## Cognitive Psychology: Theory, Process, and Methodology - Dawn M. McBride, J. Cooper Cutting 2019

Problem Solving

**Questions to Consider**

· How often and what kind of problems do you solve every day?

· How do you solve problems: through trial and error, through conscious deliberation, or do solutions just suddenly occur to you?

· Why are some problems more difficult to solve than others?

· What gets in your way when trying to solve problems?

· How do expert problem solvers differ from novices?

**Introduction: Problem Solving in Daily Life**

*Consider what might be a typical morning. Your alarm goes off and you stumble out of bed faced with a decision about what to wear that day. You’ve got a job interview in the afternoon, but you also want to hit the gym in the morning. You decide to dress in your sweats now, carry your nice clothes, and shower at the gym. You’ve got time for a quick breakfast, but you realize that you are out of milk. You are going to need to figure out a way to go to the store to buy some, but with your interview followed by your evening section of your psychology seminar course, you aren’t sure when you’ll have the time. But for now you are hungry, so you sit down to a cup of coffee and the Sudoku puzzle in the morning paper. You notice that there is a new kind of puzzle, KenKen. It looks similar to Sudoku, but as you try to solve it, you find that it is much harder for you to solve than the Sudoku you do every morning. After trying to finish your puzzles you head upstairs to pack your interview clothes, and a solution suddenly hits you. Instead of packing the clothes into your gym bag, you grab a small cooler and pack them into that. You realized that you should have time to grab some milk from the corner store before your interview, and it should keep cold in the cooler during both the interview and your seminar class. Happy with this solution, you head to the gym (and on the way begin to wonder what you will do with the cooler during your interview).*

*This opening story illustrates how our days are full of problems we try to find solutions for. The problems we face vary in size and scope: Some are little (e.g., solving the puzzle in the newspaper, figuring out what to wear), while others are larger (e.g., how to get a good job). Many people consider our problem-solving abilities to be the prototype of “higher thought,” a centerpiece of our cognitive processes. Typically, the problem-solving process has been described as a cycle of stages (e.g., Bransford & Stein, 1993; Dewey, 1910; Polya, 1957; Pretz, Naples, & Sternberg, 2003; Wallas, 1926). These stages typically include processes like the following:*

· *Recognize and identify the problem*

· *Define and mentally represent the problem*

· *Develop a solution strategy*

· *Allocate mental resources for solving the problem*

· *Monitor progress toward the goal and evaluate the solution*

*This cycle is not intended to imply serial stages of processing. Instead, it is intended to describe the kinds of cognitive processes involved in solving problems. This chapter focuses on research and theory on the first four stages of the cycle.*

**Recognizing and Identifying a Problem**

Researchers studying problem solving typically describe a problem as a situation in which there is a difference between a current state and a desired goal state. Problem solving is the process of developing a solution (or set of solutions) designed to change the state of affairs from the current state to the goal state. Consider three parts of our opening story that may be considered problems. Getting dressed in the morning involves the need to move from the state of undress (the current state) to getting dressed for the day (the goal state). On normal days you may not consider this a problem since you probably have a ready solution available. However, in our story special circumstances require a different solution, one in which you can dress appropriately for both a job interview and a workout at the gym. Rather than using your usual dressing solution, you have to come up with an alternative plan. Solving the puzzle in the morning paper is also an example of a problem. Consider the Sudoku puzzle in __Figure 11.1__ (the solution is found in __Figure 11.17__ at the end of the chapter). A Sudoku puzzle is typically a nine-by-nine grid with some of the cells blank and the others containing numbers. Your task is to complete the grid by filling in the empty cells with numbers, with the constraint that each row, column, and three-by-three cell doesn’t have any repeated numbers. Sudoku puzzles are set up so that there is only one correct solution. These features make the Sudoku a **well-defined problem**. This doesn’t mean that it is necessarily an easy problem to solve but rather that the goals and constraints are known, and by applying particular procedures a correct solution can be found. In contrast, the problems of getting a job, getting dressed for the day, or even arranging your day so that you can get milk, work out, and go to class and a job interview don’t typically have a single correct solution. Problems like these are considered **ill-defined problems**. Ill-defined problems lack clear paths between the current and goal states. As a result, ill-defined problems are often much more difficult to mentally represent, identify solution strategies for, and solve. Goel (2010) argues that performance patterns of brain-damaged patients (particularly those with frontal lobe lesions) suggest that there are neuropsychological differences between well- and ill-defined problems.

**Well-defined problem:** a problem that has a clearly defined goal state and constraints

**Ill-defined problem:** a problem that lacks a clearly defined goal state and constraints

**Figure 11.1** A Sudoku Puzzle

**Defining and Representing Problems**

Defining and representing a problem is the process of stating the scope and goal of the problem and organizing the knowledge needed for addressing the problem. This knowledge includes mentally representing the current and goal states, the rules or constraints, and the allowable operations available to solve the problem.

**Stop and Think**

· 11.1. Make a list of some of the problems you have already faced today.

· 11.2. For each problem in Stop and Think 11.1, identify the initial and goal states and how you went about solving the problem.

· 11.3. Which of the problems in Stop and Think 11.1 would you classify as well-defined and which as ill defined? What characteristics of the problems led you to classify them in that way?

For example, consider the pennies in __Figure 11.2__. The problem is to move two pennies so that all of the pennies are touching three and only three other pennies. Give the problem a try. Can you find the solution? The solution, given in __Figure 11.3__, is the same for initial states (a) and (b). People typically find the problem difficult to solve because they represent the problem in two dimensions, as if sliding the pennies on a table. As a result, they don’t consider lifting the pennies and stacking them. In other words, they don’t consider moving the pennies in the third dimension an allowable operation. Initial state (b) is usually found to be more difficult than (a) because in (a) there are no places where they can slide a penny so that it touches three other pennies. As a result, people are quicker to change their representation of the problem in (a) to allow for stacking of the coins. In contrast, people who start with (b) typically maintain their two-dimensional representation of the problem longer because there are some places where they can move the pennies that touch three others, suggesting that they are getting closer to the final goal state.

Consider another problem illustrated in __Figure 11.4__. The task is to determine whether you can cover an eight-by-eight checkerboard with dominos. Each domino can cover two checker squares. The catch is that the checkerboard has been distorted by the removal of the two diagonal corner squares. Give it a try. Can you cover the entire board with dominos (the dominos can’t hang off of the edges or be altered in any way)? If you aren’t certain of your answer, consider the same problem but with __Figure 11.5__ instead. Most people find the problem much easier to solve when they alter their representation of the problem this way. Here you can easily see that both of the removed squares are yellow and that each domino will cover one red square and one white square. However, if two white squares are removed, then there are thirty-two red squares and thirty white squares and no way to cover the entire board with the dominos.

**Figure 11.2** Pennies Problem

Photo source: Photos.com/Photos.com/Thinkstock.

**Figure 11.3** Solution to the Pennies Problem

Photo source: Photos.com/Photos.com/Thinkstock.

**Functional Fixedness**

Consider the following problem. You are hanging decorative strings of lights on your back porch. After you finally manage to screw the two strings of lights to the eaves, you climb back down to the deck only to realize that when standing on the deck you can’t reach both sets of dangling lights. To make them work you need to plug them into each other. You don’t want to climb back up with your screwdriver and take one down. Is there a way to grasp both ends of the lights without going back up the ladder? Many people are stumped by this problem. Here is a hint: The screwdriver is the key to the solution. When most people think of a screwdriver, they think about the function of turning screws. However, for this problem, the screwdriver can be used for a different function. The solution is to tie the screwdriver to one of the strings of lights and swing it back and forth (see __Figure 11.6__).

