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
Interpretation
Mathematics simplifies complex tasks by relying on established principles, allowing for efficiency in problem-solving.
George Polya's quote emphasizes the inherent efficiency in mathematics, suggesting that true mastery lies not in brute force or extensive effort, but in understanding and applying underlying principles. By leveraging these principles, one can solve problems with ease, illustrating the beauty of mathematics as a tool that reduces the labor involved in finding solutions.
In practice
A teacher might use this quote to inspire students to appreciate the problem-solving aspects of mathematics.
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!
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.
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.
Employers, like most people, tend to trust their intuitions. But when employers decide whom to hire, they trust those intuitions far more than they should.
I want my kids to graduate from high school. But that's not enough. I also want them to go to college. Why? Because rich people's kids go to college. And if that's good enough for them, it's good enough for my kids. Because you know what? College graduates don't tend to go to jail as frequently as nongraduates.
Is sloppiness in speech caused by ignorance or apathy? I don't know and I don't care.
A man only learns in two ways, one by reading, and the other by association with smarter people.
Once you educate the boys, they tend to leave the villages and go search for work in the cities, but the girls stay home, become leaders in the community, and pass on what they've learned. If you really want to change a culture, to empower women, improve basic hygiene and health care, and fight high rates of infant mortality, the answer is to educate girls.
The books that help you most are those which make you think that most. The hardest way of learning is that of easy reading; but a great book that comes from a great thinker is a ship of thought, deep freighted with truth and beauty.
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