Developing Mathematical Thinking: A Guide to Rethinking the Mathematics Classroom

In this country we have done a poor job of helping students come to see the wonder, beauty and power of mathematics. Standards can be brought into the picture, but unless we think about what it means to truly engage students in mathematics we will continue to be unsuccessful. The goal of this book is to begin to change the way students experience mathematics in the middle and high school classrooms. In this book you will find a theoretical basis for this approach to teaching mathematics, multiple guides and questions for teachers to think about in relation to their everyday teaching, and over 30 examples of problems, lessons, tasks, and projects that been used effectively with urban students.

Table of Contents:
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

About the Author :
Jonathan Katz has been involved in math education a both a teacher and math coach for 33 years. He received his doctorate from Columbia University Teachers College in 2009.

Review :
I recommend Developing Mathematical Thinking for anyone interested in the transformation of a mathematics classroom to a place of inquiry, creativity, and excitement for both teacher and students. It would be an excellent resource to build collaboration among middle and secondary in-service and preservice teachers, mathematics teacher educators, mathematics coaches, and professional development facilitators. This well-researched, amply referenced book will have teachers, coaches and school leaders feeling as passionate about mathematics instruction as Jonathan Katz so clearly does. His love of mathematics and teaching are palpable on every page, and his nagging desire to move students beyond doing well on standardized tests toward having them view mathematics as a creative endeavor, with importance in the real world, makes this a timely, welcome addition to the field. There is a notion that inquiry-based, problem solving mathematics cannot be done in higher level high school courses. This book quickly and clearly demonstrates that to be false. This approach to teaching mathematics in our high schools might be what simultaneously narrows the achievement gap in our own country while simultaneously making us globally competitive. Both practical and philosophical, Developing Mathematical Thinking, addresses the questions "What is mathematics for?", "What to do Monday?", and the areas in between. In the spirit of inquiry, Katz leads the reader through a series of questions, investigations, and assessment tools that help educators, administration, students, and families finally make sense of mathematics. Decades of teaching and coaching experience are documented in this book. Katz gives life to mathematics through joyful and challenging learning experiences, not just lesson plans. Jonathan Katz has written a timely and important book sharing best practices of promoting deep mathematical thinking for all students. Based on his extensive years of experience as a teacher and professional developer, Jonathan shares detailed examples of how inquiry-based learning can be done well in a variety of settings. This book will be of great use to schools, teachers, and professional developers alike. Jonathan's love for mathematics and interest in promoting rigorous mathematics learning for all students shine through on every page. The vision presented by Jonathan Katz revolutionized the way I think about mathematics education – in turn I have seen dramatic improvements in student engagement and attitudes towards learning. This book is full of questions and ideas to challenge teachers and students alike. It is essential for any teacher whose goal is to inspire their students and develop deep and lasting conceptual understandings. It is great delight to have the insight and inspiration Dr. Katz always brings to the mathematics classroom now in book form. The clarity of the framework he presents brings us the thoughtful tools we need to make the study of mathematics stimulating and rewarding. His vision to make mathematics a work of "wonder, power and beauty" for all different learners is valuable and compelling.