 1–6 would be for unstreamed classes. Work at higher grades was restricted mainly to streams taking more extensive courses in mathematics. Information was collected from the teachers, various tests were made, but for practical and theoretical reasons it was hardly possible to make any detailed comparison with traditional teaching. For example, in most cases only a minor part of the new teaching matter was in common with traditional teaching (NR 1967a, pp. 45–46). The teachers taking part in these experiments were assumed to comprise a positive selection of skilled and experienced staff. However, they often faced entirely new material, and, in many cases,

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they had no opportunity to study the entire series of texts in advance. Their reports show that when teaching was repeated for a second year, results were better than in the first year. Experiments started simultaneously in several grades to acquire a wide experience in a reasonable time. The same concepts belonging to modern mathematics, such as the set concept and its derived concepts, had therefore to be treated from scratch at several different levels. The comparison with traditional teaching was influenced by the fact that the experimental texts—unlike most school textbooks—were not based on extensive practical experience in teaching a known body of matter (NR 1967a, pp. 46–47). In the following, the work process is expounded in some detail according to archived documents about the activities of the NKMM (SE/RA/2717).

Grades 1–6 The working team for grades 1–6, Bundgaard (D), Bue (N), Heinonen (F), and Hultman (S), began its activity in a meeting during December 9–11, 1960. Agnete Bundgaard wrote to Sandgren (I 39) on January 2, 1961, reporting on the meeting as the contact person of the team to the committee. The members were becoming acquainted and there were different opinions. They would like to meet next time for 8 days to do some proper work. Language problems emerged. The Finn, Dr. Heinonen, did not understand much Swedish and no Danish. Matts Håstad replied on January 11 (U 42), regretting Heinonen’s case, but Mats Björkman (S) was willing to assist the team as expert. Håstad recommended that the team would meet during 3 days before the NKMM meeting in February. A directive to the writers of experimental texts for grades 1–6 seems to have been sent out in October 1961 but is documented in a revised form in early November 1961 (U 213). The main document had 14 pages and was written in Swedish. The stated main goal was to make the children confident with the number concept. To that end, children were to become confident with what it means when two [finite] sets contain the same number of elements, and when two sets contain a different number of elements. In connection to that, the concepts “more than” and “less than” were to be practiced. This would be made clear by pairing elements in the two sets. The addition was to be presented as a union of disjoint sets where the commutative law would emerge, as well as the associative law. Multiplication was to be introduced as repeated addition. Children were expected to create their own addition table and the multiplication table. Subtraction was introduced as a “fill-in” method, such as 3 + □ = 7. The number 0 would be introduced that way, and eventually, the unsolvability of the problem 3 + □ = 1. The algorithms for multi-digit multiplication and division would be more detailed and different from what was commonly used. Decimal fractions were to be introduced before common fractions which were to be introduced in practical situations. Algorithms for common fractions were to be confined to simple cases and revisited in grades 7 and 8. Attached were three appendices. There was a 12-page appendix from Torgeir Bue in Norwegian where sets were not mentioned, and the mathematics was put in context. An emphasis was on differentiating the content, related to children’s experiences, for example, using the same operations in different situations; differentiating the difficulty level; and differentiating the forms of exercises, for example, by self-controlling exercises, fill-in exercises, situation-exercises where the children must find facts themselves; exercises where children would make up texts themselves with given facts; exercises with texts without numbers; variations of picture-exercises; detective-exercises with errors for children to find; exercises in interpreting graphs, etc. Another appendix, written in Danish by Agnete Bundgaard, had 11 pages plus three pages on instruction. In the first school year, emphasis was to be laid on building up the set concept and the number concept, using related concepts, such as pairs, disjoint sets, subsets, sets of sets, mapping into and onto a set, and one-to-one correspondence mapping. This was to be taught by using pins, balls,

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apples, etc. The third appendix consisted of three pages of sample problems, written in Swedish. Some were taken from the journal Arithmetic Teacher, and 25 problems were on combinatorics. There were reactions to the main document. Torgeir Bue (I 278) had little to say, simply that the work had gone fast and well. He did not want to object to details but referred to his own document. The mathematics expert Erik Kristensen (I 285) doubted that he was knowledgeable enough about primary-level mathematics teaching. He sent, however, several comments from which document U 213 had been revised. Among his comments, he found the goals vague: “To provide understanding of mathematical connections” and “to make the mathematics teaching enjoyable.” The latter was a method rather than a goal. He expressed a pity for the children, hoping that they could endure receiving such a thorough teaching as was presented for the number concept. He commented also on new algorithms for subtraction and division, wondering if they were better than the old ones, even if it reflected a true idealism to have children understand long division in detail. In a meeting of the NKMM on October 12–13, 1961 (U 253), it was decided to continue the work for grades 1–6 in two teams and to write experimental texts for grades 1–3 and grades 4–6, respectively. Each team was to include three pedagogues and one mathematician. Experimental teaching was to begin during the school year 1962–1963 if possible. The former working team was asked to complete its directive to the textbook authors. Bue, Bundgaard, and Hultman as well as Björkman were present at the meeting, but not Heinonen. Agnete Bundgaard wrote to Sandgren on November 16, 1961 (I 286), quite upset. She found it desirable to unite the two writing teams for grades 1–6. It would not be possible to create comprehensive material if one team was to begin at grade 4 without knowledge about the material prepared for the first three years. Perhaps it would be opportune to unite the former and the new teams. She found it very important to choose topics systematically. Or maybe, she should just be quiet, and things would be organized in Sweden? She was not in agreement with Bue about including addition- and multiplication tables and the metric system. Her pupils would not like self-controlled exercises either. She wondered who was responsible for instructions to the authors. And was the original team for grades 1–6 dissolved? Seven pages of comments on the directive were attached to Bundgaard’s letter. Matts Håstad replied promptly (U 230), saying that her contribution would be included as an appendix to the directive. The original team for grades 1–6 was dissolved. Bue disappeared as did Heinonen. The only members left were Bundgaard together with Hultman and Björkman who were both titled as experts. In January 1962, a list was sent out with 13 persons (U 268) proposed to the two writing teams for grades 1–6 textbooks. The list contained four Swedes, one of them Hultman, and three Danes, one of them Agnete Bundgaard. There were two Norwegians who declined the invitation to join the teams (I 333–335). Norway had by then withdrawn from the cooperation of writing joint Nordic texts for grades 1–6. Four Finns were listed in the writing teams. According to Paavo Malinen in June 1962 (I 440), they met in Åbo (in the Swedish-speaking area of Finland) in May 1962. They had not been active earlier within the NKMM at that level. Now two members, one of them Eeva Kyttä, were ready to start a preliminary experiment on grades 1–2 the following autumn. In November 1962, Agnete Bundgaard wrote to Sandgren (I 647) and gave her opinion of the American SMSG material. She found it somewhat disjointed and had not been aware that it was being translated into Danish. She described her ideas about introducing the four arithmetic operations. Discussion of the set concept, and the concepts of subsets, mappings, and disjoint sets, created the basis for the addition of two numbers, and the determination of the number of elements in a given set’s complementary set created the basis for subtraction. By concrete examples, made visible by sets, the laws and rules for addition, subtraction, multiplication, and division could be treated; the derived operations [subtraction and division] though by solutions to the equations a + x = b and a ⋅ x = b respectively. Thereafter, the [decimal] number notation would be introduced, and then the usual calculations would be explained. Bundgaard had tested her ideas in cooperation with a young teacher at

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her school and the pupils responded positively (Figure 12.3). Old cards with pictures of numbers had been discarded by the children as they preferred the one-to-one correspondence to the number line. Bundgaard’s ideas were realized in textbooks by her and Eeva Kyttä for grades 1 and 2 which were published and later translated into Icelandic. The axioms of the number field, clearly expressed, were introduced step by step in this series, beginning with the commutative law of addition in its first volume at age 7 (see Figure 12.4), after having presented subsets and ordering (see Figure 12.5). The commutative law was followed by the associative law for addition later in grade 1 (see Figure 12.6). The distributive law was presented in grade 2 (see Figure 12.7). Additive and multiplicative identities were presented in grade 3 (at age 9) in textbooks written by Agnete Bundgaard only. Multiplicative inverses were introduced in connection with the division of fractions at age 12. The impossibility of dividing by 0 was presented already in grade 4 by discussing the impossibility of solving fill-in examples such as 7 = 0 ⋅ Υ. The additive inverse was not presented, as negative numbers were not presented in the series. On August 26, 1964 (I 1164), Agnete Bundgaard reported that 11 classes in Denmark would test the text for grade 1, and 7 classes test the text for grade 2. She had heard nothing from Hultman. On September 10, 1964 (U 839), Håstad informed the Norwegian Experiment Council (Forsøksrådet for Skolverket) that for grades 1 and 2, texts were available in Swedish, Danish, and Finnish, while the Swedish text was partly different from the others. Texts for grade 3 were being processed and would be available in autumn 1965. In October 1964, Håstad (U 866) thanked Bundgaard for her texts for grades 1 and 2 and informed her that Charles Hultman, Margareta Kristiansson, and himself were preparing to write a text for grade 3. Håstad asked which part of that text Bundgaard had thought to work on. In her reply, Bundgaard asked Håstad (I 1247) if they would like to consider if the material should be less ready-made, but rather prepared in such a way that the children could use well-defined objects to proceed themselves and discover simplifications. In other words, to let them discover algorithms on their own. On January 5, 1965 (I 1314), Agnete Bundgaard informed Matts Håstad that she had started to prepare the text for grade 3 and was excited to know what they (probably Håstad, Hultman, and Kristiansson) thought about a new version of the text for grade 1. She had also written 30 pages of

Figure 12.3  Teachers at the school on Niels Ebbesen Road, Frederiksberg, Denmark, in 1965; Agnete Bundgaard is the second from right in the middle row. (Frederiksberg Stadsarkiv)

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Figure 12.4  The commutative law. The first introduction of addition and the “+” symbol. Age 7. (Bundgaard and Kyttä 1967, Vol. 1, p. 32)

Figure 12.5  Subsets and ordering with one-to-one correspondence to the number line. Age 7. (Bundgaard and Kyttä 1967, Vol. 1, p. 29)

notes for a course to prepare teachers. Very active teachers were testing the material for first grade in two classes. In April 1965, it was clear that there were two different versions for grade 3: Swedish and Danish. Charles Hultman wrote (U 1052) about the Swedish experimental texts for the lowest levels. He said that teachers who had used the Swedish text for grades 1 and 2 expressed doubts about the use of set-­ parentheses, {…}. Goals could be reached without these symbols; less formalities were desired. The symbols , and ≠ had been well received. The insecurity that teachers felt might be due to their unfamiliarity with modern mathematics—a critique of the education of primary-level teachers. Agnete Bundgaard continued her work alone in 1966. In February 1966, she reported (I 1633) that she had contacted a publisher, Gyldendal, who offered to send material to 60 classes for free. In return, the teachers were to attend a course and a couple of meetings. In late summer 1966, she recounted (I 1734) that tests were being run in two classes in grade 5, four classes in grade 4, eight classes in grade 3, 18 classes in grade 2, and 73 classes in grade 1. Furthermore, there were eight classes in Iceland, and one class in Greenland to take care of. In October, she was preparing new material, revising texts for all these grades, and writing teacher guides, in addition to giving talks and courses, and taking care of her own teaching (I 1780). In January 1967, she had prepared a report on the Danish-Finnish version of the grade 1 text for which Matts Håstad suggested some cuts and alterations (U 1676). The final report indicated the number of classes in grades 1–3 that had trialled the

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Figure 12.6  Associative law for addition. Age 7. (Bundgaard and Kyttä 1967, Vol. 1, p. 74)

Figure 12.7  Distributive law. Age 8. (Bundgaard and Kyttä 1968, Vol. 2b, p. 72)

materials during 1966–1967, and that the material had been used in Finland in a total of 34 classes but did not mention the classes in Iceland and Greenland. The experimental texts by Bundgaard for grades 4–5, mentioned above, and their tests were not mentioned in the final report of the NKMM activities (NR 1967b, p. 104).

