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.
What old people say you cannot do, you try and find that you can. Old deeds for old people, and new deeds for new.
You are in integrity when the life you are living on the outside matches who you are on the inside.
The psychologist Gerd Gigerenzer has a simple heuristic. Never ask the doctor what you should do. Ask him what he would do if he were in your place. You would be surprised at the difference
Although I usually think I know what I'm going to be writing about, what I'm going to say, most of the time it doesn't happen that way at all. At some point I get misled down a garden path, I get surprised by an idea that I hadn't anticipated getting, which is a little bit like being in a laboratory.
What makes authentic disciples is not visions, ecstasies, biblical mastery of chapter and verse, or spectacular success in the ministry, but a capacity for faithfulness. Buffeted by the fickle winds of failure, battered by their own unruly emotions, and bruised by rejection and ridicule, authentic disciples may have stumbled and frequently fallen, endured lapses and relapses, gotten handcuffed to the fleshpots and wandered into a far county. Yet, they kept coming back to Jesus.
[In developing your potential] ... I don't develop; I am
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