problem solving in english literature

Critical Thinking with Literature: It’s Problem-Solving

By Sharon Crowley
June 29, 2015

Critical thinking tops the list of skills students need for success in the complex 21st century. When it comes to science and math, most people equate critical thinking with problem solving. In those content areas, students apply their understanding of basic concepts to a task for which the solution is not known in advance. By grappling with a challenging problem, students extend their learning. Critical thinking about literature is not so different. With a written work, the problem or task is often an open-ended, text-based question. Students use their comprehension of the text to develop interpretations—or solutions to the problem.

If you want your students to engage in higher-order thinking as they read and discuss literature, include these key elements of problem-solving activities:

Genuine, intriguing questions. To think critically, there must be something to think critically about. With literature, it’s a text that leaves your students puzzling and asking questions about a character, event, symbol, or structure. Predictable or moralistic texts with flat characters don’t generate intriguing questions. When texts are sufficiently complex, the questions that spring from them present engaging problems.

Divergent answers. Just as genuine problems in math or science allow for multiple strategies and solutions, a discussion-worthy question about a piece of literature should invite multiple interpretations or answers. In Shared Inquiry discussions, considering divergent ideas is what drives students to find deeper meaning in a text.

Ample evidence. As in math or science, for an answer or solution to be sound, there must be relevant reasons behind it. Likewise, ideas about the meaning of literary texts must be supported with the evidence from the work itself. Evidence and reasoning make ideas valid and debatable. Without evidence, ideas are simply guesses.

Opportunities to evaluate evidence. Some pieces of scientific or mathematical data are more compelling than others. The same is true when exploring a question about a rich work of literature. Collaborative discussion is a time for participants to share the evidence that supports their ideas, to weigh that evidence, and to strengthen ideas by debating each other’s assertions or suggesting additional evidence.

Collaboration. A good discussion question, or problem, is one that students want to work on together. Just as students benefit from combining their skills and perspectives when solving a math or science problem, discussing an interpretive question as a group yields more thoughtful and considered answers than if students had worked alone. Follow-up questions that ask students to clarify, elaborate, and explain their ideas help deepen and enliven the conversation.

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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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A Review of Literature

First Online: 14 March 2021

Cite this chapter

Samantha S. Reed 4 ,
Carol A. Mullen 5 &
Emily T. Boyles 6

Part of the book series: SpringerBriefs in Education ((BRIEFSEDUCAT))

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This chapter of Problem - Based Learning in Elementary Schools offers a review of the literature addressing problem-based learning (PBL) in elementary classrooms and other educational contexts. Definitions provided generally uphold PBL as a teaching method in which students gain knowledge and skills by working collaboratively to both investigate and respond to authentic and engaging open-ended questions and/or problems. Also described is the history of PBL and its instructional impact on student learning and achievement. In accordance with the current study conducted in Virginia, key strategies of PBL instruction are identified that contribute to 21st-century skills identified nationally and at the state level, which education policy in Virginia denotes as critical thinking, creativity, communication, collaboration, and citizenship (the 5Cs). While exploring the PBL phenomenon and the relevant literature, academic achievement, student impact, and patterns are discussed. At present, much of the existing research focuses on grades K–12 in general terms or postsecondary education. Elementary grades, while investigated, continue to be underrepresented in explorations of PBL, a gap that this book helps to fill.

Definitions
Elementary school
K–12 education
Literature review
Problem-based learning
Project-based learning
Student achievement
21st-century skills

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Seven valuable skills learned as an English Literature Student [Guest Post from Inspiring Interns]

Anyone taking, or considering taking, a degree in English Literature will already know of the mockery that often comes along with the subject. Heralded as one of the ‘pointless’ degrees, it is deemed useless when it comes to leading into a career by many students. But that couldn’t be further from the truth. The skills gained by English Literature majors are undeniably useful in the workplace and the doors that an English degree can therefore open are endless.

Here are just a few of the skills that English graduates can boast of:

The ability to meet deadlines English students are often required to get through a huge reading list, submit essays and conduct research on a weekly basis. They are therefore no strangers to tight deadlines. Their capability to hit those cut-off dates without needing to beg for an extension comes down to their ability to manage their time efficiently and work well under pressure.

Research skills It is sometimes believed that English essays are all about opinions, and while to a certain extent that is true, they also require a huge amount of research. For every point made, there must be a critic or reference to back it up, and so literature students have to quickly develop the research skills needed to enable them to find the required sources.

An ability to understand different points of view In an office environment, where people from all walks of life are thrown together, it is necessary to be able to understand where everyone is coming from. English students are taught to approach a text from a variety of angles in order to understand how people could read the same piece of writing in a different way to themselves. This is a skill that is easily transferred into the work environment and can often be invaluable when it comes to settling a dispute or sealing a deal.

Written communication It goes without saying that most of an English major’s time is spent reading and writing. While it might seem obvious to state that their grammar and spelling is therefore top notch, it’s still worth shouting about as those skills do not come naturally to everyone. In a day and age where text speech is now commonplace having someone on your team that can produce clear and concise copy, with minimal mistakes, is invaluable.

A creative mind Thinking outside of the box is the crux of an English literature degree so students are constantly pushed to unleash their creative side to enable them to approach a text in a unique way. Creativity and problem solving often come hand-in-hand, so having an English graduate on your team can help you to achieve better results within the workplace.

Being able to craft a civil argument There is no ‘right’ answer when it comes to English essays but that doesn’t mean that everyone is correct. Sounds odd? Bear with us. English students have to question everything they read and so they learn to carefully craft an argument, back it up with evidence and present it. However, they are also aware that the person sitting opposite them in the lecture hall may have come to a completely different conclusion to the one they did. Literature students must therefore be able to civilly defend their own point of view and be capable of taking constructive criticism from others.

Confidence There comes a time in every English Literature student’s life when they will have to write an essay on a poem, novel or play that they simply do not understand or like. But one skill that literature students acquire very early on is the ability to write with confidence about an array of topics. Fake it until you make it comes to mind, and in this case, it is true but it is still an undeniably useful life skill to have.

Shannon Clark writes for Inspiring Interns, which specialises in sourcing candidates for internships and graduate jobs . You can find more of her writing at www.shannonlclark.com and www.sweetserendipityblog.co.uk .

