Failing in the discharge of this obligation is failing in all respects.
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Replying to @MathPrinceps @stevenstrogatz and
It is certainly true that effort is needed to acquire skill, and that skill is needed for successful performance of complex tasks. But not all exertion is miserable, nor need it be.
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Replying to @MathPrinceps @stevenstrogatz and
The notion that suffering ennobles is pernicious. Merely because a form of practice is onerous does not make it valuable.
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Replying to @MathPrinceps @stevenstrogatz and
A key question, typically sadly neglected, is: what practice regimen is optimal? In professional and Olympic sports, systematic attempts to answer this question have led to vast improvements in outcomes.
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Replying to @MathPrinceps @stevenstrogatz and
Certainly not all forms of practice are equally beneficial, and excess practice of the wrong sort can be harmful. So one may reasonably question the wisdom of advising practice without specifying what sort of practice is advisable.
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Replying to @MathPrinceps @stevenstrogatz and
Only experiment can determine what practice regimen is optimal, but it seems clear that the search should be guided by the experience and intuition of experts (who, after all, have acquired their expertise through practice of a presumably more-nearly-optimal sort.)
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Replying to @MathPrinceps @stevenstrogatz and
But experts have very little incentive to grapple seriously with this question. And so we have a sort of tragedy of the commons: it's to the advantage of every individual expert to imagine this work to be somebody else's job -- with the result that no one even attempts to do it.
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Replying to @MathPrinceps @stevenstrogatz and
And with experts ignoring the job, it falls to those whose expertise is scant -- who, unsurprisingly, achieve poor results.
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Replying to @MathPrinceps @stevenstrogatz and
What happens when a real expert undertakes to devise a practice regimen based on his own experience and intuition? A conspicuous and instructive example is that of Frédéric Chopin, whose etudes are practiced by every pianist aspiring to mastery of the instrument.
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Replying to @MathPrinceps @stevenstrogatz and
But Chopin's etudes are far more than mere technical exercises; they're also compositions of great beauty, frequently programmed, and beloved by pianists and audiences alike.
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Imagine what mathematics learning would be like, if it required of young would-be mathematicians the mastery of mathematical repertoire as aesthetically rewarding as the Chopin etudes!
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It’s always a pleasure to read your thoughts. Thanks for sharing them.
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You are most kind to say so, Professor Strogatz. I am grateful (and relieved) to learn that I'm not making a pest of myself here.
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