ADDING + IT UP

HELPING CHILDREN LEARN MATHEMATICS

Mathematics Learning Study Committee

Jeremy Kilpatrick, Jane Swafford, and Bradford Findell, editors

Center for Education

Division of Behavioral and Social Sciences and Education

National Research Council

National Academy Press
Washington, DC



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Adding + It Up: Helping Children Learn Mathematics ADDING + IT UP HELPING CHILDREN LEARN MATHEMATICS Mathematics Learning Study Committee Jeremy Kilpatrick, Jane Swafford, and Bradford Findell, editors Center for Education Division of Behavioral and Social Sciences and Education National Research Council National Academy Press Washington, DC

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Adding + It Up: Helping Children Learn Mathematics NATIONAL ACADEMY PRESS 2101 Constitution Avenue, N.W. Washington, DC 20418 NOTICE: The project that is the subject of this report was approved by the Governing Board of the National Research Council, whose members are drawn from the councils of the National Academy of Sciences, the National Academy of Engineering, and the Institute of Medicine. The members of the committee responsible for the report were chosen for their special competences and with regard for appropriate balance. This study was supported by Contract/Grant No. ESI-9816818 between the National Academy of Sciences and the U.S. Department of Education and the National Science Foundation. Any opinions, findings, conclusions, or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the organizations or agencies that provided support for the project. Library of Congress Cataloging-in-Publication Data Adding it up: helping children learn mathematics/Jeremy Kilpatrick, Jane Swafford, and Bradford Findell, editors. p. cm. Includes bibliographical references and index. ISBN 0-309-06995-5 (hardcover) 1. Mathematics—Study and teaching (Elementary) —United States. 2. Mathematics—Study and teaching (Middle school) —United States. I. Kilpatrick, Jeremy. II. Swafford, Jane. III. Findell, Bradford. QA135.5 .A32 2001 372.7–dc21 2001001734 Suggesed citation: National Research Council. (2001). Adding it up: Helping children learn mathematics. J.Kilpatrick, J. Swafford, and B.Findell (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. Additional copies of this report are available from National Academy Press, 2101 Constitution Avenue, N.W., Lockbox 285, Washington, DC 20055; (800) 624–6242 or (202) 334–3313 (in the Washington metropolitan area); Internet, http://www.nap.edu Printed in the United States of America. Copyright 2001 by the National Academy of Sciences. All rights reserved.

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Adding + It Up: Helping Children Learn Mathematics THE NATIONAL ACADEMIES National Academy of Sciences National Academy of Engineering Institute of Medicine National Research Council The National Academy of Sciences is a private, nonprofit, self-perpetuating society of distinguished scholars engaged in scientific and engineering research, dedicated to the furtherance of science and technology and to their use for the general welfare. Upon the authority of the charter granted to it by the Congress in 1863, the Academy has a mandate that requires it to advise the federal government on scientific and technical matters. Dr. Bruce M.Alberts is president of the National Academy of Sciences. The National Academy of Engineering was established in 1964, under the charter of the National Academy of Sciences, as a parallel organization of outstanding engineers. It is autonomous in its administration and in the selection of its members, sharing with the National Academy of Sciences the responsibility for advising the federal government. The National Academy of Engineering also sponsors engineering programs aimed at meeting national needs, encourages education and research, and recognizes the superior achievements of engineers. Dr. Wm.A.Wulf is president of the National Academy of Engineering. The Institute of Medicine was established in 1970 by the National Academy of Sciences to secure the services of eminent members of appropriate professions in the examination of policy matters pertaining to the health of the public. The Institute acts under the responsibility given to the National Academy of Sciences by its congressional charter to be an adviser to the federal government and, upon its own initiative, to identify issues of medical care, research, and education. Dr. Kenneth I.Shine is president of the Institute of Medicine. The National Research Council was organized by the National Academy of Sciences in 1916 to associate the broad community of science and technology with the Academy’s purposes of furthering knowledge and advising the federal government. Functioning in accordance with general policies determined by the Academy, the Council has become the principal operating agency of both the National Academy of Sciences and the National Academy of Engineering in providing services to the government, the public, and the scientific and engineering communities. The Council is administered jointly by both Academies and the Institute of Medicine. Dr. Bruce M.Alberts and Dr. Wm.A.Wulf are chairman and vice chairman, respectively, of the National Research Council.

