You have a problem when you find yourself in a place where you do not want to be. In such circumstances, you try to find a way to change the situation so that you end up in a situation you do want. Let us say that it is the night before a test, and you are trying to study but just cannot settle down and focus on the material. This is a situation you do not want. So, you try to think of ways to help yourself focus. You might begin with the material that most interests you, or provide yourself with rewards, such as a music break or a trip to the refrigerator. If these activities enable you to get down to work, your problem is solved.

Two major types of problems complicate our daily lives. The first and most frequent is the ill-defined problem, one that does not have a clear goal or well-defined solution paths. Your study block is an ill-defined problem: Your goal is not clearly defined (i.e., somehow get focused), and the solution path for achieving the goal is even less clear (i.e., there are many ways to gain focus). Most everyday problems (being a better person, finding that special someone, achieving success) are ill defined. In contrast, a well-defined problem is one with clearly specified goals and clearly defined solution paths. Examples include following a clear set of directions to get to school, solving simple algebra problems, or playing a game of chess.

In 1945, German psychologist Karl Duncker reported some important studies of the problem-solving process. He presented people with ill-defined problems and asked them to “think aloud” while solving them (Duncker, 1945). Based on what people said about how they solve problems, Duncker described problem solving in terms of means-ends analysis, a process of searching for the means or steps to reduce the differences between the current situation and the desired goal. This process usually took the following steps:

Consider, for example, one of Duncker’s problems:

A patient has an inoperable tumour in his abdomen. The tumour is inoperable because it is surrounded by healthy but fragile tissue that would be severely damaged during surgery. How can the patient be saved?

The goal state is a patient without the tumour and with undamaged surrounding tissue. The current state is a patient with an inoperable tumour surrounded by fragile tissue. The difference between these two states is the tumour. A direct-means solution would be to destroy the tumour with X-rays, but the required X-ray dose would destroy the fragile surrounding tissue and possibly kill the patient. A subgoal would be to modify the X-ray machine to deliver a weaker dose. After this subgoal is achieved, a direct-means solution could be to deliver the weaker dose to the patient’s abdomen. But this solution will not work either: The weaker dose would not damage the healthy tissue but also would not kill the tumour. So, what to do? Find a similar problem that has a known solution. Let us see how this can be done.

When we engage in analogical problem solving, we attempt to solve a problem by finding a similar problem with a known solution and applying that solution to the current problem. Consider the following story:

An island surrounded by bridges is the site of an enemy fortress. The massive fortification is so strongly defended that only a very large army could overtake it. Unfortunately, the bridges would collapse under the weight of such a huge force. So, a clever general divides the army into several smaller units and sends the units over different bridges, timing the crossings so that the many streams of soldiers converge on the fortress at the same time and the fortress is taken.

Does this story suggest a solution to the tumour problem? It should. Removing a tumour and attacking a fortress are very different problems, but the two problems are analogous because they share a common structure: The goal state is a conquered fortress with undamaged surrounding bridges. The current state is an occupied fortress surrounded by fragile bridges. The difference between the two states is the occupying enemy. The solution is to divide the required force into smaller units that are light enough to spare the fragile bridges and send them down the bridges simultaneously so that they converge on the fortress. The combined units will form an army strong enough to take the fortress (see FIGURE 9.13).

This analogous problem of the island fortress suggests the following direct-means solution to the tumour problem:

Surround the patient with X-ray machines and simultaneously send weaker doses that converge on the tumour. The combined strength of the weaker X-ray doses will be sufficient to destroy the tumour, but the individual doses will be weak enough to spare the surrounding healthy tissue.

Did this solution occur to you after reading the fortress story? In studies that have used the tumour problem, only 10 percent of participants spontaneously generated the correct solution. This percentage rose to 30 percent if participants read the island fortress problem or other analogous story. However, the success climbed dramatically to 75 percent among participants who had a chance to read more than one analogous problem or were given a deliberate hint to use the solution to the fortress story (Gick & Holyoak, 1980).

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Why was the fortress problem so ineffective by itself? Problem solving is strongly affected by superficial similarities between problems, and the relationship between the tumour and fortress problems lies deep in their structure (Catrambone, 2002).

Analogical problem solving shows us that successfully solving a problem often depends on learning the principles underlying a particular type of problem and also that solving lots of problems improves our ability to recognize certain problem types and generate effective solutions. Some problem solving, however, seems to involve brilliant flashes of insight and creative solutions that have never before been tried. Creative and insightful solutions often rely on restructuring a problem so that it turns into a problem you already know how to solve (Cummins, 2012).

