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| Foreword | |
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| Introduction | |
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| What You Teach and How You Teach It | |
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| The Power of KWC: An Alternative to Key Words | |
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| Using KWC to Tap Prior Knowledge | |
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| Using KWC to Structure Group Learning | |
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| Using KWC to Deepen Connections | |
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| Extensions | |
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| Six Big Ideas | |
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| The Research on Mathematical Learning and Teaching | |
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| Engaging Prior Understanding | |
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| The Essential Role of Factual Knowledge and Conceptual Frameworks | |
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| The Importance of Self-Monitoring | |
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| Six Big Ideas: Building on Mathematical Research and Principles | |
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| Teachers Broaden Their View of Problem Solving | |
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| Making Connections Between the Problem and Their Lives | |
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| Creating Multiple Representations of Increasing Abstraction | |
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| Students Solving Problems: Same Concept, Multiple Contexts | |
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| Cognitively Based Planning for Language, Connections, Contexts, and Representations | |
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| Integrating Reading Comprehension Strategies and Math Processes via Cognitive Principles | |
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| Making Meaningful Connections Among Mathematical Concepts | |
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| The Connectedness of Strands | |
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| How Does This All Fit Together? | |
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| Numbers and Early Algebra | |
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| Algebra in the Classroom, Then and Now | |
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| Partial Products Like You've Never Seen Them | |
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| Starting Out with Base Ten Blocks and Graph Paper | |
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| Moving on to More Abstract Representations and Mental Math | |
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| Red Dots | |
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| Algebra Tiles | |
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| Partial Quotients | |
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| Andy's Inheritance | |
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| Square the Digits and Sum the Squares | |
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| Summing the Cubes | |
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| 'The Irrational Tangram | |
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| Proportional Reasoning | |
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| What Proportional Reasoning Looks and Sounds Like in the Classroom | |
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| Shampoo Bottle | |
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| Cats and Rats | |
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| Making Seismometers | |
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| Developing Students' Proportional Reasoning Skills | |
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| Understanding Differences Between Additive and Multiplicative Transformations | |
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| Understanding Ratios | |
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| Understanding Rates | |
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| Interesting Applications of Rate | |
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| Algebraic Thinking and Modeling | |
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| Line of Best Fit and Linear Combinations | |
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| Positive Slope Situations | |
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| Inverse Linear Relations | |
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| Finite Differences: Quadratic, Cubic, and Beyond | |
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| Quadratic Equations | |
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| Cubic Equations | |
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| Conclusion | |
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| Geometry and Measurement | |
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| Multiple Representations for Solving a Geometry Problem | |
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| Ordering Shapes by Two-Dimensional Size | |
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| Measuring the Area | |
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| Make My Polygon | |
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| A Great Extension: Making Dodecagons | |
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| What's Your Angle? | |
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| Tessellations: A Different Way | |
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| Pythagoras 'R' Us | |
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| Pythagoras and Similarity | |
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| Primitive Pythagorean Triples (PPT) | |
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| Geometry and the Metric System | |
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| Silent Snow, Secret Snow | |
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| Conclusion | |
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| Data Analysis and Probability | |
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| Exploring Experimental Probability | |
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| Chevalier de Mere's Game of Chance | |
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| Inference and Prediction: Probability Bags | |
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| A Plethora of Pigs | |
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| Model Building with Montana Red Dog | |
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| Exploring Possible Outcomes in Theoretical Probability | |
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| Combination Pizzas and Permutation Locks | |
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| Product Versus Square | |
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| Montana Red Dog Follow-Up | |
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| De Mere's Bets Follow-Up | |
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| Concluding Thoughts | |
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| Appendix | |
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| References | |
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| Problem Index | |
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| Index | |