Table of Contents

The Author xiii

About SEDL xv

About This Book xvii

Introduction xix

Julian’s Story xix

Rationale and Purpose xx

Who Benefits from This Book? xxii

One The Problem with Math Is English (and a Few Other Things) 1

Why Language and Symbolism? 1

What We Are Teaching 4

Turning the tide: A Sampling of Approaches 6

Mathematics Is About Relationships 8

Connecting the Pieces and Looking Ahead 9

Two Why a Language Focus in Mathematics? 11

The Convergence of Mathematics and English: More Than Just Vocabulary 11

Problems Based on the English Language 13

A Number of Problems with Number 16

Three Language and Symbolism in Traditional Instruction 21

Shortcomings of Traditional Instruction 22

More Language and Symbolism Issues: Adding Fuel to the Fire 32

Tell Me Again Why the Language Focus in Math? 38

Four So What Does Conceptual Understanding Look Like? 41

It Starts with Definitions 42

Making Connections in Math: Beyond Connecting Dots 51

The Interpretation and Translation Of Math 55

Conclusion 61

Five The Order of Operations: A Convention or a Symptom of What Ails Us? 63

The Roots of the Rules 64

The Natural Order: A Mathematical Perspective 65

Conclusion: A Conceptual Understanding of the Order of Operations 78

Six Using Multiplication as a Critical Knowledge Base 81

Understanding Key Definitions and Connections 81

Interpreting Multiplication 86

Using The Power of the Distributive Property 88

Feeling Neglected: The Units in Multiplication 100

Conclusion: Small Details, Huge Impact 103

Seven Fractions: The ‘‘F Word’’ in Mathematics 105

Defining Fractions: Like Herding Cats 105

The Fraction Kingdom 107

Interpreting Fractions 116

Conclusion 124

Eight Operations with Fractions 127

Adding and Subtracting Fractions 127

Multiplying Fractions 131

Dividing Fractions 150

Conclusion 160

Nine Unlocking the Power of Symbolism and Visual Representation 161

Symbolism 161

Visual Representation 168

The Power of Interpretation: Three Perspectives of Trapezoids 178

Conclusion 187

Ten Language-Focused Conceptual Instruction 189

Language Focus: Beyond the Definitions 190

The Secrets to Solving Word Problems 192

Suggested Instructional Strategies 197

Conclusion 216

Eleven Mathematics: It’s All About Relationships! 219

Language and Symbolism: Vehicles for Relationship Recognition 220

Relationships and Fractions 224

Proportional Reasoning 227

Relationships: Important Considerations 230

Relationships: Making Powerful Connections 234

Conclusion 249

Twelve The Perfect Non-Storm: Understanding the Problem and Changing the System 251

A Systemic Issue 251

Math Makeover 257

Conclusion 264

Bibliography 267

Index 269

About the Author

Concepcion Molina, Ed.D., is a program associate with SEDL, a private, nonprofit education research, development, and dissemination corporation based in Austin, Texas. Dr. Molina supports systemic reform efforts in mathematics and works to assist state and intermediate education agencies in their efforts to improve instruction and student achievement.