[Math Lair] Problem Solving

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"Intelligence is essentially the ability to solve problems"
—George Polya

From a mathematical perspective, what is problem solving? Well, a problem can be considered to be any situation that can be investigated using mathematical methods and analyses, and that involves elements such as number, shape, pattern, equation, and proof, and solving involves finding ways of satisfying the requirements and restrictions in the problem. So, problem solving is finding ways of finding a satisfactory answer in situations that can be investigated using mathematical methods. Problem solving is a significant part of mathematics. It is particularly important as a means of developing logical thinking and critical thinking, which can be used in everyday life.

Well-defined problems have a clear goal, a small set of information to start from, and often present a set of rules or guidelines to follow when solving the problem. Ill-defined problems don't have this sort of information spelled out. While problems found in mathematics textbooks are often well-defined, the problems encountered in everyday life are often not well-defined, and having an approach to tackling those problems is useful.

George Polya's book How to Solve It is a classic book on problem solving. In this book, he presents a list of four stages of solving a problem. These are:

  1. Understanding the problem: This is the "what" of the problem. What is the unknown? What information is provided? What restrictions are there? Is there enough information? Draw a diagram or other aid to help understand the problem.
  2. Devising a plan: This is the "how" of the problem. Have you seen a similar problem before? If you aren't sure how to solve the problem, can you solve a simpler problem? Can the problem be changed into a similar or logically equivalent problem? Can you solve part of the problem? Has all of the given information been used? Is the plan consistent with any restrictions in the problem?
  3. Carrying out the plan: Carry out the plan, step by step, checking each step. Can you verify or prove that the step is correct?
  4. Looking back: Can you check the result? Can you answer the problem in a different manner? Can you use the result to solve another problem?

One important point to remember is that these stages are meant to be flexible; in the process of solving a problem, you may not necessarily proceed through the steps in a strict linear order. You might find, for example, that a plan that you have chosen does not work out and you need to back up to understand the problem better. Also note that other books may present other stages of problem solving, but these steps are analogous to the steps listed here, possibly with the steps above combined or separated.

Polya's book also discusses heuristics, which are the methods and rules of learning, discovery, and problem solving. Being aware of and practicing the use of heuristics is another important part of being able to solve problems.

There are several blocks to problem solving. One of these is referred to in psychology as "mental set", the tendency to approach a problem in a certain way, possibly because that way worked before, instead of another, potentially more helpful, way. It is important to keep an open mind, be able to look at the problem from multiple perspectives, and search for a variety of ways of creating a solution.

If you understand the four-step method described above (or another similar method), you might find it, taken generally, to be helpful for solving problems in everyday life. Say, for example, that I want to create a web site about mathematics. I might start by asking myself what this would involve. Obviously, I can't write about every mathematical topic, so I have to choose some subset (in my case, recreational mathematics, practical topics, some fun games and puzzles, various surprising results in mathematics, and history) that connects reasonably well together, that would likely appeal to others, and which I understand well enough to be able to write about. I may also want to decide what specific topics to write about.

Now that I've decided what to write about, I need to decide how to do this. Do I already know enough about the specific topics that I want to write about? Do I have material already that I can draw on? Do I need to go to the library and do research, or get books from somewhere else, or look on the Internet for material? How can I fit the topics together well? Do I need to learn about HTML, or Javascript, or other languages? Thinking about this, I may find that perhaps I missed something important in the first step and I may need to go back and examine what I need to do.

The third step would be to do the research, put the website together, and so on. Again, I may find that what I've planned isn't adequate and need to re-plan. The fourth step would be to look back and see whether I have achieved the goals I set, whether I could make something better, whether anything needs revision, whether there is something else that could be done, and so on.

As mentioned above, the heuristics page is useful to problem-solvers. You may want to check out the list of puzzles and problems on the Math Lair.

For more information on problem solving, you can also check out my e-book for Kindle, Solving Math Problems.

Additionally, here are some external links of interest:

Sources used (see bibliography page for titles corresponding to numbers): 4, 8, 10, 11.