problem solving concepts and theories pdf

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21 Problem Solving

Miriam Bassok, Department of Psychology, University of Washington, Seattle, WA

Laura R. Novick, Department of Psychology and Human Development, Vanderbilt University, Nashville, TN

• Published: 21 November 2012
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This chapter follows the historical development of research on problem solving. It begins with a description of two research traditions that addressed different aspects of the problem-solving process: ( 1 ) research on problem representation (the Gestalt legacy) that examined how people understand the problem at hand, and ( 2 ) research on search in a problem space (the legacy of Newell and Simon) that examined how people generate the problem's solution. It then describes some developments in the field that fueled the integration of these two lines of research: work on problem isomorphs, on expertise in specific knowledge domains (e.g., chess, mathematics), and on insight solutions. Next, it presents examples of recent work on problem solving in science and mathematics that highlight the impact of visual perception and background knowledge on how people represent problems and search for problem solutions. The final section considers possible directions for future research.

People are confronted with problems on a daily basis, be it trying to extract a broken light bulb from a socket, finding a detour when the regular route is blocked, fixing dinner for unexpected guests, dealing with a medical emergency, or deciding what house to buy. Obviously, the problems people encounter differ in many ways, and their solutions require different types of knowledge and skills. Yet we have a sense that all the situations we classify as problems share a common core. Karl Duncker defined this core as follows: “A problem arises when a living creature has a goal but does not know how this goal is to be reached. Whenever one cannot go from the given situation to the desired situation simply by action [i.e., by the performance of obvious operations], then there has to be recourse to thinking” (Duncker, 1945 , p. 1). Consider the broken light bulb. The obvious operation—holding the glass part of the bulb with one's fingers while unscrewing the base from the socket—is prevented by the fact that the glass is broken. Thus, there must be “recourse to thinking” about possible ways to solve the problem. For example, one might try mounting half a potato on the broken bulb (we do not know the source of this creative solution, which is described on many “how to” Web sites).

The above definition and examples make it clear that what constitutes a problem for one person may not be a problem for another person, or for that same person at another point in time. For example, the second time one has to remove a broken light bulb from a socket, the solution likely can be retrieved from memory; there is no problem. Similarly, tying shoes may be considered a problem for 5-year-olds but not for readers of this chapter. And, of course, people may change their goal and either no longer have a problem (e.g., take the guests to a restaurant instead of fixing dinner) or attempt to solve a different problem (e.g., decide what restaurant to go to). Given the highly subjective nature of what constitutes a problem, researchers who study problem solving have often presented people with novel problems that they should be capable of solving and attempted to find regularities in the resulting problem-solving behavior. Despite the variety of possible problem situations, researchers have identified important regularities in the thinking processes by which people (a) represent , or understand, problem situations and (b) search for possible ways to get to their goal.

A problem representation is a model constructed by the solver that summarizes his or her understanding of the problem components—the initial state (e.g., a broken light bulb in a socket), the goal state (the light bulb extracted), and the set of possible operators one may apply to get from the initial state to the goal state (e.g., use pliers). According to Reitman ( 1965 ), problem components differ in the extent to which they are well defined . Some components leave little room for interpretation (e.g., the initial state in the broken light bulb example is relatively well defined), whereas other components may be ill defined and have to be defined by the solver (e.g., the possible actions one may take to extract the broken bulb). The solver's representation of the problem guides the search for a possible solution (e.g., possible attempts at extracting the light bulb). This search may, in turn, change the representation of the problem (e.g., finding that the goal cannot be achieved using pliers) and lead to a new search. Such a recursive process of representation and search continues until the problem is solved or until the solver decides to abort the goal.

Duncker ( 1945 , pp. 28–37) documented the interplay between representation and search based on his careful analysis of one person's solution to the “Radiation Problem” (later to be used extensively in research analogy, see Holyoak, Chapter 13 ). This problem requires using some rays to destroy a patient's stomach tumor without harming the patient. At sufficiently high intensity, the rays will destroy the tumor. However, at that intensity, they will also destroy the healthy tissue surrounding the tumor. At lower intensity, the rays will not harm the healthy tissue, but they also will not destroy the tumor. Duncker's analysis revealed that the solver's solution attempts were guided by three distinct problem representations. He depicted these solution attempts as an inverted search tree in which the three main branches correspond to the three general problem representations (Duncker, 1945 , p. 32). We reproduce this diagram in Figure 21.1 . The desired solution appears on the rightmost branch of the tree, within the general problem representation in which the solver aims to “lower the intensity of the rays on their way through healthy tissue.” The actual solution is to project multiple low-intensity rays at the tumor from several points around the patient “by use of lens.” The low-intensity rays will converge on the tumor, where their individual intensities will sum to a level sufficient to destroy the tumor.

A search-tree representation of one subject's solution to the radiation problem, reproduced from Duncker ( 1945 , p. 32).

Although there are inherent interactions between representation and search, some researchers focus their efforts on understanding the factors that affect how solvers represent problems, whereas others look for regularities in how they search for a solution within a particular representation. Based on their main focus of interest, researchers devise or select problems with solutions that mainly require either constructing a particular representation or finding the appropriate sequence of steps leading from the initial state to the goal state. In most cases, researchers who are interested in problem representation select problems in which one or more of the components are ill defined, whereas those who are interested in search select problems in which the components are well defined. The following examples illustrate, respectively, these two problem types.

The Bird-and-Trains problem (Posner, 1973 , pp. 150–151) is a mathematical word problem that tends to elicit two distinct problem representations (see Fig. 21.2a and b ):

Two train stations are 50 miles apart. At 2 p.m. one Saturday afternoon two trains start toward each other, one from each station. Just as the trains pull out of the stations, a bird springs into the air in front of the first train and flies ahead to the front of the second train. When the bird reaches the second train, it turns back and flies toward the first train. The bird continues to do this until the trains meet. If both trains travel at the rate of 25 miles per hour and the bird flies at 100 miles per hour, how many miles will the bird have flown before the trains meet? Fig. 21.2 Open in new tab Download slide Alternative representations of Posner's ( 1973 ) trains-and-bird problem. Adapted from Novick and Hmelo ( 1994 ).

Some solvers focus on the back-and-forth path of the bird (Fig. 21.2a ). This representation yields a problem that would be difficult for most people to solve (e.g., a series of differential equations). Other solvers focus on the paths of the trains (Fig. 21.2b ), a representation that yields a relatively easy distance-rate-time problem.

The Tower of Hanoi problem falls on the other end of the representation-search continuum. It leaves little room for differences in problem representations, and the primary work is to discover a solution path (or the best solution path) from the initial state to the goal state .

There are three pegs mounted on a base. On the leftmost peg, there are three disks of differing sizes. The disks are arranged in order of size with the largest disk on the bottom and the smallest disk on the top. The disks may be moved one at a time, but only the top disk on a peg may be moved, and at no time may a larger disk be placed on a smaller disk. The goal is to move the three-disk tower from the leftmost peg to the rightmost peg.

Figure 21.3 shows all the possible legal arrangements of disks on pegs. The arrows indicate transitions between states that result from moving a single disk, with the thicker gray arrows indicating the shortest path that connects the initial state to the goal state.

The division of labor between research on representation versus search has distinct historical antecedents and research traditions. In the next two sections, we review the main findings from these two historical traditions. Then, we describe some developments in the field that fueled the integration of these lines of research—work on problem isomorphs, on expertise in specific knowledge domains (e.g., chess, mathematics), and on insight solutions. In the fifth section, we present some examples of recent work on problem solving in science and mathematics. This work highlights the role of visual perception and background knowledge in the way people represent problems and search for problem solutions. In the final section, we consider possible directions for future research.

Our review is by no means an exhaustive one. It follows the historical development of the field and highlights findings that pertain to a wide variety of problems. Research pertaining to specific types of problems (e.g., medical problems), specific processes that are involved in problem solving (e.g., analogical inferences), and developmental changes in problem solving due to learning and maturation may be found elsewhere in this volume (e.g., Holyoak, Chapter 13 ; Smith & Ward, Chapter 23 ; van Steenburgh et al., Chapter 24 ; Simonton, Chapter 25 ; Opfer & Siegler, Chapter 30 ; Hegarty & Stull, Chapter 31 ; Dunbar & Klahr, Chapter 35 ; Patel et al., Chapter 37 ; Lowenstein, Chapter 38 ; Koedinger & Roll, Chapter 40 ).

All possible problem states for the three-disk Tower of Hanoi problem. The thicker gray arrows show the optimum solution path connecting the initial state (State #1) to the goal state (State #27).

Problem Representation: The Gestalt Legacy

Research on problem representation has its origins in Gestalt psychology, an influential approach in European psychology during the first half of the 20th century. (Behaviorism was the dominant perspective in American psychology at this time.) Karl Duncker published a book on the topic in his native German in 1935, which was translated into English and published 10 years later as the monograph On Problem-Solving (Duncker, 1945 ). Max Wertheimer also published a book on the topic in 1945, titled Productive Thinking . An enlarged edition published posthumously includes previously unpublished material (Wertheimer, 1959 ). Interestingly, 1945 seems to have been a watershed year for problem solving, as mathematician George Polya's book, How to Solve It , also appeared then (a second edition was published 12 years later; Polya, 1957 ).

The Gestalt psychologists extended the organizational principles of visual perception to the domain of problem solving. They showed that various visual aspects of the problem, as well the solver's prior knowledge, affect how people understand problems and, therefore, generate problem solutions. The principles of visual perception (e.g., proximity, closure, grouping, good continuation) are directly relevant to problem solving when the physical layout of the problem, or a diagram that accompanies the problem description, elicits inferences that solvers include in their problem representations. Such effects are nicely illustrated by Maier's ( 1930 ) nine-dot problem: Nine dots are arrayed in a 3x3 grid, and the task is to connect all the dots by drawing four straight lines without lifting one's pencil from the paper. People have difficulty solving this problem because their initial representations generally include a constraint, inferred from the configuration of the dots, that the lines should not go outside the boundary of the imaginary square formed by the outer dots. With this constraint, the problem cannot be solved (but see Adams, 1979 ). Without this constraint, the problem may be solved as shown in Figure 21.4 (though the problem is still difficult for many people; see Weisberg & Alba, 1981 ).

The nine-dot problem is a classic insight problem (see van Steenburgh et al., Chapter 24 ). According to the Gestalt view (e.g., Duncker, 1945 ; Kohler, 1925 ; Maier, 1931 ; see Ohlsson, 1984 , for a review), the solution to an insight problem appears suddenly, accompanied by an “aha!” sensation, immediately following the sudden “restructuring” of one's understanding of the problem (i.e., a change in the problem representation): “The decisive points in thought-processes, the moments of sudden comprehension, of the ‘Aha!,’ of the new, are always at the same time moments in which such a sudden restructuring of the thought-material takes place” (Duncker, 1945 , p. 29). For the nine-dot problem, one view of the required restructuring is that the solver relaxes the constraint implied by the perceptual form of the problem and realizes that the lines may, in fact, extend past the boundary of the imaginary square. Later in the chapter, we present more recent accounts of insight.

The entities that appear in a problem also tend to evoke various inferences that people incorporate into their problem representations. A classic demonstration of this is the phenomenon of functional fixedness , introduced by Duncker ( 1945 ): If an object is habitually used for a certain purpose (e.g., a box serves as a container), it is difficult to see

A solution to the nine-dot problem.

that object as having properties that would enable it to be used for a dissimilar purpose. Duncker's basic experimental paradigm involved two conditions that varied in terms of whether the object that was crucial for solution was initially used for a function other than that required for solution.

Consider the candles problem—the best known of the five “practical problems” Duncker ( 1945 ) investigated. Three candles are to be mounted at eye height on a door. On the table, for use in completing this task, are some tacks and three boxes. The solution is to tack the three boxes to the door to serve as platforms for the candles. In the control condition, the three boxes were presented to subjects empty. In the functional-fixedness condition, they were filled with candles, tacks, and matches. Thus, in the latter condition, the boxes initially served the function of container, whereas the solution requires that they serve the function of platform. The results showed that 100% of the subjects who received empty boxes solved the candles problem, compared with only 43% of subjects who received filled boxes. Every one of the five problems in this study showed a difference favoring the control condition over the functional-fixedness condition, with average solution rates across the five problems of 97% and 58%, respectively.

The function of the objects in a problem can be also “fixed” by their most recent use. For example, Birch and Rabinowitz ( 1951 ) had subjects perform two consecutive tasks. In the first task, people had to use either a switch or a relay to form an electric circuit. After completing this task, both groups of subjects were asked to solve Maier's ( 1931 ) two-ropes problem. The solution to this problem requires tying an object to one of the ropes and making the rope swing as a pendulum. Subjects could create the pendulum using either the object from the electric-circuit task or the other object. Birch and Rabinowitz found that subjects avoided using the same object for two unrelated functions. That is, those who used the switch in the first task made the pendulum using the relay, and vice versa. The explanations subjects subsequently gave for their object choices revealed that they were unaware of the functional-fixedness constraint they imposed on themselves.

In addition to investigating people's solutions to such practical problems as irradiating a tumor, mounting candles on the wall, or tying ropes, the Gestalt psychologists examined how people understand and solve mathematical problems that require domain-specific knowledge. For example, Wertheimer ( 1959 ) observed individual differences in students' learning and subsequent application of the formula for finding the area of a parallelogram (see Fig. 21.5a ). Some students understood the logic underlying the learned formula (i.e., the fact that a parallelogram can be transformed into a rectangle by cutting off a triangle from one side and pasting it onto the other side) and exhibited “productive thinking”—using the same logic to find the area of the quadrilateral in Figure 21.5b and the irregularly shaped geometric figure in Figure 21.5c . Other students memorized the formula and exhibited “reproductive thinking”—reproducing the learned solution only to novel parallelograms that were highly similar to the original one.

The psychological study of human problem solving faded into the background after the demise of the Gestalt tradition (during World War II), and problem solving was investigated only sporadically until Allen Newell and Herbert Simon's ( 1972 ) landmark book Human Problem Solving sparked a flurry of research on this topic. Newell and Simon adopted and refined Duncker's ( 1945 ) methodology of collecting and analyzing the think-aloud protocols that accompany problem solutions and extended Duncker's conceptualization of a problem solution as a search tree. However, their initial work did not aim to extend the Gestalt findings

Finding the area of ( a ) a parallelogram, ( b ) a quadrilateral, and ( c ) an irregularly shaped geometric figure. The solid lines indicate the geometric figures whose areas are desired. The dashed lines show how to convert the given figures into rectangles (i.e., they show solutions with understanding).

pertaining to problem representation. Instead, as we explain in the next section, their objective was to identify the general-purpose strategies people use in searching for a problem solution.

