Pólya, George

From WikEd

1887 - 1985

Interviewed on his 90th birthday Pólya stated, "I started studying law, but this I could stand just for one semester. I couldn't stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little 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, 1979)

1 Descriptions, definitions, synonyms, organizer terms, types of
2 Application in classrooms
3 Evidence of effectiveness
4 Critics and their rationale
5 Alternative explanations due to Diversity considerations
6 Signed “life experiences”, testimonies and stories
7 References and other links of interest
8 Beyond the credits and down the "Tube"s
9 Have any problem solving stories to share?
10

Descriptions, definitions, synonyms, organizer terms, types of

At the time of the above quote, George Pólya was 90 years of age. He was Professor Emeritus of Mathematics at Stanford University and maintained an active speaking schedule. Born and educated in Hungary, he had a PhD in Mathematics from Budapest and had taught at Stanford, Brown and Smith Universities in the United States. By the time Pólya was 90 years old, the Society for Industrial and Applied Mathematics had established the George Pólya Prize, the Mathematical Association of America had established the George Pólya Award for expository writing and he had been elected to membership in the National Academy of Sciences. Other memberships he held included the American Academy of Arts and Sciences, the California Mathematics Council which later established The George Pólya Memorial Award and corresponding membership in the Academie des Sciences in Paris. He held honorary membership in the Hungarian Academy, the London Mathematical Society and the Swiss Mathematical Society. George Pólya had written over 250 papers on mathematics education as well as mathematics and had written a number of books, three of which were very widely read: How to Solve It, Mathematics and Plausible Reasoning, and Mathematical Discovery. Pólya had influenced many teachers with his problem solving writings, lectures and films, and upset the mathematical community by writing about the importance of “guessing” and proving in mathematics. Pólya brought European models of competitive mathematics education to America as well as patterns of thinking, reasoning and problem solving. Beyond that, he made important contributions to number theory, probability theory, the theory of functions, and the calculus of variations.

Application in classrooms

In general:

Pólya found memorization tedius, though at times helpful. He was an advocate of improving the standard of teaching and soon became interested in the process of problem solving. His very first job as a tutor was with a young man struggling with his problem solving skills. Pólya came to believe problem solving was not an inborn quality and needed to be taught. Though particularly helpful in the area of mathematics, many of his ideas and suggestions for problem solving as well as guessing (with judgment) and proving, can be employed in other subject areas successfully as well. In his article "The Goals of Mathematical Education" (2001), Pólya discusses the aim of teaching mathematics and comes to this conclusion which is also true outside of mathematics:

             "The schools should turn out employable adults -- adults who can fill a job.  
              But a higher aim is to develop all the resources of the growing child in order 
              that he can fill in the job for which he is best fitted.  So the higher aim, 
              I express it so, is to develop all the inner resources of the child." (O'Brien, 2001)

In this same article Pólya wrote that teaching is an art, not a "science", with room for personal differences and creativity. He also believed that teaching should be active, as you "cannot learn just by reading" or listening or watching movies. "You must add the action of your own mind." He uses the analogy of a midwife when describing the role of a teacher, with the midwife watching over the birth of ideas but only intervening if "the labor is too long". Students should discover ideas themselves. The role of the teacher is also to help students develop good mental habits for tackling a problem. He believed in "staying power" and developing clarity. Pólya encouraged schools to move away from the old authoritative, teacher-centered methods toward new permissive, student centered classrooms.

Pólya gives some advise to teachers in his book, How to Solve It (1988), before he introduces then develops his four steps to problem solving (listed in the section below). He states that helping students is one of the most important tasks of a teacher, although it puts demands on "time, practice, devotion, and sound principles". Pólya believes the student should work independently as much as possible, with the teacher helping as necessary. He believes the students should do a "reasonable share of the work" and that the teacher needs to find a good balance between helping too much and not enough. If the teacher has to help quite a bit, he cautions that it should be done "unobtrusively and naturally".

