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xix | |
Preface |
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xxi | |
Acknowledgments |
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xxiv | |
About The Authors |
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xxvii | |
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Chapter One Introduction: Why You Need To Teach Students To Mathematize |
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1 | (13) |
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Problem-Solving Strategies Gone Wrong |
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2 | (1) |
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What Is Mathematizing? Why Is It Important? |
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3 | (3) |
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Focusing on Operation Sense |
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4 | (1) |
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Using Mathematical Representations |
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4 | (2) |
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Teaching Students to Mathematize |
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6 | (5) |
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Building Your Understanding of the Operations and Related Problem Situations |
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7 | (3) |
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Playing in the Mathematizing Sandbox: A Problem-Solving Model |
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10 | (1) |
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Final Words Before You Dive In |
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11 | (3) |
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PART I ADDITIONA AND SUBTRACTION |
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Chapter Two Add-To And Take-From: Locating The Change |
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14 | (24) |
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Thinking About Active Addition and Subtraction Situations |
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14 | (12) |
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Students and Teachers Think About the Problems |
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18 | (1) |
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Finding the Unknown, Three Story Structures |
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18 | (2) |
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Story Structures: Implications for Teaching |
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20 | (2) |
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Focus on Take-From Situations |
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22 | (2) |
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Modeling the Active Problem Situation |
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24 | (2) |
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Teaching Students to Use Concrete and Pictorial Models |
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26 | (1) |
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Moving Beyond Whole Numbers |
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27 | (6) |
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Students and Teachers Think About the Problems |
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29 | (1) |
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Teaching Opportunities in the Student Work Samples |
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30 | (1) |
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Complicating Things: The Start Unknown Variation |
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30 | (3) |
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The Power of Writing Stories |
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33 | (1) |
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34 | (1) |
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35 | (1) |
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36 | (2) |
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Chapter Three Part-Part-Whole: Understanding The Relationship |
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38 | (20) |
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Thinking About Part-Part-Whole Situations |
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38 | (7) |
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Students and Teachers Think About the Problems |
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42 | (1) |
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Defining the Part-Part-Whole Situation |
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43 | (1) |
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Modeling Relationships Versus Action |
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44 | (1) |
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Improving the Accuracy of Student Models |
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45 | (1) |
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Writing Equations: Addition or Subtraction? |
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45 | (2) |
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Moving Beyond Whole Numbers |
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47 | (8) |
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Students and Teachers Think About the Problems |
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48 | (2) |
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Finding the Equation in the Model |
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50 | (1) |
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Modeling Measurement Problems |
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51 | (3) |
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The Special Case of Both Parts Unknown |
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54 | (1) |
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55 | (1) |
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56 | (1) |
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57 | (1) |
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Chapter Four Additive Comparisons: Another Kino Op Relationship |
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58 | (22) |
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Thinking About Additive Comparison Situations |
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58 | (11) |
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Students and Teachers Think About the Problems |
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62 | (2) |
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64 | (2) |
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Building Models for Comparisons |
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66 | (2) |
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Knowing When to Add and When to Subtract |
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68 | (1) |
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69 | (1) |
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Moving Beyond Whole Numbers |
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70 | (4) |
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Students and Teachers Think About the Problems |
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71 | (1) |
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The Anchor-Jump Strategy and Bar Models |
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71 | (1) |
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Modeling the Difference in Comparison Situations |
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72 | (2) |
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Problem Posing as an Instructional Strategy |
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74 | (1) |
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75 | (1) |
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75 | (2) |
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77 | (3) |
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PART II Multiplication And Division |
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Chapter Five Equal Groups Multiplication: Two Factors, Different Joes |
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80 | (22) |
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Thinking About Equal Groups Situations |
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80 | (11) |
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Students and Teachers Think About the Problems |
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84 | (1) |
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Multiplier and Measure Factors |
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85 | (2) |
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Equal Groups and the Commutative Property |
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87 | (2) |
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Representing Larger Quantities |
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89 | (2) |
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Moving Beyond Whole Numbers |
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91 | (8) |