Connections between Reading Comprehension and Word-Problem Solving via Oral Language Comprehension: Implications for Comorbid Learning Disabilities

The Vanderbilt Learning Disabilities Innovation Hub, funded by the Eunice Kennedy Shriver National Institute of Child Health and Human Development, addresses a vulnerable, understudied, and underserved subgroup of the learning disabilities population: students with comorbid difficulty in two critical areas, reading comprehension and math word-problem solving. This form of learning disability occurs frequently ( Landerl & Moll, 2010 ; Mann Koepke & Miller, 2013 ), and as Koponen et al. (2018) observed, half of children with poor performance in one domain also have difficulty in the other domain.

Moreover, students with such comorbidity experience worse outcomes in each area than do peers with difficulty in only one of these domains ( Cirino, Fuchs, Elias, Powell, & Schumacher, 2015 ; Willcutt, Petrill, et al., 2013 ), and comorbidity is associated with inadequate response to generally effective intervention ( Fuchs, Fuchs, & Prentice, 2004 ). Nonetheless, the problem of co-occurring reading and math difficulty is understudied: The literature is small; most studies are descriptive; and most investigations focused on reading and math difficulty define comorbidity in terms of lower-order skill: calculations and word reading. In this article, we refer to difficulty across reading comprehension and word-problem solving as higher-order comorbidity .

Students with higher-order comorbidity also represent an especially vulnerable learning disability subtype for two reasons. First, reading comprehension is a strong predictor of quality of life, financial security, and life expectancy (Batty, Kivimaki, & Deary, 2010; Meneghetti, Carretti, & De Beni, 2006 ; Ritchie & Bates, 2013 ; Taraban, Rynearson, & Kerr, 2000 ). Limited reading comprehension decreases access to content knowledge and undermines learning during and after formal schooling. Second, word-problem solving is the best school-age predictor of adult employment and wages ( Every Child a Chance Trust, 2009 ; Murnane, Willett, Braatz, & Duhaldeborde, 2001 ), and it represents a major emphasis in almost every strand of the math curriculum from kindergarten through high school. So word-problem solving difficulty limits school as well as occupational success. It is therefore highly problematic that schools struggle to provide students with intervention in more than one domain. Typically, reading intervention takes priority over math intervention, which leaves students with comorbid reading and math learning disability without the math intervention they require to succeed.

In this article, we discuss the Vanderbilt Hub’s approach to studying students with higher-order comorbidity. In the first segment of this article, we describe a framework that connects reading comprehension and word-problem solving development via oral language comprehension, and we establish connections between oral language comprehension and reading comprehension (Catts, Hogan, & Adolf, 2005; Gough & Tunmer, 1986 ; Peng et al., in press ) and between language comprehension and word-problem solving ( Bernardo, 1999 ; Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, & Bryant, 2010a , 2008 ; Van der Schoot, Bakker Arkema, Horsley, & Van Lieshout, 2009 ). This first segment provides the basis for the article’s second segment, focused on the Vanderbilt Hub’s innovative approach for investigating connections between reading comprehension and word-problem solving via language comprehension. This study tests a theoretically-coordinated framework for scaffolding performance in both high-priority domains of academic development, while exploring disaggregated effects for boys versus girls and for native and non-native English speakers.

Reading comprehension and word-problem solving differ in some transparent ways. For example, most reading passages, because they are more extended than the typical word problem, make stronger demands on inferencing and background knowledge than occurs with word problems. On the other hand, in word problems, representing the text is not the final goal. Instead, students answer the word problem’s question, which requires representing the text with a number sentence that includes a missing quantity, deriving the mathematical result, evaluating whether the answer is computationally reasonable and correct, and communicating the solution ( Jiménez & Verschaffe, 2014 ).

Even so, based on theories of reading comprehension, discourse processing, and word-problem solving ( Perfetti, Yang, & Schmalhofer, 2008 ; Rapp, van den Broek, McMaster, Kendeou, & Espin, 2007 ; Verschaffel & De Corte, 1997 ), representations of texts, including reading passages and word-problem statements, necessarily have three components. The first involves constructing a coherent microstructure and deriving a hierarchical macrostructure to capture the text’s essential ideas. The second component is the situation model, which requires supplementing the text with inferences based on the reader’s prior knowledge. With the third component, the reader derives a problem model or schema to match the passage’s or word-problem statement’s content.

Kintsch and Greeno (1985) posited that although the comprehension strategies, the nature of required knowledge structures, and the form of resulting structures, inferences, and problem models differ by task, the general features of this process apply across stories, informational passages, and word problems. Reading passages and word-problem statements alike require individuals to build the propositional text structure, inference, and identify schema, and this processing makes strong demands on reasoning and working memory. In reading passages or word-problem statements, the child processes the propositional text base to sequentially fill the slots of a model that evolve over the passage. This involves coding relevant objects, actors, and actions (as well as quantities for word problems), across segments of text, while re-sorting information as new ideas alter hypotheses about the situation and the schema.

Because the processes by which stories and informational text are understood more thoroughly (e.g., Perfetti, Yang, & Schmalhofer, 2008 ; Rapp, van den Broek, McMaster, Kendeou, & Espin, 2007 ) than is the case for word-problem solving, we illustrate text comprehension processes in the context of word problems. Note, however, that the Vanderbilt Hub and its Project study address text comprehension and word-problem solving in equal measure and in parallel ways.

Consider the text processing required for a combine word problem (Part 1 plus Part 2 equals Total or P1 + P2 = T): Joe has 3 marbles. Tom has 5 marbles. Tom also has 2 balls. How many marbles do the boys have in all? A low-risk child processes sentence 1’s propositional text base to identify that object = marbles; quantity = 3; actor = Joe; but Joe’s role = to be determined (TBD). This is placed in short-term memory. In sentence 2, propositions are similarly coded and held in memory. In sentence 3, balls fails to match the object code in sentences 1 and 2, signaling the number 2 as perhaps irrelevant; this is added to memory. In sentence 4, the question, the quantitative proposition how many marbles and the phrase in all cues the child to identify the combine schema; assign the role of superset (Total) to the question; assign subset roles (Parts 1 and 2) to the TBDs in memory; and reject 2 balls. Filling in these slots of the schema triggers a set of problem-solving strategies.

