The Language of Mathematics Telling Mathematical Tales

The Language of Mathematics: Telling Mathematical Tales emerges from several contemporary concerns in mathematics, language, and mathematics education, but takes a different stance with respect to language. Rather than investigating the way language or cu

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Mathematics Education Library O VLUME 64 Managing Editor A.J. Bishop, Monash University, Melbourne, Australia

Editorial Board .JP. Becker, Illinois, U.S.A. C. Keitel, Berlin, Germany F. Leung, Hong Kong, China G. Leder, Melbourne, Australia D. Pimm, Edmonton, Canada A. Sfard, Haifa, Israel O. Skovsmose, Aalborg, Denmark

The titles published in this series are listed at the end of this volume.

Bill Barton

The Language of Mathematics Telling Mathematical Tales

123

Bill Barton University of Auckland Auckland New Z ealand b.barton@ auckland.ac.nz

Series Editor: Alan Bishop Monash University Melbourne 3800 Australia Alan.Bishop@ Education.monash.edu.au

Library of Congress Control Number: 2007936207

ISBN -13: 978-0-387-72858-2

e-ISBN-13: 978-0-387-72859-9

Printed on acid-free paper. ¤ 2008 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 987654321 springer.com

Dedication

This book is dedicated to my wife, Pip.

Acknowledgements

I wish first to acknowledge the academic inspiration and guidance I received from Ubiratan D’Ambrosio, Alan Bishop, Maria do Carmo Domite, and Andy Begg. Many of the ideas in this book emerged from their writing or in discussion with one or other of them. I am indebted to them for their mentoring and/or insights. There are many others who have contributed to my thinking about language and mathematics: my colleagues and students, reviewers of papers, and the many respondents at seminars or conference presentations. Thank you for your critical questions, original ideas, and helpful suggestions. Special thanks also go to my language advisers, in particular: Shehenaz Adam (Dhivehi); Willy Alangui (Kankana-ey); and Tony Trinick (Maori). John Mason helped me considerably with investigations into DoubleAngle Geometry. He spent many hours exploring and analysing this environment, trying to answer my questions, and providing insights. Three institutions have supported this work. My home university, The University of Auckland, has been generous in its leave and funding entitlements. The Department of Mathematics at The University of British Columbia, and the Departamento Didáctica de la Matemática of Universidad de Granada also hosted me for extended writing periods. Others helped in the writing process. Thanks to the many who read and commented on my writing: in particular Ivan Reilly