Mathematics is not a spectator sport!
Interpretation
Mathematics requires active participation and engagement, rather than passive observation.
The quote by George Polya emphasizes the importance of actively engaging with mathematical concepts instead of simply watching others perform calculations or problem-solving. It suggests that true understanding and mastery of mathematics come from hands-on practice, exploration, and participation, akin to how one must actively play a sport to appreciate and excel in it.
In practice
In a classroom setting, a teacher might say this quote to encourage students to participate in problem-solving activities.
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.
If you wish to learn swimming you have to go into the water and if you wish to become a problem solver you have to solve problems.
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.
Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself
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.
Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.
There exists one book, which, to my taste, furnishes the happiest treatise of natural education. What then is this marvelous book? Is it Aristotle? Is it Pliny, is it Buffon? No-it is Robinson Crusoe.
Ah yes, the head is full of books. The hard part is to force them down through the bloodstream and out through the fingers.
Education is an act of love, and thus an act of courage
When I was growing up, my parents told me, 'Finish your dinner. People in China and India are starving.' I tell my daughters, 'Finish your homework. People in India and China are starving for your job.'
I don't see why a book shouldn't be intellectually sound, entertaining, and fun to read. Historians who write academic history, which is unreadable, are basically wasting their time.
When I read about the way in which library funds are being cut and cut, I can only think that American society has found one more way to destroy itself.
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