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Helping Children Learn Mathematics HELPING CHILDREN LEARN MATHEMATICS Mathematics Learning Study Committee Jeremy Kilpatrick and Jane Swafford, Editors Center for Education Division of Behavioral and Social Sciences and Education National Research Council NATIONAL ACADEMY PRESS WASHINGTON, DC
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Helping Children Learn Mathematics NATIONAL ACADEMY PRESS 2101 Constitution Avenue, NW 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 ExxonMobil Foundation. Additional support was also provided by 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 authors and do not necessarily reflect the views of the organizations or agencies that provided support for the project. International Standard Book Number 0-309-08431-8 Library of Congress Catalog Control Number: 2002106187 Additional copies of this report are available from National Academy Press, 2101 Constitution Avenue, NW, 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 2002 by the National Academy of Sciences. All rights reserved. Suggested citation: National Research Council. (2002). Helping Children Learn Mathematics. Mathematics Learning Study Committee, J.Kilpatrick and J.Swafford, Editors. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
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Helping Children Learn Mathematics THE NATIONAL ACADEMIES Advisers to the Nation on Science, Engineering, and Medicine 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. Harvey V.Fineberg 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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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, ExxonMobil Corporation (Retired) HUNG HSI WU, University of California, Berkeley JANE SWAFFORD, Study Director BRADFORD FINDELL, Program Officer BRIAN McQUILLAN, Senior Project Assistant CAROLE LACAMPAGNE, Director, Mathematical Sciences Education Board
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Helping Children Learn Mathematics Acknowledgments In preparing this book, we had the good fortune of working with a number of people who shared our enthusiasm for the project, and we are indebted to them for the insights and assistance they provided. The entire Mathematics Learning Study Committee contributed valuable ideas, but we particularly acknowledge the members of the subcommittee who conceptualized and shaped the book: Thomas Carpenter, James Hiebert, Casilda Pardo, and Edgar Robinson. We are grateful to them for their inspiration, hard work, and guidance. During the outlining and preliminary drafting of the book, we benefited from the views and prose provided by Pat McNees, a writer consultant. And we are especially indebted to Steve Olson, who took our rough drafts and sketchy ideas and turned them into this final product. Without his expert editorial assistance, the project might never have been completed. We would like to thank our sponsor for this book, the ExxonMobil Foundation, for seeking to engage a much broader audience in discussion and action. In particular, we thank Edward Ahnert, president of the Foundation, and Joe Gonzales, program officer, for their support and interest. We also want to thank the sponsors of the Mathematics Learning Study, the National Science Foundation and the U.S. Department of Education, for their continued support during the production of the book. This book has been reviewed by individuals chosen for their diverse perspectives and technical expertise, in accordance with procedures approved by the National Research Council’s (NRC) 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 book as sound as possible and to ensure that the book meets institutional standards for objectivity, evidence, and responsiveness to the study charge. The review comments and draft manuscript remain
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Helping Children Learn Mathematics confidential to protect the integrity of the deliberative process. We thank the following individuals for their review of this book: Arthur J.Baroody, Department of Curriculum and Instruction, University of Illinois; Keith Devlin, Center for the Study of Language and Information, Stanford University; Nancy A.Doorey, Brandywine School District, Delaware; Andrew M.Gleason, Department of Mathematics, Harvard University; Nancy Larson, West Haven Schools, Connecticut; Mike Riley, Bellevue School District, Washington; Zalman P. Usiskin, Department of Education, University of Chicago; and Carrie L.Valentine, Madison Metropolitan School District and University of Wisconsin. 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 book before its release. The review of this book was overseen by Alfred Manaster, University of California at San Diego, and Patrick Suppes, Stanford University. Appointed by the NRC, they were responsible for making certain that an independent examination of this book was carried out in accordance with institutional procedures and that all review comments were carefully considered. Responsibility for the final content of this book rests solely with the authoring committee and the NRC. Finally, we extend our sincere thanks to a number of individuals at the NRC who made significant contributions to our work: Michael J.Feuer, director of the Center for Education (CFE), for providing key advice; Kirsten Sampson Snyder, reports officer for CFE, for guiding us through the report review process; Bradford Findell, former program officer, for his contributions to several drafts; Patricia Morison, senior program officer, for shepherding this project through its final stages; Sally Stanfield and Francesca Moghari, National Academy Press, for making our book look so nice; Carole Lacampagne for her wise advice during late stages; and Dionna Williams and Yvonne Wise for help with production. Last, we would like to express our appreciation to Brian McQuillan, senior project assistant, who provided a wealth of logistical support during the project. Jeremy Kilpatrick, Chair Jane Swafford, Study Director Mathematics Learning Study Committee
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Helping Children Learn Mathematics Contents Overview 1 Introduction 3 What Does It Mean to Be Successful in Mathematics? 9 How Does School Mathematics Need to Change for All Students to Become Mathematically Proficient? 25 What Can Parents and Caregivers Do? 35 What Can Teachers Do? 37 What Can Administrators Do? 38 What Can Policy Makers Do? 39 Notes 40
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Helping Children Learn Mathematics This page in the original is blank.