Mathematics and Beauty: Aesthetic Approaches to Teaching Children

209 Pages·2009·1.14 MB·English

Mathematics
and Beauty
AESTHETIC APPROACHES
TO TEACHING CHILDREN
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Mathematics
and Beauty
AESTHETIC APPROACHES
TO TEACHING CHILDREN
N S
ATHALIE INCLAIR
Foreword by William Higginson
Teachers College, Columbia University
New York and London
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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
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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
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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
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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
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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?
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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
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