Preface Introduction What Will You Find in This Book Chapter 1: An Explanation of the ISA Approach to Teaching and Learning Mathematics Introduction A Vision of Mathematics in an ISA Classroom Guide To Creating a Vision and Four-Year Plan ISA Mathematics Rubric Indicators of Teacher Instructional Practices That Elicit Student Mathematical Thinking Indicators of Student Demonstration of Mathematical Thinking Chapter 2: A Guide to Teaching and Learning Mathematics Using the Five Dimensions of the ISA Rubric Introduction Dimension 1: Problem Solving Problem Solving Definition and Overview Teaching Idea #1: Choosing the Appropriate Problem Teaching Idea #2A: Use Problems with Multiple Strategies Teaching Idea #2B: Selecting an Appropriate Strategy Teaching Idea #3: Value Process and Answer Teaching Idea #4: Answer Student Questions to Foster Understanding Teaching Idea #5: Error as a Tool for Inquiry Teaching Idea #6: Students Create Their Own Problems Dimension II: Reasoning and Proof Reasoning and Proof Definitions and Overview Teaching Idea #1: Conjecturing Teaching Idea #2: Evidence and Proof Teaching Idea #3: Metacognition Dimension III: Communication Communication Definition and Overview Teaching Idea #1: Writing in Journals Teaching Idea #2: Writing in Problems and Projects Teaching Idea #3: Oral Communication Dimension IV: Connections Connections Definition and OverviewX Teaching Idea #1: There Are Common Structures That Bind Together the Multiple Ideas of Mathematics Teaching Idea #2: The History of Mathematics Helps Students Make Sense of and Appreciate Mathematics Teaching Idea #3: Using Contextual Problems That Are Meaningful to Students Dimension V: Representation Representation Definition and Overview Teaching Idea #1A: Learning to Abstract - Moving from Arithmetic to Algebra Teaching Idea #1B: Learning to Abstract - Use Examples of Physical Structures Teaching Idea #2: Making Sense of Confusion to Solve Problems Teaching Idea #3: Interpreting and Explaining Teaching Idea #4A: Mathematical Modeling – Modeling Mathematical Ideas and Real World Situations Teaching Idea #4B: Mathematical Modeling – Projects of the World That Use Rich Mathematics Chapter 3: Problems, Investigations, Lessons, Projects, and Performance Tasks Introduction Example 1: Display Dilemma Problem – Using Multiple Strategies / Looking for Patterns Example 2: Shakira’s Number – Valuing Process Example 3: Crossing the River – Valuing Process Example 4: Checker Board Problem – Simplifying the Problem Example 5: When Can I Divide? – Using Errors as a Tool of Inquiry Example 6: Creating a Mathematical Situation: Three Examples – Students Create Their Own Problems Example 7: The Game of 27 – Reasoning and Conjecturing Example 8: The String Problem – Conjecturing Example 9: Congruence and Similarity – Conjecturing and Proof Example 10: The Race – Metacognition on Multiple Strategies Example 11: Murder Mystery – Evidence and Proof Example 12: The Locker Problem – Metacognition Example 13: Gaming the Dice – Writing in Problems Example 14: Does Penelope Crash Into Mars? – Problems Are Meaningful to Students Example 15: Consecutive Sums Problem – Patterns and Conjecturing Example 16: Activity to Lead to Definition and Multiple Representations of a Function – Structures in Mathematics Example 17: The Pythagorean Triplets – The History of Mathematics Example 18: Laws of Exponents – Moving From Arithmetic to Algebra Example 19: Working with Variables – Learning to Abstract: Moving from Arithmetic to Algebra Example 20: Models of the Seagram Building – Use of Physical Structures Example 21: How Tall Is Your School Building? – Use of Physical Structures Example 22: Model Suspension Bridge Project – Modeling Real World Situations Example 23: Shoe Size Problem – Modeling Real World Situations Example 24: The Peg Game – Using Games to Understand Mathematics Example 25: Concentration of Medication in a Patient’s Blood Over Time – Modeling Using Real World Data Example 26: Marcella’s Bagels – Working Backwards Example 27: What is normal? – Modeling Mathematical Ideas and Real World Situations Example 28: Can You Build the Most Efficient Container? – Mathematical Modeling Example 29: Salary Choice – Mathematical Modeling Example 30: Border Problem – Learning to Abstract: Moving from Arithmetic to Algebra Example 31: The Magical Exterior Angles – Encouraging the Use of Evidence and Proof in Daily Problem Solving Example 32: Creating a Fair Game – Projects of the World That Use Rich Mathematics Chapter 4: Various Guides for Teachers Introduction School Mathematics: A Self-Assessment What Does An Inquiry Process Look Like In Mathematics? How to Write an Inquiry Lesson Questions to Think About When Planning an Inquiry-Based Common Core Aligned Unit List of Questions to Think About When Writing a Mathematical Performance Task Guide to Writing an Inquiry Lesson Inquiry-Based Lesson Planning Template Big Ideas in Algebra Big Ideas in Geometry Big Ideas in Probability and Statistics Questions for Students to Ask Themselves When Solving a Problem An Inquiry Approach to Look at Student Work An Inquiry Approach to Look at a Teacher-Created Task, Activity, or Lesson Teacher’s Perceptions Continuum Student’s Perceptions Continuum School Mathematics: A Self-Assessment References