Search in a Problem Space: The Legacy of Newell and Simon

Newell and Simon ( 1972 ) wrote a magnum opus detailing their theory of problem solving and the supporting research they conducted with various collaborators. This theory was grounded in the information-processing approach to cognitive psychology and guided by an analogy between human and artificial intelligence (i.e., both people and computers being “Physical Symbol Systems,” Newell & Simon, 1976 ; see Doumas & Hummel, Chapter 5 ). They conceptualized problem solving as a process of search through a problem space for a path that connects the initial state to the goal state—a metaphor that alludes to the visual or spatial nature of problem solving (Simon, 1990 ). The term problem space refers to the solver's representation of the task as presented (Simon, 1978 ). It consists of ( 1 ) a set of knowledge states (the initial state, the goal state, and all possible intermediate states), ( 2 ) a set of operators that allow movement from one knowledge state to another, ( 3 ) a set of constraints, and ( 4 ) local information about the path one is taking through the space (e.g., the current knowledge state and how one got there).

We illustrate the components of a problem space for the three-disk Tower of Hanoi problem, as depicted in Figure 21.3 . The initial state appears at the top (State #1) and the goal state at the bottom right (State #27). The remaining knowledge states in the figure are possible intermediate states. The current knowledge state is the one at which the solver is located at any given point in the solution process. For example, the current state for a solver who has made three moves along the optimum solution path would be State #9. The solver presumably would know that he or she arrived at this state from State #5. This knowledge allows the solver to recognize a move that involves backtracking. The three operators in this problem are moving each of the three disks from one peg to another. These operators are subject to the constraint that a larger disk may not be placed on a smaller disk.

Newell and Simon ( 1972 ), as well as other contemporaneous researchers (e.g., Atwood & Polson, 1976 ; Greeno, 1974 ; Thomas, 1974 ), examined how people traverse the spaces of various well-defined problems (e.g., the Tower of Hanoi, Hobbits and Orcs). They discovered that solvers' search is guided by a number of shortcut strategies, or heuristics , which are likely to get the solver to the goal state without an extensive amount of search. Heuristics are often contrasted with algorithms —methods that are guaranteed to yield the correct solution. For example, one could try every possible move in the three-disk Tower of Hanoi problem and, eventually, find the correct solution. Although such an exhaustive search is a valid algorithm for this problem, for many problems its application is very time consuming and impractical (e.g., consider the game of chess).

In their attempts to identify people's search heuristics, Newell and Simon ( 1972 ) relied on two primary methodologies: think-aloud protocols and computer simulations. Their use of think-aloud protocols brought a high degree of scientific rigor to the methodology used by Duncker ( 1945 ; see Ericsson & Simon, 1980 ). Solvers were required to say out loud everything they were thinking as they solved the problem, that is, everything that went through their verbal working memory. Subjects' verbalizations—their think-aloud protocols—were tape-recorded and then transcribed verbatim for analysis. This method is extremely time consuming (e.g., a transcript of one person's solution to the cryptarithmetic problem DONALD + GERALD = ROBERT, with D = 5, generated a 17-page transcript), but it provides a detailed record of the solver's ongoing solution process.

An important caveat to keep in mind while interpreting a subject's verbalizations is that “a protocol is relatively reliable only for what it positively contains, but not for that which it omits” (Duncker, 1945 , p. 11). Ericsson and Simon ( 1980 ) provided an in-depth discussion of the conditions under which this method is valid (but see Russo, Johnson, & Stephens, 1989 , for an alternative perspective). To test their interpretation of a subject's problem solution, inferred from the subject's verbal protocol, Newell and Simon ( 1972 ) created a computer simulation program and examined whether it solved the problem the same way the subject did. To the extent that the computer simulation provided a close approximation of the solver's step-by-step solution process, it lent credence to the researcher's interpretation of the verbal protocol.

Newell and Simon's ( 1972 ) most famous simulation was the General Problem Solver or GPS (Ernst & Newell, 1969 ). GPS successfully modeled human solutions to problems as different as the Tower of Hanoi and the construction of logic proofs using a single general-purpose heuristic: means-ends analysis . This heuristic captures people's tendency to devise a solution plan by setting subgoals that could help them achieve their final goal. It consists of the following steps: ( 1 ) Identify a difference between the current state and the goal (or subgoal ) state; ( 2 ) Find an operator that will remove (or reduce) the difference; (3a) If the operator can be directly applied, do so, or (3b) If the operator cannot be directly applied, set a subgoal to remove the obstacle that is preventing execution of the desired operator; ( 4 ) Repeat steps 1–3 until the problem is solved. Next, we illustrate the implementation of this heuristic for the Tower of Hanoi problem, using the problem space in Figure 21.3 .

As can be seen in Figure 21.3 , a key difference between the initial state and the goal state is that the large disk is on the wrong peg (step 1). To remove this difference (step 2), one needs to apply the operator “move-large-disk.” However, this operator cannot be applied because of the presence of the medium and small disks on top of the large disk. Therefore, the solver may set a subgoal to move that two-disk tower to the middle peg (step 3b), leaving the rightmost peg free for the large disk. A key difference between the initial state and this new subgoal state is that the medium disk is on the wrong peg. Because application of the move-medium-disk operator is blocked, the solver sets another subgoal to move the small disk to the right peg. This subgoal can be satisfied immediately by applying the move-small-disk operator (step 3a), generating State #3. The solver then returns to the previous subgoal—moving the tower consisting of the small and medium disks to the middle peg. The differences between the current state (#3) and the subgoal state (#9) can be removed by first applying the move-medium-disk operator (yielding State #5) and then the move-small-disk operator (yielding State #9). Finally, the move-large-disk operator is no longer blocked. Hence, the solver moves the large disk to the right peg, yielding State #11.

Notice that the subgoals are stacked up in the order in which they are generated, so that they pop up in the order of last in first out. Given the first subgoal in our example, repeated application of the means-ends analysis heuristic will yield the shortest-path solution, indicated by the large gray arrows. In general, subgoals provide direction to the search and allow solvers to plan several moves ahead. By assessing progress toward a required subgoal rather than the final goal, solvers may be able to make moves that otherwise seem unwise. To take a concrete example, consider the transition from State #1 to State #3 in Figure 21.3 . Comparing the initial state to the goal state, this move seems unwise because it places the small disk on the bottom of the right peg, whereas it ultimately needs to be at the top of the tower on that peg. But comparing the initial state to the solver-generated subgoal state of having the medium disk on the middle peg, this is exactly where the small disk needs to go.

Means-ends analysis and various other heuristics (e.g., the hill-climbing heuristic that exploits the similarity, or distance, between the state generated by the next operator and the goal state; working backward from the goal state to the initial state) are flexible strategies that people often use to successfully solve a large variety of problems. However, the generality of these heuristics comes at a cost: They are relatively weak and fallible (e.g., in the means-ends solution to the problem of fixing a hole in a bucket, “Dear Liza” leads “Dear Henry” in a loop that ends back at the initial state; the lyrics of this famous song can be readily found on the Web). Hence, although people use general-purpose heuristics when they encounter novel problems, they replace them as soon as they acquire experience with and sufficient knowledge about the particular problem space (e.g., Anzai & Simon, 1979 ).

Despite the fruitfulness of this research agenda, it soon became evident that a fundamental weakness was that it minimized the importance of people's background knowledge. Of course, Newell and Simon ( 1972 ) were aware that problem solutions require relevant knowledge (e.g., the rules of logical proofs, or rules for stacking disks). Hence, in programming GPS, they supplemented every problem they modeled with the necessary background knowledge. This practice highlighted the generality and flexibility of means-ends analysis but failed to capture how people's background knowledge affects their solutions. As we discussed in the previous section, domain knowledge is likely to affect how people represent problems and, therefore, how they generate problem solutions. Moreover, as people gain experience solving problems in a particular knowledge domain (e.g., math, physics), they change their representations of these problems (e.g., Chi, Feltovich, & Glaser, 1981 ; Haverty, Koedinger, Klahr, & Alibali, 2000 ; Schoenfeld & Herrmann, 1982 ) and learn domain-specific heuristics (e.g., Polya, 1957 ; Schoenfeld, 1979 ) that trump the general-purpose strategies.

It is perhaps inevitable that the two traditions in problem-solving research—one emphasizing representation and the other emphasizing search strategies—would eventually come together. In the next section we review developments that led to this integration.

The Two Legacies Converge

Because Newell and Simon ( 1972 ) aimed to discover the strategies people use in searching for a solution, they investigated problems that minimized the impact of factors that tend to evoke differences in problem representations, of the sort documented by the Gestalt psychologists. In subsequent work, however, Simon and his collaborators showed that such factors are highly relevant to people's solutions of well-defined problems, and Simon ( 1986 ) incorporated these findings into the theoretical framework that views problem solving as search in a problem space.

In this section, we first describe illustrative examples of this work. We then describe research on insight solutions that incorporates ideas from the two legacies described in the previous sections.

Relevance of the Gestalt Ideas to the Solution of Search Problems

In this subsection we describe two lines of research by Simon and his colleagues, and by other researchers, that document the importance of perception and of background knowledge to the way people search for a problem solution. The first line of research used variants of relatively well-defined riddle problems that had the same structure (i.e., “problem isomorphs”) and, therefore, supposedly the same problem space. It documented that people's search depended on various perceptual and conceptual inferences they tended to draw from a specific instantiation of the problem's structure. The second line of research documented that people's search strategies crucially depend on their domain knowledge and on their prior experience with related problems.

Problem Isomorphs

Hayes and Simon ( 1977 ) used two variants of the Tower of Hanoi problem that, instead of disks and pegs, involved monsters and globes that differed in size (small, medium, and large). In both variants, the initial state had the small monster holding the large globe, the medium-sized monster holding the small globe, and the large monster holding the medium-sized globe. Moreover, in both variants the goal was for each monster to hold a globe proportionate to its own size. The only difference between the problems concerned the description of the operators. In one variant (“transfer”), subjects were told that the monsters could transfer the globes from one to another as long as they followed a set of rules, adapted from the rules in the original Tower of Hanoi problem (e.g., only one globe may be transferred at a time). In the other variant (“change”), subjects were told that the monsters could shrink and expand themselves according to a set of rules, which corresponded to the rules in the transfer version of the problem (e.g., only one monster may change its size at a time). Despite the isomorphism of the two variants, subjects conducted their search in two qualitatively different problem spaces, which led to solution times for the change variant being almost twice as long as those for the transfer variant. This difference arose because subjects could more readily envision and track an object that was changing its location with every move than one that was changing its size.

Recent work by Patsenko and Altmann ( 2010 ) found that, even in the standard Tower of Hanoi problem, people's solutions involve object-bound routines that depend on perception and selective attention. The subjects in their study solved various Tower of Hanoi problems on a computer. During the solution of a particular “critical” problem, the computer screen changed at various points without subjects' awareness (e.g., a disk was added, such that a subject who started with a five-disc tower ended with a six-disc tower). Patsenko and Altmann found that subjects' moves were guided by the configurations of the objects on the screen rather than by solution plans they had stored in memory (e.g., the next subgoal).

The Gestalt psychologists highlighted the role of perceptual factors in the formation of problem representations (e.g., Maier's, 1930 , nine-dot problem) but were generally silent about the corresponding implications for how the problem was solved (although they did note effects on solution accuracy). An important contribution of the work on people's solutions of the Tower of Hanoi problem and its variants was to show the relevance of perceptual factors to the application of various operators during search for a problem solution—that is, to the how of problem solving. In the next section, we describe recent work that documents the involvement of perceptual factors in how people understand and use equations and diagrams in the context of solving math and science problems.

Kotovsky, Hayes, and Simon ( 1985 ) further investigated factors that affect people's representation and search in isomorphs of the Tower of Hanoi problem. In one of their isomorphs, three disks were stacked on top of each other to form an inverted pyramid, with the smallest disc on the bottom and the largest on top. Subjects' solutions of the inverted pyramid version were similar to their solutions of the standard version that has the largest disc on the bottom and the smallest on top. However, the two versions were solved very differently when subjects were told that the discs represent acrobats. Subjects readily solved the version in which they had to place a small acrobat on the shoulders of a large one, but they refrained from letting a large acrobat stand on the shoulders of a small one. In other words, object-based inferences that draw on people's semantic knowledge affected the solution of search problems, much as they affect the solution of the ill-defined problems investigated by the Gestalt psychologists (e.g., Duncker's, 1945 , candles problem). In the next section, we describe more recent work that shows similar effects in people's solutions to mathematical word problems.

The work on differences in the representation and solution of problem isomorphs is highly relevant to research on analogical problem solving (or analogical transfer), which examines when and how people realize that two problems that differ in their cover stories have a similar structure (or a similar problem space) and, therefore, can be solved in a similar way. This research shows that minor differences between example problems, such as the use of X-rays versus ultrasound waves to fuse a broken filament of a light bulb, can elicit different problem representations that significantly affect the likelihood of subsequent transfer to novel problem analogs (Holyoak & Koh, 1987 ). Analogical transfer has played a central role in research on human problem solving, in part because it can shed light on people's understanding of a given problem and its solution and in part because it is believed to provide a window onto understanding and investigating creativity (see Smith & Ward, Chapter 23 ). We briefly mention some findings from the analogy literature in the next subsection on expertise, but we do not discuss analogical transfer in detail because this topic is covered elsewhere in this volume (Holyoak, Chapter 13 ).

Expertise and Its Development

In another line of research, Simon and his colleagues examined how people solve ecologically valid problems from various rule-governed and knowledge-rich domains. They found that people's level of expertise in such domains, be it in chess (Chase & Simon, 1973 ; Gobet & Simon, 1996 ), mathematics (Hinsley, Hayes, & Simon, 1977 ; Paige & Simon, 1966 ), or physics (Larkin, McDermott, Simon, & Simon, 1980 ; Simon & Simon, 1978 ), plays a crucial role in how they represent problems and search for solutions. This work, and the work of numerous other researchers, led to the discovery (and rediscovery, see Duncker, 1945 ) of important differences between experts and novices, and between “good” and “poor” students.

One difference between experts and novices pertains to pattern recognition. Experts' attention is quickly captured by familiar configurations within a problem situation (e.g., a familiar configuration of pieces in a chess game). In contrast, novices' attention is focused on isolated components of the problem (e.g., individual chess pieces). This difference, which has been found in numerous domains, indicates that experts have stored in memory many meaningful groups (chunks) of information: for example, chess (Chase & Simon, 1973 ), circuit diagrams (Egan & Schwartz, 1979 ), computer programs (McKeithen, Reitman, Rueter, & Hirtle, 1981 ), medicine (Coughlin & Patel, 1987 ; Myles-Worsley, Johnston, & Simons, 1988 ), basketball and field hockey (Allard & Starkes, 1991 ), and figure skating (Deakin & Allard, 1991 ).

