Table Of ContentWays of Knowing in Science and Mathematics Series
RICHARD DUSCHL, SERIES EDITOR
ADVISORY BOARD: Charles W. Anderson, Raffaella Borasi, Nancy Brickhouse,
Marvin Druger, Eleanor Duckworth, Peter Fensham, William Kyle, Roy Pea,
Edward Silver, Russell Yeany
Improving Instruction in Rational Numbers and How Students (Mis-)Understand Science and
Proportionality: Using Cases to Transform Mathematics: Intuitive Rules
Mathematics Teaching and Learning, Volume 1 RUTH STAVY AND DINA TIROSH
MARGARET SCHWAN SMITH, EDWARD A. SILVER, AND
Reading Counts: Expanding the Role of Reading in
MARY KAY STEIN, WITH MELISSA BOSTON,
MARJORIE A. HENNINGSEN, AND AMY F. HILLEN Mathematics Classrooms
RAFFAELLA BORASI AND MARJORIE SIEGEL
Improving Instruction in Algebra: Using Cases to
Transform Mathematics Teaching and Learning, Transforming Middle School Science Education
Volume 2 PAUL DEHART HURD
MARGARET SCHWAN SMITH, EDWARD A. SILVER, AND
Designing Project-Based Science: Connecting
MARY KAY STEIN, WITH MARJORIE A. HENNINGSEN,
Learners Through Guided Inquiry
MELISSA BOSTON, AND ELIZABETH K. HUGHES
JOSEPH L. POLMAN
Improving Instruction in Geometry and Measurement:
Using Cases to Transform Mathematics Teaching and Implementing Standards-Based Mathematics
Learning, Volume 3 Instruction: A Casebook for Professional Development
MARGARET SCHWAN SMITH, EDWARD A. SILVER, AND MARY KAY STEIN, MARGARET SCHWAN SMITH,
MARY KAY STEIN, WITH MELISSA BOSTON AND MARJORIE A. HENNINGSEN, AND EDWARD A. SILVER
MARJORIE A. HENNINGSEN
The New Science Teacher: Cultivating Good Practice
Standards-Based Mathematics Assessment in Middle DEBORAH TRUMBULL
School: Rethinking Classroom Practice
Problems of Meaning in Science Curriculum
THOMAS A. ROMBERG, EDITOR
DOUGLAS A. ROBERTS AND
Mathematical Development in Young Children: LEIF ÖSTMAN, EDITORS
Exploring Notations
Inventing Science Education for the New Millennium
BÁRBARA M. BRIZUELA
PAUL DEHART HURD
Inside Science Education Reform: A History of
Improving Teaching and Learning in Science and
Curricular and Policy Change
Mathematics
J MYRON ATKIN AND PAUL BLACK
DAVID F. TREAGUST, REINDERS DUIT, AND
BARRY J. FRASER, EDITORS
Connecting Girls and Science: Constructivism,
Feminism, and Science Education Reform Reforming Mathematics Education in America’s Cities:
ELAINE V. HOWES The Urban Mathematics Collaborative Project
NORMAN L. WEBB AND THOMAS A. ROMBERG, EDITORS
Investigating Real Data in the Classroom: Expanding
Children’s Understanding of Math and Science What Children Bring to Light: A Constructivist
RICHARD LEHRER AND LEONA SCHAUBLE, EDITORS Perspective on Children’s Learning in Science
BONNIE L. SHAPIRO
Free-Choice Science Education: How We Learn
Science Outside of School STS Education: International Perspectives on Reform
JOHN H. FALK, EDITOR JOAN SOLOMON AND GLEN AIKENHEAD, EDITORS
Science Teaching/Science Learning: Constructivist Reforming Science Education: Social Perspectives
Learning in Urban Classrooms and Personal Reflections
ELNORA S. HARCOMBE RODGER W. BYBEE
FROM TEACHERS COLLEGE PRESS
A comprehensive program to improve mathematics
instruction and student achievement Own the
Complete
Set!