Pólya also gives hints to teachers in his book, Mathematical Discovery: On understanding, learning, and teaching problem solving Vol. I (1966). He states that this book should give prospective and current high school mathematics teachers an opportunity to do creative work at an appropriate level. He acknowledges that they will probably not do research on an advanced level, but that they need the opportunity to do some creative work themselves to be able to recognize and encourage creative work in their students. Pólya also states that the best practice for students is offered in group work. He suggests three steps to this process. First, each student should be given a different problem (just one each) to solve during a class session. Next, the student should take that solved problem home and "check, complete, review, and, if possible, simplify his solution, look out for some other approach to the result and, by these means or any other means, master the problem as fully as he can." Then during the next class period discussion groups of four students each should spend the period taking turns acting as teacher for their group, presenting their problem, challenging their initiative and guiding them to the solution.

In the second volume of his book, Mathematical Discovery: On understanding, learning, and teaching problem solving (1966), Pólya again addresses teachers. He discusses two points of teacher training. First he states the sad fact that the knowledge of high school mathematics teachers is "on the average, insufficient". To this he suggests experience with creative problem solving work in such an environment as a Problem Solving Seminar. Second, he addresses the debate of whether methods courses are valuable or a waste of time. Pólya quotes a teacher who told him during a discussion on this topic, "The mathematics department offers us tough steak which we cannot chew and the school of education vapid soup with no meat in it." His suggestion is that methods courses be offered in close connection to subject matter or practice teaching. Pólya then offers his Ten Commandments for Teachers.

1)  Be interested in your subject.
2)  Know your subject.
3)  Know about the ways of learning: The best way to learn anything is to discover it by yourself.
4)  Try to read the faces of your students, try to see their expectations and difficulties, 
    put yourself in their place.
5)  Give them not only information, but "know-how," attitudes of mind, the habit of methodical work.
6)  Let them learn guessing.
7)  Let them learn proving.
8)  Look out for such features of the problem at hand as may be useful in solving the problems to come
    - try to disclose the general pattern that lies behind the present concrete situation.
9)  Do not give away your whole secret at once - let the students guess before you tell it - let them 
    find out by themselves as much as is feasible.
10) Suggest it, do not force it down their throats.

Problem solving:

In Pólya's How to Solve It (1988), which sold over one million copies and has been translated into 17 languages, he lists four steps in the problem solving process. He cautions against skipping any of these steps to jump to a quick conclusion. Pólya states it is important that one understand the problem and the connections or "something very undesirable and unfortunate may result". The steps below are taken from various chapters of his book, How to Solve It.

 Step 1: Understand the Problem. (or - "see clearly what is required")
           * What is the unknown?
           * What are the data?
           * What is the condition? 
           * Draw a figure.  
               Point out the unknown and the data.
               Introduce suitable notation.
           * Guess: Is it possible to satisfy the condition?
           * Is the condition sufficient to determine the unknown? (Is it a reasonable problem?)
               Or is it insufficient? Or redundant? Or contradictory?

Step 2: Devise or Make a Plan. (or - "the idea of the solution")
* Do you know a related problem?
What is the hypothesis?
What is the conclusion?
Try to think of a familiar theorem having the same or a similar conclusion.
Could you use it?
* How are the items (the data and the unknown) connected?
You may be obliged to consider auxiliary problems if an immediate connection cannot be found.
You should obtain eventually a plan of the solution.
* Look at the unknown.
Try to think of a familiar problem having the same or a similar unknown.
What is the unknown?
Do you know any problem with the same unknown?
Do you know any problem with a similar unknown?
Could you use it?
How far is the unknown then determined?
How can it vary?
Should you introduce some auxiliary element in order to make its use possible?
Could you restate the problem?
If you cannot solve the proposed problem try to solve first some related problem.
* Do you know a theorem that could be useful?
* Go back to definitions.
* Could you restate it still differently?
* Could you keep only a part of the condition, drop the other part?
* Could you solve a part of the problem?
* Could you imagine a more...
accessible related problem?
general problem?
special problem?
analogous problem?
* Did you use all the data?
* Could you derive something useful from the data?
* Could you think of other data appropriate to determine the unknown?
* Could you change the unknown or data (or both?) so that
the new unknown and the new data are nearer to each other?
* Did you use the whole condition?
Could you use its result?
Could you use its method?
* Have you taken into account all essential notions involved in the problem?
* Look at your conclusion.
Did you use the hypothesis?
What is the hypothesis?
Did you use the whole hypothesis?

