# Mathematical Discovery

## On Understanding

Learning and Teaching Problem Solving

## George Polya

## Foreword

You, the readers, have before you a book devoted to explaining how you can improve your understanding of mathematics and how you can better solve mathematical problems. In brief, its intention is that you learn how to do mathematics and how to use mathematics. The. examples are designed for those concerned with mathematics at the high school or early undergraduate level.

We claim that this is a wonderful book and that its reissue is timely and necessary. We do not believe that many would dispute this claim but nevertheless we want to take this opportunity to establish its validity. Our proposition then is that *this book is eminently suitable for the needs of all concerned teachers and interested students of mathematics* at the levels indicated above; and we will prove our proposition by contradiction!

If this book were not eminently suitable for your needs, it would have to fail for one of the following reasons, all of which I will easily demonstrate to be absurd! Let me then list eight purely hypothetical, but absolutely false, charges and quickly refute each one!

**It never was suitable.**This assertion would be so ridiculous we scarcely need to rebut it. This book had a tremendous success when it first appeared and was found invaluable by teachers and students alike. It was-and is-a most pleasant and interesting book to read, and it brought not only enlightenment but joy to the reader.**Mathematics is no longer relevant.**This assertion is equally ridiculous! It is not mathematicians but the leaders of our society who are to be heard most stridently proclaiming the importance of mathematical knowledge and understanding to today's citizens if we are to remain among the leading nations of the world. We mathematicians would also point to the great advantage to individuals in their own lives of being able to reason effectively and solve quantitative problems.**The objectives considered have ceased to be crucial.**This false view is easily disposed of, but it may be that some could believe the view correct. For it is sometimes thought that, with the availability of the hand-calculator and its big brother, the computer, the use of mathematics is reduced to pushing buttons and feeding in canned programs. This view would be quite false. It is the drudgery that has been eliminated from mathematics by these modern devices and not the need for thought. We assuredly still need to recognize when a problem is suitable-and ripe-for mathematical treatment, and we need to plan a strategy for tackling the problem; these utterly human tasks are the ones with which Polya is concerned.**The methods described are now perceived, in many cases, not to be the best or are very seriously disputed.**This is a standard reason frequently offered in justification for allowing a once valuable book to go out of print. It does not apply in this case because the author is not concerned with slick solutions but with the whole strategy of problem-solving. That strategy is so closely related to the way human beings think-when they are thinking effectively-that it is not going to undergo any dramatic changes. Polya describes the methods that work for him and other successful problemsolvers. As the old adage goes, "the proof of the pudding is in the eating. "**The particular mathematical content has been superseded.**The mathematical content of this book is drawn from the standard precalculus curriculum, that is, from arithmetic, algebra, and geometry, with some elementary combinatorics. The mathematical material is not, of course, developed systematically, since the author's purpose is to exemplify problem-solving strategies and principles with the aid of interesting and intriguing mathematical questions. It is true that the availability of the hand calculator is not assumed but, on the other hand, the nature of the problems treated is not such as to make its availability essential. Polya shows you how to think about a problem, how to look at special cases, how to generalize in interesting and important directions and how to solve a problem. These skills will never be superseded.**The content is well known.**One wishes this were true, but it is not! The content is not even well known to teachers, let alone to students. The examples Polya gives carry a fresh illumination which must surely inspire any reader. There is fascination in his Description of the mathematical ideas, in his elucidation of pedagogical principles, and in his own unique style of exposition. There is nothing in this book that bears the stamp of staleness.**The content is now obvious.**This assertion differs in an essential way from the previous one. That claimed that the content has already been conveyed to the reader from other sources. This claims that it does not need to be conveyed because what the text is saying has now become obvious as a result of a changing climate of thought in mathematical education. But this assertion is just as wrong-headed as its predecessor! The changing climate gives emphasis to the need to teach the art of problem-solving, of mathematical discovery; but we are very far from being a nation of skilled problemsolvers and discoverers! So, far from rendering this book superfluous, our current view of how to teach mathematics renders it absolutely crucial. The world has caught up with Polya; and what he and his lifetime friend, Gabor Szeg6, saw so clearly many years ago, long before this book was written, is now the cornerstone of the program of the National Council of Teachers of Mathematics' in their campaign to improve the quality of mathematics education.**The book is too expensive for those who would benefit from having it.**A mere reissue of the original edition might have led to this charge. The publishers are to be congratulated on bringing the original two volumes together into one volume and in making available a soft-cover edition. The separation of the text material into two volumes was justified at the time of original publication, since it avoided undue delay in the appearance of a much awaited book. There is no case for it today. This book is surely worth its present price to anyone genuinely interested in mathematics.

Thus there can be no possible argument against reissuing this book-and overwhelming arguments in favor of doing so. You, the reader, will surely share my enthusiasm for this wonderfully stimulating, wonderfully contemporary work, written by a man whose youthful approach to new discoveries in mathematics, or any other fields, has never deserted him. Polya succeeds better than almost anybody else in conveying, on the printed page, the excitement he himself feels in the intellectual adventures he so vividly describes. We are privileged and fortunate to be able to share the adventure with him.

The reader's attention should be drawn to two features of this new edition. The bibliography has been updated by Professor Gerald Alexanderson and the index has been expanded by Professor Jean Pedersen. Thus, while the text remains Polya's original, this new edition is, in every sense, a text for our times.

Case Western Reserve University Cleveland, Ohio | PETER HILTON |

Battelle Human Affairs Research Center Seattle, Washington |

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