These comments about analogies remind us that much depends on how a problem solver thinks about, or interprets, a problem. If she focuses on the problem’s underlying dynamic and not its surface form, she’s more likely to use an analogy. If she breaks the problem into subproblems, she’s more likely to find a relevant subroutine. If she approaches the problem with an appropriate mental set, this too can be helpful.
These points suggest that when problem solving fails, a person’s best bet may be to change her understanding of the problem—to one that highlights the problem’s sub-goals or one that suggests an analogy. This point certainly seems in line with common experience: Sometimes we’re utterly baffled by a problem; but then, later, we find an alternative way to approach the issue and quickly come up with the answer. Sometimes this restructuring of the problem is gradual, as we change our ideas about the problem one by one; but sometimes it’s quite abrupt, experienced as a flash of insight with an accompanying exclamation of “Aha!”
It should be said that these flashes of insight, when they occur, aren’t uniformly ben-eficial. Sometimes the (apparent) insights turn out to be false alarms, because the new understanding simply leads to yet another dead end (Metcalfe, 1986; Metcalfe & Weibe, 1987). Thus, it seems that the “Aha!” experience should not be understood as “I see the solution!” Instead, the experience merely implies, “I’ve discovered a new approach!” Whether this approach will turn out to be productive can only be decided after the fact. One way or another, though, and whether the changes are sudden or slow, these changes in the way a problem is defined are often essential for breaking out of an unproductive mental set and moving toward one that will lead to a solution. (The prob-lems in Figure 9.19 may also involve insight. See Figure 9.24 for their solutions.)