**Figure 11.4** Domino and Distorted-Checkerboard Problem

Photo source: Hemera Technologies/PhotoObjects.net/Thinkstock.

**Figure 11.5** Variation of the Distorted-Checkerboard Problem

Photo source: Hemera Technologies/PhotoObjects.net/Thinkstock.

**Functional fixedness** is focusing on how things are usually used, while ignoring other potential uses. Gestalt psychologists identified this bias as a common barrier to our ability to solve problems (Maier, 1931). When faced with a problem, we retrieve information about the objects in it (e.g., string lights, ladder, screwdriver) and search for similar problems involving similar objects. When we start developing potential solutions, they are based in part on what functions the objects can perform. In the case of the screwdriver, based on how we’ve used it before, the functions that we consider probably involve turning screws, not using it as a weight for a pendulum. As a result, the representation of the problem space may not even include using the screwdriver in this way as a potential solution.

**Functional fixedness:** focusing on how things are typically used and ignoring other potential uses in solving a problem

These three problems demonstrate that the way we represent problems can have a powerful impact on our ability to solve problems. In the pennies problem, if we don’t represent the problem in three dimensions, then the solution (stacking the coins) isn’t going to be a possibility we consider. In the checkerboard example, representing the problem without colors doesn’t preclude finding the correct solution, but it also doesn’t highlight the importance of considering neighboring squares as an important feature of the problem. The addition of colors to both the board and the dominos spotlights this characteristic, often making it much easier to find the correct solution. In other words, how we mentally represent a problem can both help or hinder finding the solution. The __next section__ describes how we find solutions to problems.

**Stop and Think**

· 11.4. Consider the following problem. You have a jug of apple juice and a container of water. After putting both the apple juice and the water into a large pitcher, the apple juice and water remain separate. How does this happen?

· 11.5. As you consider potential solutions to the problem in Stop and Think 11.4, think about how you mentally represent it. What assumptions about the problem does your mental representation lead you to make?

· 11.6. A solution to the problem in Stop and Think 11.4 is that the water is frozen in the form of ice cubes. Do you think that you would have thought of the solution if the problem had stated “a tray of water” instead of a “container”?

**Figure 11.6** The Two-String Problem

**Developing Solutions to Problems: Approaches and Strategies**

Consider again the Sudoku problem in __Figure 11.1__. Go ahead and try to solve the puzzle, but while you do talk out loud about how you are trying to find the solution. What sorts of things did you find yourself saying? This method of thinking aloud is a commonly used methodology in research on problem solving. Unlike many of the cognitive processes discussed in this book, many of the processes underlying problem solving may be accessible to internal introspection. By having people think aloud while solving problems, researchers may gain insights into how people represent the problem, what information they are attending to, and what strategies they attempt to use. However, one needs to be cautious and keep in mind that some of the processes may not be consciously accessible, and furthermore, people may not report all of what is consciously available (e.g., they may not report strategies they began to consider but then rejected). Consider your own verbal reports of how you tried to solve the Sudoku puzzle. Do you feel that you were able to describe everything that went through your mind as you solved the puzzle? The next sections describe some of the strategies researchers propose we use to solve problems.

**Associationist Approach: Trial-and-Error Strategy**

Early theories of problem solving focused primarily on trial and error. The idea was that when faced with a problem we try out a solution and see if it works. If it doesn’t work, then we try another. This process is repeated until the problem is solved. Over time, as we accumulate associations between problems and successful solutions, we use these associations when encountering new problems. If the new problems are similar to old ones, then we will try to apply the solutions we used for the old ones (e.g. Thorndike, 1911). This approach to problem solving is known as the associationist approach.

Generally trial-and-error approaches (see __Photo 11.1__ for an example) work well when there are relatively few possible solutions. For example, if your problem is trying out what clothes to wear to your job interview and you only have three suits, then you can try each suit on and it won’t take very long. However, if you also have three dress shirts and two pairs of nice shoes, then the number of possible combinations quickly grows. Trying each suit with each shirt and each pair of shoes results in eighteen possible combinations (three suits × three shirts × two pairs of shoes).

Now think back to the Sudoku problem. If you have not done many Sudoku problems already, you may go about trying to find the solution using the method of trial and error, filling in all of the empty spaces with numbers and then seeing if your solution fit the puzzle constraints. However, chances are that you would get frustrated with using this strategy for this problem. While it is true that using this method, trying every number in every empty square, will ultimately result in the correct solution to the puzzle, it will likely take a very long time (with forty-five empty slots, and nine possible numbers in each one, there are over a trillion possibilities to check). So while trial and error might work for relatively simple problems, usually other strategies are necessary.

**Photo 11.1** Have you ever tried to solve a Rubik’s cube? If so, what strategy did you use? Most people use trial and error at first to solve it.

©iStockphoto.com/vgajic

**Gestalt Approaches**

Psychologists in the Gestalt tradition of the 1900s argued against purely associationist theories of problem solving (e.g., Duncker, 1945; Köhler, 1959). They argued that associationist theories predicted that problem solving was generally a gradual process and that many nonsensical errors should occur when people try to solve problems. However, when researchers began using think-aloud protocols, it became apparent that problem solvers appeared to use systematic strategies, rather than trial and error. The Gestalt approach is that a problem solver goes beyond past associations, and solutions arise out of new productive processes. These productive processes include creating mental representations of information structured to achieve particular goals (for a similar perspective, see the Gestalt approach to perception in __Chapter 3__). Often the solution is the result of a sudden breaking away from past associations, resulting in a reorganization of the mental representation of the problem. Other times problems are solved by recognizing that a past problem, even one that differs in surface features, shares an underlying structure and solution with the current problem (e.g., Wertheimer, 1945).

**Stop and Think**

· 11.7. Think about some of the problems you encounter in your own life. What are some situations where you try to solve them through trial and error? Are there ways in which these different situations are similar?

· 11.8. Are there other problems that you’d never consider trying to use trial and error to solve? Why would trial and error not work well in these situations?

**Insight**

Using the think-aloud method, Gestalt psychologists noticed that people often get the feeling that the solution to a problem suddenly occurs to them, a kind of “aha” experience, rather than gradually developing over multiple attempted solutions. In our opening story the solution to the problem, to carry the clothing in a cooler, suddenly came upon the person. Think back to when you tried to solve the earlier pennies and checkerboard problems. If you were able to come up with the solution, did you experience it as an “aha” moment, or was it the result of a more gradual solution process?

**Insight:** suddenly realizing the solution to a problem

Not all problems are solved using **insight**. Insight problems are typically those in which solvers cannot initially find a solution and have often stopped consciously thinking about the problem, when suddenly the correct solution emerges into consciousness (e.g., Duncker, 1945, Maier, 1931, Metcalfe & Wiebe, 1987). Gestalt psychologists theorized that we unconsciously continued to process the problem, searching for solutions during the incubation period following initial attempts to solve the problem. Much of their research attempted to describe which conditions promoted insightful solutions. They argued that insightful solutions often result when particular barriers to problem solutions are overcome. Some of the barriers they identified are discussed later in this chapter. For example, in our pennies example, the solution required changing from representing the problem as one in a two-dimensional space (so only sliding the coins was allowed) to one in a three-dimensional space (allowing for the coins to be stacked). In this case, insight happened when you suddenly realized that you could restructure how you represent it and the solution to the problem became apparent.

