Analysis of Arithmetic for Mathematics Teaching

472 Pages·1992·24.19 MB·English

by  Gaea Leinhardt

ANALYSIS OF
A R I T H M E T I C
§ MATHEMATICS
TEACHING
Edited by
Gaea Leinhardt
Ralph Putnam
Rosemary A. Hattrup
ANALYSIS OF ARITHMETIC
FOR MATHEMATICS TEACHING

ANALYSIS OF ARITHMETIC
FOR MATHEMATICS TEACHING
Edited by
Gaea Leinhardt
University of Pittsburgh
Ralph Putnam
Michigan State University
Rosemary A. Hattrup
University of Pittsburgh
LAWRENCE ERLBAUM ASSOCIATES, PUBLISHERS
1992 Hillsdale, New Jersey Hove and London
Copyright © 1992 by Lawrence Erlbaum Associates, Inc.
Ail rights reserved. No part of the book may be reproduced in
any form, by photostat, microform, retrieval system, or any other
means, without the prior written permission of the publisher.
Lawrence Erlbaum Associates, Inc., Publishers
365 Broadway
Hillsdale, New Jersey 07642
Library of Congress Cataloging-in-Publication Data
Analysis of arithmetic for mathematics teaching / edited by Gaea
Leinhardt, Ralph Putnam, Rosemary A. Hattrup.
p. cm.
Includes bibliographical references and index.
ISBN 0-8058-0929-5
1. Arithmetic—Study and teaching. I. Leinhardt, Gaea.
II. Putnam, Ralph. 111. Hattrup, Rosemary A.
QA135.5A54 1992
513'.07—dc20 91-13053
CIP
Printed in the United States of America
10 9 8 7 6 5 4 3 2
C
ontents
ACKNOWLEDGMENTS vii
PREFACE ix
1 MAKING SENSE OF NUMBERS IN SCHOOL
MATHEMATICS 1
Judith T. Sowder
2 RESEARCH ON LEARNING AND TEACHING
ADDITION AND SUBTRACTION OF
WHOLE NUMBERS 53
Karen C. Fuson
3 SOLVING MULTIPLICATION WORD PROBLEMS 189
Pearla Nesher
4 TEACHING AND LEARNING LONG DIVISION FOR
UNDERSTANDING IN SCHOOL 221
Magdalene Lampert
5 MATHEMATICAL, COGNITIVE, AND INSTRUCTIONAL
ANALYSES OF DECIMAL FRACTIONS 283
James Hiebert
5 RATIONAL AND FRACTIONAL NUMBERS AS
MATHEMATICAL AND PERSONAL KNOWLEDGE:
IMPLICATIONS FOR CURRICULUM AND
INSTRUCTION 323
Thomas E. Kieren
VI CONTENTS
FROM PROTOQUANTITIES TO OPERATORS:
BUILDING MATHEMATICAL COMPETENCE ON A
FOUNDATION OF EVERYDAY KNOWLEDGE 373
Lauren B. Resnick
EPILOGUE 431
Marilyn Rauth and Lovely Billups
AUTHOR INDEX 441
SUBJECT INDEX 449
A
cknowledgments
This enterprise between a research institute and a teaching organization
was a social and intellectual experiment. One result of the experiment is
this volume; the other result is a set of volumes authored principally by the
teachers who participated in the 3-year effort (Thinking Mathematics, Vol­
ume 1: Foundations, Thinking Mathematics, Volume IT: Training Modules,
Thinking Mathematics, Volume 2: Extensions, and Thinking Mathematics,
Volume 2T: Training Modules, Bodenhausen et al., in preparation). Because
of the experimental nature of the effort, more people and agencies than
usual were involved, and the demands on the reviewers, teachers, re­
searchers, and publishers were heavier than normal.
The editors wish to acknowledge the patience, insights, and good humor
of all the authors who contributed to this volume. Several of them had to
wait a very long time to see their text in print. We wish to acknowledge the
tremendous and very important help of the reviewers of specific chapters:
Merlyn Behr, Michael Cole, Vicky Kouba, Stellan Ohlsson, and John P. Smith,
111. Their thoughtful comments enhanced the quality of the final product.
The combined teams of teachers that read, interpreted, cind critiqued
each chapter were also a very important and vocal audience that have
influenced the shape of this document: Louise Eggert, Angela Silva, and
Carole Tuccy, 1988 Visiting Practitioners; and Nancy Denhart, Judy
Bodenhausen, Alice Gill, Meg Kaduce, and Marcy Miller, 1989/1990/1991
Visiting Practitioners. We wish to especially thank Judy Bodenhausen for
her careful reading of the chapter by Pearla Nesher.
In addition to the editors who helped assemble this volume, many re­
searchers were present during the summer workshops and contributed
VII
