It is often said that to teach any subject well, one has to understand it “at least as well as one’s students do.” It is a paradoxical truth that to teach mathematics well, one must also know how to misunderstand it at least to the extent one’s students do!
preceding list of questions and suggestions entitled “How to Solve It.” Any question or suggestion quoted from it will be printed in italics, and the whole list will be referred to simply as “the list” or as “our list.”
The following pages will discuss the purpose of the list, illustrate its practical use by examples, and explain the underlying notions and mental operations.
The title of the very short second part is “How to Solve It.” It is written in dialogue; a somewhat idealized teacher answers short questions of a somewhat idealized student.
The third and most extensive part is a “Short Dictionary of Heuristic”; we shall refer to it as the “Dictionary.” It contains sixty-seven articles arranged alphabetically. For example, the meaning of the term HEURISTIC (set in small capitals) is explained in an article with this title on page 112. When the title of such an article is referred to within the text it will be set in small capitals. Certain paragraphs of a few articles are more technical; they are enclosed in square brackets.
The Dictionary should not be read too quickly; its text is often condensed, and now and then somewhat subtle.
The author looks at the situation sometimes from the point of view of the student and sometimes from that of the teacher (the latter case is preponderant in the first part). Yet most of the time (especially in the third part) the point of view is that of a person who is neither teacher nor student but anxious to solve the problem before him.
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.
naturally, the teacher is led to ask the same questions and to indicate the same steps again and again. Thus, in countless problems, we have to ask the question: What is the unknown? We may vary the words, and ask the same thing in many different ways: What is required? What do you want to find? What are you supposed to seek? The aim of these questions is to focus the student’s attention upon the unknown.
If the reader is sufficiently acquainted with the list and can see, behind the suggestion, the action suggested, he may realize that the list enumerates, indirectly, mental operations typically useful for the solution of problems. These operations are listed in the order in which they are most likely to occur.
Experience shows that the questions and suggestions of our list, appropriately used, very frequently help the student. They have two common characteristics, common sense and generality. As they proceed from plain common sense they very often come naturally; they could have occurred to the student himself. As they are general, they help unobtrusively; they just indicate a general direction and leave plenty for the student to do.
The student may absorb a few questions of our list so well that he is finally able to put to himself the right question in the right moment and to perform the corresponding mental operation naturally and vigorously.
Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. 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.
when the teacher solves a problem before the class, he should dramatize his ideas a little and he should put to himself the same questions which he uses when helping the students. Thanks to such guidance, the student will eventually discover the right use of these questions and suggestions, and doing so he will acquire something that is more important than the knowledge of any particular mathematical fact.
Trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem. We have to shift our position again and again. Our conception of the problem is likely to be rather incomplete when we start the work; our outlook is different when we have made some progress; it is again different when we have almost obtained the solution.
First, we have to understand the problem; we have to see clearly what is required.
Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan.
Third, we carry out our plan.
Fourth, we look back at the completed solution, we review and discuss it.
The worst may happen if the student embarks upon computations or constructions without having understood the problem. 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. Some of the best effects may be lost if the student fails to reexamine and to reconsider the completed solution.
It is foolish to answer a question that you do not understand.
Such foolish and sad things often happen, in and out of school, but the teacher should try to prevent them from happening in his class. The student should understand the problem. But he should not only understand it, he should also desire its solution.
The best that the teacher can do for the student is to procure for him, by unobtrusive help, a bright idea. The questions and suggestions we are going to discuss tend to provoke such an idea.
the details are important but we cannot go into them now.
Let us hope that the last hint was explicit enough to provoke the idea of the solution which is to introduce a right triangle, (emphasized in Fig. 1) of which the required diagonal is the hypotenuse. Yet the teacher should be prepared for the case that even this fairly explicit hint is insufficient to shake the torpor of the students;
To devise a plan, to conceive the idea of the solution is not easy. It takes so much to succeed; formerly acquired knowledge, good mental habits, concentration upon the purpose, and one more thing: good luck. To carry out the plan is much easier; what we need is mainly patience.
We may convince ourselves of the correctness of a step in our reasoning either “intuitively” or “formally.” We may concentrate upon the point in question till we see it so clearly and distinctly that we have no doubt that the step is correct; or we may derive the point in question according to formal rules.
The main point is that the student should be honestly convinced of the correctness of each step. 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?
Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. 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.
no problem whatever is completely exhausted. There remains always something to do; with sufficient study and penetration, we could improve any solution, and, in any case, we can always improve our understanding of the solution.
as we prefer perception through two different senses, so we prefer conviction by two different proofs: Can you derive the result differently? We prefer, of course, a short and intuitive argument to a long and heavy one: Can you see it at a glance?
One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else.
The students will find looking back at the solution really interesting if they have made an honest effort, and have the consciousness of having done well. Then they are eager to see what else they could accomplish with that effort, and how they could do equally well another time.
Can you use the result, or the method, for some other problem?
Our example, although fairly simple, is sufficient to show this. The teacher can ask several questions about the result which the students may readily answer with “Yes”; but an answer “No” would show a serious flaw in the result.
“Did you use all the data? Do all the data a, b, c appear in your formula for the diagonal?”
“Length, width, and height play the same role in our question; our problem is symmetric with respect to a, b, c. Is the expression you obtained for the diagonal symmetric in a, b, c? Does it remain unchanged when a, b, c are interchanged?”
“If the height c decreases, and finally vanishes, the parallelepiped becomes a parallelogram. If you put c = 0 in your formula, do you obtain the correct formula for the diagonal of the rectangular parallelogram?”
These questions have several good effects. First, an intelligent student cannot help being impressed by the fact that the formula passes so many tests. He was convinced before that the formula is correct because he derived it carefully. But now he is more convinced, and his gain in confidence comes from a different source; it is due to a sort of “experimental evidence.” Then, thanks to the foregoing questions, the details of the formula acquire new significance, and are linked up with various facts.
Of course, our list is just a first list of this kind; it seems to be sufficient for the majority of simple cases, but there is no doubt that it could be perfected. It is important, however, that the suggestions from which we start should be simple, natural, and general, and that their list should be short.
This method of questioning is not a rigid one; fortunately so, because, in these matters, any rigid, mechanical, pedantical procedure is necessarily bad.
Our method admits a certain elasticity and variation, it admits various approaches (section 15), it can be and should be so applied that questions asked by the teacher could have occurred to the student himself.
Instead of this, with the best intention to help the students, the question may be offered: Could you apply the theorem of Pythagoras? The intention may be the best, but the question is about the worst. We must realize in what situation it was offered; then we shall see that there is a long sequence of objections against that sort of “help.”
The suggestion is of too special a nature. Even if the student can make use of it in solving the present problem, nothing is learned for future problems. The question is not instructive.
Even if he understands the suggestion, the student can scarcely understand how the teacher came to the idea of putting such a question. And how could he, the student, find such a question by himself? It appears as an unnatural surprise, as a rabbit pulled out of a hat; it is really not instructive.
“The idea is not so bad. How many parallelograms have you now in your figure?” “Two. No, three. No, two. I mean, there are two of which you can prove immediately that they are parallelograms. There is a third which seems to be a parallelogram; I hope I can prove that it is one. And then the proof will be finished!” We could have gathered from his foregoing answers that the student is intelligent. But after this last remark of his, there is no doubt. This student is able to guess a mathematical result and to distinguish clearly between proof and guess. He knows also that guesses can be more or less plausible.
Where should I start? Start again from the statement of the problem. Start when this statement is so clear to you and so well impressed on your mind that you may lose sight of it for a while without fear of losing it altogether.
Analogy is a sort of similarity. Similar objects agree with each other in some respect, analogous objects agree in certain relations of their respective parts.
It is desirable to foresee the result, or, at least, some features of the result, with some degree of plausibility. Such plausible forecasts are often based on analogy.
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. It would be foolish to regard the plausibility of such conjectures as certainty, but it would be just as foolish, or even more foolish, to disregard such plausible conjectures.
An analogical conclusion from many parallel cases is stronger than one from fewer cases. Yet quality is still more important here than quantity. Clear-cut analogies weigh more heavily than vague similarities, systematically arranged instances count for more than random collections of cases.
It appears extremely unlikely that the conjectures suggested by these questions should be wrong, that such a beautiful regularity should be spoiled. 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).
There is a one-one correspondence between the objects of the two systems S and S′, preserving certain relations. That is, if such a relation holds between the objects of one system, the same relation holds between the corresponding objects of the other system. Such a connection between two systems is a very precise sort of analogy; it is called isomorphism
There is a one-many correspondence between the objects of the two systems S and S′ preserving certain relations. Such a connection (which is important in various branches of advanced mathematical study, especially in the Theory of Groups, and need not be discussed here in detail) is called merohedral isomorphism (or homomorphism
We are glad when we have succeeded in recollecting a problem related to ours and solved before. It is probable that we can use such a problem but we do not know yet how to use it.
