In How To Solve It, G. Polya describes four steps for solving problems and outlines them at the very beginning of the book for easy reference. The steps outline a series of general questions that the problem solving student can use to successfully write resolutions. Without the questions, common sense goes through the same process; the questions simply allow students to see the process on paper. Polya designed the questions to be general enough that students could apply them to almost any problem.
The four steps are:
This method is very similar to the method in Thinking Mathematically by John Mason, except Polya separates devising a plan, and carrying out the plan. This may seem silly at first, but Polya argues that it does make a difference. By first devising a plan, students can eliminate mistakes they might make by rushing into the actual execution of the plan. When they plan it out first and then do the math, it is possible to check their work as they go along.
In the first step, students should be able to state the unknown, or the thing they want to find to answer the question, the data the question gives them to work with, and the condition, or limiting circumstances they must work around. If they can identify all of these, and explain the question to other people, then they have a good understanding of what the problem is asking. Polya suggests that students draw a picture if possible, or introduce some kind of notation to visualize the question.
To devise a plan, students can start by trying to think of a related problem they have solved before to help them. If the student can think of a problem they have solved before that had a similar unknown, it could also be helpful. Students can also try to restate the problem in an easier or different way, and try to solve that. By looking at these related problems, students may be able to use the same method, or other part of the plan used. After students have decided which calculations, computations, or constructions that they need, and have made sure that all data and conditions were used, they can try out their plan. To carry out the plan, they must do all the calculations, and check them as they go along. Then they should ask themselves, "Can I see it is right?" and then, "Can I prove it is right?"
When students look back on the problem and the plan they carried out, they can increase their understanding of the solution. It is always good to recheck the result and argument used, and to make sure that it is possible to check them. Then students should ask, "Can I get the result in a different way?"and "Can I use this for another problem?" The last chapter of the book is a very helpful encyclopedia of the terms used in the explanation of the first chapter.
By Beth Nuckolls.
A few links: