John Conway calls this “a wonderful book” and so have many others. Moreover, like John Conway, many find that it takes “a long time to appreciate just HOW wonderful it is”. In Conway’s case, it was becoming a teacher that did it. Indeed, the book is addressed to both students and teachers, but Conway found that “many of Polya’s remarks that hadn’t helped me as a student now made me a better teacher of those whose problems had differed from mine. . . Perhaps his best point is that learning must be active. As he [Polya] said in a lecture on teaching, ‘Mathematics. . . is not a spectator sport.’” “How to Solve It” gives many ideas for mathematicians of all levels who find themselves out on the playing field. Perhaps his best idea is“find another related problem” -- a special case, a more GENERAL case, something “merely” ANALOGOUS in some way (like a lower dimension). Another idea, perhaps obvious-sounding but also subtly helpful, is “Check that you used ALL the data.”
The parts of the book which I enjoyed and related to most were the many good, familiar DESCRIPTIONS of what it’s like to do math research. I laughed when he said, “You are lucky if you have any idea at all.” And when he asked, “What can I do with an incomplete idea?” and began his answer, “You should consider it,” I felt encouraged. And on the next page, “Perhaps you will be led astray by some of your ideas” and “even if you do not have any appreciable new ideas for a while you should be grateful if your conception of the problem becomes more complete. . ..” And on p. 76, “a detail strikes you. .. then you concentrate upon another detail. . . after a while you again consider the object as whole but you see it now differently.” “Determination and emotions play an important role,” he wisely tells us on p. 93.
Something else enjoyable and interesting that Polya does is quote proverbs. “Take counsel of your pillow” appears in the section on “subconscious work”, and then, beginning p. 222, there’s a whole section specifically on proverbs. Related to his insistence that ”the very first thing we must do for our problem is to understand it", he quotes, “Who understands ill, answers ill” and “think on the end before you begin”. “Try all the keys in the bunch.” “Do and undo, the day is long enough.” “Second thoughts are best.” “It could be an interesting task,” he remarks, to collect and group proverbs about planning, seeking means, and choosing between lines of action, in short, proverbs about solving problems.” Indeed, he is good at making up his own “proverbs”, such as “Look around when you have got your first mushroom. . . they grow in clusters.” In general, his counsel and support can be soothing.
I feel that his organization of materials is sometimes confusing. For example, I don’t really understand why he writes, so many times, “try to solve a related problem”, along with near equivalent advice such as to look for “an auxiliary problem”. Also, since the book was written in the 40’s, it is not surprising that there is sexism throughout. In particular, on p. 71, the “intelligent student” is “he” and the “conscientious shopper” is “she”. We can probably excuse this, but not ignore it.
That said: Another very positive aspect of this book is its spirit. This spirit is perhaps best described by the last sentence in the original author’s Introduction. “. . . most of the time. . . the point of view is that of a person who is neither teacher nor student but anxious to solve the problem before him.”
Marion Cohen (Mathwoman199436@aol.com) teaches at the University of the Sciences in Philadelphia.