Contributors |
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xiii | |
Foreword: Build It and They Will Come |
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xv | |
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Preface |
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xxi | |
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1 Insights from Cognitive Science on Mathematical Learning |
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On the Nature of Theories and Models in Cognitive Psychology |
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2 | (1) |
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The Role of Theories in Cognitive Psychology |
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2 | (2) |
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Theory Testing and Validation |
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4 | (1) |
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Methodological Considerations |
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4 | (1) |
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5 | (1) |
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What, When, How, and Who? |
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6 | (5) |
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Challenges for Instruction |
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11 | (2) |
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Conclusions and Future Directions |
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13 | (1) |
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14 | (7) |
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Part I Complex Arithmetic Processing |
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2 The Understanding of Additive and Multiplicative Arithmetic Concepts |
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21 | (1) |
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What is Conceptual Knowledge of Arithmetic? |
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22 | (1) |
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The Importance of Conceptual Knowledge |
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23 | (1) |
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A Brief History of Research on Conceptual Knowledge |
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23 | (3) |
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The Importance of Multiplicative Concepts and the State of Current Research |
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26 | (1) |
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Additive Versus Multiplicative Concepts |
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26 | (1) |
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27 | (1) |
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The Associativity Concept |
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27 | (1) |
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Are Additive and Multiplicative Concepts the Same? |
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28 | (1) |
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29 | (1) |
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30 | (1) |
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Inversion Versus Associativity |
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31 | (2) |
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Individual Differences and Factors in the Use of Conceptually-Based Shortcuts |
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33 | (1) |
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34 | (2) |
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Factors Relating to Conceptually-Based Shortcut Use |
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36 | (1) |
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Computational Skills and Age |
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36 | (1) |
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37 | (1) |
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37 | (1) |
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38 | (2) |
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40 | (1) |
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Conclusions and Future Directions |
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41 | (1) |
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42 | (5) |
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3 Arithmetic Word Problem Solving: The Role of Prior Knowledge |
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47 | (1) |
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The Role of Daily Life Experience in Solving Arithmetic Word Problems |
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48 | (2) |
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The Role of Stereotypic Representations About Problem Solving in School |
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50 | (2) |
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The Role of Problem Schemata Stored in Long-Term Memory |
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52 | (6) |
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The Use of Schemata Versus Situation-Based Models |
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58 | (2) |
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60 | (2) |
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Conclusions and Future Directions |
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62 | (1) |
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63 | (4) |
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4 Neurodevelopmental Disorders as Model Systems for Understanding Typical and Atypical Mathematical Development |
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67 | (1) |
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Spina Bifida as a Model System for Understanding Mathematical Learning Disabilities |
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68 | (4) |
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Longitudinal Approaches to the Study of Mathematical Development and Disability |
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72 | (1) |
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Sources of Mathematical Disability |
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73 | (6) |
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Longitudinal Studies of Mathematical Cognition in Children with SBM and Their Typically Developing Peers |
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79 | (1) |
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Are Domain-General Cognitive Abilities Related to Number Knowledge? |
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79 | (2) |
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Longitudinal Mediation of School-Age Mathematics Achievement |
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81 | (2) |
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What is the Relation of Different Preschool Domain-General Cognitive Abilities to Different Mathematics Outcomes at School Age? |
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83 | (2) |
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Do the Longitudinal Mediators Differ for Math and Reading? |
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85 | (1) |
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What is the Relation of Early Domain-General Abilities and Domain-Specific Number Knowledge to Later Mathematical Achievement? |
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86 | (1) |
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What are the Implications of the Findings for Assessment and Intervention? |
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87 | (2) |
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Conclusions and Future Directions |
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89 | (1) |
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90 | (1) |
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90 | (12) |
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Part II Rational Number Processing |
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5 The Transition from Natural to Rational Number Knowledge |
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The Importance of Rational Numbers |
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102 | (1) |
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Rational Numbers: A Challenge for Learners and for Mathematics Education |
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102 | (1) |
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The Interference of Natural Number Knowledge in Rational Number Tasks |
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103 | (1) |
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The Size of Rational Numbers |
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104 | (1) |
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The Effect of Arithmetic Operations |
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104 | (1) |
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The Dense Structure of Rational Numbers |
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104 | (1) |
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Representation of Numbers as an Intersecting Difficulty |
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105 | (1) |
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105 | (1) |
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Theoretical Frameworks for Studying the Natural Number Bias |
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106 | (1) |
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The Conceptual Change Perspective |
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106 | (2) |