 Step 3: Carry Out the Plan.
           * Check each step.
           * Can you see clearly that the step is correct?
           * Can you prove that the step is correct?

 Step 4: Look Back at the Completed Solution. (or - "review and discuss it")
           * Reconsider and reexamine the result and the path that led to it.
           * Can you check the result?
           * Can you check the argument? 
           * Can you derive the result differently?
           * Can you see it at a glance?
           * Can you use the result, or the method, for some other problem?

On wisdom of proverbs:

Towards the end of Pólya's book, How to Solve It (1988), he states, "Solving problems is a fundamental human activity." He believed that most conscious thinking involved considering problems. Pólya was struck by the number of proverbs that address problem solving and listed a number of them in this book. While he admits many are contradictory and not necessarily an authoritative source, he states "it would be a pity to disregard the graphic description of heuristic procedures provided by proverbs."

              Who understands ill, answers ill.
              Think on the end before you begin.  (In Latin, "respice finem")
              A fool looks to the beginning, a wise man regards the end.
              A wise man begins in the end, a fool ends in the beginning.
              Where there is a will there is a way.
              Diligence is the mother of good luck.
              Perseverance kills the game.
              An oak is not felled at one stroke.
              If at first you don't succeed, try, try again.
              Try all the keys in the bunch.
              Arrows are made of all sorts of wood.
              As the wind blows you must set your sail.
              Cut your coat according to the cloth.
              We must do as we may if we can't do as we would.
              A wise man changes his mind, a fool never does.
              Have two strings to your bow.
              Do and undo, the day is long enough.
              The end of fishing is not angling but catching.
              A wise man will make more opportunities than he finds.
              A wise man will make tools of what comes to hand.
              A wise man turns chance into good fortune.
              Look before you leap.
              Try before you trust.
              A wise delay makes the road safe.
              If you will sail without danger you must never put to sea.
              Do the likeliest and hope the best.
              Use the means and God will give the blessing.
              We soon believe what we desire.
              Step after step the ladder is ascended.
              Little by little as the cat ate the flickle.
              Do it by degrees.
              What a fool does at last, a wise man does at first.
              He thinks not well that thinks not again.
              Second thoughts are best.
              It is safe riding at two anchors.

On plausible reasoning:

George Pólya's second book, Mathematics and Plausible Reasoning (1954), was written to "complete" his previous book, How to Solve It. It consists of two volumes: Volume I is subtitled Induction and Analogy in Mathematics. Volume II is subtitled Patterns of Plausible Inference. The first volume begins a discourse on "guessing" that later leads to his articles (referred to in the next section) on "Guessing and Proving". Pólya begins Volume I by discussing conjectures, some being reliable, respectable conjectures and others not. He states that we "secure our mathematical knowledge by demonstrative reasoning, but we support our conjectures by plausible reasoning". Demonstrative reasoning is final and safe. Plausible reasoning is provisional, hazardous and controversial. Mathematicians use demonstrative reasoning when writing proofs. However, to arrive at those proofs, they must use plausible reasoning. They must guess. Everything new is arrived at through plausible reasoning. Pólya goes on to state that he knows of no way to teach plausible reasoning. He states, "The efficient use of plausible reasoning is a practical skill and it is learned, as any other practical skill, by imitation and practice." That is what his book, Mathematics and Plausible Reasoning, was written to do. Both volumes provide examples, comments, underlying motives, plausible inferences and "practice" to learn the skill of plausible reasoning. At the end of Volume II Pólya again discusses the fact that plausible reasoning does not have specific rules that can be taught. He contrasts it to demonstrative logic where, "if all steps conform to the rules, the demonstration is valid, but it is invalid if there is a step violating the rules." On the other hand, Pólya states "Yet there are rules of different kinds. Logical rules are very different from legal rules." He gives several examples to illustrate this point, one being a court of law. The court will listen, but not to irrelevancies. This is regulated by rules of admissibility, which provide for an orderly administration of justice. Pólya goes on to state that plausible reasoning is similar, using rules of admissibility in scientific discussion. He states:

             "Our patterns register various points concerning such verifications that could 
              reasonably influence the weight of the evidence (as analogy, or lack of analogy,
              with former verifications, etc.).  In collecting these patterns the author's 
              intention was to list those general points that, according to the usage of good
              scientists, are admissible in a scientific discussion, with a view to reasonably
              influencing the credibility of the conjecture discussed."

