Polya's Problem - Solving Strategy
- 1. Polya’s ProblemSolving Strategy By: EG Archide
- 2. Ancient mathematicians such as Euclid and Pappus were interested in solving mathematical problems, but they were also interested in heuristics, the study of the methods and rules of discovery and invention. In the seventeenth century, the mathematician and philosopher René Descartes (1596–1650) contributed to the eld of heuristics. He tried to develop a universal problem-solving method. Although he did not achieve this goal, he did publish some of his ideas in Rules for the Direction of the Mind and his better-known work Discourse de la Methode. Another mathematician and philosopher, Gottfried Wilhelm Leibnitz (1646–1716), planned to write a book on heuristics titled Art of Invention. Of the problem-solving process, Leibnitz wrote, “Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves.”
- 3. One of the foremost recent mathematicians to make a study of problem solving was George Polya (1887–1985). He was born in Hungary and moved to the United States in 1940. The basic problem-solving strategy that Polya advocated consisted of the following four steps. Polya’s Four-Step Problem-Solving Strategy 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Review the solution Polya’s four steps are deceptively simple. To become a good problem solver, it helps to examine each of these steps and determine what is involved.
- 4. Understand the Problem This part of Polya’s four-step strategy is often overlooked. You must have a clear understanding of the problem. To help you focus on understanding the problem, consider the following questions. ■ Can you restate the problem in your own words? ■ Can you determine what is known about these types of problems? ■ Is there missing information that, if known, would allow you to solve the problem? ■ Is there extraneous information that is not needed to solve the problem? ■ What is the goal? Devise a Plan Successful problem solvers use a variety of techniques when they attempt to solve a problem. Here are some frequently used procedures. ■ Make a list of the known information. ■ Make a list of information that is needed. ■ Draw a diagram. ■ Make an organized list that shows all the possibilities. ■ Make a table or a chart. ■ Work backwards. ■ Try to solve a similar but simpler problem. ■ Look for a pattern. ■ Write an equation. If necessary, dene what each variable represents. ■ Perform an experiment. ■ Guess at a solution and then check your result.
- 5. Carry Out the Plan Once you have devised a plan, you must carry it out. ■ Work carefully. ■ Keep an accurate and neat record of all your attempts. ■ Realize that some of your initial plans will not work and that you may have to devise another plan or modify your existing plan. Review the Solution Once you have found a solution, check the solution. ■ Ensure that the solution is consistent with the facts of the problem. ■ Interpret the solution in the context of the problem. ■ Ask yourself whether there are generalizations of the solution that could apply to other problems.
- 6. Consider the map shown at the right. Allison wishes to walk along the streets from point A to point B. How many direct routes can Allison take?
- 7. Solution: Understand the Problem We would not be able to answer the question if Allison retraced her path or traveled away from point B. Thus we assume that on a direct route, she always travels along a street in a direction that gets her closer to point B. Devise a Plan The given map has many extraneous details. Thus we make a diagram that allows us to concentrate on the essential information. See the figure at the left. Because there are many routes, we consider the similar but simpler diagrams shown below. The number at each street intersection represents the number of routes from point A to that particular intersection.
- 8. The product of the ages, in years, of three teenagers is 4590. None of the teens are the same age. What are the ages of the teenagers?
- 9. Look for patterns. It appears that the number of routes to an intersection is the sum of the number of routes to the adjacent intersection to its left and the number of routes to the intersection directly above. For instance, the number of routes to the intersection labeled 6 is the sum of the number of routes to the intersection to its left, which is 3, and the number of routes to the intersection directly above, which is also 3. Carry Out the Plan Using the pattern discovered above, we see from the figure at the left that the number of routes from point A to point B is 20 + 15 = 35. Review the Solution Ask yourself whether a result of 35 seems reasonable. If you were required to draw each route, could you devise a scheme that would enable you to draw each route without missing a route or duplicating a route?
- 10. Reference Aufman RF., Lockwood JS., Nation RD., Clegg DK., Mathematical Excursions. Fourth Ed., Cengage Learning © 2018