With typical school instruction, children gradually construct the combine schema on their own, just as they devise their own strategies for handling the demands on reasoning and working memory that this text processing, problem-solving sequence involves. Errors are viewed as failures (a) to produce the intended mental representations with respect to the three components in preceding paragraph or (b) to manage demands on reasoning and working memory. Such demands are well documented for reading comprehension (e.g., Berninger et al., 2010 ; Eason, Goldberg, Young, Geist, & Cutting, 2012 ; Swanson, & Jerman, 2007 ) and word-problem solving (e.g., Fuchs et al., 2010a , 2010b ).

Unfortunately, children with reading comprehension or word-problem solving difficulty fail to discover schemas on their own, and demands often exceed their reasoning and working memory capacity. Direct skills intervention (in reading comprehension or word-problem solving) for students with reading comprehension or word-problem solving difficulty explicitly teaches step-by-step strategies to help children derive the intended mental representation and formulate connections among the propositional text base, the situation model, and the schema (or model) in ways that reduce reasoning and working memory demands.

For example, schema-based word-problem solving tutoring ( Fuchs et al., 2009 , 2014 ) explicitly teaches at-risk children step-by-step strategies that begin with identifying word-problem statements as combine, compare, or change schema. Then children are taught to build the propositional text structure. Schema-based word-problem solving tutoring facilitates connections among the situation model, schema, and productive solution strategies by making these connections explicit, while reducing demands on reasoning and working memory.

More specifically, with schema-based word-problem solving tutoring, the child RUNs through the problem: R eads it, U nderlines the question in which the object code (marbles) is revealed, and N ames the explicitly taught (combine) schema. This prompts the child to write the combine meta-equation, P1 + P2 = T (Part 1 plus Part 2 equals Total). The child then re-reads the problem statement and while re-reading, writes replacements for P1 and P2 (quantities for each relevant “part”) and crosses out irrelevant objects and numbers. This reduces the burden on reasoning and working memory and provides the equation for problem solving. This in turn prompts the counting-up adding strategy (also taught in schema-based word-problem solving tutoring). Word-problem solving and reading comprehension interventions that compensate for reasoning and working memory demands with explicit skills instruction, such as schema-based word-problem solving tutoring, have proven effective in improving reading comprehension ( D. Fuchs et al., in press ; Jitendra, Hoppes, & Xin, 2000 ; Williams, Hall, & Lauer, 2004 ; Williams et al., 2014 ) or word-problem solving ( Fuchs et al., 2014 ; Fuchs et al., 2009 ).

More pertinent to Vanderbilt Hub and the present article, Kintsch and Greeno (1985) and Rapp, van den Broek, McMaster, Kendeou, and Espin (2007) further posited that competent performance in each domain also relies heavily on language comprehension processes. As per Kintsch and Greeno, in word-problem solving, children “understand important vocabulary and language constructions prior to school entry” (p. 111) and “through instruction in arithmetic and word-problem solving, learn to treat these words in a special, task-specific way, including extensions to ordinary usage for terms (e.g., all or more ) to more complicated constructions involving sets ( in all and more than) ” (p. 111). In reading comprehension, because or therefore acquires special meaning within cause-effect passages to prime readers to search for and connect consequences with trigger events.

The assumption is that students have the necessary language comprehension abilities to understand text and problem statements to derive appropriate problem models. Yet, assuming children with limitations in language comprehension will learn to treat domain-specific words in a special task-specific manner through school exposure to reading comprehension and word-problem solving tasks is tenuous, and this may be particularly so for non-native English speakers ( Mancilla-Martinez & Lesaux, 2010 ; 2011 ) and other at-risk and diverse populations.

In the case of word problems, Cummins, Kintsch, Reusser, & Weiner (1988) computationally modeled errors with two types of defects: incorrect math problem-solving processes versus language comprehension processing errors. Correct problem representation depended more on language comprehension, and changing wording in minor ways dramatically affected accuracy. To illustrate how word-problem solving taxes language comprehension, consider this combine problem: Joe has 3 cats. Tom has 5 dogs. Tom has 2 plants. How many pets do the boys have in all? Objects in this text increase demands on language comprehension for assigning roles for the propositional text structure (despite similar demand for inducing the schema), due to more sophisticated representations of vocabulary involving taxonomic relations at superordinate levels and subtle distinctions among categories (dogs + cats = pets; plants are not pets). All this suggests that instruction focused on language comprehension processes as well as the mathematical aspects of word-problem solving is required within tutoring, with an explicit focus on strengthening word-problem language for building propositional, situational, and schema representations. These ideas apply equally well in the domain of reading comprehension.

For this reason, the Hub’s innovative approach involves embedding reading comprehension-language comprehension instruction within reading comprehension direct skills instruction and embedding parallel word-problem solving-language comprehension instruction within word-problem solving direct skills intervention. Before explaining that approach, we briefly describe some recent studies examining connections between reading comprehension, word-problem solving, and oral language comprehension.

Some Recent Studies

Some prior work examining concurrent relations between reading comprehension and word-problem solving suggest an association. For example, Vilenius-Tuohimaa, Aunola, and Nurmi (2008) reported substantial shared variance across reading comprehension and word-problem solving when controlling for foundational reading skill. Swanson, Cooney, and Brock (1993) identified reading comprehension as a correlate of word-problem solving while controlling for working memory, knowledge of operations, word-problem propositions, and calculation skill. Boonen, van der Schoot, Florytvan, de Vries, and Jolles (2013) found that reading comprehension had medium to large relations with word-problem solving. Although this relation was not evident at the word-problem item-level in Boonen, van Wesel, Jolles, and van der Schoot (2014) , the authors indicated that their word-problem items did not involve the semantic complexity that warrants strong reliance on reading comprehension.

Cirino, Child, and Mcdonald (2018) extended the concurrent literature by longitudinally assessing the role of domain-specific and general predictors in kindergarten and multiple types of reading and mathematics outcomes in first grade. The correlation between first-grade reading comprehension and math problem solving was .67. After partialling all 11 predictors, the correlation decreased to .21, which corresponds to an R 2 value of .04. This suggests commonality in the underlying sources of variance contributing to reading comprehension and word-problem solving. Surprisingly, however, language was differentially predictive of math problem solving compared to reading comprehension. This may be due to the age level of the participants, because first-grade reading comprehension relies more heavily on word reading than vocabulary knowledge ( Garcia & Cain, 2014 ).

