GEORGE POLYA HOW TO SOLVE IT PDF
Stanford University. G. POLYA. Mathematical Method. A New Aspect of. How To Solve It . you solve it by your own means, you may experience the tension and. Behind the desire to solve this or that problem that confers no material . 1 George Polya was born Gyorgy Polya (he dropped the accents sometime later) on. How to Solve It () is a small volume by mathematician George Pólya describing methods of problem solving. How to Solve It suggests the following steps.
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In George Polya published the book How To Solve It which quickly became his most In this book he identifies four basic principles of problem solving. George Polya-How to Solve It - Free download as PDF File .pdf) or read online for free. How to Solve It, by George Polya. About problem resolution. An overall framework for problem solving was described by G. Polya in a book called “How to Solve It!” (2nd Ed., Princeton University Press). Although Polya's.
The teacher may rely here upon the student's un- sophisticated familiarity with spatial relations. The teacher can make the problem interesting by making it concrete. The classroom is a rectangular paral- lelepiped whose dimensions could be measured, and can be estimated; the students have to find, to "measure indirectly," the diagonal of the classroom.
The dialogue between the teacher and the students may start as follows: Which letter should de- note the unknown? I mean, is the condition sufficient to determine the unknown?
If we know a, b, c, we know the parallele- piped. If the parallelepiped is determined, the diagonal is determined. Devising a plan. We have a plan when we know, or know at least in outline, which calculations, computa- tions, or constructions we have to perform in order to obtain the unknown.
The way from understanding the problem to conceiving a plan may be long and tortuous. In fact, the main achievement in the solution of a prob- lem is to conceive the idea of a plan.
This idea may emerge gradually. Or, after apparently unsuccessful trials and a period of hesitation, it may occur suddenly, in a flash, as a "bright idea.
The questions and suggestions we are going to discuss tend to provoke such an idea. In order to be able to see the student's position, the teacher should think of his own experience, of his diffi- culties and successes in solving problems. 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.
Good ideas are based on past experience and formerly acquired knowledge. 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 con- structing a house but we cannot construct a house with- out collecting the necessary materials. The materials necessary for solving a mathematical problem are certain relevant items of our formerly acquired mathematical knowledge, as formerly solved problems, or formerly proved theorems.
Thus, it is often appropriate to start the work with the question: Do you know a related problem? The difficulty is that there are usually too many prob- lems which are somewhat related to our present problem, that is, have some point in common with it. How can we choose the one, or the few, which are really useful?
There is a suggestion that puts our finger on an essential com- mon point: If we succeed in recalling a formerly solved problem which is closely related to our present problem, we are lucky. We should try to deserve such luck; we may de- serve it by exploiting it. Could you use it?
If they do not work, we must look around for some other appropriate point of contact, and explore the vari- ous aspects of our problem; we have to vary, to transform, to modify the problem.
Some of the questions of our list hint specific means to vary the problem, as generalization, specialization, use of analogy, dropping a part of the condition, and so on; the details are important but we cannot go into them now. Variation of the problem may lead to some appropriate auxiliary problem: Trying to apply various known problems or theorems, considering various modifications, experimenting with various auxiliary problems, we may stray so far from our original problem that we are in danger of losing it alto- gether.
Yet there is a good question that may bring us back to it: We return to the example considered in section 8. As we left it, the students just succeeded in understanding the problem and showed some mild inter- est in it. They could now have some ideas of their own, some initiative. If the teacher, having watched sharply, cannot detect any sign of such initiative he has to resume carefully his dialogue with the students.
He must be pre- pared to repeat with some modification the questions which the students do not answer. He must be prepared to meet often with the disconcerting silence of the students which will be indicated by dots Do you know a problem hav- ing the same unknown? Example 11 "The diagonal of a parallelepiped.
We have not had any problem yet about the diagonal of a parallelepiped. Did you never solve a problem whose un- known was the length of a line? For instance, to find a side of a right triangle. Would you like to use it? Could you introduce some auxiliary element in order to make its use possible?
Have you any triangle in your figure? Yet the teacher should be prepared for the case that even this fairly ex- plicit hint is insufficient to shake the torpor of the stu- dents; and so he should be prepared to use a whole gamut of more and more explicit hints.
Now, what will you do?
You have now a triangle; but have you the unknown? And the other, I think, is not difficult to find. Yes, the other leg is the hypotenuse of another right triangle. Now I see that you have a plan. Carrying out the plan. To devise a plan, to con- ceive 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: To carry out the plan is much easier; what we need is mainly patience.
