How to Solve It

How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.[1]

How to Solve It
First edition (publ. Princeton University Press)
AuthorGeorge Pólya
GenreMathematics, problem solving
Publication date
1945

Four principles

How to Solve It suggests the following steps when solving a mathematical problem:

  1. First, you have to understand the problem.[2]
  2. After understanding, make a plan.[3]
  3. Carry out the plan.[4]
  4. Look back on your work.[5] How could it be better?

If this technique fails, Pólya advises:[6] "If you can't solve a problem, then there is an easier problem you can solve: find it." 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

"Understand the problem" is often neglected as being obvious and is not even mentioned in many mathematics classes. Yet students are often stymied in their efforts to solve it, simply because they don't understand it fully, or even in part. In order to remedy this oversight, Pólya taught teachers how to prompt each student with appropriate questions,[7] depending on the situation, such as:

  • What are you asked to find or show?[8]
  • 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 understand all the words used in stating the problem?
  • Do you need to ask a question to get the answer?

The teacher is to select the question with the appropriate level of difficulty for each student to ascertain if each student understands at their own level, moving up or down the list to prompt each student, until each one can respond with something constructive.

Second principle: Devise a plan

Pólya mentions that there are many reasonable ways to solve problems.[3] 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[9]
  • Make an orderly list[10]
  • Eliminate possibilities[11]
  • Use symmetry[12]
  • Consider special cases[13]
  • Use direct reasoning
  • Solve an equation[14]

Also suggested:

  • Look for a pattern[15]
  • Draw a picture[16]
  • Solve a simpler problem[17]
  • Use a model[18]
  • Work backward[19]
  • Use a formula[20]
  • Be creative[21]
  • Applying these rules to devise a plan takes your own skill and judgement.[22]

Polya lays a big emphasis on the teachers' behavior. A teacher should support students with devising their own plan with a question method that goes from the most general questions to more particular questions, with the goal that the last step to having a plan is made by the student. He maintains that just showing students a plan, no matter how good it is, does not help them.

Third principle: Carry out the plan

This step is usually easier than devising the plan.[23] In general, 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 that much can be gained by taking the time to reflect and look back at what you have done, what worked and what did not, and with thinking about other problems where this could be useful.[24][25] 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:

HeuristicInformal DescriptionFormal analogue
AnalogyCan you find a problem analogous to your problem and solve that?Map
GeneralizationCan you find a problem more general than your problem?Generalization
InductionCan you solve your problem by deriving a generalization from some examples?Induction
Variation of the ProblemCan 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 ProblemCan 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 beforeCan you find a problem related to yours that has already been solved and use that to solve your problem?Pattern recognition
Pattern matching
Reduction
SpecializationCan you find a problem more specialized?Specialization
Decomposing and RecombiningCan you decompose the problem and "recombine its elements in some new manner"?Divide and conquer
Working backwardCan you start with the goal and work backwards to something you already know?Backward chaining
Draw a FigureCan you draw a picture of the problem?Diagrammatic Reasoning[26]
Auxiliary ElementsCan you add some new element to your problem to get closer to a solution?Extension

Influence
  • The book 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 paper Steps Toward Artificial Intelligence that "everyone should know the work of George Pólya on how to solve problems."[27]
  • Pólya's book has had a large influence on mathematics textbooks as evidenced by the bibliographies for mathematics education.[28]
  • 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.
  • How to Solve it by Computer is a computer science book by R. G. Dromey.[29] It was inspired by Pólya's work.
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See also

Notes
  1. Pólya, George (1945). How to Solve It. Princeton University Press. ISBN 0-691-08097-6.
  2. Pólya 1957 pp.6-8
  3. Pólya 1957 pp.8-12
  4. Pólya 1957 pp.12-14
  5. Pólya 1957 pp.14-15
  6. Pólya 1957 p114
  7. Pólya 1957 p33
  8. Pólya 1957 p214
  9. Pólya 1957 p99
  10. Pólya 1957 p2
  11. Pólya 1957 p94
  12. Pólya 1957 p199
  13. Pólya 1957 p190
  14. Pólya 1957 p172 Pólya advises teachers that asking students to immerse themselves in routine operations only, instead of enhancing their imaginative / judicious side is inexcusable.
  15. Pólya 1957 p108
  16. Pólya 1957 pp103-108
  17. Pólya 1957 p114 Pólya notes that 'human superiority consists in going around an obstacle that cannot be overcome directly'
  18. Pólya 1957 p105, p29-32, for example, Pólya discusses the problem of water flowing into a cone as an example of what is required to visualize the problem, using a figure.
  19. Pólya 1957 p105, p225
  20. Pólya 1957 pp141-148. Pólya describes the method of analysis
  21. Pólya 1957 p172 (Pólya advises that this requires that the student have the patience to wait until the bright idea appears (subconsciously).)
  22. Pólya 1957 pp148-149. In the dictionary entry 'Pedantry & mastery' Pólya cautions pedants to 'always use your own brains first'
  23. Pólya 1957 p.35
  24. Pólya 1957 p.36
  25. Pólya 1957 pp.14-19
  26. Diagrammatic Reasoning site
  27. Minsky, Marvin. "Steps Toward Artificial Intelligence".CS1 maint: ref=harv (link).
  28. Schoenfeld, Alan H. (1992). D. Grouws (ed.). "Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics" (PDF). Handbook for Research on Mathematics Teaching and Learning. New York: MacMillan: 334–370. Archived from the original (PDF) on 2013-12-03. Retrieved 2013-11-27.CS1 maint: ref=harv (link).
  29. Dromey, R. G. (1982). How to Solve it by Computer. Prentice-Hall International. ISBN 978-0-13-434001-2.

References
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