George Polya
1887 - 1985

George Polya was a Hungarian who immigrated to the United States in 1940. His major contribution is for his work in problem solving.

Growing up he was very frustrated with the practice of having to regularly memorize information. He was an excellent problem solver. Early on his uncle tried to convince him to go into the mathematics field but he wanted to study law like his late father had. After a time at law school he became bored with all the legal technicalities he had to memorize. He tired of that and switched to Biology and the again switched to Latin and Literature, finally graduating with a degree. Yet, he tired of that quickly and went back to school and took math and physics. He found he loved math.

His first job was to tutor Gregor the young son of a baron. Gregor struggled due to his lack of problem solving skills. Polya (Reimer, 1995) spent hours and developed a method of problem solving that would work for Gregor as well as others in the same situation. Polya (Long, 1996) maintained that the skill of problem was not an inborn quality but, something that could be taught.

He was invited to teach in Zurich, Switzerland. There he worked with a Dr. Weber. One day he met the doctor’s daughter Stella he began to court her and eventually married her. They spent 67 years together. While in Switzerland he loved to take afternoon walks in the local garden. One day he met a young couple also walking and chose another path. He continued to do this yet he met the same couple six more times as he strolled in the garden. He mentioned to his wife “how could it be possible to meet them so many times when he randomly chose different paths through the garden”.

He later did experiments that he called the random walk problem. Several years later he published a paper proving that if the walk continued long enough that one was sure to return to the starting point.

In 1940 he and his wife moved to the United States because of their concern for Nazism in Germany (Long, 1996). He taught briefly at Brown University and then, for the remainder of his life, at Stanford University. He quickly became well known for his research and teachings on problem solving. He taught many classes to elementary and secondary classroom teachers on how to motivate and teach skills to their students in the area of problem solving.

In 1945 he 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 text he identifies four basic principles .

Polya’s First Principle: 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 don’t understand it fully, or even in part. 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 a diagram that might help you understand the problem?
  • Is there enough information to enable you to find a solution?

Polya’s Second Principle: Devise a plan

Polya mentions (1957) that it are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included:
  • Guess and check
  • Make and orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation
  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be ingenious
Polya’s third Principle: Carry out the plan

This step is usually easier than devising the plan. In general (1957), all you need is care and patience, given that you have the necessary skills. Persistent with the plan that you have chosen. If it continues not to work discard it and choose another. Don’t be misled, this is how mathematics is done, even by professionals. Polya’s Fourth Principle: Look back

Polya mentions (1957) 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.

George Polya went on to publish a two-volume set, Mathematics and Plausible Reasoning (1954) and Mathematical Discovery (1962). These texts form the basis for the current thinking in mathematics education and are as timely and important today as when they were written. Polya has become known as the father of problem solving.

Contributed by A. Motter

References:

  1. Long, C. T., & DeTemple, D. W., Mathematical reasoning for elementary teachers. (1996). Reading MA: Addison-Wesley
  2. Reimer, L., & Reimer, W. Mathematicians are people too. (Volume 2). (1995) Dale Seymour Publications
  3. Polya, G. How to solve it. (1957) Garden City, NY: Doubleday and Co., Inc.

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