Polya's Method Of Problems example essay topic

Jo C Tu Contemporary Mathematics Prof. Marcelo Lla rull June 13, 2003 Mr. George Polya would seem to be an excellent teacher for an individual like me. Why? I believe I can't understand math and maybe if he was still around he can answer my questions about why math seems to be intertwined with everything in the world. Under his tutelage I believe he can devise a personal method of problem solving for me so that math wouldn't be such a scary subject.

Unfortunately, he passed away at the ripe old age of ninety-seven on September 7, 1985 (Young 401). I thought maybe if I found out a little bit more about him, I could find the meaning of math. "It is a shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between".

(Alexanderson 247). A quote of Polya's that has been echoed in many articles and books about whose most well known published work is How to Solve It. Born on December 13, 1887 in Budapest, Austria-Hungary, Polya was the second youngest of five children to Jakob and Anna Deutsch Polya (O'Conner Robertson). At a young age Polya enjoyed geography (Young 401), Greek, Latin and the modern language of German and Hungarian (O'Conner Robertson).

Persuaded by his mother (Anna Deutsch Polya), George pursued law just like his father at the University of Budapest (Young 402). Unfortunately, he could only bear with the subject for just one semester (Alexanderson 248). Instead, he turned to languages and literature earning teaching certificates in Latin and Hungarian (Alexanderson 248). At a young age Polya seemed not to be fascinated with math: "It is rather unusual that someone who went to spend their life being so fascinated by so many different branches of mathematics should not have fallen in love with the subject at school but in Polya's case this is exactly what happened.

He did not score particularly high marks in mathematics at the Gymnasium (Polya attended Daniel Berzsenyi Gymnasium before enrolling into University of Budapest), his work in geometry being graded as merely "satisfactory". He did score rather better in arithmetic, however. The reason for his lack of success in mathematics may well have been due to poor teaching, and he would later describe two of his three mathematics teacher at the Gymnasium as "despicable teachers". (O'Conner Robertson) Even Polya himself interviewed by G.L. Alexanderson described his road to mathematics in Mathematical People: Profiles and Interviews: "I was partly influenced by my teachers and by mathematical fashion of that time. Later I was influenced by my interest in discovery. I looked at a few questions just to find out how you handle this kind of question.

I was influenced also-this is farther away-because I did not come straight to mathematics. I was influenced by the tortuous way I came to mathematics". Polya's understanding of math was strengthen when he started to take an interest in philosophy (Young 402). Advised by his professor Bern at Alexander Polya took physics and mathematic courses to understand philosophy, which eventually paved the groundwork for the very beginnings of his contributions to the world of math (O'Conner and Robertson).

Polya's contribution in areas of probability, geometry, real and complex analysis, combinatorics, number theory, and mathematical physics are impressionable (Young 403) but his greatest contribution is guiding teachers in how to approach students with math. His most well known published work How to Solve It has been translated into fifteen languages (Smith 5) and later on into a total of seventeen languages and sold over one million copies (Motter). In the interview with G.L. Alexanderson for Mathematical People: Profiles and Interviews Polya said, "How to Solve It makes the fewest demands on the knowledge of the reader. A reader who knows very little mathematics can read it with some interest".

Polya later on published other works on how to solve math problems, Mathematics and Plausible Reasoning (1954), and Mathematical Discovery, which was published in two volumes (O'Conner and Robertson) Polya, a late math bloomer emphasized in solving problems one must be familiar with heuristic. Heuristic is designating a method of education or of computer programming in which the pupil or machine proceeds along empirical lines, using rules of thumb, to find solutions or answers (Neu felt 634). According to Polya "The aim of heuristic is to study the methods and rules of discovery and invention. Heuristic, as an adjective means 'serving to discover'. Its purpose is to discover the solution of the present problem. What is good education?

Systematically giving opportunity to the student to discover things by himself" (O'Conner Robertson). His interest in teaching led him to win a Blue Ribbon from the Education Film Library Association in 1968 for his film, "Let us Teach Guessing". (Young 403) Polya's perspective on teaching in general is this: "Teaching is not a science; it is an art. If teaching were a science there would be a best way of teaching and everyone would have to teach like that. Since teaching is not a science, there great latitude and much possibility for personal differences... let me tell you what my idea of teaching is. Perhaps the first point, which is widely accepted, is that teaching must be active or rather active learning... the main point in mathematics teaching is to develop the tactics of problem solving" (O'Conner Robertson) Polya's method of problems solving, is simple and direct.

This method of problem solving was published in How to Solve it that sold over one million copies and translated into seventeen languages (O'Conner Robertson). The birth of the book was written twice. According to Polya, he wrote a draft in German while in Zurich (Alexanderson 253). Once he arrived in America he observed that there was an interest in "how-to" books. Godfrey Harold Hardy even suggested to Polya to go to America if Polya was to publish a how-to book on problem solving. Hardy, an English mathematician who worked with Polya on Inequalities, Inequalities (published in 1934) suggested the title How to Solve It (Young 402).

Polya rewrote the drafts of How to Solve It in English, which he said was considerable different from his German version (Alexanderson 253). He had to try four publishers before finding one to publish the English version in the United States (O'Conner and Robertson). Polya's four basic principles of problem solving: 1) Understand the Problem 2) Devise a Plan 3) Carry out the Plan 4) Looking Back His method of problem solving was written in 1941 (Smith 7). His method is simple and to the point.

