|chapter 13||Problem Solving
General Problem-Solving Methods People solve problems all the time. Some problems are pragmatic (“I want to Tom borrowed my car. How can I get there?”). Others are social (“I really want Amy to notice me; how should I arrange it?”). Others are academic (“Im trying to prove this theorem. How can I do it, starting from these axioms?”). What these situations share, though, is the desire to figure out how to reach some goal-a configuration that defines what we call problem solving. How do people solve go to the store, but problems? Problem Solving as Search Researchers compare problem solving to a process of search, as though you were navigating through a maze, seeking a path toward your goal (see Newe & Simon, 1972; also Bassok & Novick, 2012 Mayer, 2012). To make this point concrete, consider the Hobbits and Orcs problem in Figure 13.1. For this problem, you have choices for the various moves you can make (transporting creatures back and forth), but you’re limited by the size of the boat and the requirement that Hobbits can never be outnumbered (lest they be eaten). This situation leaves you with a set of options shown graphically in Figure 13.2. The figure shows the moves available early in the solution and depicts the options as a tree. with each step leading to more branches. All the branches together form the problem space- that is, the set of all states that can be reached in solving the problem.
To solve this problem, one strategy would be to trace through the entire problem space, exploring each branch in turn. This would be like exploring every possible corridor in a maze, an approach that would guarantee that you’d eventually find the solution. For most problems, however, this approach would be hopeless. Consider the game of chess. In chess, which move is best at any point in the game depends on what your opponent will be able to do in response to your move, and then what you’ll do next. To make sure you’re choosing the best move, therefore, you need to think ahead through a few cycles of play, so that you can select as your current move the one that will lead to the best sequence. Let’s imagine, therefore, that you decide to look ahead just three cycles of play-three of your moves and three of your opponent’s. Some calculation, however, tells us that for three cycles of chess play there are roughly 700 million possibilities for how the game could go; this number immediately rules out the option of considering every possibility. If you could evaluate 10 sequences per second, you’d still need more than 2 years, on a 24/7 schedule, to evaluate the full set of options for each move. And, of course, there’s nothing special here about chess, because most real-life problems offer so many options that you couldn’t possibly explore every one. Plainly, then, you somehow need to narrow your search through a problem space, and specifically, what you need is a problem-solving heuristic. As we’ve discussed in other chapters. heuristics are strategies that are efficient but at the cost of occasional errors. In the domain of problem solving, a heuristic is a strategy that narrows your search through the problem space-but (you hope) in a way that still leads to the problem’s solution. General Problem-Solving Heuristics One commonly used heuristic is called the hill-climbing strategy. To understand this term, imagine that you’re hiking through the woods and trying to figure out which trail leads to the mountaintop. You obviously need to climb uphill to reach the top, so whenever you come to a fork in the trail, you select the path that’s going uphill. The problem-solving strategy works the same way: At each point you choose the option that moves you in the direction of your goal. This strategy is of limited use, however, because many problems require that you briefly move away from your goal; only then, from this new position, can the problem be solved. For instance, if you want Mingus to notice you more, it might help if you go away for a while; that way, he’ll be more likely to notice you when you come back. You would never discover this ploy, though, if you relied on the hill-climbing strategy. Even so, people often rely on this heuristic. As a result, they have difficulties whenever a problem requires them to “move backward in order to go forward.” Often, at these points, people drop their current plan and seek some other solution to the problem: “This must be the wrong strategy; I’m going the wrong way” (See, e.g., Jeffries, Polson, Razran, & Atwood, 1977; Thomas, 1974.) Fortunately, people have other heuristics available to them. For example, people often rely on means-end analysis. In this strategy, you compare your current state to the goal state and you ask “What means do I have to make these more alike?” Figure 13.3 offers a commonsense example. Pictures and Diagrams People have other options in their mental toolkit. For example, it’s often helpful to translate a problem into concrete terms, relying on a mental image or a picture. As an illustration, consider the problem in Figure 13.4. Most people try an algebraic solution to this problem (width of each volume multiplied by the number of volumes, divided by the worm’s eating rate) and end up with the wrong People generally get this problem right, though, if they start by visualizing the arrangement. Now, they can see the actual positions of the worm’s starting point and end point, and this usually answer. takes them to the correct answer. (See Figure 13.5; also see Anderson, 1993; Anderson & Helstrup, 1993; Reed, 1993; Verstijnen, Hennessey, van Leeuwen, Hamel, & Goldschmidt, 1998.)