Example ourselves that the details fit into the outline, and so we have to examine the details one after the other, patiently, till everything is perfectly clear, and no obscure corner remains in which an error could be hidden.
If the student has really conceived a plan, the teacher has now a relatively peaceful time. The main danger is that the student forgets his plan. This may easily happen if the student received his plan from outside, and ac- cepted it on the authority of the teacher; but if he worked for it himself, even with some help, and conceived the final idea with satisfaction, he will not lose this idea easily. Yet the teacher must insist that the student should check each step. We may convince ourselves of the correctness of a step in our reasoning either "intuitively" or "formally.
The difference between "insight" and "formal proof" is clear enough in many important cases; we may leave further discussion to philosophers. 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 "see- ing" and "proving": Can you see clearly that the step is correct? But can you also prove that the step is correct?
Let us resume our work at the point where we left it at the end of section The student, at last, has got the idea of the solution. He sees the right triangle of which the unknown x is the hypotenuse and the given height c is one of the legs; the other leg is the diagonal of a face.
The student must, possibly, be urged to introduce suitable notation. Thus, he may see more clearly the idea of the solution which is to introduce an auxiliary problem whose unknown is y. Finally, working at one right tri- angle after the other, he may obtain see Fig. Thus, the teacher rna y ask: Even in the latter case, there is some danger that the answer to an incidental question may become the main difficulty for the majority of the students. Looking back. Even fairly good students, when they have obtained the solution of the problem and writ- ten down neatly the argument, shut their books and look for something else.
Doing so, they miss an important and instructive phase of the work. Looking Back date their knowledge and develop their ability to solve problems. A good teacher should understand and impress on his students the view that no problem whatever is com- pletely 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.
The student has now carried through his plan. He has written down the solution, checking each step. Thus, he should have good reasons to believe that his solution is correct. Nevertheless, errors are always possible, especially if the argument is long and involved.
Hence, verifications are desirable. Especially, if there is some rapid and in- tuitive procedure to test either the result or the argument, it should not be overlooked. In order to convince ourselves of the presence or of the quality of an object, we like to see and to touch it. And as we prefer perception through two different senses, so we prefer conviction by two different proofs: Can you de- rive the result differently?
We prefer, of course, a short and intuitive argument to a long and heavy one: 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. We have a natural opportunity to investigate the connections of a problem when looking back at its solution. 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. In section 12, the students finally ob- tained the solution: The teacher cannot expect a good answer to this question from inexperienced stu- dents.
How to Solve It
The students, however, should acquire fairly early the experience that problems "in letters" have a great advantage over purely numerical problems; if the prob- lem is given "in letters" its result is accessible to several tests to which a problem "in numbers" is not susceptible at all. 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. Do all the data a, b, c appear in your formula for the diagonal?
Is the expression you obtained for the diagonal sym- metric in a, b, c? Does it remain unchanged when a, b, c are interchanged? Our problem is analogous to a problem of plane geometry: Is the result of our 'solid' problem anal- ogous to the result of the 'plane' problem?
Example parallelepiped becomes a parallelogram. Does your formula show this? If, in your formula, you substi- tute 12a, 12b, 12c for a, b, c respectively, the expression of the diagonal, owing to this substitution, should also be multiplied by Is that so?
These questions have several good effects. First, an in- telligent 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.
The formula has therefore a better chance of being re- membered, the knowledge of the student is consolidated.
Finally, these questions can be easily transferred to simi- lar problems. After some experience with similar prob- lems, an intelligent student may perceive the underlying general ideas: If he gets into the habit of directing his attention to such points, his ability to solve problems may definitely profit.
In the Classroom Can you check the argument? To recheck the argument step by step may be necessary in difficult and important cases. Usually, it is enough to pick out "touchy" points for rechecking. In our case, it may be advisable to discuss retrospectively the question which was less advisable to discuss as the solution was not yet attained: Can you prove that the triangle with sides x, y, c is a right tri- angle?
See the end of section Can you use the result or the method for some other problem? With a little encouragement, and after one or two examples, the students easily find applications which consist essentially in giving some concrete interpretation to the abstract mathematical elements of the problem. The teacher himself used such a concrete interpretation as he took the room in which the discussion takes place for the parallelepiped of the problem.