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Adding + It Up: Helping Children Learn Mathematics MATHEMATICS LEARNING STUDY COMMITTEE JEREMY KILPATRICK, Chair, University of Georgia DEBORAH LOEWENBERG BALL, University of Michigan HYMAN BASS, University of Michigan JERE BROPHY, Michigan State University FELIX BROWDER, Rutgers University THOMAS P.CARPENTER, University of Wisconsin-Madison CAROLYN DAY, Dayton Public Schools KAREN FUSON, Northwestern University JAMES HIEBERT, University of Delaware ROGER HOWE, Yale University CAROLYN KIERAN, University of Quebec, Montreal RICHARD E.MAYER, University of California, Santa Barbara KEVIN MILLER, University of Illinois, Urbana-Champaign CASILDA PARDO, Albuquerque Public Schools EDGAR ROBINSON, Exxon Mobil Corporation (Retired) HUNG-HSI WU, University of California, Berkeley NATIONAL RESEARCH COUNCIL STAFF JANE SWAFFORD, Study Director BRADFORD FINDELL, Program Officer GAIL PRITCHARD, Program Officer SONJA ATKINSON, Administrative Assistant SPECIAL OVERSIGHT COMMISSION FOR THE MATHEMATICS LEARNING STUDY RONALD L.GRAHAM, Chair, University of California, San Diego DEBORAH LOEWENBERG BALL, University of Michigan IRIS CARL, Houston Independent School District THOMAS P.CARPENTER, University of Wisconsin-Madison CHRISTOPHER CROSS, Council for Basic Education RONALD DOUGLAS, Texas A&M University ROGER HOWE, Yale University LYNNE REDER, Carnegie Mellon University HAROLD STEVENSON, University of Michigan PHILLIP URI TREISMAN, University of Texas, Austin

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Adding + It Up: Helping Children Learn Mathematics REVIEWERS This report has been reviewed in draft form by individuals chosen for their diverse perspectives and technical expertise, in accordance with procedures approved by the National Research Council’s Report Review Committee. The purpose of this independent review is to provide candid and critical comments that will assist the institution in making its published report as sound as possible and to ensure that the report meets institutional standards for objectivity, evidence, and responsiveness to the study charge. The review comments and draft manuscript remain confidential to protect the integrity of the deliberative process. We wish to thank the following individuals for their participation in the review of this report: JOHN ANDERSON, Carnegie Mellon University RICHARD A.ASKEY, University of Wisconsin-Madison ARTHUR BAROODY, University of Illinois, Urbana-Champaign GUNNAR CARLSSON, Stanford University JERE CONFREY, University of Texas JOHN DOSSEY, Illinois State University JEAN-CLAUDE FALMAGNE, University of California, Irvine HERBERT GINSBURG, Columbia University KENNETH KOEDINGER, Carnegie Mellon University CAROLYN MAHER, Rutgers University ALFRED MANASTER, University of California, San Diego BETHANY RITTLE-JOHNSON, Carnegie Mellon University

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Adding + It Up: Helping Children Learn Mathematics MARIA SANTOS, San Francisco Unified School District PATRICK THOMPSON, Vanderbilt University ZALMAN USISKIN, University of Chicago Although the reviewers listed above have provided many constructive comments and suggestions, they were not asked to endorse the conclusions or recommendations nor did they see the final draft of the report before its release. The review of this report was overseen by Ronald L.Graham, University of California, San Diego, and Patrick Suppes (NAS), Stanford University. Appointed by the National Research Council, they were responsible for making certain that an independent examination of this report was carried out in accordance with institutional procedures and that all review comments were carefully considered. Responsibility for the final content of this report rests entirely with the authoring committee and the institution.

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Adding + It Up: Helping Children Learn Mathematics ACKNOWLEDGMENTS Adding It Up is the product of an 18-month project in which 16 individuals with diverse backgrounds, as a committee, reviewed and synthesized relevant research on mathematics learning from pre-kindergarten through grade 8. We had the good fortune of working with a number of people outside the committee who shared our enthusiasm for this project, and we are indebted to them for the intellectual insights and support that they provided. At a time when mathematics education issues have reached a critical point, both publicly and politically, it has become clear that our nation has a responsibility to provide guidance and leadership in answering questions about how to improve mathematics learning for all students. We would like to thank our sponsors, the National Science Foundation and the U.S. Department of Education, for their foresight in providing a timely opportunity to move the debate forward. In particular, we thank Janice Earle, from the National Science Foundation; Patricia O’Connell Ross, from the U.S. Department of Education; and Judy Wurtzel and Linda Rosen, both formerly with the U.S. Department of Education, for their constant support and interest in this study. During the information-gathering phase of our work, a number of people made presentations to the committee on various topics pertaining to mathematics learning. We benefited greatly from their stimulating presentations and extend our thanks to Jo Boaler, Stanford University, School of Education; Douglas Carnine, University of Oregon, National Center to Improve the Tools of Educators; Paul Clopton, Mathematically Correct; Megan Franke, University of California, Los Angeles, Graduate School of Education and Information Studies; and Judith Sowder, San Diego State University, Center for Research in Mathematics and Science Education. Additionally, we would like to thank

