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How Students Learn MATHEMATICS IN THE CLASSROOM Committee on How People Learn, A Targeted Report for Teachers M. Suzanne Donovan and John D. Bransford, Editors Division of Behavioral and Social Sciences and Education
THE NATIONAL ACADEMIES PRESS â¢ 500 Fifth Street, N.W. â¢ Washington, D.C. 20001 NOTICE: The project that is the subject of this report was approved by the Govern- ing 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 Award No. R215U990024 between the National Acad- emy of Sciences and the U.S. Department of Education. 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 National Research Council (U.S.). Committee on How People Learn, A Targeted Report for Teachers. How students learn : history, mathematics, and science in the classroom / Committee on How People Learn, A Targeted Report for Teachers ; M. Suzanne Donovan and John D. Bransford, editors. p. cm. "Division of Behavioral and Social Sciences and Education." Includes bibliographical references and index. ISBN 0-309-07433-9 (hardcover) â ISBN 0-309-08948-4 (pbk.) â ISBN 0-309-08949-2 (pbk.) â ISBN 0-309-08950-6 (pbk.) 1. Learning. 2. Classroom management. 3. Curriculum planning. I. Donovan, Suzanne. II. Bransford, John. III. Title. LB1060.N38 2005 370.15â²23âdc22 2004026246 Additional copies of this report are available from the National Academies Press, 500 Fifth Street, 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 2005 by the National Academy of Sciences. All rights reserved. Suggested citation: National Research Council. (2005). How Students Learn: Math- ematics in the Classroom. Committee on How People Learn, A Targeted Report for Teachers, M.S. Donovan and J.D. Bransford, Editors. Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press.
The National Academy of Sciences is a private, nonprofit, self-perpetuating soci- ety of distinguished scholars engaged in scientific and engineering research, dedi- cated 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 engineer- ing programs aimed at meeting national needs, encourages education and research, and recognizes the superior achievements of engineers. Dr. Wm. A. Wulf is presi- dent 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 Academyâs purposes of furthering knowledge and advising the federal govern- ment. Functioning in accordance with general policies determined by the Academy, the Council has become the principal operating agency of both the National Acad- emy 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 chair and vice chair, respectively, of the National Research Council. www.national-academies.org
v COMMITTEE ON HOW PEOPLE LEARN: A TARGETED REPORT FOR TEACHERS JOHN D. BRANSFORD (Chair), College of Education, University of Washington SUSAN CAREY, Department of Psychology, Harvard University KIERAN EGAN, Department of Education, Simon Fraser University, Burnaby, Canada SUZANNE WILSON, School of Education, Michigan State University SAMUEL S. WINEBURG, Department of Education, Stanford University M. SUZANNE DONOVAN, Study Director SUSAN R. MCCUTCHEN, Research Associate ALLISON E. SHOUP, Senior Project Assistant ELIZABETH B. TOWNSEND, Senior Project Assistant
vii Preface This book has its roots in the report of the Committee on Developments in the Science of Learning, How People Learn: Brain, Mind, Experience and School (National Research Council, 1999, National Academy Press). That report presented an illuminating review of research in a variety of fields that has advanced understanding of human learning. The report also made an important attempt to draw from that body of knowledge implications for teaching. A follow-on study by a second committee explored what research and development would need to be done, and how it would need to be communicated, to be especially useful to teachers, principals, superinten- dents, and policy makers: How People Learn: Bridging Research and Prac- tice (National Research Council, 1999). These two individual reports were combined to produce an expanded edition of How People Learn (National Research Council, 2000). We refer to this volume as HPL. The next step in the work on how people learn was to provide ex- amples of how the principles and findings on learning can be used to guide the teaching of a set of topics that commonly appear in the K-12 curriculum. The work focused on three subject areasâ history, mathematics, and sci- enceâand resulted in the book How Students Learn: History, Mathematics, and Science in the Classroom. Each area was treated at three levels: elemen- tary, middle, and high school. This volume includes the subset of chapters from that book focused on mathematics, along with the introduction and concluding chapter of the larger volume. The full set of chapters can be found on the enclosed CD. Distinguished researchers who have extensive experience in teaching or in partnering with teachers were invited to contribute the chapters. The