The perceptual configurations that domain experts readily recognize are associated with stored solution plans and/or compiled procedures (Anderson, 1982 ). As a result, experts' solutions are much faster than, and often qualitatively different from, the piecemeal solutions that novice solvers tend to construct (e.g., Larkin et al., 1980 ). In effect, experts often see the solutions that novices have yet to compute (e.g., Chase & Simon, 1973 ; Novick & Sherman, 2003 , 2008 ). These findings have led to the design of various successful instructional interventions (e.g., Catrambone, 1998 ; Kellman et al., 2008 ). For example, Catrambone ( 1998 ) perceptually isolated the subgoals of a statistics problem. This perceptual chunking of meaningful components of the problem prompted novice students to self-explain the meaning of the chunks, leading to a conceptual understanding of the learned solution. In the next section, we describe some recent work that shows the beneficial effects of perceptual pattern recognition on the solution of familiar mathematics problems, as well as the potentially detrimental effects of familiar perceptual chunks to understanding and reasoning with diagrams depicting evolutionary relationships among taxa.

Another difference between experts and novices pertains to their understanding of the solution-relevant problem structure. Experts' knowledge is highly organized around domain principles, and their problem representations tend to reflect this principled understanding. In particular, they can extract the solution-relevant structure of the problems they encounter (e.g., meaningful causal relations among the objects in the problem; see Cheng & Buehner, Chapter 12 ). In contrast, novices' representations tend to be bound to surface features of the problems that may be irrelevant to solution (e.g., the particular objects in a problem). For example, Chi, Feltovich, and Glaser ( 1981 ) examined how students with different levels of physics expertise group mechanics word problems. They found that advanced graduate students grouped the problems based on the physics principles relevant to the problems' solutions (e.g., conservation of energy, Newton's second law). In contrast, undergraduates who had successfully completed an introductory course in mechanics grouped the problems based on the specific objects involved (e.g., pulley problems, inclined plane problems). Other researchers have found similar results in the domains of biology, chemistry, computer programming, and math (Adelson, 1981 ; Kindfield, 1993 / 1994 ; Kozma & Russell, 1997 ; McKeithen et al., 1981 ; Silver, 1979 , 1981 ; Weiser & Shertz, 1983 ).

The level of domain expertise and the corresponding representational differences are, of course, a matter of degree. With increasing expertise, there is a gradual change in people's focus of attention from aspects that are not relevant to solution to those that are (e.g., Deakin & Allard, 1991 ; Hardiman, Dufresne, & Mestre, 1989 ; McKeithen et al., 1981 ; Myles-Worsley et al., 1988 ; Schoenfeld & Herrmann, 1982 ; Silver, 1981 ). Interestingly, Chi, Bassok, Lewis, Reimann, and Glaser ( 1989 ) found similar differences in focus on structural versus surface features among a group of novices who studied worked-out examples of mechanics problems. These differences, which echo Wertheimer's ( 1959 ) observations of individual differences in students' learning about the area of parallelograms, suggest that individual differences in people's interests and natural abilities may affect whether, or how quickly, they acquire domain expertise.

An important benefit of experts' ability to focus their attention on solution-relevant aspects of problems is that they are more likely than novices to recognize analogous problems that involve different objects and cover stories (e.g., Chi et al., 1989 ; Novick, 1988 ; Novick & Holyoak, 1991 ; Wertheimer, 1959 ) or that come from other knowledge domains (e.g., Bassok & Holyoak, 1989 ; Dunbar, 2001 ; Goldstone & Sakamoto, 2003 ). For example, Bassok and Holyoak ( 1989 ) found that, after learning to solve arithmetic-progression problems in algebra, subjects spontaneously applied these algebraic solutions to analogous physics problems that dealt with constantly accelerated motion. Note, however, that experts and good students do not simply ignore the surface features of problems. Rather, as was the case in the problem isomorphs we described earlier (Kotovsky et al., 1985 ), they tend to use such features to infer what the problem's structure could be (e.g., Alibali, Bassok, Solomon, Syc, & Goldin-Meadow, 1999 ; Blessing & Ross, 1996 ). For example, Hinsley et al. ( 1977 ) found that, after reading no more than the first few words of an algebra word problem, expert solvers classified the problem into a likely problem category (e.g., a work problem, a distance problem) and could predict what questions they might be asked and the equations they likely would need to use.

Surface-based problem categorization has a heuristic value (Medin & Ross, 1989 ): It does not ensure a correct categorization (Blessing & Ross, 1996 ), but it does allow solvers to retrieve potentially appropriate solutions from memory and to use them, possibly with some adaptation, to solve a variety of novel problems. Indeed, although experts exploit surface-structure correlations to save cognitive effort, they have the capability to realize that a particular surface cue is misleading (Hegarty, Mayer, & Green, 1992 ; Lewis & Mayer, 1987 ; Martin & Bassok, 2005 ; Novick 1988 , 1995 ; Novick & Holyoak, 1991 ). It is not surprising, therefore, that experts may revert to novice-like heuristic methods when solving problems under pressure (e.g., Beilock, 2008 ) or in subdomains in which they have general but not specific expertise (e.g., Patel, Groen, & Arocha, 1990 ).

Relevance of Search to Insight Solutions

We introduced the notion of insight in our discussion of the nine-dot problem in the section on the Gestalt tradition. The Gestalt view (e.g., Duncker, 1945 ; Maier, 1931 ; see Ohlsson, 1984 , for a review) was that insight problem solving is characterized by an initial work period during which no progress toward solution is made (i.e., an impasse), a sudden restructuring of one's problem representation to a more suitable form, followed immediately by the sudden appearance of the solution. Thus, solving problems by insight was believed to be all about representation, with essentially no role for a step-by-step solution process (i.e., search). Subsequent and contemporary researchers have generally concurred with the Gestalt view that getting the right representation is crucial. However, research has shown that insight solutions do not necessarily arise suddenly or full blown after restructuring (e.g., Weisberg & Alba, 1981 ); and even when they do, the underlying solution process (in this case outside of awareness) may reflect incremental progress toward the goal (Bowden & Jung-Beeman, 2003 ; Durso, Rea, & Dayton, 1994 ; Novick & Sherman, 2003 ).

“Demystifying insight,” to borrow a phrase from Bowden, Jung-Beeman, Fleck, and Kounios ( 2005 ), requires explaining ( 1 ) why solvers initially reach an impasse in solving a problem for which they have the necessary knowledge to generate the solution, ( 2 ) how the restructuring occurred, and ( 3 ) how it led to the solution. A detailed discussion of these topics appears elsewhere in this volume (van Steenburgh et al., Chapter 24 ). Here, we describe briefly three recent theories that have attempted to account for various aspects of these phenomena: Knoblich, Ohlsson, Haider, and Rhenius's ( 1999 ) representational change theory, MacGregor, Ormerod, and Chronicle's ( 2001 ) progress monitoring theory, and Bowden et al.'s ( 2005 ) neurological model. We then propose the need for an integrated approach to demystifying insight that considers both representation and search.

According to Knoblich et al.'s ( 1999 ) representational change theory, problems that are solved with insight are highly likely to evoke initial representations in which solvers place inappropriate constraints on their solution attempts, leading to an impasse. An impasse can be resolved by revising one's representation of the problem. Knoblich and his colleagues tested this theory using Roman numeral matchstick arithmetic problems in which solvers must move one stick to a new location to change a false numerical statement (e.g., I = II + II ) into a statement that is true. According to representational change theory, re-representation may occur through either constraint relaxation or chunk decomposition. (The solution to the example problem is to change II + to III – , which requires both methods of re-representation, yielding I = III – II ). Good support for this theory has been found based on measures of solution rate, solution time, and eye fixation (Knoblich et al., 1999 ; Knoblich, Ohlsson, & Raney, 2001 ; Öllinger, Jones, & Knoblich, 2008 ).

Progress monitoring theory (MacGregor et al., 2001 ) was proposed to account for subjects' difficulty in solving the nine-dot problem, which has traditionally been classified as an insight problem. According to this theory, solvers use the hill-climbing search heuristic to solve this problem, just as they do for traditional search problems (e.g., Hobbits and Orcs). In particular, solvers are hypothesized to monitor their progress toward solution using a criterion generated from the problem's current state. If solvers reach criterion failure, they seek alternative solutions by trying to relax one or more problem constraints. MacGregor et al. found support for this theory using several variants of the nine-dot problem (also see Ormerod, MacGregor, & Chronicle, 2002 ). Jones ( 2003 ) suggested that progress monitoring theory provides an account of the solution process up to the point an impasse is reached and representational change is sought, at which point representational change theory picks up and explains how insight may be achieved. Hence, it appears that a complete account of insight may require an integration of concepts from the Gestalt (representation) and Newell and Simon's (search) legacies.

Bowden et al.'s ( 2005 ) neurological model emphasizes the overlap between problem solving and language comprehension, and it hinges on differential processing in the right and left hemispheres. They proposed that an impasse is reached because initial processing of the problem produces strong activation of information irrelevant to solution in the left hemisphere. At the same time, weak semantic activation of alternative semantic interpretations, critical for solution, occurs in the right hemisphere. Insight arises when the weakly activated concepts reinforce each other, eventually rising above the threshold required for conscious awareness. Several studies of problem solving using compound remote associates problems, involving both behavioral and neuroimaging data, have found support for this model (Bowden & Jung-Beeman, 1998 , 2003 ; Jung-Beeman & Bowden, 2000 ; Jung-Beeman et al., 2004 ; also see Moss, Kotovsky, & Cagan, 2011 ).

Note that these three views of insight have received support using three quite distinct types of problems (Roman numeral matchstick arithmetic problems, the nine-dot problem, and compound remote associates problems, respectively). It remains to be established, therefore, whether these accounts can be generalized across problems. Kershaw and Ohlsson ( 2004 ) argued that insight problems are difficult because the key behavior required for solution may be hindered by perceptual factors (the Gestalt view), background knowledge (so expertise may be important; e.g., see Novick & Sherman, 2003 , 2008 ), and/or process factors (e.g., those affecting search). From this perspective, solving visual problems (e.g., the nine-dot problem) with insight may call upon more general visual processes, whereas solving verbal problems (e.g., anagrams, compound remote associates) with insight may call upon general verbal/semantic processes.

The work we reviewed in this section shows the relevance of problem representation (the Gestalt legacy) to the way people search the problem space (the legacy of Newell and Simon), and the relevance of search to the solution of insight problems that require a representational change. In addition to this inevitable integration of the two legacies, the work we described here underscores the fact that problem solving crucially depends on perceptual factors and on the solvers' background knowledge. In the next section, we describe some recent work that shows the involvement of these factors in the solution of problems in math and science.

Effects of Perception and Knowledge in Problem Solving in Academic Disciplines

Although the use of puzzle problems continues in research on problem solving, especially in investigations of insight, many contemporary researchers tackle problem solving in knowledge-rich domains, often in academic disciplines (e.g., mathematics, biology, physics, chemistry, meteorology). In this section, we provide a sampling of this research that highlights the importance of visual perception and background knowledge for successful problem solving.

The Role of Visual Perception

We stated at the outset that a problem representation (e.g., the problem space) is a model of the problem constructed by solvers to summarize their understanding of the problem's essential nature. This informal definition refers to the internal representations people construct and hold in working memory. Of course, people may also construct various external representations (Markman, 1999 ) and even manipulate those representations to aid in solution (see Hegarty & Stull, Chapter 31 ). For example, solvers often use paper and pencil to write notes or draw diagrams, especially when solving problems from formal domains (e.g., Cox, 1999 ; Kindfield, 1993 / 1994 ; S. Schwartz, 1971 ). In problems that provide solvers with external representation, such as the Tower of Hanoi problem, people's planning and memory of the current state is guided by the actual configurations of disks on pegs (Garber & Goldin-Meadow, 2002 ) or by the displays they see on a computer screen (Chen & Holyoak, 2010 ; Patsenko & Altmann, 2010 ).

In STEM (science, technology, engineering, and mathematics) disciplines, it is common for problems to be accompanied by diagrams or other external representations (e.g., equations) to be used in determining the solution. Larkin and Simon ( 1987 ) examined whether isomorphic sentential and diagrammatic representations are interchangeable in terms of facilitating solution. They argued that although the two formats may be equivalent in the sense that all of the information in each format can be inferred from the other format (informational equivalence), the ease or speed of making inferences from the two formats might differ (lack of computational equivalence). Based on their analysis of several problems in physics and math, Larkin and Simon further argued for the general superiority of diagrammatic representations (but see Mayer & Gallini, 1990 , for constraints on this general conclusion).

Novick and Hurley ( 2001 , p. 221) succinctly summarized the reasons for the general superiority of diagrams (especially abstract or schematic diagrams) over verbal representations: They “(a) simplify complex situations by discarding unnecessary details (e.g., Lynch, 1990 ; Winn, 1989 ), (b) make abstract concepts more concrete by mapping them onto spatial layouts with familiar interpretational conventions (e.g., Winn, 1989 ), and (c) substitute easier perceptual inferences for more computationally intensive search processes and sentential deductive inferences (Barwise & Etchemendy, 1991 ; Larkin & Simon, 1987 ).” Despite these benefits of diagrammatic representations, there is an important caveat, noted by Larkin and Simon ( 1987 , p. 99) at the very end of their paper: “Although every diagram supports some easy perceptual inferences, nothing ensures that these inferences must be useful in the problem-solving process.” We will see evidence of this in several of the studies reviewed in this section.

Next we describe recent work on perceptual factors that are involved in people's use of two types of external representations that are provided as part of the problem in two STEM disciplines: equations in algebra and diagrams in evolutionary biology. Although we focus here on effects of perceptual factors per se, it is important to note that such factors only influence performance when subjects have background knowledge that supports differential interpretation of the alternative diagrammatic depictions presented (Hegarty, Canham, & Fabricant, 2010 ).

In the previous section, we described the work of Patsenko and Altmann ( 2010 ) that shows direct involvement of visual attention and perception in the sequential application of move operators during the solution of the Tower of Hanoi problem. A related body of work documents similar effects in tasks that require the interpretation and use of mathematical equations (Goldstone, Landy, & Son, 2010 ; Landy & Goldstone, 2007a , b). For example, Landy and Goldstone ( 2007b ) varied the spatial proximity of arguments to the addition (+) and multiplication (*) operators in algebraic equations, such that the spatial layout of the equation was either consistent or inconsistent with the order-of-operations rule that multiplication precedes addition. In consistent equations , the space was narrower around multiplication than around addition (e.g., g*m + r*w = m*g + w*r ), whereas in inconsistent equations this relative spacing was reversed (e.g., s * n+e * c = n * s+c * e ). Subjects' judgments of the validity of such equations (i.e., whether the expressions on the two sides of the equal sign are equivalent) were significantly faster and more accurate for consistent than inconsistent equations.

