George Polya, 1887-1985
The Father of Problem Solving in Mathematics Education
In His Own Words — from a lecture to teachers transcribed by Thomas C. O’Brien
George Polya can rightly be called the father of problem solving in mathematics education. For that distinction and his many contributions to our field, the California Mathematics Council chose to name one of our prestigious awards in his honor.
Dr. Polya was a distinguished mathematician and professor at Stanford University. Polya (1887-1985) made important contributions to probability theory, number theory, the theory of functions, and the calculus of variations. He was the author of the classic works How to Solve It, Mathematics and Plausible Reasoning, and Mathematical Discovery, which encouraged students to become thoughtful and independent problem solvers. He was an honorary member of the Hungarian Academy, the London Mathematical Society, and the Swiss Mathematical Society, and a member of the (American) National Academy of Sciences, the American Academy of Arts and Sciences, and the California Mathematics Council, as well as a corresponding member of the Academie des Sciences in Paris.
The first-person essay that follows is a slightly edited transcript of a lecture Professor Polya presented to Tom O’Brien’s inservice and preservice mathematics education students in the late 1960s. There is no better way to understand the problem solving philosophy George Polya so strongly advocated for students, then to read his own words:
I wish to talk to you about the teaching of mathematics in the elementary school. In fact my talk will consist of two parts. In the first part I will talk about the aims of teaching mathematics in the elementary school, and in the second part, how to teach it.
I must confess that I am talking about these things as an outsider. I was always interested in teaching, but most of my time, about half a century, I taught in the university, and in the last fifteen years, I was mainly concerned with teaching on the high school level. Thus I am talking to you as an outsider, but you may find one or two points in what I am saying that may be useful to you in your grade level.
What is the aim of teaching mathematics in the elementary school? It is better to consider the most general question: What is the aim of the schools? And the even better question is: What do people generally think is the aim of the schools? The first is the point of view of the parents. Your neighbor Mr. Smith has a son Jimmy. He is against Jimmy being a dropout. He says that if Jimmy drops out of school he will never get a good job. So the aim of the school, according to Mr. Smith and all the other Mr. and Mrs. Smiths in the general public, is to prepare Jimmy for a job—to prepare students to earn a living. But what is the point of view of the community? It is the same. The community, the country, the state, and the city all want people to earn a living and pay taxes and not live on public assistance. So the community also wants schools to prepare young people to have jobs.
If the parents think a little farther, and the community thinks a little farther, the aim is somewhat different. Reasonable parents, a reasonable Mr. and Mrs. Smith, want their son Jimmy to have a job for which he is well suited so he will earn more and feel happier. By the way, this is also the aim of the community – that you have jobs on one side and people on the other side, and you assign people to jobs they are best suited so that they produce the greatest output. Or even better, their happiness should be maximized.
What can the school do for that? The point is that when a child comes to school you can’t know what job he or she will have later on, nor do you know what job he or she will be best suited. So what should we do? We should prepare youngsters so they can choose between ALL possible jobs, and give them a view of the whole world so they can recognize which jobs will best suite them. You can express this in many ways. I like the following: Schools should develop all the interior resources of the child.
We have therefore two kinds of aims in the schools. We have good and narrow aims—schools should turn out employable adults who can fill a job. But a higher aim is to develop all the resources of growing children so they can fill the job for which they are best suited. The higher aim is to develop all the inner resources of the child.
Now what about mathematics teaching? Mathematics in the elementary schools has a good and narrow aim, and that is pretty clear. Everybody should be able to read and write and do some arithmetic, and perhaps a little more—an adult who is illiterate is not employable in a modern society. Therefore the good and narrow aim of the elementary school is to teach the arithmetical skills—addition, subtraction, multiplication, division, how to measure length, area, volume, fractions, percentages, rates, and perhaps even a little more. This is a good and narrow aim of the elementary schools—to transmit this knowledge—and we shouldn’t forget it.