While the work of the Gestalt psychologists described the conditions under which insightful problem solving may occur, what exactly insight is has been a controversial research question (e.g., Weisberg, 1988; Weisberg & Alba, 1981). Researchers have proposed a number of processes that may underlie insight problem solving (e.g., Kaplan & Simon, 1990; Knoblich, Ohlsson, Haider, & Rhenius, 1999). For example, Janet Davidson and colleagues (e.g., Davidson, 1995; Davidson & Sternberg, 1986) proposed three mental processes involved: selective encoding, selective combination, and selective comparison. Selective encoding contributes to restructuring so that information originally viewed as irrelevant becomes viewed as relevant. For example, in our checkerboard example, when you mentally added color to the board and domino, the critical information about the importance of neighboring squares becomes relevant. Selective combination is when a previously nonobvious framework for relevant features becomes identified. The realization that the pennies can be moved in three dimensions, not just two, is an example of selective combination. Selective comparison is when you discover a nonobvious connection between new information and prior knowledge.

**Figure 11.7** The Nine-Dot Problem: Connect the Dots With Four Straight Lines

Let’s consider another problem. Look at the dots in __Figure 11.7__. Your task is to connect all of the dots using four connected straight lines. Go ahead and give it a try. If you are having trouble, think about our solution to the pennies problem where we had to restructure the problem and break outside the boundaries of two dimensions. The key, as in the pennies problem, is to represent the problem without borders (you can find the solution in __Figure 11.8__). In the pennies problem the borders limited the problem space to two dimensions. In the nine-dot problem, the borders limit the figure to the implied edges of a square made by the arrangement of the dots. If you were able to recognize the similarity between the two solutions (to “think outside the boundaries”) to these problems, you were using the process of selective comparison.

The difficulty of the nine-dot problem has been linked to a variety of factors that influence how we mentally represent the problem. Gestalt psychologists argued that perceptual grouping principles (see __Chapter 3__ for more details) make us think of the nine dots as grouped as a single figure and that the white space around it is background (Maier, 1930). Our attempts to solve the problem are biased such that we limit the lines we draw to stay within the borders of the figure. Disrupting the likelihood of grouping the dots as a single figure can increase the solution rate (Chronicle, Ormerod, & MacGregor, 2001; Kershaw & Ohlsson, 2004). Giving problem solvers the first line or two, reducing the number of possible solutions, increases the solution rate (MacGregor, Ormerod, & Chronicle, 2001). Past experience with problems with similar solutions also can increase solution rates (Kershaw & Ohlsson, 2004; Weisberg & Alba, 1981).

**Figure 11.8** The Nine-Dot Solution: The Key Is to Represent the Problem Without Borders

Past experience can be critical for solving problems. If you have encountered a problem before, and were able to solve it, then you can probably solve the current problem using that same solution (in which case you may even think “not a problem”). You have probably done this yourself. When you are working on homework problems assigned at the end of a chapter, do you go back and look at a sample problem that was worked through earlier in the chapter?

Chi and Snyder (2012) examined the role of past experience for solving the nine-dot problem. They used a technique called transcranial direct current stimulation (tDCS) to temporarily inhibit their participants’ right anterior temporal lobe (tDCS can also be used to temporarily stimulate as well). tDCS is a noninvasive procedure in which a weak direct current is applied directly to the scalp (see __Chapter 2__). Chi and Snyder (2012) gave their participants nine minutes to solve the nine-dot problem, three minutes before stimulation, three minutes during stimulation, and three minutes immediately after stimulation. Participants were randomly assigned to either the stimulation condition or a “sham” control condition (in which they did not receive stimulation). They found that 40 percent of the participants who received stimulation were able to solve the problem. In comparison, none of those in the control condition solved it. Chi and Snyder proposed that the stimulation temporarily inhibited their participants’ reliance on past experiences, which in this case corresponded to the participants viewing the dot patterns as being bounded by a square. In other words, the stimulation essentially allowed them to think outside of the box and find the solution to the problem.

**Stop and Think**

· 11.9. Have you ever experienced that “aha” feeling when solving a problem? If you can remember what the problem and your solution was, do you think it was the result of changing the way you represented the problem?

**Mental Set**

Luchins (1942) presented people with the following problem. Imagine that you have three containers of water. You want to end up with 100 cups of water. You start out with three different-sized containers: Container A holds 21 cups, Container B holds 127 cups, and Container C holds 3 cups. How can you measure out 100 cups? The solution is to fill up Container B (127 cups), then remove water from Container B with Container C twice [127 − (2 × 3) = 121] and then use Container A once to remove water from Container B (121 − 21 = 100). Now you try a few:

· Trial 1) Target: 18 cups A: 23, B: 49, C: 4

· Trial 2) Target: 21 cups A: 9, B: 42, C: 6

· Trial 3) Target: 22 cups A: 18, B: 48, C: 4

Did you notice that all three trials could be solved using the same solution as our original problem (B − 2C − A)? Did you solve all three additional problems in the same way? Go back and look at Trial 3. Did you notice a much more direct solution? You could have just filled A and C and added them together. If you didn’t notice this, but instead used the same solution as you did for the others, then you were using a **mental set** bias. Mental set is similar to the functional fixedness bias. We tend to use the same set of solutions for similar problems, even if there are other, simpler solutions available. Both solutions are correct. While the A + C solution may be simpler than the B − 2C − A solution, it is typically more work to come up with that solution given your recent history of using the longer solution. It is faster to recall and use the method you just used than to generate a new possible solution (Bilalic, McLeod, & Gobet, 2008).

**Analogical Transfer**

In the preceding example of the water containers, all of the puzzles are so similar that you probably thought of them as essentially the same problem. A similar process can happen with problems that, on the surface, may seem like completely different problems. However, under the surface the problems might have a similar structure. This is the strategy of **analogical transfer**, using the same solution for two problems with the same underlying structure.

**Mental set:** a tendency to use the same set of solutions to solve similar problems

**Analogical transfer:** using the same solution for two problems with the same underlying structure

Try to solve a variation of a problem Karl Duncker gave his participants (Duncker, 1945).

Imagine that you are a surgeon and your patient has an inoperable stomach tumor. However, one possible surgical method you think might work is to use a beam of radiation. A high-intensity beam should destroy the tumor. However, at high intensities, the beam will also destroy the surrounding healthy tissue. How can one cure the patient with these beams and, at the same time, avoid harming the healthy tissue that surrounds the tumor?

Do you have the solution yet? If not, consider the following hints: (1) What if you could adjust the intensity of the beams? and (2) What if you had more than one beam? With these hints people often come up with the solution of using multiple low-intensity beams to converge on and destroy the tumor without harming the surrounding tissue (called the “dispersion solution”). Duncker’s radiation problem is a classic example of an insight problem. People usually have great difficulty solving the problem without revising their mental representation of the problem to include the potential of multiple adjustable-intensity beams. But if you had previously solved a problem with a similar solution, would it make it easier to solve the radiation problem? Mary Gick and Keith Holyoak tested this in a series of experiments.

Before giving people the radiation problem, Gick and Holyoak (1980) presented them with a different problem like this one.

