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Category: George Polya

George Polya 14 Most Inspiring Audio Quotes, Ideas How To Solve It

For solving problems use: Request-Response-Result, and a verification of the result. Is a method used to solve all kinds of problems: Understand the problem, make a plan, execute the plan, look back and reflect. George PΓ³lya 14 Life Changing Audio Quotes Sayings for you to Enjoy, Read, Listen as Audiobook, Motivate, Share. Thought-Provoking Ideas better than Pictures/Photos, Memorable Thoughts Inspiration from George Polya author of Books as: How to Solve It, Mathematics and Plausible Reasoning, Mathematical discovery, The Stanford mathematics problem book: with hints and solutions. Audio Mp3 Quotations / Citations and Personal Development Ideas free online on Enthusiasm, Experience, Knowledge, Mathematics, Motivation, Planning, Problem Solving, Progress, Questions, Science, Solutions, Understanding. Man Voice β™‚ + Woman ♀ Pronunciation and Accent.

β€œ First, we have to understand the problem; we have to see clearly what is required. Second, we have to see how the various items are connected, how the unknown is linked to the data, in order to obtain the idea of the solution, to make a plan. Third, we carry out our plan. Fourth, we look back at the completed solution, we review and discuss it. ” – George Polya

β€œ Trying to find the solution, we may repeatedly change our point of view, our way of looking at the problem. We have to shift our position again and again. Our conception of the problem is likely to be rather incomplete when we start the work; our outlook is different when we have made some progress; it is again different when we have almost obtained the solution. ” – George Polya

β€œ It is foolish to answer a question that you do not understand. It is sad to work for an 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. ” – George Polya

β€œ 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 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 β€œbright idea. ” – George Polya

β€œ By looking back at the completed solution, by reconsidering and reexamining the result and the path that led to it, they could consolidate their knowledge and develop their ability to solve problems. ” – George Polya

β€œ The worst may happen if the student embarks upon computations or constructions without having understood the problem. It is generally useless to carry out details without having seen the main connection, or having made a sort of plan. Many mistakes can be avoided if, carrying out his plan, the student checks each step. Some of the best effects may be lost if the student fails to reexamine and to reconsider the completed solution. ” – George Polya

β€œ If the student is lacking in understanding or in interest, it is not always his fault; the problem should be well chosen, not too difficult and not too easy, natural and interesting, and some time should be allowed for natural and interesting presentation. ” – George Polya

β€œ 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 constructing a house but we cannot construct a house without 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? ” – George Polya

β€œ Look at the unknown! Try to think of a familiar problem having the same or a similar unknown. ” – George Polya

β€œ The difference between β€œseeing” and β€œproving”: Can you see clearly that the step is correct? But can you also prove that the step is correct? ” – George Polya

β€œ Errors are always possible: verifications are desirable. Can you check the result? Can you check the argument? Can you derive the result differently? Can you see it at a glance? ” – George Polya

β€œ Imagine cases in which they could utilize again the procedure used, or apply the result obtained. Can you use the result, or the method, for some other problem? ” – George Polya

β€œ To teach effectively a teacher must develop a feeling for his subject; he cannot make his students sense its vitality if he does not sense it himself. He cannot share his enthusiasm when he has no enthusiasm to share. How he makes his point may be as important as the point he makes; he must personally feel it to be important. ” – George Polya

β€œ Math has 2 faces: Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. ” – George Polya

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Best George Polya 12 Brilliant Audio Quotes Wise Ideas Think Solve Problems

George Polya designed a 4 step process for problem solving: S-1: Understand the problem. S-2: Devise a plan (translate). S-3: Carry out the plan (solve). S-4: Look back (check and interpret). George PΓ³lya 12 Problem Solving & Life Changing Audio Quotes Sayings for you to Enjoy, Read, Listen as Audiobook, Motivate, Share. Thought-Provoking Ideas better than Pictures/Photos, Memorable Thoughts Inspiration from George Polya author of Books as: How to Solve It, Mathematics and Plausible Reasoning, Mathematical discovery, The Stanford mathematics problem book: with hints and solutions. Audio Mp3 Quotations / Citations and Personal Development Ideas free online on Achievement, Mastery, Math, Mind, Pedantry, Practice, Problem Solving, Problems, Questioning, Reflection, Trust, Understanding, Visualize. Man Voice β™‚ + Woman ♀ Pronunciation and Accent.

β€œ Where should I start? 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 … What can I gain by doing so? You should understand the problem, familiarize yourself with it, impress its purpose on your mind. ” – George Polya

β€œ Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems, and, finally, you learn to do problems by doing them. ” – George Polya

β€œ 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. ” – George Polya

β€œ 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. ” – George Polya

β€œ A mathematician who can only generalise is like a monkey who can only climb up a tree, and a mathematician who can only specialise is like a monkey who can only climb down a tree. In fact neither the up monkey nor the down monkey is a viable creature. A real monkey must find food and escape his enemies and so must be able to incessantly climb up and down. A real mathematician must be able to generalise and specialise. ” – George Polya

β€œ There is a limit beyond which we should not force the conscious reflection, when it is better to leave this problem alone for a while. But it is desirable not to set aside a problem to which we wish to come back later without the impression of some achievement; at least some little point should be settled, some aspect of the question somewhat elucidated when we quit working. Only such problems come back improved whose solution we passionately desire, or for which we have worked with great tension. Conscious effort and tension seem to be necessary to set the unconscious work going. ” – George Polya

β€œ To write and speak correctly is certainly necessary; but it is not sufficient. A derivation correctly presented in the book or on the blackboard may be inaccessible and uninstructive, if the purpose of the successive steps is incomprehensible, if the reader or listener cannot understand how it was humanly possible to find such an argument. ” – George Polya

β€œ You should not put too much trust in any unproved conjecture, even if it has been propounded by a great authority, even if it has been propounded by yourself. You should try to prove it or disprove it. ” – George Polya

β€œ Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. … To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery.” – George Polya

β€œ 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 principal parts of the problem attentively, repeatedly, and from from various sides. ” – George Polya

β€œ The elegance of a mathematical theorem is directly proportional to the number of independent ideas one can see in the theorem and inversely proportional to the effort it takes to see them. ” – George Polya

β€œ In order to translate a sentence from English into French two things are necessary. First, we must understand thoroughly the English sentence. Second, we must be familiar with the forms of expression peculiar to the French language. The situation is very similar when we attempt to express in mathematical symbols a condition proposed in words. First, we must understand thoroughly the condition. Second, we must be familiar with the forms of mathematical expression. ” – George Polya

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