Pol Ya

George Pólya

George Pólya, circa 1973 George Pólya (December 13, 1887 – September 7, 1985, in Hungarian Pólya György) was a Hungarian mathematician. Life and works He was born as Pólya György in Budapest, Hungary, and died in Palo Alto, California, USA. He was a professor of mathematics from 1914 to 1940 at ETH Zürich in Switzerland and from 1940 to 1953 at Stanford University carrying on as Stanford Professor Emeritus the rest of his life and career. 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." Pólya's four principles 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?

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What are you asked to find or show?

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Can you restate the problem in your own words?

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Can you think of a picture or a diagram that might help you understand the problem?

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Is there enough information to enable you to find a solution?

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Do you need to ask a question to get the answer?

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:

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Guess and check

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Make an orderly list

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Eliminate possibilities

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Use symmetry

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Consider special cases

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Use direct reasoning

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Solve an equation

Also suggested: •

Look for a pattern

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Draw a picture

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Solve a simpler problem

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Use a model

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Work backward

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Use a formula

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Be creative

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Use your head/noggen

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. 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.