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The Dual Process Perspective on Reasoning |
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108 | (1) |
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Combining the Conceptual Change Theory and Dual Process Perspective to Study Mathematical Thinking and Learning |
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109 | (1) |
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Overview of Our Studies Using Both Conceptual Change Theory and Dual Process Perspective |
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109 | (1) |
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110 | (1) |
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111 | (1) |
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112 | (1) |
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How are the Three Aspects Related to Each Other? |
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113 | (2) |
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Conclusions and Future Directions |
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115 | (2) |
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117 | (3) |
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120 | (6) |
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6 Fraction Development in Children: Importance of Building Numerical Magnitude Understanding |
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Integrated Theory of Numerical Development |
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126 | (2) |
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Understanding of Fractions Involves Both Conceptual and Procedural Knowledge |
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128 | (1) |
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Fraction Development in Early Childhood |
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129 | (1) |
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Early Fraction Calculation Ability |
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129 | (1) |
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130 | (1) |
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Early Knowledge of Proportionality |
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130 | (1) |
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130 | (1) |
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Fraction Development Between Third and Sixth Grade: Findings from the Delaware Longitudinal Study |
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131 | (1) |
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Predictors of Fraction Knowledge |
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132 | (2) |
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Growth in Fraction Magnitude Understanding |
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134 | (2) |
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Helping Students Who Struggle with Fractions |
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136 | (1) |
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137 | (1) |
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137 | (4) |
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7 Numbers as Mathematical Models: Modeling Relations and Magnitudes with Fractions and Decimals |
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Understanding Rational Numbers |
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141 | (1) |
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141 | (1) |
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Prior Research on Magnitude Assessment and Misconceptions About Rational Numbers |
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142 | (1) |
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142 | (1) |
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Magnitude Representations for Rational and Natural Numbers |
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143 | (1) |
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Relational Affordances of the Fraction Notation |
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144 | (1) |
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Using Mathematics to Model Relations |
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144 | (1) |
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145 | (2) |
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Modeling with Rational Numbers |
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147 | (1) |
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Alignments Between Rational Numbers and Quantity Types |
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148 | (1) |
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Discrete/Continuous Ontological Distinction |
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148 | (1) |
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Modeling Discrete and Continuous Quantities with Fractions and Decimals |
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149 | (3) |
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Modeling Magnitude with Decimals |
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152 | (2) |
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Connections Between Rational Numbers and Other Math Concepts |
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154 | (1) |
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Multiplicative Reasoning and Fraction Understanding |
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154 | (2) |
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Differential Contributions of Magnitude and Relational Knowledge to Learning Algebra |
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156 | (2) |
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Conclusions and Future Directions |
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158 | (2) |
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160 | (7) |
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Part III Algebraic, Geometric, and Trigonometric Concepts |
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8 Understanding Children's Difficulties with Mathematical Equivalence |
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167 | (1) |
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Children's Difficulties with Mathematical Equivalence Problems |
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168 | (3) |
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Popular Accounts of Children's Mathematics Learning Difficulties |
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171 | (1) |
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The Symbol Misunderstanding Account |
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171 | (3) |
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The Deficient Working Memory System Account |
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174 | (4) |
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The Poor Number Knowledge Account |
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178 | (3) |
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The Change-Resistance Account |
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181 | (6) |
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Conclusions and Future Directions |
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187 | (1) |
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188 | (1) |
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188 | (9) |
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9 Learning and Solving More Complex Problems: The Roles of Working Memory, Updating, and Prior Skills for General Mathematical Achievement and Algebra |
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197 | (2) |
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Algebra and Earlier Mathematics Skills |
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199 | (1) |
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200 | (2) |
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202 | (1) |
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Arithmetic and Algebraic Word Problems in the Singapore Curriculum |
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203 | (2) |
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General Mathematics Achievement, Algebra, and Relations with Domain-General and Domain-Specific Influences |
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205 | (4) |
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209 | (1) |
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210 | (2) |
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General Mathematical Achievement, Domain-Specific and Domain-General Precursors |
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212 | (1) |
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Mathematical Relational Skills and Arithmetic Word Problems |
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212 | (1) |
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Algebraic: Problems, Earlier Mathematical Skills, and Domain-General Capacities |
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213 | (1) |
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Conclusions and Future Directions |
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214 | (2) |
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216 | (1) |
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217 | (5) |
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10 Learning Geometry: The Development of Geometrical Concepts and the Role of Cognitive Processes |
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Classical Studies on Geometry |
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222 | (1) |
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Core Intuitive Principles of Geometry |
