Ah, now this is something I may be able to convey best with a picture. Take a look at this number, for example.pic.twitter.com/NNNbnJodbE
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Mathematical physicist and mentor to mathematically talented youth. Talent is that which bridges the gap between what can be taught and what must be learned.
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Ah, now this is something I may be able to convey best with a picture. Take a look at this number, for example.pic.twitter.com/NNNbnJodbE
This is a beautiful, classical number. Pierre de Fermat, one of the first great students of number, was enchanted by it.
Most people never realize that this number has this fascinating, multifaceted, 7-based structure. They know it only as 400.
But to refer to this marvelous creature merely as 400 is to miss all its meaning and magic and poetry. No real mathematician would do this.
It takes specialized training and extensive experience to recognize the beauty in its conventional representation, as 7^0 + 7^1 + 7^2 + 7^3.
It is easier to see it in the picture I have shown you. But symbolic representations are powerful. Just as written language is powerful.
No symbolic representations are needed, though, to appreciate the wonder of this number, which, like the best words, has multiple meanings.
And indeed some of the meaning in a number may be best appreciated by "seeing it," rather than by representing it by symbols of any kind.
Some of the meaning of a word comes from its sound when pronounced -- and some of the meaning in a number comes from its visible "shape."
I've stressed that numbers are like words, and the metaphor continues: mathematics is like language, its masterpieces like great narratives.
Unfortunately, mathematics education aims to teach the written language of mathematics before its students have learned the spoken language.
Learning a new abstract system of symbolic representation in order to write the words of a language you do not speak is naturally very hard.
We wait until children have extensive experience of spoken language before we try to teach them to read and write. By then, they're eager.
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