Heuristics For Problem Solvers†
A heuristic is a thinking strategy, something that can be used to tease out further information about a problem and that can thus help you figure out what to do when you don't know what to do. Here are twenty heuristics that can be useful when you are facing what seems intractable. They help you to monitor your thought processes: to step back and watch yourself at work, thus keeping your cool.
- Ask somebody else how to do it. This is probably the most-used strategy, world-wide, though it's not one we encourage our students to use, at least not initially. (Google it goes here too, and is never encouraged.)
- Guess and try (guess, check, and revise). Your 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's easy to try another one. And another. 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 your mind up for thinking, and perhaps you can use the organized data to generate more information.
- Draw a picture of problem information. Translate problem information into pictures, diagrams, sketches, glyphs, arrows, or...?
- 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 how or whether it changes the model.
- 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 the problem out, if it's stated in a narrative form. This can have the same effect as drawing a picture. What's more, doing 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. This 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").
- Make the problem simpler. Use easier or smaller numbers; or look at extreme cases (for example, assuming that the maximum amount of one of the varying quantities is used). Often you can use what you learn from the mini-version to help unlock the big one.
- Restate the problem yet 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 that you may be making (you may 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 these heuristics.
The above heuristics are those which are easily pointed out to students as they engage with problems in the classroom. However, real world problems are often those which are confronted many times over or on increasingly complex levels. For those, George Polya, the father of modern problem solving heuristics, identified a fifth class (E) called looking back heuristics. We include those 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 looks at a problem situation are invariably deeper and can lead to wonderful surprises.
- 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.