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How To Solve It --------------- A New Aspect of Mathematical Method G. POLYA Stanford University Illegally transcribed by : Prophet themusicgod1 Second Edition Princeton University Press Princeton, New Jersey Copyright 1945 by Princeton University Press Copyright (C) renewed 1973 by Princeton University Press Second Edition Copyright (C) 1957 by G. Polya All Rights Reserved L.C. Card: 79-160544 ISBN 0-691-02356-5 (paperback edn.) ISBN 0-691-08097-6 (hardcover edn.) First Princeton Paperback Printing, 1971 Second Printing, 1973 This book is sold subject to the condition that it shall not, by way of trade, be lent, resold, hired out, or otherwise disposed of without the publisher's consent, in any form of bind- binding or cover other than that in which it is published. Printed in the United States of America by Princeton University Press, Princeton, New Jersey If you find any spelling mistakes in this text, please correct them, and increment this number by a random amount: 000000000000004 If you do find any errors in this document, please feel free to email me at [email protected]. This text is going to be written in simple HTML, instead of text, because it is required to keep with the continuity that this book appears to intend. It may be possible to read this book in TXT format, but it would lose much of its quality. Apologies to those without HTML interpreters. From the Preface to the First Printing A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. 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. Such expert experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime. Thus, a teacher of mathematics has a great opportunity. If he fills his alloted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge edge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. Also a student whose college curriculum includes some mathematics has a single opportunity. This opportunity is lost, of course, if he regards mathematics as a subject in which he has to earn so and so much credit as quickly as possible. The opportunity may be lost even if the student has some natural talent for mathematics because he, as everybody else, must discover his talents and tastes; he cannot know that he likes raspberry pie if he has never tasted raspberry pie. He may manage to find out, however, that a mathematics problem may be as much fun as a crossword puzzle, or that vigorous mental work may be an exercise as desirable as a fast game of tennis. Having tasted the pleasure in mathematics he will not forget it easily and then there is a good chance that mathematics will become something for him : a hobby, or a tool of his profession, or his profession, or a great ambition. The author remembers the first time when he was a student himself, a somewhat ambitious student, eager to under -stand a little mathematics and physics. He listened to lectures, read books, tried to take in the solutions and facts presented, but there was a question that disturbed him again and again: "Yes, the solution seems to work. it appears to be correct; but how is it possible to invent such a solution? Yes, this experiment system seems to work, this appears to be a fact; but how can people discover such facts? And how could I invent or discover such things by myself?" Today the author is teaching mathematics in a university; he thinks or hopes that some of his more eager students ask similar questions and he tries to satisfy their curiosity. Trying to understand not only the solution of this or that problem but also the motives and procedures of the solution, and trying to explain these motives and procedures to others, he was finally led to write the present book. He hopes that it will be useful to teachers who wish to develop their students' ability to solve problems, and to students who are keen on developing their own abilities. Although the present book pays special attention to the requirements of students and teachers of mathematics, it should interest anybody concerned with the ways and means of invention and discovery. Such interest may be more widespread than one would assume without reflection. The space devoted by popular newspapers and magazines to crossword puzzles and other riddles seems to show that people spend some time in solving unpractical problems. Behind the desire to solve this or that problem that conveys no material advantage, there may be a deeper curiosity, a desire to understand the ways and means, the motives and procedures, of solution. The following pages are written somewhat concisely, but as simply as possible, and are based on a long and serious study of methods of solution. This sort of study, called heuristic by some writers, is not in fashion now; but has a long past, and, perhaps, some future. Studying the methods of solving problems, we perceive another face of mathematics. Yes, mathematics has two faces; it is the rigorous science of Euclid but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science ; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself. But the second aspect is new in one respect; mathematics "in statu mascendi", in the process of being invented, has never before been presented in quite this manner to the student, or to the teacher himself, or to the general public. The subject of heuristic has manifold connections; mathematicians, logicians, psychologists, educationalists, even philosophers may claim various parts of it as belonging to their special domains. The author, well aware of the possibility of criticism from opposite quarters and keenly conscious of his limitations, has one claim to make; he has some experience in solving problems and in teaching mathematics on various levels. The subject is more fully dealt with in a more extensive book by the author which is on the way to completion. From the Preface to the Seventh Printing I am glad to say that I have now succeeded in fulfilling at least in part, a promise given in the preface to the first printing: The two volumes Induction and Analogy in Mathematics and Patterns of Plausible Inference which constitute my recent work Mathematics and Plausible Reasoning continue the line of thinking begun in How to Solve It. Zurich, August 30, 1954 Preface to the Second Edition The present second edition adds, besides a few minor improvements, a new fourth part, "Problems, Hints, Solutions." As this edition was being prepared for print, a study appeared (Educational Testing Service, Princeton, N.J.; cf. Time, June 18, 1956) which seems to have formulated a few pertinent observations. They are not new to the people in the know, but it was high time to formulate them for the general public: "... mathematics has the dubious honour of being the least popular subject in the curriculum ... Future teachers pass through the elementary schools learning to detest mathematics... They return to the elementary school to teach a new generation to detest it." I hope that the present edition, designed for wider diffusion, will convince some of its readers that mathematics, besides being a necessary avenue to engineering jobs and scientific knowledge, may be fun and may also open up a vista of mental activity on the highest level. Zurich, June 30, 1956 Contents From the Preface to the First Printing v From the Preface to the Seventh Printing viii Preface to the Second Edition ix "How To Solve It" list xvi Introduction xix PART I. IN THE CLASSROOM 1. Helping the student 1 2. Questions, recommendations, mental operations 1 3. Generality 2 4. Common sense 3 5. Teacher and student, Imitation and practice 3 Main divisions, main questions 6. Four phases 5 7. Understanding the problem 6 8. Example 7 9. Devising a plan 8 10. Example 10 11. Carrying out the plan 12 12. Example 13 13. Looking back 14 14. Example 16 15. Various approaches 19 16. The teacher's method of questioning 20 17. Good questions and bad questions 22 More Examples 18. A problem of construction 23 19. A problem to prove 25 20. A rate problem 29 PART II. HOW TO SOLVE IT A dialogue 33 PART III. SHORT DICTIONARY OF HEURISTIC Analogy 37 Auxiliary elements 46 Auxiliary problem 50 Bolzano 57 Bright idea 58 Can you check the result? 