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The accounts for Iceland and Greenland were the first to mention experiments outside the four countries comprising the NKMM.  The tests in Iceland were the beginning of translations of the Bundgaard-series for grades 1–6 into Icelandic during 1966–1972. According to Gíslason (1978), the initial number of classes in Iceland were seven. In 1966, when the Nordic project was ending, Matts Håstad reported (U 1536) that the authors of the experimental material for grades 1–3 were Agnete Bundgaard and Eeva Kyttä for the Danish-Finnish version, while in fact only the texts translated into Icelandic for grades 1 and 2 are attributed to Eeva Kyttä. Authors of the Swedish version were Margareta Kristiansson, Carin Klaesson, and Matts Håstad. Hultman was not mentioned there, but in the final NKMM report he is listed as an expert for grades 1–6 (NR 1967b, p. 222).

Grades 4–6—SMSG Material The NKMM committee discussed early (U 1) the possibility of using material from the American School Mathematics Study Group (SMSG). In November 1962 (I 615), permission to translate the SMSG texts for grades 4–6 into Nordic languages was granted upon a request from Edward G. Begle by Stanford University, which was where Begle was now based (after moving from Yale University). Experimental teaching of SMSG texts, translated into Swedish, began in 1963 (U 1545). In September 1963, the Norwegian Halvorsen (I 917) wrote comments on the first part of the Swedish translation for grade 4 with respect to a possible translation into Norwegian. According to Halvorsen, the material was not consistent with the Norwegian plan for experiments on the 9-year compulsory school. Some topics, such as sets and place-value notation in base five, would also take time. However, it could be adjusted to the new Norwegian experimental curriculum. He proposed that a team would take the translation and adjust it to use in a unified compulsory school. The Norwegian Experiment Council was asked to decide if it could be used in a number of schools. The result was that the SMSG material was translated verbatim into Norwegian (Gjone 1983, Vol. II, p. 80; Vol. III, p. 17). The SMSG text was tested during 1964–1966 in about 20 classes in Sweden. During that period, comparisons showed that this text was the least liked in Sweden but reactions to it did improve over time. The teachers were positively surprised at how well the geometry was going. The final report does not mention that the SMSG material had also been translated into Norwegian (NR 1967b, pp. 108–111). In April 1965, Hultman (U 1052) reported that the teachers in Sweden felt insecure with the translated material for grades 4–5. That might be the reason why the pupils’ results were not as positive as they could have been. The pupils had had difficulties in reading the texts. The texts might not have been translated well enough, and teacher guides were still only available in English. The quest for understanding was more thorough in the experimental texts than in the traditional material. Comparison between the results of the experimental teaching and traditional teaching would not be possible until after grade 6 had been tested. There was, however, a reason to expect that the experimental pupils would be more successful in tasks that demanded independent mathematical thinking, while the other pupils would do better on certain mechanical calculations. The experience acquired was deemed to be so positive that it was highly desirable to continue the experimental teaching with more groups of pupils and more teachers.

Grades 7–9 At the first meeting of the NKMM in October 1960 (U 8, U 9), Bent Christiansen (D), Kay Piene (N), and Inkeri Simola (F) were appointed to a team to make proposals on mathematics contents for grades 7–9. The directive (U 81) to the authors of the experimental texts made it clear in its introduction that the topics were to be presented in terms of the set concept, its derived concepts, and their symbols—for example, the union, ∪, and the intersection of two sets, ∩, the subset, ⊆, the empty set, ∅, the symbol ∈, a complementary set, and some symbols from logic.

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The experimental texts were to be on two topics: Algebra and geometry. An algebra text in three volumes was ready for testing during the academic year 1962–1963, written by Bent Christiansen, Matts Håstad, and Ragnar Solvang. Bent Christiansen had the primary responsibility for the first volume, which had three chapters (Christiansen and Christiansen 1965a). The experiences were different in Denmark and Sweden. The committee decided that a somewhat shorter version of algebra, written in a simpler language than the first version, would be used in future experiments in Sweden (NR 1967b, p. 111). A new Danish version appeared, denoted A 7–9, version D (Christiansen and Christiansen 1965a). It became the basis for experiments in 35 grade 7 classes in Denmark during the academic year 1964–1965. The authors of the Danish version were Allan Malmberg Christiansen and Bent Christiansen. The aim was to research how learning arithmetic and calculations were affected by implementing concepts from set theory, including concepts on relations and functions. Experience showed that the Danish version’s first chapter would be suitable for an experiment that would be performed in 65 grade 6 classes during 1965–1966. A report to the Danish experimentation committee, dated June 24, 1965, signed by Allan Christiansen and Bent Christiansen (1965b) on behalf of the Danish department of the NKMM, said that it was of particular interest to study whether understanding and skills might be achieved with less training and more emphasis on insight into the basic concepts of algebra and its interplay with computations. In a letter by Bent Christiansen to Matts Håstad, dated December 14, 1965 (I 1607), it appears that there was a friction concerning the dissemination of information about the experimental activities. Bent Christiansen, who headed the Mathematics Institute at Danmarks Lærerhøjskole [the Royal Danish School of Educational Studies] in Copenhagen and played an important role internationally (Figure 12.8), regretted that the leader and secretary of the NKMM project had replied negatively to a request from UNESCO to publish in its New Trends an abstract of a report of the experimental activities in grade 7 in Denmark. Christiansen pointed out that he and his collaborators had written the Danish algebra text. Under no circumstances would he agree that his Institute’s possibilities of

Figure 12.8  ICME-3, Karlsruhe 1976, Opening lecture. On the front row from the right: Heinz Kunle, Mrs. Kunle, Shokichi Iyanaga, Mrs. Heinrich Behnke, Bent Christiansen, Hanne Christiansen, Hans-Georg Steiner, Mrs. Steiner. (Courtesy of Gert Schubring and Livia Giacardi)

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e­ xpression about achieved experiences would again be forfeited. He had arranged to tear the present experiment with 65 classes in grade 6 away from the NKMM committee. The Institute’s contribution to that matter was quite formidable. Most of the 65 teachers had attended courses there, often quite long ones, with participants being provided with special literature. Christiansen indicated that he would not dream of having the teaching of the experimental texts made by accidentally chosen teachers without professional insight or real interest in the matter. The choice of teachers was extensive: More than 1000 teachers in Denmark were attending courses and more than 4000 teachers, one-third of the number of Danish teachers, had been in contact with the new material. Furthermore, Bent Christiansen stated that he had noticed that the authors’ names had been left out in a new version of volume V of the Swedish Algebra 7–9, where many of his own thoughts had been used, while all other volumes were meticulously marked to the authors in concern. The friction seems to have been settled according to exchanges during 1966 (I 1706; I 1808), where economic transactions were explained. On October 27, 1966, a letter from Håstad (U 1536) shows that a Swedish algebra version had been developed, based on earlier versions. The authors listed were Bent Christiansen, Matts Håstad, and Ragnar Solvang, while the Danish algebra version was attributed to Allan Christiansen and Bent Christiansen only. Two geometry series of different difficulty levels were made, both written in Swedish (U 839; U 1536). The more extensive one was written by two Swedes, Bertil Nyman and John Amundsson, and the Finn Inkeri Simola. It was translated into Finnish and Norwegian and tested in all three countries. The goal was to provide pupils with an introduction and motivation. Geometrical objects were introduced by set-theoretical concepts, for example, an angle as the union of two rays with the same endpoint. In the second part, a complete axiomatic system was provided with six basic axioms, based on Gustave Choquet’s axiom system (Choquet 1969). One complaint from teachers was that some important results were only found in the exercises (NR 1967b, pp. 124–125). The less extensive geometry was written by the Swedes Gunnar Bergendal, Ove Hemer, and Nils Sander. It was used successfully in Sweden and Finland (NR 1967b, p. 126), and its volume for grade 7 was translated into Icelandic (Bergendal et al. 1970). A summary of replies to a questionnaire from eight Swedish and three Finnish teachers exists about testing geometry in grade 7 during the academic year 1961–1962 (U 376). The general opinion was to continue the experiment; seven teachers had noticed an increased interest by their pupils, some missed logical proofs, the material suited the age level, and most teachers did not feel that there was a lack of in-service courses. The NKMM experimental texts, probably the one in algebra with Solvang being co-author, and the more extensive one of geometry, were tested in 8 classes of grade 7 in Trondheim, Norway, during 1965–1966. The experiment continued in Oslo and Trondheim for 5 years. There was no formal assessment, but a report was based on the teachers’ impressions: “One feels that the pupils attack the tasks in a more independent way […]. The weaker pupils’ attitudes may have become positive and had a fortunate effect on their performance.” Another experiment of 3-year duration was made in 1969 with the experimental texts revised by Ragnar Solvang. The revision became a basis for an axiomatic, deductive version of the Norwegian experimental national curriculum (Gjone 1983, Vol. III, pp.  12–15), which in its final version in 1976 became balanced between modern and traditional approaches.

