Developing Mathematical Thinking
A Guide to Rethinking the Mathematics Classroom
Published on 9. July 2014
Book
Hardback
144 pages
978-1-4758-1056-1 (ISBN)
Description
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.
Reviews / Votes
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, NYCMore details
Language
English
Place of publication
United States
Publishing group
Bloomsbury Publishing Plc
Target group
Professional and scholarly
Dimensions
Height: 235 mm
Width: 157 mm
Thickness: 13 mm
Weight
370 gr
ISBN-13
978-1-4758-1056-1 (9781475810561)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
E-Book
07/2014
1st Edition
Rowman & Littlefield Publishers
€39.49
Available for download
E-Book
07/2014
1st Edition
Rowman & Littlefield Publishers
€39.49
Available for download
Person
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.
Content
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
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