Customer Reviews

How to Solve It: A New Aspect of Mathematical Method

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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.

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.

It's delightful to see this book is still in bookstores after 60 years, and I can still remember how much fun it was to read it 30 years ago. I came across it recently in a local bookstore, and after poring over it again, I was inspired to write a little review about it.

The most important thing about the book is Polya's little heuristic method for breaking down math problems and guiding you thru the process of solving them. Try to visualize the problem as a whole. Diagram it at first, even if you don't have all the details. Just initially try to get the most important parts of the problem down. Then try to get some sense of the relationship of the parts to the whole. Then tackle each of the component parts. If you get stuck, ask yourself if you could approach it another way, what could be missing, and so on. To this end, the questions at the back of the book are worth their weight in gold.

Polya's little heuristic and methods book is a timeless classic. This and Lancelot Hogben's "Mathematics for the Millions" have done more good for suffering math students than all the the dry textbooks put together that really don't teach you "how to solve it."

In fact, do you want to be a robot? I talked to a woman who took a whole semester in computer science and came out learning nothing. She told me this. My love affair with Real Math started with this book in a library. I was reading a book which had a bunch of interviews with the most successful programmers in the world. One was Czech and I do not remember his name. But he was asked the following question. "What in your opinion is the biggest mistake that programmers are doing in their educations or their work today?" He answered, "It's simple. They don't know how to solve problems. At our company, we have some simple books that tell you how to do this. The best is Polya's 'How to Solve It'. It has a little diagram in the back that completely runs you through a series of questions on solving math problems. But even in schools, they don't take this approach. Everything is by rote and repetition! You solve a problem and YOU DON'T KNOW WHAT YOU SOLVED! We have a lot of these little books." The late Isaac Asimov wrote a beautiful little book called "The Realm of Algebra". It's out of print. But he explains the entire realm of algebra in something like 150 pages. The best book I've ever seen about math. Math can be fun. Programming can be fun. But only if you ask Polya's questions in the back of this book. "What do I have to do to make this problem complete?" "What is missing from this problem?" "What could I add to make this problem solved?" A two page diagram in the back. And everybody knows that programming is just "crummy mathematics". BUY THE BOOK! BUY THE BOOK! BUY THE BOOK!. 2 pages in the end of this book and at least 50% of your math/programming problems are down the drain. Buy the books for your son if you are a Betty Crocker. Or your daughter. Or they will end up in the "Valley of the Dead". Solving problems in school for years and years and simply not knowing what they did! Good luck. Oh yes. One last thing. BUY THE BOOK!

Pardon the cliche, but: no home should be without this book. Polya's examples are mathematical, but the principles of heuristics and problem solving are universally applicable, whether you are a programmer, a mechanic, a TV repairman, or anyone trying to solve a problem. This is to problem solving what Strunk and White are to writing. BUY IT. I've lost track of my copy, and once I finish this review, I'll be replacing it.

i must say that this is one of the most interesting books i've ever read. it is a must for teachers of mathematics and the sciences. polya exposes with an unmatched wit and cunning the art of breaking it down and analyzing parts of parts in order to arrive at a reliable and justifiable solution.

polya is the quintessential teacher and this teacher finds his writings both compelling as well as inspiring. this is NOT the last book by this great mathematician that i plan to read.

a word of advice for teachers: read it and REREAD it until you get the point. the teaching of mathematics is not a mere monkey-see-monkey-do regurgitation process. rather, it is the art of conveying to the student the problem-solving process. once the student has mastered this talent, the sky's the limit. and speaking of limits, the sky may just be a gross underexaggeration.

LOVE IT! got to say again, "BUY THE BOOK!" and this goes doubly for TEACHERS!

Polya prescribes different forms to approaching a problem through some guide questions that a solver should ask ("Is there a related problem"). The exposition is quite short, majority of the book is devoted to a glossary of heuristic terms which prove very helpful. Polya uses common problems in high school geometry to demonstrate his point which make it easily understandable.

I'm glad I have discovered an excellent book on problem solving which would prove indispensable in my programming career. Other programming books mainly demonstrate features of an OS or a computer language but this book goes into the heart of the computer science which is problem solving.

Yes, this is a classic. Yes, I refer back to it frequently and always find something new which informs my teaching. But it is not the first book that I would recommend to a student (even the highly-selective students at a certain Norwegian liberal arts college in the upper Midwest) - the language can be quite difficult to hack through.

My favorite thing about this book is the historical perspective. Many of the ideas Polya puts forth are similar to those in the NCTM Standards, for example, and as you know there is far from unanimous support for what some perceive as the latest incarnation of "new math". Reading this book, one sees it's not new math at all. Pappus, one learns, introduced "analysis and synthesis" (what I might call "working backwards") as a follow-up to Euclid's Elements. That was a while ago...

Fascinating, makes me want to know more history so that I can weave it into my classes, but this book is not for everybody.

I got this book because I saw good reviews and heard that this was a classic...so I got it.

This is the first book that I ever encountered that teaches problem-solving. Further more, it teaches it through the use of heuristics(noun: A commonsense rule (or set of rules) intended to increase the probability of solving some problem). Half of the book cantains what the author calls The Dictionary - which contains a large number of heuristics that a problem-solver can use in his attempt to dissolve a problem. The author also describes in the first few chapters of the book on how to go about solving problems. Really gave me a new perspective on problem solving...Can't wait to apply what I have learned.

Buy it!(its cheap anyway...nothing to lose)

This is a book on problem solving, in its widest meaning, though mathematics dominates, given the author's education. It's a feast! Prepare yourself for many hours of fun and education. If you'll accept an advice, study every single book written by Polya.