Rote learning in mathematics is almost entirely focused on mastery of systems of symbolic representation -- on notations, rather than notions. Asking beginners in math to learn the manipulation of symbols first is like asking toddlers to learn to read before they learn to talk.
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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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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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