TODO: process the quotes below into the article.
In his book How to Become a Straight-A Student Cal Newport writes:
Next, try to solve the problem in the most obvious way possible. This, of course, probably won't work, because most difficult problems are tricky by nature. By failing in this initial approach, however, you will have at least identified what makes this problem hard. Now you are ready to try to come up with a real solution.
The next step is counterintuitive. After you've primed the problem, put away your notes and move on to something else. Instead of trying to force a solution, think about the problem in between other activities. As you walk across campus, wait in line at the dining hall, or take a shower, bring up the problem in your head and start thinking through solutions. You might even want to go on a quiet hike or long car ride dedicated entirely to mulling over the question at hand.
More often than not, after enough mobile consideration, you will finally stumble across a solution. Only then should you schedule more time to go back to the problem set, write it down formally, and work out the kinks. It's unclear exactly why solving problems is easier when you're on the go, but, whatever the explanation, it has worked for many students. Even better, it saves a lot of time, since most of your thinking has been done in little interludes between other activities, not during big blocks of valuable free time.
In her book A Mind for Numbers, Barbara Oakley distinguishes between "focused mode" (same thing as wikipedia:task-positive network?) and "diffuse mode" (same thing as wikipedia:task-negative network?):
Focused-mode thinking is essential for studying math and science. It involves a direct approach to solving problems using rational, sequential, analytical approaches. [...] Diffuse-mode thinking is also essential for learning math and science. It allows us to suddenly gain a new insight on a problem we’ve been struggling with and is associated with “big-picture” perspectives. Diffuse-mode thinking is what happens when you relax your attention and just let your mind wander. This relaxation can allow different areas of the brain to hook up and return valuable insights.
In his book The Mind Is Flat, Nick Chater doesn't deny the phenomenon, but disagrees that this sort of incubation-based thinking works due to unconsciously working on the problem:
Poincaré and Hindemith cannot possibly be right. If they are spending their days actively thinking about other things, their brains are not unobtrusively solving deep mathematical problems or composing complex pieces of music, perhaps over days or weeks, only to reveal the results in a sudden flash. Yet, driven by the intuitive appeal of unconscious thought, psychologists have devoted a great deal of energy in searching for evidence for unconscious mental work. In these studies, they typically give people some tricky problems to solve (e.g. a list of anagrams); after a relatively short period of time, they might instruct participants to continue, to take a break, to do another similar or different mental task, or even get a night’s sleep, before resuming their problems. According to the ‘unconscious work’ perspective, resuming after a break should lead to a sudden improvement in performance, compared with people who just keep going with the task. Studies in this area are numerous and varied, but I think the conclusions are easily summarized. First, the effects of breaks of all kinds are either negligible or non-existent: if unconscious work takes place at all, it is sufficiently ineffectual to be barely detectable, despite a century of hopeful attempts. Second, many researchers have argued that the minor effects of taking a break – and indeed, Poincaré’s and Hindemith’s intuitions – have a much more natural explanation, which involves no unconscious thought at all.
The simplest version of the idea comes from thinking about why one gets stuck with a difficult problem in the first place. What is special about such problems is that you can’t solve them through a routine set of steps (in contrast, say, to adding up columns of numbers, which is laborious but routine) – you have to look at the problem in the ‘right way’ before you can make progress (e.g. with an anagram, you might need to focus on a few key letters; in deep mathematics or musical composition, the space of options might be large and varied). So ideally, the right approach would be to fluidly explore the range of possible ‘angles’ on the problem, until hitting on the right one. Yet this is not so easy: once we have been looking at the same problem for a while, we feel ourselves to be stuck or going round in circles. Indeed, the cooperative computational style of the brain makes this difficult to avoid.