Knowing and Learning Mathematics for Teaching

Mathematics Teacher Preparation Content Workshop Program Steering Committee

Center for Education

Mathematical Sciences Education Board

National Research Council

NATIONAL ACADEMY PRESS
Washington, D.C.



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Knowing and Learning Mathematics for Teaching Knowing and Learning Mathematics for Teaching Mathematics Teacher Preparation Content Workshop Program Steering Committee Center for Education Mathematical Sciences Education Board National Research Council NATIONAL ACADEMY PRESS Washington, D.C.

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Knowing and Learning Mathematics for Teaching NATIONAL ACADEMY PRESS 2101 Constitution Avenue, N.W. Washington, D.C. 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 Grant No. DUE-9706060 between the National Academy of Sciences 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. International Standard Book Number 0-309-07252-2 Library of Congress Catalog Card Number 00-110978 Additional copies of this report are available from National Academy Press, 2101 Constitution Avenue, N.W., Lockbox 285, Washington, D.C. 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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Knowing and Learning Mathematics for Teaching 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. William 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. William A. Wulf are chairman and vice chairman, respectively, of the National Research Council.

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Knowing and Learning Mathematics for Teaching MATHEMATICS TEACHER PREPARATION CONTENT WORKSHOP PROGRAM STEERING COMMITTEE Deborah Loewenberg Ball, Chair, University of Michigan Richard Askey, * University of Wisconsin-Madison Hyman Bass, * Columbia University Genevieve Knight, Coppin State College Mark Saul, Bronxville High School Deborah Schifter, Education Development Center, Inc. Olga Garcia Torres, Tucson Unified School District Staff Rodger Bybee, Executive Director, Center for Science, Mathematics, and Engineering Education (CSMEE) Joan Ferrini-Mundy, Associate Executive Director, CSMEE Gail Burrill, Project Director Bradford Findell, Program Officer Kirsten Sampson Snyder, Reports Officer Doug Sprunger, Senior Project Assistant Danna Brennan, Project Assistant * Member of the National Academy of Sciences

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Knowing and Learning Mathematics for Teaching MATHEMATICAL SCIENCES EDUCATION BOARD (MSEB) 1998-1999 Hyman Bass, Chair, * Columbia University Jere Confrey, Vice Chair, University of Texas at Austin Richard A. Askey, * University of Wisconsin-Madison Sherry Baca, Prescott Unified School District Deborah Loewenberg Ball, University of Michigan Benjamin Blackhawk, St. Paul Academy and Summit School Richelle Blair, Lakeland Community College Patricia Campbell, University of Maryland Ingrid Daubechies, * Princeton University Karen Economopoulos, TERC Susan Eyestone, National Parent Teachers Association Lee Jenkins, Antioch Unified School District Glenda T. Lappan, Michigan State University Miriam Masulo, IBM Corporation David Moore, Purdue University Mari Muri, Connecticut Department of Education Richard Normington, TQM Services Group Mark Saul, Bronxville Public Schools Richard Schoen, Stanford University Edward A. Silver, University of Pittsburgh William Tate, University of Wisconsin-Madison Jerry Uhl, University of Illinois at Urbana-Champaign Susan S. Wood, J. Sargeant Reynolds Community College Staff Joan Ferrini-Mundy, Director, through June, 1999 Gail Burrill, Director Bradford Findell, Program Officer Sonja Atkinson, Administrative Assistant * Member of National Academy of Sciences

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Knowing and Learning Mathematics for Teaching Dedication James R. C. Leitzel was a member of the Mathematical Sciences Education Board (MSEB) from 1994 to 1997. Jim's contributions to mathematics education are vast and diverse, but one of his principal commitments was to the improvement of teacher preparation and professional development. He was an articulate advocate for the Board's initiatives in this area. Jim served on the Professional Development of Teachers of Mathematics Working Group that contributed to the National Council of Teachers of Mathematics' Professional Standards for Teaching Mathematics and was the editor of the Mathematical Association of America's Call for Change: Recommendations for the Mathematical Preparation of Teachers of Mathematics. Jim's dedication to teachers and their growth as mathematics teachers also extended to those in his classes at The Ohio State University, the University of Nebraska-Lincoln, and the University of New Hampshire and to the many mathematics educators for whom he became a mentor. Jim passed away in 1998 but left a legacy of concern, care, and nurturing for teachers and teaching. In recognition of his contributions both to the MSEB and to the mathematics education community, we dedicate this book to him.

