Mathematical Work in Educational Context: The Perspective of the Theory of Mathematical Working Spaces (Mathematics Education in the Digital Era, 18) [1st ed. 2022] 9783030908492, 9783030908508, 3030908496

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Mathematics Education in the Digital Era

Alain Kuzniak Elizabeth Montoya-Delgadillo Philippe R. Richard   Editors

Mathematical Work in Educational Context The Perspective of the Theory of Mathematical Working Spaces

Mathematics Education in the Digital Era Volume 18

Series Editors Dragana Martinovic, University of Windsor, Windsor, ON, Canada Viktor Freiman, Faculté des sciences de l’éducation, Université de Moncton, Moncton, NB, Canada Editorial Board Marcelo Borba, State University of São Paulo, São Paulo, Brazil Rosa Maria Bottino, CNR – Istituto Tecnologie Didattiche, Genova, Italy Paul Drijvers, Utrecht University, Utrecht, The Netherlands Celia Hoyles, University of London, London, UK Zekeriya Karadag, Giresun Üniversitesi, Giresun, Turkey Stephen Lerman, London South Bank University, London, UK Richard Lesh, Indiana University, Bloomington, USA Allen Leung, Hong Kong Baptist University, Kowloon Tong, Hong Kong Tom Lowrie, University of Canberra, Bruce, Australia John Mason, Open University, Buckinghamshire, UK Sergey Pozdnyakov, Saint-Petersburg State Electro Technical University, Saint-Petersburg, Russia Ornella Robutti, Università di Torino, Torino, Italy Anna Sfard, USA & University of Haifa, Michigan State University, Haifa, Israel Bharath Sriraman, University of Montana, Missoula, USA Eleonora Faggiano, Department of Mathematics, University of Bari Aldo Moro, Bari, Bari, Italy

The Mathematics Education in the Digital Era (MEDE) series explores ways in which digital technologies support mathematics teaching and the learning of Net Gen’ers, paying attention also to educational debates. Each volume will address one specific issue in mathematics education (e.g., visual mathematics and cyber-learning; inclusive and community based e-learning; teaching in the digital era), in an attempt to explore fundamental assumptions about teaching and learning mathematics in the presence of digital technologies. This series aims to attract diverse readers including: researchers in mathematics education, mathematicians, cognitive scientists and computer scientists, graduate students in education, policy-makers, educational software developers, administrators and teachers-practitioners. Among other things, the high quality scientific work published in this series will address questions related to the suitability of pedagogies and digital technologies for new generations of mathematics students. The series will also provide readers with deeper insight into how innovative teaching and assessment practices emerge, make their way into the classroom, and shape the learning of young students who have grown up with technology. The series will also look at how to bridge theory and practice to enhance the different learning styles of today’s students and turn their motivation and natural interest in technology into an additional support for meaningful mathematics learning. The series provides the opportunity for the dissemination of findings that address the effects of digital technologies on learning outcomes and their integration into effective teaching practices; the potential of mathematics educational software for the transformation of instruction and curricula; and the power of the e-learning of mathematics, as inclusive and community-based, yet personalized and hands-on. Submit your proposal: Please contact the publishing editor at Springer: [email protected] Forthcoming titles: • Mathematics Education in the Age of Artificial Intelligence- Philipe R. Richard, M. Pilar Velez and Steven van Vaerenbergh (eds.) • The Mathematics Teacher in the Digital Era: International Research on Professional Learning and Practice (2nd Edition) – Alison Clark-Wilson, Ornella Robutti and Nathalie Sinclair (eds.) • 15 Years of Mathematics Education and its Connections to the Arts and Sciences- Claus Michelsen, Astrid Berckman, Victor Freiman, Uffe Thomas Jankvist and Annie Savard (eds.) • Quantitative Reasoning in Mathematics and Science Education - Gülseren Karagöz Akar, ˙Ismail Özgür Zembat, Selahattin Arslan and Patrick W. Thompson (eds.) • The Evolution of Research on Teaching Mathematics - Agida Manizade, Nils Fredrik Buchholtz and Kim Beswick (eds.)- Open Access! • Mathematical Competencies in the Digital Era - Uffe Thomas Jankvist and Eirini Geraniou (eds.)

More information about this series at https://link.springer.com/bookseries/10170

Alain Kuzniak · Elizabeth Montoya-Delgadillo · Philippe R. Richard Editors

Mathematical Work in Educational Context The Perspective of the Theory of Mathematical Working Spaces

Editors Alain Kuzniak LDAR, Département de Mathématiques Université de Paris Paris, France

Elizabeth Montoya-Delgadillo Pontificia Universidad Católica de Valparaíso Valparaíso, Chile

Philippe R. Richard Département de Didactique Université de Montréal Montreal, QC, Canada

ISSN 2211-8136 ISSN 2211-8144 (electronic) Mathematics Education in the Digital Era ISBN 978-3-030-90849-2 ISBN 978-3-030-90850-8 (eBook) https://doi.org/10.1007/978-3-030-90850-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 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 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

Preface

Introduction Initiated in the late 1990s and developed over more than a decade, the set of methodological and theoretical tools pertaining to Mathematical Working Spaces was established and recognized as a theory in the mid-2010s. Since then, the theory has been thoroughly developed and applied through intensive collaboration between researchers of various countries, mainly Europe (Cyprus, France, Greece, Spain…) and America (Argentina, Chile, Canada, Mexico, Peru…). This collaborative and international effort has implicated a network of research projects, theses, study days and ETM-MWS symposia (the latter have been dedicated to MWS every two years since 2009). The result is a large quantity and wide diversity of publications, master’s and doctoral degree theses, with original ideas and perspectives centered on mathematical work considered the central focus for the learning and teaching of mathematics. The purpose of this book is to provide an overview of the current state of the theory for researchers, teacher trainers and curriculum developers who wish to learn about the theory and use it most effectively. In line with the spirit of the theory, this book was written by different authors to reflect the conceptual unity at the heart of the theory of MWS and, at the same time, to show the freedom and diversity of approaches given space therein. It is divided into three parts. The first part is centered on theoretical and methodological considerations closely related to the theory of MWS. The second part develops the three MWS dedicated to observing how mathematical work depends on the expectations of educational organizations, how it is formed and taught, and how individuals appropriate it. Using the terms of the theory, these three MWS are the reference MWS, suitable MWS, and personal MWS. In the third part, some applications and perspectives are discussed regarding topics of major importance today in mathematics education such as technological and digital tools, teacher training and modeling.

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Theoretical Characteristics and the Specific Methodology of the Theory of Mathematical Working Spaces This first part of the book endeavors to present the growing body of theory and methodology associated with the theory of MWS. This is an overview, the results of which are discussed and used throughout the rest of the book. In the chapter “The Theory of Mathematical Working Spaces—Theoretical Characteristics” written by Alain Kuzniak, the main issues addressed by the theory of MWS (ThMWS) and key constructs are presented: the mathematical work in relation to Mathematical Working Spaces; the semiotic, instrumental and discursive geneses associated with the MWS diagram; the different levels of MWS linked to reference, suitable and personal work, etc. This presentation is organized around what constitutes the main purpose of the theory and justifies its position in mathematics education research: describing, understanding and (trans)forming mathematical work in an educational context. A description of the methodology employed in research based on the theory of MWS is the specific subject of the chapter “Methodological Aspects in the Theory of Mathematical Working Spaces”, written by Assia Nechache and Ines GómezChacón. The evolution of the theory of Mathematical Working Spaces and its increasing use by many researchers calls for a reflection on the methodological principles of research conducted within its framework. Starting from the observation that it is important to ensure the robustness and operationality of the particular methodology employed in the theory of MWS, the authors introduce the methodological tools associated with its constructs and principles. Several examples from empirical research are evoked to demonstrate how these methodological tools can be used to address some specific issues regarding teachers’ practices and the development of learning in real classrooms. In the follow-up of the first chapter, the authors focus on the mathematical work in which students and teachers actually engage, describing and characterizing the formation and/or transformation of the work. Chapter “The Theory of Mathematical Working Spaces in Brief”, which sets out to provide a conclusion on the first part of the book and is conceived as a summary chapter, written by Alain Kuzniak, Assia Nechache and Philippe R. Richard, and pinpoints the main constructs of the theory of MWS in relation to some frequently raised questions. It helps the reader to explore the conceptual complexity of the theory. In both form and substance, its purpose is to provide the basis of an online glossary and is expected to evolve with further contributions by researchers. To keep the chapter concise, some important elements for the study of mathematical work are not included. These are, however, presented at other points in the book in relation to the themes and research which led to their introduction to the theory.

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The Three MWS Used to Observe Reference, Suitable and Personal Work The second part discusses the three different types of MWS distinguished in the study of mathematical work in its diversity and complexity: reference MWS, personal MWS and suitable MWS. This part of the book is based on numerous examples of research undertaken within the framework of the theory. It thus aims to illustrate the first theoretically oriented part of the book, by showing that the theory of MWS is not only a conceptual framework but is first and foremost a theory in mathematics education which has generated a large current of research anchored in the reality of the classroom. Developed in the chapter “The Reference Mathematical Working Space” (written by Elizabeth Montoya-Delgadillo and Claudia G. Reyes Avendaño), the reference MWS clarifies the way in which mathematical work is specified by an educational organization and should be undertaken in an educational context. The chapter looks at how the reference MWS can be identified and accessed through different approaches—institutional and organizational, epistemological and historical, didactical and cognitive. Using examples, the chapter “The Reference Mathematical Working Space” shows how it is possible characterize mathematical work. In particular, the specific issue of paradigms in the theory of Mathematical Working Spaces is addressed in detail and correlated with the reference MWS. While the reference MWS specifies the way in which students are supposed to understand and undertake the mathematical tasks set, it is essential to know how students really work. In the chapter “Personal Mathematical Work and Personal MWS”, Romina Menares-Espinoza and Laurent Vivier focus on the personal MWS. This MWS is unique to each individual and is defined by how they handle a mathematical problem using their own knowledge and cognitive capacities. For the study of personal work, the chosen approach is based on studying tasks with a twofold reflection, focusing on both the epistemological aspect and the cognitive work of the subject. The chapter particularly explores the question of how knowledge is used in different mathematical domains, especially analysis and geometry. Matching the personal mathematical work undertaken by students and the reference mathematical work with the aim of promoting student learning is one of the main purposes of mathematics education. In the theory of MWS, this transitional function is fulfilled within the suitable or idoine MWS (Chapter “The Idoine or Suitable MWS as an Essential Transitional Stage Between Personal and Reference Mathematical Work”, written by Carolina Hernandez-Rivas, Alain Kuzniak and Blandine Masselin). The suitable MWS can be seen as a connecting structure designed and implemented by a designer-actor (teacher or researcher) to help students build their personal MWS. Different examples of studies on suitable MWS are given to show how mathematical work is progressively designed, tuned and developed in a given school context and by the educational organization. These studies are based on analyses of how the cognitive components and processes of the MWS are organized

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according to the different mathematical contents and domains taught (Geometry, Probability, Calculus).

New Perspectives in Mathematics Education Generated by the Theory of Mathematical Working Spaces The purpose of the third and last part is different from the first two parts of the book. It is no longer a question of establishing the theoretical and methodological principles of the theory of MWS, but rather of focusing on current questions in mathematics education research. The first of these concerns digital tools and artifacts which force us to think about a new kind of mathematical work. The second relates to the ways in which teachers apprehend this mathematical work and are trained to teach it. Finally, how can modeling interact with mathematical work to effectively progress towards a new kind of teaching of mathematics? To address these issues, the perspective of the theory of MWS is used and related to other ways of understanding these issues. It is thus possible to point out differences and complementarities between approaches, and the new possibilities which the theory of MWS brings to these subjects. In the chapter “The Theory of Mathematical Working Spaces: Theoretical Environment, Epistemological Stance and Dialogue with Other Theories”, Alain Kuzniak, Elizabeth Montoya Delgadillo and Laurent Vivier specifically address the issue of dialogue between the theory of MWS and other theories in the field to understand some of the conditions which have been shown to favor interactions and lead to successful cooperation with other theories. To address this and understand the motives and dynamics of these exchanges, the theoretical environment in which the theory of MWS has developed is first described. An in-depth study of the mathematical work which emerges in this context is presented and the epistemological stance of the theory in the field of mathematics education research is clarified. International collaborations have undoubtedly contributed to enriching the theory of MWS, helping to identify its limits and clarify its real scope. The two subsequent chapters are mainly concerned with the present and future mathematical work implemented in numerical and digital environments which are increasingly rich and more complex. The particularities of mathematical work in the digital age are discussed in the chapter “Mathematical Work in the Digital Age. Variety of Tools and the Role of Geneses”, written by Jesús Flores Salazar, Jorge Gaona and Philippe R. Richard. This chapter is particularly concerned with how artifacts impact and transform mathematical work. After some historical considerations on mathematical work when symbolic or mechanical tools and algorithmic methods are used, the new mathematical work is defined by reflecting on interaction between humans and machines. Adaptation in a process of idoneity between the teaching and the learning project, as well as between the intention of the designer and the work carried out by the user are discussed.

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In the chapter “Instrumental Genesis in the Theory of MWS: Insight from Didactic Research on Digital Artifacts”, Jean-baptiste Lagrange and Philippe R. Richard examine the central relationship between notions introduced by Vérillon and Rabardel: artifact and instrumental genesis, and homonym notions in the theory of MWS. They underline how the two approaches derive from different research perspectives. Mathematical Working Space emerged from didactic research in geometry and Vérillon and Rabardel’s works from cognitive ergonomics, or psychology in the workplace. The point made is that there is no reason why homonym notions should carry the same meaning, and it is therefore important to work continuously on basic notions in order to specify the meanings we seek to convey in a given theory, as well as to be cautious with regard to possible semantic bias when separate theories use similar vocabulary. This book is an opportunity to discuss instrumental genesis in the two theoretical elaborations with precision, and to develop a firm basis for further research. The theory of MWS in relation to constructs drawn from other theoretical frameworks is also discussed in the chapter “Mathematics Teachers’ Knowledge and Professional Development: A Cross-Case Comparison Study” (written by Ines Gómez-Chacón), this time in relation to the professional development of teachers, and how the theory can be made operational. Ongoing research on mathematical and didactic knowledge for teacher training builds on various notions which deconstruct this knowledge into different items such pedagogical content knowledge (PCK) to complement subject content knowledge. In the chapter, some significant elements from the theory of MWS are presented regarding the role of the teacher. The notion of the development of mathematical work is used to make sense of the interplay between epistemological and cognitive planes and variables related to the specialized knowledge of mathematics and teaching (MTSK). The results from the Cross-case Analysis, which is the basis of the study, are presented through the elected representative cases. In the chapter “Modeling in Education: New Perspectives Opened by the Theory of Mathematical Working Spaces”, some of the perspectives developed from the theory of MWS on modeling education are discussed by Jean-baptiste Lagrange, Jaime Huincahue and Giorgios Psycharis. The guiding line therein is the mathematical work which students develop in modeling tasks. The complex relationship between reality and mathematics, and between modeling and mathematization are explored. The plurality of models for a given reality help to design tasks which place transitions or coordination between specific domains corresponding to different a priori suitable working spaces at stake. In these studies, mathematical work contributes towards clarifying and operationalizing models as well as giving meaning to abstract mathematical notions. A MWS perspective could thus break with a conception of modeling as an activity pursued individually and for individual competencies.

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Some Challenges Ahead for the Theory of MWS The final collaborative chapter (“Mathematical Work and Beyond”) is presented with the intention of identifying future challenges of the theory within the community of mathematics education research. The guiding idea of the chapter is both to deepen the study of mathematical work while considering how to best explore it and its limits in a wider context. In the first axis, it deals with the study and long-term construction of mathematical work in a large number of mathematical domains. The second axis then specifically addresses questions of learning and teaching by introducing an evaluative and comparative perspective. The third axis develops the interdisciplinary perspective of a new kind of mathematical work developed in connection with other disciplines and uses new and more sophisticated media. Finally, the fourth axis goes beyond the walls of the classroom to question the various contexts, notably socio-cultural, which interact with and shape the development of mathematical work. So many challenges and avenues for future research which are not all related to the theory of MWS alone and which require exchanges with academics incorporating different points of view and approaches. These interactions must nevertheless not distort the specificity and nature of each of the theories used for this research. In this broadening to other horizons—and this is a final challenge—it is important to uphold the adaptability of the theory of MWS, without denying its basic principles and long-term goals. Paris, France Valparaíso, Chile Montréal, Canada

Alain Kuzniak Elizabeth Montoya-Delgadillo Philippe R. Richard

Contents

The General Framework of the Theory of Mathematical Working Spaces The Theory of Mathematical Working Spaces—Theoretical Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alain Kuzniak

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Methodological Aspects in the Theory of Mathematical Working Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Assia Nechache and Inés M. Gómez-Chacón

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The Theory of Mathematical Working Spaces in Brief . . . . . . . . . . . . . . . . Alain Kuzniak, Assia Nechache, and Philippe R. Richard

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The Different Types of Mathematical Work and MWS The Reference Mathematical Working Space . . . . . . . . . . . . . . . . . . . . . . . . . Elizabeth Montoya-Delgadillo and Claudia G. Reyes Avendaño

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Personal Mathematical Work and Personal MWS . . . . . . . . . . . . . . . . . . . . Romina Menares Espinoza and Laurent Vivier

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The Idoine or Suitable MWS as an Essential Transitional Stage Between Personal and Reference Mathematical Work . . . . . . . . . . . . . . . . . 121 Carolina Henríquez-Rivas, Alain Kuzniak, and Blandine Masselin Development of the Theory and Interaction with Other Theoretical Approaches The Theory of Mathematical Working Spaces: Theoretical Environment, Epistemological Stance and Dialogue with Other Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Alain Kuzniak, Elizabeth Montoya-Delgadillo, and Laurent Vivier

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Mathematical Work in the Digital Age. Variety of Tools and the Role of Geneses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Jesús Victoria Flores Salazar, Jorge Gaona, and Philippe R. Richard Instrumental Genesis in the Theory of MWS: Insight from Didactic Research on Digital Artifacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 Jean-Baptiste Lagrange and Philippe R. Richard Mathematics Teachers’ Knowledge and Professional Development: A Cross-Case Comparison Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Inés M. Gómez-Chacón Modeling in Education: New Perspectives Opened by the Theory of Mathematical Working Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 Jean-Baptiste Lagrange, Jaime Huincahue, and Giorgos Psycharis Mathematical Work and Beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Jesús Victoria Flores Salazar, Alain Kuzniak, Elizabeth Montoya-Delgadillo, Assia Nechache, Philippe R. Richard, and Laurent Vivier Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

The General Framework of the Theory of Mathematical Working Spaces

The Theory of Mathematical Working Spaces—Theoretical Characteristics Alain Kuzniak

1 Introduction The theory of MWS emerged more than twenty years ago, initially based on research by Houdement and Kuzniak (1999) on geometry teaching. Their research highlighted the need to consider mathematical work as a key issue for the learning and teaching of mathematics. As a result, mathematical work, targeted and implemented by educational institutions and organizations, has become a central concern, which has naturally led to this notion being questioned and developed in mathematics education. Researchers from different countries have collaboratively sought to describe, understand and shape the mathematical work in an educational context. All this research has contributed toward a substantial theoretical development which now constitutes the theory of Mathematical Working Spaces. In this chapter, some of the theoretical constructs and issues addressed by the theory of MWS (ThMWS) are presented. The notion of mathematical work and the different kinds of MWS are first defined. Theoretical tools enabling the study of mathematical work in a school context are also introduced with, in particular, the diagram of MWS in relation to the question of geneses and circulation of work. Description of the methodology employed in research based on the theory of MWS is the specific subject of the following chapter (Chap. 2, Nechache & Gómez-Chacón). This presentation of the theoretical framework is largely based on the previous contributions published in various journals and presented in conferences. In particular, it draws on the introductory article to the special issue of ZDM Mathematics Education on MWS (Kuzniak et al., 2016). Introductions to the special issues of Relime (Kuzniak & Richard, 2014) and Bolema (Gómez-Chacón et al., 2016) as well as Vivier’s (2020) presentation given at the MWS study days in Paris are also evoked. A. Kuzniak (B) LDAR, Université de Paris, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Kuzniak et al. (eds.), Mathematical Work in Educational Context, Mathematics Education in the Digital Era 18, https://doi.org/10.1007/978-3-030-90850-8_1

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Finally, it also draws on material from Kuzniak’s plenary conference at the ETM6 symposium in Valparaiso (2019), where a “historical” account of the emergence of the theory was presented.

2 Origins and Objectives of the Theory 2.1 Two Seminal Examples The theory of MWS is anchored in the reality of educational actors with regards both their training practices and learning in the classroom. To illustrate this anchoring in different educational contexts, we refer to two examples which shed light on some of the initial motivations of the theory. These two examples are drawn from studies carried out in the domain of geometry. Studies in this particular mathematical domain are central to the idea of geometrical work and thus, by extension, of mathematical work. These examples are as follows: Example 1. The solution proposed by a pre-service primary teacher, also an undergraduate in philosophy (Houdement & Kuzniak, 1999)

This classic geometry problem asks students to demonstrate that the central quadrilateral (HEFG) is a square like the square (ABCD). The figure is plotted on a square grid (Fig. 1). Fig. 1 Elementary task in geometry (Houdement & Kuzniak, 1999)

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A pre-service teacher claims that all sides of the quadrilateral (HEFG) are equal because EH is equal to AE. To show this “false” equality, the student uses a compass to perform and compare the measurements of both sides. The deviation of the lengths and the relative uncertainty (approximately 3%) are small and can explain the error in the transfer. However, one might expect subjects to apply the rationale of triangular inequality or to lean on visual intuition which obviously contradicts the equality. What could cause such a glaring error and how can we interpret this seemingly inconsistent reasoning by the student? Moreover, this student is an adult undergraduate in philosophy, which in principle confers rational and well-organized reasoning. This and other similar cases motivated an in-depth analysis of the relationship between purely deductive proof and the signs or drawing tools used to support such reasoning. Example 2. Different conceptions of geometry among future teachers in Chile and France

Research conducted between 2003 and 2005 on geometry teaching for pre-service teachers in Chile and France was another strong trigger identifying the need for theoretical reflection for a better understanding of what mathematical work is (Guzman & Kuzniak, 2006). Two contrasting answers emerged and were characteristic of the type of relationship Chilean and French students have to geometry. In Chile: it is easy and fun, there are a lot of drawings and constructions with tools. In France: it is boring, you have to write a lot, justify everything you see.

Once again, it was necessary to explain the basic reasons which could explain such a marked difference between the feelings expressed by future teachers in these two countries. The notion of geometric paradigms (7.1) is introduced to explain these differences without making any value judgment on either of the two educational systems.

2.2 Describing, Understanding and (Trans)forming Mathematical Work The research initiated within the project between Paris (France) and Valparaiso (Chile) contributed to the development of theoretical and methodological constructs relevant to the study of mathematics education in compulsory education as well as in teacher training. The aim of these investigations was to understand the differences between teaching and learning mathematics in different countries and educational institutions, without ranking them. This is in clear contrast to the approach adopted by large international studies such as PISA or TIMSS, which are all characterized by such rankings. The research then focused on some challenging issues in mathematics education by considering the long-term goals of a subject working in mathematics. This focus served to highlight the central and unifying place of mathematical work as a research

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topic in mathematics education. An initial overview of these studies introduced the operative notion of Mathematical Working Spaces. These ensure that the mathematical work progresses in accordance with what is expected by the school institution. Different paradigms guide and lead the work undertaken by students and teachers. Current research in the theory of MWS is broadly concerned with the description, understanding and formation of mathematical work. Generally speaking, it goes beyond adopting an approach which would be satisfied with simply reporting the existing phenomena in the teaching and learning of mathematics, without attempting to influence their evolution. A description of the work, drawing on a certain number of constructs, facilitates our understanding of the phenomena observed. The objective is then to share this understanding with the people involved in developing mathematical work, in order to act on these phenomena. This approach leads to an exploration of specific questions such as: What is the mathematical work expected? What is the mathematical work implemented in a given institution? What is the function attributed to each of the acting parties in this work? In what way is it possible to form or transform students’ mathematical work? These questions refer to the characterization of mathematical work in different contexts. They also imply taking the different motivations of the people involved in education into account, i.e., students and teachers. For students, the priority may be the immediate product of their work, the solution to the task which they must submit to their teacher. For teachers, the essence of the mathematical work they expect from their students may be more related to demonstrating use of the methods associated with the processes involved in finding the solution.

3 A Didactical Perspective on Mathematical Work 3.1 The Core of the Theory: Mathematical Work1 The object of the theory of MWS is the didactic study of mathematical work in which students and teachers are effectively engaged. But what does the notion of mathematical work cover in this context? Mathematical work must be understood as an intellectual work of production, the development and finality of which are defined and supported by mathematics. Conversely, mathematics is transformed by the very fact that it is considered as specific human work. Mathematical work involves consideration of three aspects related to its execution and development: The goals of the work. This attribute helps us to distinguish work from a simple activity. This is done by assigning a general and a long-term purpose to a series of actions by clarifying the stakes.

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This topic is further developed in Chap. 7 (Kuzniak, Montoya and Vivier).

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The processes of the work. These are related to the procedures and constraints involved in implementing given tasks. The appropriation and respect of the rules of functioning are important aspects for observation of the work. The results of the work. Results should be valid and correct within the mathematical domain in question.

