How to Solve It: A New Aspect of Mathematical Method (Paperback)

by G. Polya (Author) "Helping the student..." (more)
Key Phrases: heuristic syllogism, accessible related problem, proposed problem try, Euclid's Elements, John Jones, Director Brown (more...)

Editorial Reviews

Review
Any young person seeking a career in the sciences would do well to ponder this important contribution to the teacher's art. -- Review

Review
Every prospective teacher should read it. In particular, graduate students will find it invaluable. The traditional mathematics professor who reads a paper before one of the Mathematical Societies might also learn something from the book: 'He writes a, he says b, he means c; but it should be d.' (E. T. Bell Mathematical Monthly )

[This] elementary textbook on heuristic reasoning, shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected. (Herman Weyl Mathematical Review )

I recommend it highly to any person who is seriously interested in finding out methods of solving problems, and who does not object to being entertained while he does it. (Scientific Monthly )

Any young person seeking a career in the sciences would do well to ponder this important contribution to the teacher's art. (A. C. Schaeffer American Journal of Psychology )

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Product Details

• Paperback: 224 pages
• Publisher: Princeton University Press (November 1, 1988)
• Language: English
• ISBN-10: 0691023565
• ISBN-13: 978-0691023564
• Product Dimensions: 8 x 5.2 x 0.6 inches
• Shipping Weight: 10.1 ounces
• Average Customer Review: 4.6 out of 5 stars See all reviews (35 customer reviews)
• Amazon.com Sales Rank: #346,742 in Books (See Bestsellers in Books)

Customer Reviews

Most Helpful Customer Reviews

74 of 75 people found the following review helpful:

5.0 out of 5 stars A Classic for Problem-Solvers, September 16, 2001

By	Philip Hamilton (Austin, TX USA) - See all my reviews (REAL NAME)

I found Pollya's "heuristic" approach to problem-solving applicable to both mathematical and non-mathematical problems. The goal of the heuristic approach is to study (and use!) the methods and rules of discovery and invention.

Here are just some of the questions that Pollya teaches as tools:

1. What is the unknown? What is the data? What conditions does the solution need to satisfy?
2. Do you know a related problem? Look at the unknown and try to think of a familiar problem having the same or a similar unknown.
3. Can you restate the problem? Can you solve a part of the problem.
4. Can you think of other data appropriate to determine the unknown?
5. Can you check the result?
6. Can you look back and use the result or the method for some other problem?

Overall, the author provides a systematic way to creatively solve problems. This volume has withstood the test of time for nearly 50 years. I recommend it highly.

44 of 44 people found the following review helpful:

5.0 out of 5 stars Indispensable for anyone who solves problems professionally., July 7, 1996

By A Customer

This review is from: How to Solve It: A New Aspect of Mathematical Method (Hardcover)

How to Solve It is the most significant contribution to heuristic since Descartes' Discourse on Method. The title is accurate enough, but the subtitle is far too modest: the examples are drawn mostly from elementary math, but the method applies to nearly every problem one might encounter. (Microsoft, for instance, used to and may still give this book to all of its new programmers.) Polya divides the problem-solving process into four stages--Understanding the Problem, Devising a Plan, Carrying out the Plan, and Looking Back--and supplies for each stage a series of questions that the solver cycles through until the problem is solved. The questions--what is the unknown? what are the data? what is the condition? is the condition sufficient? redundant? contradictory? could you restate the problem? is there a related problem that has been solved before?--have become classics; as a computer programmer I ask them on the job every day.

The book is short, 250 large-print pages in the paperback. Its style is clear, brilliant and does not lack in humor. Here is Polya's description of the traditional mathematics professor: "He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and turn his back on the class. He writes A; he says B; he means C; but it should be D." Behind the humor, though, lurks a serious complaint about mathematical pedagogy. Fifty years ago, when Polya was writing, and today still, mathematics was presented to the student, under the tyranny of Euclid, as a magnificent but frozen edifice, a series of inexorable deductions. Even the student who could follow the deductions was left with no idea how they were arrived at. How to Solve It was the first and best attempt to demystify math, by concentrating on the process, not the result. Polya himself taught mathematics at Stanford for many years, and one can only envy his students. But the next best thing is to read his book.

36 of 36 people found the following review helpful:

5.0 out of 5 stars A delightful and satisfying classic, December 17, 2004

By	Sandra Feder "Author of Side Effect and One o... - See all my reviews (REAL NAME)

This review is from: How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) (Paperback)

Are you like a dog with a bone when you're working on a brain teaser? After pages of scribbles, do you get a big grin on your face when you turn to the answers and say: "I'm right!" Then this book is for you.

And if you're not yet a die-hard problem-solver? You should step right up, too. You may get hooked.

G. Polya's book is based on the fact that, if we study how someone does something successfully, we can learn to do it successfully as well. How To Solve It is an application of 'heuristics' to solving problems.

There are certain mental operations useful in solving problems, any sorts of problems. Polya (who was an eminent mathematician and former Professor of Mathematics at Stanford University) describes and illustrates the most usual and useful of these operations, in a way that is irresistible and eye-opening.

These useful mental operations are organized according to when they come into play during the four steps to solving a problem. 1. You have to understand the problem. (Not as easy as it sounds.) 2. Find the connection between the data given and the unknown. Conceive the idea of a plan for the solution. 3. Carry out the plan. 4. Examine the solution obtained.

If you take some time and try to solve the problems selected to illustrate each mental operation, you will be well-rewarded. You will likely discover something surprising about your own problem-solving methods, and improve them in the process. You will definitely discover many new ideas and techniques to add to your arsenal.

For example, a first impulse when confronted with a problem is often to try to 'swallow it whole' -- to try to meet all of the conditions of the problem at once. G. Polya suggests keeping only part of the condition, and dropping the other part. This can lead you straight to a solution you might otherwise have completely missed.

His techniques help you to stand back and get to the heart of the problem, rather than getting lost in it.

Something else I liked very much about his book is his encouragement to guess, or to reason 'plausibly.' While the final proof must be strictly logical, "Anything is right that leads to the right idea." Problem-solving has every right to be fun, as well as purposeful.