The Philosophy of Mathematics Education The Philosophy of Mathematics Education Paul Ernest © Paul Ernest 1991 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without permission in writing from the copyright holder and the Publisher. First published 1991 This edition published in the Taylor & Francis e-Library, 2004. RoutledgeFalmer is an imprint of the Taylor & Francis Group British Library Cataloguing in Publication Data Ernest, Paul The philosophy of mathematics education. 1. Education. Curriculum subjects: Mathematics. Teaching. I. Title 510.7 ISBN 0-203-49701-5 Master e-book ISBN ISBN 0-203-55790-5 (Adobe eReader Format) ISBN 1-85000-666-0 (Print Edition) ISBN 1-85000-667-9 pbk Library of Congress Cataloging-in-Publication Data is available on request Contents List of Tables and Figures viii Acknowledgments ix Introduction xi Rationale xi The Philosophy of Mathematics Education xii This Book xiv Part 1 The Philosophy of Mathematics 1 1 A Critique of Absolutist Philosophies of Mathematics 3 Introduction 3 The Philosophy of Mathematics 3 The Nature of Mathematical Knowledge 4 The Absolutist View of Mathematical Knowledge 7 The Fallacy of Absolutism 13 The Fallibilist Critique of Absolutism 15 The Fallibilist View 18 Conclusion 20 2 The Philosophy of Mathematics Reconceptualized 23 The Scope of the Philosophy of Mathematics 23 A Further Examination of Philosophical Schools 27 Quasi-empiricism 34 3 Social Constructivism as a Philosophy of Mathematics 42 Social Constructivism 42 Objective and Subjective Knowledge 45 Social Constructivism: Objective Knowledge 49 A Critical Examination of the Proposals 60 4 Social Constructivism and Subjective Knowledge 68 Prologue 68 The Genesis of Subjective Knowledge 69 v Contents Relating Objective and Subjective Knowledge of Mathematics 81 Criticism of Social Constructivism 85 5 The Parallels of Social Constructivism 89 Introduction 89 Philosophical Parallels 89 Sociological Perspectives of Mathematics 93 Psychological Parallels 101 Conclusion: A Global Theory of Mathematics 106 Part 2 The Philosophy of Mathematics Education 109 6 Aims and Ideologies of Mathematics Education 111 Epistemological and Ethical Positions 111 Aims in Education: An Overview 121 7 Groups with Utilitarian Ideologies 137 Overview of the Ideologies and Groups 137 The Industrial Trainers 140 The Technological Pragmatists 151 8 Groups with Purist Ideologies 168 The Old Humanists 168 The Progressive Educators 181 9 The Social Change Ideology of the Public Educators 197 The Public Educators 197 A Critical Review of the Model of Ideologies 214 10 Critical Review of Cockcroft and the National Curriculum 217 Introduction 217 The Aims of Official Reports on Mathematics Education 219 The National Curriculum in Mathematics 224 Conclusion 230 11 Hierarchy in Mathematics, Learning, Ability and Society 232 Hierarchy in Mathematics 232 Hierarchy in Learning Mathematics 238 The Hierarchy of Mathematical Ability 243 Social Hierarchy 248 Inter-relating Mathematical, Ability, and Social Hierarchies 254 12 Mathematics, Values and Equal Opportunities 259 Mathematics and Values 259 Anti-racist and Multicultural Mathematics Education 266 Gender and Mathematics Education 274 Conclusion 279 vi Contents 13 Investigation, Problem Solving and Pedagogy 281 Mathematics Results from Human Problem Posing and Solving 281 Problems and Investigations in Education 283 The Power of Problem Posing Pedagogy 291 Conclusion 295 References 297 Index 317 vii List of Tables and Figures Table 1.1: Proof of 1+1=2 with Justification 5 Table 6.1: A Comparison of Williams’ (Modified) and Cosin’s Groups 128 Table 6.2: The Match between Five Social Groups and Ideologies 130 Table 6.3: A Model of Educational Ideology for Mathematics 134 Table 7.1: Overview of the Five Educational Ideologies 138 Table 13.1: A Comparison of Inquiry Methods for Teaching Mathematics 286 Figure 4.1: The Relationship between Objective and Subjective Knowledge of Mathematics 85 Figure 11.1: The Curriculum and Assessment Framework of the National Curriculum 247 Figure 11.2: Correspondence between Rigid Hierarchical Theories of Mathematics Curriculum, Ability and Social Class/Occupation 255 Figure 11.3: Correspondence between Progressive Hierarchical Theories of the Mathematics Curriculum, Ability and Social Class/Occupation 256 Figure 12.1: The Reproductive Cycle of Gender Inequality in Mathematics Education 276 Figure 13.1: The Relationship between Espoused and Enacted Beliefs of the Mathematics Teacher 290 ix Acknowledgments I wish to acknowledge some of the support, help and encouragement I have received in writing this book, from a large number of friends, colleagues and scholars. A number of colleagues at Exeter have been very helpful. Charles Desforges was instrumental in helping me to conceive of the book, and has made useful criticisms since. Bob Burn has been my fiercest and hence most useful critic. Neil Bibby, Mike Golby, Phil Hodkinson, Jack