A small country is ruled by a dictator living in a strong fortress situated in the middle of the country, surrounded by villages. Many roads radiate outward from the fortress like spokes on a wheel. A general vows to capture the fortress and free the country. The general knows that if his entire army could attack the fortress at once it could be captured, but the dictator had planted mines on each of the roads. The mines were set so that small bodies of men could pass over them safely; however, any large force would detonate the mines, destroying the villages. A full-scale direct attack on the fortress therefore appeared impossible. The general solved the problem by dividing his army up into small groups and dispatched each group to the head of a different road. When all was ready he gave the signal, and each group charged down a different road. All of the small groups passed safely over the mines, and the army then attacked the fortress in full strength. In this way, the general was able to capture the fortress and overthrow the dictator.

While the surface features of this problem are different from those of the radiation problem, the structure of the underlying problems is similar (see __Table 11.1__ and __Figure 11.9__). Gick and Holyoak found that 70 percent of the people who received the army problem and its solution solved the radiation problem using the dispersion solution, compared to only 10 percent of the people who didn’t get the army problem. Some participants received slightly different versions of the army problem. In one variation the general finds an unmined road to the fortress, so the solution is to send the entire army down this road to attack. This is analogous to a different solution to the tumor problem, aiming the radiation beam at the tumor in a way to bypass the tissue (e.g., aiming it down the patient’s throat). With the unmined-road initial story, only 10 percent of participants arrived at the dispersion solution for the radiation problem, and 70 percent proposed an open-passage solution. Another group of people were presented a version of the story in which the army general is ordered to parade throughout the entire country. If the dictator is not impressed by the parade, the general will be dismissed. This version of the problem has a similar solution to the radiation problem but has a much less analogous desired goal: producing an impressive parade instead of capturing a fortress. Only 50 percent of the people receiving this story solved the radiation problem with the dispersion solution.

Source: Adapted from Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. *Cognitive Psychology*, *12*, 306—355.

The Gick and Holyoak (1980) experiments demonstrate that past experience with analogous problems can be a powerful strategy for solving problems (sometimes called positive transfer). However, there was one surprising result across their experiments: Unless explicitly told that the two problems might be related, participants rarely recognized and used the analogous relationship between the problems when trying to solve the radiation problem (30 percent solved without the hint, 70 percent solved with the hint). What may underlie their difficulty? To use the analogical transfer strategy one needs to retrieve an appropriately related problem, map the pieces of the new problem onto the structure of the retrieved problem, and then correctly generalize the solution that arises out of the mapping process. These processes rely on recognizing the similarity between the different problems. However, there are two levels of similarity to consider. One is the similar surface features of the context (e.g., medical, military), objects (e.g., tumor, doctor, radiation beam, healthy tissue, dictator, army, general, villages), and actions (e.g., armies attacking, beams destroying tissue) of the problems. The other level is the underlying structure of the problem (dictator = tumor, army = radiation beam; see __Table 11.1__). As Gick and Holyoak’s studies show, the key to using analogous solutions is using problems with similar underlying structures. However, we are often strongly influenced by the similarity of the surface structure of the problems (e.g., Holyoak & Koh, 1987; Novick, 1988; Ross, 1984, 1989; Ross & Kilbane, 1997).

Sometimes the surface structure and the underlying structural relationships are strongly related. In these cases, retrieval, guided by surface similarity, can be beneficial, leading to positive transfer between problems. However, we also have experience with problems that are similar on the surface but require very different solutions. In these situations, using problems that have strong surface similarities but different underlying structures leads individuals to attempt the wrong solutions (as did the participants in Gick and Holyoak’s unmined-road story condition). This is referred to as negative transfer. In other words, if you try to use analogical transfer to solve problems, the problems that you retrieve from past experience may reflect the similarity of surface features rather than the critical structural similarities between problems. Furthermore, the strong influence of surface similarity may interfere with searching memory for problems with underlying structural similarity. For example, Gick and Holyoak’s participants rarely noticed the relevance of the army problem, even though it was presented immediately before the radiation problem. However, when they were given a hint that the army problem might be useful, encouraging them to recall its details, then they were able to make use of the analogous underlying structure.

**Figure 11.9** The Radiation and Army Problems

**Summary**

The Gestalt approach to problem solving was focused on the structure of the representation of the problem. Insight solutions arise from a restructuring of the representation. Mental sets and analogous solutions arise from using past solutions. However, most of the research was primarily descriptive, focused on describing when insight and analogical reasoning occurred. With the development of the computer and the information processing approach to cognition, a new approach to problem solving emerged that focused on the underlying processes involved in solving problems.

**Stop and Think**

· 11.10. Consider the following problem. The local baseball team holds tryouts for new pitchers and catchers. They invited sixteen players at each position to try out for the team. On the day of the tryouts, the weather is windy, so the coaches decide to have all of the players try out at the same time. However, two of the catchers have to cancel at the last moment. Can the coaches pair up the remaining players and have them all try out at the same time? Does this problem remind you of any of the problems earlier in the chapter?

· 11.11. Compare the underlying structure of the baseball player problem in Stop and Think 11.10 and the checkerboard and dominos problem (see Figures 11.4 and 11.5). List the problem statement, desired goal, problem constraints, and solution for each. Would you consider the two problems to be analogous?

**Problem Solving as Problem Space Searches**

In the early 1960s Allen Newell and Herb Simon developed a computer program for problem solving called the General Problem Solver (GPS for short). By the 1970s their approach radically changed the way cognitive psychologists theorized about human problem solving. Newell and Simon proposed that problem solving typically proceeds by dividing the larger problem into smaller problems, searching for solutions to the smaller problems, and evaluating these solutions to see if they bring you closer to solving the larger problem. Consider an example from Newell and Simon (1972, p. 416):

**Algorithm:** a prescribed problem-solving strategy that always leads to the correct solution in problems with a single correct solution

**Heuristic:** a problem-solving strategy that does not always lead to the correct solution

I want to take my son to nursery school. What’s the difference between what I have and what I want? One of distance. What changes distance? My automobile. My automobile won’t work. What is needed to make it work? A new battery. What has new batteries? An auto repair shop. I want the repair shop to put in a new battery; but the shop doesn’t know I need one. What is the difficulty? One of communication. What allows communication? A telephone ... and so on.

The overall problem is that the current state (my son is at home) does not match the goal state (my son is at school). Finding the solution to this problem consists of a guided search through a problem space. The problem space consists of a mental representation of the set of intermediate states, subgoals, and operators (the actions that can be performed to change a state). In the example, the problem solver recognizes that a car can be used to drive his son from home to school (so driving the car is an operator here). But in this case the car isn’t running. So a subgoal is to change the current state and get the car to run. This subproblem is solved by buying a new battery at the store (so buying a battery is another operator). Newell and Simon proposed that there are different ways to search through the problem state. By using an **algorithm**, you can consider the entire problem space, searching every possible solution. While this guarantees finding the solution (assuming there is one), if the problem space is especially large, we may not have the resources available to search the entire space. In contrast, **heuristic** searches consider only part of the search space. Instead of considering all possible solutions, we instead mentally consider potential chains of subproblems, evaluating how each operator changes the current state.

**Figure 11.10** The Tower of Hanoi

Consider the Tower of Hanoi problem depicted in __Figure 11.10__. This puzzle involves three different-sized discs that can be moved back and forth on three pegs. The goal of the puzzle is to move the discs from the first peg to the final peg. The rules are that you can only move one disc at a time, only the top disc of a stack may be moved, and discs may only be placed on top of either an empty peg or a larger disc. The operator in this puzzle is moving a disc from one peg to another. The problem space consists of all the possible moves that can be made.