In discussing these findings and related work with other external representations, Goldstone et al. ( 2010 ) proposed that experience with solving domain-specific problems leads people to “rig up” their perceptual system such that it allows them to look at the problem in a way that is consistent with the correct rules. Similar logic guides the Perceptual Learning Modules developed by Kellman and his collaborators to help students interpret and use algebraic equations and graphs (Kellman et al., 2008 ; Kellman, Massey, & Son, 2009 ). These authors argued and showed that, consistent with the previously reviewed work on expertise, perceptual training with particular external representations supports the development of perceptual fluency. This fluency, in turn, supports students' subsequent use of these external representations for problem solving.

This research suggests that extensive experience with particular equations or graphs may lead to perceptual fluency that could replace the more mindful application of domain-specific rules. Fisher, Borchert, and Bassok ( 2011 ) reported results from algebraic-modeling tasks that are consistent with this hypothesis. For example, college students were asked to represent verbal statements with algebraic equations, a task that typically elicits systematic errors (e.g., Clement, Lochhead, & Monk, 1981 ). Fisher et al. found that such errors were very common when subjects were asked to construct “standard form” equations ( y = ax ), which support fluent left-to-right translation of words to equations, but were relatively rare when subjects were asked to construct nonstandard division-format equations (x = y/a) that do not afford such translation fluency.

In part because of the left-to-right order in which people process equations, which mirrors the linear order in which they process text, equations have traditionally been viewed as sentential representations. However, Landy and Goldstone ( 2007a ) have proposed that equations also share some properties with diagrammatic displays and that, in fact, in some ways they are processed like diagrams. That is, spatial information is used to represent and to support inferences about syntactic structure. This hypothesis received support from Landy and Goldstone's ( 2007b ) results, described earlier, in which subjects' judgments of the validity of equations were affected by the Gestalt principle of grouping: Subjects did better when the grouping was consistent rather than inconsistent with the underlying structure of the problem (order of operations). Moreover, Landy and Goldstone ( 2007a ) found that when subjects wrote their own equations they grouped numbers and operators (+, *, =) in a way that reflected the hierarchical structure imposed by the order-of-operations rule.

In a recent line of research, Novick and Catley ( 2007 ; Novick, Catley, & Funk, 2010 ; Novick, Shade, & Catley, 2011 ) have examined effects of the spatial layout of diagrams depicting the evolutionary history of a set of taxa on people's ability to reason about patterns of relationship among those taxa. We consider here their work that investigates the role of another Gestalt perceptual principle—good continuation—in guiding students' reasoning. According to this principle, a continuous line is perceived as a single entity (Kellman, 2000 ). Consider the diagrams shown in Figure 21.6 . Each is a cladogram, a diagram that depicts nested sets of taxa that are related in terms of levels of most recent common ancestry. For example, chimpanzees and starfish are more closely related to each other than either is to spiders. The supporting evidence for their close relationship is their most recent common ancestor, which evolved the novel character of having radial cleavage. Spiders do not share this ancestor and thus do not have this character.

Cladograms are typically drawn in two isomorphic formats, which Novick and Catley ( 2007 ) referred to as trees and ladders. Although these formats are informationally equivalent (Larkin & Simon, 1987 ), Novick and Catley's ( 2007 ) research shows that they are not computationally equivalent (Larkin & Simon, 1987 ). Imagine that you are given evolutionary relationships in the ladder format, such as in Figure 21.6a (but without the four characters—hydrostatic skeleton, bilateral symmetry, radial cleavage, and trocophore larvae—and associated short lines indicating their locations on the cladogram), and your task is to translate that diagram to the tree format. A correct translation is shown in Figure 21.6b . Novick and Catley ( 2007 ) found that college students were much more likely to get such problems correct when the presented cladogram was in the nested circles (e.g., Figure 21.6d ) rather than the ladder format. Because the Gestalt principle of good continuation makes the long slanted line at the base of the ladder appear to represent a single hierarchical level, a common translation error for the ladder to tree problems was to draw a diagram such as that shown in Figure 21.6c .

The difficulty that good continuation presents for interpreting relationships depicted in the ladder format extends to answering reasoning questions as well. Novick and Catley (unpublished data) asked comparable questions about relationships depicted in the ladder and tree formats. For example, using the cladograms depicted in Figures 21.6a and 21.6b , consider the following questions: (a) Which taxon—jellyfish or earthworm—is the closest evolutionary relation to starfish, and what evidence supports your answer? (b) Do the bracketed taxa comprise a clade (a set of taxa consisting of the most recent common ancestor and all of its descendants), and what evidence supports your answer? For both such questions, students had higher accuracy and evidence quality composite scores when the relationships were depicted in the tree than the ladder format.

Four cladograms depicting evolutionary relationships among six animal taxa. Cladogram ( a ) is in the ladder format, cladograms ( b ) and ( c ) are in the tree format, and cladogram ( d ) is in the nested circles format. Cladograms ( a ), ( b ), and ( d ) are isomorphic.

If the difficulty in extracting the hierarchical structure of the ladder format is due to good continuation (which leads problem solvers to interpret continuous lines that depict multiple hierarchical levels as depicting only a single level), then a manipulation that breaks good continuation at the points where a new hierarchical level occurs should improve understanding. Novick et al. ( 2010 ) tested this hypothesis using a translation task by manipulating whether characters that are the markers for the most recent common ancestor of each nested set of taxa were included on the ladders. Figure 21.6a shows a ladder with such characters. As predicted, translation accuracy increased dramatically simply by adding these characters to the ladders, despite the additional information subjects had to account for in their translations.

The Role of Background Knowledge

As we mentioned earlier, the specific entities in the problems people encounter evoke inferences that affect how people represent these problems (e.g., the candle problem; Duncker, 1945 ) and how they apply the operators in searching for the solution (e.g., the disks vs. acrobats versions of the Tower of Hanoi problem; Kotovsky et al., 1985 ). Such object-based inferences draw on people's knowledge about the properties of the objects (e.g., a box is a container, an acrobat is a person who can be hurt). Here, we describe the work of Bassok and her colleagues, who found that similar inferences affect how people select mathematical procedures to solve problems in various formal domains. This work shows that the objects in the texts of mathematical word problems affect how people represent the problem situation (i.e., the situation model they construct; Kintsch & Greeno, 1985 ) and, in turn, lead them to select mathematical models that have a corresponding structure. To illustrate, a word problem that describes constant change in the rate at which ice is melting off a glacier evokes a model of continuous change, whereas a word problem that describes constant change in the rate at which ice is delivered to a restaurant evokes a model of discrete change. These distinct situation models lead subjects to select corresponding visual representations (e.g., Bassok & Olseth, 1995 ) and solutions methods, such as calculating the average change over time versus adding the consecutive changes (e.g., Alibali et al., 1999 ).

In a similar manner, people draw on their general knowledge to infer how the objects in a given problem are related to each other and construct mathematical solutions that correspond to these inferred object relations. For example, a word problem that involves doctors from two hospitals elicits a situation model in which the two sets of doctors play symmetric roles (e.g., work with each other), whereas a mathematically isomorphic problem that involves mechanics and cars elicits a situation model in which the sets play asymmetric roles (e.g., mechanics fix cars). The mathematical solutions people construct to such problems reflect this difference in symmetry (Bassok, Wu, & Olseth, 1995 ). In general, people tend to add objects that belong to the same taxonomic category (e.g., doctors + doctors) but divide functionally related objects (e.g., cars ÷ mechanics). People establish this correspondence by a process of analogical alignment between semantic and arithmetic relations, which Bassok and her colleagues refer to as “semantic alignment” (Bassok, Chase, & Martin, 1998 ; Doumas, Bassok, Guthormsen, & Hummel, 2006 ; Fisher, Bassok, & Osterhout, 2010 ).

Semantic alignment occurs very early in the solution process and can prime arithmetic facts that are potentially relevant to the problem solution (Bassok, Pedigo, & Oskarsson, 2008 ). Although such alignments can lead to erroneous solutions, they have a high heuristic value because, in most textbook problems, object relations indeed correspond to analogous mathematical relations (Bassok et al., 1998 ). Interestingly, unlike in the case of reliance on specific surface-structure correlations (e.g., the keyword “more” typically appears in word problems that require addition; Lewis & Mayer, 1987 ), people are more likely to exploit semantic alignment when they have more, rather than less modeling experience. For example, Martin and Bassok ( 2005 ) found very strong semantic-alignment effects when subjects solved simple division word problems, but not when they constructed algebraic equations to represent the relational statements that appeared in the problems. Of course, these subjects had significantly more experience with solving numerical word problems than with constructing algebraic models of relational statements. In a subsequent study, Fisher and Bassok ( 2009 ) found semantic-alignment effects for subjects who constructed correct algebraic models, but not for those who committed modeling errors.

Conclusions and Future Directions

In this chapter, we examined two broad components of the problem-solving process: representation (the Gestalt legacy) and search (the legacy of Newell and Simon). Although many researchers choose to focus their investigation on one or the other of these components, both Duncker ( 1945 ) and Simon ( 1986 ) underscored the necessity to investigate their interaction, as the representation one constructs for a problem determines (or at least constrains) how one goes about trying to generate a solution, and searching the problem space may lead to a change in problem representation. Indeed, Duncker's ( 1945 ) initial account of one subject's solution to the radiation problem was followed up by extensive and experimentally sophisticated work by Simon and his colleagues and by other researchers, documenting the involvement of visual perception and background knowledge in how people represent problems and search for problem solutions.

The relevance of perception and background knowledge to problem solving illustrates the fact that, when people attempt to find or devise ways to reach their goals, they draw on a variety of cognitive resources and engage in a host of cognitive activities. According to Duncker ( 1945 ), such goal-directed activities may include (a) placing objects into categories and making inferences based on category membership, (b) making inductive inferences from multiple instances, (c) reasoning by analogy, (d) identifying the causes of events, (e) deducing logical implications of given information, (f) making legal judgments, and (g) diagnosing medical conditions from historical and laboratory data. As this list suggests, many of the chapters in the present volume describe research that is highly relevant to the understanding of problem-solving behavior. We believe that important advancements in problem-solving research would emerge by integrating it with research in other areas of thinking and reasoning, and that research in these other areas could be similarly advanced by incorporating the insights gained from research on what has more traditionally been identified as problem solving.

As we have described in this chapter, many of the important findings in the field have been established by a careful investigation of various riddle problems. Although there are good methodological reasons for using such problems, many researchers choose to investigate problem solving using ecologically valid educational materials. This choice, which is increasingly common in contemporary research, provides researchers with the opportunity to apply their basic understanding of problem solving to benefit the design of instruction and, at the same time, allows them to gain a better understanding of the processes by which domain knowledge and educational conventions affect the solution process. We believe that the trend of conducting educationally relevant research is likely to continue, and we expect a significant expansion of research on people's understanding and use of dynamic and technologically rich external representations (e.g., Kellman et al., 2008 ; Mayer, Griffith, Jurkowitz, & Rothman, 2008 ; Richland & McDonough, 2010 ; Son & Goldstone, 2009 ). Such investigations are likely to yield both practical and theoretical payoffs.

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Problem-solving concepts and theories

Affiliation.

• 1 Mississippi StateUniversity, College of Veterinary Medicine, USA. [email protected]
• PMID: 14648495
• DOI: 10.3138/jvme.30.3.226

Many educators, especially those involved in professional curricula, are interested in problem solving and in how to support students' development into successful problem solvers. The following article serves as an overview of educational research on problem solving. Several concepts are defined and the transition from one theory to another is discussed. Educational theories describing problem solving in the context of behavioral, cognitive, and information-processing pedagogy are discussed. The final section of the article describes prior findings regarding expert-novice differences in problem solving of various kinds.

Publication types

• Education, Veterinary*
• Models, Educational*
• Problem Solving*

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THEORIES AND PRINCIPLES OF PROBLEM SOLVING IN MATHEMATICS

2023, Article

Doing mathematics means that students are engaged in learning mathematics through reasoning and problem solving (NCTM, 2014). Prospective mathematics teachers need to learn about how to engage students in solving and talking about tasks that can be tackled in different ways by different students. Mathematically, proficient students are able to make sense of a situation, select solution paths, consider alternative strategies and monitor their progress (CCSSO, 2010). Before we can be effective in teaching mathematics, we need to have a good knowledge about what we are supposed to be teaching and how students learn mathematics. We are familiar with why we teach mathematics at the basic and high schools.

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Problem-Solving Theory: The Task-Centred Model

• Living reference work entry
• First Online: 12 April 2022
• Cite this living reference work entry

• Blanca M. Ramos 5 &
• Randall L. Stetson 6  

Part of the book series: Social Work ((SOWO))

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This chapter examines the task-centred model to illustrate the application of problem-solving theory for social work intervention. First, it provides a brief description of the problem-solving model. Its historical development and key principles and concepts are presented. Next, the chapter offers a general overview of the crisis intervention model. The task-centred model and crisis intervention share principles and methods drawn from problem-solving theory. The remainder of the chapter focuses on the task-centred model. It reviews its historical background, viability as a framework for social work generalist practice, as well as its applicability with diverse client populations and across cultural settings. The structured steps that guide task-centred implementation throughout the helping process are described. A brief critical review of the model’s strengths and limitations is provided. The chapter concludes with a brief summary and some closing thoughts.

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Ramos, B.M., Stetson, R.L. (2022). Problem-Solving Theory: The Task-Centred Model. In: Hölscher, D., Hugman, R., McAuliffe, D. (eds) Social Work Theory and Ethics. Social Work. Springer, Singapore. https://doi.org/10.1007/978-981-16-3059-0_9-1

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7 Module 7: Thinking, Reasoning, and Problem-Solving

This module is about how a solid working knowledge of psychological principles can help you to think more effectively, so you can succeed in school and life. You might be inclined to believe that—because you have been thinking for as long as you can remember, because you are able to figure out the solution to many problems, because you feel capable of using logic to argue a point, because you can evaluate whether the things you read and hear make sense—you do not need any special training in thinking. But this, of course, is one of the key barriers to helping people think better. If you do not believe that there is anything wrong, why try to fix it?

