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| Preface | |
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| Proportional Reasoning: An Overview | |
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| Student Strategies | |
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| Introduction | |
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| The Constant of Proportionality | |
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| Reasoning: Beyond Mechanization | |
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| Invariance and Covariance | |
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| Solving Proportions Using k | |
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| Multiplicative Thinking | |
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| Critical Components of Powerful Reasoning | |
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| Getting Started | |
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| Analyzing Children's Thinking | |
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| Activities | |
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| Fractions and Rational Numbers | |
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| Student Strategies | |
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| New Units and a New Notational System | |
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| The Psychology of Units | |
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| New Operations and Quantities | |
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| Interference of Whole Number Ideas | |
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| Problems with Terminology | |
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| Development of Sets of Numbers | |
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| Kinds of Fractions | |
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| What are Fractions? | |
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| Rational Numbers | |
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| Fractions as Numbers | |
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| Fractions, Ratios and Rates | |
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| Many Sources of Meaning | |
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| Multiple Interpretations of the Fraction � | |
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| Activities | |
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| Relative Thinking and Measurement | |
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| Student Strategies | |
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| Two Perspectives on Change | |
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| Relative Thinking and Understanding Fractions | |
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| Encouraging Multiplicative Thinking | |
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| Two Meanings for "More" | |
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| The Importance of Measurement | |
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| The Compensatory Principle | |
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| The Approximation Principle | |
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| Recursive Partitioning Principle | |
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| Measuring More Abstract Qualities | |
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| Other Strategies | |
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| Activities | |
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| Quantities and Covariation | |
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| Student Strategies | |
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| Building on Children's Informal Knowledge | |
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| Quantities Unqualified | |
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| Quantifiable Characteristics | |
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| Discussing Proportional Relationships in Pictures | |
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| Visualizing, Verbalizing, and Symbolizing Changing Relationships | |
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| Covariation and Invariance | |
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| Cuisenaire Strips | |
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| Scale Factors | |
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| Similarity | |
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| Indirect Measurement | |
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| Testing for Similarity | |
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| Mockups and Pudgy People | |
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| Activities | |
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| Proportional Reasoning | |
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| Student Strategies | |
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| The Unit | |
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| Units Defined Implicitly | |
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| Using Units of Various Types | |
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| Reasoning Up and Down | |
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| Units and Unitizing | |
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| Unitizing Notation | |
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| Flexibility in Unitizing | |
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| Children's Thinking | |
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| Classroom Activities to Encourage Unitizing | |
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| Visual Activities | |
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| Reasoning with Ratio Tables | |
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| Problem Types | |
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| Ratio Tables | |
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| Increasing the Difficulty | |
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| Analyzing Relationships | |
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| Percent | |
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| Percents as an Instructional Task | |
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| Reasoning with Percents | |
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| Activities | |
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| Reasoning with Fractions | |
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| Student Strategies | |
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| Visualizing Operations | |
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| Equivalent Fractions and Unitizing | |
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| Comparing Fractions | |
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| Fractions in Between | |
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| Activities | |
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| Fractions as Part-Whole Comparisons | |
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| Student Strategies | |
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| Part-Whole Fractions: The Big Ideas | |
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| Unitizing and Equivalence | |
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| Problems in Current Instruction | |
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| Fraction Models | |
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| Fraction Strips | |
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| Comparing Part-Whole Fractions | |
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| Discrete Units | |
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| Multiplication | |
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| Partitive and Quotative Division | |
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| Division | |
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| Other Rational Number Interpretations | |
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| Activities | |
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| Fractions as Quotients | |
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| Student Strategies | |
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| Quotients | |
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| Partitioning as Fair Sharing | |
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| Partitioning Activities | |
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| Children's Partitioning | |
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| Equivalence | |
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| Should We Reduce? | |
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| Understanding Fractions as Quotients | |
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| More Advanced Reasoning | |
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| Sharing Different Pizzas | |
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| Activities | |
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| Fractions as Operators | |
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| Student Strategies | |
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| Operators | |
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| Exchange Models | |
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| Composition | |
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| Area Model for Multiplication | |
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| Area Model for Division | |
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| Compositions and Paper Folding | |
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| Understanding Operators | |
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| Activities | |
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| Fractions as Measures | |
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| Student Strategies | |
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| Measures of Distance | |
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| Static and Dynamic Measurement | |
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| The Goals of Successive Partitioning | |
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| Understanding Fractions as Measures | |
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| Units, Equivalent Fractions, and Comparisons | |
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| Fraction Operations | |
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| Activities | |
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| Ratios and Rates | |
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| Student Strategies | |
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| What is a Ratio? | |
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| Notation and Terminology | |
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| Equivalence and Comparison of Ratios | |
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| Ratios as an Instructional Task | |
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| What is a Rate? | |
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| Operations with Rates and Ratios | |
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| Linear Graphs | |
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| Comparing Ratios and Rates Graphically | |
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| Speed: The Most Important Rate | |
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| Characteristics of Speed | |
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| Students' Misconceptions about Speed | |
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| Average Speed | |
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| Distance-Speed-Time and Graphs | |
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| Activities | |
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| Changing Instruction | |
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| Student Comments | |
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| Why Change? | |
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| A Summary of Fraction Interpretations | |
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| Central Structures | |
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| Characteristics of Proportional Thinkers | |
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| Obstacles to Change | |
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| Sequencing Topics | |
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| Directions for Change | |
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| Index | |