Drawing on Experience Where do these points leave us with regard to the questions with which we began-and, in particular, the ways in which people differ from one another in their mental abilities? There’s actually little difference from one person to the next in the use of strategies like hill climbing or means-end analysis-most people can and do use these strategies. People do differ, of course, in their drawing ability and in their imagery prowess (see Chapter 11), but these points are relevant only for some problems. Where, then, do the broader differences in problem-solving skill arise? Problem Solving via Analogy Often, a problem reminds you of other problems you’ve solved in the past, and so you can rely on your past experience in tackling the current challenge. In other words, you solve the current problem by means of an analogy with other, already solved, problems. It’s easy to show that analogies are helpful (Chan, Paletz, & Schunn, 2012; Donnelly & McDaniel, 1993; Gentner & Smith, 2012; Holyoak, 2012), but it’s also plain that people under-use analogies. Consider the tumor problem (see Figure 13.6A). This problem is difficult, but people generally solve it if they use an analogy. Gick and Holyoak (1980) first had their participants read about a related situation (see Figure 13.6B) and then presented them with the tumor problem. When participants were encouraged to use this hint, 75% were able to solve the tumor problem. Without the hint, only 10% solved the problem. Note, though, that Gick and Holyoak had another group of participants read the “general and fortress” story, but these participants weren’t told that this story was relevant to the tumor problem. problem (see Figure 13.7). (Also see Kubricht, Lu, & Holyoak, Only 30% of this group solved the tumor 2017.) Apparently, then, uninstructed use of analogies is rare, and one reason lies in how people search through memory when seeking an analogy. In solving the tumor problem, people seem to ask themselves: “What else do I know about tumors?” This search will help them remember other situations in which they thought about tumors, but it won’t lead them to the “general and fortress” problem. This (potential) analogue will therefore lie dormant in memory and provide no help. (See e.g., Bassok, 1996; Cummins, 1992; Hahn, Prat-Sala, Pothos, & Brumby, 2010; Wharton, Holyoak, Downing, & Lange, 1994.) To locate helpful analogies in memory, you generally need to look beyond the superficial features of the problem and think instead about the principles governing the problem-focusing on what’s sometimes called the problem’s “deep structure.” As a related point, you’ll be able to use an analogy only if you figure out how to map the prior case onto the problem now being solved-only if you realize, for example, that converging groups of soldiers correspond to converging rays and that a fortress-to-be-captured corresponds to a tumor-to-be-destroyed. This mapping process can be difficult (Holyoak, 2012; Reed, 2017), and failures to figure out the mapping are another reason people regularly fail to find and use analogies. Strategies to Make Analogy Use More Likely Perhaps, then, we have our first suggestion about why people differ in their problem-solving ability Perhaps the people who are better problem solvers are those who make better use of analogies- plausibly, because they pay attention to a problem’s deep structure rather than its superficial traits. Consistent with these claims, it turns out that we can improve problem solving by encouraging people to pay attention to the problems’ underlying dynamic. For example, Cummins (1992) instructed participants in one group to analyze a series of algebra problems one by one. Participants in a second group were asked to compare the problems to one another, describing what the problems had in common. The latter instruction forced participants to think about the problems underlying structure; guided by this perspective, the participants were more likely, later on, to use the training problems as a basis for forming and using analogies. (Also see Catrambone, Craig, & Nersessian, 2006; Kurtz & Loewenstein, 2007; Lane & Schooler, 2004; Pedrone, Hummel, & Holyoak 2001.) Expert Problem Solvers How far can we go with these points? Can we use these simple ideas to explain the difference between ordinary problem solvers and genuine experts? To some extent, we can. We just suggested, for example, that it’s helpful to think about problems in terms of their deep structure, and this is, it seems, the way experts think about problems. In one study, participants were asked to categorize simple physics problems (Chi, Feltovich, & Glaser, 1981). Novices tended to place together all the problems involving river currents., all the problems involving springs, and so on, in each case focusing on the surface form of the problem. In contrast, experts (Ph.D. students in physics) ignored these details of the problems and, instead, sorted according to the physical principles relevant to the problems’ solution. (For more on expertise, see Ericsson & Towne, 2012.) We’ve also claimed that attention to a problem’s deep structure promotes analogy use, so if experts are more attentive to this structure, they should be more likely to use analogies-and they are (e.g., Bassok & Novick, 2012). Experts’ reliance on analogies is evident both in the laboratory (e.g., Novick and Holyoak, 1991) and in real-world settings. Christensen and Schunn (2005) recorded work meetings of a group of engineers trying to create new products for the medical world. As the engineers discussed their options, analogy use was frequent-with an analogy being offered in the discussion every 5 minutes! Setting Subgoals Experts also have other advantages. For example, for many problems, it’s helpful to break a problem into subproblems so that the overall problem can be solved part by part rather than all at once. This, too, is a technique that experts often use. Classic evidence on this point comes from studies of chess experts (de Groot, 1965, 1966; also see Chase & Simon, 1973). The data show that these experts are particularly skilled in organizing a chess game-in seeing the structure of the game, understanding its parts, and perceiving how the parts are related to one another. This skill can be revealed in many ways, including how chess masters remember board positions. In one procedure, chess masters were able to remember the positions of 20 pieces after viewing the board for just 5 seconds; novices remembered many fewer (see Figure 13.8). In addition, there was a clear pattern to the experts’ recollection: In recalling the layout of the board, the experts would place four or five pieces in their proper positions, then pause, then recall another group, then pause, and so on. In each case, the group of pieces was one that made “tactical sense”-for example, the pieces involved in a “forked” attack, a chain of mutually defending pieces, and the like. (For similar data with other forms of expertise, see Tuffiash, Roring, & Ericsson, 2007 also see Sala & Gobet, 2017.)
