Discovering Mathematics: The Art of Investigation [Paperback]

Anthony Gardiner (Author)

Book Description

Publication Date: May 14, 1987

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.

Editorial Reviews

Review

`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 Review

`This is not a book about mathematics but about doing mathematics. The explicit aim of the author is to guide the reader in thinking about, and mentally investigating a mathematical problem. In this sense the real content of the book is not contained in the book itself. The reader has to use the presented text, the hints, the exercises and so on for developing subjective experience with, and expertise in, formulating hypotheses, finding proofs, checking hypotheses and many other valuable cognitive techniques. To attain this goal different problems are presented and the process of solution is documented in a very detailed and exemplary way . . . Mathematics could be much more attractive and valuable if teachers would make use of problems like those presented here and - even more important - would implement a teaching style in accordance with this presentation.' Zentrallblatt fur Mathematik

`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

About the Author

A. Gardiner, Department of Mathematics, University of Birmingham.

Product Details

• Paperback: 220 pages
• Publisher: Oxford University Press, USA (May 14, 1987)
• Language: English
• ISBN-10: 0198532652
• ISBN-13: 978-0198532651
• Product Dimensions: 9.2 x 6.1 x 0.6 inches
• Shipping Weight: 12.6 ounces (View shipping rates and policies)
• Average Customer Review: 4.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #3,352,444 in Books (See Top 100 in Books)

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1 of 1 people found the following review helpful:

4.0 out of 5 stars A good primer of basic mathematical puzzles, solutions provided, July 21, 2008

By 

Charles Ashbacher (Marion, Iowa United States) - See all my reviews
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This review is from: Discovering Mathematics: The Art of Investigation (Dover Books on Mathematics) (Paperback)

Mathematics is the act of investigating what happens when you set down a set of objects, a set of rules on what you can do with the objects and then do things following those rules. Sometimes we put forward educated guesses as to what our result will be and at other times we simply just do what we can do to see what happens.
Discovering mathematics is nothing more than doing these things and in this book, Gardiner starts with some simple games that are easy to analyze. In the next section, a generalized three-digit number is selected, the three digits reversed and the process is repeated. List of self-referential sentences such as

"This sentence uses thirty symbols."

are analyzed next. Further topics involve geometric figures, an extensive section involving more flipping of the digits in the positions of a number, simple recurrence relations, classic jug and postage stamp problems, solving linear equations and coinage problems. Large numbers of additional exercises are given at the end of the chapters and hints followed by solutions are also included.
This book should be considered more a primer on mathematics than a book on mathematics. The problems are more in the area of mathematically based puzzles rather than what we would consider "pure" math problems. Nevertheless, it is a good source of quick problems for classes in mathematics for education majors or classes in algebra near or at the level of precalculus.