ICME-13 Monographs. Series editor Gabriele Kaiser, Faculty of Education, Didactics of Mathematics, Universität Hamburg, Hamburg, Germany

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1 ICME-13 Monographs Series editor Gabriele Kaiser, Faculty of Education, Didactics of Mathematics, Universität Hamburg, Hamburg, Germany

2 Each volume in the series presents state-of-the art research on a particular topic in mathematics education and reflects the international debate as broadly as possible, while also incorporating insights into lesser-known areas of the discussion. Each volume is based on the discussions and presentations during the ICME-13 Congress and includes the best papers from one of the ICME-13 Topical Study Groups or Discussion Groups. More information about this series at

3 Paul Ernest Editor The Philosophy of Mathematics Education Today 123

4 Editor Paul Ernest School of Education University of Exeter Exeter UK ISSN ISSN (electronic) ICME-13 Monographs ISBN ISBN (ebook) Library of Congress Control Number: Springer International Publishing AG, part of Springer Nature 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express 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. Printed on acid-free paper This Springer imprint is published by the registered company Springer International Publishing AG part of Springer Nature The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

5 Contents A Plea for a Critical Transformative Philosophy of Mathematics Education... 1 Luis Radford Part I Introduction to the Field The Philosophy of Mathematics Education: An Overview Paul Ernest Part II The Nature of Mathematics The Who and What of the Philosophy of Mathematical Practices Jean Paul Van Bendegem The Philosophy of Mathematical Education Between Platonism and the Computer Michael Otte A Dialogical Conception of Explanation in Mathematical Proofs Catarina Dutilh Novaes The Amalgam of Faith and Reason: Euclid s Elements and the Scientific Thinker Melissa Andrade-Molina, Paola Valero and Ole Ravn Part III Critical Mathematics Education Students Foregrounds and Politics of Meaning in Mathematics Education Ole Skovsmose The Struggle Is Pedagogical: Learning to Teach Critical Mathematics Eric Rico Gutstein v

6 vi Contents Some Thoughts on a Mathematics Education for Environmental Sustainability Richard Barwell Epistemological Questions About School Mathematics Margaret Walshaw The Concept of Culture in Critical Mathematics Education Brendan Larvor and Karen François The Ethics of Mathematics: Is Mathematics Harmful? Paul Ernest Part IV Philosophical Theory in Mathematics Education Research On the Need for Theory of Mathematics Learning and the Promise of Commognition Anna Sfard On the Roles of Language in Mathematics Education Ladislav Kvasz The Separation of Mathematics from Reality in Scientific and Educational Discourse Uwe Schürmann Mathematics Education Actualized in the Cyberspace: A Philosophical Essay Maria Aparecida Viggiani Bicudo Part V Philosophy of/in Teaching, Learning and Doing Mathematics Making Distinctions: A Phenomenological Exploration in Mathematics Education John Mason Using Rules for Elaborating Mathematical Concepts Michael Meyer Towards a Wider Perspective: Opening a Philosophical Space in the Mathematics Curriculum Nadia Stoyanova Kennedy Creativity Research in Mathematics Education Simplified: Using the Concept of Bisociation as Ockham s Razor Bronislaw Czarnocha, William Baker and Olen Dias Teaching of Velocity in Mathematics Classes Chances for Philosophical Ideas Regina Dorothea Möller

7 Contents vii Time for Work: Finding Worth-While-Ness in Making Mathematics Hilary Povey, Gill Adams and Colin Jackson Hades The Invisible Side of Mathematical Thinking Walther Paravicini, Jörn Schnieder and Ingrid Scharlau Developing Rules Due to the Use of Family Resemblances in Classroom Communication Jessica Kunsteller

8 Editor and Contributors About the Editor Paul Ernest is emeritus professor at Exeter University, UK where he established and ran the masters and doctoral programmes in mathematics education including distance learning modes. His research concerns questions about the nature of mathematics and how it relates to teaching, learning and society. Inevitably this raises further questions concerning mathematics and philosophy, including ethics and social justice, which he continues to address. His publications include The Philosophy of Mathematics Education (Routledge, 1991), with citations, and Social Constructivism as a Philosophy of Mathematics (SUNY Press, 1998). He founded and edits the Philosophy of Mathematics Education Journal, now in its 28th year of publication, freely accessible via education/research/centres/stem/publications/pmej/. p.ernest@ex.ac.uk. Contributors Gill Adams Sheffield Institute of Education, Sheffield Hallam University, Sheffield, UK Melissa Andrade-Molina Faculty of Engineering and Science, Aalborg University, Aalborg, Denmark William Baker Mathematics Department, Hostos Community College, City University of New York, Bronx, NY, USA Richard Barwell University of Ottawa, Ottawa, Canada Maria Aparecida Viggiani Bicudo São Paulo State University, São Paulo, Brazil Bronislaw Czarnocha Mathematics Department, Hostos Community College, City University of New York, Bronx, NY, USA ix

9 x Editor and Contributors Olen Dias Mathematics Department, Hostos Community College, City University of New York, Bronx, NY, USA Catarina Dutilh Novaes Faculty of Philosophy, University of Groningen, Groningen, The Netherlands Paul Ernest Graduate School of Education, University of Exeter, Exeter, UK Karen François Center for Logic and Philosophy of Science, Vrije Universiteit Brussel (VUB), Brussels, Belgium Eric Rico Gutstein Department of Curriculum and Instruction, University of Illinois at Chicago, Chicago, IL, USA Colin Jackson Sheffield Institute of Education, Sheffield Hallam University, Sheffield, UK Nadia Stoyanova Kennedy Mathematics Department, New York City College of Technology, City University of New York, Brooklyn, NY, USA Jessica Kunsteller Institute of Mathematics Education, University of Cologne, Cologne, Germany Ladislav Kvasz Department of Mathematics and Mathematics Education, Pedagogical Faculty, Charles University, Prague, Czech Republic Brendan Larvor School of Humanities, University of Hertfordshire, Hatfield, Herts, UK John Mason The Open University, Milton Keynes, UK; University of Oxford, Oxford, UK Michael Meyer Institute of Mathematics Education, Cologne, Germany Regina Dorothea Möller University of Erfurt, Erfurt, Germany Michael Otte Melle, Germany Walther Paravicini Universität Göttingen, Göttingen, Germany Hilary Povey Sheffield Institute of Education, Sheffield Hallam University, Sheffield, UK Luis Radford Laurentian University, Sudbury, ON, Canada Ole Ravn Department of Learning and Philosophy, Aalborg University, Aalborg, Denmark Ingrid Scharlau Paderborn University, Paderborn, Germany Jörn Schnieder Lübeck University, Lübeck, Germany Uwe Schürmann Westfälische Wilhelms-Universität, Münster, Germany Anna Sfard University of Haifa, Haifa, Israel

10 Editor and Contributors xi Ole Skovsmose Department of Learning and Philosophy, Aalborg University, Aalborg, Denmark Paola Valero Department of Mathematics and Science Education, Stockholm University, Stockholm, Sweden Jean Paul Van Bendegem Department of Philosophy, VUB, Brussels, Belgium Margaret Walshaw Institute of Education, Massey University, Palmerston North, New Zealand