Mathematics and Beauty AESTHETIC APPROACHES TO TEACHING CHILDREN SSiinnccllaaiirr ffiinnaall pprrooooffss..iinndddd FFMMii 66//2299//22000066 1100::5533::5522 AAMM SSiinnccllaaiirr ffiinnaall pprrooooffss..iinndddd FFMMiiii 66//2299//22000066 1100::5544::3333 AAMM Mathematics and Beauty AESTHETIC APPROACHES TO TEACHING CHILDREN N S ATHALIE INCLAIR Foreword by William Higginson Teachers College, Columbia University New York and London SSiinnccllaaiirr ffiinnaall pprrooooffss..iinndddd FFMMiiiiii 66//2299//22000066 1100::5544::3344 AAMM Published by Teachers College Press, 1234 Amsterdam Avenue, New York, NY 10027 The section in Chapter 4 entitled “The Mathematical Aesthetic in Action” fi rst appeared as an article of the same name by Nathalie Sinclair in the International Journal of Computers for Mathematical Learning, 7(1), 45−63 (2002). Reprinted with kind permission of Kluwer Academic Publishers. Copyright © 2006 by Teachers College, Columbia University All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, or any information storage and retrieval system, without permission from the pub- lisher. Library of Congress Cataloging-in-Publication Data Sinclair, Nathalie. Mathematics and beauty : aesthetic approaches to teaching children / Nathalie Sinclair ; foreword by William Higginson. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-8077-4722-3 (pbk. : alk. paper) ISBN-10: 0-8077-4722-X (pbk. : alk. paper) 1. Mathematics—Study and teaching—Research. I. Title. QA11.2.S55 2006 372.7—dc22 2006018105 ISBN-13: 978-0-8077-4722-3 (paper) ISBN-10: 0-8077-4722-X (paper) Printed on acid-free paper Manufactured in the United States of America 13 12 11 10 09 08 07 06 8 7 6 5 4 3 2 1 SSiinnccllaaiirr ffiinnaall pprrooooffss..iinndddd FFMMiivv 66//2299//22000066 1100::5544::3344 AAMM Contents Foreword by William Higginson vii Acknowledgments xi 1. Introduction 1 Where Beautiful Mathematics Comes From 1 The Aesthetic as a Vital Lens on Learning 4 Plan for the Book 9 A Background Note 12 PART I: BEAUTY AND PLEASURE IN HUMAN EXPERIENCE 2. Reclaiming the Aesthetic from the Arts 17 The Fluidity of Human Faculties 17 The Aesthetic Nature of Inquiry 21 Connecting the Aesthetic to Learning: First Steps 26 Fitting the Pieces Together 28 3. “Wired” for Beauty and Pleasure 29 Homo Aestheticus 29 A “Sense of Order” 33 PART II: BEAUTY AND PLEASURE IN MATHEMATICS 4. Developing a Mathematical Aesthetic Lens 39 The Aesthetic 39 The Mathematical Aesthetic 42 The Mathematical Aesthetic in Action 43 The Aesthetics of Detachment 56 5. Aesthetics and the Development of Mathematics 59 Debriefi ng the Kissing Triangles 59 The Importance of the Mathematical Aesthetic 61 Fitting the Pieces Together 65 v SSiinnccllaaiirr ffiinnaall pprrooooffss..iinndddd FFMMvv 66//2299//22000066 1100::5544::3344 AAMM vi Contents PART III: FOCUSING THE AESTHETIC LENS ON STUDENTS 6. The Motivational Role of the Aesthetic 69 The Aesthetic Dimension of Problem Selection 71 The Aesthetic Dimension of Problem Posing 74 Coloring with Numbers 83 Fitting the Pieces Together 97 7. The Generative Role of the Aesthetic 99 Three Examples of the Generative Role 100 Mindful Mathematics 108 Fitting the Pieces Together 109 8. The Evaluative Role of the Aesthetic 113 Which Solution Is Better? 117 Wonder and the Aesthetic 129 Valuing the Aesthetic 131 Fitting the Pieces Together: Aesthetics and Inquiry 133 PART IV: AESTHETIC ENCULTURATION 9. Peering Inside the Mathematics Culture 139 From Outside to Inside the Culture 139 What Do Mathematicians Value? 140 The Aesthetic Dimension of Mathematical Values 144 10. Mathematical Values in Teaching 153 Revealing Values in Topics, Tasks, and Tools 154 Communicating Values in the Classroom 162 Some Closing Words 175 Notes 177 References 183 Index 191 About the Author 196 SSiinnccllaaiirr ffiinnaall pprrooooffss..iinndddd FFMMvvii 66//2299//22000066 1100::5544::3344 AAMM Foreword Nathalie Sinclair is a gifted teacher, comfortable with learners of all ages and able to relate easily and well to students and teachers in the class- room. At the same time, as someone who has studied mathematics at the graduate level herself, she is both able to understand the world-view of the research mathematician and able to fi t easily into discussions in the research institute’s common room. Nathalie’s book is a contribution to mathematics education, a fi eld long at the forefront of vigorous discus- sions about the nature, purpose, and means of academic instruction. From “new” to “applied” to “problem-based” to “constructivist/standards/ fuzzy,” almost every decade of the last half-century seems to have had its version of ambitious curriculum reforms for school mathematics. The fi nal phases of these initiatives have had several constants. On the hu- man front, these have often included: embattled proponents and angry opponents of the reform, confused parents, frustrated teachers, and low- achieving learners. Mathematics is almost universally seen as an essential component in any well-balanced and comprehensive school curriculum for both el- ementary and secondary school learners. This position is usually rational- ized by pointing to the foundational role that the discipline plays in many fundamental aspects of contemporary society. While the real depth and sophistication of this argument is infrequently heard in curriculum dis- cussions, it seems unlikely that this view will be seriously challenged in the near future as the characteristics and implications of an “information society” become ever more evident. A small group (relative to the population as a whole) that has played a key role in many reform initiatives is that of research mathematicians. Sometimes initiating the reforms—as was the case in the “new math” ven- ture of the 1960s—but more often taking the part of opponents, their in- volvement is frequently very passionate. This refl ects and is infl uenced by their deep emotional feeling for their craft. They speak of the nature of mathematics and of the satisfaction in creating mathematics in almost exactly the same ways as devotees of the arts: Beauty is all. vii SSiinnccllaaiirr ffiinnaall