If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.
Interpretation
Take your time to fully understand concepts before attempting to prove them.
George Polya emphasizes the importance of fully comprehending a theorem before rushing into its proof. He advises a thoughtful and thorough examination of the theorem and its implications, encouraging individuals to validate the truth of their understanding before proceeding with formal proof.
In practice
In a mathematics lecture discussing the importance of foundational understanding.
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.
God's grace doesn't always come in comfortable forms. But it's still grace, and it's still evidence that He loves us.
Suppose you read about a pill that you could take once a day to reduce anxiety and increase your contentment. Would you take it? Suppose further that the pill has a great variety of side effects, all of them good: increased self-esteem, empathy, and trust; it even improves memory. Suppose, finally, that the pill is all natural and costs nothing. Now would you take it? The pill exists. It is meditation.
It takes stamina to get up like an athlete every single night, seven to eight performances a week, 20 weeks in a row. And there are many young performers who only learn their craft in the two minute bits it takes to film a scene. You never learn the arc of storytelling, the arc of a character that way.
Any man who thinks he can be happy and prosperous by letting the government take care of him better take a closer look at the American Indian.
I don't pretend we have all the answers. But the questions are certainly worth thinking about.
Until the juice ferments a while in the cask, it isn't wine. If you wish your heart to be bright, you must do a little work.
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