A dull student may propose, as application, to calculate the diagonal of the cafeteria instead of the diagonal of the classroom. If the students do not volunteer more imaginative remarks, the teacher himself may put a slightly different problem, for instance: Or they may use the method, introducing suitable right triangles the latter alternative is less obvious and somewhat more clumsy in the present case. After this application, the teacher may discuss the con- figuration of the four diagonals of the parallelepiped, and the six pyramids of which the six faces are the bases, the center the common vertex, and the semidiagonals the lateral edges.
Various Approaches back to his question: Now there is a better chance that the students may find some more interesting concrete interpretation, for instance, the following: To support the pole, we need four equal cables. The cables should start from the same point, 2 yards under the top of the pole, and end at the four corners of the top of the building.
How long is each cable? Various approaches. Let us still retain, for a while, the problem we considered in the foregoing sections 8, 10, 12, The main work, the discovery of the plan, was described in section Let us observe that the teacher could have proceeded differently.
Starting from the same point as in section 10, he could have followed a somewhat different line, asking the following questions: Could you think of a simpler analogous prob- lem of plane geometry?
What might be an analogous problem about a figure in the plane? It should be concerned with -the diagonal-of-a rectangular-" "Parallelogram. Besides, if the students are so slow, the teacher should not take up the present problem about the paral- lelepiped without having discussed before, in order to prepare the students, the analogous problem about the parallelogram. Then, he can go on now as follows: Can you use it?
It consists in conceiving the diagonal of the given parallelepiped as the diagonal of a suitable parallelogram which must be introduced into the figure as intersection of the parallelepiped with a plane passing through two opposite edges. The idea is essentially the same as before section 10 but the ap- proach is different. In section 10, the contact with the available knowledge of the students was established through the unknown; a formerly solved problem was recollected because its unknown was the same as that of the proposed problem.
In the present section analogy provides the contact with the idea of the solution. The teacher's method of questioning shown in the foregoing sections 8, 10, 12, 14, 15 is essentially this: Begin with a general question or suggestion of our list, and, if necessary, come down gradually to more specific and concrete questions or suggestions till you reach one which elicits a response in the student's mind. The Teacher's Method of Questioning 21 have to help the student exploit his idea, start again, if possible, from a general question or suggestion contained in the list, and return again to some more special one if necessary; and so on.
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 impor- tant, however, that the suggestions from which we start should be simple, natural, and general, and that their list should be short.
The suggestions must be simple and natural because otherwise they cannot be unobtrusive. The suggestions must be general, applicable not only to the present problem but to problems of all sorts, if they are to help develop the ability of the student and not just a special technique. The list must be short in order that the questions may be often repeated, unartificially, and under varying cir- cumstances; thus, there is a chance that they will be eventually assimilated by the student and will contribute to the development of a mental habit.
It is necessary to come down gradually to specific sug- gestions, in order that the student may have as great a share of the work as possible. This method of questioning is not a rigid one; for- tunately so, because, in these matters, any rigid, mechani- cal, 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.
If a reader wishes to try the method here proposed in his class he should, of course, proceed with caution. He should study carefully the example introduced in section 8, and the following examples in sections 18, 19, He should start with a few trials and find out gradually how he can manage the method, how the students take it, and how much time it takes.
Good questions and bad questions. If the method of questioning formulated in the foregoing section is well understood it helps to judge, by comparison, the quality of certain suggestions which may be offered with the in- tention of helping the students.
Let us go back to the situation as it presented itself at the beginning of section 10 when the question was asked: 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 of- fered; then we shall see that there is a long sequence of objections against that sort of "help.
Thus the question fails to help where help is most needed. 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. And how could he, the stu- dent, 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.
A Problem of Construction None of these objections can be raised against the pro- cedure described in section 10, or against that in sec- tion A problem of construction. Inscribe a square in a given triangle.
Two vertices of the square should be on the base of the triangle, the two other vertices of the square on the two other sides of the triangle, one on each.
I am not so sure. If you cannot solve the proposed problem, try to solve first some related problem. Could you satisfy a part of the con- dition? How many vertices are there? Keep only a part of the condition, drop the other part. What part of the condition is easy to satisfy? How far is the unknown now determined?
How can it vary? A Problem to Prove it can vary; the same is true of its fourth corner. Draw more squares with three comers on the perimeter in the same way as the two squares already in the figure. Draw small squares and large squares.
What seems to be the locus of the fourth corner?
If the student is able to guess that the locus of the fourth corner is a straight line, he has got it. A problem to prove. Two angles are in different planes but each side of one is parallel to the correspond- ing side of the other, and has also the same direction. Prove that such angles are equal. What we have to prove is a fundamental theorem of solid geometry. The problem may be proposed to stu- dents who are familiar with plane geometry and ac- quainted with those few facts of solid geometry which prepare the present theorem in Euclid's Elements.
