Foreword |
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xi | |
Preface |
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xiii | |
Introduction |
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xv | |
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1 | (32) |
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1 Math According to Jen, Bobby, and Others |
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3 | (8) |
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3 | (2) |
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Bobby's Experience with Multiplication and Number Arrangements |
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5 | (2) |
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Children's Descriptions about Making Sense |
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7 | (1) |
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Trevor and Jim Invent Their Own Multiplication Methods |
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7 | (1) |
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8 | (3) |
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2 Math Grammar of Nouns, Verbs, and Stories |
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11 | (12) |
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"Putting Together" Verbs for Addition and Multiplication |
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11 | (2) |
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"Taking Apart" Verbs for Subtraction and Division |
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13 | (3) |
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A Short Interlude about the Use of That Negative Sign |
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16 | (1) |
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Word Problems Put Math Verbs in Stories |
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16 | (6) |
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22 | (1) |
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3 The Choice between Two Rs: Rote Memorization or Reasoning |
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23 | (10) |
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Sound Bites That Bite Back |
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24 | (3) |
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Memorization vs. Organization |
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27 | (3) |
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Acronyms: Sense or Nonsense |
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30 | (1) |
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30 | (3) |
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33 | (36) |
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4 In Search of Like Terms: Classification Revisited |
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35 | (10) |
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Like Terms, Common Denominators, and Same Units |
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35 | (2) |
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Place Value Columns Have Like Terms Categories |
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37 | (2) |
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What's Wrong with This Picture? |
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39 | (3) |
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42 | (1) |
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A Note about Multiplication and Division |
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42 | (1) |
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43 | (2) |
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5 Artful Assembly of Operations |
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45 | (12) |
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Facts, Frogs, and Formats |
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45 | (4) |
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Jackie's Thinking about Number Operations |
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49 | (3) |
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James, Janie, and Napier's Lattice Multiplication |
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52 | (1) |
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Division Interpretations That Made Sense |
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53 | (2) |
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55 | (2) |
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6 Same Math, Same Meaning, Different Organization: New vs. Old |
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57 | (12) |
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"If It Ain't Broke, Don't Fix It" |
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58 | (1) |
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Multiplication Organization Sense |
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59 | (2) |
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Digit Alignment Continues |
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61 | (2) |
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Zeke, Jake, and Long Division |
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63 | (4) |
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67 | (2) |
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PART III RELATIONSHIPS AND PATTERNS |
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69 | (28) |
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7 "Seeing" Math Patterns with Viewfinders |
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71 | (10) |
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Multiplication Table Patterns |
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72 | (4) |
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76 | (2) |
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Kathy's and Rudy's Fraction Viewfinders |
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78 | (2) |
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80 | (1) |
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8 Spelunking for Patterns with More Viewfinders |
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81 | (10) |
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Reflections on Addition and Multiplication |
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81 | (2) |
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Subtraction, Division, and Missy's Different "Family" |
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83 | (3) |
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A Few Rules of Engagement for Working with Numbers |
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86 | (2) |
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88 | (3) |
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9 Functions, Predictability, and Balance |
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91 | (6) |
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Predictability and Functions |
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91 | (1) |
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One Answer or Many Answers, Same Balance |
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92 | (4) |
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96 | (1) |
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97 | (34) |
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10 A Multiplication Area Image for the Ages |
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99 | (8) |
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From Theon to Dienes: A Bridge across the Centuries |
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100 | (1) |
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Same Design for Mixed Numbers, Fractions, and Decimals |
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101 | (4) |
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Visual-Spatial Multiple-Digit Multiplication |
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105 | (1) |
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106 | (1) |
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11 Ratios, Proportions, and Rate of Change |
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107 | (12) |
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109 | (2) |
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David's Decluttering Fractions 109 Changing Numbers, Changing Locations, and Moving Targets in Proportions |
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111 | (5) |
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Slope as a Rate of Change |
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116 | (1) |
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117 | (2) |
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119 | (12) |
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The Shapes for an Algebra Transition |
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119 | (2) |
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From Multiplication Tables to Coordinate Tables |
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121 | (1) |
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122 | (3) |
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125 | (1) |
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125 | (2) |
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Conclusion: What Parents Can Do |
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127 | (1) |
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Problem Solving beyond Word Problems |
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128 | (1) |
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Right Tool for the Right Job: A Hard Look at Technology |
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129 | (1) |
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130 | (1) |
Glossary |
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131 | (4) |
References |
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135 | (4) |
About the Author |
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139 | |