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Adding + It Up: Helping Children Learn Mathematics Steven Stahl and Donna Alvermann, University of Georgia, and Susan Burns, George Mason University, for providing us with insights about the parallels between mathematics and reading. And we are grateful to Carne Barnett, WestEd Regional Education Laboratory; Deborah Schifter, Education Development Center; Patricia Campbell, University of Maryland, Center for Mathematics Education; Anne Morris, University of Delaware, School of Education; and Mary Kay Stein, University of Pittsburgh, Learning Research and Development Center; for providing information about specific programs in elementary mathematics or teacher development. We also wish to acknowledge the people who provided informative commissioned papers that expanded and enhanced our collective thinking. In particular, we appreciate the work of Rolf Blank, Council of Chief State School Officers; Graham Jones, Cynthia Langrall, and Carol Thornton, Illinois State University; Gloria Ladson-Billings and Richard Lehrer, University of Wisconsin-Madison; and Denise Mewborn, University of Georgia. We also thank Douglas McLeod and Judith Sowder, San Diego State University, and Les Steffe, University of Georgia, for their assistance with research reviews for specific topics on which we had questions. While writing the final draft of this report, we commissioned several chapter reviews that strengthened our research synthesis and focused our prose. Many thanks to Kathleen Cramer, University of Minnesota; James Kaput, University of Massachusetts-Dartmouth; Mary Lindquist, Columbus State University; Thomas Post, University of Minnesota; and Edward Rathmell, University of Northern Iowa. While the individuals listed above have provided many constructive comments and suggestions, responsibility for the final content of this report rests solely with the authoring committee and the National Research Council. Finally, we would like extend our sincere thanks to several individuals within the National Research Council and in other places who made significant contributions to our work: Rodger Bybee, former Executive Director for the Center, and Patrice Legro, former Division Director for Special Projects, for providing the initial impetus for this project and getting it off to a strong start; Gail Pritchard, Program Officer, for keeping us on the straight and narrow in complying with the myriad of NRC policies and procedures; Bradford Findell, Program Officer, for researching, drafting, and editing many sections of the report; Michael J.Feuer, Executive Director for the Center for Education (CFE), for providing key advice; Kirsten Sampson Snyder, Reports Officer for CFE, for guiding us through the report review process; Steve Olson and Yvonne Wise, for providing editorial assistance; Sally Stanfield, National

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Adding + It Up: Helping Children Learn Mathematics Academy Press, for making our report look so nice; Lynn Geiger and Gooyeon Kim, doctoral students at the University of Georgia, for assisting the chair in his work on this report; Mark Hoover, doctoral student at the University of Michigan, for helping on some early drafts of chapters; and Todd Grundmeier, graduate student at the University of New Hampshire, for tracking down most of our references and verifying them for appropriateness and accuracy. Lastly, we would like to express our appreciation to Sonja Atkinson, Administrative Assistant, whose agility in managing the complex arrangements, attention to detail, and cheerful attitude made our work much easier and our time together more enjoyable. Jeremy Kilpatrick, Chair Jane Swafford, Study Director Mathematics Learning Study Committee

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Adding + It Up: Helping Children Learn Mathematics PREFACE Public concern about how well U.S. schoolchildren are learning mathematics is abundant and growing. The globalization of markets, the spread of information technologies, and the premium being paid for workforce skills all emphasize the mounting need for proficiency in mathematics. Media reports of inadequate teaching, poorly designed curricula, and low test scores fuel fears that young people are deficient in the mathematical skills demanded by society. Such concerns are far from new. Over a century and a half ago, Horace Mann, secretary of the Massachusetts State Board of Education, was dismayed to learn that Boston schoolchildren could answer only about a third of the arithmetic questions they were asked in a survey. “Such a result repels comment,” he said. “No friendly attempt at palliation can make it any better. No severity of just censure can make it any worse.” In 1919, when part of the survey was repeated in school districts around the country, the results for arithmetic were even worse than they had been in 1845. Apparently, there has never been a time when U.S. students excelled in mathematics, even when schools enrolled a much smaller, more select portion of the population. Over the last half-century, however, mathematics achievement has become entangled in urgent national issues: building military and industrial strength during the Cold War, maintaining technological and economic advantage when the Asian tigers roared, and most recently, strengthening public education against political attacks. How well U.S. students are learning mathematics and what should be done about it are now matters for every citizen to ponder. And one hears calls from many quarters for schools, teachers, and students to boost their performance.