3.2 The Relationship Between Mathematical Work in the School Context and the Activity of the Mathematician Since the very first investigations in mathematics education or didactics at the beginning of the 1970s, the activity of the mathematician was considered as a possible model for students’ activity. Of what does this activity consist exactly, which could be identified as being specific to mathematicians? Intended to address the community of mathematicians and teachers, Freudenthal (1971) attempted to define the mathematician’s activity as being that of solving problems or looking for problems considered “mathematical”. It is also an activity of organizing a subject matter. In the same vein, Thurston (1994) proposed to characterize mathematics as the field produced by the work of mathematicians: “those humans who advance human understanding of mathematics”. Rather than a circular definition, this can be seen as rather a recursive definition, wherein mathematics is seen as a critical and repeated revision of the research and results generated by those who produce it. These processes are necessarily long term, and so we can assume that it is not possible to have access to a full mathematical work without being engaged in such a progressive and recursive process: the laborious dimension of the work. The students’ journey is thus difficult, and they must go through this uncertain genesis, built on reflections shaped by all previous experiences. Fortunately, their path is doubly defined, on the one hand by the previous work of all the mathematicians who have succeeded in making it practicable and comprehensible, and on the other hand by the effort of the teachers, who have arranged access to this work to help the students undertake their own mathematical work. To what extent can this idealized vision of the mathematician’s work, in line with the mathematical work expected of the student, be taken as a model for teaching? It does not, for example, consider the fact that most students are not destined to become mathematicians. They generally simply aspire to become knowledgeable users of mathematics. Moreover, it does not reflect the current content of school mathematical curricula, which organize and underpin mathematical work today. Thus, the elementary geometry taught today has no relation to the geometry practiced by contemporary mathematicians. It is therefore no longer possible to refer solely to a conception of mathematics aligned with the work of the mathematician to determine the specificities of mathematical work in a school context. However, while reference to the work of the mathematician is not exclusive, it nevertheless remains essential

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to implement sound epistemological vigilance regarding the choices made in the educational sphere. In the theory of MWS, the notion of paradigm is one of the key concepts which helps to ensure this vigilance, by characterizing the mathematical work undertaken. A paradigm stands for a combination of beliefs, convictions, techniques, methods and values shared by a community (Kuhn, 1966). In Kuhn’s work, this community is scientific, but in the context of the present study, it represents educational communities dealing with the injunctions of the institutions in which they are integrated. Among the practices and rules which structure exchanges within the community, there is a defined way of choosing the problems studied, as well as a way of communicating the solutions to these problems. This necessarily implies using a specific form of discourse—with a set of rules and codes—which Granger (1963) calls a style. By this, he means a particular way of presenting rational knowledge by submitting it to codified norms. These norms help to give mathematical objects a well-defined meaning and to guide the problem-solving work. They also make it possible to exclude certain practices by limiting the interpretations available to the reader, be they a researcher, a teacher or a student. For Kuhn, paradigms cannot cohabit, and their relations are inevitably conflicting: A new paradigm tends to eliminate former paradigms. In the context of mathematics, we have a less conflictual view according to which paradigms can coexist and support different approaches to solve problems and thus enrich the understanding of certain mathematical concepts. However, this cohabitation is not without risks and ambiguities and can be a source of difficulties.

3.3 Cognitive and Epistemological Aspects: Mathematical Tasks and Problem Solving So far, in this chapter, mathematical work has been considered mainly in terms of its epistemological coherence. But the human (or even social, and in any case shared) component of the work implies the need to consider a cognitive dimension, which must also be related to the epistemological dimension. This relationship between the two aspects, epistemological and cognitive, is particularly visible in problem-solving activities. Problem solving occupies an essential place in the work of mathematicians, as well as in mathematics education. In addressing problems, students apply techniques and knowledge in accordance with the paradigm at hand. In the theory of MWS, tasks are the medium for problem solving. In a rather broad and open way, we can define the notion of task as an adaptation of the definition proposed by Sierpinska (2004) and developed by Nechache (2017): A “mathematical task” refers to any type of mathematical exercise, question, or problem, with clearly formulated assumptions and questions, that is known to be solvable in a timely manner by students in a well-defined Mathematical Working Space.

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From Activity Theory (Vandebrouck, 2013), we retain the fruitful distinction between the tasks prescribed to individuals and the activities they develop to cope with these tasks. We must be able to observe the prescribed tasks and the activities produced, to be able to describe the work in terms of both the mathematical content and the (possible) cognitive approaches at work. This leads to a double standpoint on the task, that of the prescriber (generally the teacher) and that of the receiver (the student). The latter is not passive and can even sequence their work with a series of self-prescribed sub-tasks. The study of tasks in relation to their execution is based on two approaches which arguably complement each other. In the first, a fine-grained study of the signs, techniques and theoretical properties mobilized to perform the task enables us to identify and emphasize how the mathematical domain in question is structured. The second approach centers more on the gap between what is expected of the students and what they actually produce. This implies an acute observation of mathematical practices in the school environment. Mathematical Working Spaces are introduced and used to facilitate these observations and provide an account of the complexity of mathematical work.

4 Mathematical Working Space 4.1 Definition and Representation of the MWS In order to grasp the mathematical work of individuals confronted with mathematical tasks, it is necessary to take two closely related aspects of such work into account. The first aspect, which is epistemological in nature, is related to the mathematical content of the domain studied. The second, of a cognitive nature, relates to the processes and ways of doing adopted by the individuals solving these tasks. These two aspects are identified and described according to two planes: the epistemological plane and the cognitive plane, each being itself organized in a triadic manner. The epistemological plane has three interacting components: a set of concrete and tangible symbols (representamen); a set of artifacts such as drawing tools or software; and a theoretical system of reference (the referential) based on definitions, properties and theorems. The cognitive level is structured around three cognitive processes: visualization linked to the deciphering and interpretation of signs; construction dependent on the artifacts used and associated techniques and proof based on the referential. A Mathematical Working Space (MWS) thus refers to an abstract structure organized in such a way as to generate mathematical work and to enable individuals to perform tasks in a specific mathematical domain (geometry, probability, etc.). In the case of school mathematics, these individuals are generally not experts but students, more-or-less experienced in the mathematical fields studied. The purpose of the MWS is to highlight the mathematical work proposed by educational institutions and organizations and sometimes suggested by mathematical books or textbooks.

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But it also provides an account of the way mathematical work develops and unfolds in an educational context when individuals are confronted with mathematical tasks in practice. The question of the relationship and organization of the different components of work quickly became central in the research carried out on mathematical work in the theory of MWS. This reflection went hand-in-hand with the exploration of possible representations. The MWS diagram, in the form of a prism, was created to represent the relationships and interactions between the different components and processes. It should be seen primarily as a tool for visualizing and summarizing the detailed analyses of actions produced by subjects performing mathematical tasks. The diagram facilitates an explanation and understanding of the phenomena observed which goes beyond an initial description. In the theory, the articulation between the planes is based on a conception of mathematical work as generated by interactions between the various epistemological components and cognitive processes. This implies transformations and adaptations that are described in the theory through three fundamental geneses: the semiotic genesis, the instrumental genesis and the discursive genesis of proof (Fig. 2). Later efforts by Coutat and Richard (2011) identified the idea of vertical planes (named Sem-Ins, Ins-Dis, and Sem-Dis on Fig. 3) as media for interactions between geneses. The evolution of the work can be visualized through the flow, named circulation, between these different planes. The MWS diagram is closely associated with the development of the theory, of which it is in a way the emblem, but it should not be confused with the theory as a whole. Its use as a methodological tool requires selection of the most appropriate modes of representation to reflect the evolution of the work. This methodological question is explored in detail in Chap. 2 (Nechache & Gómez-Chacón), but also in the specific chapters on the different types of MWS (Part II-Chaps. 4, 5 and 6).

Fig. 2 Mathematical working space diagram

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Fig. 3 Three vertical planes in the MWS diagram

4.2 Epistemological Components and Cognitive Processes 4.2.1

The Epistemological Plane and Its Components

In accordance with a conception of mathematics based on semiotic representations, which goes beyond the simple consideration of representation systems, it seems relevant to use the term sign or representamen to summarize the component related to concrete and tangible objects. According to Peirce (1931, 2.228), the sign, or representamen, is "something which stands for something in some respect or capacity". Depending on the mathematical domain involved, signs can be geometrical figures, algebraic symbols, graphs or even tokens, models or pictures in the case of problems involving modeling. From a very general point of view, the notion of artifact includes everything that has undergone a transformation, however small, of human origin with a certain purpose. Its meaning is not limited to material objects, and it can include a strong symbolic dimension. In the theory of MWS, mathematical artifacts will generally be associated with material objects to avoid confusion with other components of the epistemological plane.2 These objects will be intimately linked to rules and techniques of construction or calculation (Euclidean division, “classical” constructions using ruler and compass, etc.). These techniques are based on formulas or algorithms, the validation and theoretical status of which are no longer problematic or are no longer questioned by their user. In contrast to representamen and artifacts, the set of properties, definitions, theorems and axioms refers to the theoretical part of the mathematical work, and this is why it is called the theoretical frame of reference or theoretical referential in short. 2

The material component of mathematical artifacts alone is often not enough to identify them, and it is important to know their symbolic valence: Indeed, the artifact is often recognized through observation of the technique implemented, Chap. 9 (Lagrange & Richard).

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This is not only a collection of properties: it supports the deductive discourse of proof specific to mathematics, which requires that it be organized in a coherent way, welladapted to the tasks students are called upon to solve, and that the statements through which it is expressed have been the object of some form of institutionalization. The referential is not absolute and determined in a unique way once and for all: its constitution and organization rather depend, in particular, on the mathematical paradigms and mathematical work targeted by the educational institutions or organizations.

4.2.2

The Cognitive Plane and Its Processes

Since the initial development of the MWS, it is undoubtedly through the various extensions of research to new mathematical domains that the theory has made its most convincing and fundamental progress. By leaving the field of geometry, researchers have had to conceive and extend some of the cognitive processes initially associated with geometric work such as visualization and construction. Each of these cognitive processes must be understood in a very wide scope. The process of visualization can be considered as the cognitive process associated with identifying and developing a set of manipulations and transformations of the representamen or signs—which, in this respect, can very well be perceived as acoustic images in strictly oral contexts. It is thus possible to study the role of signs in different mathematical domains, such as mathematical analysis, which give rise to a specific work of visualization based on spreadsheet data, or numerical writings. The visualization must be distinguished from the simple vision or perception of signs. The process of construction is associated with actions triggered by the use of artifacts, actions that lead to tangible results such as figures, graphs or the results of a calculation associated with an algorithm. These results are concrete realizations of some mathematical entities which are generally not isolated and form configurations to be observed and explored (Chap. 9, Lagrange & Richard). The process of proof considered here relies on deductive and logical discourse and must be based on, or lead to, assertions with a clear theoretical status. In contrast, simple empirical validation is more likely to relate to construction or visualization, as discussed above. In addition to deductively organized arguments, these assertions may well be definitions, hypotheses, conjectures or statements of counterexamples, as they are involved in Lakatos’ description of the processes of proofs and refutations (1976).

4.3 Generative Processes in MWS 4.3.1

The Semiotic, Instrumental and Discursive Geneses

Through the lens of MWS, the development by an individual, whether generic or not, of appropriate mathematical work is a gradual process which is progressively

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developed, as a process of bridging the epistemological plane and the cognitive plane. It evolves in accordance with different specific yet intertwined generative developments: semiotic, instrumental and discursive geneses. Each of these geneses can be described according to two perspectives: One stance focuses on epistemological components (top-down perspective), and a second is oriented toward cognitive processes (bottom-up perspective). The terms bottom-up and top-down are only relative to the position of the two epistemological and cognitive planes in the MWS diagram. The Semiotic genesis associates signs and representamen with visualization. It explains the dialectical relationship between syntactic and semantic perspectives on mathematical objects, represented and organized by semiotic systems. The semiotic genesis confers to the representamen their status of operational mathematical objects. In this way, it establishes the links between the function and structure of the expressed signs. The bottom-up perspective linked to this genesis can be considered as a process of decoding and interpreting signs from a given representamen through visualization. The opposite perspective, directed from visualization toward a representamen, can be evoked when a sign is produced or specified. The instrumental genesis establishes the link between the artifacts and the construction processes contributing to the completion of mathematical work. The bottom-up perspective describes the actions by which the user appropriates the various techniques associated with the artifact. The top-down perspective relates to the adequate choice of the artifact required to perform the intended actions. In the theory of MWS, the two points of view, epistemological and cognitive, are essential because if using the artifacts modifies the users’ way of doing and thinking, in return, the mathematics they develop is also transformed. Digital tools associated with computers have completely revolutionized the question of artifacts in mathematics and mathematics education (Chaps. 8 and 9). The discursive genesis of proof is the process by which the properties and results organized in the referential are actuated to be available for mathematical reasoning and discursive validations. By this, we mean validations going beyond graphic, empirical or instrumented verification, but which could nevertheless be triggered by these. In this genesis, the bottom-up genesis is related to a deductive discourse of proof supported by properties structured in the referential. Conversely, identifying properties and definitions to be included in the referential, possibly hinted at by instrumental, computational or visual treatments, is relevant to the top-down perspective. In mathematical work, not every discourse3 should be related to the discursive genesis (of proof), and, for instance, the description, characterization or designation of an object may rather be understood as pertaining to the semiotic genesis, in this specific system of signs which belong to usual language. Furthermore, in the theory, the study of genesis is based on the observation of the subjects’ actions, possibly completed by a reflexive discourse by the latter on their actions.

3

For a discussion on the different types of discourses in the theory of MWS, see Pizarro (2018) who introduces three kind of discourse in relation to each genesis.

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An Example of a Cognitive Unit (CU) Associated with a Mathematical Entity (ME): The Circle

To illustrate this previous development, we will look at the different components and processes of mathematical work as they may have appeared in a teaching sequence on the circle for Grade 4–6 students (Feniche & Taveau, 2009). In this sequence analyzed from the angle of the teacher’s intentions, the geometric work is centered on developing the concept of the circle, defined as a set of points all equidistant from a given point, its center. The designers of the sequence initially rely on use of the compass and graduated ruler to bring out the properties of the points of the circle. Then, these material tools progressively recede into the background to encourage deductive reasoning based on the property of the circle and leading to the construction of other geometric figures such as a triangle, the sides of which are given. According to the theory of MWS, the mathematical work developed around the circle is linked to the constitution of a cognitive unit associated with the “circle” mathematical entity. A mathematical entity is a triplet of elements where each element of the triplet belongs to different components of the epistemological plane. In the case studied in our example, the “circle” mathematical entity is thus composed as follows (\circle\, compass and ruler, equidistance). In this triplet, \circle\ is the representamen and constitutes a signifier for the referent “circle”. The representamen is an image or a drawing associated with a set of points and can be described with words. But its description presupposes a semiotic genesis associated with the drawing and is part of the cognitive unit4 associated with the “circle” sign. It can thus be stated that it is the union of a closed line and a point which appears in the center of this circle5 . The second element of the triplet which constitutes the circle entity is made up of material artifacts such as the compass and the graduated ruler, along with the construction techniques associated with these artifacts. Use of these techniques by a subject with the aim of constructing a figure corresponding to the circle belongs to the cognitive construction process. The third element is composed of the properties of the circle in relation to the notion of equidistance. Implementing these properties enables the subject to be sure that the object represented or described in a problem is indeed a circle. With these properties, it is also possible to propose and justify constructions of figures requiring the use of circles. The cognitive unit associated with the circle is gradually built up through the completion of a series of tasks and activities. The cognitive unit is not a ready-made construct, and it must rather be generated within the framework of the teaching. It is built on an interaction between the different geneses stimulated by the different tasks proposed by the teacher to these students. Each of the cognitive processes mobilizes triggers, or is triggered, by a second cognitive process. This leads us to specify the 4

A cognitive unit encompasses the different elements of knowledge about the mathematical entity. We can see here that the point is part of the circle in its cognitive unit, whereas from a mathematical point of view, it is not.

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interactions between the different geneses and to observe the circulation of work in the MWS throughout the learning process. A detailed study of this example is presented in Chap. 2 (Nechache & Gómez-Chacón) as an example of the study of the geneses.

4.4 Interactions and Circulation of Work in the MWS A full understanding of the different geneses and the three interacting planes between these geneses ([Sem-Ins], [Ins-Dis] and [Sem-Dis]) relates to methodological issues, but also to more specific and detailed studies of mathematical work. These studies are intended to provide an account of the progress of students’ learning and of teachers’ teaching. In particular, they serve to specify the validations used to solve tasks, i.e., the way of defining the communicative and heuristic elements of mathematical work. To illustrate interactions between geneses, we focus on the different validations that can support proof and evidence in mathematics. For this, Richard et al. (2019), propose ways in which to describe several types of proofs (discursivo-graphical, experimental or mechanical, algorithmic) in relation to the different geneses and the vertical planes of the MWS.

4.4.1

Combining the Semiotic Genesis and the Discursive Genesis: The [Sem-Dis] Plane and the Discursive-Graphic Proof

The most common proofs in school mathematics are based on a close connection between semiotic and discursive geneses. This coordination is crucial for developing work that goes beyond the simple iconic vision of objects. In geometry, for example, one can access perceptual and graphical reasoning by imagining transformations and rearrangements in a given figure (e.g., the classic Pythagorean theorem by cutting, or more generally a wordless proof). But to communicate and discuss the logical articulation of the reasoning and its epistemic validity, it is necessary to produce a discourse to detail the reasoning by focusing on the relevant elements. The mathematical work can then be conceived as taking place within the plane [Sem-Dis], and the semiotic dimension guides the phases of the proof discourse. Conversely, a formal written proof, based on properties and theorems explicitly identified a priori, can be developed and understood with the help of heuristic coding signs, added stepby-step to a given figure. In this case, the semiotic dimension helps us to understand the discursive genesis that guides the work in the plane [Sem-Dis].

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Combining the Semiotic Genesis and the Instrumental Genesis: The [Sem-Ins] Plane and Experimental or Mechanical Proof

The interaction between semiotic and instrumental geneses favors an activity of exploration and discovery which relies on digital or classical artifacts in relation to the visible results they produce. The digital artifacts enable graphical representations to be explored—for example functions or geometric configurations—and promote the emergence and understanding of a particular notion, without any epistemic validation being at stake. In elementary geometry, simple measuring tasks, using a ruler or a protractor for example, contribute to the instrumental genesis of these measuring instruments. But at the same time, they encourage the unveiling and identification of the components of the associated figurative representation—for example, locating the vertex, the sides and the interior of an angle—and they thus promote the semiotic genesis linked to the geometric object in question. The proofs of existence that result from this type of work will be considered experimental, or mechanical proofs according to Richard’s definition. Many examples of this can be found in the construction exercises with digital artifacts. Thus, for example, the construction of the three heights of a triangle shows that the three lines are concurrent, and that this property is invariant by dragging in a geometry software (DGS). The strength of these proofs is their functional and practical character, but this advantage also has its drawbacks. It leads students to avoid the standard mathematical argument based on a set of logically articulated properties. It also raises the question of the transparency of these operations from a logical point of view.

4.4.3

The [Ins-Dis] Plane, Combining the Instrumental Genesis and the Discursive Genesis of Proof: Algorithmic Proof

A connection between instrumental and discursive geneses arises typically when the construction of a figure or a graph relies on a routinized technique (possibly implemented in a software) and on a construction algorithm associated with this technique. To qualify this type of construction supported by deductive reasoning based on the theoretical referential, Richard uses the term “algorithmic proof”. Euclid’s formal proof on the construction using a ruler and compass of a pentagon is an example of such algorithmic proof, as are, more generally, classical constructions using a ruler and compass, where the students have to justify each step of the construction with appropriate geometrical reasoning. One can envisage a real deductive reasoning, relying strongly on the discursive dimension, but built progressively over steps that are suggested or verified by examples constructed using the artifacts. In the same way, execution of an algorithm by a machine belongs to the instrumental dimension, while justification of this algorithm is naturally associated with the discursive genesis.

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An Interpretation of the Role of Artifacts Through the Interplay Between Vertical Planes

In the instrumental genesis, the artifacts used can produce configurations of figures, graphs or results of a calculation. The new objects or set of objects thus produced can be assimilated to signs and then be interpreted through a semiotic genesis. This pragmatic production of an object does not require a discursive validation. The mathematical work is situated thus in the plane [Sem-Ins] and can in some cases lead to experimental proofs of existence (4.4.1). In other cases, when justification of the construction process or of the properties observed on the configurations becomes the main concern, it is the epistemic valence (Artigue, 2002) or non-iconic reasoning (Duval, 2005) that is emphasized. The work is done this time in the [Ins-Dis] plane with a type of algorithmic proof according to Richard’s definition (4.4.3). Finally, in many cases, the user’s only intention is to acquire or use the techniques associated with the artifact at hand. In this case, the work is said to be confined to the instrumental dimension.

4.4.5

Blockages, Breaks and Confinements in the Circulation of Work

The MWS diagram, with its different planes and dimensions, is a valuable methodological tool to identify the different phases of the problem-solving process. By highlighting the geneses and their different interactions, it becomes possible to describe the evolution of the mathematical work in order to identify the errors, detect the confinements or explain the blockages in terms of circulation in the MWS diagram. Confinement appears when the work remains in a vertical plane or in one dimension. Students’ blockages sometimes result from confinements and prevent them from concluding their work. The analyses in terms of circulation give clues to understand the difficulties encountered by the students and also the conditions of success based on flexibility in the use of geneses. It is then possible to design and propose adaptations relating, for example, to the choice of new specific tasks or more adapted artifacts, etc. The mediations thus proposed encourage the removal of these blockages or confinements by helping the subject’s activity to rebound. Studying the circulation of work in the MWS can also contribute to the understanding of some of the teachers’ choices by connecting them to particular theories of learning (constructivism, behaviorism, etc.). This can help the teachers to evolve in response to the difficulties observed. For example, regarding activities with strong constructivist connotations, it might be a matter of examining how discovery processes—encouraged, in some cases, by technological equipment—can avoid being locked into the [Sem-Ins] plane, where they end up stagnating. Indeed, the student’s activities are too often guided toward exploring objects or relations with the aim of formulating a conjecture but without any link to proof, other than simple empirical verification, being solicited. The activities may also be limited to manipulative techniques, disconnected from any theoretical

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justification. The question could then be how to shift the work toward the discursive genesis of proof, so that the three MWS geneses are more fully involved.

5 Variety of Mathematical Work and Diversity of MWS 5.1 The Reference, Suitable and Personal MWS The mathematical work will depend on its users and their position in the school institution and, in particular, on their role in the prescription of the curriculum and in its implementation and achievement. Are they designers or executors? For the present purpose, three types of mathematical work in relation to different MWS will be distinguished: reference, suitable and personal. The reference work, in relation to the reference MWS, is defined, normally, by people or organizations in charge of the school institution. One of the essential elements of this MWS is the theoretical referential. It is defined with respect to knowledge, ideally according to mathematical criteria. But social, economic and political criteria also contribute to shaping it, to such an extent that describing it requires thorough research: of the intended curriculum, of treatises written by mathematicians, of essays and articles by mathematics educators, etc. Moreover, mathematical work is not only the text of knowledge, it is also the practice, the spirit, the ways of doing things, etc. In short, it is everything which in the theory of MWS will be grouped under the term “paradigm” and which can be described in a precise way drawing on the different elements of the theory. For this, it will often be necessary to rely on different sources such as school textbooks, various research works in epistemology and didactics, teaching and research colloquia. Finally, it should be noted that the reference work is not given but must be identified and constructed for the needs of each study (Chap. 4, Montoya-Delgadillo & Reyes-Avendaño). The personal work relates to pupils or students and reflects the reality of the work undertaken by individuals who appropriate and handle problem solving in relation to their personal MWS (Chap. 5, Menares-Espinoza & Vivier). It is in fact very fluid and progresses according to the actions the subjects are led to perform. This does not prevent us from being able to identify (beyond the variations inherent to the proposed tasks) stable compliant or non-compliant forms of work (7.1). The suitable work and the suitable (or idoine) MWS refer to this intermediate state of transmission and mediation of knowledge where tension exists between the teacher’s expectations and the redefinition of tasks and the problem in order to for the students’ personal work to progress. The MWS in question has been named a suitable or idoine MWS precisely to emphasize these transformations which characterize its nature, in order to best adjust to the educational project and to the reality of the learners. It is therefore changing and evolving, but, once again, it can nevertheless be characterized, thanks to certain invariants which research on the suitable MWS has identified and demonstrated. This research has established the need to take two

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states of this MWS into account: a potential state, which corresponds to what the teacher foresees, and an actual or effective state, which shows the work carried out in the classroom (Chap. 6, Kuzniak, Henriquez-Rivas & Masselin). The theory of MWS provides an account of what exists, and there will be cases where analysis of these different work forms will reveal deficiencies or wrong and poorly constructed work forms. This may seem natural in the case of personal mathematical work, but it is also sometimes true for suitable work and exceptionally for reference work.

