Table Of ContentModeling
Mathematical Ideas
Modeling
Mathematical Ideas
Developing Strategic Competence
in Elementary and Middle School
Jennifer M. Suh
and Padmanabhan Seshaiyer
ROWMAN & LITTLEFIELD
Lanham • Boulder • New York • London
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Copyright © 2017 by Jennifer M. Suh and Padmanabhan Seshaiyer
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British Library Cataloguing in Publication Information Available
Library of Congress Cataloging-in-Publication Data
Names: Suh, Jennifer M., 1971- | Seshaiyer, Padmanabhan, 1972-
Title: Modeling mathematical ideas : developing strategic competence in elementary and
middle school / Jennifer M. Suh and Padmanabhan Seshaiyer.
Description: Lanham : Rowman & Littlefield, [2017] | Includes bibliographical references and
index.
Identifiers: LCCN 2016042051 (print) | LCCN 2016043048 (ebook) | ISBN 9781475817584
(cloth) | ISBN 9781475817591 (pbk.) | ISBN 9781475817607 (ebook)
Subjects: LCSH: Mathematics—Study and teaching (Elementary) | Mathematics—Study and
teaching (Middle school) | Cognition in children.
Classification: LCC QA135.6 .S84 2017 (print) | LCC QA135.6 (ebook) | DDC 372.7—dc23
LC record available at https://lccn.loc.gov/2016042051
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Printed in the United States of America
Contents
Preface ix
1 Developing Strategic Competence through Modeling
Mathematical Ideas 1
1.1 Developing Strategic Competence through Modeling
Mathematical Ideas 2
1.2 Promoting Mathematical Proficiency and Practices 5
1.3 Problem Solving and Mathematical Modeling in the Elementary
and Middle Grades 10
1.4 Multiple Representations and Strategies as Tools to Cultivate Visible
Thinking in Mathematics 13
1.5 Importance of Understanding the Vertical Learning
Progression to Deepen Students’ Mathematical Understanding
of Conceptual Models 20
1.6 Technology Integration in Problem Solving 22
1.7 More Related Rich Problems to Explore 23
2 Setting Math Norms to Promote Math Reasoning and Modeling:
Developing Mathematical Hearts and Minds 25
2.1 Developing Persistent Problem Solvers with a Productive
Disposition toward Mathematics 25
2.2 Unpacking the Mathematics for Deeper Conceptual Learning—
Concept Mapping 28
2.3 Choosing Worthwhile Tasks through Cognitive Demand Analysis 30
2.4 Promoting Effective Teaching Practices to Optimize
the Learning Environment 31
2.5 Integration Technology to Motivate and Differentiate Learning 34
2.6 Assessing Students Understanding through a Problem-based Task 36
3 Engaging in Mathematical Modeling in the Elementary
and Middle Grades 39
3.1 Mathematical Modeling in the Elementary and Middle Grades:
What are the Building Blocks? 39
v
vi Contents
3.2 Mathematical Modeling through Unstructured Real-world Problems 44
3.3 Lesson Study Focused on Mathematical Modeling: Traffic-jam Task 47
3.4 Promoting the Twenty-first-Century Skills 51
3.5 Technology Integration in Problem Posing and Problem Solving 53
3.6 A Related Rich Problem to Explore 55
4 Understanding Numbers and Operations 57
4.1 Lesson Study Vignette: Prime and Composite Numbers 57
4.2 Visible Thinking in Math: Using Multiple Representations
as Records of Students’ Strategic Thinking 59
4.3 Zooming in on the Learning Progression of Numbers and Operations 62
4.4 Teaching Strategies: Using Math Happenings to Showcase
Real-world Mathematics Applications 66
4.5 Connecting Procedural Fluency and Conceptual Understanding 67
4.6 Technology Integration in Problem Solving 74
4.7 More Related Rich Problems to Explore 75
5 Modeling Math Ideas with Patterns and Algebraic Reasoning 77
5.1 Lesson Study Vignette—Growing Staircase Problem 77
5.2 Visible Thinking in Math: Using a Modeling Math Mat 82
5.3 Patterns and Algebra: Zooming in on the Learning Progressions 85
5.4 Teaching Strategies: Promoting the Algebraic Habits of Mind 86
5.5 Lesson Vignette: What Would You Choose? Analyzing Change
in Number Patterns 87
5.6 Technology Integration in Problem Solving 90
5.7 More Related Rich Problems to Explore 93
6 Extending the Mathematical Learning Progressions for Expressions
and Equations 95
6.1 Lesson Study Vignette: Setting a Math Learning Agenda 95
6.2 Zooming in on the Learning Progressions for Algebra 97
6.3 Visible Thinking in Math: Naming, Sequencing,
and Connecting Math Strategies 99
6.4 Teaching Strategies: Using Misconceptions to Repair
Understanding and Looking for Efficiency 103
6.5 Technology Integration in Problem Solving 106
6.6 More Related Rich Problems to Explore 107
7 Developing Fraction Sense 109
7.1 Lesson Study Vignette: The Unusual Baker 109
7.2 Visible Thinking in Math: Assessing Student Learning through
Classroom Artifacts 110
7.3 Zooming in on the Learning Progressions: Fractions 112
7.4 Implementing Mathematical Tasks that Promote Reasoning
and Problem solving 114
7.5 Teaching Strategies, Using Representations, and
Overcoming Common Misconception 116
Contents vii
7.6 Technology Integration in Problem Solving 118
7.7 More Related Rich Problems to Explore 121
8 Modeling Operations with Fractions 123
8.1 Lesson Study Vignette: Stuffed with Pizza—Adding fractions 123
8.2 Visible Learning in Math—Using Tools to Prove their Thinking 124
8.3 Learning Progression in Fraction Operations—Zooming in on
Multiplying and Dividing fractions 126
8.4 Lesson Study Vignette: Share My Candy 127
8.5 Teaching Strategies: Strategy Mapping on the Board Plan 131
8.6 Use of Students’ Diversity of Strategies as Pedagogical Content Tools 134