However, we must also have a higher aim if we wish to develop all the resources of the growing child. And the part that mathematics plays is mostly about thinking. Mathematics is a good for teaching thinking. But what is thinking? The thinking that you can learn in mathematics is, for instance, to handle abstractions. Mathematics is about numbers. Numbers are an abstraction. When we solve a practical problem, in order to solve this, we must first make it into an abstract problem. Mathematics applies directly to abstractions. Some mathematics teaching should at least enable a child to handle abstractions, to handle abstract structures. Structure is a fashionable word now. It is not a bad word. I am quite for it.
But I think there is one point that is even more important: Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean doing mathematics? In the first place it means to be able to solve mathematical problems. To achieve the higher aims I am talking about, there are some general tactics of problem solving—the right attitude for problem solving and ability to attack all kinds of problems, not only simple problems that can be solved with simple arithmetic, but more complicated problems of engineering, physics and so on, which will be further developed in the high school. But the foundations of problem solving should be started in the elementary school. And so I think an essential point in the elementary school is to introduce children to the tactics of problem solving—not to solve this or that kind of problem, not to make just long divisions or some such thing, but to develop a general attitude and ability for the solution of problems.
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 is great latitude and much possibility for personal differences. In an old British manual there was the following sentence, “Whatever the subject, what the teacher really teaches is himself.” So therefore when I tell you to teach so or so, please take it in the right spirit. Take as much of my advice as fits you personally. You must teach yourself.
George Polya’s Advice to Teachers:
- Be interested in your subject.
- Know your subject.
- Know about the ways of learning: The best way to learn anything is to discover it by yourself.
- Try to read the faces of your students, try to see their expectations and difficulties, put yourself in their place.
- Give them not only information, but "know-how," attitudes of mind, the habit of methodical work.
- Let them learn guessing.
- Let them learn proving.
- 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.
- 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.
- Suggest it, do not force it down their throats.
There are as many good ways of teaching as there are good teachers. But let me tell you my idea of good teaching. Perhaps the first point, which is widely accepted, is that teaching must be active, or rather learning must active—this is the better expression.
You cannot learn just by reading. You cannot learn just by listening to lectures. You cannot learn just by watching. We all must add the action of your own mind in order to learn something. Socrates expressed it two thousand years ago very colorfully when he said that an idea should be born in the student’s mind, and the teacher should just act as a midwife. The idea should be born in the student’s mind naturally and the midwife shouldn’t interfere too much, too early. But if the labor of birth is too long, the midwife must then intervene. This is a very old principle and there is a modern name for it – discovery learning. The student learns by his own actions. The most important action of learning is to discover something by yourself. This should be the most important part in teaching because what students discover by themselves will last longer and be better understood.
There are other principles of teaching, or rules of thumb. Another principle was stated by the great educators Socrates, Plato, Comenius, and Montessori—that there are certain priorities in learning. For instance, things come before words. Kant said, “All human cognition begins with intuitions, proceeds hence to conceptions, and ends in ideas.” Let me translate this into simpler terms. Learning begins with action and perception, proceeds hence to words and concepts, and should end in good mental habits. This is the general aim of mathematics teaching—to develop in each student as much as possible the good mental habits for tackling any kind of problem.
You should develop the whole personality of the student, and mathematics teaching should especially develop thinking, clarity, and persistence. It could also develop character to some extent, but most important is the development of thinking.
I believe the most important part of thinking that is developed in mathematics is the right attitude in tackling and solving problems. We have problems in everyday life. We have problems in science. We have problems in politics. We have problems everywhere. The right attitude for problem solving may be slightly different from one domain to another, but we have only one head, and therefore it is natural that there should be one general set of tactics to tackling all kinds of problems. My personal opinion is that the main point in mathematics teaching is to develop these tactics of problem solving.
The two principles of active learning—priority of action and perception—are taken into account by almost all mathematics teaching today. However, there is allegedly a Chinese proverb that says, “I hear and I forget. I see and I remember. I do and I understand.” So “I hear and I forget.” What you just hear you forget quickly. Good advice is very quickly forgotten. What you see with your own eyes is remembered better, but you really understand it when you do it with your own hands. So there must be more than just priority of action and perception in our teaching.
