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(Redirected from György Pólya)
'George Pólya' (December 13, 1887 – September 7, 1985, in Hungarian ''Pólya György'') was a Hungarian mathematician.
He was born ''György Pólya'' in Budapest, Hungary to Jewish converts to Catholicism, and died in Palo Alto, 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, combinatorics, and probability.
In his later days, he spent considerable effort on trying to characterize the general methods that people use to solve problems, and to describe how problem-solving should be taught and learned. He wrote three books on the subject: ''How to Solve It'', ''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 simply mathematical ones. The book includes advice for teaching students 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 as a primary source in mathematical education, but is poorly received and not well understood in general education, where it is believed that many scholars may themselves have found mathematics difficult.
In 1976 The Mathematical Association of America established the George Pólya award "for articles of expository excellence published in the College Mathematics Journal."
In ''Mathematics and Plausible Reasoning Volume I'', Pólya discusses inductive reasoning in mathematics, by which he means reasoning from particular cases to the general rule. (He also includes a chapter on the technique called mathematical induction, but that technique is not his main theme.) In ''Mathematics and Plausible Reasoning Volume II'', he discusses more general forms of inductive logic that can be used to roughly determine to what degree a conjecture (in particular, a mathematical conjecture) is plausible. Volume II gives equal expression to his interests in mathematics, natural science, cognitive psychology, and most of all, community of interest. Its breadth makes it highly readable for those who share his interests and his perspective upon them. Pólya could (can) teach at once as an individual, and as a role model. His 3-book series regarding Plausible Reasoning seeks to reach an audience far beyond his formal peers of the time.
★ 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 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.
★ 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 challenges your curiosity and 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.
★ 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 Neuman)
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 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 ingenious
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.
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.
★ Multivariate Polya distribution
★ Pólya conjecture
★ Pólya enumeration theorem
★ Pólya Prize
★ Landau-Kolmogorov inequality
★ "Problems and theorems in analysis"
★ The George Pólya Award
★
★
★ PolyaPower -- an introduction to Polya's Heuristics
George Pólya ca 1973
'George Pólya' (December 13, 1887 – September 7, 1985, in Hungarian ''Pólya György'') was a Hungarian mathematician.
Life and works
He was born ''György Pólya'' in Budapest, Hungary to Jewish converts to Catholicism, and died in Palo Alto, 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, combinatorics, and probability.
In his later days, he spent considerable effort on trying to characterize the general methods that people use to solve problems, and to describe how problem-solving should be taught and learned. He wrote three books on the subject: ''How to Solve It'', ''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 simply mathematical ones. The book includes advice for teaching students 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 as a primary source in mathematical education, but is poorly received and not well understood in general education, where it is believed that many scholars may themselves have found mathematics difficult.
In 1976 The Mathematical Association of America established the George Pólya award "for articles of expository excellence published in the College Mathematics Journal."
In ''Mathematics and Plausible Reasoning Volume I'', Pólya discusses inductive reasoning in mathematics, by which he means reasoning from particular cases to the general rule. (He also includes a chapter on the technique called mathematical induction, but that technique is not his main theme.) In ''Mathematics and Plausible Reasoning Volume II'', he discusses more general forms of inductive logic that can be used to roughly determine to what degree a conjecture (in particular, a mathematical conjecture) is plausible. Volume II gives equal expression to his interests in mathematics, natural science, cognitive psychology, and most of all, community of interest. Its breadth makes it highly readable for those who share his interests and his perspective upon them. Pólya could (can) teach at once as an individual, and as a role model. His 3-book series regarding Plausible Reasoning seeks to reach an audience far beyond his formal peers of the time.
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 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.
★ 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 challenges your curiosity and 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.
★ 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 Neuman)
Polya's Four 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 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 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. 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.
Polya's fourth Principle: Review/Extend
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.
See also
★ Multivariate Polya distribution
★ Pólya conjecture
★ Pólya enumeration theorem
★ Pólya Prize
★ Landau-Kolmogorov inequality
★ "Problems and theorems in analysis"
External links
★ The George Pólya Award
★
★
★ PolyaPower -- an introduction to Polya's Heuristics
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