Improving Instruction in
Rational Numbers and Proportionality
Using Cases to Transform Mathematics Teaching
and Learning (Volume 1)
Margaret Schwan Smith, Edward A. Silver, & Mary Kay Stein with
Melissa Boston, Marjorie A. Henningsen, and Amy F. Hillen
Improving Instruction in Algebra
Using Cases to Transform Mathematics Teaching
and Learning (Volume 2)
Margaret Schwan Smith, Edward A. Silver, & Mary Kay Stein with
Marjorie A. Henningsen, Melissa Boston, and Elizabeth K. Hughes
Improving Instruction in
Geometry and Measurement
Using Cases to Transform Mathematics Teaching
and Learning (Volume 3)
Margaret Schwan Smith, Edward A. Silver, & Mary Kay Stein with
Melissa Boston and Marjorie A. Henningsen
ALSO OF INTEREST FROM TEACHERS COLLEGE PRESS
Standards-Based Mathematics Implementing Standards-Based
Assessment in Middle School Mathematics Instruction
Rethinking Classroom Practice A Casebook for Professional Development
Edited by Thomas A. Romberg Mary Kay Stein, Margaret Schwan Smith,
Marjorie A. Henningsen, and
Edward A. Silver
www.tcpress.com
Improving Using Cases to
Instruction in Transform
Geometry Mathematics
and Teaching and
Measurement Learning, Volume 3
Margaret Schwan Smith
Edward A. Silver
Mary Kay Stein
with Melissa Boston and
Marjorie A. Henningsen
Teachers College, Columbia University
New York and London
The material in this book is based on work supported by
National Science Foundation grant number ESI-9731428 for the
COMET (Cases of Mathematics Instruction to Enhance Teaching)
Project. Any opinions expressed herein are those of the
authors and do not necessarily represent the views of the
National Science Foundation.
Published by Teachers College Press, 1234 Amsterdam Avenue, New York, NY 10027
Copyright © 2005 by Teachers College, Columbia University
All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or
mechanical, including photocopy, or any information storage and retrieval system, without permission from the publisher.
Library of Congress Cataloging-in-Publication Data
Smith, Margaret Schwan.
Improving instruction in geometry and measurement / Margaret Schwan Smith,
Edward A. Silver, Mary Kay Stein, with Marjorie A. Henningsen, and Melissa Boston.
p. cm. — (Using cases to transform mathematics teaching and learning ; v. 3) (Ways of
knowing in science and mathematics series)
Includes bibliographical references and index.
ISBN 0-8077-4531-6 (pbk. : acid-free paper)
1. Geometry—Study and teaching (Middle school)—United States—Case studies. I.
Silver, Edward A., 1948– II. Stein, Mary Kay. III. Title IV. Series.
QA461.S65 2005
516'.0071'2—dc22 2004055350
ISBN 0-8077-4531-6 (paper)
Printed on acid-free paper
Manufactured in the United States of America
12 11 10 09 08 07 06 05 8 7 6 5 4 3 2 1
To the teachers in the QUASAR Project—although nearly a
decade has passed since our work together ended, we con-
tinue to draw inspiration from your work. You were true
pioneers in creating instructional environments that pro-
moted mathematics learning for all students. Thank you for
sharing your successes and struggles with us. We continue
to learn so much from you.
Contents
Acknowledgments ix Reading the Case 21
THE CASE OF ISABELLE OLSON 23
Introduction xi Analyzing the Case 35
Great Expectations for Middle Grades
Extending Your Analysis of the Case 35
Mathematics xi
Connecting to Your Own Practice 36
The Materials in This Volume xii
Exploring Curricular Materials 36
Why Geometry and Measurement? xiv
Connecting to Other Ideas and Issues 37
This Volume and Its Companions xv
4. Exploring Volume and
PART I: Surface Area—
USING CASES TO ENHANCE LEARNING THE CASE OF KEITH CAMPBELL 39
Opening Activity 39
1. Using Cases to Learn 3 Reading the Case 40
The Cases 3 THE CASE OF KEITH CAMPBELL 41
The Cases as Learning Opportunities 4 Analyzing the Case 52
Using the Case Materials 5 Extending Your Analysis of the Case 52
Connecting to Your Own Practice 53
2. Reasoning About Units for Linear
and Area Measure— Exploring Curricular Materials 53
THE CASE OF BARBARA CRAFTON 8 Connecting to Other Ideas and Issues 54
Opening Activity 8
Reading the Case 9 5. Estimating and
THE CASE OF BARBARA CRAFTON 10 Calculating Volume—