Therefore the schools, especially elementary schools, are today in an evolution. A sizable fraction, ten to twenty percent, already have the new method of teaching which can be characterized in the following way compared with the old method of teaching. The old method is authoritative and teacher-centered. The new method is student-centered. In the old time the teacher was in the center of the class or in front of the class. Everybody looked at the teacher and what he or she said. Today the individual students should be in the center of the class, and they should be allowed to do whatever good idea comes to their mind. They should be allowed to pursue ideas in their own way, each by themselves, or in small groups. If a student has a good idea in a class discussion then the teacher changes his plans and allows the class to follow this good idea.
I must tell you one name, a person who is particularly active in this direction and who is very clever, very good. This is Miss Edith Biggs. She is a particularly gifted teacher who stands in with great enthusiasm and talent for this new student-centered teaching.
In such a student-centered class, each group of kids are doing something else. They play (let’s just say that they think that they play, but really they learn). The teacher gives them various materials. A class period consists of the teacher giving kids various materials and a problem to solve. They play and they develop their own ideas in play. For instance, one of the materials is squared paper. And a good supply of cubes, cubes of one half inch and several dozens of them, maybe even a hundred. So the kids play with that. It is activity teaching—teaching by action involvement.
Let me give you an example of this activity. The class discusses little rectangles—proceeding from action and perception of things they have often seen and touched. Everybody has seen a room, and the walls of an ordinary room are rectangles, or almost rectangles. So you naturally learn what a rectangle is. The floor of the usual room is a rectangle. And any wall is a rectangle. The ceiling is a rectangle. One key idea in mathematics is to understand length and area, so you measure the length of the rectangles and come to the concept of the perimeter of rectangles. Then you deal with the area of the rectangle. You build up the rectangle from equal squares, from unit squares, and come to the notion of the area of rectangles.
We are now in a class that is somewhat familiar with the area and perimeter of rectangles. On the same sheet of paper, draw overlapping rectangles, with the same perimeter—a perimeter of twenty (see illustration at right). There are nine such rectangles. They start with width = 1 and height = 9, and then width = 2 and height = 8, and down to width = 9 and height =1.
There are many things to observe—action and perception. Some of the kids will be struck by the observation that all the corners of these rectangles are on a straight line. Then they will notice that one of these rectangles has equal sides and you can ask many questions about it. One of the interesting points is that the teacher should not ask the questions; the kids should ask the questions. They all have the same perimeter. Do they have the same area? Which one has the greatest area?
Here is another activity with rectangles. Again take square grid paper and cut out different rectangles, but this time with the same areas—let’s say area of 24 square units. Overlap them on the same paper. Now the corners opposite to the one corner in which they overlap are not on a straight line. There is some funny kind of curved line.
Kids with an imagination will join these to make a curved line. So that is another consideration. This is an example of an activity with rectangles where the kids have their own choice. They make their own remarks and the teacher just helps a little now and then with some hints. If the kids have no ideas at all, then the well-instructed teacher, who is used to this student-centered teaching, can give a few good hints.
Perhaps one point that Miss Biggs and the Nuffield Foundation do not emphasize sufficiently is the rule of guessing. Guessing comes to us naturally. Everybody tries to guess and does not have to be taught to do so. What has to be taught is reasonable guessing, and especially to not believe your own guesses, but to test them. And students’ activity will start much better if you start them by guessing.
Here is one example. One activity is to measure the length and the width of the classroom. Some students may be bored by this if they have already done it with a former teacher. You can get a little more interest if you start with a guess. You may say, “It seems to me that this classroom is twice as long as it is wide. Is it really?” I hope some of the kids will say, “No, it is longer than twice.” Others will say, “No, it is shorter.” A very few will say, “Exactly.” After they have all guessed, they will do the measuring with much more interest because everybody will be interested whether his or her guess will be true or not. This is a very special case in the tactics of problem solving. If you go farther, you will notice that guessing plays an important role. The solution to a problem naturally always starts with a guess, but not always with a good guess. On the contrary, usually the guess is never completely good. It is just a little out of center and the art of problem solving consists in great part in correcting your guesses.
I have given you my ideas about how you should teach mathematics. They are the ideas of active learning, priority of action and perception, and teaching by allowing students to start an activity by letting them guess. I hope one of these points will find a sympathetic ear with some of you.