Pólya concludes this discussion by stating that the powers of the court are divided between the judge and the jury just as judging a proposed conjecture is divided between impersonal rules and personal good sense. The impersonal rules of plausible reasoning judge what evidence deservers consideration. "Yet it is for your personal good sense to decide whether the particular piece of evidence just submitted has sufficient weight or not."

On guessing and proving:

Pólya begins his article "Guessing and Proving" (1978), by stating that Euler's theorem on polyhedra was published without proof, but with an inductive argument. Pólya believes he discovered this theorem inductively as well. He states, "Yet he does not give a direct indication of how he was led to his theorem, of how he "guessed" it, whereas in some other cases he offers suggestive hints about the ways and motives of his inductive considerations." Pólya goes on in this article to walk through a rediscovery of Euler's path to his theorem on polyhedra. At the end of this article Pólya ponders "What is scientific method?" and concludes that it is "Guess and test." For mathematicians, he clarifies it as "First guess, then prove."

That article must have stirred up some debate, as George Pólya followed it with a second article titled "More on Guessing and Proving" (1979). In it he states that mathematicians may have had something against the title "Guessing and Proving", as "It may be all right for a mathematician to talk about proving, but what business has a mathematician to talk about guessing?" Pólya encourages old-fashioned math teachers to convert to his view and consider that guessing belongs in every classroom. He believes that guessing is important and he couples that with the importance of judgment, as guesses may be

            "more or less reasonable,
             more or less reliable,
             more or less well supported,
             more or less respectable."

Pólya accompanies these thoughts on guessing and judgment with the fact that we live in a complex world full of contraditory assertions, perils and deception. He also states that guessing in the classroom makes learning mathematics more fun and productive. Pólya advises mathematics teachers to "acquaint his students with mathematical proofs". Instead of bothering students with proofs of things that are intuitive or asking them to learn proofs by heart, a good teacher should use examples of good proofs to "make him (the student) understand the role and interest of strict proofs." It is then, Pólya states, that students are prepared to "distinguish between guesses and guesses". This process may look different on the high school level than in a class for future engineers, and different yet in a class for future mathematicians. Yet learning to guess and prove well is important not only for the study of math, but to develop personality and character. Pólya ends this article by stating about guessing, proving and judgment, "And judgment may be the thing you need most among the complexities and deceptions of modern life".

Technology use and integration:

Software is available that allows students to creatively manipulate, draw and explore mathematics problems and concepts. Geometer's Sketchpad is an interactive, open ended software program that aides in the areas of algebra, geometry, trigonometry, precalculus, and calculus as users draw, manipulate and solve problems. Another software option is FX Draw made by efofex. This software takes a step further, as it is useful in a wider variety of mathematical problems and concepts. It can also be purchased as part of a broader MathPack bundle that includes FX Draw, FX Equation, FX Graph and FX Stat. Yet another piece of interactive 3D software is Cabri. Not only does the software help you explore measurements and equations, polyhedrons, space geometry, optics and physics but it also integrates with Microsoft Word and PowerPoint. For younger students in grades 3-8, Tom Snyder Productions offers GO Solve Word Problems. Its guided tour of the product speaks of a four step process: Understand, Plan, Solve and Check. Look familiar? Yes, it's basicly Pólya's four steps to problem solving. The makers of this software explain that many student skip the first step of understanding the problem. They simply try to solve the problem without really understanding it. This software uses graphic organizers to help students "draw" the problem to better understand it before solving. It also utilizes self paced instruction, guided practice, independent practice and assessment.