THE CASE OF NANCY UPSHAW 56
Analyzing the Case 18
Opening Activity 56
Extending Your Analysis of the Case 18
Reading the Case 56
Connecting to Your Own Practice 19
THE CASE OF NANCY UPSHAW 58
Exploring Curricular Materials 19
Analyzing the Case 69
Connecting to Other Ideas and Issues 20
Extending Your Analysis of the Case 69
3. Exploring Area and Perimeter— Connecting to Your Own Practice 70
THE CASE OF ISABELLE OLSON 21 Exploring Curricular Materials 70
Opening Activity 21 Connecting to Other Ideas and Issues 71
vii
viii Contents
PART II: Appendix A. Sample Responses to
FACILITATING LEARNING FROM CASES THE CASE OF BARBARA CRAFTON 109
Teacher-Generated Solutions to the
6. Using Cases to Support Learning
Opening Activity 109
About Teaching 75
Teacher-Generated Responses
Why Cases? 75
to the “Analyzing the Case”
Learning from Cases 76
Activity 110
What Can Be Learned from Our Cases 77
Preparing for and Facilitating Case
Appendix B. Sample Responses to
Discussions 78
THE CASE OF ISABELLE OLSON 112
Putting the Pieces Together 80
Teacher-Generated Solutions to the
Opening Activity 112
7. Facilitating Learning from
THE CASE OF BARBARA CRAFTON 81 Teacher-Generated Responses
Case Analysis 81 to the “Analyzing the Case”
Activity 116
Facilitating the Opening Activity 84
Facilitating the Case Discussion 85
Extending the Case Experience 86 Appendix C. Sample Responses to
THE CASE OF KEITH CAMPBELL 118
8. Facilitating Learning from Teacher-Generated Solutions to the
THE CASE OF ISABELLE OLSON 87 Opening Activity 118
Case Analysis 87
Teacher-Generated Responses
Facilitating the Opening Activity 90
to the “Analyzing the Case”
Facilitating the Case Discussion 92 Activity 119
Extending the Case Experience 93
Appendix D. Sample Responses to
9. Facilitating Learning from
THE CASE OF NANCY UPSHAW 121
THE CASE OF KEITH CAMPBELL 94
Teacher-Generated Solutions to the
Case Analysis 94
Opening Activity 121
Facilitating the Opening Activity 98
Teacher-Generated Responses
Facilitating the Case Discussion 100
to the “Analyzing the Case”
Extending the Case Experience 101
Activity 123
10. Facilitating Learning from
THE CASE OF NANCY UPSHAW 102 References 125
Case Analysis 102
Facilitating the Opening Activity 106 About the Authors 129
Facilitating the Case Discussion 107
Extending the Case Experience 108 Index 131
Acknowledgments
The ideas expressed in this book grew out of our work Linda Foreman, Susan Friel, Judith Jacobs, Jeremy
on the QUASAR Project and have developed over the past Kahan, Rebecca McGraw, Jack Moyer, Kathy Pfaendler,
decade through our interactions and collaborations Elizabeth Phillips, and Judith Zawojewski who piloted
with many teachers, teacher educators, and mathema- early versions of the cases, provided helpful feedback,
ticians. We would like to thank mathematicians and and expanded our view regarding the possible uses of
mathematics teacher educators Hyman Bass, John Beem, the cases and related materials.
Nadine Bezuk, Kathleen Cramer, George Bright, Vic- Finally we would like to acknowledge the contribu-
toria Kouba, John Moyer, John P. Smith III, Judith tions of Cristina Heffernan who developed the COMET
Roitman, and Orit Zaslavsky, who provided feedback website, provided feedback on early versions of the cases
on early versions of the cases. Your thoughtful com- and facilitation materials, and identified tasks in other
ments helped ensure that the cases were both sound and curricula that corresponded to the cases; Michael Steele
compelling. who created final versions of the figures and provided
We are indebted to Victoria Bill whose varied and technical assistance; Amy Fleeger Hillen who provided
frequent use of the cases over the past 5 years has helped feedback on all aspects of the book and assistance in the
us recognize the flexibility and power of the cases to final stages of writing the facilitation chapters and ap-
promote learning in a range of situations; and to our pendices; and Kathy Day who provided valuable assis-
colleagues Fran Arbaugh, Cathy Brown, Marta Civil, tance in preparing initial versions of the figures, locating
Gilberto Cuevas, Beatrice D’Ambrosio, Skip Fennell, data, and copying materials.
ix
