Fragments to Mosaic: The Art of Teaching Math to Shape Critical Thinking and Problem Solving Skills

By Don White and John Lowder

"Fragments to Mosaic" is a book for everyone interested in education especially math education. The principles described and explained are transferable to other subjects. Many people suffer from fragmented understanding. They often fail to clearly see the big picture. Here, students, teachers, parents, and school administrators can find hope, encouragement and direction. Many thought provoking cartoon illustrations make the message more appealing and understandable than the average book on pedagogy. The book provides the Why, What, and How of a method of teaching that begins with student perceptions, emphasizes conceptual connections, and develops holistic understanding while fostering critical thinking and problem solving skills. Care has been given to point out key differences between this method and more common methods of teaching. Readers will discover how more can be learned better in less time when teachers study, learn, and practice this "art."

• Language: English
• Publisher: Xlibris US
• Release date: May 27, 2015
• ISBN: 9781503543935

Don White

Prior to becoming an academic, Dr McManus worked within the global software industry. Dr Don White is Dean of Faculty Business and Law, University of Lincoln. Dr White has lectured extensively, on management information systems and has acted as a consultant to the National Health Service (HNS).

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Copyright © 2015 by Don White; John Lowder. 651264

ISBN:   Softcover           978-1-5035-4394-2

              Hardcover         978-1-5035-4395-9

              EBook               978-1-5035-4393-5

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner.

Rev. date: 05/26/2015

Xlibris

1-888-795-4274

www.Xlibris.com

For it is God which worketh in you both to will and to do of his good pleasure.

Philippians 2:13

DEDICATION

I dedicate this book to The Master Teacher and the people He used to will and to do this book—Monica Smith and John Lowder. Without the encouragement and inspiration of my friend and professional colleague, Monica Smith, I doubt that I would have ever written this book. It was John Lowder, my friend and collaborating author, whose countless hours of patient listening and creative expression enabled me to reverse engineer myself and articulate the ClearMath® pedagogy. You might say, Monica was the will and John was the do. I wish to express my sincere thanks to both.

Don White

CONTENTS

Foreword

The ClearMath® Premise

Introduction

Part I Why?

Chapter 1 ClearMath® to the Rescue

Chapter 2 Evolution of a Revolution … a Paradigm Shift

Part II What?

Chapter 3 Introduction to Perceptual Imperatives™

Chapter 4 Arithmetic Facts

Chapter 5 The Real Number Line

Chapter 6 Math Properties

Chapter 7 Math as a Language

Chapter 8 Generalization

Chapter 9 Inverses

Chapter 10 Equivalence

Chapter 11 Forms of One & Implied Ones

Chapter 12 Terms and Factors

Chapter 13 Rational Expressions

Chapter 14 Models

Part III How?

Chapter 15 Introduction to ClearMath® Pedagogy

Chapter 16 Focus: Begin with the Big Picture in Mind

Chapter 17 Asking: Making the Lesson Fit the Student

Chapter 18 Conversation: The Heart of Perceptual Teaching®

Chapter 19 Tying Together: Perceptual Connectors

Chapter 20 Spiraling: Review & Preview for Holistic Understanding

Conclusion/Summary

Author Biography

Glossary/Index

FOREWORD

In this day and age, we are all looking for ways to streamline and improve instruction. Society’s focus on leaving no child behind has pressured many teachers into drilling test-taking strategies at the expense of mastery learning. That kind of teaching is not what we planned for our careers when we entered the profession. In contrast, this book describes a methodology that builds conceptual understanding while developing reasoning skills.

One ninety-minute classroom observation in January 2007 marked a turning point in my teaching philosophy. I had just been hired as an Instructional Supervisor in August 2006, so I was eager to compare our district’s student performance to that of comparable districts. I noticed a neighboring district had comparable math scores in the eighth grade, but their ninth graders had consistently improved. This trend was an anomaly that made me curious. When I found out a ninth grade math teacher, Don White, was the husband of one of our own teachers, I solicited an opportunity to visit his classroom. That visit changed my view of teaching forever. After twenty-two years in education, I experienced an Algebra I class as all math classes should be taught.

After watching him guide a classroom of average freshmen through a lesson on quadratic functions, I was hooked! His Algebra I students skillfully handled problems and questions that would challenge many of our Algebra II students—and his students were in the middle of their Algebra I course! When I observed his teaching, I noticed how he insisted on clarity of definition; comparing and contrasting; connections and generalization. This in turn facilitates connected and in-depth understanding.

True to his philosophy that less is more, the homework assignment from the previous day was only one problem, but the mathematical knowledge needed to solve the assignment was staggering. The students could not wait to share their strategies for solving the problem. Their pride and confidence was affirming and empowering. The classroom was as much an immersion in the language of mathematics as any foreign language immersion program. I was enthralled with the experience. I could not wait to visit again, and I wanted my teachers to share in the experience as well.

I was exposed to ClearMath® teaching as a ninth grader. That’s when I fell in love with math, declared my college major, and determined my future profession. Today I am a high school math and physics teacher.

(Deidre Higgins,

High School Math and Physics Teacher)

The teachers who observed Don in action were convinced his methods yield great results. All aspired to replicate his methods. We yearned for a manual providing guidance. I encouraged Don to write a book to document his teaching methods. However, the nuances of his pedagogy proved to be rather difficult to identify and communicate.

While encouraging Mr. White to write a book, I did not realize he was really teaching thinking skills. With time, that became my biggest lightbulb moment. The math skills his students learned were just by-products of the higher order thinking skills he was teaching. Math was just his medium, his paint.

Don needed someone to help him translate his methods into the manual we continued to request, and John Lowder appeared. John’s son had been a student in one of Don’s Algebra I classes. Years later, while visiting the public library, John noticed Don helping a young lady with her math. John paused momentarily and said, Young lady, you are blessed. Afterwards, Don approached John about helping write this book. Thus began a conversation and collaboration continuing to this day. I am convinced this encounter was divine intervention. At long last, they have been able to capture the essence of ClearMath® in writing. I also trust you will come to appreciate that ClearMath® is not a scripted day-by-day curriculum manual, but rather, it is a pedagogy focusing on critical thinking and in-depth reasoning skills.

This book is not just for math teachers as the title might lead you to believe. It is for anyone who believes in quality education. The learning style this methodology precipitates is transferable and transformative. This is not a revision of the old paradigm. It is a shift! The philosophy and pedagogy found in this book will help develop the critical thinking skills needed for 21st century education.

I have been a cheerleader and devil’s advocate throughout the development of this book. A big revelation I have gained from my conversations with Don is most teachers struggle to deconstruct, assess, and discuss specifics of the instructional process. We are adept at discussing content curriculum and resources, but we get nervous discussing how we structure and deliver lessons. I am surrounded by dedicated teachers who work long hours preparing and delivering lessons. I believe even the best of these teachers will be able to read this book and gain ideas that will help them improve their craft.

I finish on a personal note. I have been privileged to have Don work with my son in enrichment sessions for the past three years. As a result, my child has developed critical thinking and reasoning strategies that have made him a better student in all his classes. Thus, I have witnessed the impact of ClearMath® from both a personal and a professional standpoint. It works!

Respectfully,

Monica Smith

I remember well the first time I