5.2 MWS Associated with Specific Mathematical Domains The increasing number of studies applying MWS theory to mathematical domains other than geometry has shown the need to introduce a particular MWS associated to the new domain in question. The issue of interactions between different mathematical domains or fields is central to understanding how mathematical work arises and evolves. Indeed, frequently, the paths to solving a given mathematical task require transitioning from the initial mathematical field to another one. For instance, resolving a problem stated exclusively in geometrical terms will often prompt a shift to the algebraic or arithmetic field when magnitude and measurements are at play. To clarify the relationships and boundaries between mathematical domains (or fields), Montoya and Vivier (2014) propose that a field should be organized into a consistent structure within mathematics, this consistency being assessed from an epistemological as much as a mathematical viewpoint. The best proof that a given segment of mathematics should be called a field or a domain is that it is regarded as such by the community of mathematicians. In this way, we may speak of a MWSGeometry , a MWSAlgebra , a MWSAnalysis , a MWSProbability , etc. However, a (new) mathematical field is often encountered through dealing with its specific objects, among which some may already be familiar. For instance, in analysis, students are familiar with functions well before encountering series, or the notions of “limit” or “derivative”. Indeed, schooling institutions favor working organizations that are structured inside partial MWS, dedicated to specific objects more than to global fields: MWSFunctions , MWSSequences , etc. It is also worth reflecting upon the way new transversal themes—such as Algorithms Science or Modeling—may be integrated into MWS, and what kind of reorganization of the framework is necessary to foster this integration. One of the issues pertaining to the study of MWS is precisely that of describing and analyzing these partial (or “local”) MWS, but also the articulation and coordination between them, needed to achieve comprehensive and consistent mathematical work. Another important issue relates to understanding the reasons underpinning an efficient change of domains to successfully perform a task and see how this depends on mathematical signs or artifacts or properties.

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6 Describing and Forming Mathematical Work As pointed out in 2.2, understanding observed mathematical work requires a description based on theoretical and methodological constructs which enable us to go beyond the simple observation of phenomena. In this section, we specify some of the observables and theoretical tools used to describe and shape mathematical work in a school context. In Chap. 2, Nechache and Gómez-Chacón further develop the methodological principles associated with the theory.

6.1 Mathematical Tasks and Didactic Situations Section 3.3 defined what is meant by mathematical tasks and underlined their importance in relation to the notion of activity, which accounts for what the individual does when faced with a task. Tasks are not part of a MWS, but they participate in its activation when a subject is confronted with them and must perform them. In the theory of MWS, proposal of a task in a classroom involves a situation organized by the teacher. It is this implementation that we will call a didactic situation. This definition is close to that of Brousseau (2002) who specifies that didactic situations must include a teaching intention. Ensuring that the tasks and didactic situations chosen are the right vehicles for effective mathematical work is an important aspect of the theory of MWS. Individuals who seek to perform a task produce actions with impulses as well as goals to be reached. It is thus possible to speak of a dynamic of work in the MWS and observe circulation between geneses which the diagrams attempt to explain. The first researcher to successfully capture this flow through MWS diagrams was Lebot (2011) in a study on learning the concept of “angle” in middle school. Subsequently, all the fine-grained studies of mathematical work, especially Ph.D. level, have sought to represent this circulation in the way most appropriate to their research issue. In the theory of MWS, task-achievement analysis helps identify the purpose, processes and results of work (3.1). A task is determined by its goals and the conditions necessary for its performance, which can be individual or collective. Its performance is based on a set of actions that are coordinated and organized to reach this goal. In general, it will be possible to identify in this succession of actions the intermediate goals that the individual has set for themself in order to plan the resolution of the problem presented and thus break down the initial task into a succession of sub-tasks. When observing the activity of an individual carrying out the task prescribed to them, it will thus be relevant to identify different phases which we will call episodes. More precisely, an episode is the data of a sub-task and of the sequence of actions associated with the performance of the sub-task. As will be seen in Chap. 2 on methodology (Nechache & Gómez-Chacón), decomposition of a task into episodes

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also depends on the grain size of the analysis chosen, as well as on the students’ school level.

6.2 Toward a Study of Mathematical Work in Its Globality: Programming and Emblematic Tasks The question of the relations between mathematical domains and of developing a coherent progression can in a certain way be explored through a reflection of a purely mathematical epistemological nature on the ordering of the different definitions, properties and theorems that constitute the mathematical edifice. This approach has always been favored by mathematicians with the elaboration of treatises sometimes called elements as with Euclid in antiquity or with Bourbaki in the middle of the structuralist period, sometimes cours once educational institutions appeared. In France, it was the military and engineering schools, and one of the most famous courses organizing mathematics in a teaching perspective is, for example, that of Bezout. In didactics of mathematics, this vision has been taken up and structured by Chevallard (1992) with the notion of mathematical organization associated with an epistemological model of reference. However, this approach based on content alone does not take into account the cognitive complexity and the arrangement of tasks according to their difficulty and their contribution to the constitution of mathematical work. Identification of global mathematical work is necessary, but, as is often the case in didactics, it is based on particular studies of specific and sometimes isolated tasks. This question does not only concern the theory of MWS. Indeed, any theoretical approach in didactics is confronted with this tension between the results of partial studies and their interpretation in global terms. Fortunately, studies on teaching practices show a rapid stabilization of these practices and suggest a standardization of the mathematical work implemented as a result. It was to identify the constitutive invariants of this standard mathematical work that the notion of emblematic task was introduced by Kuzniak and Nechache (2016). Emblematic tasks are specific tasks which must meet a number of conditions that can only be validated through analysis and experimentation. In particular, they must: 1.

2.

3.

Be available in the reference MWS, in other words, benefit from a recognition that shows their adequacy for the mathematical work targeted by the school institution for students; Be active in the suitable MWS, i.e., be part of the tasks that are actually proposed first in the potential suitable MWS defined by the textbooks, but also—and especially—in regular classes and Be potentially conducive to a complete mathematical work in accordance with the requirements of the dominant paradigm in the institution. The emblematic tasks must enable a complete circulation between the components and the planes by relying on work that is consistent at the process level and correct at the outcome level.

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By relying on a well-articulated choice of such emblematic tasks, it is possible to represent the mathematical work developed globally in the classrooms by the students with their teachers from a set of more local observations.

6.3 Tools and Instruments in Mathematical Work 6.3.1

Semiotic, Technological and Theoretical Tools and Instruments

Problem solving in mathematics involves the use of different mathematical objects that can be considered as tools to perform the task. We will start by elaborating in the theory of MWS the meaning of the terms tools and instruments that are often used in mathematics education. According to the usage by prehistorians or archeologists, the artifact (4.2.1) is an object transformed by humans following a certain objective. In principle, it is a neutral term, the use of which remains to be specified, particularly in a technological perspective. The notion of tool differs from that of artifact because it rather specifies the undertaking of a productive activity. As such, a cake is an artifact but not a tool. The notion of instrument has acquired a particular meaning in didactics of mathematics through the importation of certain concepts of ergonomics into educational studies. In this context, an instrument is the result of a construction of schemas by the individual performing the activity, and it thus becomes the fruit of a cognitive genesis. According to this conception, an instrument is thus made up of two components: an artifact, material or symbolic and the associated schemas of use (Chaps. 8 and 9). In the theory of MWS, and whenever it is useful, the term tool is reserved for objects of the epistemological plane that have a specific potential use in the context of problem solving. Use of the word instrument is prioritized as soon as a subject (pupil, student or teacher) has developed their use of a tool to effectively solve the proposed task. We will not integrate the idea of schemas of use except through observation of the implementation of certain techniques for using tools. This is aligned with the spirit of the theory, which favors tangible observables and avoids addressing questions related to the subject’s mental functioning (8.1). In relation to the different components of the epistemological plane and the different geneses of mathematical work, it is possible to consider three types of tools and instruments. Semiotic tools are non-material tools—like diagrams, symbols or figures—which may be used as instruments for operating on semiotic representations of mathematical objects. Technological tools are artifacts—like drawing tools or routinized techniques— which are “at hand”, based on algorithms or calculators with implemented calculation or drawing algorithms used as instruments which may be used for constructing mathematical objects such as figures, graphs and numbers.

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Theoretical tools are elements of the referential non-material tools like theorems which may be used as instruments for mathematical reasoning based on the logical and mathematical properties of mathematical objects.

6.3.2

An Example of the Flexibility of a Mathematical Tool: Probabilistic Trees

The study of probabilistic trees shows the great flexibility of the tools introduced in mathematics and the necessity for a precise consideration of their role to ensure appropriate use in their domain of application. Trees were introduced in French secondary school teaching as tools for solving probabilistic problems in the early 1990s. At that time, they were seen only as semiotic tools, the use of which was not the object of specific learning. With the contribution of research in mathematics education, probabilistic trees have been progressively extended from use as a semiotic tool essentially used for description, to a technological tool with calculation techniques justified within the developed theoretical referential. For instance, Dupuis and Rousset-Bert (1996) introduced “weighted trees with margin” to calculate results on conditional probability (Fig. 4). Thus, the probabilistic tree has become a technological tool with the rules of calculation justified by the properties of conditional probabilities. Its theoretical role as a legitimate theoretical tool for proof remained to be officially recognized. This recognition was given in the French curriculum in 2010, wherein it was indicated that probabilistic trees have a role of proof when their construction is well established: A well-constructed weighted tree constitutes a proof. This example shows how recognition of the possible roles of a tool depends on the school institution which, through its choices, favors certain modes of use of the tools, sometimes in opposition to older or more academic modes of proof. Thus, in the case of trees, we can note a strong reluctance among some teachers to use them as a tool of proof because this does not correspond to their conception of proof in probability Fig. 4 Weighted tree

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acquired at university. These variations and their interpretational difficulties are taken into account in the theory of MWS with the notion of paradigms.

7 Characterizing Mathematical Work To summarize the various constraints on mathematical work developed in the two previous sections, Kuzniak and Nechache (2021) proposed a characterization of this work based on analysis of the processes and results which emerge during task and problem resolution. These two authors thus consider that work is compliant (7.1) when the processes and procedures used are valid and conform to the expectations present in the work paradigm or paradigms favored in the reference MWS. The work is said to be correct when the results obtained are exact according to the mathematical point of view retained (7.1). Finally, the mathematical work is complete when its circulation is ensured between all dimensions and components of the MWS diagram (7.2). By using these different criteria (conformity, correctness and completeness), it is possible to specify the types of controls implemented in the observed mathematical work (7.3) and then to determine certain forms of work favored by the students (7.4).

7.1 Conformity and Correctness of Work The first research associating the notion of paradigms with working spaces appeared in geometry education (Houdement & Kuzniak, 1999). The basic idea is summarized by the authors as follows: In education, three distinct paradigms structure the field of geometry. These paradigms reflect different stages in the succession of academic cycles. Each stage is characterized by specific practices and challenges to the teaching and learning of the discipline.

Three paradigms, referred to as Geometry I, II and III, are introduced. Two of these—Geometry I and II—play an important role in current secondary education in France. Each paradigm is sufficiently comprehensive and coherent to define and structure geometry as a mathematical domain and to set up appropriate working spaces to solve a wide range of problems. Subsequently, this use of paradigms to understand some of the differences between educational institutions has been developed in several other mathematical domains, such as analysis, probability, algebra and kinematics. (Chap. 4, Montoya-Delgadillo & Reyes-Avendaño). The reference paradigm must be consistent and well-constructed, which means that the procedures implemented and the results produced within the paradigm are mathematically correct. One question which may arise is that of the mathematics serving as the epistemological support for this paradigm and which may itself evolve over time and depend

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on its field of application. This question falls under the epistemology of mathematics and must be distinguished from the question of the paradigms actually implemented by teachers in the suitable MWS or developed by students in their personal work. Studies on the reality of these school paradigms have demonstrated their complexity. These paradigms are often not homogeneous and may refer to different academic paradigms. They may also be invalid from an epistemological point of view. One of the objects of current studies on the actual mathematical work undertaken by teachers and students is precisely to ensure, on the one hand, the conformity of the work processes to a reference paradigm and on the other hand, the correctness of the products of this work. Here, we find that the emphasis in the theory of MWS is on the processes and outcomes of mathematical work. The processes must be valid in the sense that they conform to the rules of operation of the preferred paradigm in teaching. The results produced must be mathematically exact and correct in accordance with the chosen referential in the epistemological plane.

7.2 Circulation of Work and Complete Mathematical Work Use of MWS and the associated diagram enables an analysis of the activity generated by a task or a series of tasks. Verification of the efficiency of tasks as supports of didactic situations and vectors of an adequate mathematical work is an important object of study in the framework of the theory of MWS. Individuals engaged in a task produce actions driven by their mathematical knowledge and by their understanding of the goals to be reached. It is in the triggering and unfolding of these actions that the circulation of work or dynamic of the MWS lies. The MWS diagrams enable us to visualize this circulation by relying on a detailed analysis of task performance (6.1). These analyses highlight certain blockages or confinements in one dimension or plane of the MWS diagram. They also reveal cases where more varied interactions exist between all the planes of the MWS. In this case, it is therefore possible to speak of a complete mathematical work. Using the distinction between tools and instruments (6.3), mathematical work is thus complete when the following conditions are met: A real relationship between the epistemological and cognitive planes. This means that students are able to choose the relevant tools to solve a problem and then use them appropriately as instruments to solve the given task (Kuzniak et al., 2016, p. 862).

The articulation between the different geneses and the vertical planes appears in the MWS diagram (Fig. 2). This time it is a question of verifying that all the different geneses of work related to signs, artifacts and properties are taken into account. Completeness must be understood in the context of a global progression of teaching around a mathematical theme. No task should be evaluated in isolation from the whole in which it fits. Indeed, some tasks necessary for the assimilation of

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routinized techniques may be confined to a single plane or even a single genesis of the diagram. They are nonetheless necessary for the development of mathematical work. In this approach, analysis of a task or a set of tasks using MWS helps us to understand the work and to plan it in the short and medium term. The theory of MWS thus enables analysis of the mathematical work developed during a teaching session or a series of sessions by specifying its different stages. These phases are represented by a set of comic strip-like diagrams that bring together a series of diagrams on the mathematical work developed during the course (Chap. 2, Nechache & GómezChacón).

7.3 Controls on Mathematical Work The notion of control is essential in for subjects in regulating their work. Controls were introduced in mathematics education by Schoenfeld (1985) in relation to an approach to teaching based on problem solving and students’ initiative. In this framework, it is essential to ensure the vigilance of each student on the validity of their results in relation to the work processes and this independently of the teacher’s gaze, at least initially. To this end, Balacheff (2013) developed the idea of a control system based on semiotic tools, individual actions and classes of problems to be solved. In the same vein, Arzarello and Sabena (2011) introduced the notion of control to account for the fact that students make certain decisions regarding the selection and use of resources. They argue that control can be semiotic or theoretical depending on the nature of the resources they select and implement. Within the theory of MWS, the notion of control is introduced to account for the way students manage their choice and use of mathematical tools and instruments (Kuzniak & Nechache, 2021). Thus, semiotic, technological or theoretical controls are considered depending on the type of tools or instruments used (6.3.1). These controls concern the validity of the statements and the appropriateness of the various tools and instruments selected. In this way, it is possible to ensure the conformity and correctness of the work carried out. The means of control have evolved recently with the increasing importance of digital artifacts and contribute toward changing the mathematical work developed in schools (Chap. 8, Flores Salazar, Gaona & Richard).

7.4 Identifying Forms of Mathematical Work Studies drawing on the theory of MWS attempt to identify certain invariants in the ways of working implemented by individuals confronted with mathematical tasks. Observation of these ways of working then helps the development of teaching projects

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which seek to prioritize certain forms of work or, on the contrary, reduce the impact of certain others. To determine which forms of work actually exist, it is possible to use the various tools presented in this chapter. In the context of geometry teaching, Kuzniak and Nechache (2021) identified five forms of work. Each is characterized according to the procedures and results produced, assessed according to the following three criteria: conformity of the work carried out with a given geometric paradigm; the degree of completeness and the correctness of the results obtained. The identification of generic forms of work in different mathematical domains enables us to assess the distance between the teachers’ expectations and the actual forms of work implemented by the students. Thus, their identification is an important lever to enable the construction or transformation of mathematical work in a school context.

8 Conclusion In this conclusion, we discuss some specificities and limits of the theory of MWS, highlighting its originality in the field of mathematics education6 . We also consider how to develop and master its uses, given its complexity.

8.1 Epistemological Anchoring and Observation of the subject’s Actions As Radford (2017) points out, the theory of MWS is a didactic theory which maintains a strong link with mathematical contents and is thus distinguished, according to him, from mainstream approaches to mathematics education. It does this by closely combining epistemological and cognitive aspects around the unifying notion of mathematical work. Elaboration of mathematical work is seen as the harmonious networking of three types of genesis: semiotic, instrumental and discursive. This synthetic and triadic vision of mathematical work gives it coherence and facilitates its analysis. According to this conception of mathematical work, it is the arrangement and relationships between the different epistemological components and cognitive processes which ensure the coherent and complete development of mathematical work. Studying the mathematical content involved in learning is important to make sense of these geneses and more generally to identify and describe the paradigms guiding the work. In order to access the cognitive processes, the subjects’ actions are observed and analyzed. Their actions are the visible manifestation of the subjects’ cognitive work when confronted with tasks and problem solving. Observation and 6

This point is discussed in further detail in Chap. 7 (Kuzniak, Montoya-Delgadillo & Vivier).

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analysis of mathematical actions provide access to real mathematical work. It is then possible to see how, if necessary, it can be transformed or adapted to better fit mathematical, social or educational expectations. In short, the theory has been developed for the in-depth study of mathematical work in a school context, and its vocation is to describe it, to understand it, but also to form or transform it.

8.2 Points not Addressed by the Theory The theory of MWS relies on the visible effects of a subject’s actions through explicit observables to understand the cognitive aspects. In this sense, the theory adopts a materialist approach to reality. It has neither the ambition nor the means to explore certain problems which are more related to the psychology and mental functioning of the subject. Thus, for example, it does not address: • The impact on the development of a subject’s mental apparatus through information and feedback induced by the environment. • The nature of the black box in precise reference to the psychological constitution of the geneses. • The existence and possible place in the brain of all kinds of schemes that can intervene in the mathematical work. Moreover, the whole activity of teaching and learning mathematics is not limited to the mathematical work. It is important to consider the elements which are external to this work, but which participate in its undertaking in practice: affects and emotions, collaborative and collective work, language exchanges, cultural and social context and so on. These are complementary and indispensable elements which are not directly related to the core of the theory, but which can contribute to fruitful exchanges with specialists in these matters, without distorting the specificity and nature of each field of research. These exchanges and interactions are part of the theoretical and methodological innovation that has characterized MWS research to date. This partly explains the open and integrative plasticity which, according to Artigue (2016) and Bikner-Ahsbahs (2017), is one of its attractions. But, while the theory of MWS remains relatively open and evolving, it is essential to consider its distinctive epistemological orientations and choices among other theories in mathematics education research. These biases also provide insight into the possibilities for cooperation and potential complementarity with theories which share a common epistemological sensibility (Kidron, 2016).

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8.3 On the Use of Theoretical and Methodological Tools of the Theory of MWS In this chapter, we have given an introductory overview of the theory of MWS, focusing on theoretical constructs. These are introduced to provide an understanding of the mathematical work both on a global level (variety of MWS and interactions between these various spaces, paradigms, etc.) and on a more local level concerning precise analyses of tasks and didactic situations (tools, instruments, work circulation with confinements and blockages, etc.). This diversity and variety of constructs show the current richness of the theory, but, at the same time, this increasingly vast theoretical corpus makes its access and use more complex. Chapter 2 on methodology and the chapters on the different types of MWS (Part II) describe and illustrate in more detail how these different tools are used, with the aim of both facilitating their understanding and familiarizing the reader with their most appropriate application.

References Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematics Learning, 7, 245–274. Artigue, M. (2016). Mathematical working spaces through networking lens. ZDM Mathematics Education, 48(6), 935–939. https://doi.org/10.1007/s11858-016-0810-z Arzarello, F., & Sabena, C. (2011). Semiotic and theoretic control in argumentation and proof activities. Educational Studies in Mathematics, 77(2/3), 189–206. https://doi.org/10.1007/s10 649-010-9280-3 Balacheff, N. (2013). cK¢, a model to reason on learners’ conceptions. In M. V. Martinez & A. Castro (Eds.), Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (PME–NA) (pp. 2–15). PME. Bikner-Ahsbahs, A. (2017). Introduction to the papers of TWG17: Theoretical perspectives and approaches in mathematics education research. In T. Dooley, & G. Gueudet (Eds.), Proceedings of the 10th CERME (pp 2683–2691). Dublin, Ireland. Brousseau, G. (2002). Theory of didactical situations in mathematics. Dordrecht: Springer. Chevallard, Y. (1992). Concepts fondamentaux de la didactique : Perspectives apportées par une approche anthropologique. Recherches En Didactique Des Mathématiques, 12(1), 73–112. Coutat, S., & Richard, P. R. (2011). Les figures dynamiques dans un espace de travail mathématique pour l’apprentissage des propriétés mathématiques. Annales De Didactique Et De Sciences Cognitives, 16, 97–126. Dupuis, C., & Rousset-Bert, S. (1996). Arbres et tableaux de probabilités: Analyse en terme de registre de représentation. Repères-Irem, 22, 51–72. Duval, R. (2005). Les conditions cognitives de l’apprentissage de la géométrie: Développement de la visualisation, différenciation des raisonnements et coordination de leur fonctionnements. Annales De Didactique Et De Sciences Cognitives, 10, 5–53. Freudenthal, H. (1971). Geometry between the devil and the deep sea. Educational Studies in Mathematics, 3, 413–435. Fénichel, M., & Taveau, C. (2009). Enseigner les mathématiques au cycle 3. Le cercle sans tourner en rond. DVD, CRDP Créteil.

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Gómez-Chacón, I., Kuzniak, A., & Vivier, L. (2016). El rol del profesor desde la perspectiva de los Espacios de Trabajo Matemático. Boletim de Educação Matemática – Bolema, 30(54), 1–22. https://doi.org/10.1590/1980-4415v30n54a01. Guzman, I., & Kuzniak, A. (2006). Paradigmes géométriques et géométrie enseignée au Chili et en France. Irem Paris-Diderot. Granger, G.G. (1963). Essai d’une philosophie du style. Paris: Armand Colin, rééd. Paris: Odile Jacob 1987. Houdement, C., & Kuzniak, A. (1999). Un exemple de cadre conceptuel pour l’étude de l’enseignement de la géométrie en formation des maîtres. Educational Studies in Mathematics, 40(3), 283–312. https://doi.org/10.1023/A:1003851228212 Kidron, I. (2016). Epistemology and networking theories. Educational Studies in Mathematics, 91(2), 149–163. https://doi.org/10.1007/s10649-015-9666-3 Kuhn, T. S. (1966). The structure of scientific revolutions (2nd ed.). Chicago: University of Chicago Press. Kuzniak, A. (2019). La théorie des Espaces de Travail Mathématique – Développement et perspectives. In L. Vivier, & E. Montoya-Delgadillo (Eds.), Sexto Simposio sobre el Trabajo Matemático (pp. 21–60). Valparaíso: Pontificia Universidad Católica de Valparaíso. Kuzniak, A., Tanguay, D., & Elia, I. (2016). Mathematical Working Spaces in schooling: An introduction. ZDM Mathematics Education, 48(6), 721–737. https://doi.org/10.1007/s11858-0160812-x Kuzniak, A., Nechache, A., & Drouhard, J.-P. (2016). Understanding the development of mathematical work in the context of the classroom. ZDM Mathematics Education, 48(6), 861–874. https://doi.org/10.1007/s11858-016-0773-0 Kuzniak, A., & Nechache, A. (2016). Tâches emblématiques dans l’étude des ETM idoines et personnels: Existence et usages. Quinto Simposio Espacio de Trabajo Matemático - ETM5. Florina, Grecia. Kuzniak, A., & Nechache, A. (2021). Personal geometrical work of pre-service teachers: A case study based on the theory of Mathematical Working Spaces. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-020-10011-2 Kuzniak, A., & Richard, P. R. (2014). Mathematical working spaces. Viewpoints and perspectives. Relime, Revista Latinoamerica de investigacion en matematica educative, 17(4), 5–40. https:// doi.org/10.12802/relime.13.1741a. Lakatos, I. (1976). Proofs and refutations. Cambridge: Cambridge University Press. Lebot, D. (2011). Mettre en place le concept d’angle et de sa grandeur à partir de situations ancrées dans l’espace vécu: Quelles influences sur les ETG? Master de didactique des mathématiques. Irem, Université Paris-Diderot. Montoya, E., & Vivier, L. (2014). Les changements de domaine dans le cadre des Espaces de Travail Mathématique. Annales De Didactique Et De Sciences Cognitives, 19, 73–101. Nechache, A. (2017). La catégorisation des tâches et du travailleur-sujet : Un outil méthodologique pour l’étude du travail mathématique dans le domaine des probabilités. Annales deDidactique Et De Sciences Cognitives, 19, 67–90. Peirce, C. S. (1931). Collected Papers, vols. 1–6. Cambridge: Harvard University Press. Collected papers. Pizarro, A. (2018). El trabajo geométrico en clases de séptimo básico en Chile: Un estudio de casos sobre la enseñanza de los triángulos. Thèse de l’Université de Paris. Paris: Université de Paris. Radford, L. (2017). On inferentialism. Mathematics Education Research Journal, 29(4), 493–508. https://doi.org/10.1007/s13394-017-0225-3 Richard, P. R., Venant, F., & Gagnon, M. (2019). Issues and challenges about instrumental proof. In G. Hanna, D. Reid, & M. de Villiers (Eds.), Proof technology in mathematics research and teaching. Cham: Springer International Publisher. Schoenfeld, A. (1985). Mathematical problem solving. New York: Academic Press. Sierpinska, A. (2004). Research in mathematics education through a keyhole: Task problematization. For the Learning of Mathematics, 24(2), 7–15.

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Thurston, W. (1994). On proof and progress in mathematics. Bulletin of the American Mathematical Society, 30(2), 161–177. Vandebrouck, F. (Ed.). (2013). Mathematics classrooms students’ activities and teachers’ practices. Rotterdam: Sense Publishers. Vivier, L. (2020). Portée et usage du travail mathématique dans le cadre de la théorie des ETM. In M. Flores, A. Kuzniak, A. Nechache, & L. Vivier (Eds.), Regards croisés sur le travail mathématique en contexte éducatif. Cahiers du LDAR 21 (pp. 55–70). Irem, Université de Paris.

