How to Solve It
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George Pólya's 1945 book How to Solve It is a small volume describing methods of problem solving.[1]
Contents |
[edit] Overview
This book was published at Princeton University. It suggests the following steps when solving a mathematical problem:
- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look back on your work. How could it be better?
If this technique fails, Pólya advises: "If you can't solve a problem, then there is an easier problem you can solve: find it."[2] Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"
His book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:
| Heuristic | Informal Description | Formal analogue |
| Analogy | Can you find a problem analogous to your problem and solve that? | Map |
| Generalization | Can you find a problem more general than your problem? | Generalization |
| Induction | Can you solve your problem by deriving a generalization from some examples? | Induction |
| Variation of the Problem | Can you vary or change your problem to create a new problem (or set of problems) whose solution(s) will help you solve your original problem? | Search |
| Auxiliary Problem | Can you find a subproblem or side problem whose solution will help you solve your problem? | Subgoal |
| Here is a problem related to yours and solved before | Can you find a problem related to yours that has already been solved and use that to solve your problem? | Pattern recognition Pattern matching Reduction |
| Specialization | Can you find a problem more specialized? | Specialization |
| Decomposing and Recombining | Can you decompose the problem and "recombine its elements in some new manner"? | Divide and conquer |
| Working backward | Can you start with the goal and work backwards to something you already know? | Backward chaining |
| Draw a Figure | Can you draw a picture of the problem? | Diagrammatic Reasoning [3] |
| Auxiliary Elements | Can you add some new element to your problem to get closer to a solution? | Extension |
The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.
The book has achieved "classic" status because of its considerable influence (see the next section).
Other books on problem solving are often related to more creative and less concrete techniques. See lateral thinking, mind mapping, brainstorming, and creative problem solving.
[edit] Influence
- It has been translated into several languages and has sold over a million copies, and has been continuously in print since its first publication.
- Marvin Minsky said in his influential paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya [87] on how to solve problems." [4]
- Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education. [5]
- Russian physicist Zhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with Pólya's famous book.
[edit] Notes
- ^ Pólya, George (1945). How to Solve It. Princeton University Press. ISBN 0-691-08097-6.
- ^ Quotations by Polya
- ^ Diagrammatic Reasoning site
- ^ Minsky, Marvin, Steps Toward Artificial Intelligence, http://web.media.mit.edu/~minsky/papers/steps.html.
- ^ Schoenfeld, Alan H.; D. Grouws (Ed.) (1992). "Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics". Handbook for Research on Mathematics Teaching and Learning (New York: MacMillan): pp. 334–370. http://gse.berkeley.edu/faculty/ahschoenfeld/Schoenfeld_MathThinking.pdf..
[edit] See also
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Wikiquote has a collection of quotations related to: George Pólya |