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Knowing and Learning Mathematics for Teaching Acknowledgments We would like to acknowledge the staff at the National Research Council 's (NRC) Center for Science, Mathematics, and Engineering Education (CSMEE) for their efforts in putting the Workshop together. In particular, Doug Sprunger was instrumental in overseeing arrangements for the meeting and in arranging these proceedings for review and publication. Onsite support was also provided by Kirsten Sampson Snyder and Tina Winters. We are grateful to the members of the Program Steering Committee for their oversight in planning the program for the Workshop. We also wish to acknowledge the speakers and in particular the discussion group leaders for their contributions and leadership that gave substance to the discussion. 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 the 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: Shelly Ferguson, National Council of Teachers of Mathematics Kay McClain, Vanderbilt University Albert Otto, Illinois State University Tad Watanabe, Towson State University Laura van Zoost, University of Western Michigan While the individuals listed above have provided many constructive comments and suggestions, responsibility for the final content of this report rests solely with the steering committee and the National Research Council. Gail Burrill Project Director Mathematical Sciences Education Board

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Knowing and Learning Mathematics for Teaching Contents     Introduction   1     Workshop Overview: Knowing and Learning Mathematics for Teaching   3     Pre-Workshop Tasks   8     Teachers' Understanding of Fundamental Mathematics   11      Knowledge of Fundamental Mathematics for Teaching Liping Ma, Cathy Kessel   12      Elementary Teachers and Essential Mathematical Knowledge Mark Saul   17      What Is Fundamental Mathematics? Genevieve Knight   20     RECONSIDERING THE MATHEMATICS THAT TEACHERS NEED TO KNOW   23      Investigating Teaching Practice: Setting the Stage   25      What Mathematical Knowledge Is Entailed in Teaching Children to Reason Mathematically? Deborah Loewenberg Ball, Hyman Bass   26      Reaction to the Presentation by Deborah Ball and Hyman Bass James Lewis   35      Investigating Teaching Practice: What Mathematical Knowledge, Skills, and Sensibilities Does It Take?   39      Analyzing Student Thinking Virginia Bastable   40      Remodeling Mathematical Tasks Olga G. Torres   49      Analyzing Student Work Michaele F. Chappell   53      Managing Class Discussion Erick Smith   60

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Knowing and Learning Mathematics for Teaching      What Kinds of Mathematical Knowledge Matter in Teaching?   65      Perspectives from a Mathematician Alan Tucker   66      Perspectives from a Mathematics Educator Deborah Schifter   69      Perspectives from the Community College Gladys Whitehead   72     HOW CAN TEACHERS DEVELOP SUCH MATHEMATICAL KNOWLEDGE?   75      Investigating Alternative Approaches to Helping Teachers Learn Mathematics   77      Student Curriculum Materials: Japanese Teachers' Manuals Shin-ying Lee   78      Student Curriculum Materials: Investigations in Number, Data, and Space Marco Ramirez   86      Case Materials Carne Barnett   90      Programs and Practices Deborah Schifter, Virginia Bastable, and Jill Bodner Lester   94      Video as a Delivery Mechanism Bradford Findell, Deborah Loewenberg Ball   98      Promising Approaches for Helping Prospective Elementary Teachers Learn Mathematics for Teaching   105      Looking at Textbooks Richard Askey   106      The Professional Growth of a Classroom Teacher Carol Midgett   116      The Importance of Mathematical Content Alice Gill   119      NSF and Teacher Preparation Programs James Lightbourne   122      Where Are We? Moderator's Summary Joan Ferrini-Mundy   125      Concluding Remarks Deborah Loewenberg Ball   127     DISCUSSION GROUP REPORTS   129      Question #1        Discussion Group #1   131      Discussion Group #10   133      Question #2        Discussion Group #2   137      Discussion Group #9   140

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Knowing and Learning Mathematics for Teaching      Question #3        Discussion Group #3   144      Discussion Group #8   147      Question #4        Discussion Group #4   149      Discussion Group #7   152      Question #5        Discussion Group #5   155      Discussion Group #6   158     APPENDICES   163     Appendix A: Pre-Workshop Reading   165     Appendix B: Workshop Agenda   166     Appendix C: Homework Problems   169     Appendix D: Workshop Participant List   170     Appendix E: Transcript of Ball Videos   174     Appendix F: Explanation of the Unit on Weight   183     Appendix G: Excerpts from Investigations   193     Appendix H: Mathematics Case Methods Project   201     Appendix I: Biographical Information   210

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