Quoting from Sections 7 and 8 of William Thurston’s Mathematical Education essay titled “Mystery and Mastery” and “Competence and Intimidation”
Mathematics is amazingly compressible: you may struggle a long time, step by
step, to work through some process or idea from several approaches. But once you
really understand it and have the mental perspective to see it as a whole, there
is often a tremendous mental compression. You can file it away, recall it quickly
and completely when you need it, and use it as just one step in some other mental
process. The insight that goes with this compression is one of the real joys of
mathematics.
After mastering mathematical concepts, even after great effort, it becomes very
hard to put oneself back in the frame of mind of someone to whom they are mysterious.
I remember as a child, in fifth grade, coming to the amazing (to me) realization
that the answer to 134 divided by 29 is 134⁄29 (and so forth). What a tremendous
labor-saving device! To me, ‘134 divided by 29’ meant a certain tedious chore,
while 134⁄29 was an object with no implicit work. I went excitedly to my father to
explain my major discovery. He told me that of course this is so, a/b and a divided
by b are just synonyms. To him it was just a small variation in notation.
One of my students wrote about visiting an elementary school and being asked
to tutor a child in subtracting fractions. He was startled and sobered to see how
much is involved in learning this skill for the first time, a skill which had condensed
to a triviality in his mind.
Mathematics is full of this kind of thing, on all levels. It never stops.
[...]
Similarly, students at more advanced levels know many things which less advanced
students don’t yet know. It is very intimidating to hear others casually toss
around words and phrases as if any educated person should know them, when you
haven’t the foggiest idea what they’re talking about. Less advanced students have
trouble realizing that they will (or would) also learn these theories and their associated
vocabulary readily when the time comes and afterwards use them casually
and naturally. I remember many occasions when I felt intimidated by mathematical
words and concepts before I understood them: negative, decimal, long division,
infinity, algebra, variable, equation, calculus, integration, differentiation, manifold,
vector, tensor, sheaf, spectrum, etc. It took me a long time before I caught on to
the pattern and developed some immunity.
Quoting from Sections 7 and 8 of William Thurston’s Mathematical Education essay titled “Mystery and Mastery” and “Competence and Intimidation”
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