How to Solve Problems: New Methods and Ideas

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Author: Spyros Kalomitsines

In this book you will find suggestions and methods that will help you overcome many difficulties when you solve problems. These methods are given in simple and easy-to-follow ways, always illustrated by examples. If one example requires mathematical knowledge beyond your grasp, you can skip it. There is a sufficient variety of examples for you to be able to comprehend the methods described. After reading this book, your problem solving skills will have improved appreciably. Even if you still cannot solve a problem, you will have approached it in refreshing and exciting ways.

In Chapter 1, the “description method” is introduced. This method shows how to generate ideas towards solutions, based on existing knowledge. This is perhaps the most important method; it encompasses a large number of problems and is easily combined with most other methods. In Chapter 2, a method is given that helps to avoid loops when solving a problem. In Chapter 3, difficult problems are solved by a certain combination of the description method, the means-end analysis and the method of getting out of loops. In Chapter 4, additional methods or alternative techniques for certain kinds of problems are discussed, as: contradiction, working backward, defining a procedure, setting subgoals, techniques for word problems and problems in matrix algebra, etc. In Chapter 5, you will find a teaching model for problems with long or difficult solutions. You will also find a model for helping students when they solve hard problems. The model shows how to group problems according to their common characteristics and to search for useful patterns. A number of such groups from calculus are presented. In Chapter 6, you will find an answer to the question: Can the previous methods be programmed on a computer? This can be done through a formalization of the methods. Such a procedure sets the stage for a fruitful production of new problem solving techniques. This book is intended to help teachers but also to help students who want to improve their problem solving skill on their own.

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Table of Contents

Preface
Acknowledgements
Chapter 1. The Description Method
Chapter 2. The Method of Getting Out of Loops
Chapter 3. The Spiral Method for Solving Problems
Chapter 4. Other Methods
Chapter 5. Two Models for Teaching Mathematics and Problem Solving
Chapter 6. A Formal Computer Model of Our Methods
Solutions of Unsolved Problems
References
About the Author
Index

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