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. We may just feel that it is a “bright idea” to conceive the problem that way with such and such elements added.
We may have this or that reason for introducing an auxiliary element, but we should have some reason. We should not introduce auxiliary elements wantonly. 3. Example. Construct a triangle, being given one angle, the altitude drawn from the vertex of the given angle, and the perimeter of the triangle.
We may have this or that reason for introducing an auxiliary element, but we should have some reason. We should not introduce auxiliary elements wantonly.
Therefore we must introduce p. But how? We may attempt to introduce p in various ways. The attempts exhibited in Figs. 9, 10 appear clumsy. If we try to make clear to ourselves why they appear so unsatisfactory, we may perceive that it is for lack of symmetry.
Now, the sides b and c play the same role; they are interchangeable; our problem is symmetric with respect to b and c. But b and c do not play the same role in our figures 9, 10; placing the length p we treated b and c differently; the figures 9 and 10 spoil the natural symmetry of the problem with respect to b and c. We should place p so that it has the same relation to b as to
Now, the sides b and c play the same role; they are interchangeable; our problem is symmetric with respect to b and c. But b and c do not play the same role in our figures 9, 10; placing the length p we treated b and c differently; the figures 9 and 10 spoil the natural symmetry of the problem with respect to b and c. We should place p so that it has the same relation to b as to c.
If we have some little experience in solving problems of construction, we shall not fail to introduce into the figure, along with ED, the auxiliary lines AD and AE, each of which is the base of an isosceles triangle. In fact, it is not unreasonable to introduce elements into the problem which are particularly simple and familiar, as isosceles triangle.
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. The introduction of an auxiliary element is a conspicuous step. If a tricky auxiliary line appears abruptly in the figure, without any motivation, and solves the problem surprisingly, intelligent students and readers are disappointed; they feel that they are cheated.
Mathematics is interesting in so far as it occupies our reasoning and inventive powers. But there is nothing to learn about reasoning and invention if the motive and purpose of the most conspicuous step remain incomprehensible.
Human superiority consists in going around an obstacle that cannot be overcome directly, in devising a suitable auxiliary problem when the original problem appears insoluble.
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.
The profit that we derive from the consideration of an auxiliary problem may be of various kinds. We may use the result of the auxiliary problem.
In other cases, we may use the method of the auxiliary problem.
Risk. We take away from the original problem the time and the effort that we devote to the auxiliary problem. If our investigation of the auxiliary problem fails, the time and effort we devoted to it may be lost.
we should exercise our judgment in choosing an auxiliary problem. We may have various good reasons for our choice. The auxiliary problem may appear more accessible than the original problem; or it may appear instructive; or it may have some sort of aesthetic appeal.
Two problems are equivalent if the solution of each involves the solution of the other.
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;
In the foregoing we derived from an original condition (A) a sequence of conditions (B), (C), (D), . . . each of which was equivalent to the foregoing. This point deserves the greatest care. Equivalent conditions are satisfied by the same objects. Therefore, if we pass from a proposed condition to a new condition equivalent to it, we have the same solutions. But if we pass from a proposed condition to a narrower one, we lose solutions, and if we pass to a wider one we admit improper, adventitious solutions which have nothing to do with the proposed problem.
If we could solve A we could hence derive the full solution of B. But not conversely; if we could solve B, we would obtain, possibly, some information about A, but we would not know how to derive the full solution of A from that of B. In such a case, more is achieved by the solution of A than by the solution of B. Let us call A the more ambitious, and B the less ambitious of the two problems.
Bright idea, or “good idea,” or “seeing the light,” is a colloquial expression describing a sudden advance toward the solution;
every teacher knows that students achieve incredible things in this respect. Some students are not disturbed at all when they find 16,130 ft. for the length of the boat and 8 years, 2 months for the age of the captain who is, by the way, known to be a grandfather. Such neglect of the obvious does not show necessarily stupidity but rather indifference toward artificial problems.
Can you check the argument? Checking the argument step by step, we should avoid mere repetition. First, mere repetition is apt to become boring, uninstructive, a strain on the attention. Second, where we stumbled once, there we are likely to stumble again if the circumstances are the same as before. If we feel that it is necessary to go again through the whole argument step by step, we should at least change the order of the steps, or their grouping, to introduce some variation.
It requires less exertion and is more interesting to pick out the weakest point of the argument and examine it first.