Methodological Aspects in the Theory of Mathematical Working Spaces Assia Nechache and Inés M. Gómez-Chacón

1 Introduction The evolution of the theory of Mathematical Working Spaces and its increasing use among researchers has led to a reflection on the methodological principles of research conducted within the framework of this theory. When a theory is used to formulate and answer research questions, it is important to ensure the robustness of the particular methodology associated with that theory. In the present case, the methodology implemented must be consistent with the principles of the theory of MWS. It must also be operational; in other words, according to Radford (2008), it must be able to produce and process data to provide satisfactory answers to the research questions asked. The satisfactory answers may be based, for example, on methods not specific to the theory used. In this chapter, we will not discuss all the methods that are satisfactory or valid—statistical, case studies, etc. The main intention is to present the methodological principles and tools that relate directly to the theory of MWS. The theory of MWS is deeply rooted in mathematics education, and it has developed while remaining connected to teachers’ practices and the development of learning in real classrooms. The primary focus of the theory is the specific study of the mathematical work in which students and teachers are actually engaged. The theory of MWS provides methodological tools for advancing in the study of three main issues addressing mathematical work: its description, characterization and formation and/or transformation (1.2.2). By addressing these three issues, the A. Nechache (B) LDAR, CY Cergy Paris Université, Cergy, France e-mail: [email protected] I. M. Gómez-Chacón Interdisciplinary Mathematics Institute (IMI), Faculty of Mathematical Science, Complutense University of Madrid, Madrid, Spain © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Kuzniak et al. (eds.), Mathematical Work in Educational Context, Mathematics Education in the Digital Era 18, https://doi.org/10.1007/978-3-030-90850-8_2

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goal is to provide an account of what is really taught and learned when dealing with mathematical work in an educational context. The study of this work depends strongly on the position of the working subject (designer or performer of the work) in the school institution. Thus, in the case of research on the reference mathematical work, the designers of this work can be those who elaborate the programs, while the worker-subjects are often epistemic subjects. Depending on the type of work—reference, personal or suitable (1.5)—considered in the research, different types of MWS—reference, personal or suitable—will then be considered as the main methodological tool for conducting the analysis of the work. Numerous studies which have used the theory of MWS to formulate and answer questions related to the teaching and learning of mathematics have developed original and relevant methodological tools based on particular studies of specific tasks and didactic situations. The aim of this chapter is to outline this rich diversity and help researchers to develop their future investigations within the theory of MWS. The first three Sects. (2–4) outline the methodological tools and how they have been used in research to describe, characterize or (trans)form mathematical work. The last Sect. (5) presents the study of potential and actual mathematical work through the lens of MWS theory in relation to a priori and a posteriori methods of analysis.

2 Describing Mathematical Work The description of mathematical work aims to capture the work carried out in a specific educational context and has a twofold objective. These are: – to identify the role of each of the three MWS geneses in the mathematical work and the circulation of that work within a mathematical domain or domains; and – to identify possible blockages and misunderstandings that may arise during the process of the work and explain the origin of these blockages and misunderstandings. The description is based on the constructs of MWS theory and has an explanatory purpose. In the following sections, examples of the methodological tools used by researchers to describe mathematical work within one or more mathematical domains are presented.

2.1 Describing the Development of Mathematical Work Within a Mathematical Domain In their research on the teaching of geometry in compulsory school curricula, Kuzniak and Nechache (2015) used the theory of MWS to describe the different components and processes of the work as they emerged during a teaching sequence (consisting

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of five sessions) on the notion of the circle (Fénichel & Taveau, 2009) in Grade 4–6 lessons. In the sequence analyzed, from the perspective of the teacher’s intentions, the mathematical work developed mainly focuses on the notion of the circle seen as a set of points equidistant from a given point, its center. More specifically, the main objectives are to define the circle as the set of points equidistant from a given point; to use this property to solve distance problems; to link definition and property to the construction of a circle using the compass, and also to transfer lengths. For each of the five sessions, an analysis drawing on the theory of MWS identified the different geneses and vertical planes mobilized in the work. Below we detail the analysis carried out for session 1. The objective of this first session is to present the notion of a circle as being a set of all points equidistant from a given point, the center. A variety of materials and artifacts are made available to the students: white paper and tracing paper, a string, square and compass. Students are asked to draw 15 points at a given distance from a point A. To do this, they must first place a point A on the white paper, then a point B. Then, they must place 15 points “located at a distance from A which is the same as the distance of B from A.” Thus, the instrumental and semiotic geneses are mobilized here to trace the 15 points and make a circle emerge. Then, certain students’ productions are displayed on the board and discussed. The strategies they used to carry out the task are clarified and formulated. The objective is first to validate the notion of a circle and to bring out the notion of equidistance in relation to a given point. This session ends with the institutionalization, by the teacher, of the characteristic property of the circle, which then enables enrichment of the theoretical reference. The geometric work therefore begins in the [Sem-Ins] plane and ends with the statement of the characterization of the circle as a set of points equidistant from a given point. This new property enriches the theoretical referential of the MWS. The latter consists of the properties and definitions of the different figures used at this level of school education. By analyzing the following sessions in the same way, it was possible to describe the development of geometric work over the five sessions. This description, based on an identification of the geneses mobilized in the work, highlights the dynamics of geometric work over the course of the different sessions. In their study, Kuzniak and Nechache (2015) introduce the use of a comic strip-like set of diagrams based on MWS diagrams (Fig. 1) to give a global and quick vision of the development of mathematical work throughout a teaching sequence: The analysis of the work, based on this set of diagrams (Fig. 1), also named comics, shows that the three geneses of the MWS and the three vertical planes are well-mixed throughout the five teaching sessions. This leads to a circulation of mathematical work through all the components of the MWS. Consequently, the mathematical work is then qualified as complete (1.7.2). In this first example, the theory of MWS is used to describe the mathematical work planned in a class session or set of sessions. The circulation of work produced in each session is visualized by using MWS diagrams. The set of diagrams thus obtained enable us to see how the mathematical work expected during this sequence is developed.

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Session 1

Session 2

Session 3

Session 4

Session 5

Fig. 1 Dynamic evolution of mathematical work over the five sessions (Kuzniak & Nechache, 2015)

This method of analyzing a sequence was presented to a group of researchers and teacher trainers who then used it to carry out the analyses of a sequence. This enabled them to verify the robustness of this method of analysis according to the convergence of the analyses toward the comic-like strip above.

2.2 Describing the Circulation of Work Around a Concept Within a Single Mathematical Domain To describe the geometric work around the concept of the triangle in an actual suitable MWS taught in the 7th grade in Chile, Pizarro (2018) proposes a codebook considering variables that are all (except one) related to the components of the MWS. These variables relate to: – The elements of the theoretical referential of the suitable MWS, in particular those concerning the geometric content (properties, theorems, definitions, etc.) involved in performing the prescribed tasks. – The nature of the discourse used during the implementation of the suitable MWS in the classroom (actual or effective MWS). The discourse is either the explication of a definition or property (element of the theoretical referential) or an explanation of the construction technique associated with an artifact. – The process of constructing geometric figures (elements of the cognitive plane) implemented during the execution of tasks: freehand construction, construction using geometric drawing tools (graduated ruler, compass, square). – The visualization process (element of the cognitive plane) of the geometric figures evoked: decomposing and recomposing the figures (representamen). – The artifacts associated with the processes involved in constructing the figure, such as geometric construction tools, dynamic geometry software, and other tools used in everyday life (string, wool, sticks, etc.). – The resources (textbooks, video projectors, etc.) used as support to help the teacher implement the suitable MWS.

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In summary, the first five variables outlined above each relate to one of the components of the MWS and are used to observe the role of each of these components of the suitable MWS in the development of the mathematical work, while the last variable, relating to resources, is used to observe what supports implementation of the suitable MWS. To identify each of these variables when implementing the suitable MWS in classrooms, Pizarro (2018) designed what she called a frequency table. This table indicates the frequency of occurrence of each of the variables. To make it operational, the filmed session is divided into a series of 15-s (temporal) episodes. Each episode represents a moment when the teacher gives an instruction, modifies the instruction for clarity, increases the degree of difficulty, offers help or formulates an institutionalization. Use of the frequency table combined with the codebook enabled Pizarro to identify the variables, as well as precisely when and in what order the variables appear. The variables identified are then counted to determine the frequency of their appearance during a session. Identifying the frequency of appearance of these variables highlights the different components of the suitable MWS used in carrying out the geometric tasks, and consequently the geneses activated or even prioritized in the development of the geometric work. This overview of the relationships between the different components of the suitable MWS demonstrated within an episode of a session, as well as the circulation of geometric work produced in an episode, is presented using the MWS diagram. For example, Fig. 2 shows that the semiotic genesis and only the theoretical referential were mobilized in one episode. We therefore see a circulation in the semiotic-referential semi-plane and not in the semiotic-discursive plane, because the process of proof as associated with the theoretical referential was not activated in this episode. Each of the diagrams enables Pizzaro to obtain an overview of the circulation of work in each of the episodes constituting a session of the class recorded on film (the actual suitable MWS). The set of diagrams obtained provides an illustration of the global circulation of the geometric work in the observed suitable MWS. For example, in Fig. 3, we see that the class session filmed was divided into six episodes. In each episode, the circulation of the work is identical: It is in the semiotic–instrumental plane and the epistemological plane. Fig. 2 Example of work circulation in the semi-plane [Sem - ref] (Pizarro, 2018, p. 140)

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Fig. 3 Global circulation flow of mathematical work in the actual suitable MWS (Pizarro, 2018, p. 164)

Furthermore, the method used to describe geometric work enables the nature of the discourse used in the actual suitable MWS to be identified. Each of these discourses— semiotic discourse, technological discourse and theoretical discourse—is associated with one of the three semiotic, instrumental and discursive geneses of the MWS (Pizarro, 2018). These discourses are expressed in writing or orally, fall in the cognitive plane and interact with the elements (of the epistemological plane) of the genesis with which they are associated.

2.3 Describing the Circulation of Work Between Different Mathematical Domains Derouet’s (2017) work on the teaching of continued probability distributions in 12th grade high school classes aims to describe the interactions and circulations between mathematical domains and subdomains. Based on the theory of MWS, Derouet analyzes the mathematical work carried out in a class session performing a task. The “Aso volcano” task prescribed to the students uses data on the eruptions of the Aso volcano in Japan. It was divided into five subtasks highlighting the subdomains (density probability, integral calculus, descriptive statistics) used in each of these subtasks. Analysis of this work is performed through the actual suitable MWS and is based on the following observables: – The geneses of each MWS subdomain involved in the five subtasks mobilized during their completion; and – The role of the students and the teacher in completing each of the five subtasks. To perform this analysis of the work, a two-color coding system is designed: red for what is the students’ responsibility and green for the teacher. For example, in Fig. 4, the double red arrow between two red oval shapes indicates that the semiotic geneses of the descriptive statistics (SD) and integral calculus (CI) subdomains were activated and mobilized by the students. This coding also indicates a shift in the subdomains, supported by the students, which occurred between the descriptive statistics and integral calculus subdomains. The dotted oval shape means that the discursive genesis of the density probability (PaD) subdomain of the MWS is mobilized by the students, but this occurs following the teacher’s intervention to restart the work between the

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Fig. 4 Example of the roles of students and the teacher in completing a subtask (Derouet, 2017, p. 72)

two semiotic geneses of the SD and CI MWS. As for the oval shape in green, this means that the instrumental genesis of the CI MWS is mobilized and taken over by the teacher. The use of coding combined with the MWS diagram enabled Derouet to highlight the various transitions between different MWS associated with each of the three subdomains (PaD, SD and CI) during the completion of subtasks within the "Aso volcano" task. These transitions are identified through the activation and mobilization (by students or teacher) of the geneses of the MWS related to each of the subdomains. For example, in Fig. 4, we can see that transitions occur between the MWS associated with the SD subdomain and the MWS associated with the CI subdomain through students’ activation and mobilization of the semiotic geneses of each of these two MWS. These transitions observed illustrate the changes in subdomains which took place during completion of the subtasks. All of these subdomain changes are illustrated using the diagram of the MWS as in Fig. 5, where the two-way red arrow means that subdomain changes occurred between SD and CI. These changes were brought about by the students mobilizing the semiotic genesis associated with the SD and CI MWS. We also note that the mathematical work developed is essentially carried out in the semiotic genesis between SD and CI (and undertaken by the students). Nevertheless, this work is relaunched each time by the teacher who activates the discursive genesis of the PaD MWS.

2.4 Describing the Subject’s Personal Work To provide an account of the mathematical work actually produced by student teachers, Kuzniak and Nechache (2021) designed an original method of analysis inspired by cognitive task analysis (CTA) methods (Darses, 2001). These methods

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Fig. 5 Changes of subdomains during the execution of a subtask (Derouet, 2017, p. 72)

are used in the psychology of work and have been developed in particular to study the cognitive activity of people performing complex tasks in a production environment. The method of analyzing the work produced by students is based on the theory of MWS. It uses data collected during resolution of complex geometric tasks performed by students in the classroom. In this method, the choice of task is important: It must offer students a diversity of approaches enabling a better account of their choice and their personal work. The method of analysis is based on a division of the students’ activity into episodes during performance of this task. Each of these episodes comprises a sequence of precise mathematical actions which lead to completion of a subtask. All these actions are then analyzed in great detail using the theory of MWS. The MWS diagram is also used to illustrate the circulation of work during task resolution. This method consists of three steps: – The first step is a top-down analysis to identify and describe the sequences of actions and episodes planned by the student to complete the prescribed task. Their actions are defined in terms of the cognitive and epistemological components

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and processes of the MWS planes. Each episode corresponds to a student’s selfprescribed subtask. In this step, the authors particularly observed the place and role of semiotic tools, technological tools and theoretical tools. – The second step consists of a bottom-up analysis to give a summary overview of the episodes planned by a student and to deduce the logical organization of their mathematical actions and thus discover their cognitive path. The analysis seeks to identify how the work-generating processes interact within the MWS diagram. This then helps us to understand what guided the student in executing the task. – In a third step, the observations made in the previous two analyses are used to account for the states of mathematical work. It is thus a question of evaluating whether the work processes conform with all or part of one of the paradigms, whether the work is complete or is confined within a particular genesis or plane and finally whether the results given by a student are mathematically correct or not. In fact, we can consider that the third stage of the analysis proposes an initial characterization of the students’ mathematical work and goes beyond a simple description of the work.

2.5 Describing the Geneses and Their Interactions in a Subject’s Mathematical Work In this subsection, we will see that the methodological tools of the theory enable us to describe the evolution and balance achieved between mathematical content and students’ cognitive development through the design and implementation of specific tasks. To account for how mathematical work is actually carried out, Gómez-Chacón and Kuzniak (2015) introduce the idea of complete mathematical work in their research in geometry. Mathematical work is considered complete when it results from the combined use of signs to visualize, artifacts to construct and rules based on properties and theorems to develop a discourse of proof. The whole process is studied through the notion of genesis, which is used according to its definition in the theory of MWS. It does not only focus on the origin of an element of knowledge in relation to the interactions between epistemological and cognitive planes, but also includes the development and transformation of the mathematical work through interactions. The process of transformation takes the form of a structured space. The MWS diagram has been used as a tool to better describe and represent the methods of resolution produced by students. It illustrates the degree of completeness of the mathematical work. From a methodological point of view, Gómez-Chacón and Kuzniak combined a qualitative approach based on a cross-examination of the solutions and a statistical analysis of data exploitation to identify not only individual profiles and patterns of problem solving, but also more broadly, types of profiles corresponding to groups

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of individuals. The characteristics of these categories refer to cognitive and epistemological aspects. The students’ blockages are taken into account and analyses of these blockages refined by performing a breakdown of emotion in relation to the mathematical content studied and the degree of cognitive flexibility demonstrated in the use of the different dimensions and facets of geometric work in the MWS. In the same vein, and with the aim of developing multidimensional use of the tools and instruments of mathematical work, Kuzniak et al. (2016) have developed a method of analyzing work. In his epistemography of mathematic knowledge, Drouhard (2009) argues that mathematics is made up of objects, and it is more important to analyze the network of mathematical knowledge associated with these objects than to understand their ontological status. To demonstrate this, he introduces different dimensions related to this knowledge: notional, which concerns the definitions and properties of the objects, semio-linguistic, which concerns writing about the objects and instrumental, which refers to possible operations on the objects. This multidimensional vision fits particularly well with the different dimensions—discursive, semiotic, instrumental—which constitute mathematical work as it is modeled in the MWS. As a result, three types of tools in the epistemological plane of the MWS are defined: semiotic, technological and theoretical. Each of these tools is associated with one of the three geneses of the MWS: semiotic, instrumental and discursive. The method of analysis thus developed consists in identifying the type of tool implemented by the subject and the way in which they are used to carry out the prescribed task. To do this, it is necessary to identify both the component of the epistemological plan mobilized and the genesis(es) activated by a subject. To illustrate this method, let us take the example of a geometric task (Kuzniak & Nechache, 2021) prescribed to 85 students enrolled in the first year of a master’s degree in teaching for future primary school teachers (23 years and older). The task statement is provided in the form of a text to be read: Alphonse has just returned from a trip in Périgord where he saw a parcel of land in the shape of a quadrilateral that had interested his family. He would like to estimate its area. To do this, during his trip, he successively measured the four sides of the plot and found, approximately, 300 m, 900 m, 610 m, 440 m. He is finding it hard to calculate the area. Can you help him by showing him the method to use?

The objective of the task is to conclude that there was insufficient data to determine the shape of the quadrilateral. Analysis of the students’ productions using the method of analysis described in Sect. 2.4 enabled us to highlight the way in which each of the tools (semiotic, technological and theoretical) was used to carry out the task. An example of this is the work of two students (cited as Theo and Francis). From the four dimensions of the land and a choice of scale, Theo constructs a convex quadrilateral (representamen) using a graduated ruler and a compass (or technological tools) on three sides of the quadrilateral. He uses these technological tools to adjust the construction of the fourth side (Fig. 6).

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Fig. 6 Construction by student Theo

The mathematical work developed by Theo circulates in the [Sem-Ins] plane, using the support of technological tools. This work is unfinished because the student does not know what to do with the figure obtained, which is not a usual figure (square, trapezoid, parallelogram) for which he knows the formula of the area. He then decides to give up. This student chose to mobilize tools from the epistemological plan without knowing why he had made such a choice. Francis, on the other hand, by choosing a scale, uses a graduated ruler and compass (technological tools) to construct a convex quadrilateral (representamen) in such a way that it takes the form of a trapezoid (Fig. 7). This construction was guided by the nature of the figure to be obtained, i.e., the trapezoid. He then justifies the nature of the figure. To do this, he uses a geometric property: “two perpendicular lines to the same line are parallel to each other” (theoretical tool) to prove that the quadrilateral he has constructed has two parallel opposite sides. He then concludes that the quadrilateral is a trapezoid (Fig. 7) by referring to the definition of a trapezoid as “a convex quadrilateral with two parallel opposite sides” (theoretical tool). Finally, Francis measures the length of one of the heights of the trapezoid (here [IJ]) on the figure (semiotic tool) using the graduated ruler (technological tool). He Fig. 7 Construction by student Francis

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Fig. 8 Student Francis’s calculation of the area

applies the formula for calculating the area of a trapezoid (technological tool) to obtain the desired area (Fig. 8). The mathematical work produced by Francis is initiated in the [Sem-Ins] plane to construct a convex quadrilateral with the shape of a trapezoid using technological tools. These tools are used by the student as semiotic tools to collect data and make measurements. Construction of the quadrilateral is then justified with the use of theoretical tools characterizing the trapezoid. The formula for calculating the area of a trapezoid (technological tool) is applied to produce the result (in this case, the area of the land). Thus, the mathematical work is completed in the [Ins-Dis] plane. Consequently, the work circulates through all the MWS geneses and planes. Moreover, the tools of the epistemological plan are mobilized throughout the process of this work and are not chosen at random by Francis. Indeed, each of these tools is chosen either to represent the trapezoid (semiotic tool), to construct the trapezoid (compass, ruler) or to justify its use (property characterizing the trapezoid). Consequently, Francis has made a guided choice of epistemological tools, in which the mathematical object (trapezoid) takes on the role of instrument on the cognitive plane. Using the method based on identifying tools of the epistemological plane associated with each of the geneses makes it possible to identify the geneses and vertical planes actually activated and mobilized in the work process. Each genesis can be analyzed in terms of the cognitive process which enables a particular individual to transform a given epistemological plane tool into a cognitive plane instrument. Moreover, this method also highlights the circulation of work established by subjects confronted with a task. In some cases, these imply possible blockages or misunderstandings between the subjects (students and teachers) in the process of attentive work.

3 Characterizing Mathematical Work The objective of characterizing mathematical work is to identify some invariants from the description of work. This enables us to define mathematical work with precision and highlight its characteristics. This characterization may enable us to deduce forms of work where relevant. Some of the methodological tools used by researchers to characterize mathematical work are illustrated in the following paragraphs.

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3.1 Cognitive Unit Associated with a Mathematical Entity Section 2.1 demonstrated how the analysis of a teaching sequence on the notion of circle (performed according to the theory of MWS) initially relies on technological tools to identify a property of the circle, as a new theoretical tool, and enrich the existing set of theoretical tools. Then, the technological tools are progressively left aside in favor of discursive reasoning using the property of the circle. At the end of their description of the mathematical work, Kuzniak and Nechache (2015, 2019c) consider that the mathematical work developed on the circle is structured according to a cognitive unit associated with the mathematical entity, the "circle."In this way, they differentiate the cognitive and epistemological aspects of a mathematical object. The mathematical entity (1.4.3.2) associated with a mathematical object is defined as being a triplet of elements of the epistemological plane, composed of representamen or signs, artifacts and the mathematical properties and definition associated with the object. Thus, the mathematical entity “circle” is the triplet here, comprising: a representamen of the circle (image or drawing of a circle); artifacts associated with the circle (compass and graduated ruler and the construction techniques associated with artifacts); the property and definition of the circle in relation to the concept of equidistance. The subjects’ visualization of the "circle"object as being the union of a point (center of the circle) and a closed line (set of points equidistant from the center of the circle), their implementation of the construction techniques associated with the artifacts to develop the circle and their justification of these constructions by citing the properties characterizing the circle, are all part of the Cognitive Unit (1.4.3.2) associated with the circle. This Cognitive Unit is generated from the interaction of the different geneses which trigger the cognitive processes. Identifying the existence and role of the Cognitive Unit associated with the “circle” mathematical entity enables us to characterize the mathematical work in terms of its circulation within the MWS by qualifying it as complete mathematical work. In the example of the work carried out by student Francis (2.5), the “trapezoid” mathematical object is considered under different aspects during completion of the task: – the drawing aspect of the trapezoid (representamen), – the material artifact aspect (ruler, compass, square) or symbolic artifact (area calculation formula) associated with the trapezoid, and – the property and definition associated with the trapezoid. Thus, Francis’ work is based on the different epistemological components of the “trapezoid” mathematical entity. This entity is made up of the three aspects of the trapezoid associated with one of the three components of the epistemological plane. The Cognitive Unit associated with the "trapezoid" mathematical entity guided Francis’ mathematical work in his completion of the prescribed task. This Cognitive Unit contributes to understanding the reasons for the circulation of work in the different planes of the MWS. Therefore, the Cognitive Unit associated with the

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mathematical entity helps us to characterize the circulation of mathematical work within the MWS, and to understand the reasons for the circulation of this work in the different planes of the MWS.

3.2 Forms of Mathematical Work Kuzniak and Nechache (2019b, 2021) sought to characterize forms of geometric work developed by students preparing for a master’s degree in education. They asked students to solve a geometric task and to then reflect on their production. The students completed the task individually. The authors identified three criteria for assessing the process, outcome and circulation of mathematical work in the MWS: compliance and correctness (1.7.1) and completeness (1.7.2). These criteria consider the paradigm(s) guiding the mathematical work and the components of the MWS mobilized in its completion. In particular, they assessed the extent to which the work processes are shaped by all or part of one of the geometric paradigms. They also looked at whether the results given by students were mathematically correct or not. Finally, they assessed whether the work was complete and to what extent it was confined to a particular genesis or plane. Using the three aforementioned criteria, it is possible to identify five main forms of geometric work observed among the students: those of dissectors, surveyors, explorers, builders and calculators. Beyond the field of geometry, these three criteria, articulated using the tools of the theory of MWS, are expected to be used to assess the work produced by students in other mathematical fields and thus to characterize forms of work. These forms of mathematical work are not specific or fixed attributes of an individual but are rather based on characteristic invariants or ideal types (Bikner-Ahsbahs, 2015) that can be common to groups of individuals.

3.3 Type of Workers To characterize the mathematical work actually produced during the implementation of tasks in the suitable MWS, and to identify the role given by the teacher and assumed by a student when confronted with mathematical tasks to complete, Nechache (2017) defined a categorization of mathematical tasks and the worker-subject. This categorization is defined according to the complexity of the tasks: – Simple tasks, with a low level of requirement, are tasks which have the aim of reinvesting and implementing knowledge and resolution techniques indicated in the task statement, and having already been studied and assimilated by the subject. Generally, resolution of these tasks does not require interaction between the three geneses of the MWS and leads to a mathematical work often confined to one of the three.

