Gyorgy Polya
Princeton, 1945
2nd Edition, 1957
ISBN 0691023565 (paper)
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Buy it from Amazon.co.uk
I was recently asked to write a column listing my Desert Island books: those I'd like to have with me if there was little else to do but read and, well, survive, which one might have thought would take up most of the day really. The editor kindly let me see the choices of previous book reviewers, I mean castaways. I was quite unsurprised to see famous heavyweight texts like Donald Knuth's The Art of Computer Programming mentioned. But one small book was mentioned by two famous reviewers: Professors Michael Jackson and Alan Bundy: Polya's How to Solve it. That is quite a testament, so of course I bought the book immediately. Jackson wrote:
"Polya discusses heuristics for problem solving, drawing on the ideas of the ancient Greek mathematicians, from Pythagoras of Samos in the sixth century BCE to Pappus of Alexandria nearly nine hundred years later.
Although most of the examples in the book are small mathematical problems in plane geometry and elementary algebra, Polya also illustrates the ideas by exploring practical problems such as the construction of a dam. The ideas Polya discusses are of great importance for software developers. They can be applied almost directly, and they illuminate the nature and role and proper use of development methods."
Polya writes with great charm and precision. Some of the charm may be accidental: when he writes "We can scarcely overestimate the importance of mathematical notation. Modern computers, using the decimal notation, have a great advantage over the ancient computers who did not have such a convenient manner of writing the numbers.", the reader realises with a start that "modern computers" means "20th century mathematicians", not machines.
But do the claims add up?
For people trying to get computers to solve problems - such as to make robots behave with some degree of intelligence - the question of how people learn to solve problems is of direct interest. Bundy expressed disappointment in this small book by Polya - it doesn't go into much detail of possible recipes - algorithms - for problem solving, though Polya's larger books go further.
For people interested in the question of what you need to do as an instructor to encourage pupils to solve problems for themselves, to explore, to reflect, to try to work with new concepts rather than just learning them by rote and parroting phrases like "processes", "systems", "value management" and "methodology" in every other sentence, then Polya is genuinely interesting. You can't make people solve problems for themselves. At best, you can show them the kinds of thing you do; give them a problem not too difficult, not too different from ones they know how to solve already; present them with the inputs they will need; draw their attention to the boundaries of the problem; even help them with its type if they need a hint; but then they have to let go of the side of the swimming-pool and strike out for themselves. Polya is, I think, very good here. He is a careful and reflective teacher, doing his best to coax his students to go a little further than they have ever gone before. It is a scary moment for them: if success comes, that is its own reward. Polya would surely be scornful of the idea of giving out physical rewards - if the satisfaction of solving a problem for yourself does not motivate you, nothing else will.
The difficulty with a book about a practical skill is naturally that it isn't really possible to speak about what goes on behind all words. Polanyi was right with his "we know more than we can tell". A book about the very moment of discovery is attempting something still more difficult: how can you talk about learning to do something that you never did before? Each problem is unique, after all. We are very close to the creative spark here, that essence of being human. Polya is brave to attempt it. Perhaps he comes as close as any book could.
Teachers, consultants, researchers, and engineers of all kinds, whether involved in mathematics or not, will find their ideas on problem-solving enhanced by this little book.
© Ian Alexander, 2007
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