Mathematics education in the schools is a relatively recent phenomenon. Neither Euler nor Lagrange experienced any of it.
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Replying to @MathPrinceps @davidmanheim and
Neither did many of history's greatest mental calculators -- who, not infrequently, were illiterate.
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Replying to @MathPrinceps @davidmanheim and
What produces mastery is not rote repetition or practice, but passionate curiosity. An obsessed student practices extensively, without even noticing that he is doing so. Practice is a by-product of curiosity. The more passionate the curiosity, the more extensive the practice.
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I'm telling you yet again that it's not always sufficient. You can feel free to tell me I lacked passionate curiosity - but you're wrong, and if you really want to investigate, I'll be happy to put you in touch with my professors, or my high school math teachers.
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Nothing, to my knowledge, is always sufficient. I am speaking of what has been typical of great past masters. My purpose is to refute the assertion that rote is always necessary. A single example of mastery attained without school-imposed rote learning suffices for this purpose.
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Replying to @MathPrinceps @davidmanheim and
Illiterate mental calculators like Jacques Inaudi pose serious problems for any theory of arithmetic learning that insists it must be rooted in rote practice imposed by schools.
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I don't claim that it's impossible to be a mathematician without rote learning, but I do claim, based on my experience and that of others I know who went to schools that didn't force us to memorize the basics, that many students won't manage to be successful without it.
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One need not be forced to memorize the basics. One can desire to learn them.
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So you agree that actually memorizing these basic facts is necessary - at some point, in some way, whether voluntary or obligatory, and whether done by incorporation in other tasks or repetition alone?
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These conversations always seem to come back to the question of what "memorize" means. There are many ways to get something into your memory/knowledge, but I wouldn't call the all "memorizing". For example most people do not memorize their own name.
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If by "memorization" we mean a conscious, systematic effort to preserve some datum for future recall, then very little of what we remember is the result of memorization. This, by the way, is why masters often struggle to explain their methods: they did not learn them consciously.
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I don't think this claim generalizes to most people or most learning. Most early learning is due to teachers / parents intending that we learn things, and later learning and studying is similar, but self-imposed, though perhaps less consciously regarded at an attempt to memorize.
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