pprrooooffss..iinndddd FFMMvviiii 66//2299//22000066 1100::5544::3344 AAMM viii Foreword A second group of individuals is much larger: disenchanted and often unsuccessful learners of mathematics. The mathematics learning experi- ence of many children has been, on the whole, quite negative. There are at least two versions of this phenomenon. One is the observable class of very low- to non-achievers, a distressingly large group in many jurisdictions. A second, less visible group—which overlaps with the former but is still distinct—is that of the emotionally stressed learner. The classroom experience of mathematics, even for those who “suc- ceed” as judged by external paper-based criteria, can be unpleasant and psychologically damaging. Of all the subjects in the school curriculum, mathematics has the unique and dubious distinction as being the most closely linked in the public mind to the experience of “anxiety.” Recent research shows these feelings to be widespread. A British study found that contemporary lower-secondary students fi nd their mathematics classes to be “TIRED,” that is, characterized by Tedium, Isolation, Rote Learning, Elit- ism, and Depersonalization. An international study of school children’s images of mathematicians generated a very large number of unattractive individuals and unpleasant behavior, including a surprisingly high cor- relation with acts of violence. Finally, we have the harsh reality that the standards for “success” in school mathematics are in many ways set at an excessively low level. Many “good” mathematics students, for reasons not of their own making, have had only a stunted exposure to the more me- chanical aspects of a deep, powerful, and historically rich human creation. All in all, this is a depressing picture. Many very bright, highly motivated individuals with substantial resources have labored long and hard to ad- dress these concerns, with very little success. What else might be tried? This is a question that Nathalie’s book addresses, and her response is quite simple. The prevailing view about mathematical ability in general, and sensitivity to mathematical beauty in particular, is an elitist one. It takes as a given that only a few individuals have the capacity to be moved by the aesthetic dimensions of the discipline. From this tiny pool of indi- viduals come our group of passionate researchers in mathematics, and a few kindred spirits in fi elds such as physics and computer science. For the rest, it is assumed that “utility” will have to suffi ce. In a move similar to the shifting of the Darwinian perspective on evolution from “survival of the fi ttest” to “survival of the fi t,” Nathalie decided to investigate the potential of bringing “beauty” into the matrix of curriculum considerations. What might classroom experience in math- ematics look like if it was assumed that sensitivity to beauty is not the sole preserve of only a few (who are often portrayed in quite unattractive social terms), but rather was open to all? SSiinnccllaaiirr ffiinnaall pprrooooffss..iinndddd FFMMvviiiiii 66//2299//22000066 1100::5544::3344 AAMM Foreword ix Once this door was opened, a large and fascinating number of ques- tions quickly emerged, many of which came out of the world of math- ematics. Although terms like beauty, harmony, elegance, and balance can be frequently found in mathematics textbooks, relatively little research had been done to make their meaning more precise. Similarly, relatively little was known of children’s ideas about beauty, particularly with respect to school subjects. And so Nathalie’s research began, in settings as varied as school classrooms and offi ces in research institutes. In the end, this book brings together new insights and affi rmations of existing realities into an extensive and thought-provoking whole. To accomplish this, Nathalie has woven together ideas and theories from a wide range of academic fi elds. The work, on one level, is intimately in- volved with students learning mathematics in school. On a second level, the work is also fundamentally philosophical. It deals with classic ques- tions from aesthetics (What does it mean to be beautiful?) and from epis- temology (What does it mean to “know” a piece of mathematics?). It gives rich insights into the different ways in which aesthetic factors surface in the disciplinary functioning of some world-class mathematicians. Its insights into the processes of mathematical thought, particularly around motiva- tion, bring it close to classic questions in psychology and to more recent studies in cognitive science. It addresses some long-standing questions in the philosophy of education articulated many years ago by scholars like John Dewey. Its ingenious use of specially created, sophisticated, and fl exible computer-based activities gives interesting glimpses into the cog- nitive and aesthetic perspectives of learners. Overall, this work contains a provocative set of philosophical, psychological, mathematical, technologi- cal, and educational insights that makes a compelling case for the inclu- sion of aesthetic considerations in the materials and approaches we bring to mathematics students at all levels. The result is something that will be of interest to scholars in many fi elds. In addition, it has important messages for parents concerned about their child’s reaction to school mathematics, for teachers who may see glimpses of a different way of thinking about a key subject, and for curriculum writ- ers who need to think about how these fi ndings may enter the educational experience of a generation of learners. Ours is an age of hyperbole. Most often fuelled by self-interest, exag- gerated claims fl ow easily from the word-processors and microphones of ideologues, scam artists, and salesmen of various sorts. One hesitates in this general “market” to make statements that might lead readers to con- clude that they are being subjected to views from such perspectives. But sometimes risks are necessary, and having passed a moderately rigorous SSiinnccllaaiirr ffiinnaall pprrooooffss..iinndddd FFMMiixx 66//2299//22000066 1100::5544::3355 AAMM