Priestley, Andy Sparkes and Rex Stoessiger have all made helpful suggestions and criticisms. Neil first suggested the term ‘social constructivism’ to me, which I have adopted. Colleagues at the Research into Social Perspectives of Mathematics Education seminar, convened by Steve Lerman and Marilyn Nickson, have heard early versions of some of the theses of the book, and contributed, through their criticism. I have also learned a great deal from the papers given by others at this seminar, since 1986. Members of the group, especially Leone Burton, Steve Lerman and Stuart Plunkett, have made very useful criticisms of the text, helping me to reconceptualize the book and its purpose (but not always in the way suggested). I also owe the title of the book to Steve. Several others, such as Barry Cooper, deserve thanks for spending some time looking at draft chapters. I am grateful to David Bloor, Reuben Hersh, Moshe Machover and Sal Restivo who have each seen portions of the book and responded positively. This was most encouraging, especially since I hold their work in very high regard. Sal Restivo is also encouraging me to expand the first part of the book into a freestanding work for his series Science, Technology and Society, with SUNY Press. The tolerance and indulgence of Malcolm Clarkson and Christine Cox at the Falmer Press has been a great help, as I overshot self-imposed deadlines and word- counts. It would not have been possible for me to write this book without the support and encouragement of my family. Indeed, my intellectual progress from the absolutist position I held twenty years ago is in no small part due to my incessant arguments and discussions with Jill about life, the universe and everything. In addition, Jill has read much of the book critically, and helped me to improve it. Jane and Nuala have also been encouraging and supportive, in their own inimitable ways. Paul Ernest University of Exeter, School of Education May 1990 x Introduction 1. Rationale The philosophy of mathematics is in the midst of a Kuhnian revolution. For over two thousand years, mathematics has been dominated by an absolutist paradigm, which views it as a body of infallible and objective truth, far removed from the affairs and values of humanity. Currently this is being challenged by a growing number of philosophers and mathematicians, including Lakatos (1976), Davis and Hersh (1980) and Tymoczko (1986). Instead, they are affirming that mathematics is fallible, changing, and like any other body of knowledge, the product of human inventiveness. This philosophical shift has a significance that goes far beyond mathematics. For mathematics is understood to be the most certain part of human knowledge, its cornerstone. If its certainty is questioned, the outcome may be that human beings have no certain knowledge at all. This would leave the human race spinning on their planet, in an obscure corner of the universe, with nothing but a few local myths for consolation. This vision of human insignificance may be too much, or rather too little, for some to bear. Does the last bastion of certainty have to be relinquished? In the modern age uncertainty has been sweeping through the humanities, ethics, the empirical sciences: is it now to overwhelm all our knowledge? However, in relinquishing the certainty of mathematics it may be that we are giving up the false security of the womb. It may be time to give up this protective myth. Perhaps human beings, like all creatures, are born into a world of wonders, an inexhaustible source of delight, which we will never fathom completely. These include the crystal worlds and rich and ornate webs which the human imagination weaves in mathematical thought. In these are infinite worlds beyond the infinite, and wondrous long and tight chains of reasoning. But it could be that such imaginings are part of what it means to be human, and not the certain truths we took them to be. Perhaps facing up to uncertainty is the next stage of maturity for the human race. Relinquishing myths of certainty may be the next act of decentration that human development requires. xi Introduction Mathematics and Education How mathematics is viewed is significant on many levels, but nowhere more so than in education and society. For if mathematics is a body of infallible, objective knowledge, then it can bear no social responsibility. Thus, the underparticipation of sectors of the population, such as women; the sense of cultural alienation from mathematics felt by many groups of students; the relationship of mathematics to human affairs such as the transmission of social and political values; its role in the distribution of wealth and power; none of these issues are relevant to mathematics. On the other hand, if it is acknowledged that mathematics is a fallible social construct, then it is a process of inquiry and coming to know, a continually expanding field of human creation and invention, not a finished product.