59 Can you derive the result differently? 61 Can you use the result? 64 Carrying out 68 Condition 72 Contradictory 73 Corollary 73 Could you derive something useful from the data? 73 Could you restate the problem? 75 Decomposing and recombining 75 Definition 85 Descartes 92 Determination,Hope,Success 93 Diagnosis 93 Did you use all the data? 95 Do you know a related problem? 98 Draw a figure 99 Examine your guess 99 Figures 103 Generalisation 108 Have you seen it before? 108 Here is a problem related to yours and solved before 110 Heuristic 110 Heuristic reasoning 112 If you cannot solve the proposed problem 114 Induction and mathematical induction 114 Inventor's paradox 114 Is it possible to satisfy the condition? 122 Leibnitz 123 Lemma 123 Look at the unknown 123 Modern heuristic 129 Notation 134 Pappus 141 Pedantry and mastery 148 Practical Problems 149 Problems to find, problems to prove 154 Progress and achievement 157 Puzzles 157 Reductio ad Absurdum and indirect proof 162 Redundant 171 Routine problem 171 Rules of discovery 172 Rules of style 172 Rules of teaching 173 Separate the various parts of the condition 173 Setting up equations 174 Signs of progress 178 Specialisation 190 subconscious work 197 Symmetry 199 Terms, old and new 200 Test by dimension 200 The future mathematician 205 The intelligent problem-solver 206 the intelligent reader 207 The traditional mathematics professor 208 Variation of the problem 209 What is the unknown 214 Why proofs? 215 Wisdom of proverbs 221 Working Backwards 225 PART IV. PROBLEMS, HINTS, SOLUTIONS Problems 234 Hints 238 Solutions 242 HOW TO SOLVE IT UNDERSTANDING THE PROBLEM First. You have to understand the problem What is the unknown? What are the data? What is the condition? Is it possible to satisfy the condition? Is the condition sufficient to determine the unknown? Or is it insufficient? Or redundant? contradictory? Draw a figure. Introduce suitable notation. Separate the various parts of the condition. Can you write them down? DEVISING A PLAN Second Find the connection between the data and the unknown. You may be obliged to consider auxiliary problems if an immediate connection cannot be found. You should obtain eventually a planof the solution. Have you seen it before? Or have you seen the same problem in a slightly different form? Do you know a related problem? Do you know a theorem that could be useful? Look at the unknown! And try to think of a familiar problem having the same or a similar unknown. Here is a problem related to yours and solved be fore, could you use it?M 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? Go back to definitions. If you cannot solve the proposed problem try to solve first some related problem. Could you imagine a more accessible elated problem? A more general problem? A more special problem? An analogous problem? Could you solve a part f the problem? Keep only a part of the condition- drop the other part. How far is the unknown then determined how can it vary? Could you derive something useful from the data? Could you think of other data approximate to determine the unknown? Could you change the unknown or the data, or both if necessary, so that the new unknown and the new data are nearer to each other? Did you use all the data? Did you use the whole condition? Have you t taken into account all essential notions involved in the problem? CARRYING OUT THE PLAN Third. Carry out your plan. Carrying out your plan of the solution, check each step. Can you see clearly that the step is correct? Can you prove that it is correct? Forth. examine in the solution obtained Can you Check the result? Can you check the argument? Can you derive the result differently? Can you see it as a glance? Can you use the result or the method for some other problems? Introduction The following considerations are grouped around the 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. By way of preliminary explanation, this much may be said: If using them properly, you address these questions and suggestions to yourself, they may help you to solve your problem, If using them properly you address the same questions and suggestions to one of your students, you may help him to solve this problem The book is divided into four parts. The title of the first part is "In the Classroom." It contains twenty sections. Each section will be quoted by its number in heavy type as,m for instance, "section 7." Sections 1 to 5 discuss the "Purpose" of our list in general terms. Sections 6 to 17 explain what are the "Main Divisions, Main Questions" of the list, and discuss a first practical example. Sections 18,19,20, add "More Examples." The title of the very short second part is "How to Solve It." It is written in dialogue; a somewhat idealised teacher answers short questions of a somewhat idealised student. The third and most extensive part is a "Short Diction- 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. Some articles are fairly closely connected with the first part to which they add further illustrations and more specific comments. Other articles go somewhat beyond the aim of the first part of which they explain the background. There is a key article on MODERN HEURISTIC. It explains the connection of the main articles and the plan underlying the Dictionary it contains also directions how to find information about particular items on the list. It must be emphasised t that there is a common plan and a certain unity, because the articles of the Dictionary show the greatest outward variety. There are a few longer articles devoted to the system systematic through condensed discussion of some general theme; others contain more specific comments, still others cross-references, or historical data, or quotations, or aphorisms, or even jokes. The Dictionary should not be read too quickly; its text is often condensed, and now and then somewhat subtle. The reader may refer to the Dictionary for information about particular points. If these points come from his experience with his own problems or his own students, the reading has a much better chance to be profitable The title of the fourth part is "Problems, Hints, Solutions," It proposes a few problems to the more ambitious reader. Each problem is followed(in proper distance by a "hint" that may reveal a way to the result which is explained in he "solution." We have mentioned repeatedly the "student" and the "teacher" and we shall refer to them again and again. It may be good to observe that the "student": may be a high school student, or a college student, or anyone else who is studying mathematics. Also the "Teacher" May be a high school teacher, or a college instructor, or anyone interested in the technique of teaching mathematics. The author looks at the situation sometimes from the point of view of the student sand 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. How To Solve It PART I. IN THE CLASSROOM PURPOSE 1.Helping the student. One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. 