The theorem that we have stated and are going to prove is the proposition 10 of Book XI of Euclid. Not only ques- tions and suggestions quoted from our list are printed in italics but also others which correspond to them as "problems to prove" correspond to "problems to find. Each side of one is parallel to the corresponding side of the other, and has also the same direction. Say it, please, using your nota- tion. And try to think of a familiar theorem having the same or a similar conclusion.
Now here is a theorem related to yours and proved before. A Problem to Prove triangles, about a pair of congruent triangles. Have you any triangles in your figure? But I could introduce some. Let me join B to C and B' to C'.
Then there are two triangles, 6, ABC 6. But what are these triangles good for? If you wish to prove this, what kind of tri- angles do you need? Now, what do you wish to prove? Look at the conclusion! You could have chosen a worse one. Thus, what is your aim? What is the hypothesis? Yes, of course, I must use that. Is that all that you know about these lines? They are parallel and equal to each other. And so are AC and A'C'. How many parallelograms have you now in your figure?
No, three. No, two. A Rate Problem 29 which you can prove immediately that they are paral- lelograms. There is a third which seems to be a parallelo- gram; I hope I can prove that it is one. And then the proof will be finished! 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.
Really, he did profit something from his mathematics classes; he has some real experience in solving problems, he can conceive and exploit a good idea. A rate problem. Water is flowing into a conical vessel at the rate r. The vessel has the shape of a right circular cone, with horizontal base, the vertex pointing downwards; the radius of the base is a, the altitude of the FIG.
Find the rate at which the surface is rising when the depth of the water is y. The students are supposed to know the simplest rules of differentiation and the notion of "rate of change. The statement of the problem seems to sug- gest that you should disregard, provisionally, the numeri- cal values, work with the letters, express the unknown in terms of a, b, r, y and only finally, after having obtained the expression of the unknown in letters, substitute the numerical values.
I would follow this suggestion. Now, what is the unknown? Could you say it in other terms? But what is the rate of change? Go back to the definition. Now, is y a function? As we said before, we disregard the numerical value of y. Can you imagine that y changes?
How would you write the 'rate of change of y' in mathematical symbols? Thus, this is your unknown. You have to ex- press it in terms of a, b, r, y. By the way, one of these data is a 'rate.
A Rate Problem "r is the rate at which water is flowing into the vessel. How would you write it in suitable notation? Thus, you have to express 1e in terms of a, b, dV ,y. H ow Wl"II you d o It dt. If you do not see yet the con- nection between 1e and the data, try to bring in some simpler connection that could serve as a stepping stone. For instance, are y and V independent of each other? When y increases, V must increase too. What is the connection?
Polya G. How to Solve It: A New Aspect of Mathematical Method
But I do not know yet the radius of the base. Call it something, say x. Now, what about x? Is it independent of y? When the depth of the water, y, increases the radius of the free surface, x, increases too. I would not miss profiting from it. Do not forget, you wished to know the connection between V and y. This looks like a stepping stone, does it not?
But you should not forget your goal. What to do? Of course! Here it is. And what about the numerical values? Start from the statement of the problem.
What can I do?
Visualize the problem as a whole as clearly and as vividly as you can. Do not concern your- self with details for the moment. What can I gain by doing so? You should understand the problem, familiarize yourself with it, impress its pur- pose on your mind.
The attention bestowed on the prob- lem may also stimulate your memory and prepare for the recollection of relevant points. Working for Better Understanding 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 alto- gether.
Isolate the principal parts of your problem. The hypothesis and the conclusion are the principal parts of a "problem to prove"; the unknown, the data, and the conditions are the principal parts of a "problem to find.
You should prepare and clarify details which are likely to play a role afterwards. Hunting for the Helpful Idea Where should I start? Start from the consideration of the principal parts of your problem. Start when these principal parts are distinctly arranged and clearly con- ceived, thanks to your previous work, and when your memory seems responsive.
Consider your problem from various sides and seek contacts with your formerly acquired knowledge. Consider your problem from various sides. Emphasize different parts, examine different details, examine the same details repeatedly but in different ways, combine the details differently, approach them from different sides.