__Figure 11.11__ shows the problem space for the first move. There are two possible moves, to move the red disc to the middle peg or the final peg. After either of these moves there are several possible moves. We could move the red disc again, either to the other empty peg (taking us to the alternative-position option on the first move) or back on top of the green disc (putting us back to the initial state). Neither of these options gets us closer to our final goal state. The other options are to move the green disc to the empty peg (which one is empty depends on which move we made in Step 1). __Figure 11.12__ shows some of the problem space (not all of the possible moves and intermediate states are shown in the figure). While there are many paths that will result in the final solution, the most efficient solution path is indicated by the red arrows.

Newell and Simon argued that we solve problems by mentally working our way through the problem space. However, as we can see, even relatively simple problems can result in very large problem spaces. Rather than search every possible path through the space, Newell and Simon proposed that we guide our search through the space using particular heuristic strategies. The following section briefly describes three of the many heuristic search processes that have been proposed.

**Figure 11.11** The Problem Space for the First Move of the Tower of Hanoi Puzzle

**Figure 11.12** Some of the Problem Space for the Tower of Hanoi Puzzle

**Means-Ends Strategy**

Newell and Simon’s GPS computer program used the **means-ends strategy**. The means-ends strategy guides the search through the problem space by repeatedly comparing the current state of the problem to the goal state, identifying the differences and developing subgoals. As each subgoal is achieved, the intermediate state gets closer to the goal state. Newell and Simon’s story about driving their son to school is an example of using this strategy. The Tower of Hanoi problem can be broken down in a similar way. All of the discs need to be moved onto the final peg, but only one peg can be moved at a time. To move the blue disc, the green disc needs to be removed. To move the green disc, the red disc needs to be moved. So the first subgoal is to move the red disc. Once the red disc is moved, then the green disc can be moved onto the empty peg. However, now there is not a spot to move the blue disc. To free up a peg for the blue disc, the red disc can be moved onto the green disc (not onto the blue disc because that would prevent the blue disc from being moved). Once the red disc is on the green disc, then the blue disc can be moved onto the third peg. Continuing the search through the space in this manner arrives at the goal state, providing the solution to the puzzle.

**Hill-Climbing Strategy**

You may have noticed that the means-ends strategy provided a straightforward solution that worked, if in the initial step the red disc was moved onto the third peg (see the left side of __Figure 11.12__). However, if the red disc was initially moved onto the middle peg, which satisfies the subgoal of freeing up the green disc, then the search through the problem space will take much longer to reach the final goal state (see the right side of __Figure 11.12__). So how do we decide which move to make on our first turn? One possibility is to look ahead at the impact of making the two choices. If the problem space is small enough, this may be possible, but even in this small problem, that would require thinking through many possible solution paths. An alternative is to use the heuristic of hill climbing. The **hill-climbing strategy** is to select the operator that results in a change most similar to the goal state. On the first move, the red disc could be moved to either the middle or third peg. Moving it to the third peg is more similar to the goal state than moving it to the middle peg. In this case, this turns out to lead to the shortest path through the problem space.

**Means-ends strategy:** a problem-solving strategy that involves repeated comparisons between the current state and the goal state

**Hill-climbing strategy:** a problem-solving strategy that involves continuous steps toward the goal state

**Stop and Think**

· 11.12. Think back to how you tried to solve the Sudoku problem in __Figure 11.1__ How did you decide on your first move? Did you focus on the overall solution to the problem, or did you identify a subgoal to try to solve?

· 11.13. Think about the problem-solving strategies described in this section. Can you think of some examples from your own life where you used these strategies to solve a problem?

**Working-Backward Strategy**

Another strategy is to try searching through the problem space backward, starting from the goal state (**working-backward strategy**). Again, consider the Tower of Hanoi problem. The final state has the blue disc on the final peg. To get it there we need to move the green and red discs to the middle peg, so that the blue disc can be moved to the final peg. If the green and red discs are both in the middle, the green disc needs to be on the bottom, so the green disc needs to be moved onto it when it is empty. So the first move should be to move the red disc to the final peg, so that the green disc can be placed in the middle. Then place the red disc onto the green disc, which frees up the final peg for the blue disc.

**Working-backward strategy:** a problem-solving strategy that involves beginning with the goal state and working back to the initial state

**Summary of Approaches and Strategies**

There is no single problem-solving mechanism for all problems. This is critical because of the vast variety of problems we are faced with on a day-to-day basis. When faced with a problem, we have many potential strategies available to solve it. If it is a relatively simple problem, then trial and error may yield the solution. If we have solved similar problems before, we may be able to use past solutions. Sometimes using one strategy doesn’t work, so we try another one. Regardless of the strategy we use, we use it within our system of cognitive processes. The __next section__ reviews how the processes of perception, attention, memory, language, and knowledge impact how we solve problems.

**Allocating Mental Resources for Solving the Problem**

Think back to when you tried to solve the Sudoku puzzle. Chances are that if you tried a trial-and-error strategy of filling in numbers randomly, you got frustrated quickly. The problem space is too large. Rather than thinking about the final goal (getting all of the empty slots filled), most people instead focus on subgoals, trying to complete rows and columns. For example, you can quickly rule out a 2 in the upper-left empty box; it cannot be correct because there is another 2 already in the row (and the column too; see __Figure 11.13a__). You may then try a 3, which seems to be a good solution because it fits the constraints: There are no other 3s in the row, column, or three-by-three box (see __Figure 11.13b__). Using this sort of approach involves your attention system (see __Chapter 4__ for more discussion of attentional processes). You have to search the rows and columns, looking for other 2s and 3s, while ignoring the other numbers. Using the same constraints you can rule out the numbers 1, 2, 7, 8, and 9, leaving 3, 4, 5, and 6 as possible numbers for that square. Often, when trying to solve a Sudoku, people will write down these possible numbers for that square rather than trying to keep all of these possibilities in mind. This is because holding the possibilities for each square quickly surpasses the limits of our working memory system (see __Chapter 5__ for a discussion of working memory).

**Figure 11.13** Potential Solutions in the Problem Space of the Sudoku Puzzle Given in __Figure 11.1__

Our ability to solve problems is constrained by our cognitive systems. Consider the effects of our long-term memory processes (Ohlsson, 1992). When we encounter a problem, we retrieve knowledge that we bring to bear on the problem. The information we retrieve determines the way we initially represent the problem. How the problem is presented, what words are used (e.g., Salomon, Magliano, & Radvansky, 2013), and whether it is presented with a diagram (e.g., Larkin & Simon, 1987) impact what knowledge is retrieved. This knowledge includes conceptual information about the features and functions of parts of the problems, as well as past solutions. We use this retrieved information to define the problem space. This information also impacts how we allocate our attention (e.g., Grant & Spivey, 2003; Wiley & Jarosz, 2012), guiding what information to focus on and what information to ignore. Our working-memory capacity places limits on how much information about the problem we can process. These limits constrain how much information about the problem is available and the search through the problem space (Chein & Weisberg, 2014; Thomas, 2013).