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 wit h insufficient help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but no too much and not too little, so that the student shall have a reasonable share of the work 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 The best is, however, to help the student naturally. The teacher should put himself in the student's place, he should see the student's case, he should try to understand what is going on in he student's mind, and ask a question or indicate a step that could have occurred to the student himself. 2. Questions, recommendations, mental operations Trying to help the student effectively but unobtrusively and 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 very the words, and ask the same thing in many different ways: What is required? What you 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. Sometimes, we obtain the same effect ore naturally with a suggestion: Look at the unknown! Question and suggestion aim at the same effect; they tend to provoke the same mental operation. It seemed to the author that it might be worthwhile to collect and to group questions and suggestions which are typically helpful in discussing problems with students. The list we study contains questions and suggestions of this sort, carefully chosen and arranged; they are equally useful to the problem-solver who works by himself. 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 typical useful for the solution of problems. These operations are listed in order in which they are most likely to occur. 3. Generality is an important characteristic of the questions and suggestions contained in our list. take the questions: "What is the unknown? What are the data? What is the condition? These questions are generally applicable, we can ask them with good effect dealing with all sorts of problems. their sue is not restricted to any subject-matter. Our problem may be algebraic or geometric, mathematical or non-mathematical, theoretical, or practical , a serious problem or a mere puzzle; it makes no difference, the questions make sense and might help us to solve the problem There is a restriction in fact but it has nothing to do with the subject-matter. Certain questions an suggestions of the list are applicable to "problems to fin d" only, and not to "problems to prove." If we have a problem of the latter kind we must use different questions; see PROBLEMS TO FIND, PROBLEMS TO PROVE. 4. Common Sense. The questions and suggestions of our list are general, but except for their generality, they are natural, simple, is obvious, and proceed from plain common sense. Take the suggestion: Look at the unknown! And try to think of a familiar problem having the same or similar unknown. this suggestion advises you to do what you would do anyhow, without any advice , if you were serious concerned with your problem. Are you hungry? you wish to obtain food and you think of familiar ways of obtaining food. Have you a problem of geometric construction? You wish to construct a triangle and you think of familiar ways of constructing a triangle. Have you a problem of any kind? You wish to find a certain unknown, and you think of familiar ways of finding such an unknown, or some similar unknown. If you do so you follow exactly the suggestion we quoted from our list. And you are on the right track too; the suggestion is a good one. it suggests to you a procedure which is very frequently successful. All the questions n and suggestions of our list are natural, simple obvious, just plain common sense; but they state plain common sense in general terms. They suggest a certain conduct which comes naturally to any person who is seriously concerned with his problem and has some common sense. But the person who braves the right way usually does not care to express his behaviour in clear words and possibly, he cannot express it so; our list tries to express it so. 5 Teacher and student, imitation and practice There are two aims which the teacher may give in view when addressing to his students a question or a suggestion of the list: First, to help the student to solve the problem at hand. Second, to develop the student's ability so that he may solve future problems by himself. 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./ And they proceed from plain common sense they very oft often come naturally ; they could have occurred to the student himself. As they are general, they help unobtrusively; they just indicate a general direction add leave plenty of the student to do. But the two aims we mentioned before are closely connected. If the student succeeds in solving the problem at hand. he adds a little to his ability to solve problems. Then we should not forget that our questions are general applicable in many cases. If the same question is repeatedly helpful, the student will scarcely ail to notice it and he will be induced to ask the question by himself in a similar situation. Asking the question repeatedly, he may succeed once in eliciting the right idea. By such a success, he discovers the right way of using the question and then eh as really assimilated it. The student may absorb a few questions of our list so well that he is finally able to put himself the right question in the right moment and to perform the core corresponding mental operation naturally and vigorously such a student has certainly derived the greatest possible p[profit from our list. What can the teacher do in order to obtain this best possible result? Solving problems is a practical skill like, like, 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 practising 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. The teacher who wishes to develop his a students' ability to do problems must instill some interest for the problems into their minds and give them plenty of opportunity for imitation and practice. If the teacher wishes to develop in his students the mental operations which correspond so the questions and suggestion of our list, he puts these questions and suggestions to the students as often as he can do so naturally. Moreover, when the teacher solves problem before the class, he house dramatise his ideas a little and he sh should put to himself the saw 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 then the know knowledgeable of any particular mathematical fact. MAIN DIVISIONS, MAIN QUESTIONS. 