Contents

 Introduction: Why Do We Need to Declutter Math? 1

Introduction 1

Calling Out the Clutter 3

Connecting the Disconnected 5

Focusing on the Important 9

The Math Attic 10

Reflection Questions for Educators and Leaders 12

What’s Ahead 13

1. What Does It Really Mean to Learn Mathematics? 15

Introduction 15

What Does It Really Mean to Learn Mathematics? 16

The Conundrum We Currently Face 19

Putting the Ideas Together 20

Back to the Very Beginning: Base-10 Arithmetic 21

Focusing on the Important: What Really Matters 35

Focusing on the Important: How Do We Get There? 37

Reflection Questions for Educators and Leaders 38

Conclusion 38

2. Measurement: What Does It Really Mean to Measure Something? 41

Introduction 41

Why Does Measurement Matter? 42

The Conundrum We Currently Face 47

Putting the Ideas Together 51

Focusing on the Important: What Really Matters? 55

Focusing on the Important: How Do We Get There? 56

The Math Attic: What Content Can Go Up in the Attic? 57

Reflection Questions for Educators and Leaders 59

Conclusion 59

3. Comparisons: What?!? There Are Two Ways to Compare Quantities? 63

Introduction 63

Why Do Different Ways of Comparison Matter? 64

The Conundrum We Currently Face 65

Putting the Ideas Together 66

Focusing on the Important: What Really Matters? 74

Focusing on the Important: How Do We Get There? 76

The Math Attic: What Content Can Go Up in the Attic? 77

Reflection Questions for Educators and Leaders 79

Conclusion 79

4. Proportional Relationships: How Does It Take Two to Tango? 81

Introduction 81

Why Do Proportional Relationships Matter? 82

The Conundrum We Currently Face 82

Putting the Ideas Together 85

Focusing on the Important: What Really Matters? 87

Focusing on the Important: How Do We Get There? 93

The Math Attic: What Content Can Go Up in the Attic? 94

Reflection Questions for Educators and Leaders 95

Conclusion 96

5. Equivalence: What’s the Same Here? 99

Introduction 99

Why Does Equivalence Matter? 100

The Conundrum We Currently Face 101

Putting the Ideas Together 103

Focusing on the Important: What Really Matters? 104

Focusing on the Important: How Do We Get There? 109

The Math Attic: What Content Can Go Up in the Attic? 110

Reflection Questions for Educators and Leaders 113

Conclusion 113

6. Graphical Representations: What Story Do Graphs Tell and How Do They Unleash Thinking? 117

Introduction 117

Why Does Thinking About Graphs Matter? 121

The Conundrum We Currently Face 121

Putting the Ideas Together 123

Focusing on the Important: What Really Matters? 140

Focusing on the Important: How Do We Get There? 140

The Math Attic: What Content Can Go Up in the Attic? 142

Reflection Questions for Educators and Leaders 143

Conclusion 144

7. Putting the Fundamental Understandings into Practice: How Can Meaningful Student Thinking Be Unleashed? 147

Introduction 147

The Five Fundamental Understandings in Play 149

What’s Next? 151

Recognizing Linear Data 155

Addendum: Testing Your Mettle 162

Conclusion 168

 Conclusion: Decluttering Mathematics: What Do I Do Now? 169

Yet One More Thing? 169

Summary of Fundamental Understandings 170

What Does the Research Community Say? 173

What Can You Do? 173

What About Pushback? 177

And Finally 178

References 181

Index 185

About the Authors 189