Polya taught teachers to teach their students this method so as to aid them in understanding the problem. Understanding the problem, to discover the solution of the problem. His method was praised by the Los Angeles Times: "He has given a new dimension to problem solving by emphasizing the organic building up of elementary steps into a complex proof, and conversely, the decomposition of mathematical invention into smaller steps. Solving problem a la Polya serves not only to develop mathematical skill but also teaches constructive reasoning in general".

(Young 402 403). This method of problem solving was aided by his difficulty in understanding mathematics (Alexanderson 251). According to Polya a good education is "systematically giving the opportunity to the student to discover things by himself" (O'Conner Robertson). The chart below displays the thought process to solve a problem.

Polya's Four Basic Principles of Problem Solving Understanding the Problem. What is the unknown? What are the data? What is the condition? Is the condition sufficient to determine the unknown? Or is it insufficient?

Is there a diagram / picture that might help you understand the problem? Is their enough information to enable you to find a solution? Devising a Plan Questions to ask: . Have you seen it before? Do you know a related problem? Is the problem related to one you have solved before?

Could you use its result? Could you use its method? Could you restate the problem? If you can't solve the problem can you imagine a more accessible related problem?

Have you taken account all essential notions involved in the problem? Strategies: . Guess and Check. Make an 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 Carry out the plan.

Carry out the plan of the solution, check each step... Can you see clearly that the step is correct? Can you prove it is correct? Looking Back. Can you check the result?

Can you check the argument? Can you derive the result differently? Can you see it at a glance? Taken from The Nature of Mathematics By Karl J. Smith 9th edition Taken from web Polya himself gave words of advice on solving problems: If you can't solve a problem, there is an easier problem you can't solve: find it. (O'Conner Robertson) Taken from The Nature of Mathematics by Karl J. Smith (9th Ed) page 11, shows how Polya's method is put to work for a math problem. EXAMPLE 4 - POLYA'S METHOD A jokester tells you that he has a group of cows and chickens and that he Counted 13 heads and 36 feet.

How many cows and chickens does he have? Solution: Let's use Polya's problem solving guidelines UNDERSTAND THE PROBLEM: A good way to make sure you under- Stand a problem is to attempt top phrase it in a simpler setting: One chicken and one cow: 2 heads and 6 feet (chickens have two; Cows have four) Two chickens and one cow 3 heads and 8 feet One chicken and two cows: 3 heads and 10 feet DEVISE A PLAN: How you organize the material is often important in Problem solving. Let's organize the information into a table: No. of chickens No. of cows No. of heads No. of feet 0 13 13 52 Do you see why we started here? The problem says we must have 13 heads.

There are other possible starting places (13 chickens and 0 cows, for example), But an important aspect of problem solving is to start with some plan. No. of chickens No. of cows No. of heads No. of feet 1 12 13 50 2 11 13 48 3 10 13 46 4 9 13 44 CARRY OUT THE PLAN: Now, look for patterns. Do you see that as the Number of cows decreases by one and the number of chickens by one, the Number of feet must decrease by two? Does this make sense to you? Remember, step 1 requires that you not just push numbers around, but that you understand what you are doing. Since we need 36 feet for the solution to this problem, we see 44-36 = 8 so the number of chickens must increase by an additional four.

The answer is 8 chickens and 5 cows. LOOK BACK: No. of chickens No. of cows No. of heads No. of feet 8 5 13 36 CHECK: 8 chickens have 16 feet and 5 cows have 20 feet, so the total Number of heads is 8 + 15 = 13 and the number of feet is 36 Polya made many contributions to the world of math, that you could see where his impact lies. "Polya's criterion" and the "Polya distribution " in probability theory; "Polya peaks", the "Polya representation", and the "Polya gap theorem" in complex theory all are concepts that bear his name (Young 403). Yet, his impact lies in educating other educators about teaching math. His contributions and experience are endless. Numerous awards for aspiring young math students, bear his name.

In 1978, the National Council of Teachers of Mathematics held problem-solving competitions in The Mathematics Student; they named the awards the Polya Prizes (Young 403), the Polya Prize for Expository Writing from the Mathematical Association of America (Young 403) and the Polya Prize in Combinatorial Theory and Its Applications from The Society for Industrial and Applied Mathematics (Young 403). He even collaborated with Jean Pederson and Peter Hilton to discuss "How to and How Not to Teach Mathematics" at the 1978 meeting at Seattle at the American Mathematical Society and the Mathematical Association of America (Dale). In 1953 Polya retired from Stanford but still was active in the world of mathematics (O'Conner Robertson). He received numerous accolades for his contributions such as honorary degrees from University of Wisconsin, University of Alberta, University of Waterloo, and the Swiss Federal Institute of Technology (Young 403). He also was elected as honorary members of the "Hungarian Academy, the London Mathematical Society, the Mathematical Association of Great Britain, the Swiss Mathematical Society" (O'Conner Robertson), "American Academy of Arts and Sciences, the National Academy of Sciences of the United States of America, the Academie Internationale de Philosophie des Sciences in Brussels, and a corresponding member of the Academie Royale des Sciences in Paris" (Young 403). Polya lived quite a full and long life, long after his retirement in from Stanford University.

Despite his retirement he still taught a class in 1978 at Stanford on a course called Combinatorics in the Computer Science Department (Young 403). Even though this charismatic man died from a stroke at age ninety-seven, his accomplishments still can be seen today. REFERENCES /

Bibliography

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