6. Four phases trying to find the solution, we may repeatedly change our point of few, 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 look is different when we have made some progress ; it is again different when we have almost obtained the solution. In order to group conveniently the questions and suggestions of our list, we shall distinguish four phases of the work. First we have to understand the problem ; we have to see clearly what is required. Second, we have to see how th the various items are connected, how the unknown-known 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. Each of these phases has its importance. It may hap- pen that the student hits upon the exceptionally bright idea and jumping all preparations blurts out with the solution. Such lucky ideas, of course, are most desirable, but something very undesirable and unfortunate may result if the student leaves out any of the four phases without having a good idea. The worst may happen if the student embarks upon computations or constructions without having understood he problem. It is generally is 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 this 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 7. Understanding the problem, it is foolish to answer a question that you do not understand. It is sad to work for and end that you do not desire. 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 . 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. Some time should be allowed form natural and interesting presentations. first of all the verbal statement of the problem must bee understood. The teacher check this, up to a certain extent. He asks the student to repeat the statement and the student should be able to state the problem fluently. the student should also be able to point out the principle parts of the problem, the unknown, the data, the condition, hence, the teacher can seldom afford to miss the questions, what is the unknown, what are the data? what is the condition The student should consider the principle parts of the problem attentively, repeatedly, and form various sides. If there is a figure connected with the problem he shoulddraw a figure and point out ion it the unknown and the data. If it is necessary to give names to these objects he should introduce suitable table notation devoting some attention to the appropriate choice of signs, he is obliged to consider the objects for which the signs have to be chosen. There is another question which may be useful in this preparatory stage provided that we do not expect a definite answer but just a provisional answer, a guess: is it possible to satisfy the condition?" (in the exposition of Part II [p33.] "Understanding the problem" is subdivided into two stages": "Getting Acquainted" and "Working for better understanding .") 8. Example Let us illustrate some of the points explained in the foregoing section. we take the following simple problem: find the diagonal of a rectangular parallelepiped of which the length, the width, and the height are known. In order to discuss this problem profitably, the students, must be familiar with the theorem of Pythagoras, and with some of its applications in plane geometry, but they may have very systematic knowledge in solid geometry. The teacher may rely here upon the students' unsophisticated familiarity with spacial relations. The teacher can make the problem interesting by making it concrete. The classroom is a rectangular parallelepiped whose dimensions could be measured and can be estimated; the students have to find, to "measure indirectly," the diagonal of the classroom. The teacher points out the length, the width, and the hight of the classroom, indicates the diagonal with the gesture and then enlivens his figure, drawn on the blackboard, by referring repeatedly to the classroom. The dialogue between the teacher may start as follows: "What is the unknown?" "The length of the diagonal of a parallelepiped." "What are the data? "The length, the width, and the height of the parallelepiped." "introduce suitable notation. Which letter should note the unknown?" "x." "Which letters would you chose for the length, width, and the height?" "a,b,c" "What is the condition, linking a, b, c, and x?" "x is the diagonal of the parallelepiped of which a, b,and c are the length, the width, and the height. "Is it a reasonable problem? I mean, is the condition sufficient to determine the unknown?" "Yes it is, if we know a, b, and c, we know the parallelepiped. If the parallelepiped is determined, the diagonal is determined." 9. Devising a plan. We have a plan when we know, or know at least in outline, which calculations, computations, 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 problem is to conceive the idea of the plan. This idea may emerge gradually, or after apparently unsuccessful trials and a period of hesitation, it may concur suddenly in a flash, as a "bright idea." the best that a teacher can do for the student is to procure for him, by unobtrusive help, a bright idea. The questions and suggestion we are going to discuss tend to provoke such an idea. In order to be able to see the students position, the teacher should think of his own experience, of his difficulties 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 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 We should try to deserve such luck: we may deserve it by exploiting it. Here is a problem related to yours and solved before, could you use it? the forgoing questions, well understood and seriously considered, very often help to start the right train of ideas, but they cannot help always, they cannot work magic. If they do not work, we must look around for some other appropriate point of contact, and explore the various aspects of our problem ; we have to vary, to transform,to modify the problem. Could you restate the problem? Some of the questions of our list hint specific means to vary the problem, as generalisation, specialisation, use of analogy, dropping a part of the condition, and so on; the details are important but we cannot go into the them now. Variation of the problem may lead to some appropriate auxiliary problem: If you cannot solve the proposed problem, try to solve first some related 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 altogether. Yet there is a good question that may bring us back to it: Did you use all the data? Did you use the whole condition? 10.Example. We return to the example consider in section 8. As we left it, the students just succeed in understanding the problem and showed some mild interest 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 dialog with the students. He must be prepared to repeat with some modification the questions which their 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 related problem?" ..... "Look at the unknown! do you know a problem having the same unknown?" ..... "Well, what is the unknown?" "The diagonal of a parallelepiped.: "Do you know any problem with the same unknown2n?" "No. We have not had any problem yet about the diagonal of a parallelepiped." "Do you know any problem with a similar unknown?" ..... "You see, the diagonal is a segment, the segment of a straight line. Did you never solve a problem whose unknown was the length of a line?" "Of course, we have solved such problems. For instance, to find the side of aright triangle." "Good! Here is a problem related to yours and solved before. Could you use it?" ..... "You were lucky enough to remember a problem which is related to your present one and which you solved ../~~~~~~~~~~~~~~~~~~/ ./|......