How to Solve It

how to solve it

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

He suggests four steps when solving a mathematical problem: 1) First, understand the problem; 2) After understanding, then make a plan; 3) Carry out the plan; and; 4) Look back on your work — how could it be better? If this technique fails, Pólya advises: ‘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?’

The first principle, ‘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, depending on the situation, such as: What are you asked to find or show? 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? And, 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.

As to planning, Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. A partial list of strategies is included: Guess and check; Make an orderly list; Eliminate possibilities; Use symmetry; Consider special cases; Use direct reasoning; and Solve an equation. Also suggested: Look for a pattern; Draw a picture; Solve a simpler problem; Use a model; Work backward; Use a formula; Be creative; and Use your head/noggin.

The third step, execution, is usually easier than devising the plan. In general, all that is needed is care and patience, given the necessary skills. This principle requires persistence with the chosen plan. If after repeated attempts it continues not to work discard it and choose another. Don’t be misled; this is how mathematics is done, even by professionals. The final step is touted as important in predicting and preempting future problems.

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