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Students and Teachers Think About the Problems |
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92 | (1) |
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Equal Groups and Repeated Addition |
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92 | (1) |
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Understanding a Portion of a Group |
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93 | (1) |
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Starting With an Expression to Build Multiplicative Reasoning |
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94 | (2) |
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96 | (3) |
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99 | (1) |
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100 | (1) |
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101 | (1) |
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Chapter Six Equal Groups Division: When A Factor Is Missing |
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102 | (26) |
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Thinking About Equal Groups Division |
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104 | (9) |
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Students and Teachers Think About the Problems |
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106 | (1) |
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106 | (2) |
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Measurement (Quotitive) Division |
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108 | (1) |
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The Unknowns in Partitive and Measurement Division |
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109 | (1) |
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110 | (1) |
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Matching Models to Contexts |
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111 | (2) |
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Moving Beyond Whole Numbers |
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113 | (9) |
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Students and Teachers Think About the Problems |
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114 | (1) |
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Entering Challenging Division Problems |
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115 | (1) |
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Translating the Five Representations: Try It Out |
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116 | (2) |
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Tracking the Unit Whole and the Referent Whole |
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118 | (4) |
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Notes on Working With Manipulatives |
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122 | (1) |
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123 | (1) |
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124 | (2) |
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126 | (2) |
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Chapter Seven Multiplicative Comparisons: Another Asymmetric Relationship |
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128 | (22) |
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Thinking About Multiplicative Comparisons |
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128 | (13) |
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Students and Teachers Think About the Problems |
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132 | (1) |
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Additive Versus Multiplicative Comparison |
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133 | (1) |
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134 | (2) |
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136 | (2) |
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Multiplicative Comparison, Measurement, and Conversion |
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138 | (3) |
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Moving Beyond Whole Numbers |
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141 | (4) |
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Students and Teachers Think About the Problems |
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142 | (1) |
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Absolute and Relative Change |
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142 | (1) |
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Helping Students Transition to Multiplicative Thinking |
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143 | (2) |
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Can Multiplication Make a Quantity Smaller? |
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145 | (2) |
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147 | (1) |
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147 | (2) |
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149 | (1) |
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Chapter Eight Area And Array: Two Factors, Same Job |
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150 | (623) |
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Thinking About Area and Array Situations |
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150 | (8) |
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Students and Teachers Think About the Problems |
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154 | (1) |
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Symmetric Versus Asymmetric Multiplication |
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154 | (3) |
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Another Symmetric Problem Situation: The Fundamental Counting Principle |
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157 | (1) |
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Modeling Arrays: Situations or Computation? |
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158 | (3) |
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Arrays as a Mathematical Structure |
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160 | (1) |
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Models of and Models for Thinking |
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161 | (2) |
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Moving Beyond Whole Numbers |
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163 | (6) |
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Students and Teachers Think About the Problems |
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164 | (2) |
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When Models Match and When They Don't |
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166 | (1) |
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Representing Fraction and Decimal Division |
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166 | (3) |
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169 | (1) |
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169 | (2) |
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171 | (2) |
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Chapter Nine Changing How You Teach Word Problems |
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173 | (1) |
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Getting Into the Mathematizing Sandbox |
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173 | (2) |
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Eight Shifts in Instruction for Building Students' Problem-Solving Skills |
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175 | (1) |
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Do Word Problems for Sense-Making |
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175 | (1) |
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Treat Context and Computation Separately |
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176 | (1) |
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Create More and Varied Representations |
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176 | (1) |
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Explore All the Work Operations Can Do |
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177 | (1) |
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Add Operation Sense Routines to the School Day |
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178 | (1) |
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Offer Students Experiences With a Variety of Problem Situations |
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178 | (1) |
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Listen to Students and Be Curious |
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179 | (1) |
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Make Time for Mathematizing in the Sandbox |
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179 | (1) |
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Guidance for Moving Forward: FAQs |
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180 | (4) |
Appendix: Situation Tables |
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184 | (3) |
References |
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187 | (3) |
Index |
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190 | |