George Pólya's 1945 book How to Solve It is a small volume describing methods of problem solving.[1]
Contents |
Four principles
How to Solve 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?"
First principle: Understand the problem
This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they don't understand it fully, or even in part. Pólya taught teachers to ask students questions such as:
- Do you understand all the words used in stating the problem?
- What are you asked to find or show?
- Can you restate the problem in your own words?
- Can you think of a picture or a diagram that might help you understand the problem?
- Is there enough information to enable you to find a solution?
- Do you need to ask a question to get the answer?
Second principle: Devise a plan
Pólya mentions (1957[citation needed]) that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:
- Guess and check
- Make an orderly list
- Eliminate possibilities
- Use symmetry
- Consider special cases
- Use direct reasoning
- Solve an equation
Also suggested:
- Look for a pattern
- Draw a picture
- Solve a simpler problem
- Use a model
- Work backward
- Use a formula
- Be creative
- Use your head/noggen
Third principle: Carry out the plan
This step is usually easier than devising the plan. In general (1957[citation needed]), all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don't be misled, this is how mathematics is done, even by professionals.
Fourth principle: Review/extend
Pólya mentions (1957[citation needed]) that much can be gained by taking the time to reflect and look back at what you have done, what worked and what didn't. Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.
Heuristics
The 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.
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.
- Russian inventor Genrich Altshuller developed an elaborate set of methods for problem solving known as TRIZ, which in many aspects reproduces or parallels Pólya's work.
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..
See also
 |
Wikiquote has a collection of quotations related to: George Pólya |
External links
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