Look at the problem in __Figure 11.14__. Imagine that the problem is made up of matchsticks arranged like a math problem. However, the math problem as stated is wrong and needs to be fixed. How can you fix the problem so that it is true by only moving a single matchstick? The solution to the problem is given below it. Gunther Knoblich and colleagues (1999) have examined how people solve matchstick arithmetic problems like these. When we first encounter a problem like this we retrieve information about math and Roman numerals. Part of our math knowledge includes rules about how we can or cannot manipulate numbers and formulae. Based on our past experience, we construct our initial problem space in a way that is constrained by these rules. Due to working-memory constraints, we probably also initially represent the elements of the problems as meaningful information chunks (see __Chapter 5__ for more details). For example, rather than representing VII as four matchsticks, we think of it as the number 7. The same is true for the mathematical operators for equals, addition, and subtraction. However, some chunks may be “tighter” than others. For example, the Roman numerals III and VI are compositional, made up of three ones and a five and a one, respectively. In contrast, the numerals V and X cannot be decomposed in the same way. Knoblich et al. (1999) argued that the inclusion of these math rules and chunks is what makes solving these problems difficult. Furthermore, they predicted that the difficulty of the problems should vary as a function of how much the solution depends on the ease with which we can relax our representations of the rules and decompose the chunks.

**Figure 11.14** Matchstick Problem and Solution

__Figure 11.15__ offers a few more of these problems for you to try (some solutions are presented in __Figure 11.16__). Chances are that you will be able to solve the first problem in (a) fairly quickly. Problems like this one require decomposing a loosely chunked Roman numeral and relaxing a fairly low-level rule of math. Additionally, the solution is similar to the first example you saw. The second problem in (b) was probably harder because it requires decomposing the equals sign, which is more tightly chunked. The third problem in (c) is the hardest, requiring the decomposition of a tightly chunked representation of X. Knoblich et al. (1999) had participants solve problems like these, systematically varying the level of rule and degree of chunking needed for the solution. These two variables correctly predicted how quickly their participants were able to solve the problems.

**Figure 11.15** Matchstick Problems

In a follow-up study, Knoblich, Ohlsson, and Raney (2001) examined the eye movements of participants trying to solve matchstick math problems. Since people tend to stare (fixate) at things they are thinking about, the researchers predicted that their eye movements would reflect how their participants were trying to solve the problems. Eye movements tended to look similar when participants first encountered the problems. They tended to focus on the Roman numerals, rather than the mathematical operators, suggesting that their initial problem spaces were biased to consider only some elements of the problems. Additionally, for the difficult problems, such as (b) and (c), the longer they worked on a problem, the longer their fixations became, suggesting that they had reached an impasse and were considering fewer potential solutions. However, at later stages of problem solving the eye movements of participants who were able to solve the problems changed. These participants shifted their gazes to the critical elements of the problems (e.g., the plus sign or the individual parts of decomposable Roman numerals). The researchers interpreted these patterns of data as consistent with the theory that the initial representation of the problems led to an inability to focus attention on the critical aspects of the problem, leading to an impasse. However, participants who were able to relax the constraints imposed by typical mathematical rules and could decompose the initially chunked representations could re-represent the problems. The re-represented versions of the problem then allowed them to attend to the critical parts of the problem and find the solution.

Traditionally, neuropsychologists studying problem solving have focused on measuring the impact of brain damage (via injury or disease), documenting the correlated deficits with localization of brain lesions. With the development of modern neural imaging techniques, this focus has begun to shift away from localization of function toward understanding cognitive mechanisms. However, many traditional problem-solving tasks are difficult to study using neural imaging techniques (Luo & Knoblich, 2007). For example, the nine-dot problem, the water jug problem, and matchstick arithmetic problems rely on different pieces of information and vary with difficulty. Also, many imaging techniques require multiple trials, but for many problems once you have an insight and discover the solution, future versions of the problem may no longer be insightful. However, cognitive neuroscientists have begun to develop new problem-solving tasks better suited for investigation with these imaging technologies.

**Figure 11.16** Solutions to Matchstick Problems

John Kounios and Mark Beeman performed a series of experiments using both EEG and fMRI to examine insight problem solving (e.g., Bowden, Jung-Beeman, Fleck, & Kounios, 2005; Jung-Beeman et al., 2004; Kounios & Beeman, 2009). Generally their studies indicate that insight is the result of a series of brain states that operate at different time scales. In particular, their results implicate an important role of the anterior temporal lobe (ATL) for solving insight problems. Chi and Snyder (2011) further investigated the role of the right ATL by having participants solve matchstick arithmetic problems. They randomly assigned their participants to one of three conditions and used tDCS (described earlier in the chapter) to selectively stimulate different regions of their brains. One group had their right ATL excited and their left ATL inhibited (R+L-), another had their left ATL excited and their right ATL inhibited (R-L+), and the third group served as a control comparison group. They found that the R+L- participants solved more of the insight problems than the other two groups. They suggested that this was probably due to diminishing top-down information, the interruption of mental set, and potentially improved participants’ set-switching abilities.

**Stop and Think**

· 11.14. Imagine that the GPS function on your phone is not working and you are driving to Disney World. What cognitive processes are you likely to rely on as you navigate your trip?

· 11.15. Think back to when you were trying to do the Sudoku problem in __Figure 11.1__. Where were you focusing your attention as you were considering options? Do you find it easier to write down options, or do you try to keep them in memory?

It should be apparent from the research reviewed in this section that solving problems happens within our cognitive architecture. How we identify, represent, solve, and evaluate problems must involve many (if not all) aspects of our cognitive processes. In other words, potentially all of the research and theory discussed in the other chapters of this book impact our ability to solve our day-to-day problems.

**Expertise**

Given that past experience has such a dramatic impact on how we solve problems, you may ask yourself whether you can become an expert problem solver. The answer is yes, at least within particular domains. If you practice doing Sudoku, you will become a better Sudoku player. The same is true for other domains, like playing chess, doing physics problems, or coaching gymnasts.

**Experts Versus Novices**

What makes an expert problem solver so much better than a novice? Given the complexity of problem solving already outlined earlier in the chapter, it should come as no surprise that it is a combination of factors.

**Perception and Attention**

As we gain experience with different types of problems, we learn which details of the problems are relevant and which are not (e.g., Haider & Frensch, 1999). For example, Moreno, Reina, Luis, and Sabido (2002) monitored the eye movements of expert and novice gymnastic coaches while they viewed gymnastic routines (see __Photo 11.2__). They found that the expert coaches had longer fixations on regions critical to the performances and fewer, shorter fixations on nonrelevant areas. Lesgold et al. (1988) compared the ability of radiologists with over ten years of experience to medical residents in finding tumors in X-rays. While both groups were able to find the main problems, experts were also able to detect a greater number of critical features and subtle cues and the relationships between these.

**Memory**

Experts also mentally group aspects of the problems differently than novices do. For example, chess experts can remember where virtually all of the pieces of a chessboard are during a game. They can do this because their past experience of thousands of games allows them to chunk the pieces in meaningful ways (e.g., in terms of defensive structures). To further support this idea, Reingold, Charness, Pomplun, and Stampe (2001) measured the eye movements of chess experts and novices. They showed that expert players spent more time looking between pieces than at individual pieces, suggesting that they were focused not on the individual pieces but rather on the overall structures on the board. This difference allows experts to focus on higher-order problem goals, reducing the problem space. However, if experts are presented with a board on which the pieces have been arranged randomly, their memory performance is similar to that of less experienced players (Chase & Simon, 1973; de Groot, 1966).

