Mathematics consists in proving the most obvious thing in the least obvious way.
Interpretation
Mathematics involves demonstrating seemingly simple truths through complex reasoning.
George Polyaβs quote highlights the essence of mathematical thinking, where what appears to be evident often requires intricate proof and exploration. This reflects the beauty and challenge of mathematics, emphasizing that deeper understanding often lies beneath the surface of obviousness.
In practice
This quote can be used in a lecture to inspire students about the beauty of mathematical proofs.
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 not a spectator sport!
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.
Our youth should also be educated with music and physical education.
No race can prosper till it learns that there is as much dignity in tilling a field as in writing a poem.
We need to align the incentives so that colleges have an incentive to keep down their costs... to graduate students on time with degrees in areas where they're going to be able to get jobs and going to be able to pay back those loans.
So far, we do not seem appalled at the prospect of exactly the same kind of education being applied to all the school children from the Atlantic to the Pacific, but there is an uneasiness in the air, a realization that the individual is growing less easy to find; an idea, perhaps, of what standardization might become when the units are not machines, but human beings.
We should spend less time at universities filling our students' minds with content by lecturing at them, and more time igniting their creativity β¦ by actually talking with them.
It took 10 months for me to learn to tie a lace; I must have howled with rage and frustration. But one day I could tie my laces. That no one can take from you. I profoundly distrust the pedagogy of ease.
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