How to Solve It explained

How to Solve It
Genre:Mathematics, problem solving
Pub Date:1945
Isbn:9780691164076

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

This book has remained in print continually since 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 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:

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:

Also suggested:

Pólya 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:

Heuristic Informal Description Formal analogue
Can you find a problem analogous to your problem and solve that? Map
Auxiliary Elements Can you add some new element to your problem to get closer to a solution? Extension
Can you find a problem more general than your problem? Generalization
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
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
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[26]

Influence

See also

References

External links

Notes and References

  1. Book: Pólya , George . George Pólya . How to Solve It . 1945 . Princeton University Press . 0-691-08097-6.
  2. pp. 6–8
  3. pp. 8–12
  4. pp. 12–14
  5. pp. 14–15
  6. p. 114
  7. p. 33
  8. p. 214
  9. p. 99
  10. p. 2
  11. p. 94
  12. p. 199
  13. p. 190
  14. p. 172 Pólya advises teachers that asking students to immerse themselves in routine operations only, instead of enhancing their imaginative / judicious side is inexcusable.
  15. p. 108
  16. pp. 103–108
  17. p. 114 Pólya notes that 'human superiority consists in going around an obstacle that cannot be overcome directly'
  18. p. 105, pp. 29–32, 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. p. 105, p. 225
  20. pp. 141–148. Pólya describes the method of analysis
  21. p. 172 (Pólya advises that this requires that the student have the patience to wait until the bright idea appears (subconsciously).)
  22. pp. 148–149. In the dictionary entry 'Pedantry & mastery' Pólya cautions pedants to 'always use your own brains first'
  23. p. 35
  24. p. 36
  25. pp. 14–19
  26. http://zeus.cs.hartford.edu/~anderson/ Diagrammatic Reasoning site
  27. Web site: Minsky . Marvin . Marvin Minsky . Steps Toward Artificial Intelligence . .
  28. Schoenfeld . Alan H. . Alan H. Schoenfeld . D. Grouws . Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics . Handbook for Research on Mathematics Teaching and Learning . MacMillan . New York . 334–370 . 1992 . 2013-11-27 . https://web.archive.org/web/20131203023917/http://gse.berkeley.edu/sites/default/files/users/alan-h.-schoenfeld/Schoenfeld_1992%20Learning%20to%20Think%20Mathematically.pdf . 2013-12-03 . dead. .
  29. Book: Dromey, R. G.. How to Solve it by Computer. Prentice-Hall International. 978-0-13-434001-2. 1982.