Evidence of effectiveness

If a good indicator of effectiveness is continued use, George Pólya's work meets that standard. In Mathematics Teaching 2006, Francis Lopez-Real wrote an article titled "A New Look at a Pólya Problem". Before analyzing a Pólya problem, he states of study and research into problem solving and mathematical investigating, "...most of the later work has been a development and refinement of Pólya's fundamental ideas, all of which remain relevant today." In September of 2007 the National Association for the Education of Young Children contained an article entitled "The Classroom That Math Built: Encouraging Young Mathematicians to Pose Problems". In it the author acknowledges that "the idea of problem solving is not new". The article chronicles five days in a second grade classroom where students are challenged to work with one problem solving activity, basically using Pólya's steps. The teacher concluded that mathematical thinking is important, that "the experience served the needs of a wide variety of mathematical learners" stimulating a variety of children with different intellectual needs and desires, and plans to broaden her use of it. I was not surprised to see Pólya's book, How to Solve it, listed in the references. This article states that The National Council of Teachers of Mathematics "has called on teachers to set up problem-solving experiences that encourage children to devise and solve their own problems". A search today (Feb. 08) on NCTM's web site of Pólya brought up 31 articles referring to his work in connection with teaching mathematics today.

A videodisc-based mathematical problem solving series called "The Adventures of Jasper Woodbury" was the subject of a study (Hickey, Moore & Pellegrino 2001) on its effects in a constructivist-inspired reform of one school district. Although it was not a study specifically on Pólya's work, it utilized similar problem solving techniques with problems requiring four to five collaborative class periods and 15 to 20 steps to solve. The series was meant to raise student interest in the math, involve them in problems of real world significance, deepen understanding of mathematical concepts, provide a variety of positive role models and incorporate the above mentioned NCTM standards. Fifth grade classroom teachers were trained in the first three of twelve Jasper adventures and used them in a specified manner in their classroom. First students viewed the video and did some brainstorming. Next they worked in groups to devise a plausible solution (guess!). Once solved, each group presented their solution to the class to be reviewed by peers before watching the end of the video containing the actual solution. As might be expected in a real classroom setting, not all of the teachers "followed the rules" equally. Of the 10 teachers using the Jasper series, the 5 that best adhered to the guidelines of the study achieved the largest gains on the Mathematical Problem-solving subtest of the ITBS. The 10 "Jasper classrooms" all increased their scores on the ITBS. The 10 "non-Jasper classroom" students all remain the same or decreased their scores. Also, students took an inventory to indicate motivation levels and all in the Jasper series indicated increased motivation levels. An earlier "Jasper" study had similar positive results but was conducted with teachers who would normally "choose" a constructivist type classroom as their preference. This study differed in two ways from the first. The classes were split about 50/50 in terms of economic strata. Also, the teachers were not all predisposed to have a constructivist classroom, but were in a school district that had implemented these characteristics throughout. Students in both economic strata participating in the "Jasper classrooms" displayed "enhanced math self-concept and interest in math" and "greater subjective competence in mathematics throughout the school year".

Richard E. Mayer, Learning and Instruction (2003):

Mayer (2003) cites various research studies in the area of problem solving. The first is in the area of mathematics, while the rest are outside that area of study. All support Pólya’s work and teaching. First, Mayer discusses Schoenfeld (1979) who taught heuristics for mathematical problem solving to college students. Though a small study, it showed a 45% gain in the trained group opposed to no gain in the control group. Certainly it seems that heuristics, or problem solving, can be taught.

Next, Mayer states that problem solving should be taught within a specific subject area. There is no compelling evidence that problem solving skills transfer from one domain or subject area to another, thus Pólya’s approach to teach problem solving as part of the mathematics curriculum was again confirmed. Pólya emphasized the process (Four Step Plan) in problem solving, not the product. Mayer confirms this approach. His text cites a Bloom and Broder (1950) study of University of Chicago students answering exam questions. Instead of looking at correct responses (product) they instead targeted successful thinking strategies (process). A ”thinking out loud” strategy was used and all students recorded their thoughts as they worked through exam questions. Remedial students looked at and analyzed differences between their thought processes and those of successful students. These remedial students increased their scores by about .5 to .7 points.

