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.
Interpretation
Understanding the source is crucial for accurate translation, whether in language or mathematics.
George Polya's quote emphasizes the importance of comprehension in the translation process, applicable both in linguistics and mathematics. To effectively translate a sentence from one language to another or to convert verbal conditions into mathematical symbols, one must first have a thorough understanding of the original content and then be adept at the specific forms of expression used in the target language or discipline. This underscores the broader principle that deep understanding is essential in any field of knowledge for clear and accurate communication.
In practice
This quote could be used in a lecture about the importance of understanding concepts before teaching them.
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
Mathematics is the cheapest science. Unlike physics or chemistry, it does not require any expensive equipment. All one needs for mathematics is a pencil and paper.
By its very definition, civic responsibility means taking a healthy role in the life of one's community. That means that classroom lessons should be complemented by work outside the classroom. Service-learning does just that, tying community service to academic learning.
People who grow up with two or more languages understand that each can express certain aspects of reality better than the other.
The authority of those who teach is often an obstacle to those who want to learn.
I've spoken in every state in the union, meeting and hugging the people who later bought my books. I spoke to anybody who wanted to hear me, including 1,000 nuns who could pay me only with homemade bread.
I come to writing the same way I come to teaching, which is that my goal is always to create life-long readers.
For fifteen years, I was a teacher of youth. They were years out of the fullness and bloom of my younger manhood. They were years mingled of half breathless work, of anxious self-questionings, of planning and replanning, of disillusion, or mounting wonder.
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