George Pólya

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George Pólya, around 1973
George Pólya, around 1973

George Pólya (b. December 13, 1887 – d. September 7, 1985, in Hungarian Pólya György) was a Hungarian mathematician.

Contents

[edit] Life and works

He was born as Pólya György in Budapest, Hungary, and died in Palo Alto, California, USA. For most of his career in the United States, he was a professor of mathematics at Stanford University. He worked on a great variety of mathematical topics, including series, number theory, mathematical analysis, geometry, algebra, combinatorics, and probability[1] In his later days, he spent considerable effort on trying to characterize the methods that people use to solve problems, and to describe how problem-solving should be taught and learned. He wrote four books on the subject: How to Solve It, Mathematical Discovery: On Understanding, Learning, and Teaching Problem Solving; Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics, and Mathematics and Plausible Reasoning Volume II: Patterns of Plausible Reasoning.

In How to Solve It, Pólya provides general heuristics for solving problems of all kinds, not only mathematical ones. The book includes advice for teaching students of mathematics and a mini-encyclopedia of heuristic terms. It was translated into several languages and has sold over a million copies. Russian physicist Zhores I. Alfyorov, (Nobel laureate in 2000) praised it, saying he was very pleased with Pólya's famous book. The book is still referred to in mathematical education. Douglas Lenat's Automated Mathematician and Eurisko artificial intelligence programs were inspired by Pólya's work.

In 1976 The Mathematical Association of America established the George Pólya award "for articles of expository excellence published in the College Mathematics Journal."

[edit] Quotes

  • To be a good mathematician, or a good gambler, or good at anything, you must be a good guesser.
  • Observe also (what modern writers almost forgot, but some older writers, such as Euler and Laplace, clearly perceived) that the role of inductive evidence in mathematical investigation is similar to its role in physical research.
  • How I need a drink, alcoholic of course, after the heavy chapters involving quantum mechanics (This is a mnemonic for the first fifteen digits of π; the lengths of the words are the digits.)
  • If you can't solve a problem, then there is an easier problem you can solve: find it.
  • Wishful thinking is imagining good things you don't have...[It] may be bad as too much salt is bad in the soup and even a little garlic is bad in the chocolate pudding. I mean, wishful thinking may be bad if there is too much of it or in the wrong place, but it is good in itself and may be a great help in life and in problem solving.
  • He was the only student that ever scared me (in reference to John von Neumann)
  • Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment.
  • A Great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery (from "Faces of Mathematics", page 3, Robert, A. W., Macalester College).

[edit] Pólya's four principles

[edit] 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. Pólya 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?
  • Do you need to ask a question to get the answer?

[edit] Second principle: Devise a plan

Pólya mentions (1957) that there 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 an orderly list
  • Eliminate possibilities
  • Use symmetry
  • Consider special cases
  • Use direct reasoning
  • Solve an equation

Also suggested:

  • Look for a pattern
  • Draw a picture
  • Solve a simpler problem
  • Use a model
  • Work backward
  • Use a formula
  • Be creative
  • Use your head

[edit] 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. Persist 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.

[edit] Fourth principle: Review/extend

Pólya 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, if these relate to the original problem.

[edit] See also

[edit] References

  1. ^ Roberts, A. Wayne (1995). Faces or Mathemathics, Third Edition. New York, NY USA: HaperCollins College Publishers, 479. 

[edit] External links