It seems, then, that the masters-experts in chess-memorize the board in terms of higher-order units, defined by their strategic function within the game. This perception of higher-order units helps to organize the experts’ thinking. By focusing on the units and how they’re related to one another, the experts keep track of broad strategies without getting bogged down in the details. Likewise, these units set subgoals for the experts. Having perceived a group of pieces as a coordinated attack, an expert sets the subgoal of preparing for the attack. Having perceived another group of pieces as the early deevelopment of a pin (a situation in which a player cannot move without exposing a more valuable piece to an attack), the expert creates the subgoal of avoiding the pin. It turns out, though, that experts also have other advantages, including the simple fact that they know much more about their domains of expertise than novices do. Experts also organize their knowledge more effectively than novices. In particular, studies indicate that experts’ knowledge is heavily cross-referenced, so that each bit of information has associations to many other bits (e.g., Bédard & Chi, 1992; Bransford, Brown & Cocking, 1999; Reed, 2017). As a result, experts have better access to what they know. It’s clear, therefore, that there are multiple factors separating novices from experts, but these factors all hinge on the processes we’ve already discussed-with an emphasis subproblems, and memory search. Apparently, then, we can use our theorizing so far to describe on analogies, how people (in particular, novices and experts) differ from one another. e. Demonstration 13.1: Analogies Analogies are a powerful help in solving problems, and they are also an excellent way to convey new information. Imagine that you’re a teacher, trying to explain some points about astronomy. Which of the following explanations do you think would be more effective?
Collapsing stars spin faster and faster as they fold in on themselves and their size decreases. This principle called phenomenon of spinning faster as the star’s size shrinks occurs because of a “conservation of angular momentum.”
Collapsing stars spin faster as their size shrinks. Stars are thus like ice skaters, who pirouette faster as they pull in their arms. Both stars and skaters operate by a principle called “conservation of angular momentum”
Which version of the explanations would make it easier for students to answer a question like the following one?
What would happen if a star “expanded” instead of collapsing?
a) Its rate of rotation would increase.
b) Its rate of rotation would decrease.
c) Its orbital speed would increase.
d) Its orbital speed would decrease.
