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.
The longer I live the more I realize the impact of attitude on life. Attitude, to me, is more important than facts. It is more important than the past, than education, than money, than circumstances, than failures, than successes, than what other people think or say or do. It is more important than appearance, giftedness or skill. It will make or break a company . . . a church . . . a home.
Doubt is invariably the result of want or weakness of faith.
If a man does not know to what port he is steering, no wind is favorable to him. Ignoranti quem portum petat, nullus suus ventus est.
I found the emotion that as an athlete you block out, and it really helped me to understand myself as a person. I'm a really emotional person and it helped make me a better person.
Both wit and understanding are trifles without integrity; it is that which gives value to every character. The ignorant peasant, without fault, is greater than the philosopher with many; for what is genius or courage without a heart?
So, my unsolicited advice to women in the workplace is this. When faced with sexism, or ageism, or lookism, or even really aggressive Buddhism, ask yourself the following question: “Is this person in between me and what I want to do?” If the answer is no, ignore it and move on. Your energy is better used doing your work and outpacing people that way. Then, when you’re in charge, don’t hire the people who were jerky to you.
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