Rowman & Littlefield Publishers / Institute for Student Achievement

Pages: 150
•
Trim: 6½ x 9¼

978-1-4758-1056-1 • Hardback • July 2014 • $91.00 • (£70.00)

978-1-4758-1057-8 • Paperback • July 2014 • $47.00 • (£36.00)

978-1-4758-1058-5 • eBook • July 2014 • $44.50 • (£35.00)

**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.

**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

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.

**— ****National Council of Teachers of Mathematics**

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.

**— ****Denise Stavis Levine, Ph.D.**

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.

**— ****Paul C. Jablon, Ph.D., Lesely University**

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.

**— ****Grace O’Keeffe, Hudson High School of Learning Technologies, New York**

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.

**— ****Erica Walker, professor, math education, Teachers College, Columbia University**

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.

**— ****Eric Marintsch, mathematics teacher, United States and Germany**

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.

**— ****Kate Burch, principal, Harvest Collegiate High School, NYC**