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– Standard tasks, with a medium level of requirement, are simplified or in-depth tasks that require the identification and application of knowledge or techniques that are not indicated in the task statement. These knowledge and techniques are already known by the subject. Generally, their resolution involves interactions between the different geneses of the MWS and involves mathematical work often elaborated in at least one of the three planes of the MWS. – Rich tasks, with a high level of requirement, are tasks with the aim of developing critical thinking, reasoning and methodology, and which require the use of knowledge and resolution techniques which have not necessarily been learned by the subject. In some cases, their resolution requires changes in mathematical domains (Montoya & Vivier, 2014), registers of semiotic representation and modeling. These changes are the responsibility of the subject performing the task. Moreover, their resolution mobilizes the geneses and vertical planes of the MWS. This then allows for the development of a complete mathematical work (Kuzniak & Nechache, 2014). According to their category, resolving the prescribed tasks generates different types of activities, thus leading subjects to develop particular forms of work. As a result, each of the three categories of tasks is associated with a category of workersubject: Pieceworkers, Technicians and Engineers (Nechache, 2017, p.75). Indeed, the often-repetitive tasks which restrict the work to a single dimension favor a pieceworker’s method of working. Tasks which rely mainly on the instrumental dimension underpinned by an understanding of the choice of techniques and often confined within a vertical plane of the MWS favor the technician’s method of working. And complete tasks which mobilize all interactions between geneses generate the more inventive engineer’s method of working. This categorization of the worker-subject constitutes a methodological tool for identifying the role given by the teacher and adopted by a student when confronted with the performance of mathematical tasks. The analysis of the mathematical work produced in the suitable MWS enables us to characterize the circulation of work according to the category of task proposed by the teacher to the students and to understand the personal mathematical work which the teacher aims to develop in these students.

3.4 Controls on Mathematical Work Controls (1.7.3) are important in the processes of decision making, guiding and regulating mathematical work. To characterize mathematical work various studies using the MWS theory have indicated that certain meta-cognitive processes and affective processes play an important role in determining both the more intuitive responses in reasoning and more deliberate and reflective decision making (GómezChacón et al., 2016a; Gómez-Chacón, 2018; Kuzniak & Nechache, 2021). Studying middle school classrooms, authors such as Gómez-Chacón et al. (2016b) have studied

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the intertwining of regulatory factors which occur in the field of proof in geometric reasoning with technology based on the geneses of the theory of MWS. To study regulation (both at the individual level and in collaborative work between two students), cognitive categories were considered in relation to the proof process, as were mathematical attitudes (affective dimension), as a social interaction between teacher–student and student–student. Particular focus was placed on diagnosing the interaction in transitions between the construction and testing processes at the instrumental–discursive [Ins-Dis] plane of the MWS. We note that mathematical attitudes have a strong cognitive component and refer to the deployment of a general mathematical disposition and mind habits. Disposition refers not simply to attitudes, but also to a tendency to think and act positively. Students’ mathematical attitudes are manifested in the way they approach tasks such as flexibility of thought, critical thinking, perseverance, independence, accuracy and thoroughness, etc., which are important in mathematics. The methodology was developed on two levels, macroscopic and microscopic. For the macroscopic study, two groups of 9th grade middle school students (ages 14–15) in a public school were studied. The overall progress of all students was analyzed in terms of their development of mathematical competence, and their attitudes toward mathematics before and after GeoGebra was introduced in the classroom. Data were collected declaratively, from questionnaires and whole-class interviews and through observation. The study of mathematical competence included the following categories: representation, use of tools and resources (associated with the [Sem-Ins] plane) and argumentation-proof (associated with the [Ins-Dis] plane). For the affective domain, the mathematical attitudes studied were: flexibility of thought, critical thinking, perseverance, precision and rigor, creativity, autonomy and systematization. For the microscopic study, a micro-analysis was drawn from the case studies with the aim of showing that the transition from instrumental to discursive genesis is not independent of the semiotic genesis. Transitions from instrumental to discursive genesis are brought about in different ways, either through changes in the levels of demonstration or through interactions and affective factors. Figure 9 summarizes the analysis of one of the class episodes where two students worked together with the categories analyzed. The method of analysis applied in this research highlights the factors which impact the transition between the two geneses as well as the stability of the work in the [InsDis] plane. These factors are: (a) the reflections of the teacher and heuristic mediation derived through the students’ use of the Dynamic Geometric System (DGS) tool and (b) the mathematical attitude of the students, mediated by their use of the DGS tool. Working with the DGS in proof construction ([Ins-Dis] plane) in middle school classes, in turn, presents difficulties which impede the completeness of geometric reasoning. The methodology used (micro- and macroscopic study) explains how stagnation between visualization and construction blocks transition to the proof processes inherent to the discursive genesis in MWS. The source of this mental block is not purely cognitive, but also lies in the interplay between cognition, affect and social interaction. From this perspective, the MWS approach shows its usefulness in identifying the typologies of cognitive processes which underpin students’ mathematical

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Fig. 9 Reasoning in inter-genesis and inter-plan transitions, episode 2 (Gómez-Chacón et al., 2016b)

attitudes and also shows how positive attitudes determine access to decision making in proofs.

4 (Trans)forming Mathematical Work Research on the formation or transformation of mathematical work aims to build a new structure for the epistemological plane of work or to provide a new structure for the cognitive plane. Transforming work thus amounts to profoundly modifying and shaping it to make it more adequate to educational expectations, epistemological requirements and cognitive competencies in the subjects. In some cases, the transformation of work sought is radical and implies an important change of tools and methods used in the same mathematical domain. For example, in her thesis work, Reyes identified the difficulties students enrolled in engineering courses at university (in Mexico) experienced in trying to make sense of the notion of “function.” She then sought to investigate whether the study of motion phenomena (related to kinematics) through mathematical modeling processes would help students to better understand the concept of function, designing a set of tasks which she then gave to the students to undertake. From a task design perspective, Reyes (2020) used the Kinematics Working Spaces (KWS) which helped her to gain a clear view of the cognitive processes generated by students. She focused on how they use the properties, semiotic representations and artifacts associated with the concept of function during task completion.

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Using the components of the KWS and the modeling cycle (Blum & Leiss, 2007), Reyes designed a set of tasks which combine mathematics and kinematics. The goal of these tasks is then to study motion phenomena. These tasks are designed in such a way that resolving them necessarily requires mobilizing semiotic representations, artifacts and properties of the theoretical referential associated with the notion of function. Thus, the tasks are conceived in such a way as to mobilize each of the components of the MWS epistemological plane relating to the notion of function and to associate the components of the epistemological plane. The registers of semiotic representation associated with the function object brought into play in these tasks are: natural language, value table, graph and algebra. Another register of representation from the field of physics is also considered: that of physical movement. The latter is fundamental to initiate the processes of mathematical work in students with the function object. The artifacts associated with the function object are chosen so as to favor the transition between the different registers of representation: These are software programs GeoGebra and Tracker. GeoGebra is used in mathematics and enables us to generate graphs with mathematical notations. It then allows transition between the graphic register and the algebraic register. Tracker (Fig. 10) is used in physics and enables relationships to be established between the real movement and the variables related to distance and time (for example, speed and acceleration). These relationships in turn trigger a process of coding the motion which subsequently generates symbolic and graphic representations closely related to physical perceptions of motion. This

Fig. 10 Tracker interface (Reyes, 2018, p. 155)

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software therefore has an important role in the transition from the physical movement register to that of graphic representation. These two technological tools have a facilitating role in the transition between the representation registers of a kinematic-like nature and the other of mathematics. In terms of the students’ task completion, Reyes analyzed the mathematical work using the modeling cycle and the KWS diagram. For each of the steps in the modeling cycle, the circulation of mathematical work is described using the MWS diagram. As, for example, in Fig. 11 which describes the circulation of mathematical work within the KWS components during the first stage of the modeling cycle. Numbers were used to specify the direction of the work circulation. Thus, the mathematical work was initiated in the instrumental genesis (number 1) by the use of a material artifact (here, a model of a windmill) to complete the movement described in the task statement. The process of visualizing the semiotic genesis (number 2) exercised in the movement led the students to use the representation of this movement in the physical movement (PM) representation register. This representation is analyzed (number 3) using the elements of the theoretical referential to decide whether or not to keep it to address the problem at hand. Therefore, in the first stage of the modeling cycle, the mathematical work was initiated in the instrumental genesis, transitioning to the semiotic genesis and finally to the discursive genesis. A summary overview of the mathematical work circulation in each step of the modeling cycle is presented using comic strip-like sets of diagrams and the modeling cycle diagram (Fig. 12). This form of overview illustrates the overall dynamics of the circulation of mathematical work during the execution of the modeling cycle.

Fig. 11 Circulation of mathematical work in stage 1 of the modeling cycle (Reyes, 2020, p. 262)

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Fig. 12 Analysis of mathematical work during completion of a modeling task (Reyes, 2020, p. 274)

The method used by Reyes, to develop and form mathematical work around the notion of “function” to make it more accessible to students, is based on task design using the theory of MWS. These tasks are specifically designed so that each component of the epistemological plane, as well as the geneses, are mobilized by the students during completion of these tasks. In addition, Reyes uses the MWS diagram to account for the circulation of mathematical work actually produced during students’ completion of the tasks. Thus, the theory of MWS is used here both to form work and to analyze how work was completed.

5 Analysis of Mathematical Work: The Distinction Between Potential and Actual Mathematical Work In many studies using the theory of MWS, we notice use of the expression a priori analysis which is very popular in the didactics of mathematics and which often has a Brousseau’ TDS (Theory of Didactical Situations) connotation. However, these so-called a priori analyses as they appear in this research are different from those of TDS. These analyses are simply considered as a first stage of analysis of the mathematical content at hand.

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We consider that two kinds of analysis must be carried out to study mathematical work according to the theory of MWS. One of these is an analysis of the potential and expected mathematical work which helps us to anticipate and foresee how this work will potentially be elaborated. A second is that of the actual mathematical work, i.e., analysis of the work actually produced by an individual (or group of individuals) during task performance. In some cases, the gap between potential and actual mathematical work can be studied more specifically.

5.1 Potential Mathematical Work First of all, a task (or tasks) analysis is necessary, because tasks have a crucial role to play in activating the MWS and thus giving access to the mathematical work. To perform this analysis, it is important to describe the task by explicitly presenting the elements of the three poles of the epistemological plane (representamen, artifacts and theoretical referential) brought into play in completion of the task. This requires identifying the tools of the epistemological plane (semiotic, technological and theoretical) which are potentially useful for solving the task. Indeed, the elements of each component of the epistemological plane constitute observables (see Pizzaro’s method, 2.2) of the work, which will guide the observation of tasks/sessions and thus identify the cognitive processes activated in relation to each component of the epistemological plane. This task analysis must enable descriptions of the different processes and results of the mathematical work which can potentially be observed in the subject performing the task. These processes encompass the mathematical procedures and reasoning potentially mobilized by the subject. From this, we can develop an idea of the cognitive processes which potentially occur. From there, we can induce the paradigm likely to guide the mathematical work. Moreover, this analysis must be able to account for the different phases of the development of mathematical work and its circulation within the potential suitable MWS. Local circulation of the work within each of the phases can be described a priori using the MWS diagram. The analysis of the potential mathematical work thus serves as a working hypothesis which can be confirmed or modified later by data analysis.

5.2 Actual Mathematical Work The analysis of actual mathematical work is based on the mathematical actions observed in individuals confronted with mathematical tasks. These actions belong to the components of the MWS (see top-down and bottom-up methods in 2.4). This leads us then to consider modalities corresponding to different observable data

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requiring specific treatments: oral statements must be transcribed from sound recordings, written productions must be collected and sometimes rewritten to make them more legible and manipulations of objects must be filmed and then described. This analysis of the work involves, on the one hand, identification of the tools of the epistemological plane associated with each of the geneses of the MWS mobilized by the subject during completion of the task. On the other hand, it involves describing how these tools are used by the individual. Identification of the tools and their uses provides information on the geneses mobilized and prioritized by the worker-subject in completing the task and of the work in the MWS. Analysis of the processes and results of the work enables us to specify the paradigm which has effectively guided the actual mathematical work. Comparison of the analysis of the expected mathematical work with that of the mathematical work actually produced enables us to identify and discuss the adequacy or distance between the potential and actual mathematical work. This comparison highlights blockages, misunderstandings or rebounds, which can appear during the completion of the task by the subjects. It also generates reflection about possible adjustments of the mathematical work.

6 Conclusion This chapter has presented methodological tools used to deal with the three types of questions regarding mathematical work in the theory of MWS. One of the methodological tools covered is that of tasks. Indeed, the analysis of mathematical work drawing on the theory of MWS is performed using particular studies on mathematical tasks. It is then a question of describing this work from the perspective of the mathematical content used in these tasks, but also in terms of the possible cognitive processes at play. However, the tasks are not designed independently of the didactic situations in which they are implemented. Therefore, the choice of mathematical tasks (and their implementation in didactic situations) studied through the theory of MWS enables us to highlight the invariants constituting the mathematical work (complete, compliant, correct). They also enable us to characterize generic forms of work in different mathematical domains, providing insight into the distance between the work expected by the designer subject and the work actually produced by the performer subject. The tasks are used as a methodological tool in the theory of MWS to identify and analyze the breaks and confinements of mathematical work in some of the MWS geneses, as well as to identify blockages and misunderstandings which can occur during task execution. Therefore, the tasks and their implementation in didactic situations enable us to observe the mathematical actions of a subject performing the task. As such, the three elements: the prescribed tasks, the didactic situations in which these tasks are implemented and the mathematical actions of a subject, form a set of inherent observables of mathematical work.

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Analysis of mathematical work is based on identifying the functioning of each of the three geneses of the MWS. This implies identifying the tools of the epistemological plane associated with its geneses: semiotic, technological and theoretical, and the way in which these tools are used and transformed by individuals faced with mathematical tasks into instruments of the cognitive plane. These tools and associated processes are observables of mathematical work which enable us to identify the geneses of the MWS which are prioritized in the development of work and to infer the circulation of mathematical work through the different components of the MWS. This circulation is also visualized using the MWS diagram. The latter constitutes another interesting methodological tool to describe, through detailed analysis of performance of the tasks, the relationships and interactions between the various components of the MWS and thus describe the circulation of mathematical work during the completion of a task or set of tasks by individuals. It also enables us to identify the origin of blockages or misunderstandings during the performance of the task. These analyses of the work are completed using methods and tools which are not specific to the theory of MWS, such as: statistics, interviews, discourse analysis, class episodes, etc. On the other hand, some researchers may use the theory of MWS to formulate questions, yet draw on other theories for their methodology because the tools of the theory of MWS are insufficient alone. As such, we need to develop a methodology which makes it possible to coherently articulate the tools borrowed from each of the theories involved. This coherence of tools within the methodology is important to justify the answers given to the research questions formulated. Conversely, some research which does not draw on the theory of MWS directly, may nevertheless draw on its methodology and tools to investigate their research questions, which also requires articulation of the tools borrowed from each of the theories. This type of networking-based methodology is discussed more specifically in Part II of this book.

References Blum, W., & Leiss, D. (2007). How do students and teachers deal with mathematical modelling problems? The example sugaloaf und the DISUM project. In C. Haines, P. L. Galbraith, W. Blum, & S. Khan (Eds.), Mathematical modelling (ICTMA12)-education, engineering and economics. Chichester: Horwood. Bikner-Ahsbahs, A. (2015). Empirically grounded building of ideal types. A methodical principle of constructing theory in the interpretative research in mathematics education. In A. BiknerAhsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to qualitative research in mathematics education (pp. 105–135). Dordrecht: Springer. https://doi.org/10.1007/978-94-017-9181-6. Darses, F. (2001). Providing ergonomists with techniques for cognitive work analysis. Theoretical Issues in Ergonomics Science, 2, 268–277. Derouet, C. (2017). Circulations entre trois domaines mathématiques: les probabilités, la statistique et l’analyse. In K. Nikolantonakis (Ed.), Proceedings of 5th Conference Espace de Travail Mathématique (pp. 63–78). Macedonia: University of Macedonia.

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Drouhard, J.-P. (2009). Epistemography and algebra. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of CERME6. Lyon, France. http://ife.ens-lyon.fr/publications/ edition-electronique/cerme6/wg4-07-drouhard.pdf. Fénichel, M., & Taveau, C. (2009). Enseigner les mathématiques au cycle 3. Le cercle sans tourner en rond. DVD, CRDP Créteil. Gómez-Chacón, I. M., & Kuzniak, A. (2015). Geometric work spaces: Figural, instrumental and discursive geneses of reasoning in a technological environment. International Journal of Science and Mathematics Education, 13(1), 201–226. https://doi.org/10.1007/s10763-013-9462-4 Gómez-Chacón, I. M., Botana, F., Escribano, J., & Abánades, M. A. (2016a). Concepto de Lugar Geométrico. Génesis de Utilización Personal y Profesional con Distintas Herramientas. Bolema– Mathematics Education Bulletin, 30(54), 67–94. Gómez-Chacón, I. M., Romero, I. M., & Garcia, M. M. (2016b). Zig-zagging in geometrical reasoning in technological collaborative environments: A mathematical working space-framed study concerning cognition and affect. ZDM Mathematics Education, 48(6), 909–924. https:// doi.org/10.1007/s11858-016-0755-2 Gómez-Chacón, I. M. (2018). Hidden connections and double meanings: A mathematical viewpoint of affective and cognitive interactions in learning. In G. Kaiser, H. Forgasz, M. Graven, A. Kuzniak, E. Simmt, & B. Xu (Eds.), Invited Lectures from the 13th International Congress on Mathematical Education. ICME-13 Monographs (pp. 155–174). Cham: Springer. https://doi.org/ 10.1007/978-3-319-72170-5_10 Kuzniak, A., & Nechache, A. (2015). Using the geometric working spaces in order to plan the teaching of geometry. In K. Krainer (Ed.), Proceedings 9th Conference European Research in Mathematics Education. Prague, Czech Republic. https://hal.archives-ouvertes.fr/hal-01287007/ document. Kuzniak, A., Nechache, A., & Drouhard, J. P. (2016). Understanding the development of mathematical work in the context of the classroom. ZDM Mathematics Education, 48(6), 861–874. Kuzniak, A., & Nechache, A. (2019a). Tâches emblématiques dans l’étude des ETM idoines et personnels: Existence et Usages. In K. Nikolantonakis (Ed.), Proceedings of 5th Conference Espace de Travail Mathématique (pp. 145–155). Macedonia: University of Macedonia. Kuzniak, A., & Nechache, A. (2019b). Personal geometrical work of pre-service teachers: a case study based on the theory of mathematical working spaces. In Procedings of Cerme11. Utrecht, Nederlands. https://hal.archives-ouvertes.fr/hal-02402239/document. Kuzniak, A., & Nechache, A. (2019c). Une méthodologie pour analyser le travail personnel d’étudiants dans la théorie des Espaces de Travail Mathématique. In L. Vivier & E. MontoyaDelgadillo (Eds.), Sexto Simposio sobre el Trabajo Matemático (pp. 61–70). Valparaíso: Pontificia Universidad Católica de Valparaíso. Kuzniak, A., & Nechache, A. (2021). On forms of geometric work: A study with pre-service teachers based on the theory of mathematical working spaces. Educational Studies in Mathematics, 106(2), 271–289. https://doi.org/10.1007/s10649-020-10011-2. Montoya, E., & Vivier, L. (2014). Les changements de domaine dans le cadre des Espaces de Travail Mathématique. Annales De Didactique Et De Sciences Cognitives, 19, 73–101. Nechache, A. (2017). La catégorisation des tâches et du travailleur-sujet: Un outil méthodologique pour l’étude du travail mathématique dans le domaine des probabilités. Annales De Didactique Et De Sciences Cognitives, 19, 67–90. Pizarro, A. (2018). El trabajo geométrico en clases de séptimo básico en Chile: Un estudio de casos sobre la enseñanza de los triángulos. Thèse de l’Université de Paris. Paris: Université de Paris. Radford, L. (2008). Connecting theories in mathematics education: challenges and possibilities. ZDM Mathematics Education, 40, 317–327. https://doi.org/10.1007/s11858-008-0090-3. Reyes, C. (2020). Enseignement et apprentissage des fonctions numériques dans un contexte de modélisation et travail mathématiques. Thèse de l’Université de Paris. Paris: Université de Paris. https://tel.archives-ouvertes.fr/tel-03211997.

The Theory of Mathematical Working Spaces in Brief Alain Kuzniak, Assia Nechache, and Philippe R. Richard

Main Purpose of the Theory Describing, understanding and (trans)forming mathematical work in a school context.

1 Mathematical Work In the theory of MWS, mathematical work must be understood as an ongoing human intellectual process of production, the orientation and finality of which are defined and supported by mathematics and more generally by mathematical culture. Mathematical work is intended to perform a task, solve a problem or overcome an obstacle in relation to mathematics. It requires the sustained mobilization and development of material or intellectual resources derived from mathematical culture. A reasonable goal of mathematics education is that, at the end of their studies, students should have a good idea and practice of mathematical work.

A. Kuzniak (B) LDAR, Université de Paris, Paris, France e-mail: [email protected] A. Nechache LDAR, CY Cergy Paris Université, Cergy, France P. R. Richard Université de Montréal, Montréal, Canada © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Kuzniak et al. (eds.), Mathematical Work in Educational Context, Mathematics Education in the Digital Era 18, https://doi.org/10.1007/978-3-030-90850-8_3

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1.1 Purpose, Processes and Results Mathematical work involves consideration of three aspects related to its performance and development: The purpose of the work. This attribute helps us to distinguish the notion of “work” from a simple activity. This is done by assigning a goal to an action, clarifying the stakes associated with it and inscribing it in the long term by demonstrating its general relevance. The processes of the work. These are related to the procedures and constraints of implementing given tasks. Appropriation of the goals of the task and compliance with the rules of functioning are important aspects for observing and characterizing the work. The results of the work. Results should be valid and coherent within the mathematical domain in question.

2 Epistemological and Cognitive Aspects of Mathematical Work Through the lens of MWS, the development by an individual, whether generic or not, of appropriate mathematical work is a gradual and progressively constructed process of bridging epistemological and cognitive planes.

2.1 The Epistemological Plane and Its Components The epistemological plane, organized according to purely mathematical criteria, has three interacting components: a set of concrete and tangible objects (representamen); a set of artifacts such as drawing tools or software; and a theoretical system of reference (the referential) based on definitions, properties and theorems. Signs or representamen The sign, or representamen, is “something which stands to somebody for something in some respect or capacity” (Peirce). Signs or representamen summarizes the component using concrete and tangible symbols. Depending on the mathematical field at stake, the signs used can be geometrical images, algebraic symbols, graphics or even tokens, models or photographs in the case of problems involving modeling. Artifacts An artifact includes everything that has undergone a transformation, however, small, of human origin. Its meaning is not limited to material objects and can include a strong

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symbolic dimension. In the theory of MWS, mathematical artifacts will generally be associated with material objects to avoid confusion with other components of the epistemological plane. These objects will be intimately linked to rules and techniques of construction or calculation (Euclidean division, "classical" constructions with ruler and compass, etc.). These techniques are based on algorithms, the validation and theoretical status of which are no longer problematic or are no longer questioned by their user. Theoretical frame of reference or referential The referential includes the set of properties, definitions, theorems and axioms and refers to the theoretical part of the mathematical work. This set is not only a collection of properties: It supports the deductive discourse of proof specific to mathematics, and this requires that it should be organized in a coherent way, well-adapted to the tasks the students are requested to solve and model. The referential is not absolute and determined in a unique way once and for all; its constitution and organization depend, in particular, on the mathematical paradigms and work targeted by the educational institutions or organizations.

2.2 The Cognitive Plane and Cognitive Processes The cognitive plane relates to the processes and procedures used by individuals in task-solving activities. It is structured around three cognitive processes: visualization, construction and proving. Visualization The process of visualization can be considered as the cognitive process associated with identifying and developing a set of treatments and transformations of the representamen or signs—which, in this respect, can very well be perceived as acoustic images in strictly oral contexts. It is, thus, possible to study the role of signs in different mathematical domains, such as mathematical analysis, which gives rise to a specific work of visualization based on spreadsheet data or numerical writings. This visualization must be distinguished from the simple vision or perception of signs. Construction The process of construction is associated with actions triggered by the use of artifacts, actions which lead to tangible results such as figures, graphs or the results of a calculation based on algorithms or produced by a numerical machine. These results are concrete products of some mathematical entities which are generally not isolated and form configurations to be observed and explored. Proving The process of proof considered here relies on deductive and logical discourse and must be based on, or lead to, assertions with a clear theoretical status. In contrast, simple empirical validation is more likely to relate to construction or visualization,

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as discussed above. In addition to deductively organized arguments, these assertions may well be definitions, hypotheses, conjectures or statements of counterexamples.

3 Mathematical Working Spaces—MWS A Mathematical Working Space refers to an abstract structure organized in such a way as to account for mathematical work and enable individuals to conduct operations with problems in a specific domain (geometry, probability, etc.). The theory of MWS aims to highlight the mathematical work proposed by educational institutions and organizations involved in the teaching and learning of mathematics.

3.1 The MWS Diagram The MWS diagram gives a general overview of the different relationships and interactions existing between the components of the epistemological plane and cognitive plane of the Mathematical Working Space. The diagram is closely associated with the development of the theory, of which it is in a way the emblem, but it should not be confused with the theory as a whole. Its use as a methodological tool requires selection of the most appropriate modes of representation to reflect the evolution of the work.

3.2 The Three Mathematical Work Geneses in the MWS Development by an individual, whether generic or not, of appropriate mathematical work is a gradual process which is progressively developed, as a process of bridging the epistemological and cognitive planes. It evolves in accordance with different specific yet intertwined generative developments: semiotic, instrumental and discursive geneses. Each of these geneses can be visualized through one of the dimensions of the MWS diagram (Fig. 1).

