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 best tournament players actually try to avoid risky plays altogether. They prefer to wait for their opponents to make the risky moves. They'll wait patiently until they catch a strong hand. When they do, they'll take down their overly aggressive foes.
No psychologist should pretend to understand what he does not understand... Only fools and charlatans know everything and understand nothing.
Whipping and abuse are like laudanum: you have to double the dose as the sensibilities decline.
There will be books written about Harry. Every child in the world will know his name.
Bear in mind that brains and learning, like muscle and physical skill, are articles of commerce. They are bought and sold. You can hire them by the year or by the hour. The only thing in the world not for sale is character.
The Man who says he can, and the man who says he can not.. Are both correct
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