It is certain that your knowledge, or my knowledge, or your students’ knowledge in mathematics is not based on formal proofs alone. If there is any solid knowledge at all, it has a broad experimental basis, and this basis is broadened by each problem whose result is successfully tested.
When the solution that we have finally obtained is long and involved, we naturally suspect that there is some clearer and less roundabout solution: Can you derive the result differently? Can you see it at a glance?
To find the solution of a problem by our own means is a discovery. If the problem is not difficult, the discovery is not so momentous, but it is a discovery nevertheless.
Having made some discovery, however modest, we should not fail to inquire whether there is something more behind it, we should not miss the possibilities opened up by the new result, we should try to use again the procedure used.
To find a new problem which is both interesting and accessible, is not so easy; we need experience, taste, and good luck. Yet we should not fail to look around for more good problems when we have succeeded in solving one. Good problems and mushrooms of certain kinds have something in common; they grow in clusters. Having found one, you should look around; there is a good chance that there are some more quite near.
All these problems are interesting but only the one obtained by specialization can be solved immediately on the basis of the solution of the original problem.
To conceive a plan and to carry it through are two different things. This is true also of mathematical problems in a certain sense; between carrying out the plan of the solution, and conceiving it, there are certain differences in the character of the work.
We may use provisional and merely plausible arguments when devising the final and rigorous argument as we use scaffolding to support a bridge during construction. When, however, the work is sufficiently advanced we take off the scaffolding, and the bridge should be able to stand by itself.
Devising the plan of the solution, we should not be too afraid of merely plausible, heuristic reasoning. Anything is right that leads to the right idea. But we have to change this standpoint when we start carrying out the plan and then we should accept only conclusive, strict arguments.
The more painstakingly we check our steps when carrying out the plan, the more freely we may use heuristic reasoning when devising it.
We should not omit any detail, we should understand the relation of the detail before us to the whole problem, we should not lose sight of the connection of the major steps. Therefore, we should proceed in proper order. In particular, it is not reasonable to check minor details before we have good reasons to believe that the major steps of the argument are sound. If there is a break in the main line of the argument, checking this or that secondary detail would be useless anyhow.
The order in which we work out the details of the argument may be very different from the order in which we invented them; and the order in which we write down the details in a definitive exposition may be still different.
The Euclidean way of exposition, however, cannot be recommended without reservation if the purpose is to convey an argument to a reader or to a listener who never heard of it before.
THE INTELLIGENT READER can easily see that each step is correct but has great difficulty in perceiving the source, the purpose, the connection of the whole argument. The reason for this difficulty is that the Euclidean exposition fairly often proceeds in an order exactly opposite to the natural order of invention.
Carrying out our plan, we check each step. Checking our step, we may rely on intuitive insight or on formal rules. Sometimes the intuition is ahead, sometimes the formal reasoning. It is an interesting and useful exercise to do it both ways. Can you see clearly that the step is correct? Yes, I can see it clearly and distinctly. Intuition is ahead; but could formal reasoning overtake it? Can you also PROVE that it is correct?
Trying to prove formally what is seen intuitively and to see intuitively what is proved formally is an invigorating mental exercise.
If you go into detail you may lose yourself in details. Too many or too minute particulars are a burden on the mind. They may prevent you from giving sufficient attention to the main point, or even from seeing the main point at all.
In almost all cases it is advisable to start the detailed examination of the problem with the questions: What is the unknown? What are the data? What is the condition?
Primitive peoples believe that words and symbols have magic power. We may understand such belief but we should not share it. We should know that the power of a word does not reside in its sound, in the “vocis flatus,” in the “hot air” produced by the speaker, but in the ideas of which the word reminds us and, ultimately, in the facts on which the ideas are based.
Determination, hope, success. It would be a mistake to think that solving problems is a purely “intellectual affair”; determination and emotions play an important role. Lukewarm determination and sleepy consent to do a little something may be enough for a routine problem in the classroom. But, to solve a serious scientific problem, will power is needed that can outlast years of toil and bitter disappointments.
“You can undertake without hope and persevere without success.”
When a student makes really silly blunders or is exasperatingly slow, the trouble is almost always the same; he has no desire at all to solve the problem, even no desire to understand it properly, and so he has not understood it. Therefore, a teacher wishing seriously to help the student should, first of all, stir up his curiosity, give him some desire to solve the problem. The teacher should also allow some time to the student to make up his mind, to settle down to his task.
Teaching to solve problems is education of the will. Solving problems which are not too easy for him, the student learns to persevere through unsuccess, to appreciate small advances, to wait for the essential idea, to concentrate with all his might when it appears.