The human brain is indeed a remarkable thinking machine, capable of amazing, complex, creative, logical thoughts. Why, then, are we telling you that you need to learn how to think? Mainly because one major lesson from cognitive psychology is that these capabilities of the human brain are relatively infrequently realized. Many psychologists believe that people are essentially “cognitive misers.” It is not that we are lazy, but that we have a tendency to expend the least amount of mental effort necessary. Although you may not realize it, it actually takes a great deal of energy to think. Careful, deliberative reasoning and critical thinking are very difficult. Because we seem to be successful without going to the trouble of using these skills well, it feels unnecessary to develop them. As you shall see, however, there are many pitfalls in the cognitive processes described in this module. When people do not devote extra effort to learning and improving reasoning, problem solving, and critical thinking skills, they make many errors.

As is true for memory, if you develop the cognitive skills presented in this module, you will be more successful in school. It is important that you realize, however, that these skills will help you far beyond school, even more so than a good memory will. Although it is somewhat useful to have a good memory, ten years from now no potential employer will care how many questions you got right on multiple choice exams during college. All of them will, however, recognize whether you are a logical, analytical, critical thinker. With these thinking skills, you will be an effective, persuasive communicator and an excellent problem solver.

The module begins by describing different kinds of thought and knowledge, especially conceptual knowledge and critical thinking. An understanding of these differences will be valuable as you progress through school and encounter different assignments that require you to tap into different kinds of knowledge. The second section covers deductive and inductive reasoning, which are processes we use to construct and evaluate strong arguments. They are essential skills to have whenever you are trying to persuade someone (including yourself) of some point, or to respond to someone’s efforts to persuade you. The module ends with a section about problem solving. A solid understanding of the key processes involved in problem solving will help you to handle many daily challenges.

7.1. Different kinds of thought

7.2. Reasoning and Judgment

7.3. Problem Solving

READING WITH PURPOSE

Remember and understand.

By reading and studying Module 7, you should be able to remember and describe:

• Concepts and inferences (7.1)
• Procedural knowledge (7.1)
• Metacognition (7.1)
• Characteristics of critical thinking:  skepticism; identify biases, distortions, omissions, and assumptions; reasoning and problem solving skills  (7.1)
• Reasoning:  deductive reasoning, deductively valid argument, inductive reasoning, inductively strong argument, availability heuristic, representativeness heuristic  (7.2)
• Fixation:  functional fixedness, mental set  (7.3)
• Algorithms, heuristics, and the role of confirmation bias (7.3)
• Effective problem solving sequence (7.3)

By reading and thinking about how the concepts in Module 6 apply to real life, you should be able to:

• Identify which type of knowledge a piece of information is (7.1)
• Recognize examples of deductive and inductive reasoning (7.2)
• Recognize judgments that have probably been influenced by the availability heuristic (7.2)
• Recognize examples of problem solving heuristics and algorithms (7.3)

Analyze, Evaluate, and Create

By reading and thinking about Module 6, participating in classroom activities, and completing out-of-class assignments, you should be able to:

• Use the principles of critical thinking to evaluate information (7.1)
• Explain whether examples of reasoning arguments are deductively valid or inductively strong (7.2)
• Outline how you could try to solve a problem from your life using the effective problem solving sequence (7.3)

7.1. Different kinds of thought and knowledge

• Take a few minutes to write down everything that you know about dogs.
• Do you believe that:
• Psychic ability exists?
• Hypnosis is an altered state of consciousness?
• Magnet therapy is effective for relieving pain?
• Aerobic exercise is an effective treatment for depression?
• UFO’s from outer space have visited earth?

On what do you base your belief or disbelief for the questions above?

Of course, we all know what is meant by the words  think  and  knowledge . You probably also realize that they are not unitary concepts; there are different kinds of thought and knowledge. In this section, let us look at some of these differences. If you are familiar with these different kinds of thought and pay attention to them in your classes, it will help you to focus on the right goals, learn more effectively, and succeed in school. Different assignments and requirements in school call on you to use different kinds of knowledge or thought, so it will be very helpful for you to learn to recognize them (Anderson, et al. 2001).

Factual and conceptual knowledge

Module 5 introduced the idea of declarative memory, which is composed of facts and episodes. If you have ever played a trivia game or watched Jeopardy on TV, you realize that the human brain is able to hold an extraordinary number of facts. Likewise, you realize that each of us has an enormous store of episodes, essentially facts about events that happened in our own lives. It may be difficult to keep that in mind when we are struggling to retrieve one of those facts while taking an exam, however. Part of the problem is that, in contradiction to the advice from Module 5, many students continue to try to memorize course material as a series of unrelated facts (picture a history student simply trying to memorize history as a set of unrelated dates without any coherent story tying them together). Facts in the real world are not random and unorganized, however. It is the way that they are organized that constitutes a second key kind of knowledge, conceptual.

Concepts are nothing more than our mental representations of categories of things in the world. For example, think about dogs. When you do this, you might remember specific facts about dogs, such as they have fur and they bark. You may also recall dogs that you have encountered and picture them in your mind. All of this information (and more) makes up your concept of dog. You can have concepts of simple categories (e.g., triangle), complex categories (e.g., small dogs that sleep all day, eat out of the garbage, and bark at leaves), kinds of people (e.g., psychology professors), events (e.g., birthday parties), and abstract ideas (e.g., justice). Gregory Murphy (2002) refers to concepts as the “glue that holds our mental life together” (p. 1). Very simply, summarizing the world by using concepts is one of the most important cognitive tasks that we do. Our conceptual knowledge  is  our knowledge about the world. Individual concepts are related to each other to form a rich interconnected network of knowledge. For example, think about how the following concepts might be related to each other: dog, pet, play, Frisbee, chew toy, shoe. Or, of more obvious use to you now, how these concepts are related: working memory, long-term memory, declarative memory, procedural memory, and rehearsal? Because our minds have a natural tendency to organize information conceptually, when students try to remember course material as isolated facts, they are working against their strengths.

One last important point about concepts is that they allow you to instantly know a great deal of information about something. For example, if someone hands you a small red object and says, “here is an apple,” they do not have to tell you, “it is something you can eat.” You already know that you can eat it because it is true by virtue of the fact that the object is an apple; this is called drawing an  inference , assuming that something is true on the basis of your previous knowledge (for example, of category membership or of how the world works) or logical reasoning.

Procedural knowledge

Physical skills, such as tying your shoes, doing a cartwheel, and driving a car (or doing all three at the same time, but don’t try this at home) are certainly a kind of knowledge. They are procedural knowledge, the same idea as procedural memory that you saw in Module 5. Mental skills, such as reading, debating, and planning a psychology experiment, are procedural knowledge, as well. In short, procedural knowledge is the knowledge how to do something (Cohen & Eichenbaum, 1993).

Metacognitive knowledge

Floyd used to think that he had a great memory. Now, he has a better memory. Why? Because he finally realized that his memory was not as great as he once thought it was. Because Floyd eventually learned that he often forgets where he put things, he finally developed the habit of putting things in the same place. (Unfortunately, he did not learn this lesson before losing at least 5 watches and a wedding ring.) Because he finally realized that he often forgets to do things, he finally started using the To Do list app on his phone. And so on. Floyd’s insights about the real limitations of his memory have allowed him to remember things that he used to forget.

All of us have knowledge about the way our own minds work. You may know that you have a good memory for people’s names and a poor memory for math formulas. Someone else might realize that they have difficulty remembering to do things, like stopping at the store on the way home. Others still know that they tend to overlook details. This knowledge about our own thinking is actually quite important; it is called metacognitive knowledge, or  metacognition . Like other kinds of thinking skills, it is subject to error. For example, in unpublished research, one of the authors surveyed about 120 General Psychology students on the first day of the term. Among other questions, the students were asked them to predict their grade in the class and report their current Grade Point Average. Two-thirds of the students predicted that their grade in the course would be higher than their GPA. (The reality is that at our college, students tend to earn lower grades in psychology than their overall GPA.) Another example: Students routinely report that they thought they had done well on an exam, only to discover, to their dismay, that they were wrong (more on that important problem in a moment). Both errors reveal a breakdown in metacognition.

The Dunning-Kruger Effect

In general, most college students probably do not study enough. For example, using data from the National Survey of Student Engagement, Fosnacht, McCormack, and Lerma (2018) reported that first-year students at 4-year colleges in the U.S. averaged less than 14 hours per week preparing for classes. The typical suggestion is that you should spend two hours outside of class for every hour in class, or 24 – 30 hours per week for a full-time student. Clearly, students in general are nowhere near that recommended mark. Many observers, including some faculty, believe that this shortfall is a result of students being too busy or lazy. Now, it may be true that many students are too busy, with work and family obligations, for example. Others, are not particularly motivated in school, and therefore might correctly be labeled lazy. A third possible explanation, however, is that some students might not think they need to spend this much time. And this is a matter of metacognition. Consider the scenario that we mentioned above, students thinking they had done well on an exam only to discover that they did not. Justin Kruger and David Dunning examined scenarios very much like this in 1999. Kruger and Dunning gave research participants tests measuring humor, logic, and grammar. Then, they asked the participants to assess their own abilities and test performance in these areas. They found that participants in general tended to overestimate their abilities, already a problem with metacognition. Importantly, the participants who scored the lowest overestimated their abilities the most. Specifically, students who scored in the bottom quarter (averaging in the 12th percentile) thought they had scored in the 62nd percentile. This has become known as the  Dunning-Kruger effect . Many individual faculty members have replicated these results with their own student on their course exams, including the authors of this book. Think about it. Some students who just took an exam and performed poorly believe that they did well before seeing their score. It seems very likely that these are the very same students who stopped studying the night before because they thought they were “done.” Quite simply, it is not just that they did not know the material. They did not know that they did not know the material. That is poor metacognition.

In order to develop good metacognitive skills, you should continually monitor your thinking and seek frequent feedback on the accuracy of your thinking (Medina, Castleberry, & Persky 2017). For example, in classes get in the habit of predicting your exam grades. As soon as possible after taking an exam, try to find out which questions you missed and try to figure out why. If you do this soon enough, you may be able to recall the way it felt when you originally answered the question. Did you feel confident that you had answered the question correctly? Then you have just discovered an opportunity to improve your metacognition. Be on the lookout for that feeling and respond with caution.

concept :  a mental representation of a category of things in the world

Dunning-Kruger effect : individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

inference : an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

metacognition :  knowledge about one’s own cognitive processes; thinking about your thinking

Critical thinking

One particular kind of knowledge or thinking skill that is related to metacognition is  critical thinking (Chew, 2020). You may have noticed that critical thinking is an objective in many college courses, and thus it could be a legitimate topic to cover in nearly any college course. It is particularly appropriate in psychology, however. As the science of (behavior and) mental processes, psychology is obviously well suited to be the discipline through which you should be introduced to this important way of thinking.

More importantly, there is a particular need to use critical thinking in psychology. We are all, in a way, experts in human behavior and mental processes, having engaged in them literally since birth. Thus, perhaps more than in any other class, students typically approach psychology with very clear ideas and opinions about its subject matter. That is, students already “know” a lot about psychology. The problem is, “it ain’t so much the things we don’t know that get us into trouble. It’s the things we know that just ain’t so” (Ward, quoted in Gilovich 1991). Indeed, many of students’ preconceptions about psychology are just plain wrong. Randolph Smith (2002) wrote a book about critical thinking in psychology called  Challenging Your Preconceptions,  highlighting this fact. On the other hand, many of students’ preconceptions about psychology are just plain right! But wait, how do you know which of your preconceptions are right and which are wrong? And when you come across a research finding or theory in this class that contradicts your preconceptions, what will you do? Will you stick to your original idea, discounting the information from the class? Will you immediately change your mind? Critical thinking can help us sort through this confusing mess.

But what is critical thinking? The goal of critical thinking is simple to state (but extraordinarily difficult to achieve): it is to be right, to draw the correct conclusions, to believe in things that are true and to disbelieve things that are false. We will provide two definitions of critical thinking (or, if you like, one large definition with two distinct parts). First, a more conceptual one: Critical thinking is thinking like a scientist in your everyday life (Schmaltz, Jansen, & Wenckowski, 2017).  Our second definition is more operational; it is simply a list of skills that are essential to be a critical thinker. Critical thinking entails solid reasoning and problem solving skills; skepticism; and an ability to identify biases, distortions, omissions, and assumptions. Excellent deductive and inductive reasoning, and problem solving skills contribute to critical thinking. So, you can consider the subject matter of sections 7.2 and 7.3 to be part of critical thinking. Because we will be devoting considerable time to these concepts in the rest of the module, let us begin with a discussion about the other aspects of critical thinking.

Let’s address that first part of the definition. Scientists form hypotheses, or predictions about some possible future observations. Then, they collect data, or information (think of this as making those future observations). They do their best to make unbiased observations using reliable techniques that have been verified by others. Then, and only then, they draw a conclusion about what those observations mean. Oh, and do not forget the most important part. “Conclusion” is probably not the most appropriate word because this conclusion is only tentative. A scientist is always prepared that someone else might come along and produce new observations that would require a new conclusion be drawn. Wow! If you like to be right, you could do a lot worse than using a process like this.

A Critical Thinker’s Toolkit 

Now for the second part of the definition. Good critical thinkers (and scientists) rely on a variety of tools to evaluate information. Perhaps the most recognizable tool for critical thinking is  skepticism (and this term provides the clearest link to the thinking like a scientist definition, as you are about to see). Some people intend it as an insult when they call someone a skeptic. But if someone calls you a skeptic, if they are using the term correctly, you should consider it a great compliment. Simply put, skepticism is a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided. People from Missouri should recognize this principle, as Missouri is known as the Show-Me State. As a skeptic, you are not inclined to believe something just because someone said so, because someone else believes it, or because it sounds reasonable. You must be persuaded by high quality evidence.

Of course, if that evidence is produced, you have a responsibility as a skeptic to change your belief. Failure to change a belief in the face of good evidence is not skepticism; skepticism has open mindedness at its core. M. Neil Browne and Stuart Keeley (2018) use the term weak sense critical thinking to describe critical thinking behaviors that are used only to strengthen a prior belief. Strong sense critical thinking, on the other hand, has as its goal reaching the best conclusion. Sometimes that means strengthening your prior belief, but sometimes it means changing your belief to accommodate the better evidence.

Many times, a failure to think critically or weak sense critical thinking is related to a  bias , an inclination, tendency, leaning, or prejudice. Everybody has biases, but many people are unaware of them. Awareness of your own biases gives you the opportunity to control or counteract them. Unfortunately, however, many people are happy to let their biases creep into their attempts to persuade others; indeed, it is a key part of their persuasive strategy. To see how these biases influence messages, just look at the different descriptions and explanations of the same events given by people of different ages or income brackets, or conservative versus liberal commentators, or by commentators from different parts of the world. Of course, to be successful, these people who are consciously using their biases must disguise them. Even undisguised biases can be difficult to identify, so disguised ones can be nearly impossible.