VIII ACKNOWLEDGMENTS
much to the effort. Two of the most significant are Barbara Grover and
William Bickel, both of whom carried major burdens of the social and
intellectual experiment part of this enterprise. Other members of our re­
search community also played important roles. Among these we wish to
thank Lauren Resnick, Ed Silver, and Orit Zaslavsky, who reviewed and
commented on the chapters with great care and led as well as participated
in discussions around the research with the Visiting Practitioners. Gina
Beavers, Peg Emmerling, Joyce Fienberg, Sister Madeliene Gregg, Susan
Honacki, Diana Jeske, Leda Mance, Judith McQuaide, Swapna Mukhopad-
hyay, Elizabeth Odoroff, and Kate Stainton were central members of the
support team that facilitated the work of all.
We want to thank the American Federation of Teachers, especially Lovely
Billups and Marilyn Rauth, for without their leadership this collaboration
would not have occurred. We also wish to acknowledge the University of
Pittsburgh and the Learning Research and Development Center, which pro­
vided not only the concrete home where many of these chapters were
worked on, but also the intellectual environment that permitted and en­
couraged the occurrence of activities that enabled this volume. Finally, the
editors thank the National Science Foundation for having the vision and
faith to believe that real dialogue between teachers and researchers was
desirable and achievable and supporting the workshops, the reviewers, and
the authors through grant number TPE-8751494.
REFERENCES
Bodenhausen, J., Denhart, N., Gill, A., Kaduce, M., Miller, M., Grover, B., Resnick, L., Leinhardt,
G., Bill, V., Rauth, M., & Billups, L. (in preparation). Thinking mathematics: Vol. I. Founda­
tions. Washington, DC: American Federation of Teachers.
Bodenhausen, J., Denhart, N., Gill, A., Kaduce, M., Miller, M., Grover, B., Hojnacki, S., Resnick, L.,
Leinhardt, G., Bill, V., & Billups, L. (in preparation). Thinking mathematics: Vol 2. Exten­
sions. Washington, DC: American Federation of Teachers.
P
reface
This book emerges from a partnership between a teaching organization
(American Federation of Teachers) and a research institute (Learning Re­
search and Development Center at the University of Pittsburgh) and repre­
sents another step in taking mathematics research through the door of the
classroom and taking classroom experience through the door of academe.
This partnership has brought researchers together with expert teachers for
intensive dialogue sessions focusing on what is known in each community
about effective mathematical learning and instruction. The dialogue has
been predicated on the belief that both the research and teaching com­
munities bring critically important perspectives to thinking about mathe­
matics learning and instruction and that special attention needs to be paid
to increasing communication between and within these communities if
teaching is to be informed by current research. This book represents one
side of an attempt to openly shcire our dialogue with the research and
teaching communities. A second side is represented in another product of
the partnership, Thinking Mathematics—a synthesis of the central ideas of
this mathematics research written by teachers for teachers.
Analysis of Arithmetic for Mathematics Teaching is intended to serve cis a
tool for researchers, particularly educational researchers who are inter­
ested in research advances in subject matters directly connected to
schooling and cognitive scientists who are interested in a broad but em­
bedded view of school subjects. The syntheses presented in this volume
represent current research that will be of use to researchers who have an
interest in both clcissroom learning and instruction. It is our hope that the
chapters in this book will also be used by teachers and teacher educators
IX