Failure to check the result at all is very frequent; the student is glad to get an answer, throws down his pencil, and is not shocked by the most unlikely results.
Although it is forbidden to use the same symbol for different objects it is not forbidden to use different symbols for the same object.
The procedure that we have just applied has a certain interest; solving problems of geometric construction, we can often follow successfully its pattern: Reduce the problem to the construction of a point, and construct the point as an intersection of two loci. But a certain step of this procedure has a still more general interest; solving “problems to find” of any kind, we can follow its pattern: Keep only a part of the condition, drop the other part. Doing so, we weaken the condition of the proposed problem, we restrict less the unknown. How far is the unknown then determined, how can it vary? By asking this, we set, in fact, a new problem.
are accepted as primitive terms and are not defined. Others are
The mathematician is not concerned with the current meaning of his technical terms, at least not primarily concerned with that. What “circle” or “parabola” or other technical terms of this kind may or may not denote in ordinary speech matters little to him. The mathematical definition creates the mathematical meaning.
A notation expressing more than another may be termed more pregnant. The modern notation for similitude of triangles is more pregnant than the older one, reflects the order and connection of things more fully than the older one, and therefore, it may serve as basis for more consequences than the older one.
There is always something arbitrary and artificial about notation; to learn a new notation is a burden for the memory. The intelligent student refuses to assume the burden if he does not see any compensation for it. The intelligent student is justified in his aversion for algebra if he is not given ample opportunity to convince himself by his own experience that the language of mathematical symbols assists the mind.
To solve such a problem, we must know, at least, the definition but it is better to know some theorems too.
We do not foresee such things with certainty, only with a certain degree of plausibility.
The first rule of discovery is to have brains and good luck. The second
The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.
The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach.
it should not be forgotten that a teacher of mathematics should know some mathematics, and that a teacher wishing to impart the right attitude of mind toward problems to his students should have acquired that attitude himself.
In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression.
I have a chess problem. I have to mate the black king in, say, two moves. On the chessboard there is a white knight, quite a distance from the black king, that is apparently superfluous. What is it good for? I am obliged to leave this question unanswered at first. Yet after various trials, I hit upon a new move and observe that it would bring that apparently superfluous white knight into play. This observation gives me a new hope. I regard it as a favorable sign: that new move has some chance to be the right one. Why? In a well-constructed chess problem there is no superfluous piece. Therefore, we have to take into account all chessmen on the board; we have to use all the data.
to solve a problem is, essentially, to find the connection between the data and the unknown.
Moreover we should, at least in well-stated problems, use all the data, connect each of them with the unknown. Thus, bringing one more datum into play is quite properly felt as progress, as a step forward.
When we work intently, we feel keenly the pace of our progress: when it is rapid we are elated; when it is slow we are depressed. We feel such differences quite clearly without being able to point out any distinct sign. Moods, feelings, general aspects of the situation serve to indicate our progress.
I have a plan. I see pretty clearly where I should begin and which steps I should take first. Yet I do not quite see the lay-out of the road farther on; I am not quite certain that my plan will work; and, in any case, I have still a long way to go. Therefore, I start out cautiously in the direction indicated by my plan and keep a lookout for signs of progress. If the signs are rare or indistinct, I become more hesitant. And if for a long time they fail to appear altogether, I may lose courage, turn back, and try another road. On the other hand, if the signs become more frequent as I proceed, if they multiply, my hesitation fades, my spirits rise, and I move with increasing confidence, just as Columbus and his companions did before sighting the island of San Salvador.
The main advantage of the exceptionally talented may consist in a sort of extraordinary mental sensibility. With exquisite sensibility, he feels subtle signs of progress or notices their absence where the less talented are unable to perceive a difference.
Past ages regarded a sudden good idea as an inspiration, a gift of the gods. You must deserve such a gift by work, or at least by a fervent wish.
The future mathematician learns, as does everybody else, by imitation and practice. He should look out for the right model to imitate. He should observe a stimulating teacher. He should compete with a capable friend. Then, what may be the most important, he should read not only current textbooks but good authors till he finds one whose ways he is naturally inclined to imitate. He should enjoy and seek what seems to him simple or instructive or beautiful. He should solve problems, choose the problems which are in his line, meditate upon their solution, and invent new problems.
The traditional mathematics professor of the popular legend is absentminded. He usually appears in public with a lost umbrella in each hand. He prefers to face the blackboard and to turn his back on the class. He writes a, he says b, he means c; but it should be d.