Problem Solving, Multiple Approaches To | Research & Encyclopedia Articles

Most of the mathematical problems that one encounters have already been solved and documented. Thus, the most important problem-solving tools are references. A reference can be anything—a book, a person, or a past experience—that aids in understanding and solving a problem. References enable one to solve problems independently when no other source of help is available.

The following are some general problem-solving strategies: break the problem into smaller parts, find a new perspective, work backward, guess. In all of these strategies, simple logic is used.

Logic. Mathematics is internally consistent. If at any point a false statement is generated, like 0 = 1, it is at once apparent that an error or a false assumption has been made. A false statement can be generated intentionally, as in a proof by contradiction.

Mutually exclusive dichotomies and trichotomies categorize the universe. Perhaps the most well-known dichotomy is true versus false. If x is true, then x is not false, and vice versa. The Trichotomy Axiom is a mathematical truism: If a and b are numbers, then either a = b, a > b, or a < b. Dividing all possible solutions into mutually exclusive categories can quickly eliminate incorrect solutions.

The Whole Is Equal to the Sum of Its Parts. When one is overwhelmed by a problem, it is a good strategy to break the problem down into smaller parts. As a general rule, the whole is equal to the sum of its parts, so, if all the pieces of a problem are solved, then the entire problem is solved.

Sometimes, the whole is greater than the sum of its parts. For example, "proper completed need in parts, may be the to assembled order once," makes no sense, but "once completed, the parts may need to be assembled in the proper order," does. After solving all of the pieces of a big problem, it must be determined that the pieces fit together into a form that makes sense.

Change of Perspective. Mathematical equations and concepts often have different forms that may be better suited to specific situations, and the internal consistency of mathematics guarantees that different forms of the same thing are equally valid. Here are 3 ways to represent 8: 2³, , .

Manipulatives and pictures can also provide a different perspective. When large amounts of information are presented, a visual or tangible connection can aid organization and understanding. For example:

Work Backward. In mathematical proofs in which the solution is known and the problem is proving the solution, working backward determines prerequisites for the solution. A series of questions may need to be asked of the task: What must be true (or false) in order that the solution is true (or false)? Do these prerequisites have prerequisites? And so on. In this manner, one continues working backward until the correct problem–solving strategy is determined.

When in Doubt… If one is confronted with a completely unfamiliar problem, guessing may well be the best strategy. Trial and error are an essential part of science and often the only way to proceed when charting new territory. A methodical approach, including meticulous recording of data and a careful search for patterns, makes guesses more informative and more accurate.

Proof.

Challenging Mathematical Problems with Elementary Solutions. San Francisco: Holden Day, 1967.

Polya, George. Mathematics and Plausible Reasoning. Princeton, NJ: Princeton University Press, 1954.

Wickelgren, Wayne A. How to Solve Mathematical Problems. New York: Dover, 1995.

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