How to solve math problems

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Polya's 4 phase approach to problem solving

In his book "How To Solve It - A new aspect of mathematical method", Polya outlined four phases of problem solving. Many are familiar with the 4 phases described by Polya but less "publicity" was given to Polya's views on how to teach the problem solving process to pupils.

Polya believed that pupils "should acquire as much experience of independent work as possible". The pupil learns nothing if too much help is given. The teacher should try to think at the level of the pupil and ask key questions or make suggestions. Questions and suggestions offered should be general and proceed from common sense such that the pupil could have thought of them himself. They should "just indicate a general direction and leave plenty for the student to do".

General questions, such as 'What are the unknowns?' and 'Have you ever solved a problem with similar unknowns?' are applicable in many cases. With repetition, the pupil may learn to ask himself the appropriate questions in similar situations. The teacher should put these questions and suggestions to the pupil as often as can be done naturally. The teacher should also ask himself the same questions when solving problems in front of the pupil. Through imitation and practise, "the student will eventually discover the right use of these questions and suggestions, and doing so he will acquire something that is more important than the knowledge of any particular mathematical fact".

Polya collected questions and suggestion that were helpful in problem solving and grouped them under the four phases of problem solving. A brief description of Polya's 4 phases of problem solving, including some examples of the suggested questions is given below:

1. Understanding the problem;
   • - Knowing what data is given, what is asked for and what are the conditions
   • - What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Draw a figure.
2. Devising a plan;
   • - Finding the connection between the data and the unknown. Deciding what actions to take to get what is asked for (the answer is worked out at the next phase.)
   • - Have you seen it before? Do you know a related problem? Look at the unknown and try to think of a familiar problem having the same or a similar unknown. Could you restate the problem? Could you solve a part of the problem? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Did you use all of the data?
3. Carrying out the plan;
   • - Executing the actions planned to get the answer.
   • - Check each step. Can you see clearly that the step is correct? Can you prove that it is correct?
4. Looking back;
   • - Checking the correctness of the answer and reviewing the solution to gain better understanding
   • - Can you check the result? Can you check the argument? Can you derive the solution differently? Can you use the result, or the method, for some other problem?

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