9.6 Problem Solving: Working It Out

You have a problem when you find yourself in a place where you don’t 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’s say that it’s the night before a test, and you are trying to study but just can’t settle down and focus on the material. This is a situation you don’t 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 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 isn’t 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.

Means–Ends Analysis

Even though it may not be easy to put together this Lego model, having instructions for assembly makes it a well-defined problem.

CORBIS COLLECTION/ALAMY

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:

1. Analyze the goal state (i.e., the desired outcome you want to attain).
2. Analyze the current state (i.e., your starting point, or the current situation).
3. List the differences between the current state and the goal state.
4. Reduce the list of differences by
   • Direct means (a procedure that solves the problem without intermediate steps).
   • Generating a subgoal (an intermediate step on the way to solving the problem).
   • Finding a similar problem that has a known solution.

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

The goal state is a patient without the tumor and with undamaged surrounding tissue. The current state is a patient with an inoperable tumor surrounded by fragile tissue. The difference between these two states is the tumor. A direct-means solution would be to destroy the tumor 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 won’t work either: The weaker dose wouldn’t damage the healthy tissue but also wouldn’t kill the tumor. So, what to do? Find a similar problem that has a known solution. Let’s see how this can be done.

Analogical Problem Solving

Figure 9.13: Analogical Problem Solving Just as smaller, lighter battalions can reach the fortress without damaging the bridges, many small X-ray doses can destroy the tumor without harming the delicate surrounding tissue. In both cases, the additive strength achieves the objective.

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:

Does this story suggest a solution to the tumor problem? It should. Removing a tumor 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 tumor problem:

Did this solution occur to you after reading the fortress story? In studies that have used the tumor problem, only 10% of participants spontaneously generated the correct solution. This percentage rose to 30% if participants read the island fortress problem or other analogous story. However, the success climbed dramatically to 75% 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).

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

Creativity and Insight

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).

Genius and Insight

Figure 9.14: Genius and Insight Young Friedrich Gauss imagined the scheme shown here and quickly reduced a laborious addition problem to an easy multiplication task. Gauss’s early insight led him to realize later an intriguing truth: This simple solution generalizes to number series of any length.

After Wertheimer, 1945/1982.

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.

Figure 9.15: Insightful Solutions Are Really Incremental Participants were asked to find a fourth word that was associated with the other three words in each series. Even if I they couldn’t find a solution, they could reliably choose which series of three words were solvable and which were not. Try to solve these.

After Bowers et al., 1990.

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 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.

Even if participants couldn’t 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).

Functional Fixedness

Figure 9.16: Functional Fixedness and the Candle Problem How can you use these objects–a box of matches, thumbtacks, and a candle–to mount the candle on the wall so that it illuminates the room? Give this problem some thought before you check the answer in Figure 9.19.

If insight is a simple incremental process, why isn’t its occurrence more frequent? In the research discussed previously, participants produced insightful solutions only 25% 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, your task is to use the items on the table to find a way to hold on to a string hanging from the ceiling and at the same time reach a second string that is too far away to grasp.

Figure 9.17: Functional Fixedness and the String Problem The strings hung from hooks on either side of the ceiling are long enough to be tied together, but they are positioned too far apart to reach one while holding on to the other. Using the tools shown on the table (nails, matches and a hammer), how can you accomplish the task? Compare your answer to that in Figure 9.20.

Figure 9.18: The Nine-Dot Problem Connect all nine dots with four straight lines without lifting your pencil from the paper. Compare your answer to those in Figure 9.21.

Difficulty solving these problems derives from our tendency to think of the objects only in terms of their normal, typical, or “fixed” functions. We don’t think to use the matchbox for a candleholder because boxes typically hold matches, not candles. Similarly, using the hammer as a pendulum weight doesn’t 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.)

Figure 9.19: The Solution to the Candle Problem What makes this problem difficult is that the usual function of the box (to hold matches) interferes with recognizing that it can be tacked to the wall to serve as a candleholder.

Figure 9.20: The Solution to the String Problem The usual function of the hammer (to pound things) interferes with recognizing that it can also serve as a weighted pendulum to swing the string into the person’s grasp.

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).

HOT SCIENCE: Sudden Insight and the Brain

The “aha!” moment that accompanies a sudden flash of insight is a compelling experience, one that highlights that solving a problem based on insight feels radically different from solving it through step-by-step analysis or trial and error. This difference in subjective experience suggests that something different is going on in the brain when we solve a problem using insight instead of analytic strategies (Kounios & Beeman, 2009).