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Adding + It Up: Helping Children Learn Mathematics During the new math era of the mid-1950s to mid-1970s, reformers emphasized changes in the mathematics curriculum; today’s reformers want changes in mathematics teaching and assessment as well. In the mathematician E.G.Begle’s laconic formulation, the problem is no longer so much teaching better mathematics as it is teaching mathematics better. Almost everyone today agrees that elementary and middle school mathematics should not be confined to arithmetic but should also include elements from other domains of mathematics, such as algebra, geometry, and statistics. There is much less consensus, however, on how these elements should be organized and taught. Different people urge that school mathematics be taken in different directions. A claim used to advocate movement in one direction is that mathematics is bound by history and culture, that students learn by creating mathematics through their own investigations of problematic situations, and that teachers should set up situations and then step aside so that students can learn. A countervailing claim is that mathematics is universal and eternal, that students learn by absorbing clearly presented ideas and remembering them, and that teachers should offer careful explanations followed by organized opportunities for students to connect, rehearse, and review what they have learned. The trouble with these claims is not that one is true and the other false; it is that both are incomplete. They fail to capture the complexity of mathematics, of learning, and of teaching. Mathematics is at the same time inside and beyond culture; it is both timely and timeless. The theorem attributed to Pythagoras was known in various forms in the civilizations of ancient Babylon and China, and it is still true the world over today even though systems of geometry now exist in which it does not hold. Mathematics is invented, and it is discovered as well. Students learn it on their own, and they learn it from others, most especially their teachers. If students are to become proficient in mathematics, teaching must create learning opportunities both constrained and open. Mathematics teaching is a difficult task under any circumstances. It is made even more complicated and challenging when teachers are paying attention simultaneously, as they should, to the manifold paths mathematics learning can take and to the multifaceted nature of mathematics itself. In this report, we have attempted to address the conflicts in current proposals for changing school mathematics by giving a more rounded portrayal of the mathematics children need to learn, how they learn it, and how it might be taught to them effectively. In coming up with that portrayal, we have drawn on the research literature as well as our experience and judgment.

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Adding + It Up: Helping Children Learn Mathematics Early on, we decided to concentrate primarily on the mathematics of numbers and their operations—for reasons spelled out in chapter 1. We wanted readers to understand that we were using the topic to illustrate what might be done throughout the curriculum. Nonetheless, we recognize the ease with which some may conclude that attention equals advocacy, that we think arithmetic must constitute the mathematics curriculum from pre-kindergarten to eighth grade. Such a conclusion would be wrong: The emphasis on numbers and operations in the research literature and the even greater emphasis in this report say nothing about what the emphasis should be in school. We support a comprehensive curriculum that draws on many domains of mathematics. The mathematician George Pólya, poking fun at the new math textbooks being assembled by platoons of mathematicians and teachers, once proposed a mock word problem something like the following: If one person can write a book in 12 months, how many months will 30 people need? Producing the present book in 18 months demanded something other than proportional reasoning; it took a superb committee of talented, dedicated people. The committee members were truly diverse, with different sorts of expertise. None of us knew all the others before we began. We brought many views, some opposing, on the issues before us. Yet we set to work immediately to develop a report we could all support, eventually meeting eight times from January 1999 to June 2000. Small groups of two or three met occasionally between committee meetings to draft sections of the report, and we engaged in countless e-mail exchanges to work out thorny details. The process worked because each of us valued the others’ opinions, we listened to one another thoughtfully and respectfully, and we worked hard together to reach our common goal. No matter how many months more or less than 18 it might have taken, none of us could have written this report alone. Whatever merits it has lie not only in the messages it contains but also in how it was produced. We offer the report in the hope that it will enable others to address the problems of school mathematics in a more balanced, informed way than is common today and in the same spirit we had of cooperation and mutual regard. Jeremy Kilpatrick, Chair Mathematics Learning Study Committee

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Adding + It Up: Helping Children Learn Mathematics TABLE OF CONTENTS     Executive Summary   1 1   Looking at Mathematics and Learning   15 2   The State of School Mathematics in the United States   31 3   Number: What Is There to Know?   71 4   The Strands of Mathematical Proficiency   115 5   The Mathematical Knowledge Children Bring to School   157 6   Developing Proficiency with Whole Numbers   181 7   Developing Proficiency with Other Numbers   231 8   Developing Mathematical Proficiency Beyond Number   255 9   Teaching for Mathematical Proficiency   313 10   Developing Proficiency in Teaching Mathematics   369 11   Conclusions and Recommendations   407     Biographical Sketches   433     Index   441

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Adding + It Up: Helping Children Learn Mathematics ADDING + IT UP

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