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 SCHABLE, 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 Transform
in Rational Mathematics
Numbers and Teaching and
Proportionality Learning, Volume 1
Margaret Schwan Smith
Edward A. Silver
Mary Kay Stein
with Melissa Boston,
Marjorie A. Henningsen,
and Amy F. Hillen
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 rational numbers and proportionality / Margaret Schwan Smith,
Edward A. Silver, Mary Kay Stein, with Melissa Boston, Marjorie A. Henningsen, and
Amy F. Hillen.
p. cm. — (Using cases to transform mathematics teaching and learning ; v. 1) (Ways of
knowing in science and mathematics series)
Includes bibliographical references and index.
ISBN 0-8077-4529-4 (pbk. : acid-free paper)
1. Mathematics—Study and teaching (Middle school)—United States—Case studies. I.
Silver, Edward A., 1948– II. Stein, Mary Kay. III. Title IV. Series.
QA13.S58 2005
513'.071'2—dc22 2004055390
ISBN 0-8077-4529-4 (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 27
THE CASE OF MARIE HANSON 28
Introduction xi Analyzing the Case 40
Great Expectations for Middle Grades Extending Your Analysis of the Case 40
Mathematics xi
Connecting to Your Own Practice 41
The Materials in This Volume xii
Exploring Curricular Materials 41
This Volume and Its Companions xiv
Connecting to Other Ideas and Issues 42
4. Comparing Similar Figures—
PART I
THE CASE OF MARCIA GREEN 44
USING CASES TO ENHANCE LEARNING
Opening Activity 44
1. Using Cases to Learn 3 Reading the Case 45
The Cases 3 THE CASE OF MARCIA GREEN 46
The Cases as Learning Opportunities 4 Analyzing the Case 54
Using the Cases 5 Extending Your Analysis of the Case 54
Connecting to Your Own Practice 55
2. Connecting Fractions, Decimals, and Exploring Curricular Materials 56
Percents—THE CASE OF RANDY HARRIS 8
Connecting to Other Ideas and Issues 56
Opening Activity 8
Reading the Case 8 5. Exploring Problems Involving
THE CASE OF RANDY HARRIS 10 Ratios and Percents—
Analyzing the Case 23 THE CASE OF JANICE PATTERSON 58
Extending Your Analysis of the Case 23 Opening Activity 58
Connecting to Your Own Practice 24 Reading the Case 58
Exploring Curricular Materials 24 THE CASE OF JANICE PATTERSON 60
Connecting to Other Ideas and Issues 25 Analyzing the Case 69
Extending Your Analysis of the Case 69
3. Introducing Ratios and Proportions— Connecting to Your Own Practice 70
THE CASE OF MARIE HANSON 26 Exploring Curricular Materials 70
Opening Activity 26 Connecting to Other Ideas and Issues 71
vii
viii Contents
PART II Appendix A: Sample Responses to
FACILITATING LEARNING FROM CASES THE CASE OF RANDY HARRIS 109
Teacher-Generated Solutions to the
6. Using Cases to Support Learning
Opening Activity 109
About Teaching 75
Facilitator-Generated List of Supporting
Why Cases? 75
and Inhibiting Factors 110
Learning from Cases 76
Teacher-Generated List of Supporting
What Can Be Learned from Our Cases 77 and Inhibiting Factors 110
Preparing for and Facilitating Case
Discussions 78 Appendix B: Sample Responses to
THE CASE OF MARIE HANSON 114
Putting the Pieces Together 80
Teacher-Generated Solutions to the
7. Facilitating Learning from Opening Activity 114
THE CASE OF RANDY HARRIS 81 Facilitator- and Teacher-Generated List of
Case Analysis 81 Ways in Which Marie Hanson Supported
Her Students’ Learning of Mathematics 116
Facilitating the Opening Activity 85
Facilitating the Case Discussion 86
Appendix C: Sample Responses to
Extending the Case Experience 87
THE CASE OF MARCIA GREEN 119
Teacher-Generated Solutions to the
8. Facilitating Learning from
Opening Activity 119
THE CASE OF MARIE HANSON 88
List of Marcia Green’s Questions 120
Case Analysis 88
Teacher-Generated Criteria for
Facilitating the Opening Activity 93
Good Questions 120
Facilitating the Case Discussion 94
Extending the Case Experience 95
Appendix D: Sample Responses to
THE CASE OF JANICE PATTERSON 123
9. Facilitating Learning from
Teacher-Generated Solutions to the
THE CASE OF MARCIA GREEN 96
Opening Activity 123
Case Analysis 96
Facilitator-Generated Responses to the
Facilitating the Opening Activity 100
Question, “Did Janice Patterson Really
Facilitating the Case Discussion 101 Teach?” 127
Extending the Case Experience 102 Teacher-Generated Responses to the
Question, “Did Janice Patterson
10. Facilitating Learning from
Really Teach?” 127
THE CASE OF JANICE PATTERSON 103
Case Analysis 103 References 131
Facilitating the Opening Activity 106
About the Authors 135
Facilitating the Case Discussion 107
Extending the Case Experience 108 Index 137
Acknowledgments
The ideas expressed in this book grew out of our work Gilberto Cuevas, Beatrice D’Ambrosio, Skip Fennell,
on the QUASAR Project and have developed over the past Linda Foreman, Susan Friel, Judith Jacobs, Jeremy
decade through our interactions and collaborations with Kahan, Rebecca McGraw, Jack Moyer, Kathy Pfaendler,
many teachers, teacher educators, and mathematicians. Elizabeth Phillips, and Judith Zawojewski, who piloted
We would like to thank mathematicians and mathemat- early versions of the cases, provided helpful feedback,
ics teacher educators Hyman Bass, John Beem, Nadine and expanded our view regarding the possible uses of
Bezuk, Kathleen Cramer, George Bright, Victoria Kouba, the cases and related materials.
John Moyer, John P. Smith III, Judith Roitman, and Orit Finally, we would like to acknowledge the contribu-
Zaslavsky, who provided feedback on early versions of the tions of Cristina Heffernan, who developed the COMET
cases. Your thoughtful comments helped ensure that the website, provided feedback on early versions of the cases
cases were both sound and compelling. and facilitation materials, and identified tasks in other
We are indebted to Victoria Bill whose varied and curricula that corresponded to the cases; Michael Steele,
frequent use of the cases over the past 5 years has helped who provided technological assistance in preparing the
us recognize the flexibility and power of the cases to final manuscript; and Kathy Day, who provided valu-
promote learning in a range of situations; and to our able assistance in preparing figures, locating data, and
colleagues Fran Arbaugh, Cathy Brown, Marta Civil, copying materials.
ix
