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.
If people can be educated to see the lowly side of their own natures, it may be hoped that they will also learn to understand and to love their fellow men better. A little less hypocrisy and a little more tolerance towards oneself can only have good results in respect for our neighbor; for we are all too prone to transfer to our fellows the injustice and violence we inflict upon our own natures.
When kids know that you refuse to let them fail ... they don't give up as easy. So sometimes they don't have it inside, [but] they're like,'You know, I don't want to do this, but I know my mother's going to be mad.'That matters to kids, and it helps get them through.
The best morals kids get from any book is just the capacity to empathize with other people, to care about the characters and their feelings. So you don't have to write a preachy book to do that. You just have to make it a fun book with characters they care about, and they will become better people as a result.
Your example, even more than your words, will be an eloquent lesson to the world.
Better guide the young than reclaim them when old For the voice of true wisdom is calling "To rescue the fallen is good, but tis best To prevent other people from falling" Better close up the source of temptation and crime Than deliver from dungeon or gallery Better put a strong fence round the top of the cliff Than an ambulance down in the valley.
NASA's been one of the most successful public investments in motivating students to do well and achieve all they can achieve, and it's sad that we are turning the program in a direction where it will reduce the amount of motivation it provides to young people.
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