Finally, Mayer describes three popular problem solving courses used in schools that have proven to produce successful results: Productive Thinking Program, Instrumental Enrichment, and Project Intelligence. All of these have four key points in common, all of which again support Pólya’s work and teaching. The first is to focus on a few well-defined skills. Pólya felt students of mathematics should have experiences with a variety of types of mathematics problems. Students then could use that prior knowledge to problem solve mathematics problems that are new to them but similar to others. He also used a list of specific types of questions for students to ask themselves as they worked through the problem solving process. The second key point is to contextualize the skills within authentic tasks. Pólya had students do this in the context of solving mathematics problems and working through proofs. The third is to personalize the skills through social interaction and discussion. One of Pólya’s popular teaching methods was to have students teach each other concepts, defend them and question them together. He also felt the teacher should give just enough help as is necessary, while having students work through the rest, often including appropriate questioning and discussion. Finally, the fourth key point is to accelerate the skills so they are taught along with lower-level skills. Pólya did not feel that mathematics teachers should wait until students had mastered math skills to introduce problem solving. Instead he advocated the use of problem solving skills along side the curriculum while teaching basic math skills. Thus, Mayer’s entire chapter 12 on problem solving confirms Pólya’s approach to the teaching of heuristics in mathematics.

Critics and their rationale

According to Tibor Frank (2001), Pólya critic Alan H. Schoenfeld called Pólya's work in the area of problem solving scientifically problematic. He questioned whether problem solving or a thinking process could be "taught" and felt that a heuristics-based classroom approach did not actually improve problem solving performance. Artificial intelligence researchers claimed they were unable to write code for programs using Pólya's heuristics and called his problem solving strategies "epiphenomenal rather than real". More recent studies in cognitive science have shown that students can indeed improve problem solving performance through the use of heurisitics. It may also now be possible to create computer programs that support Pólya's problem solving approach.

Pólya practiced mathematics during the time that the "new math" was becoming popular and replacing the "old" traditional way of teaching math. Students needed more than formulas, they needed to understand the concepts and meaning behind them, the very structure of mathematics itself. Pólya agreed that reform was necessary. Some like Morris Kline were critical of the new math reforms. However, even among mathematicians that agreed with Pólya that mathematics education reform was necessary, there were disagreements about what changes should be made. One of them was Ed Begle. Oddly enough, according to Jeremy Kilpatrick from the University of Georgia, while at Stanford University he worked with both Pólya and Begle. Pólya was on his dissertation committee and Begle was his major professor for doctoral studies. In a symposium, Reflecting on Sputnik, Kilpatrick (1997) describes them as "two eminent scholars who, while respecting each other's views, could not have had more divergent ideas as to what changes were needed in school mathematics." Leigh Hunt's Chapter 9 of Why Johnny Can't Add (2001) states of the new mathematics movement, "The sad fact is that most of the groups undertook almost no experimental work." Professor Max Beberman of the University of Illinois Committee on School Mathematics began research with classes at the university high school. That was cut short when in 1960 the National Council of Teachers of Mathematics conducted conferences across the United States in effect launching the use of "new and improved" mathematics programs. Thus Hunt (2001) explains "experimental work practically vanished and the rush to secure leadership took precedence over all other activity." Because of this rush for change, mathematicians who supported reforms differed in opinions as what changes should be made. A few college professors got together and wrote a protest against the specifics of the current movement titled On the Mathematics Curriculum of the High School. Pólya himself signed the document. Hunt's criticism of the new math is softened somewhat when she discusses attempts to study the effects of mathematics reforms. Chapter 9 discusses the difficulties in testing the effectiveness of the new math, as math tests typically simply look for a correct answer and do not "test" what changes the new math offered. Comparisons between the U.S. and other countries are problematic because typically only the best enroll at universities in other countries, while in the U.S. the enrollment is larger and thus encompasses more than "just the best". The need for a college education in today's society has also increased enrollment by students with lesser mathematics abilities.

Alternative explanations due to Diversity considerations

Pólya's ideas on problem solving and guessing and proving as well his advice to classroom teachers and avocation of student centered active learning environments all support diversity considerations. He advocates discussion of diverse ideas in an open classroom environment where differences are not only accepted but used towards educational and personal character growth. His is an environment where all students are encouraged to learn and question with appropriate support from the classroom teacher. Pólya was interested in changing negative attitudes towards math into positive ones for all students. He wrote about looking at the individual student and giving each the appropriate types of problems as well as a balanced amount of help, depending on the student's ability level and past experiences. Finally, Pólya encouraged group work where students share leadership, knowledge, methods and ideas. All of these are current diversity concerns as described by Donnell Butler of Princeton University on his web page titled "Diversity in the Classroom: Links on the Web". George Pólya was an advocate for all students as well as teachers. Mathematics is a universal language that can bring diverse peoples together.