When experts are reminded of past solutions, as in the case of analogical transfer, they are more likely than novices to focus on the underlying structural features of the problems. To illustrate this, Novick (1988) presented students with a series of math word problems, some of which were structurally similar, others of which were only similar on the surface. She demonstrated that experts (those scoring from 690 to 770 on the math SAT test) showed greater positive transfer between analogous problems relative to novices (those scoring from 500 to 650 on the math SAT). Furthermore, experts showed less negative transfer between nonanalogous problems that shared surface features. This result demonstrates that experts and novices differ in their initial problem representations. Chi, Feltovich, and Glaser (1981) found similar results for physics problems comparing advanced doctoral students (experts) to undergraduate physics majors (novices). They found that the advanced students were able to see past surface features and perceive the underlying structure of the problems.

**Photo 11.2** Moreno et al.’s (2002) study demonstrated that coaches looked at different places while watching gymnasts perform based on their own expertise level.

©iStockphoto.com/ilkercekic

**Better Strategies**

Experts also generally spend more time analyzing problems, adding relevant knowledge to their representation and planning their solutions (Chi, 2006). While novices and experts may have the same strategies available to them, experts are better at predicting and using more effective strategies. For example, chess experts and novices both look ahead for approximately the same number of moves (e.g., if I move my pawn, then she moves her bishop, then I move the knight), but the look-ahead of experts is typically focused on more knowledge-based searches focused on higher-order goals (e.g., good defensive structure rather than just reacting to a particular move of an opponent; Gobet & Simon, 1996).

One of the most persistent findings in the research on expertise is that experts’ problem-solving advantages are restricted to problems within their domain of expertise. While chess masters can remember where all of the pieces on a chessboard are, they can only do it if the pieces are arranged in a meaningful game-related way. Furthermore, if you measure chess experts’ ability to remember the location of checkers or cards played in a hand of bridge, their chunking capacities will look like those of novices (with comparable experience with checkers or bridge). So one question you might have is, outside of becoming an expert in a particular field, is there a way to improve your general problem-solving abilities?

**Becoming a Better Problem Solver**

Given what we know about the processes that underlie problem solving, what can we do to become better problem solvers? A search of bookstores and the Internet yields a vast selection of advice. One research-motivated approach is the **IDEAL framework** proposed by John Bransford and Barry Stein (1993). It is based on the same basic problem-solving cycle that has guided the structure of this chapter. *IDEAL* stands for Identify problems and opportunities, Define goals, Explore possible strategies, Anticipate outcomes and act, and Look back and learn. They suggest that effective problem solvers view problems as opportunities and actively seek them out. In other words, they gain practice recognizing and identifying problems. Defining refers to representing the problem and identifying the goals and potential operations. Good problem solvers recognize that how they represent a problem has an impact on how they try to solve it. One effective approach is a willingness to “think outside of the box” and try different ways of representing the problem. Additionally, they recognize the possibility of multiple strategies that can be explored to search the problem space for a way to achieve the goal. Good problem solvers are willing to try to actively evaluate the effects of these strategies. Understanding how and why solutions work is also important because it helps encode the underlying structural components of problems rather than the surface features. Becoming more aware of the cycle of problem solving and employing strategies targeting these stages can lead to better general problem solving.

**IDEAL framework:** a step-by-step description of problem-solving processes

**Figure 11.17** Sudoku Solution From __Figure 11.1__

**Thinking About Research**

As you read the following summary of a research study in psychology, think about the following questions:

1. Which of the approaches to the study of cognition do you think these researchers used in their experiments on problem solving: representationalist, embodied, or biological (see __Chapter 1__ for a review of these approaches)?

2. What are the independent variables in this study?

3. What are the dependent variables in this study?

4. In what way might these results be useful for everyday problem solving?

**Study Reference**

Grant, E. R., & Spivey, M. J. (2003). Eye movements and problem solving: Guiding attention guides thought. *Psychological Science*, *14*(5), 462—466.

*Purpose of the study:* The researchers examined whether participants’ ability to solve Duncker’s radiation problem (see the Analogical Transfer section earlier in the chapter) could be improved by manipulating how and where they look at the problem.

*Method of the study:* In the first experiment, the researchers examined the eye movements of participants looking at the diagram in __Figure 11.18__, while trying to solve Duncker’s radiation problem. They compared the fixation patterns of participants who were able to solve the problem without hints to those who needed hints. The second experiment again examined Duncker’s radiation problem. This experiment compared three groups of participants: One group examined the diagram used in Experiment 1; the other two groups examined an animated version of the figure. One version of the animated figure was constructed to highlight the regions of the figure that the results of Experiment 1 identified as a critical feature (i.e., the oval perimeter that represented the skin subtly pulsing). The other version of the animated figure highlighted a feature (the tumor subtly pulsing) that Experiment 1 results suggested were noncritical.

*Results of the study:* Eye fixation data from Experiment 1 showed that in the last 30 seconds of problem solving, subjects who looked at the skin in the diagram were more likely to solve the problem than to be unsuccessful. This result shows that the skin is the most relevant feature of the diagram for solving the problem. This difference did not occur in subjects looking at other parts of the diagram during this time (see __Figure 11.19__). As the results in __Table 11.2__ illustrate, successful performance in solving the problem across both experiments occurred most often when the diagram was animated to highlight the most critical feature for solving the problem (i.e., the pulsing skin).

*Conclusions of the study:* From the results of the two experiments, the researchers concluded that problem solving is enhanced when one focuses attention on the visual aspects of a problem relevant for finding a solution.

**Figure 11.18** Diagram Shown to Subjects in the Grant and Spivey (2003) Study

Source: Grant and Spivey (2003, figure 1).

**Figure 11.19** Results From Grant and Spivey’s (2003) Experiment 1

Source: Grant and Spivey (2003, figure 2).

Source: Grant and Spivey (2003, table 1).

**Chapter Review**

Summary

· **What kind of problems do you solve every day?**

Some of the problems we solve every day are well defined, with clearly stated goals and strategies for achieving those goals. Others are ill defined, with fuzzier goals and fewer clear pathways to their solutions.

· **How do you solve problems: through trial and error, through conscious deliberation, or do solutions just suddenly occur to you?**

Trial and error works as a strategy for relatively simple problems, but we typically use other strategies for more complex problems. Often we break the problem down into subproblems, working on solving those to achieve our larger goal. Sometimes we get stuck until we change how we represent the problem and a solution emerges.

· **Why are some problems more difficult to solve than others?**

Problems with clearly defined goals and constraints are typically easier to solve than those that are less clear. Problems that we have had past experience with are typically easier than those that are new to us. Problems that require us to represent relevant information in a way different from how we usually think of things are also typically difficult.

· **What gets in your way when trying to solve problems?**

We solve problems within our cognitive systems, and sometimes those systems have limitations that impact our ability to solve problems. We have limits on how much information we can attend to and hold in working memory at one time. To overcome this, we often chunk information together. Sometimes the information is chunked in a way that facilitates finding a solution. However, other times the information is grouped together in a way that interferes with finding a solution. Sometimes the problem has so many potential paths to achieving a goal that we can’t consider them all and as a result miss the right one.

· **How do expert problem solvers differ from novices?**

We all draw upon our past experiences to solve problems. Within their domain of expertise, experts have a much larger array of experiences compared to novices. This experience allows experts to focus their attention on the most relevant aspects of a problem, to focus on the underlying structure of a problem instead of surface features, to represent a problem in the most efficient way, and to retrieve past solutions to similar problems.

