Some Helpful Problem-Solving Heuristics1
A heuristic is a thinking strategy, something that can be used to tease out further information about a problem and thus help you figure out what to do when you don't know what to do. Here are 25 heuristics that can be useful in solving problems. They help you monitor your thought processes, to step back and watch yourself at work, and thus keep your cool in a challenging situation.
- Ask somebody else how to do the problem. This strategy is probably the most used world-wide, though it is not one we encourage our students to use, at least not initially.
- Guess and try (guess, check, and revise). Your first guess might be right! But incorrect guesses can often suggest a direction toward a solution. (N.B. A spreadsheet is a powerful aid in guessing and trying. Set up the relationships and plug in a number to see if you get what you want. If you don't, it is easy to try another number. And another.)
- Restate the problem using words that make sense to you. One way to do this is to explain the problem to someone else. Often this is all it takes for the light to dawn.
- Organize information into a table or chart. Having it laid out clearly in front of you frees up your mind for thinking. And perhaps you can use the organized data to generate more information.
- Draw a picture of the problem. Translate problem information into pictures, diagrams, sketches, glyphs, arrows, or some other kind of representation.
- Make a model of the problem. The model might be a physical or mental model, perhaps using a computer. You might vary the problem information to see whether and how the model may be affected.
- Look for patterns, any kind of patterns: number patterns, verbal patterns, spatial/visual patterns, patterns in time, patterns in sound. (Some people define mathematics as the science of patterns.)
- Act out the problem, if it is stated in a narrative form. Acting it out can have the same effect as drawing a picture. What's more, acting out the problem might disclose incorrect assumptions you are making.
- Invent notation. Name things in the problem (known or unknown) using words or symbols, including relationships between problem components.
- Write equations. An equation is simply the same thing named two different ways.
- Check all possibilities in a systematic way. A table or chart may help you to be systematic.
- Work backwards from the end condition to the beginning condition. Working backwards is particularly helpful when letting a variable (letter) represent an unknown.
- Identify subgoals in the problem. Break up the problem into a sequence of smaller problems ("If I knew this, then I could get that").
- Simplify the problem. Use easier or smaller numbers, or look at extreme cases (e.g., use the minimum or maximum value of one of the varying quantities).
- Restate the problem again. After working on the problem for a time, back off a bit and put it into your own words in still a different way, since now you know more about it.
- Change your point of view. Use your imagination to change the way you are looking at the problem. Turn it upside down, or pull it inside out.
- Check for hidden assumptions you may be making (you might be making the problem harder than it really is). These assumptions are often found by changing the given numbers or conditions and looking to see what happens.
- Identify needed and given information clearly. You may not need to find everything you think you need to find, for instance.
- Make up your own technique. It is your mind, after all; use mental actions that make sense to you. The key is to do something that engages you with the problem.
- Try combinations of the above heuristics.
These heuristics can be readily pointed out to students as they engage problems in the classroom. However, real-world problems are often confronted many times over or on increasingly complex levels. For those kinds of problems, George Polya, the father of modern problem-solving heuristics, identified a fifth class (E) of looking-back heuristics. We include these here for completeness, but also with the teaching caveat that solutions often improve and insights grow deeper after the initial pressure to produce a solution has been resolved. Subsequent considerations of a problem situation are invariably deeper than the first attempt.
- Check your solution. Substitute your answer or results back into the problem. Are all of the conditions satisfied?
- Find another solution. There may be more than one answer. Make sure you have them all.
- Solve the problem a different way. Your first solution will seldom be the best solution. Now that the pressure is off, you may readily find other ways to solve the problem.
- Solve a related problem. Steve Brown and Marion Walter in their book, The Art of Problem Posing, suggest the "What if not?" technique. What if the train goes at a different speed? What if there are 8 children, instead of 9? What if . . .? Fascinating discoveries can be made in this way, leading to:
- Generalize the solution. Can you glean from your solution how it can be made to fit a whole class of related situations? Can you prove your result?
Adapted from Meiring, S. P. (1980). Problem solving — A basic mathematics goal
. Columbus: Ohio Department of Education.