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223 | (2) |
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Academic Achievement in Geometry |
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225 | (5) |
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The Development of Geometrical Knowledge |
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230 | (2) |
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Cognitive Processes Involved in Geometry |
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232 | (4) |
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236 | (4) |
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Conclusions and Future Directions |
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240 | (1) |
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241 | (7) |
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11 The Unit Circle as a Grounded Conceptual Structure in Precalculus Trigonometry |
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Grounded Conceptual Structures in Mathematical Cognition |
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248 | (4) |
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The Unit Circle as a Grounded Conceptual Structure for Trigonometry |
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252 | (2) |
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Preliminary Investigations |
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254 | (3) |
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Preliminary Study: Observing Use and Success of the Unit Circle |
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257 | (2) |
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Study 2: Comparing a Unit Circle Lesson to a Rules Lesson and Baseline Knowledge |
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259 | (2) |
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Challenges in Learning the Unit Circle |
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261 | (1) |
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Unit Circle Instruction for Students Without Prior Precalculus Trigonometry |
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262 | (2) |
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Internalizing the Unit Circle |
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264 | (1) |
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The Role of Epistemic Belief in Acquiring an Integrated Conceptual Representation |
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265 | (1) |
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Conclusions and Future Directions |
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266 | (1) |
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266 | (7) |
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Part IV Instructional Approaches |
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12 The Power of Comparison in Mathematics Instruction: Experimental Evidence from Classrooms |
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273 | (1) |
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Short-Term, Researcher-Led Classroom Research |
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274 | (1) |
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274 | (2) |
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Studies on Comparing Methods |
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276 | (4) |
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Studies on Comparing Problems |
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280 | (3) |
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Summary of Researcher-Led Classroom Studies and Proposed Guidelines |
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283 | (1) |
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Year-Long Study Helping Teachers Use Comparison in Algebra I Classrooms |
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284 | (1) |
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Supplemental Curriculum Materials |
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285 | (3) |
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Implementation and Evaluation |
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288 | (2) |
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Discussion of Year-Long Study |
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290 | (1) |
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Conclusions and Future Directions |
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291 | (1) |
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291 | (1) |
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292 | (5) |
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13 Evidence for Cognitive Science Principles that Impact Learning in Mathematics |
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297 | (2) |
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299 | (1) |
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Evidence from Laboratory Studies |
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300 | (1) |
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Evidence from Classroom Studies |
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300 | (1) |
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Recommendations for Further Research |
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300 | (1) |
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Distributed Practice Effect |
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301 | (1) |
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Evidence from Laboratory Studies |
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302 | (1) |
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Evidence from Classroom Studies |
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302 | (1) |
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Recommendations for Further Research |
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302 | (1) |
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303 | (1) |
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Evidence from Laboratory Studies |
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303 | (1) |
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Evidence from Classroom Studies |
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304 | (1) |
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Recommendations for Further Research |
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304 | (1) |
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304 | (1) |
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Evidence from Laboratory Studies |
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305 | (1) |
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Evidence from Classroom Studies |
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305 | (1) |
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Recommendations for Further Research |
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306 | (1) |
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306 | (1) |
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Evidence from Laboratory Studies |
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306 | (1) |
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Evidence from Classroom Studies |
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307 | (1) |
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Recommendations for Further Research |
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308 | (1) |
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Abstract and Concrete Representations Principles |
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308 | (1) |
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Evidence from Laboratory Studies |
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308 | (1) |
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Evidence from Classroom Studies |
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309 | (1) |
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Recommendations for Further Research |
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310 | (1) |
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Error Reflection Principle |
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310 | (1) |
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Evidence from Laboratory Studies |
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311 | (1) |
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Evidence from Classroom Studies |
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311 | (1) |
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Recommendations for Further Research |
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312 | (1) |
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Analogical Comparison Principle |
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312 | (1) |
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Evidence from Laboratory Studies |
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313 | (1) |
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Evidence from Classroom Studies |
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313 | (1) |
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Recommendations for Further Research |
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314 | (1) |
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Conclusions and Future Directions |
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314 | (2) |
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316 | (1) |
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317 | (1) |
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317 | (10) |
Index |
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