_____....../.| |~|~~~~x~____==iiii|..| |.|--eeeeiiiiiiiiiiiC_| |/a................|./ |----------b--------/ Fig 1; before. Would you like to use it? Could you introduce some auxiliary element in order to make its use possible?" ..... "Look here, the problem you remembered is about a triangle. Have you any triangle in your figure?" Let us hope that the last hint was explicit enough to provoke the idea of a solution which is to introduce a right triangle (emphasised 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; and so he should be prepared to use a whole gamut of more and more explicit hints. "Would you like to have a triangle in the figure?" "What sort of triangle would you like to have in the figure?" "You cannot find yet the diagonal; but you said that you could find the side of a triangle. Now what will you do?" "Could you find the diagonal, if it were the side of a triangle?" When eventually, with more or less help, the students succeed in introducing the decisive auxiliary element, the right triangle emphasised in fig. 1 the teacher should convince himself that the students are sufficiently far ahead before encouraging them to go into actual calculations. "I think that it was a good idea to draw that triangle. You have now a triangle, but have you the unknowns?" "The unknown is the hypotenuse of the triangle, we can calculate it by the theorem of Pythagoras." "You can if both legs are known; but are they?" "One leg is given, it is c. And the other, I think, is not difficult to find. Yes, the other leg is the hypotenuse of another right triangle." Very good! Now i see that you have a plan." 11. Carrying out the plan. To devise a plan to conceive the idea of the solution is not easy. It takes so much to succeed; formerly acquired knowledge, a good mental habits, concentration upon the purpose, and once more thing; good luck: To carry out the plan is much easier; what we need is mainly patience. The plan gives a general outline; we have to connive ourselves that the details fit into the outline and so we have to examine the details one after an 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 the outside, and accepted it on the authority of the a teacher; but if he worked for it himself, even with some help, and conceived the final idea with satisfaction, he will not solve 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". We may concentrate upon the point in question till we see it so dearly and distinctly that we have no doubt that the step is correct; or we ay derive the point in question according to formal rules(The difference between "insight" and "formal proof" is clear enough in may important cases; we may leave further discussion to the philosophers.) The main point is that the student should be honestly convinced of the correctness of each step. In certain cases the teacher may emphasise the difference between "Seeing" and "proving" Can you see clearly that the st step is correct? but can you also prove that the step is correct? 12. Example let us resume our work past the point where we left it at the end of exaction 10. the student, at last has got the idea of the solution. He sees the right triangle which the unknown x the hypotenuse e and the given height c is one of the legs; the other leg is the diagonal of the face. The student must, possibly be urged to introduce suitable notation. He should chose to denote that the other leg, the diagonal of the face whose sides are a and b. Thus he may see more clearly the idea of the solution which is to introduce an auxiliary problem whose unknown in y. Finally working at one right triangle after the other, he may obtain (see fig 1.) x ^ 2 = y ^ 2 + c ^ 2 y ^ 2 = a ^ 2 + b ^ 2 and hence eliminating the auxiliary unknown y x ^ 2 = a ^ 2 + B ^ 2 + c ^ 2 x = sqrt ( a ^ 2 + b ^ 2 + c ^ 2 ) The teacher has no reason to interrupt the student if he carries out these details correctly except possibly to warn him that he should check each step. Thus the teacher may ask? "Can you See clearly that the triangle with sides x,y,c, is a right triangle?" 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 his question unless the class has had a good initiation in solid geometry. Even in the later case, there is some danger that the answer to an incidental question may become the main difficulty for the majority of the students. 13. Looking back. Even fairly good students dents, when they have obtained the solution of the problem and were 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, reconsidering it and reexamining the result and the path that led to it, they could consolidate 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 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. 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 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? In order to convince ourselves of the presence or the quality of an object, we like to see and to touch it. And as we prefer perception through two different sense, so we prefer conviction but 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 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 connection of a problem within .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 consciousness of having done well they are eager to see what else they could accomplish with that effort, and how they could do equally well another time the teacher should encourage the students to imagine cases in which they could utilise again the procedure used. or apply the result obtained, an you use the result or the method for another problem? 14.Example in section 12, the student finally obtained the solution: if the three edges of a rectangular parallelogram issued from the same corner, are a,b,c the diagonal is sqrt( a^2+ b^2 + c+2) Can you check the result? The teacher cannot expect a good answer to this question from inexperienced students. The students, however should acquire fairly early the experience that problems "in letters" have a great advantage over purely numerical problems; if the problem 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. "Did you use all the data?" Do all the data, a, b, c, x, 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? "Our problem is a problem of solid geometry, to find the diagonal of a parallelepiped with given dimensions a,b,c. Our problem is analogous to a problem of plane geometry : to find the diagonal of a rectangle with given dimensions, a,b. is the result of our 'solid' problem analogous to the result of the 'plane' problem"" "If the height c decreases, and finally finishes, the parallelepiped becomes a parallelogram. If you put c=0 in your formula, do you obtain the correct formula of the diagonal of the rectangular parallelogram?" "If the height c increases, the diagonal increases. Does your formula show this?" "If all three measures, a,b,c, of the parallelepiped increase the same proportion, the diagonal also increases in the same proportion. If in your form of a as you substitute 12a, 12b, 12c for a,b,c, respectively, the expression of the diagonal , owing to this sub substitution should also be multiplied by 12. Is that so?