Developing Mathematical Thinking

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