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.
It is a great piece of skill to know how to guide your luck even while waiting for it.
I respect the man who knows distinctly what he wants.
God has given me a mule-like stubbornness to stick with a difficult problem and the intuitive powers to conceptualize complex hypothetical situations in my mind.
When Dr. King was murdered, I had no idea who he was. But as soon as I heard his words on television that night when I was 9 years old, I was dumbstruck, awestruck by their power.
Pain in this life is not avoidable, but the pain we create avoiding pain is avoidable.
I'm in no position to hand down any advice," he said, "but there's a rule I follow when I don't know what to do." "A rule?" "If you have to choose between something that has form and something that doesn't, go for the one without form. That's my rule. Whenever I run into a wall I follow that rule, and it always works out. Even if it's hard going at the time.
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