3.3 Top-Down and Bottom-Up Perspectives on Geneses Each of these geneses can be described according to two perspectives: One stance focuses on epistemological components (top-down perspective) and a second oriented toward cognitive processes (bottom-up perspective). The terms bottom-up

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Fig. 1 Mathematical working space diagram

and top-down are only relative to the position of the epistemological and cognitive planes in the MWS diagram.

3.4 The Semiotic Genesis The semiotic genesis relates signs and representamen with visualization. It explains the dialectical relationship between syntactic and semantic perspectives on mathematical objects, represented and organized by semiotic systems. The semiotic genesis confers to the representamen their status of operational mathematical objects. In this way, it establishes the links between the function and structure of the expressed signs. The bottom-up perspective associated with this genesis can be considered as a process of decoding and interpreting signs from a given representamen through visualization. The top-down perspective, directed from visualization toward a representamen, can be evoked when a sign is produced or specified.

3.5 The Instrumental Genesis The instrumental genesis establishes the link between the artifacts and the construction processes contributing to the completion of mathematical work. The bottomup perspective describes the actions by which the user appropriates the various

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techniques associated with the artifact. The bottom-down perspective relates to the adequate choice of the artifact required to perform the intended actions. In the theory of MWS, the two perspectives, epistemological and cognitive, are essential because if using the artifacts modifies the users’ way of doing and thinking, in return, the mathematics they develop will also be transformed.

3.6 The Discursive Genesis of Proof The discursive genesis of proof is the process by which the properties and results organized in the referential are actuated in order to be available for mathematical reasoning and discursive validations. By this, we mean validations going beyond graphic, empirical or instrumented verification, but which could nevertheless be triggered by these. In this genesis, the bottom-up perspective is related to a deductive discourse of proof supported by properties structured into the referential. Conversely, the identification of properties and definitions to be included in the referential, possibly suggested by instrumental, computational or visual processes, relate to the top-down perspective.

3.7 Interactions Between Geneses and Vertical Planes The vertical planes (named Sem-Ins, Ins-Dis and Sem-Dis) make interactions between the different geneses visible: In particular, they enable expression of the main coordination games involved in the mathematical work and the geneses that are temporarily subordinated to such work (Fig. 2). A deep understanding of the different geneses and the three planes of interaction between these geneses ([Sem-Ins], [Ins-Dis] and [Sem-Dis]) is gained from looking Fig. 2 Three vertical planes in the MWS diagram

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at methodological investigation, but also at more specific and detailed studies of the proofs and validations used to solve tasks. From a methodological perspective, recognition of active geneses and interactions between geneses is supported by the identification of tools (in the epistemological plane) and instruments (in the cognitive plane) used to perform the task.

4 Diversity of Mathematical Work Associated with Various MWS 4.1 The Reference Mathematical Work in Connection with the Reference MWS The reference mathematical work corresponds to the work normally expected by educational institutions and organizations in charge of teaching mathematics. The reference MWS is naturally defined according to mathematical criteria. But cultural, economic, social and political criteria also contribute toward shaping it, to such an extent that describing it requires thorough studies: of the intended curriculum, of treatises written by mathematicians, of essays and articles by mathematics educators. This aspect in turn supposes that epistemological vigilance is required. It means that the operating rules involved in this MWS enable knowledge to be organized into a well-defined and coherent mathematical field. The set of rules, procedures and questioning which help to define the reference work refers to the notion of a paradigm. Pragmatically in educational contexts, the constitution of a MWS does not rely on a single paradigm, but rather on the interplay between different paradigms.

4.2 The Personal Mathematical Work and Personal MWS The personal work and personal MWS will depend on the actors and subjects implicated in the work—a pupil or student, a teacher or teacher trainer. This MWS reflects the reality of the work of individuals who perform a task with their own knowledge and cognitive capacities. The personal work is very fluid, and its evolution depends on the tasks the subjects are led to perform.

4.3 The Suitable MWS This MWS refers to an intermediate state of transmission and mediation of knowledge where tension exists between the teacher’s expectations and the redefinition of the task and problems in order to enable students’ personal work to progress. The MWS

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in question is designated as suitable, or idoine, MWS precisely to emphasize these transformations by which it is characterized in order to best adjust to the educational project and to the reality of the learners. It can be seen as a connecting structure from reference MWS to personal MWS mediated by designer-actors (teacher or researcher) to help receptor-actors (students) build their personal mathematical work. It is necessary to consider two states of this MWS: a potential state, which corresponds to what the teacher foresees, and an actual or effective state, which comprises the work produced in the classroom. These three MWS are not independent but are rather interrelated (4.2). Each requires a specific methodology which must be understood (Part II).

4.4 Mathematical Domain and Specific MWS A mathematical domain should be organized into a consistent structure within mathematics. The best proof that a given segment of mathematics should be called a domain is that it is regarded as such by the community of mathematicians. In this way, a specific MWS is associated with each domain. Thus, we may speak of a MWSGeometry , MWSAlgebra , MWSAnalysis , MWSProbability , etc. It is also worth reflecting upon the way new transversal themes—such as algorithmics or modeling—may be integrated into MWS and what kind of reorganization of the framework is necessary to foster this integration.

5 Paradigms in the Theory of MWS A paradigm stands for the combination of beliefs, convictions, techniques, methods and values shared by a community. This community can be scientific, but in the theory of MWS, it will be educational organizations engaged in mathematics education. In the theory, different paradigms can exist and coexist. Identifying these paradigms and their interactions helps us to understand and orient mathematical work. Paradigms ensure the conformity of work processes and the accuracy of the results produced. Mathematical work is said to be compliant when the processes and procedures used are valid and consistent with the expectations present in the preferred paradigm(s) of work. The search for paradigms associated with the MWS specific to a mathematical domain is an important focus for the theory of MWS.

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6 Mathematical Tasks 6.1 Mathematical Tasks and the Learner’s Work In the theory of MWS, tasks must be interpreted in a broad and varied sense. They can, thus, be associated with traditional problems formulated with clear assumptions and known to be solvable in a predictable amount of time by students. They can also be given with relatively vague outlines as support for open and rich problematization or modeling situations. In this way, a range of tasks, from simple to rich tasks, exists. Performance of the tasks depends on their level of mathematical sophistication and on the degree of initiative the students are free to use. Tasks are the essential units which trigger the subject’s activity. More precisely, mathematical actions with well-determined goals are seen as the visible manifestation of a subject’s cognitive activity (or mathematical thinking) when actively performing tasks. From analysis of the mathematical actions identified being performed by subjects, it becomes a matter of identifying their mathematical work with the idea of adapting or transforming it.

6.2 Task Environment and Mathematical Work A task is determined by its goals and the conditions necessary for its performance which can be individual or collective. In mathematics education, tasks are conceived and implemented in various environments, which can relate to different didactical and theoretical traditions such as task design, lesson studies or didactical engineering. The tasks within their environment and the mathematical actions performed form a set of observables which enable the mathematical work to be investigated. Studying these observables through the lens of the MWS enables generic forms of work in different mathematical domains to be characterized. In the long term, this characterization of forms serves as a solid basis for forming or transforming mathematical work to conform to the expectations of the given paradigm.

7 Tools and Instruments of Mathematical Work Solving a mathematical problem implies the use of different mathematical objects which can be considered as tools or instruments with which to perform the task. In the theory, and whenever it is useful, the term tool is reserved for objects of the epistemological plane that have a specific potential use in the context of problem solving. Tools have a productive purpose for performing the task.

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Use of the word instrument is prioritized as soon as a subject (pupil, student or teacher) has developed their use of a tool to effectively solve the proposed task. It supposes implementation of techniques by the active subject and thus concerns the cognitive aspects of the work. In relation to the different components of the epistemological plane and the different geneses of mathematical work, it is possible to consider, a priori, three types of tools and instruments: Semiotic tools are non-material tools—like diagrams, symbols or figures—that may be used as instruments for operating on semiotic representations of mathematical objects. Technological tools are artifacts—like drawing tools or routinized techniques— which are “to hand,” based on algorithms or calculators with implemented calculation or drawing algorithms used as instruments for constructing mathematical objects such as figures, graphs, numbers. Theoretical tools are elements of the referential non-material tools like theorems which may be used as instruments for mathematical reasoning based on the logical and mathematical properties of mathematical objects.

8 The MWS Diagram: A Methodological Tool to Describe, Characterize and Transform Mathematical Work 8.1 Describing the Circulation of Work The individuals involved in performing a task to produce a series of actions oriented by the goals to be reached and triggered by knowledge. The set of dynamics, thus, created enables us to speak of a circulation of work. The MWS diagram makes it possible to describe and visualize the evolution and dynamics of the various interactions between geneses during the performance of a task or set of tasks by individuals. This dynamic is called circulation of work.

8.2 Characterizing Mathematical Work When the circulation of work engages all the geneses and the different planes of the diagram, the mathematical work will be considered complete. The diagram can be used to demonstrate the completeness of the mathematical work. It also helps to identify the reasons for a blockage which may be due to confinements in a vertical plane or in a genesis. It provides an insight into the rebounds of this work to break through blockages and confinements.

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8.3 (Trans)forming the Circulation of Work By focusing on the various interactions which help or hinder mathematical work, the diagram can serve as a support for thinking about and rethinking the circulation of work (by tuning tasks and their implementation). In the long term, this can contribute toward forming or transforming the mathematical work.

9 Further Aspects on Specificity and Originality of the Theory 9.1 Epistemological Anchoring The theory of MWS is a didactic theory strongly supported by mathematical content. It achieves this by closely combining epistemological and cognitive aspects around the unifying notion of mathematical work.

9.2 A Triadic Approach The synthetic and triadic vision of mathematical work gives it coherence and facilitates its analysis. In this conception of mathematical work, it is the arrangement and relationships between the different epistemological components and cognitive processes which ensure coherent and complete development of mathematical work. The elaboration of the latter is seen as harmonious interactions of three types of genesis: semiotic, instrumental and discursive.

9.3 Processes and Results Work is based on a set of organized and finalized actions, which emphasizes the duality between, on the one hand, the processes of work and, on the other hand, the results of work.

9.4 Importance of the Idea of “idonéité” The “suitability” hallmark of a MWS is gradually constructed and adapted along the paths and vagaries of the teaching route. This explains our fondness for the French

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term “idoine,” the translation of which to “suitable” in English may not fully convey the meaning suggested by Gonseth (1945–1952, our translation): A metaphorical way of presenting suitability in the theory of MWS is to think about it as resulting from recursive feedback loops, converging toward a stable state when one exists. Instead of trying to reduce the uncertainty related to any new experience, Gonseth proposes a recursive adaptation to it.

9.5 The Link with Learning Theories Accordingly, one of the main strengths of studies based on MWS is their investigation of the interactions between the cognitive and epistemological dimensions. The underlying learning model is not, as in many theories which emerged in the 1970s, the socioconstructivist model. As far as the MWS is concerned, there is no prescribed learning model. Studies can look at other learning principles such as Socrates’ maieutics or behaviorism, or even approaches which emphasize the imitation of models presented to students.

9.6 Link to Institutional Choices The theory of MWS does not give priority to a determined institutional model with regards to the development of an epistemological basis, if only for the purpose of expecting—and this is no small matter—an approach which is globally coherent, in particular with respect to validation within a given paradigm or network of paradigms, meant to be consistent and rational. The notion of a paradigm is essential here, and in a certain way, mathematical work could be seen as a quest for harmonious articulation between coherent paradigms.

9.7 Points not Addressed by the Theory • The impact on the development of the mental apparatus of a subject of information and retroactions induced by the environment. • The nature of the black box in precise reference to the psychological constitution of the geneses. • The existence and possible place in the brain of all kinds of schemas which can intervene in the mathematical work. These are issues which the theory has neither the ambition nor the means to explore. The theory relies on the visible effects of a subject’s actions through explicit

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observables in order to understand cognitive processes. In this sense, it can be seen as a materialist-type scientific approach.

9.8 Dialog with Other Theories The theory of MWS is not a holistic theory and has close links with theories which share its materialist approach on people’s way of thinking when they are engaging in mathematics. Moreover, the whole activity of teaching and learning mathematics is not limited to mathematical work. Many complementary and indispensable elements are external to this work yet participate in its undertaking in practice: affects and emotions, collaborative and collective work, language exchanges, cultural and social context and so on. These represent many points and perspectives which are not directly related to the core of the theory, but which can contribute to fruitful exchanges with specialists in these areas. These exchanges and interactions are part of the theoretical and methodological innovation which has characterized MWS research to date.

The Different Types of Mathematical Work and MWS

The Reference Mathematical Working Space Elizabeth Montoya-Delgadillo and Claudia G. Reyes Avendaño

1 Introduction Throughout Chapter “The Theory of Mathematical Working Spaces—Theoretical Characteristics”, it has been assumed that the purpose of the theory of Mathematical Working Spaces (MWS) is the didactic study of mathematical work. In other words, on the one hand, the didactic study involves phenomena related to knowledge, teaching and learning, and on the other hand, mathematical work is an intellectual undertaking which is guided and supported by mathematics. In order to achieve this purpose, the theory of MWS is founded on several constructs which are both theoretical and methodological. Among these, the reference MWS, suitable MWS and personal MWS can characterize the mathematical work which emerges in an educational environment by considering epistemological and cognitive aspects. But, why are these three MWS considered in the theory? To answer this question, it is necessary to remember that the didactic study of mathematical work is far from being simple, with many variables involved. The coherent completion of mathematical work relies both on the knowledge which educational institutions aim to impart to learners and on educational organizations with teachers who must supervise the day-to-day teaching in their classroom and students who are there to understand and learn. To be successful, we must be aware that the process is complex for different reasons. Firstly, the knowledge taught has evolved and changed over time depending on people and contexts, etc. which have made it what it is today. Secondly, this knowledge has been organized in different ways on several occasions by various E. Montoya-Delgadillo (B) Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile e-mail: [email protected] C. G. Reyes Avendaño Universidad Nacional Autónoma de México, México, México e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Kuzniak et al. (eds.), Mathematical Work in Educational Context, Mathematics Education in the Digital Era 18, https://doi.org/10.1007/978-3-030-90850-8_4

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institutions and organizations with the aim of being taught and fulfilling the desired academic requirements. Finally, learners, in general, have different skills, needs and capacities. In this sense and with the aim of encompassing the diverse types of mathematical work that can emerge in an educational environment, the reference MWS, personal MWS and suitable MWS are introduced and defined. This chapter concentrates on the reference MWS. Beginning with a definition and explanation of how it can be described and accessed through different approaches (2), we then further highlight the role it plays within the network of the three MWS of the theory: reference, suitable and personal (3). This leads us to think more about its users and developers. In Sect. 4 and based on some examples, we show how it can be used to characterize mathematical work. Finally, we address the specific issue of paradigms in the theory of Mathematical Working Spaces and the correlation with the reference MWS (5).

2 The Reference MWS: Definition and Access 2.1 Definition The reference MWS is generated in an environment where educational institutions and organizations play a fundamental role into promoting the emergence and production of mathematical work. According to Chap. The Theory of Mathematical Working Spaces in Brief, Sect. 4.1, the reference mathematical work corresponds to the work normally expected by these institutions and organizations in charge of teaching mathematics. The reference MWS is naturally defined according to mathematical criteria. But, cultural, economic, social and political criteria also contribute toward shaping it, to such an extent that describing it requires thorough studies. For this reason, description of the reference MWS is difficult and requires consideration of a complex network of influence based on institutional, social and human contexts and needs and considering epistemological and cognitive aspects. Finally, it should be noted that the reference work is not given but must be identified and constructed for the specific needs of each research project.

2.2 Different Ways of Accessing the Reference MWS As set out above, the reference MWS is the Mathematical Working Space which ideally should be described and available in institutions organized around knowledge. But, how, in reality, do individuals in charge of teaching have access to it? Generally, official documents from the Ministry of Education, curricula, textbooks recommended by educational institutions, documents with instructions for teachers, books on the history of mathematics, among others can be an entry to the reference

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MWS. These texts usually provide, on the one hand, access to the knowledge to be taught, the order to be followed, the objectives and competences to be developed in the students and, on the other hand, to the development of the mathematical concept itself on the other, thus providing access to the reference mathematical working. Study of the reference MWS will also depend on the objective and the needs of the people or organizations in charge of teaching. Another point of interest is the possible influence of the reference MWS at university, which may differ from the reference MWS at high school. It is part of the hidden curriculum, especially for beginner teachers. Indeed, the mathematical work serving as a reference for these teachers is often out of step with, or even in opposition to, the work they must implement in secondary school. We will come back to this question in 4.4 with the notion of paradigm. In summary, study of the reference MWS implies clarifying the points on which the reflection will focus: mathematical domains and contents, the work undertaken by students or teachers. The objective is to access the MWS, in its complexity and diversity, in order to understand the resulting mathematical work. In this regard, we will draw on different perspectives and methods.

2.2.1

The Historical–Epistemological Approach

This approach to the reference MWS delves into the knowledge and explores the development of mathematical objects over time. In other words, it closely explores the needs, ideas, procedures, setbacks, successes, contexts and conditions in which the mathematical object in question arose, with the aim of taking a more extensive look at this knowledge, which supports the structure and order of the content proposed by the institution. This approach provides a more critical perspective, firstly, on the mathematical object of the domain in question (geometry, calculus, probability, etc.) and, secondly, on the configuration of the reference MWS. The historical–epistemological approach is linked to the epistemological plane of the reference MWS, and for this reason, the representamen, the artifacts (material, technological or symbolic) and the theoretical referential linked to the mathematical object in question are considered. It is important to know the history and development of science, in particular mathematics, the work and interest shown by different mathematicians who have contributed to the evolution of mathematics in specific domains, along with the definitions and properties in a corpus of knowledge, a grasp of which is necessary to understand and characterize mathematical work. In the domain of geometry, the reference MWS (Kuzniak, 2006, 2007) shows the importance of looking at Euclid’s geometry to understand the functioning of what one wants to teach. In particular, Borel (1905) confirms the importance of this teaching in high school, because of the value of transformations and movement in new geometry (Kuzniak, 2007, p. 15). Various mathematicians (Borel, Lebesgue, Klein and more recently Choquet, Dieudonné and Lang) have produced theses dedicated to high school students and teachers. They argue in favor of undertaking mathematical work

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as close to the mathematician’s way as possible. This influence culminated in the “modern math reform” of the 1960s. Today, it is still essential to review what has happened over time with the development of algebra and the influence this has had on the teaching of geometry. Moreover, this viewpoint proposed by mathematicians ensures epistemological vigilance concerning the content.

2.2.2

The Institutional and Organizational Approach

In this second approach, the evolution of the concept itself over time is no longer directly addressed, as in the historical–epistemological approach. This time, the focus is on how the reference MWS is transformed to make the teaching and learning responsive to educational, social and human needs. This approach can be seen from two perspectives, institutional and organizational. In the theory of MWS and following North D.C.’s distinctions (1990), organizations are considered as “groups of individuals bound by some common purpose to achieve goals,” and institutions as “any form of constraint that humans devise to shape human interaction.” Less formally, organizations can be viewed as the players in a game and institutions are the rules of the game. The relationship between institutions and the individuals who are economically and administratively dependent on them is not one of the submissions as supposed in other theories such as the Anthropological Theory of the Didactic where individuals are considered as submissive to institutions. In the theory of MWS, individuals and the organizations in which they are embedded (class, school, etc.) are naturally influenced by the institutional framework. In turn, they influence how the institutional framework evolves and change. On the one hand, the institutional approach focuses on identifying the rules established by the institution directing the work. In an educational environment, curricula, official documents which dictate the way in which teaching should be conducted, the order in which subjects should be taught, the competences to be developed, etc., all lead us into the institutional reference MWS. On the other hand, the organizational aspect includes, for example, the habits, professional associations and interactions between teachers, their pedagogical and scientific training, the documents and noninstitutional resources available, especially on the Internet, which are not included in the institutional documents, but do influence the teaching–learning process. It is important to keep in mind that educational needs are often strongly influenced by what students are socially expected to know at different educational levels (standardized tests, e.g., PISA, TIMSS).

2.2.3

The Cognitive Approach

In relation to didactic issues, the cognitive dimension in the theory of MWS is a fundamental aspect. As such, it is important to be able to access the reference MWS through the processes of visualization, construction and proof that emerge from the mathematical work found in the historical–epistemological approach or, ideally,

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proposed in the institutional and organizational approach. In other words, the cognitive approach observes the processes of development, teaching and/or learning of mathematical objects and identifies cognitive processes which characterize the mathematical work. The difficulty here lies in identifying subjects concerned by this work. Generally, they can be considered as epistemic students who are potentially able to mobilize the expected cognitive skills required to correctly produce the mathematical work expected of them.

2.2.4

Combining the Three Approaches

The three approaches are strongly influenced by the mathematical domain to which the concepts manipulated belong. In other words, each domain has generated, over time, a form of work to which epistemological, academic, social and educational factors specific to mathematics have contributed. As such, taking domain in question into consideration is important for all three approaches. It is important to point out, as we have done previously, that the reference MWS depends on the domain studied, the academic level and the individual or individuals being analyzed. In other words, the interpretation of the reference MWS of a given mathematical notion or topic will depend on the teachers’ degree of expertise. Are they in initial training or experienced teachers? Likewise, the interpretation of the reference MWS may vary according to the person in charge of its implementation. Do they have a high academic level in mathematics or not? These differences in the teachers’ own personal MWS account for the variety of interpretations given to the reference MWS and furthermore explain the difficulties encountered in properly capturing it in research.

2.2.5

Three Examples

To illustrate the above, we present the following research in which an analysis of the reference MWS was carried out. Different educational contexts were studied for the three investigations: density function with twelfth grade students in France; continuous functions with trainee teachers in Chile; and numerical functions with 12 students in Mexico. In each case, the historical–epistemological, institutional and cognitive approaches complement each other to access the reference MWS, and this access was specially designed for each of the investigations. It is worth mentioning that when working in an educational environment governed by curricula, textbooks and documents issued by the Ministry of Education, which provide guidelines, content, objectives and competences, study of the reference MWS may lean mostly toward analyzing these documents, focusing on the institutional and cognitive approaches and leaving aside the historical–epistemological approach. However, within the theory of MWS, all three approaches are equally important.

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Example 1 The density function (twelfth grade, France) Derouet (2016), in her PhD thesis, dealt with the articulation between probability and integral calculus in the last grade of high school in France (Grade 12). To obtain a comprehensive analysis of the reference MWS, she conducted a historical–epistemological study of the origin of the notion of probability density function which enabled her to identify the importance of statistics. She then analyzed institutional documents and textbooks and found that the link between density probabilities and integral calculus is imposed on students and is little used in the different tasks they have to perform. Finally, in collaboration with an experienced teacher, Derouet designed, implemented and studied an original introductory tasks using a research methodology which she named collaborative didactic engineering. As we can observe, in her work, Derouet conducted a study on the reference MWS considering the three approaches, historical–epistemological, institutional–organizational and cognitive, which provided the information and foundations necessary to formulate, develop and validate her research. Example 2 Continuous functions in teacher training (Chile) Menares Espinoza (2016) conducted a study of the reference MWS on continuous functions in mathematics teacher training in Chile. She studied the role played by the teacher in three scenarios: as a student in initial training, as an in-service teacher and as an individual mathematics problem solver. To do so, she designed a study of the reference MWS through phases: Phase 1, a study of university programs and texts used in initial training; Phase 2, study of the guideline standards for secondary school teachers (eighth to twelfth grades); Phase 3, study of the school curriculum; Phase 4, study of the texts used by secondary school teachers. For Menares Espinoza (2016), this study was very important to determine the reference MWS, through an institutional–organizational and cognitive approach. Likewise, Menares Espinoza also conducted a historical–epistemological study, less exhaustive, in order to find points in common between the evolution of the continuous functions and the educational objectives of the institution. This example shows how the study of the reference MWS depends on the research objectives and how it can be deepened through the three coordinated approaches we have outlined. In Menares Espinoza’s research, an important historical–epistemological study is produced, but the central part is the institutional–organizational approach of the reference MWS regarding the teacher in her three roles. Example 3 Numerical functions and modeling of movement phenomena (twelfth grade, Mexico) In the work by Reyes-Avendaño (2020), a study of the reference MWS of numerical functions was carried out considering the three approaches, the historical–epistemological, institutional–organizational and cognitive approach. This decision was made based on the needs of the research. In other words, Reyes-Avendaño (2020) worked on the notion of “function” which emerges from the modeling of movement phenomena (kinematics). Consequently, this work with kinematics, a domain

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in which physics and mathematics cross paths, led the author to a historical–epistemological study with the aim of finding points of intersection to gain insight into the use and development of the notion of function in both disciplines. An analysis of the curricula of the National Preparatory School (UNAM, Mexico) and textbooks provided valuable information on how the notion of function is currently taught in the case of differential and integral calculus and in physics classes. Study of the reference MWS from the three approaches provides rounded information for this research to generate, analyze and validate the modeling activities provided by the author. In this research, we can see that the three approaches were of utmost importance, since all of them together provided the author with valuable information on the development of functions over time in two disciplines from historical–epistemological, institutional–organizational and cognitive perspectives.

