Introduction 1
Low-Floor, High-Ceiling Tasks 2
Youcubed Summer Camp 3
Memorization versus Conceptual Engagement 4
Mathematical Thinking, Reasoning, and Convincing 5
Big Ideas 9
Structure of the Book 10
Activities for Building Norms 17
Encouraging Good Group Work 17
Paper Folding: Learning to Reason, Convince, and Be a Skeptic 21
Big Idea 1: Solving Problems with Data 23
Visualize: Tongues, Tails, and in Between 25
Play: Inspector Graph-It 36
Investigate: Data Tells Us about Ourselves 41
Big Idea 2: Thinking around Shapes 51
Visualize: Get Your Arms around It 53
Play: 36-Unit Walk 59
Investigate: Shapes on a Plane 65
Big Idea 3: Thinking in Equal Groups 75
Visualize: Sharing Crackers 78
Play: Dozens of Dice 85
Investigate: Playing with Pairs 91
Big Idea 4: Tiling to Understand Area 99
Visualize: Cover Up 101
Play: A Whole New Alphabet 111
Investigate: Sharing an Area 119
Big Idea 5: Seeing Multiplication as Area 125
Visualize: Rods Around 127
Play: Squares and Near-Squares 132
Investigate: Connecting Area and Perimeter 142
Big Idea 6: Understanding 1 2 149
Visualize: I Spy 1 2 151
Play: Spotting 1 2 160
Investigate: The Many Shapes of 1 2 171
Big Idea 7: Seeing Fractions: The Parts and the Wholes 183
Visualize: Seeing Parts and Wholes 185
Play: Cover, Cut, and Sort 191
Investigate: Taking a Fractional Walk 202
Big Idea 8: Being Flexible with Numbers 209
Visualize: How Many Do You See? 211
Play: How Close to 100? 217
Investigate: Tile and Table Patterns 222
Appendix 231
¼˝ Grid Paper 232
Grid Paper 233
1˝ Grid Paper 234
¼˝ Dot Paper 235
Centimeter Dot Paper 236
Isometric Dot Paper 237
Hundred Grids 238
About the Authors 239
Acknowledgments 241
Index 243
JO BOALER is a professor of mathematics education at Stanford University and co-founder and faculty director of youcubed. She serves as an advisor to several Silicon Valley companies and is a White House presenter on girls and STEM (Science, Technology, Engineering, and Math). The author of seven books, including Mathematical Mindsets, and numerous research articles, she is a regular contributor to news and radio in the United States and England.
JEN MUNSON is a postdoctoral fellow at Northwestern University, a professional developer, and a former classroom teacher. She received her PhD in mathematics education from Stanford University. Her research focuses on responsive, equitable mathematics instruction.
CATHY WILLIAMS is the co-founder and the executive director of youcubed at Stanford University. Before working at youcubed, she was a high school math teacher and worked in mathematics curriculum and administration at the county and district levels in California.
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