G. Polya, How to Solve It -- a new aspect of mathematical method
, Princeton University Press, 1973
At EuroPlop97 at least one participant was searching for methods (and patterns) for teaching mathematics.
Not exactly patterns, but a great book on HeuristicRule
s, problem solving and teaching mathematics. Fun to read and not aged at all (written in 1945).
From the Preface to the first print:
- "Thus a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of independent thinking.".
- "The author remembers the time when he was a student himself, a somewhat ambitious student, eager to understand a little mathematics and physics. He listened to lectures, read books, tried to take in the solutions and facts presented, but there was a question that disturbed him again and again: "Yes, the solution seems to work, it appears to be correct; but how is it possible to invent such a solution? Yes, this experiment seems to work, this appears to be a fact; but how can people discover such facts?..."
Trying to understand not only the solution of this or that problem but also the motives and procedures of the solution, and trying to explain these motives and procedures to others, Polya wrote this book.
I've had it on my bookshelves for 20 years and 3 jobs. I've looked into using it as part of a ComputerAidedThinking
(CATH) project I once worked on.
I loaned it to a student taking a math class, and watched her grade go up 1 point.
Look up "Traditional Mathematics Teacher" (If I recall correctly): you'll laugh and then stop for thought.
The latter half of this book, the dictionary, contains quite a few entries
that certainly qualify as patterns. They are not phrased in the fashions used here (or in PatternLanguage
), but exact phrasing isn't a prerequisite. The elements are there.
For example, the entry on specialization contains a counter-example pattern. The context is a need to refute a statement. The forces include the difficulty of finding a single counter-example out of many possibilities and the inability to prove the statement. The resolution? Examine the extreme cases, and/or use a failed proof to ferret out a counter-example. (Ok, so there are two possible resolutions, and neither of them is guaranteed. If you require a single resolution in patterns, as well, then I'd bet that either the resolution will be too vague to help generally or the context will be too over-specified to apply as widely as the concepts within the pattern should allow.)
Volume 2 of his Mathematics and Plausible Reasoning series is entitled Patterns of Plausible Inference
. I've yet to read this, but I wouldn't be surprised if this holds more examples.
Bought this cheaply after reading about it here. I'm about halfway through (1/3 through the dictionary) and am both enjoying it and wishing I was taught math in the way Polya describes. -- JoeWeaver
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