3 Reference MWS: Users, Targets and Usage Before presenting the tools of the theory of MWS which facilitate our understanding of the reference MWS (4.4–4.5), we will first look at who are its users. The reference MWS pilots the mathematical work that educational institutions and organizations have the charge to develop. Consequently, the main actors concerned are the teachers, researchers, authors of textbooks, curiculum makers and teachers trainers. The targets of the reference MWS are, therefore, the learners. This shows how important dependencies between the three types of MWS are. In order to determine the role of the reference MWS in networking with the two other (suitable and personal) MWS, we must bear in mind that in research conducted on the theory of MWS, the aim is to study the mathematical work presented in a school environment. For this reason, it is important to consider three aspects. The first has to do with the aims and subjects of analysis of the research: the mathematical subject and the target population. The second has to do with the available artifacts (calculators, software, drawing tools, etc.) since this can generate significant differences in the programs and their implementation. Finally, the third concerns the way in which the three types of MWS are considered. For example, we look at one of the ways in which we can access the reference MWS: through textbooks. In these textbooks, the order of the subjects to be taught, the objectives and even the competences to be developed are explicit (they can often be considered as an extension of the syllabus). However, textbooks can also be used to study the suitable MWS by analyzing the development of the subject, the exercises, activities, procedures suggested to the student and even the teaching guidelines proposed to the teacher. As such, the researcher exploring the theory of Mathematical Working Spaces should, based on the definitions of the three MWS, specify what is being analyzed and how. Another significant element is the importance and depth with which the MWS is developed. In 2.2.5, we show three investigations wherein the analysis of the reference MWS is different, and the difference lies in the nature and objectives of each investigation. In other words, in a research paper, it is not always necessary to develop all three types of MWS in the same depth. The analysis, detail and choice of one, two or all three MWS will depend on the objectives of the work or research undertaken.

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Fig. 1 Hierarchical view of the three types of MWS

To illustrate these points and uncover the interaction of the reference MWS with the other MWS, we present some diagrams. In principle (Fig. 1), we can say that a coherent and effective connection between the reference MWS and the students’ personal MWS is sought via the suitable MWS. Therefore, ideally, in the syllabus, the institutional method is to be taught. The content can be made explicit according to a historical–epistemological, institutional and/or cognitive approach to the reference MWS. Content is processed and adapted by teachers in order to be taught (suitable MWS). And finally, it is possible to analyze how learners use this content through their personal MWS. The linear behavior shown in Fig. 1 is intended to illustrate, in a general way, how to proceed in a very hierarchical educational environment without feedback. However, we know that most of the time, this is not the case, and as such, the importance of the research carried out lies in the different interactions which emerge in MWS when a phenomenon is observed. Below, we present some examples of interactions which may emerge between the MWS depending on the research conducted. In particular, we focus on the interaction between the reference MWS and the other MWS, for example, when we consider the student’s personal MWS (Fig. 2). Studying how learners understand a mathematical concept or object is the main aim of numerous studies. When research focuses on the way teachers teach a certain topic (suitable MWS), we can observe how the reference MWS guides the mathematical work, with all the richness and complexity of this space, toward what should be taught. We can also analyze the teachers personal MWS which influences how they will approach the subject in class. Consequently, although the study focuses on the suitable MWS

Fig. 2 Reference MWS and the MWS network

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Fig. 3 Reference MWS and its influence on the suitable MWS

(Chapter “The Idoine or Suitable MWS as an Essential Transitional Stage between Personal and Reference Mathematical Work”), it is important to observe how the other MWS influence it, as in Fig. 3. When using the theory of Mathematical Working Spaces, it is clear that one, two or all three types of MWS can be considered, depending on the needs of the researcher and the scope of the research. However, here, we wanted to show that it is possible to identify and characterize the mathematical work of the reference MWS a school institution wishes to develop, and with this, the influences on the other working spaces such as the suitable MWS and the student’s personal MWS. In this sense, the examples we have shown are only intended to illustrate some configurations that can be generated. However, in an investigation, these will depend on the needs and objectives of the work at hand.

4 Paradigms, Mathematical Domains and Components of MWS In the theory of MWS, describing and characterizing the mathematical working spaces that emerge in an educational environment are a fundamental process. However, its scope is not only limited to those activities, it also allows us to influence, shape and transform mathematical work through the implementation of paradigms. This section, therefore, aims to provide an understanding of how the description, characterization and design of the mathematical work in the reference MWS can be undertaken. To characterize the work, we rely on the three criteria (conformity or compliance, correctness and completeness) developed in (1.7). Mathematical work is compliant (1.7.1) when the processes and procedures used to solve tasks and problems are valid and conform to the expectations present in the work paradigm or paradigms favored in the reference MWS. The work is said to be correct when the results obtained are exact according to the mathematical point of view retained (1.7.1). Finally, the mathematical work is complete when its flow is ensured between all dimensions and components of the MWS diagram (1.7.2).

These criteria are defined according the paradigms and circulation of work within and between domains.

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This section describes the paradigms, mathematical domains and components of the MWS diagram.

4.1 The Notion of Paradigm in MWS Theory To understand how the concept of paradigm is used in the theory of Mathematical Working Spaces, we first set out some concepts proposed by Kuhn, then discuss how they are considered and transformed in the MWS. Kuhn (1966) defines two important concepts he adapts to the MWS (Reyes-Avendaño, 2020): paradigm and scientific community. The concept of paradigm for Kuhn (1966, pp. 175) has two meanings. The first concerns "the set of beliefs, values, techniques, etc. shared by the members of a given community," and the second refers to a kind of element belonging such a community. For example, the concrete solutions to a problem which, when used as models or examples, can replace explicit rules as a basis for solving the remaining problems of normal science.1 The scientific community for Kuhn (1966) consists of individuals practicing a scientific specialty. These individuals have developed in a similar academic environment and professional initiation. As such, the members of this community have absorbed the same technical literature and drawn many identical lessons from it. It is important to mention that, for Kuhn, the process of paradigm change occurs when the current paradigm is no longer capable of explaining new phenomena or of generating knowledge. It is at that moment that a "crisis" is generated in which what he calls “new schools” emerge: communities of scientists who leave the paradigm to explore new theories. The new paradigm is consolidated when one or more of these schools contribute one or more new theories which engender the development of more knowledge. For Kuhn, when one moves from one paradigm to another, the first paradigm disappears because the new paradigm can explain all the problems of the previous paradigm as well as what may arise within the new paradigm. It should not be forgotten that Kuhn’s philosophy is framed in the sciences, principally in physics, and the question of whether it can be applied to mathematics has been the subject of much debate (Gillies, 1992) in which ideas, and especially his conflicting and revolutionary views on paradigms, are contested. Reyes Avendaño (2020) describes how the concepts of paradigm and scientific community are adapted within the theory of Mathematical Working Spaces which, and it is important to keep in mind, is used within an educational and nonscientific environment. Moreover, communities and organizations involved in both environments—education and research—are different in their composition, aims and purposes. This implies that scientific development does not occur in the same way as educational development. In this sense, the paradigm in MWS will be the set of ideas, values, types of thinking, techniques, etc. that determine a certain stage of the domain 1

Normal science for Kuhn (1966) is the scientific development produced within the framework of the dominant paradigm.

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with which one is working. In other words, paradigms in the MWS are not eliminated when one moves from one paradigm to another, as in Kuhn’s concept. In the MWS theory, the paradigms have been proposed according to a historical–epistemological and didactic study of the domain being worked on (geometry, analysis, probability, kinematics, etc.), with the aim of grouping knowledge and cognitive processes (visualization, construction and proof) in each paradigm. The purpose of the paradigms, in MWS, is to define more precise characterizations of the work (mathematical or from other disciplines) which emerges in an educational environment. In the MWS, the scientific community is considered responsible for generating and developing the knowledge upon which some of the paradigms of this didactic theory are based. But, the MWS also considers the school community, made up of teachers and students who, within the MWS, make use of the knowledge in the paradigms for educational purposes. Thus, the paradigms in the theory of MWS are not the same as Kuhn’s, and they help us to study and characterize the mathematical work, or that of another discipline, which is under analysis, based on the type of developments, artifacts, proofs, symbols and constructions, etc.

4.2 Changes and Articulation Between Domains In 1.5.2, the need to consider relationships and changes between mathematical domains was highlighted. Indeed, a mathematical task is frequently set across one or two mathematical domains, and above all, it may be necessary in the course of the work to change domain. This may be for reasons related to the programming of the teaching or simply for mathematical reasons in order to be able to carry out the task. The existence and possibility for these changes to occur are closely associated with the reference MWS. This can be seen, for example, in geometry, with the introduction of analytic geometry which favors a combination of geometric and algebraic work. In Montoya Delgadillo and Vivier (2014), the authors introduce the idea of a source domain, the main domain of the statement and a resolution domain. They also insist on the necessary division of the major domains of mathematics into smaller subdomains that will help structure the larger domain. It is, thus, possible to distinguish different levels of possible change: • a change of partial MWS within the same domain: MWS of functions and MWS of sequences in analysis • a change of MWS each associated with a domain or more simply a change of domain; MWS of geometry, MWS of algebra • a wider change referring to fields or disciplines on the margins of mathematics such as kinematics or algorithmics. These changes have become increasingly important due to both institutional choices, modeling as well as the societal impact of new digital tools

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Montoya Delgadillo and Vivier limit the scope of their research to two domains, but more complex cases exist involving three domains, as in Derouet’s study (Example 2).

5 Describing and Characterizing the Reference MWS The theory of MWS is made up of several components represented in a diagram (Fig. 4) which facilitates the visualization of the processes which emerge during the mathematical work. On the one hand, there is the epistemological plane (3.2.1) made up of elements proper to mathematics itself: representamen; artifacts (material, technological and symbolic) and the theoretical referential which is the theoretical support of the mathematical work. On the other hand, there is the cognitive plane, which emerges upon entering a teaching–learning process. The cognitive plane (3.2.2) is made up of: visualization, a cognitive process associated with the identification and development of a set of procedures and transformations of the representations; construction, which is linked to the actions triggered by the use of artifacts (material, technological or symbolic); and proving, which is based on deductively organized arguments. These components can interact, giving rise to three geneses: semiotic, instrumental and discursive. To visualize the characterization of the mathematical work which emerges in the reference MWS, we will use each of the elements shown in the diagram (Fig. 4)

Fig. 4 Diagram of the MWS

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and the notion of paradigm used in this theory. In principle, it is important to understand that a type of Mathematical Working Space is generated in each paradigm where the epistemological elements, cognitive processes, genesis and vertical planes are different. Therefore, in very general2 terms, paradigm I is usually composed of processes linked to reality and tangible aspects; i.e., the work triggered is not very abstract. Paradigm II introduces theoretical elements, giving an account of reality, more or less accurately, with an operational aim (to make calculations or geometric constructions or graphs). There is a certain distance from reality, however, the objects originated therefrom. Finally, Paradigm III proposes a theorization which can have a significant distance with reality. Moreover, this paradigm does not necessarily have the ambition to rely to reality but focus on logical organization. It is pertinent to say that this description of the paradigms is very general and is only intended to give an idea of their evolution and their influence on the characterization of the reference MWS. In other ways, each paradigm3 generates characterizations of a different nature because the representations, artifacts and theoretical referential, in each of the paradigms, evolve and, for this reason, trigger different cognitive processes and geneses.

5.1 A Global Vision on Mathematical Work To show how the paradigms structure the reference MWS, we explore some of the works developed. The same effort is made in geometry and analysis in Chapter “Personal Mathematical Work and Personal MWS” (Menares and Vivier). Parzysz (2014) highlighted how, in the teaching of probability in France, the reference MWS is based on paradigms. In Junior High School up to ninth grade, teaching of probability is based on the Probability 1 (P1) paradigm. The aim of this is to describe a precise experimental protocol associated with a particular experiment, thus ensuring the reproducibility of the experiment under the same conditions "leading to observations that allow a probability of occurrence to be attributed to each of the different outcomes." The tools associated with this paradigm are diagrams such as: trees, double-entry tables and descriptive statistical diagrams (bar charts, histograms, etc.). It gives a horizon of experiments and generates hypotheses consistent with the results of the experiment. Then, in Senior High School (Grades 10–12), the leading paradigm is Paradigm 2 (P2). The generic random experiment and the concept of probability are introduced and defined. The study of the properties of this probability uses concepts and properties which are similar in appearance, to classical models (urn model) and the main laws of probability (binomial, exponential, geometric, etc.). The tools associated 2

In order to identify the exact evolution of the cognitive elements and processes considered in the paradigms of a particular domain, it is necessary to verify them directly in the theory of MWS theory for the domain in question. 3 Paradigms and their evolution depend on the domain being worked on.

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with this paradigm are computational techniques integrated in the models or specific to the various representation registers (trees, double-entry tables, weighted, etc.). Descriptive statistics (DS) and inferential statistics (IS) provide additional tools. This probability gives a theoretical horizon to the experiment and modeling in a proto-axiomatic system.4 Finally, the third paradigm (PS) is introduced to mathematics students at university. It is based on a Kolmogorov-type axiom with a hierarchy of properties. This probability gives an axiomatic and formal horizon. It is fundamentally related to analysis and especially the Lebesgue measure.

5.2 Artifacts, Signs and Paradigms in Kinematics Reyes-Avendaño (2020) defines the paradigms of kinematics, which she uses, like Blum and Leiss (2007), in a modeling cycle to design and analyze four activities for modeling motion phenomena. In this research work, the paradigms facilitated the transition between the phases of the modeling cycle to be gradual. As such, ReyesAvendaño makes use of three types of artifacts, each engendering work in a certain paradigm: • Material artifacts, such as balls, toy cars, ropes, balloons, were used to perform the movements and encourage work in the Kinematics-I paradigm (real paradigm, KI). In this paradigm, explanations are expected from a non-mathematical cognitive structure, rather based on essentially sensory aspects (perception, manipulation, etc.) recreated from the physical phenomenon, for example, how is the movement? Is it moving fast? Too slow? Is it uniform? • Technological artifacts, Tracker and GeoGebra, which were used by the author to lead students to abstract physical movement in two stages. The first (Tracker) is using graphs of position versus time, velocity and acceleration. Use of this artifact triggered work essentially in the Kinematics-II paradigm (measurement and quantification paradigm, KII) in which it is important to have the corresponding information of the situation studied to be able to express or analyze it in terms of elements, variables, properties, laws, etc. belonging to the field of physics (speed, acceleration, trajectory, etc.). The second (GeoGebra), by obtaining the parametric equations of motion and the trajectory equation, is providing entry to mathematical work (Paradigm KIII). • Symbolic artifacts were used by the author to promote mathematical work, paradigm Kinematics-III (paradigm of formalization, KIII). This paradigm is characterized by reasoning that includes chains of arguments in a symbolic language promoting intellectual proofs (Balacheff 1987). 4

In the recently updated French curricula, programming with Python (or other programming languages) is prescribed to address probability, which generates a change in the MWS reference in this domain.

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The author points out that the cognitive processes triggered do not always depend on the artifact used but on the learner’s knowledge and, probably, on other educational and social factors. However, artifacts can encourage certain types of work and cognitive processes. Considering all of the above, Reyes-Avendaño (2020) designed these modeling activities for her thesis with the aim of encouraging the transition between the three geneses (semiotic, instrumental and discursive) of each of the paradigms.

5.3 In Search of the Theoretical Referential in Geometry Cyr (2021) explores the geometry referentials used in secondary schools in Quebec with a view to implementing a referential in geometry learning software (QEDTutor). The official Quebec curricula provide only few indications on this point, which has led Cyr to study all the textbooks currently in use to identify the reference system used. Knowledge of the reference system enables access to the reference work, particularly by determining the favored geometrical paradigms. Cyr examines all the textbooks in a comprehensive manner, extracting the properties and definitions. From this study, he was able to extract what he calls “primary concepts” (e.g., angle, as there is a first definition of this concept in the textbook) and secondary concepts associated with the primary concept (measurement of the angle, half-lines defining the angle, etc.). He also looked at the link made between the figure and the text, from which he observed the semiotic dependence. He also introduced the epistemic value, which indicates whether the property stated has been justified and in what way. Based on this comprehensive study, he explored three questions: What are the geometric referentials used in secondary school textbooks? What are the discursive and paradigmatic characteristics of these referentials? How can these referentials be adapted to a theoretical referential frame of reference implemented in the learning software? He was thus able to identify several types of referentials which he ranks, from a reference system completely set out in the course part of the book, to a minimalist reference system which must be extracted from the exercises. Among these different referentials, he focuses on an axiomatic referential composed only of axioms, which exercises enrich with proprieties. There is also a conjectural referential without any formalization and based solely on conjectures of properties, validated by technological tools such as dynamic geometry software. In this case, the construction of the referential is left to the learner who does not necessarily demonstrate the properties in a classical and deductive manner. Finally, he identified the implicit referential: For this, there is no referential in the book, and it is assumed that this knowledge is known in advance. This variety of referentials is reflected in great diversity between the paradigms of geometry introduced by Houdement and Kuzniak (1999):

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Geometry I (natural geometry), in which there is a relationship with the real world and the source of validation is based on tangible aspects. Geometry II (natural axiomatic geometry), in which the relationship with reality is lost and the work is rather based on the geometrical model, while the source of validation is based on axiomatic rules. Geometry III (formal axiomatic geometry), where a complete separation from the real world is generated and the sources of validation are based on the chosen axiomatics.

5.4 Articulating Domains to Transform Mathematical Work In the research described in Example 3 (4.2.2.5), Derouet (2019) proposes an analysis of the reference MWS around the notion of density probabilities. She studied the official programs set out by the Ministry of Education, but in order to go further and prepare her own engineering teaching, she systematically studied the way in which the notion of density was introduced in French final year textbooks. In each case, she looked at: • the three mathematical sub-domains involved and the possible changes of subdomains • the tasks and sub-tasks set • the different representations used On this basis, she studied the geneses likely to be mobilized to bring about changes between the three MWS associated with the mathematical sub-domains in question. This analysis enabled her to demonstrate the important role of the notion of “area” in the approach to density probabilities. The use of the area was different according to the choices made by the authors of the textbooks, as these choices are not explicit in the official programs. For example, they may have started by introducing density probabilities before integral calculus. But sometimes, the reference MWS does not guide interpretation on the order, e.g., students studying sports at university must know and be able to use the normal distribution, but the integral calculus is not on the syllabus. Therefore, the reference MWS promotes entry by area and the use of tables or software which give the values of the normal law.

6 Conclusion This chapter has served to explore and clarify the reference MWS. It has situated its role within the theory, uses and interactions with other MWS and paradigms. The aim has been to give users a clear idea of the role of the reference MWS and thus enhance its use and application. One of the purposes of developing this chapter was to explain that access to the reference MWS does not necessarily occur through an institutional

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approach, as is usually seen in research papers. In the field of mathematics education, we deal with phenomena which emerge in an educational environment governed by institutions. As such, the most immediate and easy thing to do is to observe the Mathematical Working Space proposed by these. However, as explained throughout this chapter, the reference MWS is much broader than merely the institutional dimension even completed by the organizational viewpoint. To be fully accessed, historical–epistemological and cognitive studies are needed to characterize it according to paradigms, genesis, etc. Throughout this chapter, we have seen how the reference MWS in a given mathematical domain is difficult to identify. Now, we outline some ways for the community of researchers in mathematics education to participate in the evolution of the reference mathematical work. We suggest starting from the study of the development of the theoretical referential, which reflects the epistemology of the field being taught. From there, it is possible to develop the network of artifacts and signs most adequate for shaping mathematical work according to the context and the educational level concerned. The structuration of the cognitive plane and of the associated MWS then requires the design and implementation of a succession of emblematic tasks. Under these conditions, the reference MWS can be changed, as shown by Reyes’ research in kinematics, or Derouet’s work on the link between probality and integration or Montoya Delgadillo and Vivier’s (2016) study of the introduction of the exponential from modeling, etc. A more ambitious aim is to influence further mathematical domains globally. Last but not least, the theory of MWS continues to evolve and has expanded to other disciplines such as physics, algorithmics, engineering and statistics, engendering the development of different working spaces, fields and paradigms and opening up horizons and research perspectives which further enrich this theoretical framework.

References Balacheff, N. (1987). Processus de preuve et situations de validation. Educational Studies in Mathematics, 18(2), 147–176. Blum, W., & Leiss, D. (2007). How do students and teachers deal with modelling problems? In Mathematical modelling (pp. 222–231). Elsevier. Borel, E. (1905). Géométrie. Paris: Armand Colin. Cyr, S. (2021). Étude des référentiels de géométrie utilisés en classe de mathématiques au secondaire. Université de Montreal, Montreal, Canada. Derouet, C. (2016). La fonction de densité au carrefour entre probabilités et analyse en terminale S. Etude de la conception et de la mise en œuvre de tâches d’introduction articulant lois à densité et calcul intégral (Doctoral dissertation, Université Paris Diderot (Paris 7) Sorbonne Paris Cité). Derouet, C. (2019). Introduire la notion de fonction de densité de probabilité : Dynamiques entre trois domaines mathématiques. Recherches En Didactique Des Mathématiques, 39(2), 213–266. Gillies, D. (Ed.). (1992). Revolutions in mathematics. Oxford: Oxford Science Publications, The Clarendon Press.

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Houdement, C., & Kuzniak, A. (1999). Un exemple de cadre conceptuel pour létude de l’enseignement de la géométrie en formation des maîtres. Educational Studies in Mathematics, 40(3), 283–312. https://doi.org/10.1023/A:1003851228212 Kuzniak, A. (2006). Paradigmes et espaces de travail géométriques. Éléments d’un cadre théorique pour l’enseignement et la formation des enseignants en géométrie. Canadian Journal of Science, Mathematics and Technology Education, 6(2), 167–187. https://doi.org/10.1080/149261506095 56694 Kuzniak, A., Tanguay, D., & Elia, I. (2016). Mathematical working spaces in schooling: An introduction. ZDM Mathematics Education. https://doi.org/10.1007/s11858-016-0812-x Kuhn, T. S. (1966). The structure of scientific revolutions (2nd ed.). Chicago: University of Chicago Press. https://doi.org/10.1119/1.1969660 Kuzniak, A. (2007). Sur la nature du travail géométrique dans le cadre de la scolarité obligatoire. In G. Gueudet (Ed.), Nouvelles perspectives en didactique des mathématiques. La pensée sauvage. Menares Espinoza, R. (2016). Estudio del Espacio de Trabajo del Análisis de profesores de matemáticas en Chile: El caso de las funciones continuas. Vaparaíso: Pontificia Universidad Católica de Vaparaíso. Montoya Delgadillo, E., & Vivier, L. (2014). Les changements de domaines dans le cadre des Espaces de Travail Mathématique. Annales De Didactique Et De Sciences Cognitives, 19, 73–101. Montoya Delgadillo, E., & Vivier, L. (2016). Mathematical working space and paradigms as an analysis tool for the teaching and learning of analysis. ZDM Mathematics Education, 48(6), 739–754. https://doi.org/10.1007/s11858-016-0777-9 North, D. C. (1990). Institutions. Cambridge: Cambridge University Press. https://doi.org/10.1017/ CBO9780511808678 Parzysz, B. (2014). Espaces de travail en simulation d’expérience aléatoire au lycée : Une étude de cas. Relime, 17(4), 65–82. https://doi.org/10.12802/relime.13.1743 Reyes Avendaño, C. (2020). Enseignement et apprentissage des fonctions numériques dans un contexte de modélisation et de travail mathématique. PhD thesis. Paris: Université de Paris.

Personal Mathematical Work and Personal MWS Romina Menares Espinoza and Laurent Vivier

1 Introduction: What is the Personal MWS? The theory of MWS distinguishes three levels for understanding the complexity of mathematical work in a school context. The reference MWS (Chapter “The Reference Mathematical Working Space”) is a characterization of mathematics and mathematical work specified by an educational organization: The epistemological plane is strongly based on mathematics, and the cognitive plane specifies the way in which students are supposed to understand and undertake the mathematics required within the organization in which they study. The suitable MWS (Chapter “The Idoine or Suitable MWS as an Essential Transitional Stage Between Personal and Reference Mathematical Work”), on the other hand, organizes the mathematical work, in line with the reference MWS, to be proposed to students. The aim of the teachers who shape the suitable MWS is that their students carry out mathematical work corresponding to the reference work. It is at this level of the suitable MWS that mathematical learning takes place. Indeed, students face a mathematical task with their own knowledge and cognitive processes, which are shaped by their personal MWS. But, the work performed may not be in line with what is expected and defined by the educational organization. The aim of implementing the suitable MWS is, therefore, to align the students’ personal MWS more closely with the expectations of the teaching organization. The work carried out, which produces tangible outputs, enables assessment of its conformity with the expected work (Chapters “The Reference Mathematical Working Space” and “The Idoine or Suitable MWS as an Essential Transitional Stage Between Personal and Reference Mathematical Work”). It is, therefore, a gradual R. Menares Espinoza Universidad de Valparaíso, Valparaíso, Chile e-mail: [email protected] L. Vivier (B) LDAR, Université de Paris, Paris, France e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 A. Kuzniak et al. (eds.), Mathematical Work in Educational Context, Mathematics Education in the Digital Era 18, https://doi.org/10.1007/978-3-030-90850-8_5

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process which assumes that the personal MWS is constantly evolving—the opposite should be avoided, as this would mean no learning is taking place. How can we take into account the dynamics, the geneses, of personal MWS? This is a central issue in mathematics education which often arises regarding questions of temporality, as learning often requires a long period of time. Nevertheless, having photographs or snapshots of a MWS, even partial, is very useful for understanding how students conceive the mathematical work proposed. This enables the identification of certain learning, assessment of compliance and insight into implicit elements in the reference MWS. The personal MWS does not necessarily have epistemological or cognitive coherence, unlike the reference and suitable MWS, if these are well designed. Of course, there may be gaps, such as not knowing a definition, not knowing how to use a spreadsheet, not knowing algebraic writing, not visualizing the fact that one-third has an unlimited decimal expansion, not being able to perform a two-digit addition with carry over, not knowing that the sum of the angles of a triangle has a measure equal to 180°. But, there may also be internal inconsistencies in the personal MWS or elements of the personal MWS which are not coherent with the reference or suitable MWS. There may also be disconnections1 between different parts of the personal MWS. Identifying personal MWS is crucial in order to identify obstacles, misunderstandings, misconceptions, theorem-in-act, etc., so that we can understand students’ difficulties and adapt the suitable MWS to develop a coherent, consistent and conforming personal MWS. Compliance of personal MWS with the reference MWS (and thus with the suitable MWS) is one aim of the educational organization. However, for the same mathematical task, the mathematical work carried out by the subjects, and thus their personal MWS, significant differences in mathematical richness can be observed: On the one hand, for subjects who, as a result of previous teaching, recognize the task, it may be a standard, routine operation, conforming to the reference MWS, or on the other hand, for those subjects unfamiliar with the type of task, the work is potentially richer and more complete. We can hypothesize that—and this is what makes mathematics so interesting—only complete work is meaningful and engenders learning. This has consequences on the succession of phases of the suitable MWS: first, propose a complete, possibly partial, exploratory work, before routinizing the procedures for solving standard tasks. This chapter begins precisely with a reflection on the influence of reference MWS on personal MWS. It is relevant to refer at this point to paradigms as a tool for comparing reference MWS with personal MWS: We specifically refer to paradigms in geometry and calculus. The chapter then presents two examples of research wherein mutual influence between those MWS can be observed and an example of the conformity of personal MWS.

