Rote learning tables shouldn't mean chanting them in unison. We played pair racing games filling a randomised grid with a wax pencil - great fun!
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Absolutely - gamification is a fantastic tool for encouraging fun involvement! (And spaced repetition is much better, and more effective, than chanting.)
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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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I'd love to see an example, then. But it seems that Euler learned arithmetic from his father and took private lessons as a kid. And if you havbe a source that Lagrange wasn't in a grammar school, I'd love to see it - but it would amaze me, given his acceptance to a law school.
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What is at issue is the role of rote practice in Euler's early learning. Euler himself says that his father, a student of Jacob Bernoulli (!), "tried to impart to me the first principles of mathematics," using Rudolph's "Coss." This does not sound like rote practice to me.
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