Here are some common sources of biases:

• Personal values and beliefs.  Some people believe that human beings are basically driven to seek power and that they are typically in competition with one another over scarce resources. These beliefs are similar to the world-view that political scientists call “realism.” Other people believe that human beings prefer to cooperate and that, given the chance, they will do so. These beliefs are similar to the world-view known as “idealism.” For many people, these deeply held beliefs can influence, or bias, their interpretations of such wide ranging situations as the behavior of nations and their leaders or the behavior of the driver in the car ahead of you. For example, if your worldview is that people are typically in competition and someone cuts you off on the highway, you may assume that the driver did it purposely to get ahead of you. Other types of beliefs about the way the world is or the way the world should be, for example, political beliefs, can similarly become a significant source of bias.
• Racism, sexism, ageism and other forms of prejudice and bigotry.  These are, sadly, a common source of bias in many people. They are essentially a special kind of “belief about the way the world is.” These beliefs—for example, that women do not make effective leaders—lead people to ignore contradictory evidence (examples of effective women leaders, or research that disputes the belief) and to interpret ambiguous evidence in a way consistent with the belief.
• Self-interest.  When particular people benefit from things turning out a certain way, they can sometimes be very susceptible to letting that interest bias them. For example, a company that will earn a profit if they sell their product may have a bias in the way that they give information about their product. A union that will benefit if its members get a generous contract might have a bias in the way it presents information about salaries at competing organizations. (Note that our inclusion of examples describing both companies and unions is an explicit attempt to control for our own personal biases). Home buyers are often dismayed to discover that they purchased their dream house from someone whose self-interest led them to lie about flooding problems in the basement or back yard. This principle, the biasing power of self-interest, is likely what led to the famous phrase  Caveat Emptor  (let the buyer beware) .  

Knowing that these types of biases exist will help you evaluate evidence more critically. Do not forget, though, that people are not always keen to let you discover the sources of biases in their arguments. For example, companies or political organizations can sometimes disguise their support of a research study by contracting with a university professor, who comes complete with a seemingly unbiased institutional affiliation, to conduct the study.

People’s biases, conscious or unconscious, can lead them to make omissions, distortions, and assumptions that undermine our ability to correctly evaluate evidence. It is essential that you look for these elements. Always ask, what is missing, what is not as it appears, and what is being assumed here? For example, consider this (fictional) chart from an ad reporting customer satisfaction at 4 local health clubs.

Clearly, from the results of the chart, one would be tempted to give Club C a try, as customer satisfaction is much higher than for the other 3 clubs.

There are so many distortions and omissions in this chart, however, that it is actually quite meaningless. First, how was satisfaction measured? Do the bars represent responses to a survey? If so, how were the questions asked? Most importantly, where is the missing scale for the chart? Although the differences look quite large, are they really?

Well, here is the same chart, with a different scale, this time labeled:

Club C is not so impressive any more, is it? In fact, all of the health clubs have customer satisfaction ratings (whatever that means) between 85% and 88%. In the first chart, the entire scale of the graph included only the percentages between 83 and 89. This “judicious” choice of scale—some would call it a distortion—and omission of that scale from the chart make the tiny differences among the clubs seem important, however.

Also, in order to be a critical thinker, you need to learn to pay attention to the assumptions that underlie a message. Let us briefly illustrate the role of assumptions by touching on some people’s beliefs about the criminal justice system in the US. Some believe that a major problem with our judicial system is that many criminals go free because of legal technicalities. Others believe that a major problem is that many innocent people are convicted of crimes. The simple fact is, both types of errors occur. A person’s conclusion about which flaw in our judicial system is the greater tragedy is based on an assumption about which of these is the more serious error (letting the guilty go free or convicting the innocent). This type of assumption is called a value assumption (Browne and Keeley, 2018). It reflects the differences in values that people develop, differences that may lead us to disregard valid evidence that does not fit in with our particular values.

Oh, by the way, some students probably noticed this, but the seven tips for evaluating information that we shared in Module 1 are related to this. Actually, they are part of this section. The tips are, to a very large degree, set of ideas you can use to help you identify biases, distortions, omissions, and assumptions. If you do not remember this section, we strongly recommend you take a few minutes to review it.

skepticism :  a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

bias : an inclination, tendency, leaning, or prejudice

• Which of your beliefs (or disbeliefs) from the Activate exercise for this section were derived from a process of critical thinking? If some of your beliefs were not based on critical thinking, are you willing to reassess these beliefs? If the answer is no, why do you think that is? If the answer is yes, what concrete steps will you take?

7.2 Reasoning and Judgment

• What percentage of kidnappings are committed by strangers?
• Which area of the house is riskiest: kitchen, bathroom, or stairs?
• What is the most common cancer in the US?
• What percentage of workplace homicides are committed by co-workers?

An essential set of procedural thinking skills is  reasoning , the ability to generate and evaluate solid conclusions from a set of statements or evidence. You should note that these conclusions (when they are generated instead of being evaluated) are one key type of inference that we described in Section 7.1. There are two main types of reasoning, deductive and inductive.

Deductive reasoning

Suppose your teacher tells you that if you get an A on the final exam in a course, you will get an A for the whole course. Then, you get an A on the final exam. What will your final course grade be? Most people can see instantly that you can conclude with certainty that you will get an A for the course. This is a type of reasoning called  deductive reasoning , which is defined as reasoning in which a conclusion is guaranteed to be true as long as the statements leading to it are true. The three statements can be listed as an  argument , with two beginning statements and a conclusion:

Statement 1: If you get an A on the final exam, you will get an A for the course

Statement 2: You get an A on the final exam

Conclusion: You will get an A for the course

This particular arrangement, in which true beginning statements lead to a guaranteed true conclusion, is known as a  deductively valid argument . Although deductive reasoning is often the subject of abstract, brain-teasing, puzzle-like word problems, it is actually an extremely important type of everyday reasoning. It is just hard to recognize sometimes. For example, imagine that you are looking for your car keys and you realize that they are either in the kitchen drawer or in your book bag. After looking in the kitchen drawer, you instantly know that they must be in your book bag. That conclusion results from a simple deductive reasoning argument. In addition, solid deductive reasoning skills are necessary for you to succeed in the sciences, philosophy, math, computer programming, and any endeavor involving the use of logic to persuade others to your point of view or to evaluate others’ arguments.

Cognitive psychologists, and before them philosophers, have been quite interested in deductive reasoning, not so much for its practical applications, but for the insights it can offer them about the ways that human beings think. One of the early ideas to emerge from the examination of deductive reasoning is that people learn (or develop) mental versions of rules that allow them to solve these types of reasoning problems (Braine, 1978; Braine, Reiser, & Rumain, 1984). The best way to see this point of view is to realize that there are different possible rules, and some of them are very simple. For example, consider this rule of logic:

therefore q

Logical rules are often presented abstractly, as letters, in order to imply that they can be used in very many specific situations. Here is a concrete version of the of the same rule:

I’ll either have pizza or a hamburger for dinner tonight (p or q)

I won’t have pizza (not p)

Therefore, I’ll have a hamburger (therefore q)

This kind of reasoning seems so natural, so easy, that it is quite plausible that we would use a version of this rule in our daily lives. At least, it seems more plausible than some of the alternative possibilities—for example, that we need to have experience with the specific situation (pizza or hamburger, in this case) in order to solve this type of problem easily. So perhaps there is a form of natural logic (Rips, 1990) that contains very simple versions of logical rules. When we are faced with a reasoning problem that maps onto one of these rules, we use the rule.

But be very careful; things are not always as easy as they seem. Even these simple rules are not so simple. For example, consider the following rule. Many people fail to realize that this rule is just as valid as the pizza or hamburger rule above.

if p, then q

therefore, not p

Concrete version:

If I eat dinner, then I will have dessert

I did not have dessert

Therefore, I did not eat dinner

The simple fact is, it can be very difficult for people to apply rules of deductive logic correctly; as a result, they make many errors when trying to do so. Is this a deductively valid argument or not?

Students who like school study a lot

Students who study a lot get good grades

Jane does not like school

Therefore, Jane does not get good grades

Many people are surprised to discover that this is not a logically valid argument; the conclusion is not guaranteed to be true from the beginning statements. Although the first statement says that students who like school study a lot, it does NOT say that students who do not like school do not study a lot. In other words, it may very well be possible to study a lot without liking school. Even people who sometimes get problems like this right might not be using the rules of deductive reasoning. Instead, they might just be making judgments for examples they know, in this case, remembering instances of people who get good grades despite not liking school.

Making deductive reasoning even more difficult is the fact that there are two important properties that an argument may have. One, it can be valid or invalid (meaning that the conclusion does or does not follow logically from the statements leading up to it). Two, an argument (or more correctly, its conclusion) can be true or false. Here is an example of an argument that is logically valid, but has a false conclusion (at least we think it is false).

Either you are eleven feet tall or the Grand Canyon was created by a spaceship crashing into the earth.

You are not eleven feet tall

Therefore the Grand Canyon was created by a spaceship crashing into the earth

This argument has the exact same form as the pizza or hamburger argument above, making it is deductively valid. The conclusion is so false, however, that it is absurd (of course, the reason the conclusion is false is that the first statement is false). When people are judging arguments, they tend to not observe the difference between deductive validity and the empirical truth of statements or conclusions. If the elements of an argument happen to be true, people are likely to judge the argument logically valid; if the elements are false, they will very likely judge it invalid (Markovits & Bouffard-Bouchard, 1992; Moshman & Franks, 1986). Thus, it seems a stretch to say that people are using these logical rules to judge the validity of arguments. Many psychologists believe that most people actually have very limited deductive reasoning skills (Johnson-Laird, 1999). They argue that when faced with a problem for which deductive logic is required, people resort to some simpler technique, such as matching terms that appear in the statements and the conclusion (Evans, 1982). This might not seem like a problem, but what if reasoners believe that the elements are true and they happen to be wrong; they will would believe that they are using a form of reasoning that guarantees they are correct and yet be wrong.

deductive reasoning :  a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

argument :  a set of statements in which the beginning statements lead to a conclusion

deductively valid argument :  an argument for which true beginning statements guarantee that the conclusion is true

Inductive reasoning and judgment

Every day, you make many judgments about the likelihood of one thing or another. Whether you realize it or not, you are practicing  inductive reasoning   on a daily basis. In inductive reasoning arguments, a conclusion is likely whenever the statements preceding it are true. The first thing to notice about inductive reasoning is that, by definition, you can never be sure about your conclusion; you can only estimate how likely the conclusion is. Inductive reasoning may lead you to focus on Memory Encoding and Recoding when you study for the exam, but it is possible the instructor will ask more questions about Memory Retrieval instead. Unlike deductive reasoning, the conclusions you reach through inductive reasoning are only probable, not certain. That is why scientists consider inductive reasoning weaker than deductive reasoning. But imagine how hard it would be for us to function if we could not act unless we were certain about the outcome.

Inductive reasoning can be represented as logical arguments consisting of statements and a conclusion, just as deductive reasoning can be. In an inductive argument, you are given some statements and a conclusion (or you are given some statements and must draw a conclusion). An argument is  inductively strong   if the conclusion would be very probable whenever the statements are true. So, for example, here is an inductively strong argument:

• Statement #1: The forecaster on Channel 2 said it is going to rain today.
• Statement #2: The forecaster on Channel 5 said it is going to rain today.
• Statement #3: It is very cloudy and humid.
• Statement #4: You just heard thunder.
• Conclusion (or judgment): It is going to rain today.

Think of the statements as evidence, on the basis of which you will draw a conclusion. So, based on the evidence presented in the four statements, it is very likely that it will rain today. Will it definitely rain today? Certainly not. We can all think of times that the weather forecaster was wrong.

A true story: Some years ago psychology student was watching a baseball playoff game between the St. Louis Cardinals and the Los Angeles Dodgers. A graphic on the screen had just informed the audience that the Cardinal at bat, (Hall of Fame shortstop) Ozzie Smith, a switch hitter batting left-handed for this plate appearance, had never, in nearly 3000 career at-bats, hit a home run left-handed. The student, who had just learned about inductive reasoning in his psychology class, turned to his companion (a Cardinals fan) and smugly said, “It is an inductively strong argument that Ozzie Smith will not hit a home run.” He turned back to face the television just in time to watch the ball sail over the right field fence for a home run. Although the student felt foolish at the time, he was not wrong. It was an inductively strong argument; 3000 at-bats is an awful lot of evidence suggesting that the Wizard of Ozz (as he was known) would not be hitting one out of the park (think of each at-bat without a home run as a statement in an inductive argument). Sadly (for the die-hard Cubs fan and Cardinals-hating student), despite the strength of the argument, the conclusion was wrong.

Given the possibility that we might draw an incorrect conclusion even with an inductively strong argument, we really want to be sure that we do, in fact, make inductively strong arguments. If we judge something probable, it had better be probable. If we judge something nearly impossible, it had better not happen. Think of inductive reasoning, then, as making reasonably accurate judgments of the probability of some conclusion given a set of evidence.

We base many decisions in our lives on inductive reasoning. For example:

Statement #1: Psychology is not my best subject

Statement #2: My psychology instructor has a reputation for giving difficult exams

Statement #3: My first psychology exam was much harder than I expected

Judgment: The next exam will probably be very difficult.

Decision: I will study tonight instead of watching Netflix.

Some other examples of judgments that people commonly make in a school context include judgments of the likelihood that:

• A particular class will be interesting/useful/difficult
• You will be able to finish writing a paper by next week if you go out tonight
• Your laptop’s battery will last through the next trip to the library
• You will not miss anything important if you skip class tomorrow
• Your instructor will not notice if you skip class tomorrow
• You will be able to find a book that you will need for a paper
• There will be an essay question about Memory Encoding on the next exam

Tversky and Kahneman (1983) recognized that there are two general ways that we might make these judgments; they termed them extensional (i.e., following the laws of probability) and intuitive (i.e., using shortcuts or heuristics, see below). We will use a similar distinction between Type 1 and Type 2 thinking, as described by Keith Stanovich and his colleagues (Evans and Stanovich, 2013; Stanovich and West, 2000). Type 1 thinking is fast, automatic, effortful, and emotional. In fact, it is hardly fair to call it reasoning at all, as judgments just seem to pop into one’s head. Type 2 thinking , on the other hand, is slow, effortful, and logical. So obviously, it is more likely to lead to a correct judgment, or an optimal decision. The problem is, we tend to over-rely on Type 1. Now, we are not saying that Type 2 is the right way to go for every decision or judgment we make. It seems a bit much, for example, to engage in a step-by-step logical reasoning procedure to decide whether we will have chicken or fish for dinner tonight.