Consider the exceptional mind of the mathematician Friedrich Gauss (1777–1855). One day, Gauss’s elementary schoolteacher asked the class to add up the numbers 1 through 10. While his classmates laboriously worked their sums, Gauss had a flash of insight that caused the answer to occur to him immediately. Gauss imagined the numbers 1 through 10 as weights lined up on a balance beam, as shown in FIGURE 9.14. Starting at the left, each “weight” increases by 1. In order for the beam to balance, each weight on the left must be paired with a weight on the right. You can see this by starting at the middle and noticing that 5 + 6 = 11, then moving outward, 4 + 7 = 11, 3 + 8 = 11, and so on. This produces five number pairs that add up to 11. Now the problem is easy. Multiply. Gauss’s genius lay in restructuring the problem in a way that allowed him to notice a very simple and elegant solution to an otherwise tedious task—a procedure, by the way, that generalizes to series of any length.

According to Gestalt psychologists, insights such as these reflect a spontaneous restructuring of a problem. A sudden flash of insight contrasts with incremental problem-solving procedures in which one gradually gets closer and closer to a solution. Early researchers studying insight found that people were more likely to solve a non-insight problem if they felt they were gradually getting “warmer” (incrementally closer to the solution). But whether someone felt warm did not predict the likelihood of their solving an insight problem (Metcalfe & Wiebe, 1987). The solution for an insight problem seemed to appear out of the blue, regardless of what the participant felt.

Later research, however, suggests that sudden insightful solutions may actually result from unconscious incremental processes (Bowers et al., 1990). In one study, research participants were shown paired, three-word series like those in FIGURE 9.15 (on the next page) and asked to find a fourth word that was associated with the three words in each series. However, only one series in each pair had a common associate. Solvable series were termed coherent, whereas those with no solution were called incoherent.

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Even if participants could not find a solution, they could reliably decide, more than by chance alone, which of the pairs was coherent. However, if insightful solutions actually occur in a sudden, all-or-nothing manner, their performance should have been no better than chance. Thus, the findings suggest that even insightful problem solving is an incremental process—one that occurs outside of conscious awareness. The process works something like this: The pattern of clues that constitute a problem unconsciously activates relevant information in memory. Activation then spreads through the memory network, recruiting additional relevant information (Bowers et al., 1990). When sufficient information has been activated, it crosses the threshold of awareness and we experience a sudden flash of insight into the problem’s solution.

Finding a connection between the words strawberry and traffic might take some time, even for someone motivated to figure out how they are related. But if the word strawberry activates jam in long-term memory (see the Memory chapter) and the activation spreads from strawberry to traffic, the solution to the puzzle may suddenly spring into awareness without the thinker knowing how it got there. What would seem like a sudden insight would really result from an incremental process that consists of activation spreading through memory, adding new information as more knowledge is activated. Nonetheless, studies of brain activity during problem solving reveal striking differences between problems solved by sudden insight and those solved by using more deliberate strategies (see the Hot Science box).

If insight is a simple incremental process, why is not its occurrence more frequent? In the research discussed previously, participants produced insightful solutions only 25 percent of the time. Insight is rare because problem solving (like decision making) suffers from framing effects. In problem solving, framing tends to limit the types of solutions that occur to us. Functional fixedness—the tendency to perceive the functions of objects as fixed—is a process that constricts our thinking. Look at FIGURES 9.16 and 9.17 and see if you can solve the problems before reading on. In Figure 9.16, your task is to illuminate a dark room using the following objects: some thumbtacks, a box of matches, and a candle. In Figure 9.17, you need to tie the two strings together, but they are too far apart to reach one while holding on to the other. Using the items on the table, you are expected to reach the string that is too far away to grasp.

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Difficulty solving these problems derives from our tendency to think of the objects only in terms of their normal, typical, or “fixed” functions. We do not think to use the matchbox for a candleholder because boxes typically hold matches, not candles. Similarly, using the hammer as a pendulum weight does not spring to mind because hammers are typically used to pound things. Did functional fixedness prevent you from solving these problems? (The solutions are shown in FIGURES 9.19 and 9.20.)

Sometimes framing limits our ability to generate a solution. Before reading on, look at FIGURE 9.18. Without lifting your pencil from the page, try to connect all nine dots with only four straight lines.To solve this problem, you must allow the lines you draw to extend outside the imaginary box that surrounds the dots (see FIGURE 9.21). This constraint does not reside in the problem but in the mind of the problem solver (Kershaw & Ohlsson, 2004). Despite the apparent sudden flash of insight that seems to yield a solution to problems of this type, research indicates that the thought processes people use when solving even this type of insight problem are best described as an incremental, means–ends analysis (MacGregor, Ormerod, & Chronicle, 2001).

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