Also relying on a longitudinal design, Fuchs, Fuchs, Compton, Hamlett, and Wang (2015) assessed second graders early in the year on general language comprehension, working memory, nonlinguistic reasoning, processing speed, and foundational reading and math skill. At the end of the year, the children were assessed on word-problem specific language, reading comprehension, and word-problem solving. Path analytic mediation analysis indicated the effect of general language comprehension on reading comprehension was entirely direct, whereas the effect of general language comprehension on word-problem solving was partially mediated by word-problem specific language comprehension. Yet, across both domains, effects of working memory and reasoning operated in parallel ways. These findings are in line with Kintsch and Greeno (1985) , who suggested that operating on word problems and conventional reading passages require parallel processes that tax language comprehension, working memory, and reasoning.

More recently, Fuchs, Gilbert, Fuchs, Seethaler, and Martin (2018) extended this literature by testing effects of initial reading comprehension using a broad-based measure of reading comprehension ( Gates–MacGinitie Reading-Comprehension [ MacGinitie et al., 2002 ]), on year-end word-problem solving. We examined the specificity of effects by contrasting the contribution of the Gates to later word-problem solving against Gates’s effects on later calculations. Based on studies indicating (a) shared concurrent variance between reading comprehension and word-problem solving ( Boonen et al., 2013 , 2014 ; Swanson et al., 1993 ; Vilenius-Tuohimaa et al., 2008 ), (b) substantially similar patterns of cognitive and linguistic predictors across reading comprehension and word-problem solving ( Fuchs et al., 2015 ), and (c) shared but some distinctive predictive patterns for word-problem solving versus calculations ( Fuchs et al., 2008 ; Swanson & Beebe-Frankenberger, 2004 ), we hypothesized that the effects of Gates are stronger on word-problem solving than on pure calculations. Conversely, we expected simple, initial arithmetic skill to predict year-end, more complex calculations more strongly than either of the year-end word-problem measures.

More highly relevant to the Vanderbilt Hub’s focus, we also explored the role of start-of-year language comprehension in word-problem solving while controlling for start-of-year Gates performance. Consistent with Kintsch and Greeno (1985) and given studies documenting connections between language comprehension and reading comprehension (Catts, Hogan, & Adolf, 2005; Gough & Tunmer, 1986 ; Peng et al., in press ) and between language comprehension and word-problem solving ( Bernardo, 1999 ; Fuchs et al., 2010a , 2008 ; Van der Schoot et al., 2009 ), we expected the effects of start-of-second-grade language comprehension to be stronger on both end-of-second-grade word-problem solving outcomes than on calculations.

On one hand, we found that language comprehension strongly and uniquely predicted later word-problem solving. We also found that Gates performance was a strong predictor of later word-problem solving. On the other hand, we found that Gates was not a specific predictor of word-problem solving (i.e., it also predicted later calculations), which raises questions about what general measures of reading comprehension actually assess. (A discussion of this finding is beyond the scope of the present article. See instead Fuchs et al., 2018 .)

More pertinently, finding a stronger role for language comprehension in word-problem solving than in calculations, while controlling for effects of reading comprehension (which is expected to share variance with language comprehension and therefore compete with language comprehension as a predictor of word-problem solving), strengthens prior evidence for the importance of language comprehension within word-problem solving. Moreover, a common role for language comprehension across word-problem solving and reading comprehension represents an important connection between the two academic domains

A Theoretically-Coordinated Framework for Synergistically Improving Performance in Both High-Priority Domains of Academic Development

Finding that the cognitive, linguistic, and academic predictors of word-problem solving are separable from those involving pure calculations, while finding a stronger role for language comprehension in word-problem solving over calculations, also indicates that word-problem solving is connected to reading comprehension. At the same time, the literature convincingly demonstrates the critical role language plays in reading comprehension (e.g., Catts et al., 2005 ; Gough & Tunmer, 1986 ; Peng et al., in press ). Together, this suggests an important role for reading comprehension instruction within word-problem solving instruction as well as the need for a focus on language instruction in improving performance in both domains.

For word problems, a strong focus on language includes but is not limited to word-problem-specific vocabulary and syntactic knowledge (e.g., understanding the distinction between more than and then there were more ; that the cause and effect in change word-problem solving may be presented in either order within word-problem statements). Such an approach is consistent with recent calls ( Catts & Kamhi, 2017 ; Ukrainetz, 2017 ) to intimately connect an instructional focus on oral language to specific reading comprehension task demands, even as that instruction targets the subset of learners with language deficits for such embedded language comprehension instruction.

An integrated approach with a deliberate focus on the reading comprehension demands of word-problem solving and the language that connects reading passage and word-problem processing may also include methods to assist students in constructing explicit text-level representations, generating text-connecting inferences, retrieving general as well as math-specific background knowledge, and integrating that knowledge with information in text-level representations ( Perfetti et al., 2008 ; Rapp et al., 2007 ; Verschaffel & De Corte, 1997 ). All this is in the service of building the situation and the problem model or schema of the reading passage or word problem statement.

In this vein, the Vanderbilt Hub’s research project innovatively tests whether the effects of conceptual scaffolding designed to connect reading comprehension, word-problem solving, and language comprehension may ultimately provide direction for a theoretically coordinated approach for simultaneously improving performance across reading comprehension and word-problem solving. This includes, for example, addressing cause-effect informational text structure (a topic typically reserved for reading comprehension) in conjunction with change word problems (in which an event serves to increase or decrease a starting amount, thereby creating a new ending amount) or connecting compare-contrast informational text structure (again, a topic typically reserved for reading comprehension) in conjunction with word problems that compare quantities.