{PO If a,b,c are measured in feet, your formula gives the diagonal measured in feet too; but if you change all the measures into inches the formula should remain correct. Is that so? The two last questions are essentially equivalent; see TEST BY DIMENSION 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. The formula has therefor a better chance of being remembered, the knowledge of the student is consolidated. Finally these questions can be easily transferred to similar problems. After some experience with similar problems, and intelligent student may perceive the underlying general ideas: use all relevant data, variation of the data, symmetry, analogy. If he gets into the habit of directing his attention to such points, his ability to solve problems may definitely profit. Can you check the argument? to recheck the argument step by step may be necessary in difficulty. 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 right triangle? (see the end of section 12.) Can you use the result or the methods 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 my put a slightly different problem for instance" being given the light the width and the height of a rectangular parallelepiped find the distance of the centre from one of the corners." The students may use the result of the problem they just solved, observing that the distance required is our half to he diagonal they just calculated or they may use the method, introducing suitable right triangles(the later alternative less obvious and somewhat more clumsy in the present case). After this application, the teacher may discuss the configuration of the four diagonals of the parallelepiped and the six pyramids of which the six faces are the bases. the censer the common vertex, and the semi-diagonals the lateral edges. When the geometric imagination of the students is sufficiently enlivened. The teacher should come back to his question: Can you use the result, or the method of for some other problem? Now there is a better chance that the students may find some more interesting concrete interpretation, for instance, the following: "In the centre of the flat rectangular top of a building which is 21 yards long and 16 yards wide, a flagpole is to be erected, 8 yards high. 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 at the top of the building. How long is each cable? The students may use the method of the problem they solved in detail introducing a right triangle in the vertical plane, and another one in the horizontal plane. Or they may use the result, imagining a rectangular parallelepiped of which the diagonal, x is one of the four cables and the edges are a=10.5 b-8 c=6 by straight forward application of the formula x=14.5 or more examples see CAN YOU USE THE RESULT 15. Various approaches. Let us still retain, for awhile the problem we considered in the foregoing sections 9,10,, 12, 14. The main work, the discovery of the plan, was described in section 10. Let us observe that the teacher could have proceeded differently. Starting from the same point as in section10, he could have followed as somewhat different line, asking the following questions: "Do you know any related problem?" "Do you know an analogous problem?" "You see the proposed problem is a problem of solid geometry. Could you think of a simpler analogous problem of plane geometry? You see the proposed problem is about a figure in space, it is concerned with the diagonal of a rectangle parallelepiped. What might be an analogous problem about a figure in the plane? It should be concerned with --the diagonal--of--a-rectangular --" "Parallelogram." The students even if they are very slow and indifferent, and were not able to guess anything before, are obliged finally to contribute at least a minute part of the idea. Besides, if the students are so slow, the teacher should not take up the present problem about the parallelepiped without having discussed before, in order to prepare the students, the analogous problem bout the parallelogram. Then he can go on now as follows. Here is a problem related to yours and solved before. Can you use it? Should you introduce some auxiliary element in order to make its use possible?" Eventually the teacher may succeed in suggesting to the students the derivable idea. It consists in conceiving the diagonal of the given parallelepiped as the diagonal of a suitable parallelogram which must be introduced to the figure (as intersection of the the parallelepiped with a plane passing through two opposite edges). The idea is essentially the same as before (section 10) but the approach is different. In section 10 the contact with the available knowledge of the students was established through the unknown; a former 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. 16. The teachers method of questioning shown in the foregoing sections 8, 10,14,15, is essentially this: begin with a general question or suggestion of our list, and if necessary, come down gradually to spend more specific and concrete questions or suggestions till you reach one which elicits a response in the students mind. If you have to help the student exploit hi side, start again, if possible from the general question or suggestion contained in the list, and return again to more special; one if necessary and so on. Of course, our list is just a first list of this kind; it seems ems to be sufficient for the majority of simple cases, but there is no doubt that it could be perfected. It is important whoever that the suggestion from which we start housed 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 as special technique. The list must be short in order that the questions may be often repeated, unartificially, and under varying circumstances; thus there is 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 suggestions, 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; fortunately so, because in these matters, any rigid, mechanical, pedantical procedure is necessarily bad. Our method admits a certain elasticity and variation, and it admits various approaches (section 15), it can be and should be so applied that the 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 18,19,20, He should prepare carefully the examples which he intends to discuss, considering also various approaches. He should start with a few trials and find out gradually how he can manage the method, and how students take it, and how much time it takes. 17. Good questions and bad questions. If the method for questioning formulated in the forgoing section is well understood it helps to judge, by comparison, the quality of certain suggestions which may be offered with the intention of helping the students. Let us go back to the situation as it was presented itself at the beginning of section 10 when the question was asked Do you know a related problem? 