Does your intuition tell you that the analogy version would be better as a teaching tool? If so, then your intuition is in line with the data! Participants in one study analogy version. Later, they were asked questions about these were presented with materials just like these, in either a literal or an materials, and those instructed via analogy reliably did better. Do you think your teachers make effective use of analogy? Can you think of ways they can improve their use of analogy? Defining the Problem Experts, we’ve said, define problems in their area of expertise in terms of the problems’ underlying dynamic. As a result, the experts are more likely to break a problem into meaningful parts, more likely to realize what other problems are analogous to the current problem, and so more likely to benefit from analogies. Clearly, then, there are better and worse ways to define a problem-ways that will lead to a problem? And what solution and ways that will obstruct it. But what does it mean to “define” a determines how people define the problems they encounter? III-Defined and Well-Defined Problems For many problems, the goal and the options for solving the problems are clearly stated at the start: Get all the Hobbits to the other side of the river, using the boat. Solve the math problem, using the axioms stated. Many problems, though, are rather different. For example, we all hope for peace in the world, but what will this goal involve? There will be no fighting, of course, but what other traits will the goal have? Will the nations currently on the map still be in place? How will disputes be settled? It’s also unclear what steps should be tried in an effort toward reaching this goal. Would diplomatic negotiations work? Or would economic measures be more effective? Problems like this one are said to be ill-defined, with no clear statement at the outset of how the goal should be characterized or what operations might serve to reach that goal. Other examples of ill-defined problems include “having a good time while on vacation” and “saving money for college (Halpern, 1984; Kahney, 1986; Schraw, Dunkle, &Bendixen, 1995) When confronting ill-defined problems, your best bet is often to create subgoals, because many ill-defined problems have reasonably well-defined parts, and by solving each of these you can move toward solving the overall problem. A different strategy is to add some structure to the problem by including extra constraints or extra assumptions. In this way, the problem becomes well-defined instead of ill-defined-perhaps with a narrower set of options in how you might approach it, but with a clearly specified goal state and, eventually, a manageable set of operations to try. Functional Fixedness Even for well-defined problems, there’s usually more than one way to understand the problem. Consider the problem in Figure 13.9. To solve it, you need to cease thinking of the box as a container and instead think of it as a potential platform. Thus, your chances of solving the problem depend on how you represent the box in your thoughts, and we can show this by encouraging one representation or another. In a classic study, participants were given the equipment shown in Figure 13.9A: some matches, a box of tacks, and a candle. This configuration (implicitly) underscored the box’s conventional function. As a result, the configuration increased functional fixedness-the tendency to be rigid in how one thinks about an object’s function. With fixedness in place, the problem was rarely solved (Duncker, 1945; Fleck & Weisberg, 2004).
Other participants were given the same tools, but configured differently. They were given some matches, a pile of tacks, the box (now empty), and a candle. In this setting, the participants were less they to solve the problem (Duncker, 1945). (Also see were less likely to think of the box likely to think of the box as a container for the tacks, and so likely as a container. As a result, they were more Figure 13.10; for more on fixedness, see McCaffrey, 2012.)
“Thinking outside the Box” A related obstacle derives from someone’s problem-solving set-the collection of beliefs and assumptions a person makes about a problem. One often-discussed example involves the nine-dot problem (see Figure 13.11). People routinely fail to solve this problem, because-according to some interpretations-they (mistakenly) defined by the dots. In fact, this problem is probably the source of the cliché “You need to think assume that the lines they draw need to stay inside the “square” outside the box.”
Ironically, though, this cliché may be misleading. In one study, participants were told explicitly that to solve the problem their lines would need to go outside the square. The hint provided little participants still failed to find the solution (Weisberg & Alba, 1981). Apparently, benefit, and most beliefs about “the box” aren’t the obstacle. Even when we eliminate these beliefs, performance remains poor. Nonetheless, the expression “think outside the box” does get the broad idea right, because to most people assume solve this problem people do need to jettison their initial approach. Specifically, begin and end on dots. People also have the idea that they’ll need to that the lines they draw must maximize the number of dots “canceled” with each move; as a result, they seek solutions in which each line cancels a full row or column of dots. It turns out, though, that these assumptions are guided by these mistaken beliefs, people find this problem quite hard. (See Kershaw & Ohlsson, 2004; MacGregor, Ormerod, & Chronicle, 2001; also Öllinger, Jones, Faber, & Knoblich, wrong; and so, 2013.) In the nine-dot problem, people seem to be victims of their own problem-solving set; to find the solution, they need change that set. This phrasing, however, makes it sound like a set is a bad thing, blocking the discovery of a solution. Let’s emphasize, though, that sets also provide a benefit. as you seek most problems offer a huge number of options This is because (as we mentioned earlier) the solution-an enormous number of moves you might try or approaches you might consider. A problem-solving set helps you, therefore, by narrowing your options, which in turn eases the search for a solution. Thus, in solving the nine-dot problem, you didn’t waste any time wondering whether e holding the pencil between your toes or whether the problem you should try drawing the lines sitting down while you worked on it instead of standing up. These are was hard because you were foolish ideas, so you brushed past them. But what identifies them as foolish? It’s your problem- plausible, which ones are solving set, which tells you, among other things, which options physically possible, and so on. are In this way, a set can blind you to important options and thus be an obstacle. But a set can also blind you to a wide range of futile strategies, and this is a good thing: It enables you to focus, much more productively, on options that are likely to work out. Indeed, without a set, you might be so distracted by silly notions that even the simplest problem would become insoluble. e. Demonstration 13.2: Verbalization and Problem Solving Research on problem solving has attempted to determine what factors help problem solving (making it faster or more effective) and what factors hinder problem solving. Some of these factors are surprising. You’ll need a clock or a timer for this one. Read the first problem below, and give yourself 2 minutes to find the solution. If you haven’t found the solution in this time, take a break for 90 seconds; during that break, turn away from the problem and try to say out loud, in as much detail as possible, everything you can remember about how you’ve been trying to solve the problem. Provide information about your approach, your strategies, any solutions you tried, and so on. Then, go back and try working on the problem for another 2 minutes.
