How to Solve It Quotes

“The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“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 Pólya, How to Solve It: A New Aspect of Mathematical Method

“Mathematics, you see, is not a spectator sport. To understand mathematics means to be able to do mathematics. And what does it mean [to be] doing mathematics? In the first place, it means to be able to solve mathematical problems.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Nothing is more important than to see the sources of invention which are, in my opinion, more interesting than the inventions themselves.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“The first rule of discovery is to have brains and good luck. The second rule of discovery is to sit tight and wait till you get a bright idea.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“It is generally useless to carry out details without having seen the main connection, or having made a sort of plan.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Quite often, when an idea that could be helpful presents itself, we do not appreciate it, for it is so inconspicuous. The expert has, perhaps, no more ideas than the inexperienced, but appreciates more what he has and uses it better.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Good problems and mushrooms of certain kinds have something in common; they grow in clusters.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“If you cannot solve the proposed problem...try to solve first some related problem.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Analogy pervades all our thinking,”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“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 Pólya, How to Solve It: A New Aspect of Mathematical Method

“Here is a typical story about Mr. John Jones. Mr. Jones works in an office. He had hoped for a little raise but his hope, as hopes often are, was disappointed. The salaries of some of his colleagues were raised but not his. Mr. Jones could not take it calmly. He worried and worried and finally suspected that Director Brown was responsible for his failure in getting a raise. We cannot blame Mr. Jones for having conceived such a suspicion. There were indeed some signs pointing to Director Brown. The real mistake was that, after having conceived that suspicion, Mr. Jones became blind to all signs pointing in the opposite direction. He worried himself into firmly believing that Director Brown was his personal enemy and behaved so stupidly that he almost succeeded in making a real enemy of the director. The trouble with Mr. John Jones is that he behaves like most of us. He never changes his major opinions. He changes his minor opinions not infrequently and quite suddenly; but he never doubts any of his opinions, major or minor, as long as he has them. He never doubts them, or questions them, or examines them critically—he would especially hate critical examination, if he understood what that meant. Let us concede that Mr. John Jones is right to a certain extent. He is a busy man; he has his duties at the office and at home. He has little time for doubt or examination. At best, he could examine only a few of his convictions and why should he doubt one if he has no time to examine that doubt? Still, don’t do as Mr. John Jones does. Don’t let your suspicion, or guess, or conjecture, grow without examination till it becomes ineradicable. At any rate, in theoretical matters, the best of ideas is hurt by uncritical acceptance and thrives on critical examination. 2. A mathematical example. Of all quadrilaterals with”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“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.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“No idea is really bad, unless we are uncritical. What is really bad is to have no idea at all.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Problems “in letters” are susceptible of more, and more interesting, tests than “problems in numbers”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Pedantry and mastery are opposite attitudes toward rules. 1. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. Some pedants are poor fools; they never did understand the rule which they apply so conscientiously and so indiscriminately. Some pedants are quite successful; they understood their rule, at least in the beginning (before they became pedants), and chose a good one that fits in many cases and fails only occasionally. 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 Pólya, How to Solve It: A New Aspect of Mathematical Method

“The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“The worst may happen if the student embarks upon computations or constructions without having understood the problem.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“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.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Perseverance kills the game.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“A good notation should be unambiguous, pregnant, easy to remember.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“The best of ideas is hurt by uncritical acceptance and thrives on critical examination.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Such neglect of the obvious does not show necessarily stupidity but rather indifference toward artificial problems.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“there is nothing to learn about reasoning and invention if the motive and purpose of the most conspicuous step remain incomprehensible.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“teaching of calculus to engineers and physicists, could be essentially improved if the nature of heuristic reasoning were better understood,”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method

“Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. ... A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted.”
― George Pólya, How to Solve It: A New Aspect of Mathematical Method