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 deferred; then we shall see that there is a long sequence of objections against this sort of "help." 1) if the student is near to the solution he may understand the suggestion implied by the question; but if he is not, he quite possibly will not see at all the point at which the question is driving. Thus the question fails to help where help is most needed. 2) if the suggestion is understood it gives the whole secret away, very little remains for the student to do. 3) the suggestion is too special a nature. Even if the student can make use of it solving the present problem nothing is learned for future problems. The question is not instructive. 4) 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 unnatural surprise, as a rabbit pulled our of a hat. It's really not instructive. None of these objections can be raised against the procedure described in section 10 or against that in section 15,. MORE EXAMPLES 18. 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 other vertices of the square on the other two sides of the triangle, one on each. "what is the unknown"?" "A square" " what are the data" "A triangle is given, nothing else" what is the condition "The four corners of the square should be on the perimeter of the triangle, two corners on the base, on corner on each of the other two sides" "Is it possible to satisfy the condition?" "I think so, I am not so sure." "You do not seem to find the problem too easy." If you cannot solve the proposed problem, try to solve first some related problem. Could you satisfy a Part of the condition?" "What do you mean by a part of the condition?" "You see, the condition is concerned with all the he vertices from the square. How man many vertices are there?" "four" "A part of the condition would be concerned with less th than four vertices. Keep only a part of the condition, drop the other part. What part of the condition is easy to satisfy?" "It is easy to draw a square with two vertices on the perimeter of the triangle, or even one with three vertices on the perimeter." "Draw a figure!" The student draws Fig.2 "You kept only a part of the condition and you dropped the other part. How far is the unknown to be determined?" .~ _ / ~-_ /|~~~~~|-_ / | | -_ / | | - ~~~~~~~~~~~~~~~~ Fig. 2 "The square is not determined if it has only three vertices on the perimeter of the triangle." "Good! Draw a figure." The student draws fig. 3. .~ _ / ~-_ /|~~~~~|-_ /_| | -_ /| | | - ~~~~~~~~~~~~~~~~ Fig. 3 "The square, as you said, is not determined by the part of the condition you kept. How can it vary?" ..... "Three corners of your square are on the perimeter of the triangle but the fourth corner is not yet there where it should be. Your square, as you said, is undetermined, it can vary; the same is true of its fourth corner. How can it vary?" ..... "Try it experimentally, if you wish. Draw more squares with three corners 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? How can it vary?" ..... "Try it experimentally, if you wish. Draw more squares with three corners 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? How can it vary?" The teacher brought the student very near to the idea of the solution. If the student is able to guess that the locus of the fourth corner is a straight line, he has got it. 19. A problem to prove. Two angles are in different planes but each side of one is parallel to the corresponding 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 students who are familiar with plane geometry and acquainted 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 questions 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." (The correspondence is worked out systematically in PROBLEMS TO FIND, PROBLEMS TO PROVE, 5, 6.) "What is the first hypothesis?" "Two angles are in different planes. Each side of one is parallel to the corresponding side of the other, has also the same direction. "What is the conclusion?" "The angles are equal." "Draw a figure. Introduce suitable notation." The student draws the lines of Fig. 4 and chooses, helped more or less by the teacher, the letters as in Fig.4 . "What is the hypothesis? Say it, please, using your notation." "A,B,C are not in the same plane as A', B', C'. And AB||A'B', AC||A'C'. Also AB has the same direction as A'B', and AC the same as A'C'." .-~ C' _-~ A' .-~ ~-_ ~~-__ B' .-~ C _-~ A .-~ ~-_ ~~-__ B Fig.4 "What is the conclusion?" "/_BAC = /_B'A'C'." "Look at the conclusion! And try to think of a familiar theorem having the same or similar conclusion." "If two triangles are congruent, the corresponding angles are equal." "Very good! Now here is a theorem related to yours and proved before. Could you use it?" "I think so but I do not see yet quite how." "Should you introduce some auxiliary element in order to make its use possible?" ..... "Well, the theorem which you quoted so well is about triangles, about a pair of congruent triangles. Have you any triangles in your figure?" "No, But I could introduce some. Let me join B to C and B' to C'. Then there are two triangles /\ABC, /\A'B'C'." "Well done. But what are these triangles good for?" "To prove the conclusion, /_BAC = /_B'A'C'." "Good! If you wish to prove this, what kind of triangles do you need?" .-~ C' _-~ . A' .-~ . ~-_ . ~~-__. B' .-~ C _-~ . A .-~ . ~-_ . ~~-__. B Fig.5 "Congruent triangles. Yes, of course, I may chose B, C, B' C' so that AB = A'B' , AC = A'C'." "Very good! Now, what do you wish to prove?" "I wish to prove that the triangles are congruent, /\ABC = /\A'B'C' "If I could prove this, the conclusion /_BAC = /_B'A'C' would follow immediately. " "Fine! You have a new aim, you aim at a new conclusion Look at the conclusion And try to think of a familiar theorem having the same or a similar conclusion." "Two triangles are congruent if--if the three sides of the one are equal respectively to the three sides of the other." "Well done. You could have chosen a worse one. Now Here is a theorem related to yours and proved before. Could you use it?" "I could use it if I knew that BC = B'C'." "That is right! Thus what is your aim?" "To prove that BC = B'C'." "Try to think of a familiar theorem having the same or a similar conclusion." "Yes, I know a theorem finishing: ' . . . then the two lines are equal.' But it does not fit in." "Should you introduce some auxiliary element in order to make its use possible?" ..... "You see, how could you prove BC = B'C' when there is no connection in the figure between BC and B'C'?" ..... "Did you use the hypothesis? What is the hypothesis?" "We suppose that AB||A'B',AC||A'C'. Yes, of course, I must use that." "Did you use the whole hypothesis? You say that AB||A'B'. Is that all that you know about these lines?" "No; AB is equal to A'B', by construction. They are parallel and equal to each other. And so are AC and A'C'." "Two parallel lines of equal length -- it is an interesting configuration. Have you seen it before?" "Of course! Yes! Parallelogram! Let me join A to A', B to B', C to C'." "The idea is not so bad. How many parallelograms have you now made 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. 