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Grades 10–12 The gymnasium level was the main concern of the NKMM (Gjone 1983, Vol. II, p. 91) and the cooperation on the material for that level was the least controversial. In October 1966 (U 1536), texts were available, written by six Swedes (S), two Danes (D), and one Norwegian (N). The texts were on the following: • • • • •

Algebra by Gunnar Bergendal (S) and Per Häggmark (S) Geometry by Carl Hyltén-Cavallius (S) and Ib Schauffuss (D) Functions and calculus by Matts Håstad (S) and Haakon Waadeland (N) Statistics and probability by Björn Ajne (S) and Lennart Råde (S) Differential equations by Lars Mejlbo (D) and Carl Hyltén-Cavallius (S)

The algebra text was used in experimental teaching, beginning in 1962 (U 1545) at the first gymnasium grade in Norway and Sweden. The teachers found the text quite suitable for that level, but it was theoretical and best suited for the hard-working students. The text was difficult for the students to study on their own. Two thirds of the teacher group found that the student’s calculation skills had declined (NR 1967b, pp. 128–130). Experimental teaching of the geometry text began in 1961 (U 1545). It was used in Denmark, Norway, and Sweden, and in several Swedish-speaking schools in Finland. All the teachers agreed that vectors should be taught at gymnasium level, combining synthetic and analytic geometry, and vectors in three-dimensional space were easy for the students. The treatment of the scalar product was much criticized, while trigonometry was well received, and recommended to be treated before the scalar product (NR 1967b, pp. 130–132). The text on functions and calculus was used for experimental teaching in Sweden, and some Swedish-speaking schools in Finland from 1963. It started in 1965 in Norway, from where no experiences were reported. The text was used in the two uppermost grades together with other texts, especially in geometry. There were mixed opinions if skills had declined. 50% of the teacher group felt so, while 10% felt that they had improved. Still, students performed well in their final examinations, written and oral. Teachers found, however, the material too voluminous (NR 1967b, pp.  97, 136–137). The text on statistics and probability was used in the second or third gymnasium grade since 1961–1962 in all four countries, most extensively in Sweden. The students were generally quite interested in these topics, where knowledge from other topics in the mathematics course could be used in a new way (NR 1967b, pp. 138–140). The text on differential equations was taught in all the four countries, beginning in 1961. The teachers found it useful to work on that topic, where the calculus could be applied and trained, also as there were applications from physics (NR 1967b, p. 138). The NKMM committee published in its report a proposal on content of mathematical syllabuses, based on the experimental texts and teaching. Compared to the syllabus of grades 1–9, the syllabus for grades 10–12 was more detailed. Looking at the syllabus for grade 12, there were many items from the mathematics studies that were taught at universities in the Nordic countries in the 1960s. This can be interpreted as an experiment to create coherence between mathematics teaching at universities and gymnasia (Gjone 1983, Vol. II, p. 91).

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Aftermath The formal cooperation was concluded in 1967 when the report was published. It was not the role of the committee to publish textbooks, only to write and test experimental texts. In the following years, several of the authors of experimental texts published textbooks that were to make an impact for the following decades. Meanwhile, all four countries, Denmark, Finland, Norway, and Sweden, were working on new legislation and national curricula according to the new 9-year unified compulsory school system, followed by a revision of the gymnasium level, grades 10–12. The experimental texts and teaching were important factors in that development.

Denmark All four Danish members of the NKMM found channels for their new ideas of a wider pedagogical coherence through a multilateral production of textbooks. At the gymnasium school level, the reform was strongly reflected in the first round by the textbook series Matematik 1–3 by Erik Kristensen and Ole Rindung (1962–1964) (Skovsmose 1980). This series was published in several editions and reprints until the 1980s by Gads Forlag and had a dominating position on the market. Issue 3–4 of Nordisk Matematisk Tidskrift (NMT) in 1967, contained a very positive review of the Kristensen and Rindung series, where this clause appeared: There are immensely many things to enjoy when reading the authors’ treatment of this voluminous amount of material. The perceptive use of symbols, especially from set theory and logic, contributes to create a clarity in the presentation, which is praiseworthy2. (Møller 1967, p. 111)

In 1965, a large proportion of the group of mathematics teachers at the gymnasium level in Denmark attended summer in-service courses on modern mathematics at Aarhus University where Svend Bundgaard was a mathematics professor. The courses were reported as quite demanding (Mogens Niss, personal communication, December 1, 2020; Henrik Stetkær, personal communication, August 24, 2022). Danmarks Lærerhøjskole [The Royal Danish School of Educational Studies] played an important role in disseminating ideas about modern mathematics under the leadership of Bent Christiansen. Two books were central in this process: Almene Begreber fra Logik, Mængdelære og Algebra [General Concepts from Logic, Set Theory and Algebra] by Bent Christiansen, Jonas Lichtenberg, and Johs. Pedersen (1964), and Matematik 65 by Bent Christiansen and Jonas Lichtenberg (1965), both aimed at teacher students and teachers at in-service training, providing wide and thorough coverage of the new concept world (Skovsmose 1980). Ragnar Solvang (1966) said in his review in NMT that Matematik 65 was a very fine book that deserved a large circulation. He found it more accessible than Almene Begreber fra Logik, Mængdelære og Algebra to teachers who were not well oriented in advance about the topic. Bent Christiansen was a prolific writer. In his book on goals and methods in mathematics teaching (Christiansen 1967), he laid down a possible approach for grades 6–9, realized in the textbooks Matematik 7 1–2 of 1967–1968 in cooperation with Allan Christiansen and Jonas Lichtenberg, and Matematik 8G 1–2 of 1969–1971 in cooperation with Johs. Petersen (Skovsmose 1980, p. 37). All the textbooks authored by Christiansen were published by Munksgaard. Bent Christiansen also made a television program about modern mathematics in 1968 (Moon 1986, p. 185). Gyldendal published Agnete Bundgaard’s series, of which the first two volumes were written together with the Finnish Eeva Kyttä. The series came out during 1965–1971. Jens Høyrup (1979)  All translations were made by the author.

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deemed the series written strictly according to the rules set by the professorial reform’s demands to the Danish school system. The Blue Memorandum curriculum guidelines of 1960–1961, following the 1958 Act, could be said to incorporate many things but definitely not modern mathematics. New guidelines, following the 1975 Act, aimed directly at modern mathematics. Agnete Bundgaard had, however, been a consultant to the Blue Memorandum. The Bundgaard series was not the only series on the market for grades 1–6. Indeed, textbook publishing flourished in this period. Skovsmose (1980) mentioned six series, among them the briskly selling Matematik by Jørgen Cort and Erik Johannessen, which Jens Høyrup (1979) found marked by imagination, abounding with ideas and practical teaching experience. While the pedagogical problems with the Bundgaard series were that it was too dry and formal, Cort and Johannessen created tasks with a surplus of sprightliness. These authors were only loosely connected to the experimental activities. Cort had been one of the teachers that had taught the algebra experimental text. There was also Hej Matematik, by Matts Håstad, Curt Öreberg, and Leif Svensson, translated and adapted from Swedish.

Sweden The Swedish members of the writing teams also found publishers for their products, in particular for the gymnasium level. In Nordisk Matematisk Tidskrift, a review was published of six new series for the first-year mathematics course according to a new plan of 1966 for the gymnasium mathematics in Sweden (Hanner 1967). One of the two best-received series was Matematik för Gymnasiet, by Gunnar Bergendal, Matts Håstad, and Lennart Råde (1966–1968), all authors of experimental texts, published by Biblioteksförlaget. It was written both for the mathematics-physics stream in the gymnasium in three volumes, and for its social science stream. The series lasted in print through the 1970s. In his review, Hanner (1967) states: This is a very well processed book, which appears to be suitably concise (except possibly the chapter on numbers which is somewhat too extensive and therefore difficult to overview). A detail that elevates the value of the books is the regularly repeating summaries of the previous sections. These provide a good overview and must be of special value at revision. (p. 162)

Carl Hyltén-Cavallius and Lennart Sandgren (1958, 1969) also revised their textbook on mathematical analysis for beginning university studies with respect to modern mathematics, by writing a special introductory volume about set theory, logic, and functions, and adjusting their main text. Modern mathematics was introduced on a broad scale in grades 1–9 in Sweden when the national curriculum of 1969 took effect. The overall difference from previous curricula was that topics were moved to earlier grades, such as equations, statistics, functions, and the coordinate system. New topics included place-value number notation with bases other than ten in grade 3, vectors in grade 7, and trigonometry in grade 9. A step in the preparation was to educate all teachers in modern mathematics through a distance course called Delta. In total, 47 600 teachers were registered, a great majority of those teaching mathematics (Prytz 2018). In 1970, only the pupils in grades 1, 4, and 7 were affected by the reform. This meant that the reform involved all students and teachers only in 1972. By 1972 or 1973, the person in charge of mathematics for grades 1–9 at the central school authorities in the period of 1972–1977, Sven-Erik Gode, seems to have given up on central parts of the reform, and from the beginning felt compelled to address deficiencies in modern mathematics. As a motivation for this, teachers had indicated that the syllabus was too comprehensive. Moreover, the results of the first national tests related to the new syllabus, conducted in 1973, showed that the efforts had partially failed; comparisons revealed that students following the former syllabus had better skills in arithmetic. In the materials issued by the

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central school administration as early as 1973, ideas central to modern mathematics were made peripheral (Prytz 2017). Several textbook series for the compulsory school were published from 1969, including the modern mathematics ideas, while traditional series were in a clear minority. For example, Matts Håstad, in cooperation with Curt Öreberg and Leif Svensson, wrote a textbook series for grades 1–9, called Hej Matematik, published from 1970. It was republished several times, while from 1977, it did not contain set theory for grades 7–9. After 1974, fewer textbooks included set theory, especially for that level (Prytz 2018). However, from an elaborative analysis of data, collected from the special reports U 1369, U 1371, U 1425, and U 1431, not published in the official report on the NKMM cooperation, Prytz and Karlberg (2016) concluded that it was possible to use the modern mathematics material and attain acceptable results. Pupils in the experimental classes performed at a level equal to those who received traditional instruction. Still, the fact could not be dismissed that the material needed an experienced teacher with a special interest in modern mathematics. This analysis also contradicts a claim that the modern mathematics material per se had negative effects on students in general. Moreover, Prytz and Karlberg found no support for the claim that students who found mathematics difficult were especially disadvantaged by the modern mathematics material.

Finland In 1968, the Finnish parliament introduced legislation to reform the education system with free comprehensive schools for children between 7 and 16 years old. In 1970, a national modern mathematics curriculum was implemented for all 12 years of comprehensive schools and gymnasia (Malaty 2009). There were only two Finnish authors in the writing teams authoring experimental publications. Eeva Kyttä’s cooperation with Agnete Bundgaard on the material for grades 1–2 has been mentioned. Inkeri Simola, who was a co-author of the experimental text on geometry for grades 7–9, was the first woman to obtain a doctorate in mathematics in Finland. She was appointed as editor of the Nordisk Matematisk Tidskrift on behalf of Finland in 1953. She wrote several articles in the journal, but nothing on modern mathematics. Yrjö Juve, a member of NKMM, was the leader of the experimental project in Finland from 1960, while other obligations carried him away from that area, and he died prematurely in 1967 at age 47 (Lyytikäinen 1967).