Testing effects of an approach that is designed to focus on reading comprehension and word problems in coordinated fashion would extend theoretical understanding of both domains. This line of work is potentially important for three additional reasons, as discussed at the beginning of this article. First, students with comorbid learning disorders across reading comprehension and word-problem solving represent an especially vulnerable subset of the learning disabilities population; reading comprehension is a strong predictor of quality of life, financial security, and life expectancy ( Batty et al., 2010 ; Meneghetti et al., 2006 ; Ritchie & Bates, 2013 ); and word-problem solving is the best school-age predictor of later employment and wages ( Every Child a Chance Trust, 2009 ; Murnane et al., 2001 ). Second, students with concurrent difficulty perform lower in each domain than students with difficulty in just one academic domain ( Willcutt et al., 2013 ). Third, schools experience substantial challenges in providing students with intervention on more than one academic domain. Because reading often takes priority over math in the early grades, many comorbid students receive reading intervention as they fall further behavior in math. A coordinated approach for addressing reading comprehension and word-problem solving would alleviate this logistical problem.

In the Vanderbilt Hub’s study, we randomly assign second-grade children with comorbid difficulty across reading comprehension and word-problem solving to three study conditions. The first condition focuses on conceptual scaffolding on reading comprehension with an embedded focus on language comprehension that spans reading comprehension and word-problem solving. The second condition provides conceptual scaffolding on word-problem solving with an embedded focus on language comprehension that spans word-problem solving and reading comprehension. The third condition is a business-as-usual control group (the classroom and interventions schools typically provide). This study ran its first cohort of participants during the 2017–2018 academic year. At the time of this publication, the study was still in progress.

The study’s specific aim is to test what we refer to as the higher-order comorbidity hypothesis: Language comprehension plays a critical role in reading comprehension and word-problem solving and provides direction for understanding higher-order comorbidity, thus offering a coordinated approach for improving both outcomes and lending support for the validity of reading comprehension and word-problem solving comorbidity as an LD subtyping framework.

We ask two questions. The first is whether conceptual scaffolding on one academic domain with an embedded focus on language comprehension that spans both academic domains (reading comprehension and word-problem solving (conditions 1 and 2) produces aligned and reciprocal advantages over the control group. We expect aligned effects favoring reading comprehension scaffolding over control on reading comprehension outcomes and favoring word-problem solving scaffolding over control on word-problem solving outcomes. More central to the comorbidity hypothesis, we also hypothesize reciprocal effects, with stronger performance over control for embedded language comprehension within reading comprehension on word-problem solving outcomes and with stronger performance over control for embedded language comprehension within word-problem solving on reading comprehension outcomes. Our second question is whether aligned and reciprocal effects in part occur indirectly, via improved understanding of the language of reading comprehension or the language of word-problem solving. We hypothesize indirect effects via the aligned form of domain-specific language comprehension on both outcomes.

Thus, our focus on language comprehension is the vocabulary and syntax addressed in the reading comprehension and word-problem solving scaffolding. This focus maps onto prior studies documenting relations of vocabulary and syntax with various forms of mathematics. For example, Hornburg, Schmitt, and Purpura (2018) found that preschoolers’ mathematical language was more strongly related than general expressive vocabulary to word-problem performance and other numeracy skills. Chow and Ekholm (2019) identified concurrent relations between receptive syntax and addition operations and understanding at first and second grade. Gjicali, Astuto, and Lipnevich (2019) examined longitudinal relations between language and numeracy skills in predominantly high-poverty children. Children were 1–4 years when language skills were assessed; 4–7 years when number skills were assessed. Language comprehension was indirectly related to number identification and number relations via oral counting.

Exploratory Hypotheses and Development of a Neuroimaging Paradigm

We also consider two exploratory issues to provide insight into the robustness of effects and to address issues of diversity and inclusiveness: whether aligned and reciprocal effects for comorbid students differ for boys versus girls or for non-native English speakers versus native English speakers and whether pretest English proficiency among non-native English speakers moderates intervention effects.

Across studies on the mathematics performance for girls versus boys, performance differences between males versus females appear to increase with three variables: as grade level increases, with more complex mathematics demands on visual-spatial resources, and among higher-achievers ( Stoet, Bailey, Moore, & Geary, 2016 ). On this basis, we do not expect a main effect for girls versus boys in second-grade students with comorbid learning disability. Our exploratory issue is, however, about an interaction : whether the pattern of word-problem solving scaffolding effects differs for boys versus girls. We did not locate any study disaggregating word-problem solving intervention effects for boys versus girls. Because differential effects of word-problem solving intervention for girls versus boys have been tested infrequently if at all, we examine scaffolding effect sizes for boys versus girls to explore the robustness of effects.

In reading comprehension intervention research, the issue of differential effects for girls versus boys also has received little attention. Here, we also did not locate any intervention study separating effects. In a synthesis, however, Suggate (2016) reported a correlation between percentage of boys in the studies and outcome (outcome was aggregated across reading constructs, i.e., it was not specific to reading comprehension). The correlation was −.15 at posttest; −.31 at follow up (favoring girls). As with word-problem solving, because few relevant analyses are available and none for comorbid samples, we examine effect sizes for boys versus girls to explore the robustness of effects.

The literature on differential word-problem solving or reading comprehension intervention effects for non-native English speakers versus native English speakers is similarly thin. It fails to provide the basis for hypotheses, especially for comorbid non-native English speaking samples. Yet, the school-age population of non-native English speakers is growing rapidly ( National Center for Education Statistics, 2017 ; McFarland et al., 2017 ), and databases reveal a main effect, in which non-native English speakers’ math and reading performance falls substantially below that of native English speakers ( National Center for Education Statistics, 2017 ). Moreover, in this population, studies document the relation of vocabulary and/or syntax with word-problem solving (e.g., Foster, Anthony, Zucker, & Branum-Martin, 2019 ; Méndez, Hammer, Lopez, & Blair, 2019 ) and reading comprehension ( Mancilla-Martinez & Lesaux, 2017 ; Nakamoto, Lindsey, & Manis, 2008 ; Proctor, Carlo, August, & Snow, 2005 ).

It is of theoretical and clinical importance to explore (a) whether the pattern of effect sizes differs for non-native English speakers versus native English speakers for comorbid children and (b) a pattern of moderation in which the relation between English proficiency and performance is expected to be less strong with embedded language comprehension instruction in word-problem solving scaffolding and in reading comprehension scaffolding than in control. This is because language comprehension-embedded scaffolding is designed to compensate for poor language comprehension. We note that the Hub’s Project is not powered for formally assessing these robustness and diversity questions, but our hope is that it will provide insight into whether effects for English language learners differs from effects for monolingual counterparts.

