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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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7 | (7) |
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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 | (6) |
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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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Leveraging the Power of Children's Literature |
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11 | (1) |
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Final Words Before You Dive In |
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12 | (2) |
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Chapter Two Moving From Countingrto Addition And Subtraction |
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14 | (26) |
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Thinking About Counting, Addition, and Subtraction |
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14 | (12) |
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Sandbox Notes: Explore Your Thinking |
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16 | (2) |
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Students and Teachers Think About the Problem |
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18 | (1) |
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The Development of Counting |
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19 | (2) |
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21 | (2) |
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Representing Problem Situations or Representing Answers? |
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23 | (3) |
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Using Children's Literature to Explore Add-To and Take-From Situations |
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26 | (1) |
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Transcribe the Action or Relationship |
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26 | (1) |
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27 | (9) |
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Students and Teachers Think About the Problems |
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29 | (1) |
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Representing Problem Situations Multiple Ways |
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29 | (3) |
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Translating the Five Representations: Try It Out - |
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32 | (2) |
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Teaching Students to Use Concrete and Pictorial Models |
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34 | (2) |
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36 | (1) |
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37 | (2) |
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39 | (1) |
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Chapter Three Add-To Problems: Locating The Change |
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40 | (30) |
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Thinking About Active Addition Situations |
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40 | (15) |
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Sandbox Notes: Explore Your Thinking |
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42 | (2) |
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Students and Teachers Think About the Problems |
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44 | (4) |
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Finding the Unknown: Three Story Structures |
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48 | (1) |
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Story Structures: Implications for Teaching |
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49 | (1) |
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Modeling the Active Problem Situation |
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50 | (5) |
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Using Children's Literature to Explore Add-To Situations |
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55 | (2) |
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55 | (1) |
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Transcribe the Action or Relationship |
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56 | (1) |
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Digging Deeper Into Start Unknown Situations |
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57 | (9) |
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Complicating Things: The Start Unknown Variation |
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58 | (2) |
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Strategies for Making Sense of Start Unknown Problem Situations |
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60 | (2) |
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Separating Computations From Operations |
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62 | (2) |
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64 | (2) |
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66 | (1) |
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67 | (1) |
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68 | (2) |
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Chapter Four take-prom problems: locating the change |
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70 | (24) |
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Thinking About Active Subtraction Situations |
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70 | (11) |
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Sandbox Notes: Explore Your Thinking |
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72 | (2) |
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Students and Teachers Think About the Problems |
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74 | (1) |
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Finding the Unknown: Three Story Structures |
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75 | (1) |
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Story Structures: Implications for Teaching |
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75 | (3) |
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Modeling the Active Problem Situation |
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78 | (3) |
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Digging Deeper Into the Start and Change Unknown Situations |
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81 | (8) |
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Moving From Concrete to Symbolic Representations |
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83 | (6) |
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Using Children's Literature to Explore Take-From Situations |
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89 | (1) |
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89 | (1) |
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Find the Unknown Quantity |
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89 | (1) |
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90 | (1) |
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91 | (1) |
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92 | (2) |
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Chapter Five Tart `part' whole: Understanding The Relationship |
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94 | (26) |
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Thinking About Part-Part-Whole Situations |
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94 | (13) |
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Sandbox Notes: Explore Your Thinking |
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96 | (2) |
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Students and Teachers Think About the Problems |
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98 | (1) |
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Defining the Part-Part-Whole Situation |
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99 | (1) |
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Modeling Relationships Versus Action |
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100 | (1) |
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Moving From Counters to Bar Models |
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101 | (4) |
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The Special Case of Both Addends Unknown |
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105 | (1) |
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A Note About the Commutative Property |
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106 | (1) |
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Using Children's Literature to Explore Part-Part-Whole Situations |
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107 | (1) |
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107 | (1) |
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108 | (8) |
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Students and Teachers Think About the Problems |
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109 | (1) |
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Modeling Measurement Problems |
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110 | (1) |
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Writing Equations: Addition or Subtraction |
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111 | (2) |
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Finding the Equation in the Model |
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113 | (3) |
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116 | (1) |
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117 | (1) |
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118 | (2) |
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Chapter Six Additive comparisons: another. kind of relationship |
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120 | (26) |
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Thinking About Additive Comparison Situations |
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120 | (8) |
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Sandbox Notes: Explore Your Thinking |
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122 | (2) |
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Students and Teachers Think About the Problems |
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124 | (1) |
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125 | (1) |
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Additive Comparison Situations |
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126 | (2) |
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Building Models for Comparisons |
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128 | (4) |
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Students and Teachers Think About the Problems |
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129 | (1) |
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Making Use of Models for Thinking |
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130 | (2) |
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The Language of Comparison Problems |
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132 | (8) |
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135 | (2) |
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Building Bar Models for Comparisons |
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137 | (1) |
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From Counters to Bar Models |
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138 | (1) |
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Problem Posing as an Instructional Strategy |
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139 | (1) |
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Using Children's Literature to Explore Additive Comparison Situations |
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140 | (1) |
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140 | (1) |
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141 | (1) |
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142 | (2) |
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144 | (2) |
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Chapter Seven Early twltipucation and division: patterns and predictions |
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146 | (27) |
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Thinking About Early Multiplicative Thinking |
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146 | (8) |
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Sandbox Notes: Explore Your Thinking |
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148 | (2) |
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Students and Teachers Think About the Problem |
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150 | (1) |
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Modeling Even and Odd Numbers |
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151 | (1) |
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Equal Groups Problem Situation |
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152 | (1) |
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Using Patterns to Identify Even and Odd Numbers |
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153 | (1) |
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Using Children's Literature to Explore Early Multiplication |
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154 | (1) |
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Transcribe the Action or Relationship |
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155 | (1) |
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Bringing in Another Dimension With Arrays |
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155 | (4) |
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Recognizing Area/Array as a Problem Situation |
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158 | (1) |
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Building an Understanding of Arrays as Structures |
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158 | (1) |
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Using Children's Literature to Explore Area and Array Situations |
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159 | (1) |
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Transcribe the Action or Relationship |
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160 | (1) |
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Setting the Stage for Division |
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160 | (7) |
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163 | (3) |
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166 | (1) |
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Using Children's Literature to Explore Early Multiplication and Division |
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167 | (1) |
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Transcribe the Action or Relationship |
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167 | (1) |
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A Final Word on Introducing Early Multiplication and Division |
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168 | (1) |
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168 | (1) |
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169 | (2) |
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171 | (2) |
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Chapter Eight Changing How You Teach Woro Problems |
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173 | (12) |
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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 | (5) |
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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 | (3) |
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Finding Unexpected Mathematics in Stories |
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183 | (2) |
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183 | (2) |
Appendix: Situation Tables |
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185 | (3) |
References |
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188 | (3) |
Index |
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191 | |