viii PREFACE committee shaped the goals for the volume, and commentedâsometimes extensivelyâon the draft chapters as they were written and revised. The principles of HPL are embedded in each chapter, though there are differ- ences from one chapter to the next in how explicitly they are discussed. Taking this next step to elaborate the HPL principles in context poses a potential problem that we wish to address at the outset. The meaning and relevance of the principles for classroom teaching can be made clearer with specific examples. At the same time, however, many of the specifics of a particular example could be replaced with others that are also consistent with the HPL principles. In looking at a single example, it can be difficult to distinguish what is necessary to effective teaching from what is effective but easily replaced. With this in mind, it is critical that the teaching and learning examples in each chapter be seen as illustrative, not as blueprints for the ârightâ way to teach. We can imagine, by analogy, that engineering students will better grasp the relationship between the laws of physics and the construction of effec- tive supports for a bridge if they see some examples of well-designed bridges, accompanied by explanations for the choices of the critical design features. The challenging engineering task of crossing the entrance of the San Fran- cisco Bay, for example, may bring the relationship between physical laws, physical constraints, and engineering solutions into clear and meaningful focus. But there are some design elements of the Golden Gate Bridge that could be replaced with others that serve the same end, and people may well differ on which among a set of good designs creates the most appealing bridge. To say that the Golden Gate Bridge is a good example of a suspension bridge does not mean it is the only, or the best possible, design for a suspension bridge. If one has many successful suspension bridges to com- pare, the design features that are required for success, and those that are replaceable, become more apparent. And the requirements that are uni- form across contexts, and the requirements that change with context, are more easily revealed. The chapters in this volume highlight different approaches to address- ing the same fundamental principles of learning. It would be ideal to be able to provide two or more âHPL compatibleâ approaches to teaching the same topic. However, we cannot provide that level of specific variability in this volume. We encourage readers to look at chapters in other disciplines as well in order to see more clearly the common features across chapters, and the variation in approach among the chapters. This volume could not have come to life without the help and dedica- tion of many people, and we are grateful to them. First and foremost, the committee acknowledges the contributions of Robbie Case, who was to have contributed to the mathematics chapters in this volume. Robbie was at
ix PREFACE the height of a very productive career when his life came to an unexpected end in May 2000. Robbie combined the very best in disciplinary research and attention to the incorporation of research findings into classroom tools to support teaching and learning. In this respect, he was a model for re- searchers interested in supporting improved educational practice. The math- ematics chapters in this volume are marked by Robbie Caseâs influence. The financial support of our sponsors, the U.S. Department of Educa- tion and the Presidentâs Circle of the National Academy of Sciences, was essential. We appreciate both their support and their patience during the unexpectedly long period required to shape and produce so extensive a volume with so many different contributors. Our thanks to C. Kent McGuire, former assistant secretary of the Office of Education Research and Improve- ment for providing the initial grant for this project, and to his successor and now director of the National Institute for Education Sciences, Grover J. Whitehurst; thanks are due as well to Patricia OâConnell Ross, Jill Edwards Staton, Michael Kestner, and Linda Jones at the Department of Education for working with us throughout, and providing the time required to produce a quality product. This report is a somewhat unusual undertaking for the National Re- search Council in that the committee members did not author the report chapters, but served as advisers to the chapter authors. The contributions of committee members were extraordinary. In a first meeting the committee and chapter authors worked together to plan the volume. The committee then read each draft chapter, and provided extensive, and remarkably pro- ductive, feedback to chapter authors. As drafts were revised, committee members reviewed them again, pointing out concerns and proposing poten- tial solutions. Their generosity and their commitment to the goal of this project are noteworthy. Alexandra Wigdor, director of the Division on Education, Labor, and Human Performance when this project was begun, provided ongoing guid- ance and experienced assistance with revisions. Rona BriÃ¨re brought her special skills in editing the entire volume. Our thanks go to Allison E. Shoup, who was senior project assistant, supporting the project through much of its life; to Susan R. McCutchen, who prepared the manuscript for review; to Claudia Sauls and Candice Crawford, who prepared the final manuscript; and to Deborah Johnson, Sandra Smotherman, and Elizabeth B. Townsend, who willingly provided additional support when needed. Kirsten Sampson Snyder handled the report review process, and Yvonne Wise handled report productionâboth challenging tasks for a report of this size and complexity. We are grateful for their help. This report has been reviewed in draft form by individuals chosen for their diverse perspectives and technical expertise, in accordance with proce- dures approved by the National Research Councilâs Report Review Commit-
