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Their backers also say that teachers’ freedom of speech is imperiled when we penalize their out-of-school remarks. But these defenders have it backward. The truly scary restrictions on teacher speech lie inside the schoolhouse walls, not beyond them.

For years school curriculums have emphasized top-down instruction, especially for topics like math and science. Learn the rules first — the theorems, the order of operations, Newton’s laws — then make a run at the problem list at the end of the chapter. Yet recent research has found that true experts have something at least as valuable as a mastery of the rules: gut instinct, an instantaneous grasp of the type of problem they’re up against. Like the ballplayer who can “read” pitches early, or the chess master who “sees” the best move, they’ve developed a great eye.

Now, a small group of cognitive scientists is arguing that schools and students could take far more advantage of this same bottom-up ability, called perceptual learning