Figure 9.21: Two Solutions to the Nine-Dot Problem Solving this problem requires “thinking outside the box,” that is, going outside the imaginary box implied by the dot arrangement. The limiting box isn’t really there; it is imposed by the problem solver’s perceptual set.

To examine brain activity associated with insight, researchers used a procedure called compound remote associates that is similar in some respects to the three-word problems displayed in Figure 9.15. Each compound remote associates problem consists of three words, such as crab, pine, and sauce. Sometimes people solve these problems with a flash of insight: The solution word (apple!) suddenly pops into their minds, seemingly out of nowhere. Other times, people solve the problem by using analytic strategies that involve trying out different alternatives generating a compound word for crab and then assessing whether they fit pine and sauce. Crabgrass, works, but grass doesn’t work with pine and sauce. Crabapple? Problem solved.

Participants are instructed to press a button the moment a solution comes to mind, then describe whether they arrived at the solution via insight or through an analytic strategy. In an initial study, researchers used the electroencephalograph (EEG; see the Neuroscience chapter) to measure brain electrical activity as participants attempted to solve the problems. What they observed was striking: Beginning about one-third of a second before participants came up with a solution, there was a sudden and dramatic burst of high-frequency electrical activity (40 cycles per second or gamma band) for problems solved via insight as compared with problems solved via analytic strategies (Jung-Beeman et al., 2004). This activity was centered over the front part of the right temporal lobe, slightly above the right ear. The researchers then performed a similar study using fMRI to measure brain activity, and found that this right temporal area was the only region in the entire brain that showed greater activity for insight solutions compared with solutions based on analytic strategies.

The great French scientist Louis Pasteur once stated: “Chance favors only the prepared mind.” Inspired by this observation, researchers asked whether brain activity occurring just before presentation of a problem influenced whether that problem was solved via insight or analytic strategies (Kounios et al., 2006). It did. In the moments before a problem was solved with an insight solution, there was increased activity deep in the frontal lobes, in a part of the brain known as the anterior cingulate, which controls cognitive processes such as the ability to switch attention from one thing to another. The researchers suggested that this increased activity in the anterior cingulate allowed participants to attend to and detect associations that were only weakly activated, perhaps at a subconscious level, and that facilitate sudden insight.

A related study with the compound remote associates task revealed that when people were in a positive mood, they solved more problems with insight than people who were in a less positive mood (Subramaniam et al., 2009). Moreover, as shown in the figure, positive mood was associated with heightened activity in the anterior cingulate during the moments before a problem was presented–suggesting that being in a positive mood helps to prepare the brain for sudden insight by “turning on” the anterior cingulate and thereby increasing one’s ability to detect associations that aid problem solution. This link between positive mood and insight might help explain a striking recent finding: Moderate levels of alcohol intoxication resulted in enhanced performance on the remote associates task, and also produced a stronger feeling that problem solutions are the result of sudden insight (Jarosz, Colflesh, & Wiley, 2012). Alternatively, the beneficial effects of alcohol could be attributable to a reduction in controlled, focused processing, which frees up the ability to generate distant associations.

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Brain activity prior to problem solving provides clues about which individuals are more likely to rely on insight over analytic strategies to solve compound remote associates (Kounios et al., 2008). Using EEG to measure resting brain activity, researchers found that insight problem solvers showed more resting activity in the right cerebral hemisphere than did analytic problem solvers, which is consistent with other research linking creativity with right-hemisphere activity (Folley & Park, 2005; Howard-Jones et al., 2005).

The results of these studies suggest that the familiar image of a light bulb going off in your head when you experience an “aha!” moment is on the mark: Those moments are indeed accompanied by something like an electrical power surge in the brain, and are preceded by specific types of electrical activity patterns. It seems likely that future research will tell us much more about how to turn on the mental light bulb and keep it burning bright.

(a) Being in a positive mood was associated with increased activity in the anterior cingulate (yellow area) in the moments before people solved a problem via sudden insight. (b) Positive mood (Positive Affect-Negative Affect) is associated with heightened activity in the anterior cingulate, suggesting an increase in one’s ability to create associations needed for problem solving.

SUBRAMANIAM ET AL., 2008