x PREFACE tee. 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 re- view comments and draft manuscript remain confidential to protect the in- tegrity of the deliberative process. We thank the following individuals for their review of this report: Jo Boaler, Mathematics Education, School of Edu- cation, Stanford University; Miriam L. Clifford, Mathematics Department, Carroll College, Waukesha, Wisconsin; O.L. Davis, Curriculum and Instruction, The University of Texas at Austin; Patricia B. Dodge, Science Teacher, Essex Middle School, Essex Junction, Vermont; Carol T. Hines, History Teacher, Darrel C. Swope Middle School, Reno, Nevada; Janis Lariviere, UTeachâ Science and Mathematics Teacher Preparation, The University of Texas at Austin; Gaea Leinhardt, Learning Research and Development Center and School of Education, University of Pittsburgh; Alan M. Lesgold, Office of the Provost, University of Pittsburgh; Marcia C. Linn, Education in Mathematics, Science, and Technology, University of California, Berkeley; Kathleen Metz, Cognition and Development, Graduate School of Education, University of California, Berkeley; Thomas Romberg, National Center for Research in Math- ematics and Science Education, University of WisconsinâMadison; and Peter Seixas, Centre for the Study of Historical Consciousness, University of British Columbia. Although the reviewers listed above have provided many constructive comments and suggestions, they did not see the final draft of the report before its release. The review of this report was overseen by Alan M. Lesgold, University of Pittsburgh. Appointed by the National Research Council, he was 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 authors, the committee, and the institution. John D. Bransford, Chair M. Suzanne Donovan, Study Director
xi Contents 1 Introduction 1 M. Suzanne Donovan and John D. Bransford A Fish Story, 2 Learning Environments and the Design of Instruction, 12 Putting the Principles to Work in the Classroom, 20 Intent and Organization of This Volume, 21 Notes, 25 References, 26 Part I History (on enclosed CD; not printed in this volume) 2 Putting Principles into Practice: Understanding History 31 Peter J. Lee History and Everyday Ideas, 33 Substantive Concepts, 61 History That Works, 65 Notes, 73 References, 74 3 Putting Principles into Practice: Teaching and Planning 79 Rosalyn Ashby, Peter J. Lee, and Denis Shemilt The Reality Test, 80 Working with Evidence: Pilgrim Fathers and Native Americans, 84 Working with Evidence: The St. Brendanâs Voyage Task, 119
xii CONTENTS Appendix 3A: Implications for Planning, 164 Notes, 177 References, 177 4 âThey Thought the World Was Flat?â: Applying the Principles of How People Learn in Teaching High School History 179 Robert B. Bain Where to Begin? Transforming Topics and Objectives into Historical Problems, 181 Designing a âHistory-Considerateâ Learning Environment: Tools for Historical Thinking, 199 Conclusion, 209 Acknowledgments, 210 Notes, 211 References, 212 Part II Mathematics 5 Mathematical Understanding: An Introduction 217 Karen C. Fuson, Mindy Kalchman, and John D. Bransford Principle #1: Teachers Must Engage Studentsâ Preconceptions, 219 Principle #2: Understanding Requires Factual Knowledge and Conceptual Frameworks, 231 Principle #3: A Metacognitive Approach Enables Student Self-Monitoring, 236 Next Steps, 243 Notes, 246 References, 246 Suggested Reading List for Teachers, 256 6 Fostering the Development of Whole-Number Sense: Teaching Mathematics in the Primary Grades 257 Sharon Griffin Deciding What Knowledge to Teach, 259 Building on Childrenâs Current Understandings, 267 Acknowledging Teachersâ Conceptions and Partial Understandings, 279 Revisiting Question 2: Defining the Knowledge That Should Be Taught, 281 How Can This Knowledge Be Taught?: The Case of Number Worlds, 282 What Sorts of Learning Does This Approach Make Possible?, 302 Summary and Conclusion, 305
xiii CONTENTS Acknowledgments, 306 Notes, 306 References, 306 7 Pipes, Tubes, and Beakers: New Approaches to Teaching the Rational-Number System 309 Joan Moss Rational-Number Learning and the Principles of How People Learn, 312 Instruction in Rational Number, 319 Conclusion: How Students Learn Rational Number, 341 Notes, 343 References, 345 8 Teaching and Learning Functions 351 Mindy Kalchman and Kenneth R. Koedinger Addressing the Three Principles, 359 Teaching Functions for Understanding, 373 Summary, 389 Acknowledgments, 391 Notes, 392 References, 392 Other Relevant Readings, 393 Part III Science (on enclosed CD; not printed in this volume) 9 Scientific Inquiry and How People Learn 397 John D. Bransford and M. Suzanne Donovan Principle #1: Addressing Preconceptions, 399 Principle #2: Knowledge of What It Means to âDo Science,â 403 Principle #3: Metacognition, 407 The How People Learn Framework, 411 Conclusion, 415 Notes, 416 References, 416 10 Teaching to Promote the Development of Scientific Knowledge and Reasoning About Light at the Elementary School Level 421 Shirley J. Magnusson and Annemarie Sullivan Palinscar The Study of Light, 422 The Study of Light Through Inquiry, 426 Supporting Learning Through Cycles of Investigation, 460
xiv CONTENTS The Role of Subject-Specific Knowledge in Effective Science Instruction, 467 Conclusion, 469 Notes, 470 References, 472 11 Guided Inquiry in the Science Classroom 475 James Minstrell and Pamela Kraus The Unit: The Nature of Gravity and Its Effects, 477 Summary, 511 Notes, 512 12 Developing Understanding Through Model-Based Inquiry 515 James Stewart, Jennifer L. Cartier, and Cynthia M. Passmore Genetics, 516 Developing Darwinâs Model of Natural Selection in High School Evolution, 540 Classroom Environments That Support Learning with Understanding, 555 Summary, 561 Notes, 562 References, 563 A Final Synthesis: Revisiting the Three Learning Principles 13 Pulling Threads 569 M. Suzanne Donovan and John D. Bransford Engaging Resilient Preconceptions, 569 Organizing Knowledge Around Core Concepts, 575 Supporting Metacognition, 577 Principles of Learning and Classroom Environments, 586 Notes, 588 References, 589 Other Resources, 590 Biographical Sketches of Committee Members and Contributors 591 Index 597
How Students Learn MATHEMATICS IN THE CLASSROOM