Norway In Norway, the atmosphere still supported reforms in 1967, when the SMSG project from the United States, and the Swedish texts for grades 1–3 were adapted to Norwegian situations. The Swedish texts were tested in Oslo in 1967–1969. From 1967 onward, there was a connection between the reform projects and the project of creating a new national curriculum, culminating in 1971 (Gjone 1983, Vol. VIII, pp. 8–9). Already in 1967, the planning of a television program on modern mathematics began, supplemented by books and a correspondence course. The program was planned as an in-service course for teachers and others interested, such as parents. One of the persons preparing the course was Ragnar Solvang. The program was sent out in 1972–1973 in 35 episodes, of which five were aimed at the lowest level (Gjone 1983, Vol. V, pp. 33–37). A period of reaction and discussion followed after 1971, terminating in 1973 when the experiments were about to end. After 3 more years, in 1976, the national mathematics curriculum for compulsory

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level was complete, with a balance being achieved between modern and traditional approaches. The curriculum had swayed away from the most orthodox form of the modern mathematics movement. At the gymnasium level, the reform movement led to a necessary adjustment to reforms implemented at the university level during the 1950s. There had been controversies, mainly at the compulsory level; the mathematical knowledge necessary for an individual leaving school was a central question throughout the whole period (Gjone 1983, Vol. VIII, pp. 7–11). Publications related to the NKMM-activities include Matematikk for reallinjen : Algebra og funksjonslære 1, by Ingebrigt Johansson, Ragnar Solvang, and Ottar Ytrehus, published by Cappelens Forlag in 1968, and Logikk og mengdelære by Solvang and Ytrehus, published in 1973. Kay Piene was a member of the NKMM and of the editorial team of Nordisk Matematisk Tidskrift. He wrote several articles about modern mathematics and reviews in that journal but died in 1968 while the reform was still in progress.

IMU—An Individual Mathematics Teaching Project An interesting side effect of the redefinition of school mathematics in the Nordic countries is the Individualiserad Matematik Undervisning (IMU) [Individualized Mathematics Teaching Project], which was under development in grades 7–9 in Sweden from 1964. Its goals were as follows: 1. To construct and test self-instructional study material in mathematics; 2. To find suitable teaching methods and ways of work for using this material; 3. To try out different ways of grouping pupils and making use of teachers to achieve the maximal effect for the material and methods; and 4. To measure the effects of individualized teaching (in comparison to conventional teaching). Since the IMU material was to be introduced in connection with a new national curriculum of 1969, it was adapted to the coming curriculum. Thus, the IMU material included much from the NKMM project. One of the authors of the material was Matts Håstad, a key figure in the NKMM project (Prytz 2017). The IMU project was trialled in Norway in 1967, and tests of it began there in 1968. Its first version was a direct translation from Swedish, and the project was led by the Norwegian Experiment Council. Ragnar Solvang objected to it in 1969 and said that it would only be testing a Swedish project in Norway. “Whose interest is that for anyone else than the Swedes?” he asked (Gjone 1983, Vol. III, p. 11). The project was revised comprehensively in 1970, and later versions were published commercially by Dreyers Forlag in Oslo (Gjone 1983, Vol. III, p. 15). The IMU project can be classified as behaviorist (Howson et al. 1982, pp. 202–203). Some Norwegians objected to the idea that the IMU project belonged to the modern mathematics movement, but in Sweden it had a close connection to this movement through Matts Håstad (Gjone 1983, Vol. III, p. 16).



A Case Study: Modern Mathematics in Iceland

The Reykjavík Gymnasium belonged to the Danish gymnasium system until 1918. Danish cultural influences persisted as further studies in mathematical sciences were sought at universities and polytechnic schools in Denmark. Mathematics textbooks used at the gymnasium were in Danish, and teachers were educated in Denmark well into the twentieth century. In 1964, Guðmundur Arnlaugsson, a mathematics teacher at the Reykjavík Gymnasium, who had studied mathematics in Copenhagen at the same time as Svend Bundgaard, stayed in Denmark during World War II, and was in contact with

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Danish colleagues, began acting toward implementing modern mathematics in Iceland. Becoming a consultant in mathematics teaching at the Ministry of Education, he organized a week-long in-service course for teachers on modern mathematics using Almene Begreber fra Logik, Mængdelære og Algebra (Christiansen et al. 1964) as a basis for the course. He subsequently published his own textbook for grade 9, Tölur og mengi [Numbers and Sets] (Arnlaugsson 1966), which provided an introduction to numbers and set theory. Matematik 65 (Christiansen and Lichtenberg 1965) was used for teacher-education students. Arnlaugsson forwarded news from Svend Bundgaard about the textbook series by Agnete Bundgaard and Eeva Kyttä and suggested that the series be translated. The material for grade 1 was intended for children who could not read, so there was little to translate in that first volume. The series was tested in seven classes of grade 1 in two schools in Reykjavík during 1966–1967, with regular meetings of project leaders and teachers, and meetings with parents. The following spring, the project was presented to school leaders in Reykjavík, who became quite enthusiastic. When the decision was made, only the course material for grade 1 had been finalized in Danish, while the texts for grades 2 and 3 were still in draft form (Gíslason 1978). In 1967, school leaders, impressed by presentations of the new ideas, decided to send 86 teachers to attend a preparatory course for teaching the new textbooks. The majority of first-grade pupils in Reykjavík primary schools were to be introduced to modern mathematics (NN1 1969). The project leaders did not have the capacity to keep in touch with all the teachers and arrange information for parents to the same extent as for the first group. The first presentation of modern mathematics to the wider public was characterized by optimism. Articles were written, and Arnlaugsson made a television program, introducing modern ideas (Bjarnadóttir 2011). The television program, supplemented by Arnlaugsson’s textbook, was expected to reach a larger number of parents than would otherwise be possible. In an interview, one of the two project leaders remarked that only teaching methods were being changed, not the content, while various topics were introduced earlier than before. Mechanical working methods had been overly emphasized at the cost of the time that teachers had available to discuss basic mathematical concepts and to train logical thinking and accuracy in presentation. The children were to have no homework (Stefánsson 1967). At the time the decision was made, the content of the latter part of the series was not known. It turned out to be extremely theoretical (Høyrup 1979). The axioms of the number system, the commutative, associative, and distributive laws, place-value number notation in base five, prime numbers, permutation of three digits, and the transverse sum and its relation to the nine times table were all introduced before the end of grade 3. Set theory with pairing, subsets, intersection, and union, more place-value number notation systems, and geometry with points, lines, and planes in a set-theoretical framework were added in grade 4 (Bundgaard 1969–1972; Bundgaard and Kyttä 1967–1968). Topics such as time unit computations, listed in the Icelandic national curriculum of 1960, were not mentioned in the Bundgaard series, and monetary units were only marginally discussed in connection with the metric system. These topics were probably considered to be applications and thought to emerge naturally as a consequence of the pupils’ training in mathematical thinking. In 1970, a newspaper published an interview with Agnete Bundgaard and her colleague, Karen Plum, who had visited Iceland to give a course to 65 teachers (NN2 1970). By that time, the Bundgaard series was used in 141 classes in grades 1–4. Bundgaard said that the main emphasis was on promoting pupils’ understanding of the nature of the tasks and on training them to use their own judgment in solving tasks and problems. Modern mathematics had been introduced in many countries and was influencing how mathematics was taught. Other nations’ experiences suggested that its concepts and symbols would be of great use in training pupils in clarity of thinking and communicating. Agnete Bundgaard said:

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Modern mathematics is like a new language, totally different from the mathematics the parents of modern schoolchildren learnt themselves. Many parents have a hard time accepting not being able to know exactly what their children are working on at school and assist them. But it can have very bad consequences for the child if its parents are trying to help, more being willing than able to guide the child. This can only lead to confusion. Therefore, it has been decided not to assign homework to the children and not even allow them to bring their books home. However, to increase the parents’ understanding of what their children are working on, special books have been published, admittedly not available in Icelandic, where the new mathematics is explained, and it should pacify the parents until the moment when the children have reached enough understanding of the project to be able to explain to their parents what is happening there. (NN2 1970, p. 23)

Karen Plum continued that in many places modern mathematics had generated dispute and that many years will pass until its advantages can be proved statistically, as all comparison is difficult. But surely modern mathematics teaches children to think logically, and however the world will change, logical thinking will always be necessary. Besides, children have proved to like the modern mathematics and they show more interest in it than children at the same age who learn by the old methods, and these two items weigh not so little. (NN2 1970, p. 24)

By the time of Bundgaard’s and Plum’s visit, authorities had realized that things were not going well, the mathematics teaching experiments in primary schools had become far too voluminous, too difficult to run with respect to guiding teachers, and even in a few cases the effects had been close to being disastrous (Ragnhildur Bjarnadóttir, personal communication, September 16, 2003). The Ministry of Education and Culture had established a school development department that laid down a procedure for adopting school reforms: To set goals, write national curricula and from there, develop learning materials on an experimental basis. In the crisis that had emerged, the department decided in 1971 to skip the step of setting goals and writing a national curriculum in mathematics but go directly ahead to create a new set of homemade mathematics textbooks within the department (Andri Ísaksson, personal communication, March 10, 2003). In their final editions, sets were hardly mentioned. Enthusiasm for the modern mathematics seems to have reached its peak at the primary level in Iceland before 1972 (Bjarnadóttir 2007). The cohort born in 1965, entering grade 1 in 1972, and completing grade 6 in 1978, was the last large cohort, with about 40% of the Icelandic population studying the Bundgaard material from grade 1 to grade 6 (see Figure 12.9). After that, authorities began to pull it out gradually, while the new material was being introduced after careful testing, keeping in mind the difficulties of the rapid implementation of modern mathematics. For grades 7–9, the Swedish versions of algebra and the shorter version of geometry (Bergendal et al. 1970) were translated and used for 3–5 years until new Icelandic texts were developed, c­ ontaining

Figure 12.9  Percentage of year cohort taking Bundgaard material up through grade 6

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fewer set-theoretical concepts and symbols. Textbooks and syllabuses were on an experimental level until a new national curriculum was published in 1989. Up to the 1980s, mathematics textbooks in the vernacular were not published in Iceland for the tiny market of grades 11–12 or university. The tradition was textbooks in Danish, considered a window to the international scene of education. For grades 11–12, the Swedish textbook series by Bergendal, Håstad, and Råde (1966–1968), was introduced in 1969. The series endured for most of the 1970s in upper secondary schools. It turned out to be more accessible to Icelandic students than the Danish Kristensen and Rindung series (1962–1964), even if Swedish was less familiar to them than Danish. The Kristensen and Rindung series was used for the mathematics-physics stream of students (Bjarnadóttir 2007).