A final component of the Hub’s work is development of a neuroimaging paradigm to capture the brain mechanisms that underlie connections among reading comprehension, word-problem solving, and language comprehension. Development of a paradigm may provide the foundation for future studies focused on how the brain mechanisms associated with language comprehension serve as a potential link between reading comprehension and word-problem solving.

The overall hope is that the Vanderbilt Hub’s Project will impact science by (a) deepening understanding about language comprehension as a process involved in higher-order comorbidity; (b) providing theoretically-guided and empirical bases for the link between reading comprehension and word-problem solving; (c) strengthening support for the validity of this form of comorbidity as a learning disabilities subtyping framework; and (d) providing insight into the robustness of this approach (by gender and English proficiency status). Also, our focus on scaffolding language comprehension within reading comprehension and word-problem solving tutoring provides a platform for transdisciplinary work across learning sciences, second language learning, learning disabilities, and developmental psychology. This is critical because evidence is growing that language comprehension difficulty contributes to poor responsiveness to reading comprehension or word-problem solving intervention among students with developmental learning disabilities (e.g., Catts et al., 2008 ; D. Fuchs et al., in press ).

In sum, the Vanderbilt Hub’s science is innovative in four important ways. First, we focus on an especially vulnerable subset of the learning disabilities population, which is understudied, may suffer from disproportionately poor reading comprehension and word-problem solving outcomes and may have a distinctive set of cognitive deficits: students with comorbid difficulty with reading comprehension and word-problem solving. Second, with this understudied population, we adopt an innovative approach: conceptual scaffolding across language comprehension and one academic domain, with parallel structure what is required in the second academic domain. If this approach reveals improved performance in both academic domains, it would offer an innovative direction for treating this understudied, vulnerable population. Third, the Hub’s study provides the most stringent test to date on the connection among language comprehension, reading comprehension, and word-problem solving. Fourth, our exploratory issues address the robustness of effects as well as diversity and inclusiveness. The long-term goal of this proof-of-concept science is to impact clinical practice by offering an innovative direction aimed at improving learning for this understudied, vulnerable population.

Acknowledgments

This research was supported by 2 P20 HD075443 and Core Grant HD15052 from the Eunice Kennedy Shriver National Institute of Child Health & Human Development to Vanderbilt University. The content is solely the responsibility of the authors and does not necessarily represent the official views of the Eunice Kennedy Shriver National Institute of Child Health & Human Development or the National Institutes of Health.

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Open access
Published: 17 February 2022

Effectiveness of problem-based learning methodology in undergraduate medical education: a scoping review

Joan Carles Trullàs ORCID: orcid.org/0000-0002-7380-3475 1 , 2 , 3 ,
Carles Blay ORCID: orcid.org/0000-0003-3962-5887 1 , 4 ,
Elisabet Sarri ORCID: orcid.org/0000-0002-2435-399X 3 &
Ramon Pujol ORCID: orcid.org/0000-0003-2527-385X 1

BMC Medical Education volume 22 , Article number: 104 ( 2022 ) Cite this article

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Problem-based learning (PBL) is a pedagogical approach that shifts the role of the teacher to the student (student-centered) and is based on self-directed learning. Although PBL has been adopted in undergraduate and postgraduate medical education, the effectiveness of the method is still under discussion. The author’s purpose was to appraise available international evidence concerning to the effectiveness and usefulness of PBL methodology in undergraduate medical teaching programs.

The authors applied the Arksey and O’Malley framework to undertake a scoping review. The search was carried out in February 2021 in PubMed and Web of Science including all publications in English and Spanish with no limits on publication date, study design or country of origin.

The literature search identified one hundred and twenty-four publications eligible for this review. Despite the fact that this review included many studies, their design was heterogeneous and only a few provided a high scientific evidence methodology (randomized design and/or systematic reviews with meta-analysis). Furthermore, most were single-center experiences with small sample size and there were no large multi-center studies. PBL methodology obtained a high level of satisfaction, especially among students. It was more effective than other more traditional (or lecture-based methods) at improving social and communication skills, problem-solving and self-learning skills. Knowledge retention and academic performance weren’t worse (and in many studies were better) than with traditional methods. PBL was not universally widespread, probably because requires greater human resources and continuous training for its implementation.

PBL is an effective and satisfactory methodology for medical education. It is likely that through PBL medical students will not only acquire knowledge but also other competencies that are needed in medical professionalism.

Peer Review reports

There has always been enormous interest in identifying the best learning methods. In the mid-twentieth century, US educator Edgar Dale proposed which actions would lead to deeper learning than others and published the well-known (and at the same time controversial) “Cone of Experience or Cone of Dale”. At the apex of the cone are oral representations (verbal descriptions, written descriptions, etc.) and at the base is direct experience (based on a person carrying out the activity that they aim to learn), which represents the greatest depth of our learning. In other words, each level of the cone corresponds to various learning methods. At the base are the most effective, participative methods (what we do and what we say) and at the apex are the least effective, abstract methods (what we read and what we hear) [ 1 ]. In 1990, psychologist George Miller proposed a framework pyramid to assess clinical competence. At the lowest level of the pyramid is knowledge (knows), followed by the competence (knows how), execution (shows how) and finally the action (does) [ 2 ]. Both Miller’s pyramid and Dale’s cone propose a very efficient way of training and, at the same time, of evaluation. Miller suggested that the learning curve passes through various levels, from the acquisition of theoretical knowledge to knowing how to put this knowledge into practice and demonstrate it. Dale stated that to remember a high percentage of the acquired knowledge, a theatrical representation should be carried out or real experiences should be simulated. It is difficult to situate methodologies such as problem-based learning (PBL), case-based learning (CBL) and team-based learning (TBL) in the context of these learning frameworks.

