George Pólya’s Four Steps to Problem Solving

Hungarian mathematician George Pólya (1887-1985) published the book How to Solve It in 1945.  It has been translated into over 17 languages. In this text he introduced a basic strategy for problem-solving, including four steps:

1.   Understand the problem. Restate the problem in your own words, clarifying any points that need clarification and stating any assumptions you make about the problem.  Use the following questions to have a better understanding of the problem.

·        Do you understand all the words? Can you explain the problem statement in your own words?

·        Do you know what is given?  Do you know what the goal is?

·        Is there enough information to enable you to find a solution?

·        Can you think of a picture or diagram that might help you understand the problem?

·        Does this problem remind you of another problem you have solved?

*** (my recommendation) At this point, make an initial guess. When you have solved the problem, compare this guess to your solution.

2. Devise a plan. How do you get started? What approaches should you try? What strategies can be used?

A partial list of strategies:

• Guess, check, revise.
• Set up an equation.
• Make a picture/diagram.
• Solve an equivalent problem.
• Look for a pattern.
• Use properties of numbers.
• Make an orderly list.
• Work backward.
• Use symmetry.
• Eliminate possibilities.
• Solve a simpler problem.
• Act it out.
• Use a formula.

·        Consider special cases.

• Break the problem into simpler ones.
• Be ingenious.

3. Carry out the plan. Implement your strategy or strategies you have chosen until the problem is solved or until you decide to use a different strategy. Be persistent:  Give yourself a reasonable amount of time for solving the problem (you may have a flash of insight when you least expect it!). If you are not successful, go back to step two (or sometimes to step one).   Don't be afraid to start over. Often, a fresh start or a new strategy will lead to success.

4. Look back. Check the results and interpret the solution in terms of the original problem.  Explain how you know the solution is correct, if you think it is. Is the answer reasonable? Does it make sense? Is it close to your estimate? Why?

Reflect on your problem solving process. Doing this may enable you to predict what strategy will be successful on another occasion.

Consider the following questions—What worked?  What didn’t? Why? Was the problem hard or easy? Was it a worthwhile mathematical task? Where did you get stuck? Do you know why you got stuck at that point? Did you get discouraged? Did you try something else? How did you feel when you had solved the problem? Write down some problems that are related to this problem, and some extensions.

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