Problem 1: The drawing below shows 10 pennies. Can you make the triangle point downward by moving only 3 of the pennies?
Now, do the same for the next problem-2 minutes of trying a solution, 90 seconds of describing out loud everything you’ve thought of so far in working on the problem, and then another 2 minutes of working on the problem.
Problem 2: Nine sheep are kept together in a square pen. Build two more square enclosures that will isolate each sheep, so that each is in a pen all by itself.
Do you think the time spent describing your efforts so far helped you, perhaps by allowing you to organize your thinking, or hurt you? In fact, in several studies, this sort of “time off, to describe your efforts so far” makes it less likely that people will solve these problems. In one study, participants solved 36% of the problems if they had verbalized their efforts so far, but 46% of the problems if they didn’t go through the verbalization step. Does this fit with your experience? Why might verbalization interfere with this form of problem solving? One likely explanation is that verbalization focuses your attention on steps you’ve already tried, and this may make it more difficult to abandon those steps and try new approaches. The verbalization will also focus your attention on the sorts of strategies that are conscious, deliberate, and easily described in words; this focus might interfere with strategies that are unconscious, not deliberate, and not easily articulated. In all cases, though, the pattern makes it clear that sometimes “thinking out loud” and trying to communicate your ideas to others can actually be counterproductive! Exactly why this is, and what this implies about problem solving, is a topic in need of further research. Demonstration adapted from Schooler, J., Ohlsson, S., & Brooks, K. (1993). Thoughts beyond words: When language overshadows insight. Journal of Experimental Psychology: General, 122, 166-183. Expand the bar below for the solutions to Demonstration 13.2. Creativity There is no question, though, that efforts toward a problem solution are sometimes hindered by someone’s set, and this observation points us toward another way in which people differ. Some people are remarkably flexible in their approaches to problems; they seem easily able to “think outside the box.” Other people, in contrast, seem far too ready to rely on routine, so they’re more vulnerable to the obstacles we’ve just described. How should we think about these differences? Why do some people reliably produce novel and unexpected solutions, while other people offer only familiar solutions? This is, in effect, a question of why some people are creative and others aren’t-a question that forces us to ask: What is creativity? Case Studies of Creativity One approach to this issue focuses on individuals who’ve been enormously creative-artists like Pablo Picasso and Johann Sebastian Bach, or scientists like Charles Darwin and Marie Curie. By studying these giants, perhaps we can draw hints about the nature of creativity when it arises, on a much smaller scale, in day-to-day life-when, for example, you find a creative way to begin a conversation or to repair a damaged friendship. As some researchers put it, we may be able to learn about “little-c creativity” (the everyday sort) by studying “Big-C Creativity” (the sort shown by people we count as scientific or artistic geniuses-Simonton & Damian, 2012). Research suggests, in fact, that highly creative people like Bach and Curie tend to have certain things in common, and we can think of these elements as “prerequisites” for creativity (e.g., Hennessey & Amabile, 2010). These individuals, first of all, generally have great knowledge and skills in their domain. (This point can’t be surprising: If you don’t know a lot of chemistry, you can’t be a creative chemist. If you’re not a skilled storyteller, you can’t be a great novelist.) Second, to be creative, you need certain personality traits: a willingness to take risks, a willingness to ignore criticism, an ability to tolerate ambiguous findings or situations, and an inclination not to “foliow the crowd” Third, highly creative people tend to be motivated by the pleasure of their work rather than by the promise of external rewards. With this, highly creative people tend to work extremely hard on their endeavors and to produce a lot of their product, whether these products are poems, paintings, or scientific papers. Fourth, these highly creative people have generally been “in the right place at the right time”-that is, in environments that allowed them freedom, provided them with the appropriate supports, and offered them problems “ripe” for solution with the resources available. Notice that these observations highlight the contribution of factors outside the person, as well as the person’s own capacities and skills. The external environment, for example, is the source of crucial knowledge and resources, and it often defines the problem itself. This is why many authors have suggested that we need a systematic “sociocultural approach” to creativity-one that considers the social and historical context, as well as the processes unfolding inside the creative