Mathematical Discovery, Volume II, On Understanding, Learning, and Teaching Problem Solving

by George Pólya

Combining standard Volumes I and II into one soft cover edition, this helpful book explains how to solve mathematical problems in a clear, step-by-step progression. It shows how to think about a problem, how to look at special cases, and how to devise an effective strategy to attack and solve the problem. Covers arithemetic, algebra, geometry, and some elementary combinatorics. Includes an updated bibliography and newly expanded index.

Genres: Mathematics, Nonfiction

215 pages, Hardcover

First published January 1, 1965

Reviews

[Nia · Rating 4 out of 5]

I have long wanted to read some of Polya's work, since my undergrad days in the Math/CS department with a prof. insisting that Polya was fundamental reading both to algorithm prooving (which, in real life software engineering, I never got to do, sadly) and also to grammars.
So now, I find myself reading with great pleasure his work on constructions, problem solving and motivation, and discipline of the mind. Very nice work, and though classic, still up with the times (or ahead of them even) pedagogically.
Well worth reading even for the non-mathematically minded.

[The Zahra. · Rating 3 out of 5]

برای آن‌هایی که می‌خواهند فکرهایشان را مانند ریاضی‌دان ها در سرشان بپرورانند. و برای آن‌هایی که زیبایی ریاضیات را درک می‌کنند.

[Dorum · Rating 3 out of 5]

The first volume is pretty good. The second one not so much. That is why I gave it a general rating of 3. I would have given it a 2 for the second volume and a 4 for the first volume. It IS however good for mathematics teachers I think who would do well to follow Mr. Polya's advice. I would generally recommend his books to programmers and engineers as well. These books are worth reading much more than any of the current "design patterns" shit that seems to be so popular these days.

[Jerry · Rating 3 out of 5]

…a heap of stones is not yet a house.

There is a lot I don’t remember about why I read this book. One is that I could have sworn I’d read that it had been released into the public domain; but while there are a lot of places that mark it as open source or public domain, I cannot now find anywhere that explains why it’s there. In any case, it’s available for download on the Internet Archive, among other places.

The other is why it was on my list of things to read to begin with. I was under the impression that the book was about general problem solving. In fact, it is specifically about mathematical problem solving, both how to do it and, for teachers, how to instill a love of doing it.

The teacher should regard himself as a salesman: he wants to sell some mathematics to the youngsters… The lad who refuses to learn mathematics may be right; he may be neither lazy nor stupid, just more interested in something else—there are so many interesting things in the world around us.

Pólya does give lip service to the benefit of knowing problem-solving techniques in everyday life, but quickly slides back into mathematical examples whenever suggesting general applicability. To get the most benefit from the book, it should be read at a place where you can do complex mathematics. I did not do this; it was still interesting, but not nearly as helpful as it was meant to be.

Much of what he writes is very good advice for solving technical problems in computer programming, though often so general that it’s only recognized in retrospect.

We need helpful ideas, we naturally desire to have helpful ideas at our service. But, in fact, they are our masters and they are capricious and self-willed. They may flash upon us unexpectedly, but more often they are long in coming, and sometimes they just keep us waiting and do not turn up at all… Waiting for ideas is gambling.

Because our minds must be tricked into coming up with good ideas, and everyone’s mind works a little differently, “A problem solver must know his mind and an athlete must know his body in about the same way as a jockey knows his horses.”

He sometimes brushes up against the scientific method, or, as he describes it “in three syllables”,

GUESS AND TEST

Multiple times he suggests guessing at the start and then successively zeroing in on the solution from multiple ends, “a series of trials” or “successive approximations” that could as well be described as “groping and muddling through”.

A lot of his recommendations for prospective high school mathematics teachers is not in vogue today.

Know-how is the more valuable part of mathematical knowledge… Yet how should we teach know-how? The students can learn it only by imitation and practice.

I do think this would be a great book for teachers of high school mathematics; but it is a great deal of work to go through it the way it was meant to be read. I don’t know whether it is a great book to read without doing the problems, as I did, although it was certainly interesting in parts.

Contemporary American adults do many more miles driving a car than walking. Therefore, we must teach a baby to drive a car before he can walk.

[Ed · Rating 3 out of 5]

I've been dipping in and out of this book for the last few years.

The examples are thought-provoking but the prose tends to ramble and speculate too much. E.g., is the way we think really best represented as a square (Fig 11.1) with vertices being "Mobilization", "Isolation", "Organization" and "Combination"? Iirc, How to Solve It is a little more actionable and direct.

[Latchezar Tomov · Rating 5 out of 5]

If teachers use all book of Polya to learn from them and to apply truly the heuristic method in school, there will be revolution in education. These books are also for the mathematicians who like to produce new theorems and proofs, to create, not only for the teacher (especially "Mathematics and Plausible reasoning").

[Richard]

Grabbed out of a stack of books from a retiring math teacher.