Discussion The process of introducing modern mathematics involving cooperation among the Nordic countries was an extraordinary experiment in international collaboration and partnership. As the project went on, the individual countries mostly went their own way. Jeremy Kilpatrick (2012), writing about the international modern mathematics or New Math movement, said that from a distance, school mathematics looked much the same everywhere. However, each country has a unique school mathematics—a product of its history, culture, and traditions, and conforming to its social, political, and educational systems. Instructional materials and practices in school mathematics cannot be transported across borders as if they were a common currency. The new math era taught us the paradox of curriculum change: The more school mathematics is internationalized, the more clearly its national character is revealed. (pp. 569–570)

Kilpatrick’s observation applies very much to the Nordic countries despite their common heritage. A presumed aim of the Nordic cooperation on modernizing mathematics teaching, namely the preparation of common textbooks that might be translated into the various languages, did not happen to any large degree. Among the obstacles were national situations, such as creating new national curricula around a new 9-year compulsory school period in all the four Nordic countries, was something which would have required lengthy discussions by each country’s authorities. This does not mean that the countries were not influenced by the cooperation. The final report recounts, among many things that might be mentioned, effects on the teaching of negative numbers and trigonometry in grades 7–9 in Finland, and approximations in Norway (NR 1967b, p. 218). Another obstacle was language problems, and they probably hindered the Finns from participating in the cooperative ventures as actively as the others. Only three Finns authored parts of the final product, among them Eeva Kyttä for grades 1–2, and Inkeri Simola on geometry for grades 7–9. Twelve Swedes were among the writers of the final product. In particular, the executive director Matts Håstad authored three experimental texts: For grades 1–3, on algebra for grades 7–9, and on functions and calculus for grades 11–12. In addition, he was one of the main authors of the IMU project, and he edited the final report on the cooperation (NR 1967b), besides being one of three authors of the textbook series Hej Matematik, published in Sweden and Denmark for the compulsory level in the 1970s. Håstad gradually became a central figure in the NKMM cooperation as being most of the time the only employee, who must have had to take many decisions on his own. In Norway, the experimental activities did not appear to have much effect. The modern mathematics influences, there, seemed to be mainly in the creation of new curricula (Gjone 1983, Vol. III, p. 19). Three authors were Norwegians, one of them Ragnar Solvang, co-author of the Swedish version on algebra for grades 7–9.

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In an interview, Mogens Niss (Karp 2015) stated that Denmark was one of the countries that went furthest when it came to introducing the Bourbaki tradition, the modern mathematics approach, into university programs and high school programs. Niss mentioned the influence of the mathematics professor Svend Bundgaard, who said after having spent some time abroad: “This New Math is something we must do in Denmark. We really have to revamp the entire program and modernize it” (p. 59). Svend Bundgaard was an invited guest speaker at the Royaumont Seminar. He organized an international meeting on modern mathematics in Aarhus in 1960 as a follow-up to the Royaumont Seminar (Behnke et al. 1960), but it is not known whether he participated in the Nordic cooperation. Svend Bundgaard’s sister, Agnete, was very active in the NKMM committee in the early years of the project, but her influence diminished as her activities drifted away from the mainstream. Her later works were published commercially and were not recorded among the products of the NKMM cooperation, so they do not seem to have been accepted by the committee. Presented at first as deputy primary school inspector, she was exceptionally well versed in the Bourbaki system of set theory, logic, and structure of number sets. According to her letters, she was teaching young children. She was very definite about how she intended to introduce mathematics from the first grade according to the modern mathematics doctrine. One wonders if Svend Bundgaard cooperated with her and had something to say about her approaches. Strong personal opinions were conspicuous among the writers of the experimental texts. Agnete Bundgaard and Bent Christiansen clearly had their own opinions which they followed till the end, and these often differed from the track followed by their Nordic colleagues. One of the most noteworthy incidences in the cooperative venture was the disagreement between Agnete Bundgaard and Torgeir Bue on the directive for grades 1–6. Both submitted their own appendices, so neither of them supported the directive fully. Agnete Bundgaard expressed her disagreement on particular items in Bue’s appendix, while Bue, who was a leader of a teacher training college, resigned from the working team, and in 1963, from the NKMM committee.



Concluding Remarks

The modern mathematics reform in the Nordic countries began from a perceived need to create coherence between mathematics at the universities and the schools. At the uppermost level, the gymnasia, the reform went fairly well. In addition to making students acquainted with set-theoretical concepts and notations, new topics, such as vectors, statistics, probability, and differential equations, were introduced. Results of the introduction of modern mathematics at lower school levels were more questionable. Grades 7–9 were under revision for social reasons. Comprehensive 9-year compulsory education was being implemented in all the Nordic countries. Therefore, a new syllabus, suitable for all pupils, was necessary. The suitability of modern mathematics in this respect for that level will be left unanswered. At the lowest level, grades 1–6, the syllabus had remained unquestioned for ages. The four arithmetic operations with natural numbers, common and decimal fractions, some mensuration, the metric system, and applications from a pre-War agricultural society, were traditionally the topics for those who left compulsory school for work. Modern mathematics for this level proved to be controversial. Not only teachers but also parents and the public had to be informed and convinced. It was only natural that the program would raise questions and discussions. Traditional algorithms were so ingrown that many people did not pay attention to why they were as they were and could not argue logically for them. The aim of modern mathematics was to introduce understanding through logically based arguments. The question remains if that effort was successful. Kristensen’s comment (I 285) that it

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reflected a true idealism to have children understand long division in detail, indicates a doubt that a full understanding of all common algorithms was altogether reachable. The interplay in the Nordic countries between legislation and national curricula on the one hand, and the implementation of modern mathematics on the other, is notable. In Norway, steps were taken cautiously while new and new drafts of a national curriculum were prepared, until 1973, when a new curriculum, with a balance between the “modern” and the “traditional” approach, was adopted, and the peak of enthusiasm for modern mathematics had been left behind. In Denmark, where the enthusiasm was driven by Bent Christiansen and the Royal School of Educational Studies, the official curriculum guidelines, the Blue Memorandum, became increasingly irrelevant, and textbooks were published without adhering to official guidelines (Høyrup 1979; Moon 1986). Sweden may be seen as “in-between”: Textbooks were published according to national curricula, first to a modern curriculum, and only a short while later, to a more traditional one. The Swedish experimental texts were not as orthodox at any school level as in Denmark. In Iceland, the modern mathematics materials were used in an experiment involving a large proportion of the compulsory school population. The creation of a national mathematics curriculum was delayed until 1989, long after the experimental period in the 1960s and 1970s. A positive effect of the modern mathematics reform was that it turned attention to school mathematics and initiated a discussion and a redefinition of what future citizens could and should learn to prepare for a future in an unknown world. Possibly, Agnete Bundgaard went too far in her effort to systematize arithmetic and tie together arithmetic and geometry by set theory. Still, as in Iceland, it turned teachers’ attention to the possibility that beginners’ mathematics could be presented in many ways. It also awoke an initiative by teachers to create their own teaching materials, incorporating what was noteworthy in the old into the new syllabus. The interview with Bundgaard and Plum in Iceland had doubtlessly the aim to inform parents and the broader public. However, the idea that information in a foreign language should pacify the parents demonstrated a lack of sense of the situation and respect for the parents, and the same could be said of the decision not to let the pupils bring their textbooks home as it might “have very bad consequences for the child if its parents are trying to help, more by being willing than by being able to guide the child.” That, and remarks that “the modern mathematics teaches children to think logically” to a higher degree than earlier, and that parents should wait “until the moment when the children have reached enough understanding of the project to be able to explain to their parents” witness unrealistic convictions of the quality of the program. Discussions emerged gradually in the Nordic countries on whether implementation of the modern mathematics in schools was a reform, or a disaster resulting in whole cohorts being unable to do the simplest arithmetic (Bjarnadóttir 2006; Gjone 1983; Prytz and Karlberg 2016). However, the emerging pocket calculators were soon to reduce the need for rapid computational skills, mentally or by paper and pencil. In some cases, however, the set-theoretical concepts and their associated symbols became goals in themselves. They were to be learned thoroughly and may have overshadowed the mathematics itself, taking too much time from the number work, and distracting pupils’ attention by their distant usefulness. One wonders if a proper evaluation of the modern mathematics experiments on texts and teaching could not have provided better guidance to authorities about the quality of the new syllabuses and textbooks. One problem was that the goals and contents of the modern mathematics materials were so different from the traditional ones; there were no common standards for an evaluation. Teachers were extraordinarily important factors in the implementation. Bent Christiansen knew that and put great effort into preparing them. An effort was made in Sweden too. But inevitably, unprepared teachers took over at some point in the implementation process. Prytz and Karlberg (2016) mentioned that the modern material might have required skilled, experienced, and motivated teachers. If the teacher did not share the goals of the new syllabus, teaching might promote learning as merely a meaningless collection of concepts and symbols. This could happen when a new teacher, who had

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not been prepared or had not internalized the modern ideas, took over teaching in a modern mathematics class but had not acquainted himself/herself with methods and concepts that the pupils had had before. Pupils might also change schools when their families moved to a different area. A new teacher might not have the capacity to follow up on the pupil’s experience from the previous school. It became a still more difficult case if the parents were not allowed to help their child as was the case in Iceland. In the Nordic directives for grades 1–6, new algorithms were presented, different from the traditional methods, deeply rooted in the national heritages. There were examples of teachers, trying to turn the pupils back to the conventional algorithms in the middle of the process. There were also teachers—Agnete Bundgaard and her collaborators, for example—who told about how their little pupils were excited and jubilant about the new items they learned from her text. It is well known that a good teacher can lead pupils into a new topic of almost any kind if he/she knows the topic thoroughly and presents it with sincere enthusiasm. As Jeremy Kilpatrick (2012) said: At the crux of any curriculum change is the teacher. The teacher needs to understand the proposed change, agree with it, and be able to enact it with his or her pupils—all situated in a specific educational and cultural context. (p. 569)

Looking away from the implementation problems and critiques on the various details of the modern mathematics movement, it can also be seen as a valuable, long-needed reform. The proponents of the modern mathematics arrived back from Royaumont, enamored by the new ideas, and enthusiastic to create new teaching materials. The enthusiasm spread and opened people’s eyes to new possibilities in a reformed curriculum. Ole Skovsmose’s (1979) expression about modern mathematics in Denmark has a wider reference in the Nordic countries: The 1960s mathematics is a clear and radical breakthrough of the mathematics teaching that over a long period had stagnated around a limited set of methods and problems […]. The mathematics reform has slit the teaching out of a dead and archaic tradition, radically changed its content and enlarged its sphere. (p. 152)



Sources in Archives

SE/RA/2717, Nordiska kommittén för modernisering av matematikundervisningen [Nordic Committee for the Modernization of School Mathematics] (1960–1968). Swedish National Archives.