Try to see some new meaning in each detail, some new interpretation of the whole. Seek contacts with your formerly acquired knowledge. Try to think of what helped you in similar situations in the past. Try to recognize something familiar in what you examine, try to perceive something useful in what you recognize. What could I perceive? A helpful idea, perhaps a de- cisive idea that shows you at a glance the way to the very end. How can an idea be helpful? It shows you the whole of the way or a part of the way; it suggests to you more or less distinctly how you can proceed.
Ideas are more or less complete. You are lucky if you have any idea at alL What can I do with an incomplete idea? You should consider it. If it looks advantageous you should consider it longer. The situa- tion has changed, thanks to your helpful idea. Consider the new situation from various sides and seek contacts with your formerly acquired knowledge. What can I gain by doing so again? You may be lucky and have another idea. Perhaps your next idea will lead you to the solution right away.
Perhaps you need a few more helpful ideas after the next. Perhaps you will be led astray by some of your ideas. Nevertheless you should be grateful for all new ideas, also for the lesser ones, also for the hazy ones, also for the supplementary ideas add- ing some precision to a hazy one, or attempting the cor- rection of a less fortunate one.
Even if you do not have any appreciable new ideas for a while you should be grateful if your conception of the problem becomes more complete or more coherent, more homogeneous or better balanced. Carrying Out the Plan Where should I start? Start from the lucky idea that led you to the solution. Start when you feel sure of your grasp of the main connection and you feel confident that you can supply the minor details that may be wanting.
Make your grasp quite secure. Carry through in detail all the algebraic or geometric opera- tions which you have recognized previously as feasible.
Convince yourself of the correctness of each step by for- mal reasoning, or by intuitive insight, or both ways if you can. If your problem is very complex you may distin- guish "great" steps and "small" steps, each great step being composed of several small ones. Check first the great steps, and get down to the smaller ones afterwards.
A presentation of the solution each step of which is correct beyond doubt. From the solution, complete and correct in each detail. Consider the solution from various sides and seek contacts with your formerly acquired knowledge. Consider the details of the solution and try to make them as simple as you can; survey more extensive parts of the solution and try to make them shorter; try to see the whole solution at a glance.
Try to modify to their advantage smaller or larger parts of the solution, try to improve the whole solution, to make it intuitive, to fit it into your formerly acquired knowledge as naturally as possible. Scrutinize the method that led you to the solution, try to see its point, and try to make use of it for other problems. Scrutinize the result and try to make use of it for other problems. You may find a new and better solution, you may discover new and interesting facts.
In any case, if you get into the habit of surveying and scrutinizing your solutions in this way, you will acquire some knowledge well ordered and ready to use, and you will develop your ability of solving problems.
Similar objects agree with each other in some respect, analogous objects agree in certain relations of their respective parts. A rectangular parallelogram is analogous to a rec- tangular parallelepiped. In fact, the relations between the sides of the parallelogram are similar to those be- tween the faces of the parallelepiped: Each side of the parallelogram is parallel to just one other side, and is perpendicular to the remaining sides.
Each face of the parallelepiped is parallel to just one other face, and is perpendicular to the remaining faces. This is where you solve the equation you came up with in your 'devise a plan' step.
The equations in this tutorial will all be linear equations.
If you need help solving them, by all means, go back to Tutorial 7: Linear Equations in One Variable and review that concept. You may be familiar with the expression 'don't look back'. In problem solving it is good to look back check and interpret..
Basically, check to see if you used all your information and that the answer makes sense. If your answer does check out, make sure that you write your final answer with the correct labeling. Do you know a theorem that could be useful? Look at the unknown! Try to think of a familiar problem having the same or a similar unknown. Here is a problem related to yours and solved before. Could you use it? Could you use its result? Could you use its method? Should you introduce some auxiliary element in order to make its use possible?
Could you restate the problem? Could you restate it still differently? Could you imagine a more accessible related problem?
A more general problem? A more special problem? An analogous problem? Could you solve a part of the problem? Keep only a part of the condition, drop the other part; how far is the unknown then determined, how can it vary?Between and , he published papers on a wide range of mathematical subjects, such as series, number theory, com- binatorics, voting systems, astronomy, and probability.
Clear-cut analogies weigh more heavily than vague similarities, systematically arranged instances count for more than random collec- tions of cases.
Polya held a professorship at Stanford from until his re- tirement in , and it was there, in , that he taught his last course, in combinatorics; he died on September 7, 5, at the age of ninety-seven. I mean, is the condition sufficient to determine the unknown? Find the area S of the lateral surface of the frustum of a right circular cone, being given the radius of the lower baseR, the radius of the upper baser, and the altitude h.