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. 20. A rate problem. Water is flowing into a conical vessel at 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 cone b. ------~~~~~~~~~~~------ - a - ~ T~~~~~~~~~~~~/ \~-------___|____-----~~ \ | /| \ | / | \ | / | \.----~~T~--X-_ / | ~----__LL~~~~~~ | b \ | / | | \ | / | | \ | / | | \ | / | y | \ |/ | | \|______L____| Fig. 6 Find the rate at which the surface is rising when the depth of the water is y. Finally obtain the numerical value of the unknown supposing that a=4 ft., b=3 ft, r= 2 cu. ft. per minute and y=1 ft. The students are supposed to know the simplest rules of differentiation and the notion of "rate of change." "What are the data?" "The radius of the base of the cone a=4 ft. the altitude- of the cone b=3 ft., the rate at which the water is flowing into the vessel r=2 cu. ft. per minute, and the depth of the water at a certain moment, y=1 ft." "Correct. The statement of the problem seems to suggest that you should disregard, provisionally, the numerical 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 letters, substitute the numerical values. I would follow this suggestion. Now, "what is the unknown?" "the rate at which the surface is rising when the depth of the water is y." "What is that Could you say it in other terms?" "The rate at which the depth of the water is increasing." "What is that? Could you restate it still differently?" "The rate of change of the depth of the water." "That is right, the rate of change of y. But what is the rate of change? Go back to the definition." "The derivative is the rate of change of a function." "Correct. Now is y a function? As we said before we disregard the numerical values of y. Can you imagine that y changes?" "Yes, y, t he depth of the water, increases as the time goes by." "Thus y is a function of what?" "Of the time t." "Good. introduce suitable notation. How would you write the 'rate of change of y' in mathematical symbols?" " dy /dt " "Good. Thus, this is your unknown. You have to express it in terms of a,b,r,y. By the way, one of those data is a 'rate.' Which one?" "r is the rate at which water is flowing into the vessel." "What is that? Could you say it in other terms??" "R is the rate of change of the volume of the water in the vessel." "What is that? Could you restate it still differently? How would you write it in suitable notation?" " r= dV /dt. " "What is V?" "The volume of the water in the vessel at the time t." "Good. Thus, you have to express dy/dt in terms of a,b, dV/dt, y. How will you do it?" .... If you cannot solve the proposed problem try to solve first some related problem. If you do not see yet the connection between dy/dt and the data, try to bring in some simpler connection that could serve as a stepping stone." ..... "Do you not see that there are other connections? For instance, are y and V independent of each-other?" "No. When y increases, V must increase too." "Thus there is a connection. What is the connection? " "Well, V is the volume of a cone of which the altitude is y. But I do not know yet the radius of the base." "You may reconsider it, nevertheless. Call it something, say x." "V=pix^2y/3." "Correct. Now, what about x? Is it independant of y?" No. When the depth of the water, y, increases the radius of the free surface, x, increases too." Thus, there is a connection. What is the connection?" "Of course, similar triangles. x:y=a:b." "One more connection, you see. I would not miss profiting from it. Do not forget, you wished to know the connection between V and y." "I have x=ay/b V=pia^2y^3/(3b^2) "Very good. This looks like a stepping stone, does it not? But you should not forget your goal. What is the unknown?" "Well, dy/dt." "You have to find a connection between dy/dt, dV/dt and other quantities. And here you have one between y, V, and other quantities. What to do?" "Differentiate! Of course! dV/dt = pia^2y^2/b^2 dy/dt . Here it is." "Fine! And what about the numerical values?" "If a=4, b=3, dV/dt=r=2, y=1, then 2= pi*16*1/9 dy/dt PART II. HOW TO SOLVE IT A DIALOGUE ----------------------------------------- Getting Acquainted Where should I start? Start from the statement of the problem. What can I do? Visualise 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, familiarise yourself with it, impress its purpose on your mind. The attention bestowed on the problem 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 altogether. What can I do? 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 and the principal parts of a "problem to find." Go through the principal parts of your problem, consider them one by one, consider them in turn, consider them in various combinations, relating each detail to other details and each to the whole of the problem. " What can I gain by doing so?" 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 conceived, thanks to your previous work, and when your memory seems responsive. "What can I do?" Consider your problem from various sides and seek contacts with your formerly acquired Knowledge. Consider your problem from various sides. Emphasise 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 interpretations on the 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 recognise something familiar in what you examine. Try to perceive something useful in what you recognise. What could I perceive?" A helpful idea, perhaps a decisive 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. If it looks reliable you should ascertain how far it leads you, and reconsider the situation. The situation 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 idea. Nevertheless you should be grateful for all new ideas, also for the lesser ones, also for the hazy ones, also for the supplementary ideas adding some precision to a hazy one, or attempting the correction 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 and you feel confident that you can supply the minor details that may be wanting. What can I do?" Make your grasp quite secure. Carry through in detail all the algebraic or geometric operations which you have recognised previously as feasible. Convince yourself of the correctness of each step by formal reasoning, or by intuitive insight, or both ways if you can. If your problem is very complex you may distinguish "great" steps and "small all" steps, each great step being composed of several small ones. Check the first the great steps, and get down to the smaller ones afterwards. What can I gain by doing so? A presentation of the solution each step of which is correct beyond doubt. Where should I start? From the solution, complete and correct in each detail. What can I do? 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. Scrutinise the method that led you to the solution, try to see its point, and try to make use of it for other problems. Scrutinise the result and try to make use of it for other problems. What can I gain by doing so?" 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 scrutinising your solutions in this way, you will acquire some knowledge well ordered and ready to use, and you will develop your ability to solve problems.