You should read this very interesting article about daydreaming by Jessica Lahey in the Atlantic.

I’ve been reading about daydreaming extensively lately, and it has caused me regret every time I roused one of my students out of their reverie so they would start working on something “more productive.” Daydreaming has been found to be anything but counter-productive. It may just be the hidden wellspring of creativity and learning in the guise of idleness.

Not all mental downtime is alike, of course. Downtime spent playing a video game or zoning out with a television show may have its charms, but the kind of downtime I am talking about is different. I’m talking about the kind of mind-wandering that happens when the brain is free of interruption and allowed to unhook from the runaway train of the worries of the day. When the mind wanders freely between random thoughts and memories that float through our consciousness, unbidden. Television, videogames, and other electronic distractions prevent this kind of mental wandering because they interrupt the flow of thoughts and memories that cement the foundation of positive, productive daydreaming.

This is a point I make in my book. Eventually concentration fails, and your capacity to direct your attention on a problem flags; but the smart thing to do isn't to spend an hour surfing the Web or watching Home Alone 4: The Homecoming, but instead do things that allow your mind to unfocus and wander. In that state your mind can continue to turn over problems without your active management. Sometimes your mind will arrive at answers that have been eluding you, and it'll feel as if they've just popped into your head; at other times a new approach to a problem will dawn. But either way, it's essential to allow the mind to unfocus itself a bit, to let it loose to work on its own.

In other words, Lahey continues,

daydreaming only appears lazy from the outside, but viewed from the inside—or from the perspective of a psychologist, such as Kaufman, or a neuroscientist, such as Mary Helen Immordino-Yang—a complicated and extremely productive neurological process is taking place. Viewed from the inside, our children are exploring the only space where they truly have autonomy: their own minds.

Immordino-Yang’s work on the virtue of mental downtime includes the paper “Rest is not Idleness: Implications of the Brain’s Default Mode for Human Development and Education.” The title quotes a 19th-century British banker named John Lubbock, who wrote, “Rest is not idleness, and to lie sometimes on the grass under trees on a summer's day, listening to the murmur of the water, or watching the clouds float across the sky, is by no means a waste of time.”

Lubbock, according to Immordino-Yang, was way ahead of his time in understanding the value of idleness to our essential neurological functioning. What Lubbock called rest, Immordino-Yang calls “constructive internal reflection,” and she considers it is vital to learning and emotional well-being.

John Lubbock is not very well-known now, except among historians of Victorian science; but all nine of us recognize him as more than a banker. In his day he was a politician, occasional university chancellor, and the son of noted mathematician and statistician Sir John Lubbock. The elder Lubbock (the 3rd Baronet, to distinguish him from the first two baronets, who confusingly were also named John), was a member of the X Club, a circle of elite Victorian scientists, a member of the British Association for the Advancement of Science, and an officer in the Royal Society.

From 1842, the Lubbocks were also Charles Darwin's next-door neighbors. Both were products of fairly nouveau families that had prospered during the Industrial Revolution, both were Cambridge men, T. H. Huxley was their mutual friend, they were members of some of the same scientific societies. Darwin moved from London to the village of Downe, and bought Down House, where he lived the rest of his life. The Lubbock estate, High Elms, was adjacent to Down House, and Lubbock leased to Darwin a piece of land on which Darwin built his Sandwalk, a walking path where he would go every day to walk and think.

Darwin would use the Sandwalk as a break from the lab or writing, but often in the course of his walks, as he looked at the landscape or the trees (he had the path planted with trees and shrubs, and there were several well-established trees there as well), his mind would turn over a problem he was working on. He had the habit of setting a few stones on the side of the path, and moving them to the other side as he passed, to keep track of how far he'd walked; his sons would sometimes test to see how deep in thought he was by moving them around, and seeing if he would notice.

The point is, Darwin's walks were exactly the kind of productive rest, or constrtuctive internal reflection, that Immordino-Yang advocates, and that the younger Lubbock was thinking of when he distinguished between rest and idleness. Indeed, he might have had Darwin in mind: as a child he was a regular visitor to Down in his youth, and probably would have seen Darwin on the Sandwalk.

I'm growing convinced that one quiet but serious problem we have today is that we've unlearned the real value of rest. As Diarmaid MacCulloch puts it in his wonderful Silence: A Christian History, the Sabbath as described in Genesis "was a vital part of the creative process rather than simply an end of it." Likewise, we should recognize– in our educational policies, and in our own lives– that we can choose forms of rest that are "a vital part of the creative process," rather than merely distractions.

Update: I'd not realized when I first wrote this that the author of the "rest is not idleness" quote was the fourth baronet, not the third, and so previously had conflated the two. I've updated the piece to separate them.