1

It is difficult to know whether there was never a connection or whether a connection existed and was lost.

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The next Sect. 3 is dedicated to further elaborating on the personal work and dynamics generated in the MWS diagram. In this section, we refer to the idea of directed vertical planes and gaps that can be found in the study of the personal MWS, specifically inconsistencies, oppositions and disconnections which can occur. In the fourth Sect. 4, we look at the diversity of work that can be found for a given task analyzed in an investigation, showing the richness of the mathematical work produced by students during completion of a non-routine task. Section 5 ends by discussing questions about the personal MWS of pre-service and in-service teachers, such as the influence of reference and personal MWS on their work, and finally, a study on the analysis by pre-service teachers of school students’ MWS is presented.

2 Influence of Reference MWS on Personal MWS It is clear that the choice of curriculum, and thus the content taught, influences the tasks and the type of work required: Solving a task on derivation can only be asked if the notion of derivative has been introduced! Thus, this issue relates more to comparison of reference MWS (Chapter “The Reference Mathematical Working Space”), which is not addressed in this chapter. This section focuses on learning and especially on the work actually undertaken, with two examples where an influence of the reference MWS on the personal MWS is observed. This section begins with the notion of paradigm (Chapters “The Theory of Mathematical Working Spaces—Theoretical Characteristics” in Sect. 3.2 and Sect. 7.1 and “The Reference Mathematical Working Space” in Sect. 4.1), enabling us to explicitly set out the expectations of mathematical work in relation to the reference MWS and to identify gaps in the students’ actual mathematical work. Then, two types of influences are illustrated through a task given to students of two different countries; hence, each has a distinctly different reference MWS: (1) For the same geometrical task, French and Greek students produce different work demonstrating different knowledge; (2) For the same task using the same knowledge, absolute value, Cypriot and Turkish students demonstrate different degrees of success due to different definitions. This section ends with the notion of compliance of students’ personal MWS to the reference MWS.

2.1 Paradigms: A Tool to Compare Reference, Suitable and Personal MWS Paradigms help us to understand mathematical work in a global way, but we must insist upon the fact that their purpose is not to place individuals in pigeonholes, because often a combination of two or more paradigms, more or less connected (with possible inconsistencies), emerges in the work. These are also a source of possible

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misunderstandings between teachers and students, when the former and latter think according to different paradigms.

2.1.1

Geometrical Paradigms

We present geometrical paradigms using the “Charlotte and Marie” task (Kuzniak & Rauscher, 2011) (Fig. 1). The question in this task is: “Why can we assert that the quadrilateral OELM is a rhombus? Marie maintains that OELM is a square. Charlotte is sure this is not true. Who is right?”. Obviously, one perceives a square. This can be justified by using a square on the figure to check the right angle. However, Pythagoras theorem may be used to conclude that it is “not a square,” since the square root of 32 is not equal to 5.6. The first answer is typical of the Geometry-I paradigm, while the second is typical of the Geometry-II paradigm: Geometry-I, or natural geometry, finds its validation in the material and tangible world. In this geometry, valid assertions are generated using arguments based upon perception, experiment and deduction. Geometry-II, natural axiomatic geometry from Euclidean geometry, is built on a model of reality in which axioms are linked to the perception of space. There is also a third paradigm, Geometry-III, formal axiomatic geometry which is not discussed in this chapter.

Of course, the aim is not to classify students’ work according to Geometry-I or Geometry-II. Their work can change depending on the tasks they are set, but paradigms help in identifying mathematical works generally, in relation to the reference MWS. For geometry, in France, Geometry-I is the paradigm of elementary school and Geometry-II the paradigm of the end of junior high school. It is interesting to investigate the GI-GII transition, with a necessary overlapping between the two paradigms. Fig. 1 Charlotte and Marie figure

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Analysis Paradigms

Many tasks and methods deal with solutions of equations such as f (x) = 2, for a real function f . The work may be grounded in a graphic representation of f , visualizing how the curve and line y = 2 have a common point, or visualizing a table of values of f . But, one may conclude, using the intermediate value theorem (IVT) if f is a continuous function. Note that, it is possible to use this theorem without knowing the mathematical proof, relying on completeness of R. If so, properties and theorems could be used without knowing the nature of the mathematical objects in question. The first type of answer is typical of the A-I paradigm, while the second is typical of the A-II paradigm, and the proof of the IVT theorem relies on the A-III paradigm2 (Montoya Delgadillo & Vivier, 2016): [A-I] enables interpretations with implicit assumptions based on geometry, arithmetic calculations or the real world. [A-II] wherein the rules of calculation are defined more or less explicitly and are applied independently of reflection on the existence and nature of the objects introduced. [A-III] is characterized by work involving approximation and neighborhoods, even topological characteristics; definition and properties are set theoretically, engendering “ε work” specific to this paradigm: boundaries, inequality, “negligible” quantities.

2.2 First Type of Influence: Different Knowledge Within the Reference MWS In order to solve a task, students must be able to identify a method or strategy in their personal MWS. We propose here a study (Nikolantonakis & Vivier, 2014) on geometrical proofs with identical tasks given to pre-service elementary schoolteachers in France and Greece to complete, while the reference MWS and suitable MWS were different. How does this influence the work of students in both countries? The geometrical tasks were designed for Greek students and translated for French students. So, there was a good match with the Greek reference MWS, less so with the French reference MWS. The didactical contract, at this level, is clear, and the expected response lies in the Geometry-II paradigm. The experiment involved 26 French students, two weeks before the competitive examination to become a schoolteacher, and 100 Greek students—the tasks were proposed during an examination. The reference MWS in Greece strongly emphasizes use of theorems about isometric and similar triangles, while transformations play a minor role. The opposite was true for the reference MWS in France. We focus on the following two exercises3 (Figs. 2 and 3). The task of comparing triangles is typical in Greece, and the student is expected to implement one of the three congruence of triangles cases, using opposite angles. 2 3

In this paper, p.742, these three paradigms are referred to as AG, AC and AR. In Nikolantonakis and Vivier (2014), Exercises A and B are named Exercises 3 and 4.

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On the median (AM) of a triangle ABC, let us consider point S so that MS=MA. a. b.

Compare triangles ABM and SCM. Compare segments [AB] and [SC].

Fig. 2 Exercise A, figure not given to students

ABC is an isosceles triangle (AB=AC). Let us consider two points D and E on the straight line (BC) so that BD=CE and do not belong to [BC]. Prove that the triangle ADE is an isosceles triangle. Fig. 3 Exercise B, figure not given to students

For students in France, as this task comprised an unusual question of “comparing” triangles, the students had to first interpret the question in order to answer it and were rather expected to use the parallelogram ABSC or the central inversion with respect to M, quite representative of the reference MWS. In this exercise B, there is no reference to cases of congruence. However, one can use them (with triangles ACE and ABD) or the properties of axial symmetry (perpendicular bisector of triangle ABC from A) or the characteristic property of isosceles triangles with the perpendicular bisector from the apex. This exercise is more difficult than the previous one, notably because using these notions does not directly give the answer, in contrast to Exercise A. Table 1, which presents the students’ answers, shows significant use of the congruence of triangle cases by the Greek students, even more so in Exercise B than in Exercise A. This statistically reflects the knowledge within the reference MWS. French students used more varied procedures involving diverse knowledge. This is interpreted by the authors as a flexibility of the personal MWS among the French students compared to the stereotyped personal MWS among the Greek students. However, the Greek students had more success with this task than the French, especially for Exercise B. We can see an influence of the reference MWS here:

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Table 1 Students’ answers Exercise A

NA

OK

Congruence

Central inv

Parallelogram

Greek (100)

12%

77%

86%

0%

0%

other 0%

French (26)

3(12%)

18(69%)

14(54%)

14(54%)

8(31%)

2(8%)

Exercise B

NA

OK

Congruence

Axial sym

Perp. bisector

other

Greek (100)

12%

50%

88%

0%

1%

1%

French (26)

4(15%)

5(19%)

3(12%)

2(8%)

4(15%)

14(54%)

NA no answer, OK good answer; Congruence use of triangle congruence; Central inv. use of a central inversion; Axial sym. use of an axial symmetry; Parallelogram use of a parallelogram; Perp. bisector use of a perpendicular bisector

• the task being well identified in the Greek reference MWS, with adequate knowledge which leaves no possibility for other knowledge than the congruence of triangles; • the task not being clearly identified in the French reference MWS, the French students, therefore, chose knowledge in their personal MWS, resulting in a diversity of approaches.

2.3 Second Type of Influence: Different Definitions of Absolute Value in the Reference MWS What is the influence on personal MWS when, for the same mathematical notion, the curricula of two countries make different choices? To discuss this point, we present the work of Elia et al. (2016), who focused on students’ personal MWS on absolute value in two different countries, each having its own definition. Elia et al. (2016) compared Turkish ninth grade and Cypriot eleventh grade students’ personal MWS on absolute value. The reference MWS (and thus the suitable MWS) is very different for each. In Turkey, absolute value is first introduced as the distance to zero, while in Cyprus, it is first introduced as a number without a sign. The formal definition |x| = x if x ≥ 0 and −x if x < 0 is taught to the students of the study. These authors explored the influence of reference MWS on personal MWS. A questionnaire was developed, comprising questions A to I, and distributed to 289 Turkish and 135 Cypriot students. The authors found that 41% of the Turkish students gave the first definition taught in Turkey compared to only 16% who gave “a number without sign.” Similarly, 59% of Cypriot students gave the first definition taught in Cyprus, and only 12% gave “the distance to zero.” The formal definition, introduced during the year, was given by 7% of Turkish students and 10% of Cypriot students. Without too much surprise, then, the theoretical referential is shown to be statistically different between the students of the two populations, in accordance with the reference MWS. However, we note the strong presence of the first definitions taught and the weak presence of the new definition. But, it is rather the influence on all personal MWS and on the work done

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Table 2 Rate of correct answers to Questions B to I, Question E asked for the sign of −x; this is not discussed here (Elia et al., 2016, p. 900) Sample

Def1 Def2 Def3 B

Turkish 9th grade students

7

Cypriot 11th grade students 10

C

D

E

41

16

64 29 29 9

12

59

79 39 7

F

G

H

I

n

55 30 19 20 289

31 56 18 2

26 135

which is interesting, showing that the choices made in the reference MWS are not neutral in their impact on students’ mathematical work, their learning and on their personal MWS (Table 2). Question B is an equation (|x + 3| = 2), F, C and I are inequations, the difficulty of which can a priori be ranked, since for F (|x − 3| < 3), the solution is an interval and for C and I, a union of two intervals (|x| > 5; |x − 3| > 0), which is for I, R{3}. The rate of correct answers is in line with the increased difficulty, with Cypriot students, who are older, demonstrating better success. |x − 4| = 2; |x + 2| + |x + 6| = 0) Questions D, G and H ( constitute two equations and an inequation without solution which the authors call non-typical: “Students in both countries performed much better on the typical items which refer to the definition of absolute value, rather than on the non-typical items which require discursive reasoning.“ (p. 904) For these items, invoking the positivity of the absolute value immediately led to the result for D and G, with a little more work required for H. It is noticeable that despite the two-year gap between the two populations, it was the Turkish students who responded most accurately to these three items with only slightly more difficulty than the others (they had more success with D and G than I). Conversely, the rate of correct answers by Cypriot students was extremely low. The authors explain this difference by the fact that Turkish students used an adequate definition, while Cypriot students used an inadequate one to solve nonroutine tasks requiring discursive work. This is a direct effect of a chosen reference MWS on students’ personal MWS through the choice of definitions to be taught in each country. Students are more or less successful with less technical tasks requiring a more conceptual approach depending on the definition they have been taught.

2.4 Compliance of the Personal MWS—An Example in Argentina Over the course of their higher education, individuals certainly acquire some useful techniques for dealing with a set of tasks sharing similar characteristics. When an individual is faced with a task, they generally search in their background knowledge for strategies learnt in the past during their education. We are interested in addressing the influence that the educational system has on the individual’s personal MWS and the way in which it, in turn, determines certain ways of developing a task. Learning,

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We have a rectangular metal sheet of 30 cm width and large in length. We perpendicularly fold up the edges on each side to make a gutter (see dotted lines on the figure below). For obvious reasons, both side edges of the gutter will be the same size.

How should we fold up the metal sheet to obtain a gutter with a maximum flow? Fig. 4 Gutter task

from the perspective of the reference MWS, is the possibility for students to perform a mathematical task in a way that is consistent with the expectations of the educational system. Therefore, the personal MWS developed, or rather learned, is expected to be closely aligned to the reference and suitable MWS. In the absence of a well-identified MWS, the diversity prevails for certain tasks (Sect. 4). In contrast, for a well-identified and standard task with a specific MWS, compliance prevails. The study by Montoya Delgadillo et al. (2017) focused on the work of students in a high school teacher training institute in Córdoba, Argentina, regarding an optimization task set in geometry. The task, below in Fig. 4, was given to students at three levels of study: first year (24 students), second year (15 students) and fourth year (12 students). Only the fourth year students received a course in differential calculus. The first- and second-year students submitted productions demonstrating very little formalization. Sixteen students mobilized a geometrical MWS with a computational approach to the magnitudes involved: The optimum was sought by producing several computations, sometimes with a table of values, which equates to the A-I paradigm of analysis. Other students (18) relied on a false property: Nine gave an equitable distribution and proposed a folding solution defined by the same length of the three sides (10 cm each; seven claimed that there is no variation in volume (or area); and two proposed an extreme folding (large base, small edges or small base and large edges). All of the fourth-year students mobilized an MWS of functions. They introduced a variable and a quadratic function to find its optimum: seven by derivation and five using the formula of the vertex of the parabola, for work in the A-II paradigm. The work varied very little from student to student, with a nearly identical personal MWS for this task, except for the choice between derivation and vertex. The work is standardized, fully compliant with the reference MWS. Between the second and the fourth year, such conformity is achieved on this type of problem. It is then no longer perceived as belonging to the field of geometry but rather to analysis and more particularly to an MWS of functions.

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3 The Personal MWS in More Depth When subjects take the position of solving a mathematical task, they begin to weave a varyingly complex web of processes or mechanisms to which they may have access, which may vary in terms of ways of visualizing objects, formulas they may know, definitions of mathematical objects they may consider, theorems they may remember and so on. Once the subjects put their resources into practice, selecting and articulating them, extracting results, we consider that they are developing a mathematical activity, which, depending on each case, can reach varyingly complex levels. When we analyze the personal MWS, we try to detect in detail the processes which the individual develops with the resources at their disposal. Thus, we gather information both on cognitive aspects of their activity and on aspects of their own epistemology. In the following section, we refer to the analyses in the diagram of the personal MWS, the study of which enables us to identify the geneses activated in the work, as well as the interaction between them (Chapter “The Theory of Mathematical Working Spaces—Theoretical Characteristics” in Sect. 4, Fig. 2). We also refer to the vertical planes (Chapter “The Theory of Mathematical Working Spaces—Theoretical Characteristics” in Sect. 4, Fig. 3) as a tool of analysis to study the quality of mathematical work in greater depth and the possible inconsistencies or disconnections that may appear in an individual’s mathematical work.

3.1 MWS Diagram When we study personal MWS, we try to deepen what is deemed as the interaction between a subject and an epistemological environment. More specifically, we try to appreciate the dynamics between the geneses of the MWS, which can be quite laborious as it can involve trying to understand an intricate web of the processes at work. The MWS diagram helps us to understand how the mathematical work is actually performed by individuals, by reconstructing their personal MWS. It shows different resolution strategies or deviations from what an individual is expected to perform in their mathematical activity. Advances in MWS research have shown that certain methodological decisions on the analysis of the diagram enable us to describe the activation and interaction of the geneses in a personal MWS more clearly, which leads to a greater degree of accuracy in the studies of the mathematical work being developed. A valuable tool to fully appreciate the personal MWS being carried out is the analysis undertaken prior to the task being developed (Chapter “Methodological Aspects in the Theory of Mathematical Working Spaces” in Sect. 5). This analysis considers, among other elements, strategies that individuals could hypothetically adopt, as well

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Fig. 5 Resolution by decomposition of auxiliary figure and segment

as possible difficulties they may encounter in their work. In the diagram, we identify the dynamics of successful strategies and the possible obstacles, incoherence or disconnections both in each genesis and in the interaction between them, associated with the difficulties the individual may have. An example of the dynamics which can occur in a MWS is provided by Henríquez’s doctoral thesis (2014). In her work, the author presents a previous analysis of the dynamics which could be observed when completing geometry tasks. The tasks she sets allow for resolutions using both synthetic and analytical geometry. For example, for the task: “to prove that the lines that join the midpoints of the sides of any quadrilateral form a parallelogram,” Henríquez establishes an expected strategy in synthetic geometry, where an auxiliary line is added to the geometric configuration (segment AC in Fig. 5), declaring that although it is important to construct a supplementary line, the fundamental action in this phase is to continue toward discursive reasoning. With the use of vertical planes (Chapter “The Theory of Mathematical Working Spaces—Theoretical Characteristics” in Sect. 4.1), Henríquez (2014) determines, in a first phase, a possible activity which articulates elements of visualization and artifacts, with the incorporation of auxiliary traces and the decomposition of the input figure. The author argues that this indicates the activation of the plane [SemIns]. Following this, properties of the triangle emerge, located in the theoretical referential component. In a second phase, visualization elements are coordinated, as demonstrated by the decomposition of the figure, and elements belonging to the proof component, using the properties of the triangles formed. Visualization and theoretical referential components are identified as being of greater relevance in the work to obtain an answer to the task (decomposition of the figure and use of properties of secondary elements of the resulting triangle, such as properties of segments which join the midpoints of its sides).

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Study of the dynamics in the MWS diagram and identification of the activation and interaction of the geneses raises certain questions which merit investigation. One concerns the conclusions which, as researchers, we can draw about the personal MWS the understanding of the mathematical objects used: Is it true that if the individuals manage to activate all the geneses and vertical planes and to articulate them consistently in a task in which this is required, they are demonstrating their level of understanding of the objects at hand? This leads to a further question which has been explored in some research on MWS, which is the idea of the construction of a complete MWS and the implications its development and potentiality may have on the construction of mathematical knowledge.

3.2 Identification of Personal MWS in Geometry The general question of describing and identifying personal mathematical work in geometry (and in other domains) is presented in detail in Chapter “Methodological Aspects in the Theory of Mathematical Working Spaces”. The method they develop is based on a cognitive analysis of the actions of a subject performing a geometry task (Kuzniak & Nechache, 2021). In this section, we deal with this issue again using the example of students working in pairs. Coutat (2014) studied work produced on a task of reproduction of a CabriGéomètre figure by students at the end of elementary school in Switzerland, working in pairs. The task requires them to use the software to reproduce a figure consisting of a quadrilateral and two circles whose centers are one of the vertices of the quadrilateral and whose radius is the length of one of the sides not containing the centers (Fig. 6). The constraint is instrumental: Properties of the figure must not change with dragging.

Rebuild the drawing as below. Your construction must move like the model. Write down how you did it.

Fig. 6 Image of the computer screen, with the figure and the available tools

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Coutat did not analyze the students’ work with the MWS vertical planes, but these arguably provide a better understanding of the work carried out. We, therefore, specify these in the following. The first step is to identify the properties of the initial figure by dragging it using the dynamical geometry software (DGS) and visualizing it (action in the [Sem-Ins] plane and then interpretation in the [Sem-Dis] plane). Coutat interprets this phase as work with an instrument called Di, which consists of dragging to identify the invariants of the figure. Following this, students must reproduce the figure on a new Cabri-Géomètre document, particularly by choosing the way of drawing a circle, by the center and the radius or by the center and a point of the circle—the latter not being adequate. Finally, the construction must be validated by dragging in order to justify how the expected properties are indeed integrated in the figure—Coutat interprets this phase as work with an instrument called Dc, which consists of dragging in order to validate a construction. The statements given to the students specified that the new figure should behave like the initial one. In addition to a personal MWS of geometry, MWSG , producing the expected work, Coutat identifies four other personal MWSG . She, thus, highlights the importance of non-iconic visualization in relation to the use of DGS tools and the available properties of the theoretical referential. She also highlights the interest of such activities for the evolution of the MWSG . One pair of students did not use Di, so they could not identify the properties of the initial figure. They, therefore, reproduced a figure in a perceptive and iconic way which seems to be identical, with well-centered circles, but adjusted with the tool (center, point).4 Use of Dc invalidated the construction, but the pupils did not manage to identify the geometric relations of the initial figure. Two pairs of students managed, after some back and forth between the two figures, to identify that there are relationships in the initial figure, but without managing to identify which objects are linked (the link between the radius of the circles and the sides of the quadrilateral). In addition, they used the instrument (center, point) which was probably a blocking factor. Three pairs of students identified the relationships in the initial figure, using the Di, and used the instrument (center, radius).5 Nevertheless, they took the radii of the initial figure to construct their circle in the new figure. Use of Dc invalidated their constructions. Coutat concluded: Use of the circle tool with (radius, center) shows that students are unable to turn the perceptual connections of the model into relationships between geometric objects in the same figure. […] Their use of the circle tool with (radius, center) shows that students are unable to consider properties independently of the model. (Coutat, 2014, p. 133, our translation).

One pair of students perceived the properties of the initial figure, using Di, but without trying to identify them precisely. They worked on the initial figure by drawing a circle (center, radius) with any point, the radius being a length of one of the sides of the quadrilateral. They then move the circle to see whether or not it overlapped 4 5

That is, use of the DGS tool to draw a circle with its center and a point of the circle. That is, use of the DGS tool to draw a circle with its center and its radius.

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with the initial circle. If not, they changed the side to a new radius, and if yes, they reproduced the circle with the correct radius in their figure. Coutat states that the visualization: seems to rely more on the elements of the real space than on the relationships they may have. The discovery process shows an instrumented reflection on the figure. (Coutat, 2014, p. 133, our translation).

In this example, we refer to the MWSG resulting from collaboration between a pair of students as a personal MWSG . Of course, this is not to minimize the interactions between students. This importance can be particularly well seen in (Mithalal, 2014) or in (Gómez-Chacón et al., 2016). Nevertheless, we consider the MWSG identified by Coutat as personal in the sense that it might be the personal MWSG of one of the two students. This question is important and highlights one relevant to the theory of MWS: How can we talk about the MWS of a group without identifying which cognitive processes we are talking about? What are the influences of the suitable and personal MWS on the work produced by the group?

3.3 Directed Vertical Planes One of the main objectives of using the notion of vertical planes of the MWS in the study of personal MWS is to analyze the quality of mathematical activity in the twoby-two articulation of geneses. The idea here is not only to identify which geneses are activated, but also to understand how they are interrelated and the role they play. In her work, Menares (2016, 2019) proposes to study the genesis in each vertical plane by identifying a genesis, which directs the work, and thus achieve a more detailed characterization of the mathematical work. For this reason, she uses the name directed plane. The purpose here is to unravel the nature of the link, identifying a genesis that not only predominates in the work, but also commands it. Naturally, if one genesis acts as the director in a vertical plane, the other will act as the directed, with the possibility that in another phase of the work, they may exchange roles. The notion of the directed vertical plane can be illustrated through an example discussed in Menares’s doctoral thesis (2016). In this, the following calculus task is given to a group of teachers in their initial training: Does the equation sin(x) = 1–x have a solution x 0 where x0 ∈]0, π/2[? Justify your answer. Figure 7a, b show the productions of two students. For the work of Student E1, a semiotic manipulation of the equation can be identified, and with it, the formulation of a function fulfills certain characteristics. The objective of formulating such a function is to obtain the hypotheses of the intermediate value theorem, so the construction of discursive reasoning is the driving force of this work. Thus, discursive genesis is identified as the director and semiotic genesis as the directed. It should be noted that the analyses may lead us to consider the function f as a symbolic artifact, in which case it would be the instrumental genesis that is activated. This is a discussion that has not yet been settled, but in both

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Transcription Let f(x)=1–x–sin(x) a continuous function. f(0)=1–0–0>0 f(π/2)=1– π/2–1