Riksarkivet (RA B) [Swedish National Archives] B1 Utgående skrivelser [Outgoing Letters] U 1. Memorandum about Scandinavian cooperation concerning mathematics teaching. U 2. Letter from L. Sandgren to S. Bundgaard, dated December 31, 1959. U 3. Letter from L. Sandgren to I. Johansson, dated December 31, 1959. U 8. Minutes from the first meeting of the NKMM committee in Stockholm on October 3–4, 1960. U 9. Memorandum about the NKMM committee, dated October 9, 1960. U 23. Letter from L. Sandgren to O. Rindung, dated November 15, 1960. U 42. Letter from M. Håstad to A. Bundgaard, dated January 11, 1961. U 81. Directives to writers of experimental texts for grades 7–9 from the NKMM. U 213. Directives to writers of experiment texts for grades 1–6 from the NKMM. U 230. Letter from [M. Håstad] to A. Bundgaard, dated November 23, 1961.

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U 253. Minutes from a meeting of the NKMM committee in Helsingfors on October 12–13, 1961, dated October 14. U 268. List of members in writing teams for grades 1–6 for the NKMM, dated in January 1962. U 376. Summary of 11 reports about experiments in geometry for grades 7, dated in May 1962. U 839. Letter from M. Håstad to Forsøksrådet for Skolverket, Oslo, dated September 10, 1964. U 866. Letter from M. Håstad to A. Bundgaard, dated October 1, 1964. U 1052. Report by C. Hultman on the experimental texts for grades 1–6, dated in April 1965. U 1369. Comparative tests in mathematics in grade 9. U 1371. Standardized tests in mathematics used in the experimental classes in grade 8. U 1425. Test for grade 6. U 1431. Standardized tests for grade 3. U 1536. Letter to Dipl. Ing. M. K. Manoussokis with a list of authors, dated October 27, 1966. U 1545. Draft to the NKMM committee’s final report on the experiment texts, dated November 1966. U 1676. Letter from M. Håstad to A. Bundgaard, dated January 3, 1967. B2 Experimental texts Christiansen, A., & Christiansen, B. (1965a). A 7–9. Preface. NKMM.

Riksarkivet (RA E) (Swedish National Archives) E1 Inkomna skrivelser [Incoming Letters] Christiansen, A., & Christiansen, B. (1965b). A report to the Danish experiment committee, dated June 24, 1965. I 39. Letter to L. Sandgren from A. Bundgaard, dated January 2, 1962. I 278. Letter to the central office of NKMM from T. Bue, dated November 13, 1961. I 285. Letter to NKMM from E.  Kristensen with comments on directives for grades 1–6, dated November 16, 1961. I 286. Letter to L. Sandgren from A. Bundgaard, dated November 16, 1961. I 333. Letter to M. Håstad from K. Piene, Oslo, dated January 18, 1962. I 334. Letter to K. Piene from F. F. Ask, Trondheim, dated January 15, 1962. I 335. Letter to K. Piene from H. Rörvik and F. F. Ask, dated January 15, 1962. I 440. Letter to M. Håstad from P. Malinen Helsingfors, Finland, dated June 4, 1962. I 615. Letter signed by D. B. Adams, Stanford University, dated November 15, 1962. I 647. Letter to L. Sandgren from A. Bundgaard, dated November 22, 1962. I 917. Letter from H.L. Halvorsen Notodden, Norway, to NKMM, dated September 26, 1963. I 1164. Letter to M. Håstad from A. Bundgaard, dated August 26, 1964. I 1247. Letter to M. Håstad from A. Bundgaard, dated October 11, 1964. I 1314. Letter to M. Håstad from A. Bundgaard, dated January 5, 1965. I 1607. Letter to M. Håstad from B. Christiansen, dated December 14, 1965. I 1633. Letter to M. Håstad from A. Bundgaard, dated February 22, 1966. I 1706. Letter to M. Håstad from B. Christiansen, dated June 21, 1966. I 1734. Letter to M. Håstad from A. Bundgaard, dated in August 1966. I 1780. Letter to M. Håstad from A. Bundgaard, dated October 6, 1966. I 1808. Letter to M. Håstad from B. Christiansen, dated October 12, 1966.

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References Arnlaugsson, G. (1966). Tölur og mengi [Numbers and sets]. Reykjavík, Iceland: RN (Ríkisútgáfa námsbóka [State publishing company for textbooks]). Behnke, H., Choquet, G., Dieudonné, J., Fenchel, W., Freudenthal, H., Hajós, G., & Pickert, G. (1960). Lectures on modern teaching of geometry and related topics. Aarhus, Denmark: Matematisk Institut (Aarhus Universitet), Elementær Afdeling. Nr. 7. Bergendal, G., Håstad, M., & Råde, L. (1966–1968). Matematik för gymnasiet [Mathematics for the gymnasium]. Stockholm, Sweden: Biblioteksförlaget. Bergendal, G., Hemer, O., & Sander, N. (1970). Rúmfræði [Geometry]. Reykjavík Iceland: RN. Bjarnadóttir, K. (2006). From isolation and stagnation to “modern” mathematics in Iceland: A reform or confusion? Paedagogica Historica, 42(4–5), 547–558. Bjarnadóttir, K. (2007). Mathematical education in Iceland in historical context—Socio-economic demands and influences. PhD. thesis nr. 456-2007, Roskilde University. Available at https://rucforsk.ruc.dk/ws/files/2051540/ IMFUFA_456.pdf. Bjarnadóttir, K. (2011). Implementing “modern math” in Iceland—Informing parents and the public. In M.  Pytlak, T. Rowland, & E. Swoboda (Eds.), Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education, 9th–13th February, 2011 (pp. 1670–1679). Rzeszów, Poland: University of Rzeszów. Bundgaard, A. (1969–1972). Stærðfræði. Reikningur [Mathematics. Arithmetic] 3a–6. (Translated into Icelandic by K. Gíslason). Reykjavík, Iceland: RN. Bundgaard, A., & Kyttä, E. (1967–1968). Stærðfræði. Reikningur [Mathematics. Arithmetic] 1–2b. (Translated into Icelandic by K. Gíslason). Reykjavík, Iceland: RN. Choquet, G. (1969). Geometry in modern setting (Translation of L’enseignement de la géometrie, first published by Hermann in 1964). Paris, France: Hermann. Christiansen, B. (1967). Mål og midler i den elementære matematikundervisning [Goals and methods in the elementary mathematics teaching]. Copenhagen, Denmark: Munksgaard. Christiansen, B., & Lichtenberg, J. (1965). Matematik 65. Copenhagen, Denmark: Munksgaard. Christiansen, B., Lichtenberg, J., & Pedersen, J. (1964). Almene begreber fra logik, mængdelære og algebra. [General concepts from logic, set theory and algebra]. Copenhagen, Denmark: Munksgaard. Gíslason, K. (1978). Nýja stærðfræðin [The new mathematics]. An unprinted report to Reykjavík Director of Education. Reykjavík, Iceland. Gjone, G. (1983). “Moderne matematikk” i skolen. Internasjonale reformbestrebelser og nasjonalt læreplanarbeid [“Modern mathematics” in school. International reform efforts and national curriculum work] (Vols. I–VIII). Oslo, Norway: University of Oslo. Hanner, O. (1967). Nya svenska gymnasieböcker [New Swedish gymnasium books]. Nordisk Matematisk Tidskrift, 15(4), 157–166. Howson, A. G., Keitel, C., & Kilpatrick, J. (1982). Curriculum development in mathematics (First paperback edition). Cambridge, United Kingdom: Cambridge University Press. Hyltén-Cavallius, C., & Sandgren, L. (1958, 1969). Matematisk analys: För nybörjarstadiet vid universitet och högskolor. [Mathematical analysis: For beginners at universities and colleges]. Lund, Sweden: Lund Studentlitteratur. Høyrup, J. (1979). Historien om den nye matematik i Danmark—En skitse. [The history of the new mathematics in Denmark—A sketch]. In P. Bollerslev (Ed.), Den ny matematik i Danmark (pp. 49–65). Copenhagen, Denmark: Gyldendal. Karp, A. (2015). Interview with Mogens Niss. The International Journal for the History of Mathematics Education, 10(1), 55–76. Kilpatrick, J. (2012). The new math as an international phenomenon. ZDM–The International Journal on Mathematics Education, 44(4), 563–571. Kristensen, E., & Rindung, O. (1962–1964). Matematik I. Copenhagen, Denmark: Gads Forlag. Krüger, K. (2019). Functional thinking: The history of a didactical principle. In H.-G.  Weigand, W.  McCallum, M. Menghini, M. Neubrand, & G. Schubring (Eds.), The legacy of Felix Klein. ICME-13 monographs (pp. 35–53). Cham, Switzerland: Springer. Lyytikäinen, V. (1967). Yrjö Juve in memoriam. Nordisk matematisk tidskrift, 15(4), 145–147. Malaty, G. (2009). Mathematics and mathematics education development in Finland: The impact of curriculum changes on IEA, IMO and PISA results. Dresden, Germany: HTW. Møller, S. (1967). [Litteratur]. Nordisk Matematisk Tidskrift, 15(2–3), 108–114. Moon, B. (1986). The “New Maths” curriculum controversy: An international story. Barcombe, United Kingdom: Falmer Press. NN1. (1969). Frá Fræðslumálaskrifstofunni [From the Education Department of the Ministry of Education and Culture]. Menntamál [Educational matters], 41(1), 94–97.

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NN2. (1970). Nýja stærðfræðin er eins og nýtt tungumál [The modern mathematics is like a new language]. Morgunblaðið, September 13, 23–24. NR. Nordisk råd. (1967a). New school mathematics in the Nordic countries. Stockholm, Sweden: Nordisk råd. NR. Nordisk råd. (1967b). Nordisk skolmatematik [Nordic school mathematics]. Stockholm, Sweden: Nordisk råd. Organisation for European Economic Cooperation (OEEC). (1961). New thinking in school mathematics. Paris, France: OEEC. Prytz, J. (2017). Governance of Swedish school mathematics—Where and how did it happen? A study of different modes of governance in Swedish school mathematics, 1910–1980. Espacio, Tiempo y Educación, 4(2), 43–72. Prytz, J. (2018). The new math and school governance: An explanation of the decline of the New Math in Sweden. In F. Furinghetti & A. Karp (Eds.), Researching the history of mathematics education. An international overview (pp. 189–216). Cham, Switzerland: Springer. Prytz, J., & Karlberg, M. (2016). Nordic school mathematics revisited—On the introduction and functionality of New Math. Nordic Studies in Mathematics Education, 21(1), 71–93. Skolelovgivningen i Danmark, efter 1521. [School legislations in Denmark, after 1521]. Aarhus University. Retrieved September 20, 2021, from https://danmarkshistorien.dk/ Skovsmose, O. (1979). 60’er-matematikken—Idé og virkelighed [The mathematics of the 1960s—Ideas and reality]. In P.  Bollerslev (Ed.), Den ny matematik i Danmark (pp.  152–166). Copenhagen, Denmark: Gyldendals Matematikbibliotek. Skovsmose, O. (1980). Forandringer i matematikundervisningen. Didaktiske arbejdspapirer 1 [Changes in the mathematics teaching. Didactical papers 1]. Copenhagen, Denmark: Gyldendal. Solvang, R. (1966). Bokmeldinger [Book announcements]. Nordisk Matematisk Tidskrift, 14(3–4), 118–119. Stefánsson, E. (1967). Yngsti skólinn: Nýjungar í reikningskennslu. [The school for the youngest: Innovations in arithmetic teaching]. Foreldrablaðið [Parents’ journal], 23(1), 7–13.

