Synopses & Reviews
The word 'mathematics' usually conjures up a world of more-or-less familiar problems to be solved by more-or-less familiar techniques. This book examines a very different aspect of mathematics, namely how one can begin to explore unfamiliar, fresh ideas and chance observations, how one can pursue them through various stages until the light eventually begins to dawn, and how this whole process invariably throws up other interesting questions one would otherwise never have thought of.
The items have been designed to provide certain basic experiences from struggling to explain simple, but puzzling mathematical phenomena, to discovering for oneself some new and totally unexpected bit of mathematics. Readers should have no difficulty with the mathematical techniques which are required. They should therefore be in a position to reflect on some of the more striking features of the process whereby unsuspected mathematical relationships are uncovered, and on the way they emerge.
The problems studied avoid the kind of sophistication which would put them out of reach of ordinary students, yet are sufficiently complex to capture the essential features of the process of mathematical discovery. No attempt has been made to reduce this process to the level of checklists of catchwords. Readers will want to identify and reflect on the significance of these essential features for themselves.
"All aspiring mathematicians should work through this inspiring book....This is a marvellous account of beginning with a problem/challenge, of negotiating numbers and doodling diagrams to become familiar with what aspects are important/unimportant, of using mathematical knowledge, of searching for patterns, of making predictions, of checking, of realising frustration, of the reward for perseverance, of seeing the complex become simple, of explaining and proving results, and of final triumph and pleasure. The author is to be congratulated for not only writing such an account, but for writing it with clarity, wit and enthusiasm." Mathematical Gazette
One of the most striking characteristics of mathematics is that thoughtful and persistent mathematical analysis often provokes totally unexpected insights into what may at first have looked like an uninteresting or intractable problem. This book gives students an opportunity to discover the nature and process of mathematics by developing their ability to investigate problems without relying on the standardized methods usually taught. The techniques required are elementary, allowing the student to concentrate on the way the material is explored and developed, and on the strategies for addressing questions whose answers are not immediately obvious. The book will challenge high school and college mathematics students as well as interested general readers.
Table of Contents
PART I: Short Investigations
Advice to the Reader
1. Analyzing Games
2. From Rhyme to Reason
3. Thinking Things Through
4. Asking Questions
PART II: Extended Investigations; Investigation 1: Flips
5. Introducing 9-flips
6. The Art of Guessing
7. Other Flips
8. Flips in Other Bases
9. Counting 9-flips with n Digits
10. Recurrence Relations
11. The End of the Road.
Investigations II: The Postage Stamp Problem
12. Getting Stuck In
13. Coming Unstuck: The Search for Proof
14. One Picture is Worth a Thousand Words
15. The Final Licking
16. The Coin Problem