In 1945 George Polya published the book How To Solve It which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. In this book he identifies four basic principles of problem solving.

Polya’s First Principle: Understand the problem

This seems so obvious that it is often not even mentioned, yet studetns are often stymied in their efforts to solve problems simply because they don’t understand it fully, or even in part. Polya 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 diagram that might help you understand the problem?
  • Is there enough information to enable you to find a solution?

Polya’s Second Principle: Devise a plan

Polya mentions 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
Look for a pattern
Draw a picture
Solve a simpler problem
Use a model
Work backwards
Use a formula
Be ingenious

Polya’s Third Principle: Carry out the plan

This step is usually easier than devising the plan. 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.

Polya’s Fourth Principle: Look back

Polya mentions 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

 This is taken from the book,
How To Solve It, by George Polya, 2nd ed., Princeton University Press, 1957, ISBN
0-691-08097-6

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