Author: G. Polya
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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.
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before trying to solve a problem, the student should demonstrate his or her understanding of its statement, preferably to a real teacher, but in lieu of that, to an imagined one.
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“If you can’t solve a problem, then there is an easier problem you can’t solve: find it.”
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One of the most important tasks of the teacher is to help his students.
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The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help or with insufficient help, he may make no progress at all.
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If the student is not able to do much, the teacher should leave him at least some illusion of independent work. In order to do so, the teacher should help the student discreetly, unobtrusively.
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Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice.
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Trying to solve problems, you have to observe and to imitate what other people do when solving problems and, finally, you learn to do problems by doing them.
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Trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem.
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The worst may happen if the student embarks upon computations or constructions without having understood the problem.
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It is generally useless to carry out details without having seen the main connection, or having made a sort of plan.
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It is generally useless to carry out details without having seen the main connection, or having made a sort of plan. Many mistakes can be avoided if, carrying out his plan, the student checks each step.
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It is foolish to answer a question that you do not understand.
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If the student is lacking in understanding or in interest, it is not always his fault; the problem should be well chosen, not too difficult and not too easy, natural and interesting, and some time should be allowed
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If the student is lacking in understanding or in interest, it is not always his fault; the problem should be well chosen, not too difficult and not too easy, natural and interesting, and some time should be allowed for natural and interesting presentation.
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If there is a figure connected with the problem he should draw a figure and point out on it the unknown and the data.
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In order to be able to see the student’s position, the teacher should think of his own experience, of his difficulties and successes in solving problems.
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We know, of course, that it is hard to have a good idea if we have little knowledge of the subject, and impossible to have it if we have no knowledge.
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Good ideas are based on past experience and formerly acquired knowledge.
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Mere remembering is not enough for a good idea, but we cannot have any good idea without recollecting some pertinent facts; materials alone are not enough for constructing a house but we cannot construct a house without collecting the necessary materials.
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He must be prepared to repeat with some modification the questions which the students do not answer.
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certain cases, the teacher may emphasize the difference between “seeing” and “proving”: Can you see clearly that the step is correct? But can you also prove that the step is correct?
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In certain cases, the teacher may emphasize the difference between “seeing” and “proving”: Can you see clearly that the step is correct? But can you also prove that the step is correct?
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To this question the student may answer honestly “Yes” but he could be much embarrassed if the teacher, not satisfied with the intuitive conviction of the student, should go on asking: “But can you prove that this triangle is a right triangle?” Thus, the teacher should rather suppress this question unless the class has had a good initiation in solid geometry.
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By looking back at the completed solution, by reconsidering and reexamining the result and the path that led to it, they could consolidate their knowledge and develop their ability to solve problems.
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Especially, if there is some rapid and intuitive procedure to test either the result or the argument, it should not be overlooked. Can you check the result? Can you check the argument?
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Knowing this, we may conjecture that the center of gravity of a homogeneous tetrahedron coincides with the center of gravity of its four vertices.
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This conjecture is an “inference by analogy.” Knowing that the triangle and the tetrahedron are alike in many respects, we conjecture that they are alike in one more respect.
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Inference by analogy appears to be the most common kind of conclusion, and it is possibly the most essential kind. It yields more or less plausible conjectures which may or may not be confirmed by experience and stricter reasoning.
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The chemist, experimenting on animals in order to foresee the influence of his drugs on humans, draws conclusions by analogy. But so did a small boy I knew. His pet dog had to be taken to the veterinary, and he inquired: “Who is the veterinary?” “The animal doctor.” “Which animal is the animal doctor?”
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Again, the center of gravity of a homogeneous rod divides the distance between its end-points in the proportion 1 : 1. The center of gravity of a triangle divides the distance between any vertex and the midpoint of the opposite side in the proportion 2 : 1. Should we not suspect that the center of gravity of a homogeneous tetrahedron divides the distance between any vertex and the center of gravity of the opposite face in the proportion 3: 1?
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The feeling that harmonious simple order cannot be deceitful guides the discoverer both in the mathematical and in the other sciences, and is expressed by the Latin saying: simplex sigillum veri (simplicity is the seal of truth).
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An element that we introduce in the hope that it will further the solution is called an auxiliary element.
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In general, having recollected a formerly solved related problem and wishing to use it for our present one, we must often ask: Should we introduce some auxiliary element in order to make its use possible?
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Trying to use known results and going back to definitions are among the best reasons for introducing auxiliary elements;
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Trying to use known results and going back to definitions are among the best reasons for introducing auxiliary elements; but they are not the only ones. We may add auxiliary elements to the conception of our problem in order to make it fuller, more suggestive, more familiar although we scarcely know yet explicitly how we shall be able to use the elements added.
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Teachers and authors of textbooks should not forget that the intelligent student and THE INTELLIGENT READER are not satisfied by verifying that the steps of a reasoning are correct but also want to know the motive and the purpose of the various steps.
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Mathematics is interesting in so far as it occupies our reasoning and inventive powers.
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Auxiliary problem is a problem which we consider, not for its own sake, but because we hope that its consideration may help us to solve another problem, our original problem.
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An insect tries to escape through the windowpane, tries the same again and again, and does not try the next window which is open and through which it came into the room. A man is able, or at least should be able, to act more intelligently.
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To devise an auxiliary problem is an important operation of the mind. To raise a clear-cut new problem subservient to another problem, to conceive distinctly as an end what is means to another end, is a refined achievement of the intelligence.
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If our investigation of the auxiliary problem fails, the time and effort we devoted to it may be lost. Therefore, we should exercise our judgment in choosing an auxiliary problem.
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Sometimes the only advantage of the auxiliary problem is that it is new and offers unexplored possibilities; we choose it because we are tired of the original problem all approaches to which seem to be exhausted.
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Unhappily, there is no infallible method of discovering suitable auxiliary problems as there is no infallible method of discovering the solution. There are, however, questions and suggestions which are frequently helpful, as LOOK AT THE UNKNOWN. We are often led to useful auxiliary problems by VARIATION OF THE PROBLEM.
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Convertible reductions are, in a certain respect, more important and more desirable than other ways to introduce auxiliary problems, but auxiliary problems which are not equivalent to the original problem may also be very useful;