Stephen R. Campbell - Learning and Teaching Number Theory: Research in Cognition and Instruction (Mathematics, Learning, and Cognition: Monograph Series of the Journal of Mathematical Behavior)

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LEARNING AND TEACHING NUMBER THEORY

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Recent Titles in Mathematics, Learning, and Cognition

Monograph Series of the Journal of Mathematical Behavior

Carolyn A. Maher and Robert Speiser, Series Editors

Five Women Build a Number System

Robert Speiser and Chuck Walter

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Research in Cognition and Instruction

Edited by

Stephen R. Campbell and Rina Zazkis

Mathematics, Learning, and Cognition
Monograph Series of the Journal of Mathematical Behavior, Volume 2
Carolyn A. Maher and Robert Speiser, Series Editors

Ablex Publishing

Westport, Connecticut London

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Library of Congress Cataloging-in-Publication Data

Learning and teaching number theory : research in cognition and instruction / edited by
Stephen R. Campbell and Rina Zazkis.
p. cm.(Mathematics, learning, and cognition)
Includes bibliographical references and index.
ISBN 1567506526 (alk. paper)ISBN 1567506534 (pbk. : alk. paper)
1. Number theoryStudy and teaching. I. Campbell, Stephen R. II. Zazkis, Rina. III.
Series.
QA241.L43 2002
510 sdc21
[512'.7071] 2001031649

British Library Cataloguing in Publication Data is available.

Copyright 2002 by Stephen R. Campbell and Rina Zazkis

All rights reserved. No portion of this book may be
reproduced, by any process or technique, without the
express written consent of the publisher.

Library of Congress Catalog Card Number: 2001031649

ISBN: 1-56750-652-6
1-56750-653-4 (pbk.)

First published in 2002

Ablex Publishing, 88 Post Road West, Westport, CT 06881

An imprint of Greenwood Publishing Group, Inc.
www.ablexbooks.com

Printed in the United States of America

The paper used in this book complies with the
Permanent Paper Standard issued by the National
Information Standards Organization (Z39.481984).

10 9 8 7 6 5 4 3 2 1

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Stephen R. Campbell and Rina Zazkis

Since the beginning of the ancient Greek Pythagorean tradition over two and a half millennia ago, striving for a conceptual understanding of numbers and their properties, patterns, structures, and forms has constituted the heart, if not the soul, of mathematics and mathematical thinking. Today, a constellation of activities involving various operations, procedures, functions, relations, and applications associated with numbers occupy the main bulk of the K12 mathematics curriculum. These activities are typically conducted under the auspices of arithmetic and algebra in various guises, such as counting and measuring with numbers, tabulating and graphing collections of numbers, and formulating and solving equations applied in less formal and more familiar day-to-day, real-world, situational contexts.

With all of the attention that has been given to informal meanings and familiar contexts in mathematics education these days, it seems that not too much consideration has been given to formal contexts concerned with properties and structures of number per se. Mathematical meaning is not just a matter of grounding concepts in familiar day-to-day real-world experiences. It is also a matter of developing the conceptual foundations for making clear and general abstract distinctions. A case in point has to do with understanding differences between whole numbers and rational numbers and the different kinds of procedures involved in operating with them. It is well known that learners have many procedural and semantic difficulties in this regard (e.g., Durkin & Shire, 1991; Greer, 1987; Mack, 1995; Silver, 1992). Teachers need to have and learners

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need to develop better structural understandings of these kinds of fundamental mathematical ideas (Kieran, 1992; Ma, 1999). Perhaps a more explicit and focused emphasis on number theory can help.

Carl Friedrich Gauss, the prince of mathematicians, purportedly considered number theory, or higher arithmetic, as the queen of mathematics. Gauss, were he alive today, would likely be surprised, if not dismayed, to hear that despite the heavy emphasis on numbers and operations with numbers,