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MATHEMATICS IN THE CLASSROOM
Committe e on How People Learn, A Targ eted Report for Teac hers
M. Suzanne Donovan and John D. Bransford, Editors
Division of Behavioral and Social Sciences and Education
NATIONAL RESEARCH COUNCIL
OF THE NATIONAL ACADEMIES
THE NATIONAL ACADEMIES PRESS
Washington, D.C.
www.nap.edu
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ntE NATIONAL ACADEMIES PRESS . 500 Eildh stress N.W. . Washlng~n, D.C. 20001
NOT CE The project that is the subject of this ¢port was approved by the Govern-
ing Board of the National Research Council, whose members am drawn from the
counci s of the National Academy of Sciences, the National Academy of Engineering,
and the in aitute of Medicine. The members of the committee responsible for the
nape t wed chosen for their special competences and with Regard for appropriate
balance
This study was supported by Award No R215U990024 between the Nationa Acad-
emy of Sciences and the U.s. Department of Education. Any opinions, findings,
conclusions, or recommendations ezpnessed in this publication am those of the
author s) and do not necessari y reflect the views of the organizations or agencies
that provided support for the project.
Llbrary of Congress CamlogJng-in-Publlcatlon Dam
National Research Council (U.s ) Committee on How People Learn, A Targeted
Report for Teachets.
How students learn: history, mathematics, and science in the c assnoon1 /
Committee on How People Learn, A Targeted Report for Teachets; M. Suza me
Donovan and John D. Btansford, editors.
p. cm.
"Division of Behavioral and Social Sciences and Education."
Includes bib lag aphical references and index.
ISBN 0-309 07433-9 (hardcover)—ISBN 0-309-08948-4 (pbk.)—
ISBN 0-309 08949-2 (pbk.)—ISBN 0-309 08950-6 (pbk.) 1. Learning. 2.
Class oom management. 3. Curriculum planning. 1. Donovan, Suzanne.
11 Bransford, John 111 Title
LB1060.N38 2005
370.15'23 dc22
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 a¢a); Internet, http://www.nap.edu
printed in the United States of Amenca
Copy ight 2005 by the Natkmal Academy of Sciences All rights reserved
Suggested citation: National Research Council. (2005). How Students Learn: Matl~-
ematScs In the Classroom. Committee on How People Learn, A Targeted Report for
Teachets, M.S. Donovan and J.D. Bransford, Editors. Division of Behavioral and
Socia Sciences and Education. Washington, DC: The National Academies PASS.
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THE NATIONAL ACADEMIES
Advisers to the Notion on Srienre, Engineering, and Medicine
The National Academy of Shier Ices 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
we fare. Upon the authority of the charter granted to it by the Congress in 1863, the
Academy has a mandate that Sequins it to advise the federal government on
scientific and technical matters. Dr. Bruce M. Alberts is president of the National
Academy of Sciences.
The NatlonalAcademyofEngineeringwas 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.
shanag with the National Academy of Sciences the responsibility for advising the
federal govemrrent. The National Academy of Engineering also sponsors engineer-
ing prog ams aimed at meeting national needs, encourages education and research,
and recognizes the supenor achievements of engineers. Dr. Wm. A. Wu f is presi-
dent of the National Academy of Engineenng.
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 ¢sponsibi ity given to the National Academy of Sciences by
its congressional charter to be an adviser to the fede al 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 furthenng knowledge and advising the federal govern-
ment. Functioning in accordance with general policies determined by the Academy,
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emy of Sciences and the National Academy of Engineenng in providing services to
the govemment, the public, and the scientific and engineenng communities. The
Council is administered joint y by both Academies and the institute of Medicine. Dr.
Bruce M. A berts and Dr. Wm. A. Wu f am chair and vice chair, respectively, of the
National Research Council.
www.nedional-nendemies.or9
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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, Depattment of Education, Stanford University
M. SUZANNE DONOVAN, Study Director
SUSAN R. MCCI TCH EN, Research Associate
ALLISON E. SHOUP, StniorProjectAssutant
ELIZABETH B. TOWNSEND, Senior Project Assistant
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vii
Preface
This book has its roots in the report of the Comr ittee on t)evelopments
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 corer ittee explored what research
and development would need to be done, and how it would need to be
communicated, to be especially useful to teachers, pnncipals, supennten-
dents, and policy makers: How Peop/e 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 areas history, mathematics, and sci-
ence—and 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 Cl ).
t)istinguished researchers who have extensive experience in teaching
or in partnering with teachers were invited to contribute the chapters. The
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PREFACE
committee shaped The goals for the volume, and commented—sometimes
extensively—on 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 dhat 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 odhers 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. Widh tills in mind, it is critical dhat the teaching and learning
examples in each chapter be seen as illustrative, not as blueprints for The
"nght" way to teach.
We can imagine, by analogy, dhat engineering students will better grasp
The relationship between The laws of physics and The construction of effec-
tive supports for a blidge if they see some examples of well-designed budges,
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 Budge that
could be replaced widh 0. h e - s dhat serve The same end, and people may well
differ on which among a set of good designs creates the most appealing
blidge.
To say that the Golden Gate Bridge is a good example of a suspension
blidge 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 dhat are required for success, and Those that are
replaceable, become more apparent. And The requirements dhat are uni-
form across contexts, and The requirements dhat change with context, are
more easily revealed.
The chapters in tills volume highlight different approaches to address-
ing the same fundamental principles of teaming. It would be ideal to be able
to provide two or more "HPL compatible" 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 o her 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 madhematics chapters in this volume Robbie was at
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PREFACE iX
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 Case's influence.