In the last 50 years, various university education models have emerged and have attempted to reconcile teaching with learning, according to the principle that students should lead their own learning process. Perhaps one of the most successful models is PBL that came out of the English-speaking environment. There are many descriptions of PBL in the literature, but in practice there is great variability in what people understand by this methodology. The original conception of PBL as an educational strategy in medicine was initiated at McMaster University (Canada) in 1969, leaving aside the traditional methodology (which is often based on lectures) and introducing student-centered learning. The new formulation of medical education proposed by McMaster did not separate the basic sciences from the clinical sciences, and partially abandoned theoretical classes, which were taught after the presentation of the problem. In its original version, PBL is a methodology in which the starting point is a problem or a problematic situation. The situation enables students to develop a hypothesis and identify learning needs so that they can better understand the problem and meet the established learning objectives [ 3 , 4 ]. PBL is taught using small groups (usually around 8–10 students) with a tutor. The aim of the group sessions is to identify a problem or scenario, define the key concepts identified, brainstorm ideas and discuss key learning objectives, research these and share this information with each other at subsequent sessions. Tutors are used to guide students, so they stay on track with the learning objectives of the task. Contemporary medical education also employs other small group learning methods including CBL and TBL. Characteristics common to the pedagogy of both CBL and TBL include the use of an authentic clinical case, active small-group learning, activation of existing knowledge and application of newly acquired knowledge. In CBL students are encouraged to engage in peer learning and apply new knowledge to these authentic clinical problems under the guidance of a facilitator. CBL encourages a structured and critical approach to clinical problem-solving, and, in contrast to PBL, is designed to allow the facilitator to correct and redirect students [ 5 ]. On the other hand, TBL offers a student-centered, instructional approach for large classes of students who are divided into small teams of typically five to seven students to solve clinically relevant problems. The overall similarities between PBL and TBL relate to the use of professionally relevant problems and small group learning, while the main difference relates to one teacher facilitating interactions between multiple self-managed teams in TBL, whereas each small group in PBL is facilitated by one teacher. Further differences are related to mandatory pre-reading assignments in TBL, testing of prior knowledge in TBL and activating prior knowledge in PBL, teacher-initiated clarifying of concepts that students struggled with in TBL versus students-generated issues that need further study in PBL, inter-team discussions in TBL and structured feedback and problems with related questions in TBL [ 6 ].

In the present study we have focused on PBL methodology, and, as attractive as the method may seem, we should consider whether it is really useful and effective as a learning method. Although PBL has been adopted in undergraduate and postgraduate medical education, the effectiveness (in terms of academic performance and/or skill improvement) of the method is still under discussion. This is due partly to the methodological difficulty in comparing PBL with traditional curricula based on lectures. To our knowledge, there is no systematic scoping review in the literature that has analyzed these aspects.

The main motivation for carrying out this research and writing this article was scientific but also professional interest. We believe that reviewing the state of the art of this methodology once it was already underway in our young Faculty of Medicine, could allow us to know if we were on the right track and if we should implement changes in the training of future doctors.

The primary goal of this study was to appraise available international evidence concerning to the effectiveness and usefulness of PBL methodology in undergraduate medical teaching programs. As the intention was to synthesize the scattered evidence available, the option was to conduct a scoping review. A scoping study tends to address broader topics where many different study designs might be applicable. Scoping studies may be particularly relevant to disciplines, such as medical education, in which the paucity of randomized controlled trials makes it difficult for researchers to undertake systematic reviews [ 7 , 8 ]. Even though the scoping review methodology is not widely used in medical education, it is well established for synthesizing heterogeneous research evidence [ 9 ].

The specific aims were: 1) to determine the effectiveness of PBL in academic performance (learning and retention of knowledge) in medical education; 2) to determine the effectiveness of PBL in other skills (social and communication skills, problem solving or self-learning) in medical education; 3) to know the level of satisfaction perceived by the medical students (and/or tutors) when they are taught with the PBL methodology (or when they teach in case of tutors).

This review was guided by Arksey and O’Malley’s methodological framework for conducting scoping reviews. The five main stages of the framework are: (1) identifying the research question; (2) ascertaining relevant studies; (3) determining study selection; (4) charting the data; and (5) collating, summarizing and reporting the results [ 7 ]. We reported our process according to the PRISMA Extension for Scoping Reviews [ 10 ].

Stage 1: Identifying the research question

With the goals of the study established, the four members of the research team established the research questions. The primary research question was “What is the effectiveness of PBL methodology for learning in undergraduate medicine?” and the secondary question “What is the perception and satisfaction of medical students and tutors in relation to PBL methodology?”.

Stage 2: Identifying relevant studies

After the research questions and a search strategy were defined, the searches were conducted in PubMed and Web of Science using the MeSH terms “problem-based learning” and “Medicine” (the Boolean operator “AND” was applied to the search terms). No limits were set on language, publication date, study design or country of origin. The search was carried out on 14th February 2021. Citations were uploaded to the reference manager software Mendeley Desktop (version 1.19.8) for title and abstract screening, and data characterization.

Stage 3: Study selection

The searching strategy in our scoping study generated a total of 2399 references. The literature search and screening of title, abstract and full text for suitability was performed independently by one author (JCT) based on predetermined inclusion criteria. The inclusion criteria were: 1) PBL methodology was the major research topic; 2) participants were undergraduate medical students or tutors; 3) the main outcome was academic performance (learning and knowledge retention); 4) the secondary outcomes were one of the following: social and communication skills, problem solving or self-learning and/or student/tutor satisfaction; 5) all types of studies were included including descriptive papers, qualitative, quantitative and mixed studies methods, perspectives, opinion, commentary pieces and editorials. Exclusion criteria were studies including other types of participants such as postgraduate medical students, residents and other health non-medical specialties such as pharmacy, veterinary, dentistry or nursing. Studies published in languages other than Spanish and English were also excluded. Situations in which uncertainty arose, all authors (CB, ES, RP) discussed the publication together to reach a final consensus. The outcomes of the search results and screening are presented in Fig. 1 . One-hundred and twenty-four articles met the inclusion criteria and were included in the final analysis.

Study flow PRISMA diagram. Details the review process through the different stages of the review; includes the number of records identified, included and excluded

Stage 4: Charting the data

A data extraction table was developed by the research team. Data extracted from each of the 124 publications included general publication details (year, author, and country), sample size, study population, design/methodology, main and secondary outcomes and relevant results and/or conclusions. We compiled all data into a single spreadsheet in Microsoft Excel for coding and analysis. The characteristics and the study subject of the 124 articles included in this review are summarized in Tables 1 and 2 . The detailed results of the Microsoft Excel file is also available in Additional file 1 .

