How to Solve It (1945) is a small volume by mathematician George Pólya describing methods of problem solving.[1]
Contents |
How to Solve It suggests the following steps when solving a mathematical problem:
If this technique fails, Pólya advises:[6] "If you can't solve a problem, then there is an easier problem you can solve: find it."[7] Or: "If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?"
"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,[8] depending on the situation, such as:
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.
Pólya[10] mentions that there are many reasonable ways to solve problems.[3] 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:
Also suggested:
This step is usually easier than devising the plan.[25] 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.
Pólya[26] 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.[27] Doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.
The book contains a dictionary-style set of heuristics, many of which have to do with generating a more accessible problem. For example:
| Heuristic | Informal Description | Formal analogue |
| Analogy | Can you find a problem analogous to your problem and solve that? | Map |
| Generalization | Can you find a problem more general than your problem? | Generalization |
| Induction | Can you solve your problem by deriving a generalization from some examples? | Induction |
| Variation of the Problem | Can you vary or change your problem to create a new problem (or set of problems) whose solution(s) will help you solve your original problem? | Search |
| Auxiliary Problem | Can you find a subproblem or side problem whose solution will help you solve your problem? | Subgoal |
| Here is a problem related to yours and solved before | Can you find a problem related to yours that has already been solved and use that to solve your problem? | Pattern recognition Pattern matching Reduction |
| Specialization | Can you find a problem more specialized? | Specialization |
| Decomposing and Recombining | Can you decompose the problem and "recombine its elements in some new manner"? | Divide and conquer |
| Working backward | Can you start with the goal and work backwards to something you already know? | Backward chaining |
| Draw a Figure | Can you draw a picture of the problem? | Diagrammatic Reasoning [28] |
| Auxiliary Elements | Can you add some new element to your problem to get closer to a solution? | Extension |
The technique "have I used everything" is perhaps most applicable to formal educational examinations (e.g., n men digging m ditches) problems.
The book has achieved "classic" status because of its considerable influence (see the next section).
Other books on problem solving are often related to more creative and less concrete techniques. See lateral thinking, mind mapping, brainstorming, and creative problem solving.