The financial support of our sponsors, the U. s L)epartment of Educa-
tion and the President's 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 contnbutors. Our thanks to C. Kent McGuire,
former assistant secretary of the Of fice 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 O'Connell 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 comr ittee
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, corer ittee
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 t)ivision on Education, Labor, and
Human Performance when this project was begun, provided ongoing guid-
ance and experienced assistance with revisions Rona Bnere 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
production—both 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 Council's Report Review Commit-
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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 Boater, Mathematics Education, School of Edu-
cation, Stanford University; Miriam L. Clifford, Mathematics Department, Carroll
College, Waukesha, Wisconsin; O.L. Davis, Cumculum and Instruction, The
University of Texas at Austin; Patricia B L)odge, Science Teacher, Essex
Middle School, Essex Junction, Vermont; Carol T. Hines, History Teacher,
Barrel C. Swope Middle School, Reno, Nevada; Janis Lanviere, UTeach
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, Of Lice 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 Wisconsin-Madison; and Peter
Seixas, Centre for the Study of Histoncal 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 1>. so ansfo~ d. Chair
M. Suzanne t)onovan, Study Director
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Contents
1 Introduction
M. Suzanne Donovan andJohn D. Bransford
A Fish Story, 2
Learning Environments and the Design of instruction, 12
Putting the Pnnciples to Work in the Classroom, 20
Intent and Organization of This Volume, 21
Notes, 2S
References, 26
Part I History
(on enclosed CD; not printed in this volume)
2 Putting Pnnciples into Practice: Understanding History
PeterJ Lee
History and Everyday ideas, 33
Substantive Concepts, 61
History That Works, 65
Notes, 73
References, 74
3 Putting Pnnciples into Practice: Teaching and Planning
Rosalyn. Ashby, PeferJ Lee, and Dents Shem fit
The Reality Test, 80
Working with Evidence: Pilgrim Fathers and Native Americans, 84
Working with Evidence: The St. Brendan's Voyage Task, 119
1
31
79
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X;; CONTENTS
Appendrx 3A: Implicat ons for Planning, 164
Notes, 177
References, 177
4 "They Thought the World Was Hats": Applying the Principles of
How People Learn in Teaching High School History
Rober,TB. Bain
Where to Begin? Transforming Topics and Ob ectives into
Histoncal Problems, 181
U.esigning a "History-Considerate" Leaming Environment:
Tools for Historical Thinking, 199
Conclusion, 209
Acknowledgments, 210
Notes, 211
References, 212
Pan 11 Mathematics
5 Mathematical Understanding: An Introduction
Karen C. Fuson, Mindy Ka.7chman, and John D. Bransford
Pnnciple #1 Teachers Must Engage Students' Preconceptions, 219
Pnnciple #2: Understanding Requires Factual Knowledge and
Conceptual Frameworks, 231
Pnnciple #3: A Metacognitive Approach Enables Student
SeU-Monitonng, 236
Next Steps, 243
Notes, 246
References, 246
Suggested Reading List for Teachers, 256
6 Fostenng the Development of Whole-Number Sense:
Teaching Mathemat cs in the Pmmary Grades
Sharon Griff n
L)eciding What Knowledge to Teach, 259
Building on Children's Current Understandings, 267
Acknowledging Teachers' Conceptions and Partial
Understandings, 279
Revisiting Queston 2: L)ehming the Knowledge That
Should Be Taught, 281
How Can This Knowledge Be Taught?:
The Case of Number Worlds, 282
What Sorts of Leaming Does This Approach Make Possible?, 302
Sumr ary and Conclusion, 305
179
71
17
257
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CONTENTS Xiii
Ackno\vledgnnents, 306
Notes, 306
References, 306
7 Pipes, Tubes, and Beakers: New Approaches to Teaching the
Rational-Number System
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
MindyKalchman and Kenneth R. Koedinger
Addressing the Three Principles, 359
Teaching Functions for Understanding, 373
SUITllTlaly~ 389
Acknowledgments, 391
Notes, 392
References, 392
Other Relevant Readings, 393
Partill Science
(on enclosed CD; not printed in this volume)
9 Scientific Inquiry and How People Learn
John D. Bransford and M. Suzanne Donovan
PlinCiple #1 Addressing Preconceptions, 399
Principle #2: Knowledge of What it Means to "L)o Science," 403
Principle #3: Metacognition, 407
The How People Learn Framework, 411
Conclusion, 415
Notes, 416
References, 416
309
351
397
~ ~ Teaching to Promote the Development of Scientific Knowledge
and Reasoning About Light at the Elementary School Level 421
ShirleyJ Magnusson and Annemane Sullivan Palinscar
The Study of Light, 422
The Study of Light Through inquiry, 426
Supporting Leaming Through Cycles of investigation, 460
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X;V CONTENTS
The Role of Sublect-Specific Knowledge in Effective
Science Instruction, 467
Conclusion, 469
Notes, 470
References, 472
11 Guided Inquiry in the Science Classroom
James Minsf sell and Pamela Kraus
The Unit: The Nature of Gravity and Its Effects, 477
Sumr ary, 511
Notes, 512
12 Developing Understanding Through Model-Based inquiry
James Stewart, Jennifer L. Cartier, and Cynthia M. Passmore
Genetics, 516
L)eveloping L)arwin's 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
M. Suzanne Donovan and /ohn 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
515
569
Biographical Sketches of Committee Members and Contributors 591
Index
597
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MATHEMATICS IN THE CLASSROOM
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