Many bad decisions in some very important contexts, however, can be traced back to poor judgments of the likelihood of certain risks or outcomes that result from the use of Type 1 when a more logical reasoning process would have been more appropriate. For example:

Statement #1: It is late at night.

Statement #2: Albert has been drinking beer for the past five hours at a party.

Statement #3: Albert is not exactly sure where he is or how far away home is.

Judgment: Albert will have no difficulty walking home.

Decision: He walks home alone.

As you can see in this example, the three statements backing up the judgment do not really support it. In other words, this argument is not inductively strong because it is based on judgments that ignore the laws of probability. What are the chances that someone facing these conditions will be able to walk home alone easily? And one need not be drunk to make poor decisions based on judgments that just pop into our heads.

The truth is that many of our probability judgments do not come very close to what the laws of probability say they should be. Think about it. In order for us to reason in accordance with these laws, we would need to know the laws of probability, which would allow us to calculate the relationship between particular pieces of evidence and the probability of some outcome (i.e., how much likelihood should change given a piece of evidence), and we would have to do these heavy math calculations in our heads. After all, that is what Type 2 requires. Needless to say, even if we were motivated, we often do not even know how to apply Type 2 reasoning in many cases.

So what do we do when we don’t have the knowledge, skills, or time required to make the correct mathematical judgment? Do we hold off and wait until we can get better evidence? Do we read up on probability and fire up our calculator app so we can compute the correct probability? Of course not. We rely on Type 1 thinking. We “wing it.” That is, we come up with a likelihood estimate using some means at our disposal. Psychologists use the term heuristic to describe the type of “winging it” we are talking about. A  heuristic   is a shortcut strategy that we use to make some judgment or solve some problem (see Section 7.3). Heuristics are easy and quick, think of them as the basic procedures that are characteristic of Type 1.  They can absolutely lead to reasonably good judgments and decisions in some situations (like choosing between chicken and fish for dinner). They are, however, far from foolproof. There are, in fact, quite a lot of situations in which heuristics can lead us to make incorrect judgments, and in many cases the decisions based on those judgments can have serious consequences.

Let us return to the activity that begins this section. You were asked to judge the likelihood (or frequency) of certain events and risks. You were free to come up with your own evidence (or statements) to make these judgments. This is where a heuristic crops up. As a judgment shortcut, we tend to generate specific examples of those very events to help us decide their likelihood or frequency. For example, if we are asked to judge how common, frequent, or likely a particular type of cancer is, many of our statements would be examples of specific cancer cases:

Statement #1: Andy Kaufman (comedian) had lung cancer.

Statement #2: Colin Powell (US Secretary of State) had prostate cancer.

Statement #3: Bob Marley (musician) had skin and brain cancer

Statement #4: Sandra Day O’Connor (Supreme Court Justice) had breast cancer.

Statement #5: Fred Rogers (children’s entertainer) had stomach cancer.

Statement #6: Robin Roberts (news anchor) had breast cancer.

Statement #7: Bette Davis (actress) had breast cancer.

Judgment: Breast cancer is the most common type.

Your own experience or memory may also tell you that breast cancer is the most common type. But it is not (although it is common). Actually, skin cancer is the most common type in the US. We make the same types of misjudgments all the time because we do not generate the examples or evidence according to their actual frequencies or probabilities. Instead, we have a tendency (or bias) to search for the examples in memory; if they are easy to retrieve, we assume that they are common. To rephrase this in the language of the heuristic, events seem more likely to the extent that they are available to memory. This bias has been termed the  availability heuristic   (Kahneman and Tversky, 1974).

The fact that we use the availability heuristic does not automatically mean that our judgment is wrong. The reason we use heuristics in the first place is that they work fairly well in many cases (and, of course that they are easy to use). So, the easiest examples to think of sometimes are the most common ones. Is it more likely that a member of the U.S. Senate is a man or a woman? Most people have a much easier time generating examples of male senators. And as it turns out, the U.S. Senate has many more men than women (74 to 26 in 2020). In this case, then, the availability heuristic would lead you to make the correct judgment; it is far more likely that a senator would be a man.

In many other cases, however, the availability heuristic will lead us astray. This is because events can be memorable for many reasons other than their frequency. Section 5.2, Encoding Meaning, suggested that one good way to encode the meaning of some information is to form a mental image of it. Thus, information that has been pictured mentally will be more available to memory. Indeed, an event that is vivid and easily pictured will trick many people into supposing that type of event is more common than it actually is. Repetition of information will also make it more memorable. So, if the same event is described to you in a magazine, on the evening news, on a podcast that you listen to, and in your Facebook feed; it will be very available to memory. Again, the availability heuristic will cause you to misperceive the frequency of these types of events.

Most interestingly, information that is unusual is more memorable. Suppose we give you the following list of words to remember: box, flower, letter, platypus, oven, boat, newspaper, purse, drum, car. Very likely, the easiest word to remember would be platypus, the unusual one. The same thing occurs with memories of events. An event may be available to memory because it is unusual, yet the availability heuristic leads us to judge that the event is common. Did you catch that? In these cases, the availability heuristic makes us think the exact opposite of the true frequency. We end up thinking something is common because it is unusual (and therefore memorable). Yikes.

The misapplication of the availability heuristic sometimes has unfortunate results. For example, if you went to K-12 school in the US over the past 10 years, it is extremely likely that you have participated in lockdown and active shooter drills. Of course, everyone is trying to prevent the tragedy of another school shooting. And believe us, we are not trying to minimize how terrible the tragedy is. But the truth of the matter is, school shootings are extremely rare. Because the federal government does not keep a database of school shootings, the Washington Post has maintained their own running tally. Between 1999 and January 2020 (the date of the most recent school shooting with a death in the US at of the time this paragraph was written), the Post reported a total of 254 people died in school shootings in the US. Not 254 per year, 254 total. That is an average of 12 per year. Of course, that is 254 people who should not have died (particularly because many were children), but in a country with approximately 60,000,000 students and teachers, this is a very small risk.

But many students and teachers are terrified that they will be victims of school shootings because of the availability heuristic. It is so easy to think of examples (they are very available to memory) that people believe the event is very common. It is not. And there is a downside to this. We happen to believe that there is an enormous gun violence problem in the United States. According the the Centers for Disease Control and Prevention, there were 39,773 firearm deaths in the US in 2017. Fifteen of those deaths were in school shootings, according to the Post. 60% of those deaths were suicides. When people pay attention to the school shooting risk (low), they often fail to notice the much larger risk.

And examples like this are by no means unique. The authors of this book have been teaching psychology since the 1990’s. We have been able to make the exact same arguments about the misapplication of the availability heuristics and keep them current by simply swapping out for the “fear of the day.” In the 1990’s it was children being kidnapped by strangers (it was known as “stranger danger”) despite the facts that kidnappings accounted for only 2% of the violent crimes committed against children, and only 24% of kidnappings are committed by strangers (US Department of Justice, 2007). This fear overlapped with the fear of terrorism that gripped the country after the 2001 terrorist attacks on the World Trade Center and US Pentagon and still plagues the population of the US somewhat in 2020. After a well-publicized, sensational act of violence, people are extremely likely to increase their estimates of the chances that they, too, will be victims of terror. Think about the reality, however. In October of 2001, a terrorist mailed anthrax spores to members of the US government and a number of media companies. A total of five people died as a result of this attack. The nation was nearly paralyzed by the fear of dying from the attack; in reality the probability of an individual person dying was 0.00000002.

The availability heuristic can lead you to make incorrect judgments in a school setting as well. For example, suppose you are trying to decide if you should take a class from a particular math professor. You might try to make a judgment of how good a teacher she is by recalling instances of friends and acquaintances making comments about her teaching skill. You may have some examples that suggest that she is a poor teacher very available to memory, so on the basis of the availability heuristic you judge her a poor teacher and decide to take the class from someone else. What if, however, the instances you recalled were all from the same person, and this person happens to be a very colorful storyteller? The subsequent ease of remembering the instances might not indicate that the professor is a poor teacher after all.

Although the availability heuristic is obviously important, it is not the only judgment heuristic we use. Amos Tversky and Daniel Kahneman examined the role of heuristics in inductive reasoning in a long series of studies. Kahneman received a Nobel Prize in Economics for this research in 2002, and Tversky would have certainly received one as well if he had not died of melanoma at age 59 in 1996 (Nobel Prizes are not awarded posthumously). Kahneman and Tversky demonstrated repeatedly that people do not reason in ways that are consistent with the laws of probability. They identified several heuristic strategies that people use instead to make judgments about likelihood. The importance of this work for economics (and the reason that Kahneman was awarded the Nobel Prize) is that earlier economic theories had assumed that people do make judgments rationally, that is, in agreement with the laws of probability.

Another common heuristic that people use for making judgments is the  representativeness heuristic (Kahneman & Tversky 1973). Suppose we describe a person to you. He is quiet and shy, has an unassuming personality, and likes to work with numbers. Is this person more likely to be an accountant or an attorney? If you said accountant, you were probably using the representativeness heuristic. Our imaginary person is judged likely to be an accountant because he resembles, or is representative of the concept of, an accountant. When research participants are asked to make judgments such as these, the only thing that seems to matter is the representativeness of the description. For example, if told that the person described is in a room that contains 70 attorneys and 30 accountants, participants will still assume that he is an accountant.

inductive reasoning :  a type of reasoning in which we make judgments about likelihood from sets of evidence

inductively strong argument :  an inductive argument in which the beginning statements lead to a conclusion that is probably true

heuristic :  a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

availability heuristic :  judging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

representativeness heuristic:   judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

Type 1 thinking : fast, automatic, and emotional thinking.

Type 2 thinking : slow, effortful, and logical thinking.

• What percentage of workplace homicides are co-worker violence?

Many people get these questions wrong. The answers are 10%; stairs; skin; 6%. How close were your answers? Explain how the availability heuristic might have led you to make the incorrect judgments.

• Can you think of some other judgments that you have made (or beliefs that you have) that might have been influenced by the availability heuristic?

7.3 Problem Solving

• Please take a few minutes to list a number of problems that you are facing right now.
• Now write about a problem that you recently solved.
• What is your definition of a problem?

Mary has a problem. Her daughter, ordinarily quite eager to please, appears to delight in being the last person to do anything. Whether getting ready for school, going to piano lessons or karate class, or even going out with her friends, she seems unwilling or unable to get ready on time. Other people have different kinds of problems. For example, many students work at jobs, have numerous family commitments, and are facing a course schedule full of difficult exams, assignments, papers, and speeches. How can they find enough time to devote to their studies and still fulfill their other obligations? Speaking of students and their problems: Show that a ball thrown vertically upward with initial velocity v0 takes twice as much time to return as to reach the highest point (from Spiegel, 1981).

These are three very different situations, but we have called them all problems. What makes them all the same, despite the differences? A psychologist might define a  problem   as a situation with an initial state, a goal state, and a set of possible intermediate states. Somewhat more meaningfully, we might consider a problem a situation in which you are in here one state (e.g., daughter is always late), you want to be there in another state (e.g., daughter is not always late), and with no obvious way to get from here to there. Defined this way, each of the three situations we outlined can now be seen as an example of the same general concept, a problem. At this point, you might begin to wonder what is not a problem, given such a general definition. It seems that nearly every non-routine task we engage in could qualify as a problem. As long as you realize that problems are not necessarily bad (it can be quite fun and satisfying to rise to the challenge and solve a problem), this may be a useful way to think about it.

Can we identify a set of problem-solving skills that would apply to these very different kinds of situations? That task, in a nutshell, is a major goal of this section. Let us try to begin to make sense of the wide variety of ways that problems can be solved with an important observation: the process of solving problems can be divided into two key parts. First, people have to notice, comprehend, and represent the problem properly in their minds (called  problem representation ). Second, they have to apply some kind of solution strategy to the problem. Psychologists have studied both of these key parts of the process in detail.

When you first think about the problem-solving process, you might guess that most of our difficulties would occur because we are failing in the second step, the application of strategies. Although this can be a significant difficulty much of the time, the more important source of difficulty is probably problem representation. In short, we often fail to solve a problem because we are looking at it, or thinking about it, the wrong way.

problem :  a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

problem representation :  noticing, comprehending and forming a mental conception of a problem

Defining and Mentally Representing Problems in Order to Solve Them

So, the main obstacle to solving a problem is that we do not clearly understand exactly what the problem is. Recall the problem with Mary’s daughter always being late. One way to represent, or to think about, this problem is that she is being defiant. She refuses to get ready in time. This type of representation or definition suggests a particular type of solution. Another way to think about the problem, however, is to consider the possibility that she is simply being sidetracked by interesting diversions. This different conception of what the problem is (i.e., different representation) suggests a very different solution strategy. For example, if Mary defines the problem as defiance, she may be tempted to solve the problem using some kind of coercive tactics, that is, to assert her authority as her mother and force her to listen. On the other hand, if Mary defines the problem as distraction, she may try to solve it by simply removing the distracting objects.

As you might guess, when a problem is represented one way, the solution may seem very difficult, or even impossible. Seen another way, the solution might be very easy. For example, consider the following problem (from Nasar, 1998):

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 miles per hour. At the same time, a fly that travels at a steady 15 miles per hour starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner until he is crushed between the two front wheels. Question: what total distance did the fly cover?

Please take a few minutes to try to solve this problem.

Most people represent this problem as a question about a fly because, well, that is how the question is asked. The solution, using this representation, is to figure out how far the fly travels on the first leg of its journey, then add this total to how far it travels on the second leg of its journey (when it turns around and returns to the first bicycle), then continue to add the smaller distance from each leg of the journey until you converge on the correct answer. You would have to be quite skilled at math to solve this problem, and you would probably need some time and pencil and paper to do it.

If you consider a different representation, however, you can solve this problem in your head. Instead of thinking about it as a question about a fly, think about it as a question about the bicycles. They are 20 miles apart, and each is traveling 10 miles per hour. How long will it take for the bicycles to reach each other? Right, one hour. The fly is traveling 15 miles per hour; therefore, it will travel a total of 15 miles back and forth in the hour before the bicycles meet. Represented one way (as a problem about a fly), the problem is quite difficult. Represented another way (as a problem about two bicycles), it is easy. Changing your representation of a problem is sometimes the best—sometimes the only—way to solve it.

Unfortunately, however, changing a problem’s representation is not the easiest thing in the world to do. Often, problem solvers get stuck looking at a problem one way. This is called  fixation . Most people who represent the preceding problem as a problem about a fly probably do not pause to reconsider, and consequently change, their representation. A parent who thinks her daughter is being defiant is unlikely to consider the possibility that her behavior is far less purposeful.