8.7 Technology Integration in Problem Solving 138
8.8 More Related Rich Problems to Explore 141
9 Modeling Math Ideas with Ratio and Proportional Reasoning 143
9.1 Lesson Study Vignette: The Leaky Bathtub 143
9.2 Zooming in on the Learning Progressions on Proportional Reasoning 144
9.3 Visible Thinking in Math: Using Representational Models for
Proportional Reasoning 148
9.4 Lesson Study Vignette: The Cathedral Problem 153
9.5 Deepening Teacher Knowledge and their Strategic Competence 157
9.6 Promoting Reasoning to Rich tasks 158
9.7 Technology Integration in Problem Solving 161
9.8 More Related Rich Problems to Explore 161
10 Pulling it All Together: Strengthening Strategic Competence through
Modeling Mathematics Ideas 165
10.1 Practice-based Activities to Focus on Models and Modeling
within our Standards 165
10.2 Modeling Math with Tools and Representations
to Strengthen Strategic Competence 167
10.3 Understanding Conceptual and Interpretative Models
of Math Ideas to Strengthen Strategic Competence 167
10.4 Modeling Math through Problem Solving to Strengthen
Strategic Competence 168
10.5 Posing Tasks for Mathematical Modeling through
Unstructured Real-world Problems to Strengthen Strategic
Competence 168
10.6 Strengthening Strategic Competence for Modeling
Mathematical Ideas 169
List of Math Happenings
A Math Happening 1a: The Handshake Problem 1
A Math Happening 2: Measuring Cups 37
A Math Happening 3a: Planning for the Table Tennis Championship 39
A Math Happening 3b: Proposal for a Sport Stadium 43
A Math Happening 3c: The Traffic Jam 47
viii Contents
A Math Happening 4: The Locker Problem 57
A Math Happening 5a: Building a Staircase 77
A Math Happening 5b: Growing Tower Structures 80
A Math Happening 5c: Your Dream Job 88
A Math Happening 6a: Saving for Mom’s Present 95
A Math Happening 6b: Bikes and Trikes 98
A Math Happening 7a: Unusual Baker 109
A Math Happening 7b: Is it Fair? 115
A Math Happening 8a: Stuffed with Pizza 123
A Math Happening 8b: Share My Candy 127
A Math Happening 8c: The Turtle Race 131
A Math Happening 8d: The Mango Problem 134
A Math Happening 9a: The Leaky Bathtub Problem 143
A Math Happening 9b: The Cathedral Problem 154
A Math Happening 9c: The Robot Problem 159
Appendix A: Modeling Mathematical Ideas Toolkit 177
MMI Toolkit 1.0—Developing Strategic Competence through
Modeling Math Ideas 177
MMI Toolkit 1.1—Promoting Mathematical Practices 178
MMI Toolkit 1.2—Performance-based Assessment 179
MMI Toolkit 3.0—Self-Reflection on Twenty-first-Century Skills
(Partnership for 21st Century Skills, 2011) 181
MMI Toolkit 3.1—The Math Modeling Cycle 183
MMI Toolkit 3.2—Planning and Debriefing a Math Modeling Lesson 184
MMI Toolkit 3.3—Evaluating the Mathematical Modeling Process 185
MMI Toolkit 4—Sharing Math Happenings 187
MMI Toolkit 5 Visible Thinking Strategies 189
MMI Toolkit 6 Nameable Problem-solving Strategy Cards 192
MMI Toolkit 7 Family of Problems 193
References 197
Index 207
About the Authors 215
Preface
This book is for preservice and in-service teachers’ teaching elementary and middle-
school mathematics. The focus of our book is to develop mathematical ideas by shar-
ing ways students respond to modeling mathematical ideas and allowing teachers to
discuss student strategies through case scenarios from lesson studies taken from the
elementary and middle-grade classrooms.
With the recent release of the Common Core State Standards (CCSS) (NGACBP &
CCSSI, 2010), researchers and mathematics educators are looking at the standards
to help teachers map out the learning progression that will guide the sequence of
mathematical concepts crucial in building mathematical understanding. Learning pro-
gression extends “previous learning while avoiding repetition and large gaps” (Hunt
Institute, 2012, p. 8) and guides “a path through a conceptual corridor in which there
are predictable obstacles and landmarks” (Confrey, 2012, p. 4).
Understanding the learning progression is important for teachers for it serves as the
guiding post for analyzing student learning and tailoring their teaching sequence. This
essential instructional practice leads to important professional development focused
on vertical articulation around important math concepts and worthwhile tasks with
cognitive demand at multiple levels. Teachers need to be engaged in understanding
the learning progression across standards. Our proposed book will provide the support
with the focus on the learning progression for grades 3–8 by presenting rich problem-
solving tasks that model mathematics across the vertical progression. We hope this
book will provide a guide for navigating through the mathematics learning progres-
sion through meaningful conceptual tasks and formative and summative assessments.
All of the Lesson Study Vignettes are from our past lesson studies with classroom
teachers in the past eight years. These lesson studies were supported in part by the
Virginia Department of Education, State Council for Higher Education in Virginia,
and the National Science Foundation through a variety of Mathematics Science
Partnership grants. We have conducted multiple professional development courses
focused on these important math practices and strands and essential themes includ-
ing computational fluency, building number sense, rational numbers and proportional
reasoning, STEM in high schools and have used this curriculum.
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