**Chapter Quiz**

1. The problem-solving cycle includes all but the following stages:

1. recognize and identify the problem

2. define and mentally represent the problem

3. monitor progress toward the goal and evaluate the solution

4. create alternative kinds of problems

5. develop a solution strategy

6. allocate mental resources for solving the problem

2. Researchers typically describe a problem as

1. the difference between past problems and the current problem.

2. the difference between a current state and a desired state.

3. the difference between an insight and a representation.

4. the similarity between past problems and the current problem.

5. the similarity between attention and working memory.

3. The checkerboard and dominos problem illustrates that

1. games are a kind of problem-solving task.

2. how we represent a problem can have an impact on our ability to find a solution.

3. functional fixedness can make finding solutions easier.

4. monitoring progress toward the goal is rarely done.

5. the trial-and-error strategy is a fast and efficient method for finding a solution.

4. The associationist approach describes most problem solving as involving

1. insight.

2. analogy.

3. chunking.

4. trial and error.

5. searching through a problem space.

5. Gestalt psychologists proposed that problem solving

1. often involved unconscious processing of a problem.

2. sometimes involved insight.

3. involves thinking aloud.

4. is impacted by past experience.

5. All of the above answers are correct.

6. Successfully solving a problem using the analogy transfer strategy typically results from

1. focusing on the surface features of the problem.

2. focusing on the underlying structure of the problem.

3. focusing on both the surface features and underlying structure of the problem.

4. ignoring both the surface features and underlying structure and instead relying on insight to solve the problem.

7. Newell and Simon proposed that problem solving involves a search through a problem space. What is a problem space?

1. the part of memory where we store all of our past experience with problems

2. a mental representation of the set of intermediate states, subgoals, and operators

3. the combination of the articulatory loop and spatial sketchpad components of working memory

4. the mental set of typical functions that objects usually are used for

8. A hill-climbing strategy for problem solving is

1. an approach that starts at the top of a set of potential solutions and works down the set.

2. an approach in which operators are selected if they result in changing the current state to something that is closer to the goal state.

3. an approach in which you work through the problem space in reverse, starting with the goal state and working backward to the initial state.

4. an approach that factors in the amount of effort required to use a particular operator.

9. Experts are often much better (faster and more accurate) problem solvers within their domain of expertise because they

1. have more experience with the typical problems in the domain.

2. are usually more intelligent than novices.

3. are able to focus on the underlying structure of the problem better than novices.

4. both (a) and (c)

5. Answers (a), (b), and (c) are all correct.

10. Bransford and Stein proposed the IDEAL framework of problem solving. ** IDEAL** stands for:

1. Identify past solutions, Determine good strategies, Explore alternative methods, Always keep trying, Look back and learn

2. Inhibit surface features, Discover underlying structure, Explore possible solutions, Activate relevant knowledge, Learn from past mistakes

3. Identify potential representations, Decode chunked information, Examine past assumptions, Anticipate outcomes and act, Leap forward with intuition and insight

4. Interpret and comprehend, Define underlying assumptions, Elaborate, Activate relevant chunks, Learn from past mistakes

5. Identify problems, Define goals, Explore strategies, Anticipate outcomes and act, and Look back and learn

11. Think of situations where you have overcome functional fixedness to solve a problem (e.g., using a shoe to squish a bug). Think back to what it felt like to come up with that solution. Did it involve having an “insight?”

12. What kind of problem-solving strategies do you use in your college courses? Do you mostly use algorithmic or heuristic methods?

**Key Terms**

· Algorithm 301

· Analogical transfer 297

· Functional fixedness 293

· Heuristic 301

· Hill-climbing strategy 303

· IDEAL framework 309

· Ill-defined problem 289

· Insight 295

· Means-ends strategy 303

· Mental set 297

· Well-defined problem 289

· Working-backward strategy 304

**Stop and Think Answers**

· 11.1. Make a list of some of the problems you have already faced today.

Answers will vary.

· 11.2. For each problem in Stop and Think 11.1, identify the initial and goal states and how you went about solving the problem.

Answers will vary.

· 11.3. Which of the problems in Stop and Think 11.1 would you classify as well-defined and which as ill defined? What characteristics of the problems led you to classify them in that way?

Answers will vary.

· 11.4. Consider the following problem. You have a jug of apple juice and a container of water. After putting both the apple juice and the water into a large pitcher, the apple juice and water remain separate. How does this happen?

Answers will vary.

· 11.5. As you consider potential solutions to the problem in Stop and Think 11.4, think about how you mentally represent it. What assumptions about the problem does your mental representation lead you to make?

Answers will vary.

· 11.6. A solution to the problem in Stop and Think 11.4 is that the water is frozen in the form of ice cubes. Do you think that you would have thought of the solution if the problem had stated “a tray of water” instead of a “container”?

Answers will vary, but this changed wording may have led to a solution more frequently.

· 11.7. Think about some of the problems you encounter in your own life. What are some situations where you try to solve them through trial and error? Are there ways in which these different situations are similar?

Answers will vary.

· 11.8. Are there other problems that you’d never consider trying to use trial and error to solve? Why would trial and error not work well in these situations?

Answers will vary.

· 11.9. Have you ever experienced that “aha” feeling when solving a problem? If you can remember what the problem and your solution was, do you think it was the result of changing the way you represented the problem?

Answers will vary.

· 11.10. Consider the following problem. The local baseball team holds tryouts for new pitchers and catchers. They invited sixteen players at each position to try out for the team. On the day of the tryouts, the weather is windy, so the coaches decide to have all of the players try out at the same time. However, two of the catchers have to cancel at the last moment. Can the coaches pair up the remaining players and have them all try out at the same time? Does this problem remind you of any of the problems earlier in the chapter?

This is similar to the dominoes checkerboard problem.

· 11.11. Compare the underlying structure of the baseball player problem in Stop and Think 11.10 and the checkerboard and dominos problem (see Figures 11.4 and 11.5). List the problem statement, desired goal, problem constraints, and solution for each. Would you consider the two problems to be analogous?

Problem statement: Missing two catchers out of sixteen and want to pair up pitchers and catchers

Desired goal: Pairing up the remaining players

Problem constraints: Each pair must contain a pitcher and catcher

Solution: You cannot pair up the remaining players and have each pair consist of one catcher and one pitcher.

The problems are analogous.

· 11.12. Think back to how you tried to solve the Sudoku problem in __Figure 11.1__. How did you decide on your first move? Did you focus on the overall solution to the problem, or did you identify a subgoal to try to solve?

Answers will vary.

· 11.13. Think about the problem-solving strategies described in this section. Can you think of some examples from your own life where you used these strategies to solve a problem?

Answers will vary.

· 11.14. Imagine that the GPS function on your phone is not working and you are driving to Disney World. What cognitive processes are you likely to rely on as you navigate your trip?

Answers will vary, but some possibilities are using working memory to keep track of where you are in reality and the position on the map, imagining the route in order to locate it on the map, and using retrieval from long-term memory to locate the appropriate map to use (e.g., Disney World is in Florida; where did you put the map of Florida?).

· 11.15. Think back to when you were trying to do the Sudoku problem in __Figure 11.1__. Where were you focusing your attention as you were considering options? Do you find it easier to write down options, or do you try to keep them in memory?

Answers will vary.

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