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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Mastery may be necessary to become a mathematician, but for the rest, who mostly lack the curiosity, competence is still necessary. Rote learning through gamification makes that possible
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For what purposes is mathematical competence necessary? Vast numbers of persons seem to manage quite well without it. How about musical competence? How about artistic competence? Do you routinely feel a need for "color facts"? How about for "perspective facts"?
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Basic literacy and numeracy is necessary if we are to continue liberal democracy. People who can't do numbers cannot assess probabilities, like the chance of someone like Trump actually getting elected. They can't even balance a household budget, and tend to deny all science.
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Please read Liping Ma's KNOWING AND TEACHING ELEMENTARY MATHEMATICS. American teachers of mathematics typically lack the sort of numeracy you insist is necessary to all. Of course their students do as well. And yet the republic endures.
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Like a frog in a warming pot, you have a strange definition of "endure". The internet means a reduction in the degree to which you are controlled by an educated elite... and it shows.
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I have a PhD in mathematical physics. I am the educated elite.
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Your education is irrelevant to the public's need and propensity for it, except inasmuch as your position of privilege may have blinded you to that
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