Chapter 13

Reforms Inspired by Mathématique Moderne in Poland, 1967–1980 Zbigniew Semadeni

Abstract  Efforts to modernize Polish mathematics education began in the first half of the twentieth century. After 1957, the moving spirit of the Polish reforms was Zofia Krygowska; a description of her role in Poland and in international fora is augmented with explicit quotations from her books and articles. In 1967, under a strong influence of the French and Belgian versions of New Math, a radical reform of Polish secondary mathematics education was introduced, followed by an equally radical reform of primary education. Unfortunately, the implementation of the latter was combined with a fundamental change of the whole 12-year schooling system to an unclear 10-year system. In 1980, when the reform reached grade 4, the Solidarity movement forced the government to abandon it; yet, some changes were irreversible. In 2008, the last remains of New Math disappeared from the Polish core curriculum. In the concluding part of this chapter, the unique phenomena of New Math reforms and their ideology are discussed. Keywords  Algebraic errors · Axiomatic geometry · CIEAEM · Curricula · Curriculum reforms · Educational ideology · Geometry textbooks · Georges Papy · Hans Freudenthal · ICMI · Jean Piaget · Mathematical structures · Mathematics teachers · New Math · Polish education · Primary mathematics education · Sets · Stefan Straszewicz · Zofia Krygowska



Early Polish Efforts to Modernize Mathematics Education1

When the Fourth International Congress of Mathematicians (held in Rome in 1908) resolved to establish the International Commission on Mathematical Instruction (ICMI)2 and Felix Klein became its president, the state of Poland did not exist. Its territory was divided among the empires of Germany, Russia and Austro-Hungary; Polish students attended three different school systems. Yet, when ICMI invited educators from each member state to prepare a report on the practice of mathematics teaching in their country, Polish mathematicians managed to prepare their own report, which was published in  In this chapter, we make use of the material gathered in Semadeni (2020).  Until the 1950s mainly indicated with the French acronym CIEM (Commission Internationale de l’Enseignement Mathématique) or the German IMUK (Internationale Mathematische Unterrichtskommission). This history is described in Howson (1984), where the case of Poland before 1914 is mentioned. 1 2

Z. Semadeni (*) University of Warsaw, Warsaw, Poland e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 D. De Bock (ed.), Modern Mathematics, History of Mathematics Education, https://doi.org/10.1007/978-3-031-11166-2_13

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Figure 13.1  Stefan Straszewicz

Figure 13.2  Zofia Krygowska

the journal L’Enseignement Mathématique in July 1911 (Cercle mathématico-physique de Varsovie 1911). Officially, it was a separate part of the Russian report. In independent Poland (1918–1939), Witold Wilkosz and some other mathematicians were engaged in improving secondary education. With Klein’s ideas in mind, they discussed the question of logical precision in textbooks, and tried to keep balance between abstract mathematical structures and their concrete examples. Otto Nikodym (1930) argued that modern mathematics should be seen as a renaissance of Greek thought, incorporating an axiomatic approach and reasoning based on logic, while elementary mathematics education, seen from a modern viewpoint, was replete with errors. A report on the university preparation of Polish prospective mathematics teachers was submitted to ICMI at a meeting in Zurich in 1932. Among authors of textbooks for 11- to 17-year-olds there were top mathematicians: Wacław Sierpiński and Stefan Banach. Their textbooks were well written, in traditional style, with moderate scope, avoiding novelties.

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After the 1939 German invasion, secondary and tertiary education systems were not allowed in Poland; only a part of the elementary and vocational school systems continued to operate, with practice-­oriented reduced curricula. However, a huge system of underground Polish education was organized. It was an amazing network of clandestine classes, held in small groups of up to 5–10 students, hidden in people’s homes or disguised as another activity allowed by the Nazis. It is estimated that over a million Polish children learned in covert primary schools, about a hundred thousand students secretly attended secondary schools, and ten thousand studied at the university level. Some 15% of teachers lost their lives during that period. After 1945, contacts of Polish citizens with Western Europe were strictly limited by communist authorities. The 1956 political thaw made foreign trips possible, but it was still difficult to obtain a passport, while lack of hard currency was another obstacle. For half a century, a leader in Polish mathematics education was Stefan Straszewicz (1889–1983) (Figure 13.1), a professor at Warsaw Technical University. His PhD from the University of Zurich in 1914 was supervised by Ernst Zermelo, the founder of axiomatic set theory. In 1932, Straszewicz became the Polish national representative to ICMI and continued in that role until 1972 after the reconstitution of ICMI in 1952; during 1963–1966, he was a vice-president of the Executive Committee of ICMI. From 1949 to 1969 he was the chairman of the Commission on Mathematics Curricula of the Ministry of Education. In 1950 he organized the Polish Mathematical Olympiad for secondary school students and for the next 20 years he was its chairman. From 1953 to 1957, he was the president of the Polish Mathematical Society. However, it was Zofia Krygowska (Figure 13.2) who played the key role in reforms of mathematics education in Poland inspired by New Math; she was also very active internationally.



Krygowska’s Role in Poland and in International Reform Debates

Zofia Krygowska (officially Anna Zofia Krygowska,3 1904–1988) attended primary and secondary schools in Zakopane in the Polish highlands. She was eager to read books, also in French and German, including original versions of classics like Goethe’s Faust and Der Zauberberg [The Magic Mountain] by Thomas Mann. As a mathematics graduate from the Jagiellonian University in Cracow, she became a teacher in 1927 and taught in primary schools (from grade 1 on) and in secondary schools. Her passions were school and mountains. During the War she acted as a liaison (under the cover of a bookkeeper visiting sawmills) between the Polish Underground Teaching Organization and secret local groups in the highlands, helping them, supervising, organizing examinations, carrying textbooks and other materials in a backpack, and teaching. She often experienced a great fear; 40 years later she recalled it as a nightmare (Krygowska 1985). After the War, she worked at the Teachers Training Centre in Cracow. In 1952, she received a PhD in mathematics from the Jagiellonian University; her thesis was on the limits of precision in teaching elementary geometry. She became a lecturer at the College of Teacher Training in Cracow, which later evolved into what is now the Pedagogical University of Cracow.4 She worked on many issues, such as educating teachers of mathematics, developing didactics of mathematics to make it a scientific discipline, making use of concepts of developmental psychology, and discussing reforms to the mathematics curriculum. In 1958, her school resolved to organize a Chair of Methodology of  In her books and articles published in Poland, she always wrote only her middle name “Zofia”; she also celebrated her name day as Zofia. When she was abroad, Anna Zofia was used according to what was written in her passport. 4  For many years it was named Wyższa Szkoła Pedagogiczna, literally “Higher School of Education,” corresponding to the German term Pädagogische Hochschule. In 1999 the school became Akademia Pedagogiczna and in 2008 it became a university. 3

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Mathematics Teaching, later renamed as Chair of Didactics of Mathematics. This was the first such chair in Poland. It was part of the Faculty of Mathematics and Physics, not of the Faculty of Education (Nowecki 1992). In 1956, Straszewicz and Krygowska were members of the official delegation of the Polish Ministry of Education to the UNESCO International Conference in Geneva. The conference included a part devoted to mathematics (regarded then as the most conservative of all school subjects). Jean Piaget participated in it and expressed his view that from a psychological viewpoint, the fundamental structures of mathematics (i.e., algebraic structures, order structures, and topological structures) could and should be a basis of a new organization of school curricula (Krygowska 1957). A significant result of that conference was that Krygowska opened up contacts with mathematicians in Western Europe (Félix 1986). She was a good friend of many of them, in particular Willy Servais, Hans Freudenthal, Hans-Georg Steiner, and Tamás Varga. In 1957, she joined 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. It was established in the years 1950–1952 as an independent body and consisted of enthusiastic people.5 In the first period (till 1960) the Commission was dominated by three personalities: Its president (the mathematician Gustave Choquet, who stressed the role of fundamental structures of modern mathematics in education), its vice-president (the psychologist Jean Piaget, who opened new perspectives for research on logico-mathematical constructions of the mind), and its secretary (the educationist Caleb Gattegno, who worked on connecting the above ideas with a didactical context) (see also Chap. 3 in this volume). Piaget, Choquet, and André Lichnerowicz influenced Krygowska’s structuralist approach to secondary school mathematics. She wrote: The modest content of school curricula gives many opportunities to reveal to students fundamental algebraic structures, analogies and isomorphisms of various fragments of school mathematics which are isolated in students’ minds; this can be done in a way which is natural, simple and accessible to them. It makes possible to break down rigid barriers dividing particular branches of elementary mathematics and to show students that there are concepts which intervene—in particularly interesting ways—in completely different areas. One can easily make a general concept of an operation available to students, show examples of some “unusual” operations, non-­ commutative or non-associative. The school subject matter is pervaded by the concepts of operation, inverse operation, group, field, equivalence relation, order relations, transformations, invariants, and isomorphisms. Neglecting these concepts is completely unnatural, is an unnecessary curtain separating the student’s thought from contemporary mathematical thought. (Krygowska 1959, p. 18)

Krygowska was influenced by Piaget’s idea of relations between fundamental structures of modern mathematics and structures of cognitive development. The first operations used by children, derived from a general coordination of their actions on objects, are precisely divided into three wide categories. One of them consists of those operations where reversibility is obtained in a way akin to algebraic structures; the second—those where reversibility is obtained by reciprocity (as in structures of order, of seriation); and the third—those related to neighborhoods and continuity, that is, topological structures, genetically prior to metric structures. He compared this with Bourbaki’s analogical concept of three mother structures (Piaget 1970). In the second period 1960–1970, CIEAEM was dominated by its president Georges Papy (Figure 13.3), a professor of algebra at the University of Brussels, Belgium (see also Chap. 10 in this volume). His goals were as follows: a curriculum with a strong component of algebraic structures, modernization of geometry teaching, and modernization of primary education (Castelnuovo 1981). The 14th rencontre [meeting] of CIEAEM was held in Cracow in August 1960. It was heavily influenced by Papy and his idea of mathématique de base [“base mathematics”]. He stressed that in  The legal setting of CIEAEM should be contrasted with that of ICMI, which officially is a commission of the International Mathematical Union (IMU). Members of ICMI are states (one of them is Poland), while members of CIEAEM are people. 5

