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| Acknowledgments | |
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| Preface | |
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| About This Book | |
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| "Mathematics" or "Mathematizing"? | |
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| Learning and Teaching in the Classroom | |
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| What Is Revealed | |
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| Back to the Classroom | |
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| Teaching and Learning as Development | |
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| The Math Congress | |
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| What Is Revealed | |
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| Strategies, Big Ideas, and Models in a Teaching/Learning Framework | |
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| Strategies as Schemes | |
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| Big Ideas as Structures | |
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| Models as Tools for Thought | |
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| Walking the Edge | |
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| Summing Up ... | |
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| The Landscape of Learning | |
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| Describing the Journey | |
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| Linear Frameworks | |
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| Learning Trajectories | |
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| The Role of Context | |
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| Word Problems vs. Truly Problematic Situations | |
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| Finding Situations for Mathematizing | |
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| Building in Constraints | |
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| Open vs. Closed Situations | |
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| Word Problems vs. Context Problems | |
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| Context-Based Investigations and Inquiries | |
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| Turning Classrooms into Mathematical Communities | |
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| The Edge Between the Individual and the Community | |
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| Facilitating Dialogue | |
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| Structuring Math Workshop | |
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| Investigations | |
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| Math Congress | |
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| Minilessons | |
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| Summing Up ... | |
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| Equivalence on the Horizon | |
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| Fair Sharing: The Historical Roots of Equivalence | |
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| Early Number Systems | |
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| Unit Fractions | |
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| Measurement Contexts | |
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| Sexagesimals | |
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| The Rise of Common Fractions | |
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| The Rise of Decimal Fractions | |
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| Expanding Commercial Markets as a Context | |
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| Children's Strategies for Equivalence | |
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| Math Congress | |
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| Extending Equivalence to Decimals and Percents | |
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| Summing Up ... | |
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| Developing Big Ideas and Strategies | |
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| Big Ideas on the Horizon | |
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| Part/Whole Relation | |
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| Equivalency vs. Congruency | |
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| Connecting Multiplication and Division to Fractions | |
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| The Whole Matters: Making Things Equivalent | |
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| Relations on Relations | |
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| Decimals and Percentage Equivalents | |
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| Place Value | |
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| Strategies on the Horizon | |
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| Summing Up ... | |
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| Developing Mathematical Models | |
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| What Are Mathematical Models? | |
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| Modeling Actions and Situations | |
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| Facilitating the Development of Modeling: The Role of Context | |
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| Partitive and Quotative Models | |
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| Generalizing the Models, Generalizing the Operations | |
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| From Models of Thinking to Models for Thinking | |
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| Ratio Tables | |
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| Double Number Line | |
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| Double Number Line and Percentages | |
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| The Clock | |
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| Arrays | |
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| Summing Up ... | |
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| Algorithms Versus Number Sense | |
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| The History of Algorithms | |
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| Teaching for Number Sense | |
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| Summing Up ... | |
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| Developing Efficient Computation with Minilessons | |
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| Minilessons with Mental Math Strings | |
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| Choosing the Strategies, Choosing the Numbers, Choosing the Models | |
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| Finding a Familiar, Landmark Whole | |
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| Choosing a Common Whole | |
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| Multiplying Numerators and Denominators | |
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| Swapping Numerators and Denominators | |
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| Getting Rid of the Fraction | |
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| Developing Strategies for Computation with Decimals | |
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| Using Money | |
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| Using Fractions and Decimals Interchangeably | |
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| Summing Up ... | |
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| Assessment | |
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| Performance-Based Evaluation and Assessment | |
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| Portfolio-Based Documentation | |
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| Assessing Mathematizing | |
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| Capturing Genuine Mathematizing | |
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| Linking Contexts to Reality | |
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| Providing for Various Levels of Mathematizing | |
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| Informing Teaching | |
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| Assessment in Reformed Classrooms | |
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| Assessing the Landscape of Learning | |
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| Assessing in the Moment | |
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| Portfolio Assessment | |
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| Paper-and-Pencil Assessments | |
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| Assessment Results | |
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| Mathematizing vs. Traditional Instruction | |
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| Standardized Tests | |
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| Summing Up ... | |
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| Teachers As Mathematicians | |
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| Teacher Preparation | |
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| Learning to Mathematize | |
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| Exploring the Edge | |
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| Formulating a Vision | |
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| Living in a Mathematical World | |
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| Summing Up ... | |
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| References | |
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| Index | |