Historically, most mathematicians have learned arithmetic not by memorizing multiplication tables but by extensive self-directed investigation of numbers as experimental data. Which numbers arise as divisors of sums of two square numbers? One way to find out: do the experiment.
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Historically, most mathematicians had solid fundamental rote training in schools stricter and less forgiving than any you'll see today, and then they supplemented that education with "extensive self-directed investigation of numbers."
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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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And, again, I know of no pedagogical approach capable of succeeding (whatever "success" means) with everyone.
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