Signed “life experiences”, testimonies and stories

I became interested in George Pólya because of his work in problem solving and guessing. As a teacher in 3rd through 6th grade classrooms, I saw students struggle with how to approach and correctly solve story problems. I also observed educators' efforts to teach students these skills. Many resorted to teaching key words for students to look for in story problems that would indicate what mathematical operation to use to solve the problem. Phrases such as "in all", "altogether", "much more", "each" and "left" were discovered in a story problem, then the corresponding "correct" operation was used on the numbers provided. Instead of helping students try to understand the problem at hand, it simply gave them a "crutch" or "quick fix" to use. When working with students on problems not in the math textbook, I found that this approach was limited in two ways. It didn't produce the correct answer all the time, and students did not understand the problem at hand. They simply plugged numbers into a formula. I began drawing story problems on the board in an attempt to help students better understand the problem. I did that naturally because, when confronted with a difficult math problem, that's what I do. After some success with that, I moved to having students draw the problem as well. Although some students resisted this, as they were used to a "quick fix" and were reluctant to spend more time and effort on a problem, I found it helpful to those who truly tried to use this approach. I am no longer in the classroom. If I were, I would add group work to this scenario. While reading Chapter 5 in Richard E. Mayer's Learning and Instruction (Mayer 2003), I felt a sudden kinship as I began to read about exactly what I have been describing! George Pólya was mentioned for his work in problem solving. Upon closer investigation I discovered that George Pólya included drawing a picture and "educated guesses" in his writings about problem solving and the teaching of mathematics. His problem solving steps in his book How to Solve It make absolute sense. In fact, one must wonder when looking at them why we needed to be told to use these steps. They seem logical, practical, intuitive and simply good common sense. I agree with him that it is a skill that needs to be taught. Outside my classroom experiences, I can attest to the fact that both of my children did not come naturally to drawing and thinking about a story problem in an attempt to understand and then solve it. I had to model it, advocate for it and teach it. - Pam Olivito

References and other links of interest

References:

Alexanderson, G.L. (January, 1979). George Pólya Interviewed on His Ninetieth Birthday. The Two-Year College Mathematics Journal. Vol.10 No.1, 13-19.

Frank, Tibor (2001, March 22). George Pólya and the Heuristic Tradition Fascination with Genius in Central Europe. Retrieved February 1, 2008, from The Polanyiana, Volume 6, Number 2 Web site: http://www.kfki.hu/~cheminfo/polanyi/9702/frank.html

Hickey, Daniel T., Moore, Allison L., & Pellegrino, James W. (2001). The Motivational and Academic Consequences of Elementary Mathematics Environments: Do Constructivist Innovations and Reforms Make a Difference?. American Educational Research Journal. 38, 611-652.

Hunt, Leigh (2001, March 22). Why Johnny Can't Add: Chapter 9 The Testimony of Tests. Retrieved February 26, 2008, from Marco Learning Systems Web site: http://www.marco-learningsystems.com/pages/kline/johnny/johnny-chapt9.html

Kilpatrick, Jeremy (1997). Five Lessons from the New Math Era. Retrieved February 15, 2008, from Reflecting on Sputnik: Linking the Past, Present, and Future of Educational Reform Web site: http://www.nationalacademies.org/sputnik/kilpatin.htm

Lieberherr, K. (2006, September 18). George Pólya. MathGym Notes. Retrieved January 18, 2008, from George Polya web site: http://www.ccs.neu.edu/home/lieber/courses/materials/Polya_Father_of_ProblemSolving.pdf

Lopez-Real, Francis (2006). A New Look at a Polya Problem. Mathematics Teaching. 196, 12-15.

Mayer, R.E. (2003). Learning and Instruction. Upper Saddle River, New Jersey: Pearson Education, Inc.

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