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Figure 13.3  Papy, presented as the “pope of modern mathematics.” (Caricature by Leon Jeśmanowicz, 1971)

each epoch of the history of mathematics there were certain concepts, relations between them, ­theorems and rules forming the “base mathematics”; in modern mathematics, this fundamental role is played by sets and relations. Papy claimed that it should be much easier to teach these fundamental ideas when the learners were not yet educated and their brains were fresh, “without bad habits.” One day he made an impressive show with children aged 8–12 years. Schools were closed, and children were found in a summer play center and brought to the conference venue. The theme of those activities was relations (a university subject). Papy spoke French and his words were ad hoc translated to Polish for the children. Children were first asked to represent themselves on the blackboard, in some very simple way. They talked about how to do this and drew small marks ×, augmented later with numbers chosen by themselves. As one of the next steps, the children were asked to make marks for their sisters and brothers; when they were asked to indicate these relationships, they drew arrows from children to their sisters and other arrows to brothers. After that, Papy drew a separate graph representing some other children and arrows showing sisters and brothers; he then asked what additional arrows must be there for sure (e.g., in the case of a brother of a brother, this meant the transitivity of the relation). Teachers who watched this were astonished: The children were able to use simple graphical schemes and argued correctly (Korczowski 1961; Krygowska 1961). For a long period of time Papy’s ideas influenced Krygowska’s thinking about mathematics education. For her, three ideas were particularly important: (a) stressing the role of fundamental algebraic structures; (b) starting with mathématique de base very early; and (c) advocating Venn diagrams and arrow graphs as non-verbal language representing two main concepts of this “base mathematics.” She believed in the explanatory power of Venn diagrams for children and recommended using them even in the case where the closed curves symbolized intersecting straight lines (Krygowska 1977b). Her structuralist attitude was extended to primary grades: The structure of the ring ℤ of integers is the main subject in those experimental classes [devised by Krygowska]; the study of it is the core of a topic rich in problems of algebra, arithmetic, geometry … Examples of finite rings facilitate an approach to a general idea of an operation (example of a preconcept). (Krygowska 1971b, p. 103)

Henryk Moroz, in his thesis which was supervised by Krygowska, presented this idea in the following way: In the project of a new mathematics curriculum for grades I–IV, worked out by Z. Krygowska and the present author, the central theme is the structure of the ring of integers. Results of an experimental study irrefutably show that the structure of the ring of integers is for the pupil of age 7–11 definitely more intelligible, more interesting and more instructive than the structure of the set of non-negative rational numbers. (Moroz 1972, p. 36)

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Even the Cuisenaire rods, teaching tools providing the child with visual, muscular, and tactile images, were seen in structural terms: In order to stress the role of clear scientific conception played in the conception of activity-based teaching, let us mention another approach to elements of arithmetic, namely that based on the Cuisenaire material. Theoretically, we start with an algebraic structure of the arithmetic of non-negative integers, that is, a semigroup. Children solve various problems. (Krygowska 1977a, p. 97) Using numbers in color, the pupil of grade 1 in the preparatory period, before introducing numbers, solves various problems, and doing that he/she becomes aware of the structure of a semigroup. In the set of numbers in color, the child discovers an equivalence relation: Blocks of the same color are equally long. (Moroz 1986, p. 33)

In those times, many advanced mathematical ideas were identified by observers in children’s activities. However, later the attitude changed: “There is a long distance to cover between the spontaneous, unconscious use of structures and their becoming conscious” (Piaget and Garcia 1989, p. 25). In 1964, at a conference of Polish mathematics educators, a decisive idea of didactics of mathematics as a separate multidisciplinary academic subject was formulated by Krygowska. She presented inspiring visions and her arguments sounded convincing (Nowecki 1990a). At that conference Zdzisław Opial, an outstanding mathematician from the Jagiellonian University, said: A compromise between mathematics of the second part of the 20th century and that of the 19th century is impossible. Consequently, a compromise between modern mathematics and the present school mathematics is impossible. … links between school and university mathematics were broken many years ago and it is necessary to fix them and to raise the school mathematics back to the height of contemporary mathematics. (quoted by Nowecki 1984, p. 23) In contemporary mathematics there are numerous simple elements that organize this difficult subject, elements common to all branches of mathematics, and these elements should be transferred to the school. They should be used widely, so that school mathematics becomes a reasonable introduction to the activities of the contemporary man. One has to do this necessarily, even if this is impossible! (Opial 1966, p. 7)

These radical statements expressed the feelings of many mathematicians at that time, and reflect the atmosphere which preceded the 1967 reform (see the next section). In 1965, Krygowska spent two months at the UNESCO center in Paris as an editor of the first volume of Tendances Nouvelles de 1’Enseignement des Mathématiques/New Trends in Mathematics Teaching (published in 1966). She also contributed to volumes II (1970), III (1972), and IV (1979) of the Tendances. At the 16th International Congress of Mathematicians (ICM) held in Nice (1970), Krygowska delivered an invited address on problems of modern teacher education in the section Histoire et Enseignement (Krygowska 1971a). She also actively participated in two other ICMs: Moscow (1966) and Warsaw (1983). At ICME-1, the First International Congress on Mathematical Education held in 1969 in Lyon (France), Krygowska delivered a plenary lecture Le texte Mathématique dans l’Enseignement [Mathematical text in the teaching]; in particular, she explained in detail how to help a reader understand Bourbaki’s introductory statements initiating a series of definitions leading to that of a scheme S of construction of an echelon on n terms from a theory stronger than set theory (Bourbaki 1957). This definition started with the words: “A scheme of construction of echelons is a sequence c1, c2, …, cn of pairs of natural numbers ci = (ai , bi) satisfying the following conditions ...” (Krygowska 1969, p. 361). At ICME-2 (1972) in Exeter (UK), she worked in the section on The Professional Training of Mathematics Teachers. At ICME-3 (1976) in Karlsruhe (Germany), she was a member of the Program Committee and was the reporter in Section A2 Mathematics Education at Upper Primary and Junior High School Level (see Figure 13.4). She also lectured at numerous conferences in various countries and visited many groups working on problems of mathematics education (Ciosek 2005; Siwek 2004). In 1970 Papy resigned as the president of CIEAEM and Krygowska became his successor. She remained in that position during 1970–1975; after 1975 she was an honorary president. Her disciple, Stefan Turnau was the president of CIEAEM during 1981–1982 (Gellert et al. 2015). In August 1971 she organized the 23rd rencontre of CIEAEM in Cracow, a meeting which considered questions related to the teaching of logic in schools.

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Figure 13.4  Hans-Georg Steiner, Zofia Krygowska, and Hans Freudenthal at a meeting in Oberwolfach (Germany) preparing for ICME-3 on December 8, 1975. (Photo collection H.-G. Steiner)

During the period 1963–1972, Krygowska was a member of the Main Council of Higher Education, an advisory board of the Minister of Education in Poland. She succeeded in convincing authorities of the need to grant PhDs in academic disciplines (mathematics, biology, etc.), including cases where these concentrated on educational problems of disciplines. Krygowska actively promoted her vision of mathematics education as a field of research (see Nowecki 1984). She maintained that first one had to establish a theoretical and methodological basis of the field, emerging from mathematics itself, psychology, pedagogy, sociology, philosophy, and methodology of research, and also to work out a new conception of elementary mathematics reflecting the present stage of the development of mathematics, overwhelming the historic, anachronic structure of school curricula. Second, it was necessary to set up a suitable legal and organizational framework for educating teachers and to specify the psychological and pedagogical conditions needed for implementing it. Third, modernization of the subject content, teaching methods and activities needed to be carried out by the teacher, teaching resources, working out theoretical concepts, curricula and experimental verifications of them. She also stressed that didactics of mathematics was an independent scientific discipline, albeit interdisciplinary and still at the beginning of its development. Moreover, in this field, one would never have “absolute truths,” independent of the time, country and of general culture which forms the minds independently of teaching (Félix 1986). Krygowska created a vivid center of research on mathematics education in Cracow, and many foreigners came to join the activities. During the period 1968–1986, she was the principal supervisor of 26 PhDs in mathematics granted for research theses which were basically in the field of mathematics education (Turnau 1983). In 1972, she became professor emeritus, but her activity did not diminish at all. The list of her publications is in Nowecki (1990b). Her great success was the creation in 1982 of a research journal Dydaktyka Matematyki (now Didactica Mathematicae, with articles in English), published by the Polish Mathematical Society. She also served on editorial boards of several other journals: Educational Studies in Mathematics (1968– 1978), Recherches en Didactique des Mathématiques (1980–1988), the Polish journal Wiadomości Matematyczne (1963–1988), and the Belgian Nico (1969–1975). In 1977, she received a honoris causa doctorate from her school and a honorary membership of the Polish Mathematical Society. Hans Freudenthal praised her with words: “Among many people carrying on research in mathematics education only a few represent suitable level of mathematical and

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pedagogical quality. Madame Krygowska is one of them, and perhaps the most important and most active, and the only one who has created a school with outstanding achievements. Cracow’s school of educational research is internationally recognized” (quoted by Nowecki 1984, p. 9). In 1984 many people came to Cracow for a conference celebrating Krygowska’s 80th birthday, including Josette Adda, Emma Castelnuovo, Hans Freudenthal, Claude Gaulin, Georges Glaeser, Colette Laborde, and Tamás Varga. Hans-Georg Steiner sent a paper to be read.



The 1967 Change in the Mathematics Curriculum for Secondary Schools in Poland

In 1962, the Polish government decided to change the whole structure of general education from a 7 + 4 system to 8 + 4. During the period 1963–1971, the new curriculum was successively implemented in grades 5–8 and then in the secondary school grades 1–4. The Polish Mathematical Society organized debates on the anticipated changes. Official documents had been worked out by the ministerial commission on mathematical curricula chaired by Straszewicz (with Krygowska, Opial, Leon Jeśmanowicz, and other mathematicians as members). Straszewicz was in favor of modernizing Polish curricula, but regarded New Math as too radical. The curriculum and textbooks for grades 5–8 were then modified but remained rather traditional. In the accompanying textbooks for grades 7–8, an intuitive concept of a set was used occasionally, for example as the set of numbers satisfying the inequality 2x – 5