Andreas Wagner in Nautilus:
Mind-wandering is often considered a harmless quirk, as in the cliché of the scatter-brained professor. But it has real consequences. Let’s begin with the bad ones. Absentminded people perform less well on tests that require focused attention, such as reading comprehension tests. More worrisome, they also perform more poorly on tests that you better not flunk if you have any career aspirations. Among them is the Scholastic Aptitude Test that many colleges require for admission. But mind-wandering also has an upside—at least for well-trained minds. Indeed, many anecdotes of creators like Einstein, Newton, and eminent mathematician Henri Poincaré, report that these scientists solved important problems while not actually working on anything. The common wisdom that the best ideas arrive in the shower is exemplified by Archimedes’s discovery of how to measure an object’s volume. (OK, he was in a bathtub.) But while Archimedes’s discovery was triggered by the rising water as he entered the tub, other breakthroughs surface apropos of nothing. Take this well-known quote from the Poincaré describing a period in his life when he had worked without success on a mathematical problem:
Disgusted with my failure, I went to spend a few days at the seaside, and thought of something else. One morning, walking on the bluff, the idea came to me, with … brevity, suddenness, and immediate certainty, that the arithmetic transformations of indeterminate ternary quadratic forms were identical with those of non-Euclidean geometry.
The apparently idle period before such insights arrive has a name: incubation. If hard and seemingly futile work on a difficult problem is followed up with a less demanding activity that does not require complete focus—walking, showering, cooking—a mind is free to wander. And when that mind incubates the problem, it can stumble upon a solution.
More here.