Problem-solving fixation was examined by a group of German psychologists called Gestalt psychologists during the 1930’s and 1940’s. Karl Dunker, for example, discovered an important type of failure to take a different perspective called  functional fixedness . Imagine being a participant in one of his experiments. You are asked to figure out how to mount two candles on a door and are given an assortment of odds and ends, including a small empty cardboard box and some thumbtacks. Perhaps you have already figured out a solution: tack the box to the door so it forms a platform, then put the candles on top of the box. Most people are able to arrive at this solution. Imagine a slight variation of the procedure, however. What if, instead of being empty, the box had matches in it? Most people given this version of the problem do not arrive at the solution given above. Why? Because it seems to people that when the box contains matches, it already has a function; it is a matchbox. People are unlikely to consider a new function for an object that already has a function. This is functional fixedness.

Mental set is a type of fixation in which the problem solver gets stuck using the same solution strategy that has been successful in the past, even though the solution may no longer be useful. It is commonly seen when students do math problems for homework. Often, several problems in a row require the reapplication of the same solution strategy. Then, without warning, the next problem in the set requires a new strategy. Many students attempt to apply the formerly successful strategy on the new problem and therefore cannot come up with a correct answer.

The thing to remember is that you cannot solve a problem unless you correctly identify what it is to begin with (initial state) and what you want the end result to be (goal state). That may mean looking at the problem from a different angle and representing it in a new way. The correct representation does not guarantee a successful solution, but it certainly puts you on the right track.

A bit more optimistically, the Gestalt psychologists discovered what may be considered the opposite of fixation, namely  insight . Sometimes the solution to a problem just seems to pop into your head. Wolfgang Kohler examined insight by posing many different problems to chimpanzees, principally problems pertaining to their acquisition of out-of-reach food. In one version, a banana was placed outside of a chimpanzee’s cage and a short stick inside the cage. The stick was too short to retrieve the banana, but was long enough to retrieve a longer stick also located outside of the cage. This second stick was long enough to retrieve the banana. After trying, and failing, to reach the banana with the shorter stick, the chimpanzee would try a couple of random-seeming attempts, react with some apparent frustration or anger, then suddenly rush to the longer stick, the correct solution fully realized at this point. This sudden appearance of the solution, observed many times with many different problems, was termed insight by Kohler.

Lest you think it pertains to chimpanzees only, Karl Dunker demonstrated that children also solve problems through insight in the 1930s. More importantly, you have probably experienced insight yourself. Think back to a time when you were trying to solve a difficult problem. After struggling for a while, you gave up. Hours later, the solution just popped into your head, perhaps when you were taking a walk, eating dinner, or lying in bed.

fixation :  when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

functional fixedness :  a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

mental set :  a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

insight :  a sudden realization of a solution to a problem

Solving Problems by Trial and Error

Correctly identifying the problem and your goal for a solution is a good start, but recall the psychologist’s definition of a problem: it includes a set of possible intermediate states. Viewed this way, a problem can be solved satisfactorily only if one can find a path through some of these intermediate states to the goal. Imagine a fairly routine problem, finding a new route to school when your ordinary route is blocked (by road construction, for example). At each intersection, you may turn left, turn right, or go straight. A satisfactory solution to the problem (of getting to school) is a sequence of selections at each intersection that allows you to wind up at school.

If you had all the time in the world to get to school, you might try choosing intermediate states randomly. At one corner you turn left, the next you go straight, then you go left again, then right, then right, then straight. Unfortunately, trial and error will not necessarily get you where you want to go, and even if it does, it is not the fastest way to get there. For example, when a friend of ours was in college, he got lost on the way to a concert and attempted to find the venue by choosing streets to turn onto randomly (this was long before the use of GPS). Amazingly enough, the strategy worked, although he did end up missing two out of the three bands who played that night.

Trial and error is not all bad, however. B.F. Skinner, a prominent behaviorist psychologist, suggested that people often behave randomly in order to see what effect the behavior has on the environment and what subsequent effect this environmental change has on them. This seems particularly true for the very young person. Picture a child filling a household’s fish tank with toilet paper, for example. To a child trying to develop a repertoire of creative problem-solving strategies, an odd and random behavior might be just the ticket. Eventually, the exasperated parent hopes, the child will discover that many of these random behaviors do not successfully solve problems; in fact, in many cases they create problems. Thus, one would expect a decrease in this random behavior as a child matures. You should realize, however, that the opposite extreme is equally counterproductive. If the children become too rigid, never trying something unexpected and new, their problem solving skills can become too limited.

Effective problem solving seems to call for a happy medium that strikes a balance between using well-founded old strategies and trying new ground and territory. The individual who recognizes a situation in which an old problem-solving strategy would work best, and who can also recognize a situation in which a new untested strategy is necessary is halfway to success.

Solving Problems with Algorithms and Heuristics

For many problems there is a possible strategy available that will guarantee a correct solution. For example, think about math problems. Math lessons often consist of step-by-step procedures that can be used to solve the problems. If you apply the strategy without error, you are guaranteed to arrive at the correct solution to the problem. This approach is called using an  algorithm , a term that denotes the step-by-step procedure that guarantees a correct solution. Because algorithms are sometimes available and come with a guarantee, you might think that most people use them frequently. Unfortunately, however, they do not. As the experience of many students who have struggled through math classes can attest, algorithms can be extremely difficult to use, even when the problem solver knows which algorithm is supposed to work in solving the problem. In problems outside of math class, we often do not even know if an algorithm is available. It is probably fair to say, then, that algorithms are rarely used when people try to solve problems.

Because algorithms are so difficult to use, people often pass up the opportunity to guarantee a correct solution in favor of a strategy that is much easier to use and yields a reasonable chance of coming up with a correct solution. These strategies are called  problem solving heuristics . Similar to what you saw in section 6.2 with reasoning heuristics, a problem solving heuristic is a shortcut strategy that people use when trying to solve problems. It usually works pretty well, but does not guarantee a correct solution to the problem. For example, one problem solving heuristic might be “always move toward the goal” (so when trying to get to school when your regular route is blocked, you would always turn in the direction you think the school is). A heuristic that people might use when doing math homework is “use the same solution strategy that you just used for the previous problem.”

By the way, we hope these last two paragraphs feel familiar to you. They seem to parallel a distinction that you recently learned. Indeed, algorithms and problem-solving heuristics are another example of the distinction between Type 1 thinking and Type 2 thinking.

Although it is probably not worth describing a large number of specific heuristics, two observations about heuristics are worth mentioning. First, heuristics can be very general or they can be very specific, pertaining to a particular type of problem only. For example, “always move toward the goal” is a general strategy that you can apply to countless problem situations. On the other hand, “when you are lost without a functioning gps, pick the most expensive car you can see and follow it” is specific to the problem of being lost. Second, all heuristics are not equally useful. One heuristic that many students know is “when in doubt, choose c for a question on a multiple-choice exam.” This is a dreadful strategy because many instructors intentionally randomize the order of answer choices. Another test-taking heuristic, somewhat more useful, is “look for the answer to one question somewhere else on the exam.”

You really should pay attention to the application of heuristics to test taking. Imagine that while reviewing your answers for a multiple-choice exam before turning it in, you come across a question for which you originally thought the answer was c. Upon reflection, you now think that the answer might be b. Should you change the answer to b, or should you stick with your first impression? Most people will apply the heuristic strategy to “stick with your first impression.” What they do not realize, of course, is that this is a very poor strategy (Lilienfeld et al, 2009). Most of the errors on exams come on questions that were answered wrong originally and were not changed (so they remain wrong). There are many fewer errors where we change a correct answer to an incorrect answer. And, of course, sometimes we change an incorrect answer to a correct answer. In fact, research has shown that it is more common to change a wrong answer to a right answer than vice versa (Bruno, 2001).

The belief in this poor test-taking strategy (stick with your first impression) is based on the  confirmation bias   (Nickerson, 1998; Wason, 1960). You first saw the confirmation bias in Module 1, but because it is so important, we will repeat the information here. People have a bias, or tendency, to notice information that confirms what they already believe. Somebody at one time told you to stick with your first impression, so when you look at the results of an exam you have taken, you will tend to notice the cases that are consistent with that belief. That is, you will notice the cases in which you originally had an answer correct and changed it to the wrong answer. You tend not to notice the other two important (and more common) cases, changing an answer from wrong to right, and leaving a wrong answer unchanged.

Because heuristics by definition do not guarantee a correct solution to a problem, mistakes are bound to occur when we employ them. A poor choice of a specific heuristic will lead to an even higher likelihood of making an error.

algorithm :  a step-by-step procedure that guarantees a correct solution to a problem

problem solving heuristic :  a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

confirmation bias :  people’s tendency to notice information that confirms what they already believe

An Effective Problem-Solving Sequence

You may be left with a big question: If algorithms are hard to use and heuristics often don’t work, how am I supposed to solve problems? Robert Sternberg (1996), as part of his theory of what makes people successfully intelligent (Module 8) described a problem-solving sequence that has been shown to work rather well:

• Identify the existence of a problem.  In school, problem identification is often easy; problems that you encounter in math classes, for example, are conveniently labeled as problems for you. Outside of school, however, realizing that you have a problem is a key difficulty that you must get past in order to begin solving it. You must be very sensitive to the symptoms that indicate a problem.
• Define the problem.  Suppose you realize that you have been having many headaches recently. Very likely, you would identify this as a problem. If you define the problem as “headaches,” the solution would probably be to take aspirin or ibuprofen or some other anti-inflammatory medication. If the headaches keep returning, however, you have not really solved the problem—likely because you have mistaken a symptom for the problem itself. Instead, you must find the root cause of the headaches. Stress might be the real problem. For you to successfully solve many problems it may be necessary for you to overcome your fixations and represent the problems differently. One specific strategy that you might find useful is to try to define the problem from someone else’s perspective. How would your parents, spouse, significant other, doctor, etc. define the problem? Somewhere in these different perspectives may lurk the key definition that will allow you to find an easier and permanent solution.
• Formulate strategy.  Now it is time to begin planning exactly how the problem will be solved. Is there an algorithm or heuristic available for you to use? Remember, heuristics by their very nature guarantee that occasionally you will not be able to solve the problem. One point to keep in mind is that you should look for long-range solutions, which are more likely to address the root cause of a problem than short-range solutions.
• Represent and organize information.  Similar to the way that the problem itself can be defined, or represented in multiple ways, information within the problem is open to different interpretations. Suppose you are studying for a big exam. You have chapters from a textbook and from a supplemental reader, along with lecture notes that all need to be studied. How should you (represent and) organize these materials? Should you separate them by type of material (text versus reader versus lecture notes), or should you separate them by topic? To solve problems effectively, you must learn to find the most useful representation and organization of information.
• Allocate resources.  This is perhaps the simplest principle of the problem solving sequence, but it is extremely difficult for many people. First, you must decide whether time, money, skills, effort, goodwill, or some other resource would help to solve the problem Then, you must make the hard choice of deciding which resources to use, realizing that you cannot devote maximum resources to every problem. Very often, the solution to problem is simply to change how resources are allocated (for example, spending more time studying in order to improve grades).
• Monitor and evaluate solutions.  Pay attention to the solution strategy while you are applying it. If it is not working, you may be able to select another strategy. Another fact you should realize about problem solving is that it never does end. Solving one problem frequently brings up new ones. Good monitoring and evaluation of your problem solutions can help you to anticipate and get a jump on solving the inevitable new problems that will arise.

Please note that this as  an  effective problem-solving sequence, not  the  effective problem solving sequence. Just as you can become fixated and end up representing the problem incorrectly or trying an inefficient solution, you can become stuck applying the problem-solving sequence in an inflexible way. Clearly there are problem situations that can be solved without using these skills in this order.

Additionally, many real-world problems may require that you go back and redefine a problem several times as the situation changes (Sternberg et al. 2000). For example, consider the problem with Mary’s daughter one last time. At first, Mary did represent the problem as one of defiance. When her early strategy of pleading and threatening punishment was unsuccessful, Mary began to observe her daughter more carefully. She noticed that, indeed, her daughter’s attention would be drawn by an irresistible distraction or book. Fresh with a re-representation of the problem, she began a new solution strategy. She began to remind her daughter every few minutes to stay on task and remind her that if she is ready before it is time to leave, she may return to the book or other distracting object at that time. Fortunately, this strategy was successful, so Mary did not have to go back and redefine the problem again.

Pick one or two of the problems that you listed when you first started studying this section and try to work out the steps of Sternberg’s problem solving sequence for each one.

a mental representation of a category of things in the world

an assumption about the truth of something that is not stated. Inferences come from our prior knowledge and experience, and from logical reasoning

knowledge about one’s own cognitive processes; thinking about your thinking

individuals who are less competent tend to overestimate their abilities more than individuals who are more competent do

Thinking like a scientist in your everyday life for the purpose of drawing correct conclusions. It entails skepticism; an ability to identify biases, distortions, omissions, and assumptions; and excellent deductive and inductive reasoning, and problem solving skills.

a way of thinking in which you refrain from drawing a conclusion or changing your mind until good evidence has been provided

an inclination, tendency, leaning, or prejudice

a type of reasoning in which the conclusion is guaranteed to be true any time the statements leading up to it are true

a set of statements in which the beginning statements lead to a conclusion

an argument for which true beginning statements guarantee that the conclusion is true

a type of reasoning in which we make judgments about likelihood from sets of evidence

an inductive argument in which the beginning statements lead to a conclusion that is probably true

fast, automatic, and emotional thinking

slow, effortful, and logical thinking

a shortcut strategy that we use to make judgments and solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

udging the frequency or likelihood of some event type according to how easily examples of the event can be called to mind (i.e., how available they are to memory)

judging the likelihood that something is a member of a category on the basis of how much it resembles a typical category member (i.e., how representative it is of the category)

a situation in which we are in an initial state, have a desired goal state, and there is a number of possible intermediate states (i.e., there is no obvious way to get from the initial to the goal state)

noticing, comprehending and forming a mental conception of a problem

when a problem solver gets stuck looking at a problem a particular way and cannot change his or her representation of it (or his or her intended solution strategy)

a specific type of fixation in which a problem solver cannot think of a new use for an object that already has a function

a specific type of fixation in which a problem solver gets stuck using the same solution strategy that has been successful in the past

a sudden realization of a solution to a problem

a step-by-step procedure that guarantees a correct solution to a problem

The tendency to notice and pay attention to information that confirms your prior beliefs and to ignore information that disconfirms them.

a shortcut strategy that we use to solve problems. Although they are easy to use, they do not guarantee correct judgments and solutions

Introduction to Psychology Copyright © 2020 by Ken Gray; Elizabeth Arnott-Hill; and Or'Shaundra Benson is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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