Problem Solving Strategies

One of the primary reasons people have trouble with problem solving is that there is no single procedure that works all the time — each problem is slightly different. Also, problem solving requires practical knowledge about the specific situation. If you misunderstand either the problem or the underlying situation you may make mistakes or incorrect assumptions. One of our main goals here is to help you become a better problem solver.

In this article, we will discuss a framework developed by George Polya for thinking about problem solving. In 1945, George Polya published the book “How To Solve It” which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. In this book, he identifies four basic principles of problem solving.

Polya’s four principles

1. Understand the problem

This seems so obvious that it is often not even mentioned, yet students are often stymied in their efforts to solve problems simply because they do not fully understand the problem—or sometimes even part of it. Polya taught teachers to ask students questions such as:

• Do you understand all the words used in stating the problem?
• What are you asked to find or show?
• Can you restate the problem in your own words?
• Can you think of a picture or diagram that might help you understand the problem?
• Is there enough information to enable you to find a solution?

2. Devise a plan

Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. Choosing a strategy becomes easier with practice. Here are some strategies to consider:

3. Carry out the plan

This step is usually easier than devising the plan. 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. Do not be discouraged: trying an approach, checking it and changing direction is how mathematics is done, even by professionals.

4. Look back

Polya 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. Doing this will enable you to predict what strategy to use to solve future problems.

A detailed problem-solving checklist

The following checklist draws on Polya’s own questions, with some interpretations, to help you work through problems in mathematics class. The source is How To Solve It, by George Polya, 2nd ed., Princeton University Press, 1957, ISBN 0-691-08097-6.

1. Preparation: Understand the problem

• What is the unknown? What are the data? What is the condition?
• Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? Or contradictory?
• Draw a figure. Introduce suitable notation.
• Separate the various parts of the condition. Can you write them down?
• Rephrase the problem in your own words.
• Write down specific examples of the conditions given in the problem.

2. Thinking time: Devise a plan

• Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should eventually arrive at a plan for the solution.
• Have you seen it before? Or have you seen the same problem in a slightly different form?
• Do you know a related problem? Do you know a theorem that could be useful?
• Look at the unknown! Try to think of a familiar problem having the same or a similar unknown.
• Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
• Could you restate the problem? Could you restate it still differently? Go back to definitions.
• If you cannot solve the proposed problem, first try to solve a related problem. Could you imagine a more accessible related problem? A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary? Could you derive something useful from the data? Could you think of other data appropriate to determine the unknown? Could you change the unknown or data, or both if necessary, so that the new unknown and the new data are nearer to each other?
• Did you use all the data? Did you use the whole condition? Have you taken into account all essential notions involved in the problem?
• Once you understand what the problem is, if you are stumped or stuck, set the problem aside for a while. Your subconscious mind may keep working on it.
• Moving on to think about other things may help you stay relaxed, flexible, and creative rather than becoming tense, frustrated, and forced in your efforts to solve the problem.

3. Insight: Carry out the plan

• You must start somewhere so try something. How are you going to attack the problem?
• Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct?
• Once you have an idea for a new approach, jot it down immediately. When you have time, try it out and see if it leads to a solution.
• If the plan does not seem to be working, then start over and try another approach. Often the first approach does not work. Do not worry, just because an approach does not work, it does not mean you did it wrong. You actually accomplished something, knowing a way does not work is part of the process of elimination.
• Once you have thought about a problem or returned to it enough times, you will often have a flash of insight: a new idea to try or a new perspective on how to approach solving the problem.
• The key is to keep trying until something works.

4. Verification: Look back

• Once you have a potential solution, check whether it works.
• Did you answer the question?
• Is your result reasonable?
• Check that all conditions in the problem are satisfied.
• Double-check the computations used to find your solution.
• If you find that your solution does not work, there may only be a simple mistake. Try to fix or modify your current attempt before scrapping it. Remember what you tried—it is likely that at least part of it will end up being useful.
• Is there another way of doing the problem which may be simpler? (You need to become flexible in your thinking. There usually is not one right way.)
• Can the problem or method be generalized so as to be useful for future problems?

Basic mathematical principles to keep in mind

These principles help you reason carefully and check whether a mathematical argument is sound:

• The Always Principle: Unlike many other subjects, when we say a mathematical statement is true, we mean that it is true 100 percent of the time. We are not dealing with the uncertainty of statements that are “usually true” or “sometimes true”.

• The Counterexample Principle: Since a mathematical statement is true only when it is true 100% of the time, we can prove that it is false by finding a single example where it is not true. Such an example is called a counterexample. Of course, when we say a mathematical statement is false, this does not mean that it is never true — it only means that it is not always true. It might be true some of the time.

• The Order Principle: In mathematics, order usually matters. In a multi-step mathematical process, if we carry the steps out in a different order, we often get a different result. For example, putting your socks on first and then your shoes is quite different from putting your shoes on first and then your socks.

• The Splitting Hairs Principle: In mathematics, details matter. Two terms or symbols that look and sound similar may have mathematical meanings that are significantly different. For example, in English, we use the term equal and equivalent interchangeably, but in mathematics, these terms do not mean the same thing. For this reason, learning and remembering the precise meaning of mathematical terms is essential.

• The Analogies Principle: Often the formal terminology used in mathematics has been drawn from words and concepts used in everyday life. This is not a coincidence. Associating a mathematical concept with its “real world” counterpart can help you remember both the formal (precise) and intuitive meanings of a mathematical concept.

• The Three Way Principle: When approaching a mathematical concept, it often helps to use three complementary approaches:

  • Verbal: Make analogies, put the problem in your own words, and compare the situation to something you have seen in another area of mathematics.
  • Graphical: Draw a graph or diagram.
  • Examples: Use specific examples to illustrate the situation.

  By combining one or more of these approaches, you can often get a better idea of how to think about and solve a given problem.