Stage 5: Collating, summarizing and reporting the results

As indicated in the search strategy (Fig. 1 ) this review resulted in the inclusion of 124 publications. Publication years of the final sample ranged from 1990 to 2020, the majority of the publications (51, 41%) were identified for the years 2010–2020 and the years in which there were more publications were 2001, 2009 and 2015. Countries from the six continents were represented in this review. Most of the publications were from Asia (especially China and Saudi Arabia) and North America followed by Europe, and few studies were from Africa, Oceania and South America. The country with more publications was the United States of America ( n = 27). The most frequent designs of the selected studies were surveys or questionnaires ( n = 45) and comparative studies ( n = 48, only 16 were randomized) with traditional or lecture-based learning methodologies (in two studies the comparison was with simulation) and the most frequently measured outcomes were academic performance followed by student satisfaction (48 studies measured more than one outcome). The few studies with the highest level of scientific evidence (systematic review and meta-analysis and randomized studies) were conducted mostly in Asian countries (Tables 1 and 2 ). The study subject was specified in 81 publications finding a high variability but at the same time great representability of almost all disciplines of the medical studies.

The sample size was available in 99 publications and the median [range] of the participants was 132 [14–2061]. According to study population, there were more participants in the students’ focused studies (median 134 and range 16–2061) in comparison with the tutors’ studies (median 53 and range 14–494).

Finally, after reviewing in detail the measured outcomes (main and secondary) according to the study design (Table 2 and Additional file 1 ) we present a narrative overview and a synthesis of the main findings.

Main outcome: academic performance (learning and knowledge retention)

Seventy-one of the 124 publications had learning and/or knowledge retention as a measured outcome, most of them ( n = 45) were comparative studies with traditional or lecture-based learning and 16 were randomized. These studies were varied in their methodology, were performed in different geographic zones, and normally analyzed the experience of just one education center. Most studies ( n = 49) reported superiority of PBL in learning and knowledge acquisition [ 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 , 40 , 41 , 42 , 43 , 44 , 45 , 46 , 47 , 48 , 49 , 50 , 51 , 52 , 53 , 54 , 55 , 56 , 57 , 58 , 59 ] but there was no difference between traditional and PBL curriculums in another 19 studies [ 60 , 61 , 62 , 63 , 64 , 65 , 66 , 67 , 68 , 69 , 70 , 71 , 72 , 73 , 74 , 75 , 76 , 77 , 78 ]. Only three studies reported that PBL was less effective [ 79 , 80 , 81 ], two of them were randomized (in one case favoring simulation-based learning [ 80 ] and another favoring lectures [ 81 ]) and the remaining study was based on tutors’ opinion rather than real academic performance [ 79 ]. It is noteworthy that the four systematic reviews and meta-analysis included in this scoping review, all carried out in China, found that PBL was more effective than lecture-based learning in improving knowledge and other skills (clinical, problem-solving, self-learning and collaborative) [ 40 , 51 , 53 , 58 ]. Another relevant example of the superiority of the PBL method over the traditional method is the experience reported by Hoffman et al. from the University of Missouri-Columbia. The authors analyzed the impact of implementing the PBL methodology in its Faculty of Medicine and revealed an improvement in the academic results that lasted for over a decade [ 31 ].

Secondary outcomes

Social and communication skills.

We found five studies in this scoping review that focused on these outcomes and all of them described that a curriculum centered on PBL seems to instill more confidence in social and communication skills among students. Students perceived PBL positively for teamwork, communication skills and interpersonal relations [ 44 , 45 , 67 , 75 , 82 ].

Student satisfaction

Sixty publications analyzed student satisfaction with PBL methodology. The most frequent methodology were surveys or questionnaires (30 studies) followed by comparative studies with traditional or lecture-based methodology (19 studies, 7 of them were randomized). Almost all the studies (51) have shown that PBL is generally well-received [ 11 , 13 , 18 , 19 , 20 , 21 , 22 , 26 , 29 , 34 , 37 , 39 , 41 , 42 , 46 , 50 , 56 , 58 , 63 , 64 , 66 , 78 , 82 , 83 , 84 , 85 , 86 , 87 , 88 , 89 , 90 , 91 , 92 , 93 , 94 , 95 , 96 , 97 , 98 , 99 , 100 , 101 , 102 , 103 , 104 , 105 , 106 , 107 , 108 , 109 , 110 ] but in 9 studies the overall satisfaction scores for the PBL program were neutral [ 76 , 111 , 112 , 113 , 114 , 115 , 116 ] or negative [ 117 , 118 ]. Some factors that have been identified as key components for PBL to be successful include: a small group size, the use of scenarios of realistic cases and good management of group dynamics. Despite a mostly positive assessment of the PBL methodology by the students, there were some negative aspects that could be criticized or improved. These include unclear communication of the learning methodology, objectives and assessment method; bad management and organization of the sessions; tutors having little experience of the method; and a lack of standardization in the implementation of the method by the tutors.

Tutor satisfaction

There are only 15 publications that analyze the satisfaction of tutors, most of them surveys or questionnaires [ 85 , 88 , 92 , 98 , 108 , 110 , 119 ]. In comparison with the satisfaction of the students, here the results are more neutral [ 112 , 113 , 115 , 120 , 121 ] and even unfavorable to the PBL methodology in two publications [ 117 , 122 ]. PBL teaching was favored by tutors when the institutions train them in the subject, when there was administrative support and adequate infrastructure and coordination [ 123 ]. In some experiences, the PBL modules created an unacceptable toll of anxiety, unhappiness and strained relations.

Other skills (problem solving and self-learning)

The effectiveness of the PBL methodology has also been explored in other outcomes such as the ability to solve problems and to self-directed learning. All studies have shown that PBL is more effective than lecture-based learning in problem-solving and self-learning skills [ 18 , 24 , 40 , 48 , 67 , 75 , 93 , 104 , 124 ]. One single study found a poor accuracy of the students’ self-assessment when compared to their own performance [ 125 ]. In addition, there are studies that support PBL methodology for integration between basic and clinical sciences [ 126 ].

Finally, other publications have reported the experience of some faculties in the implementation of the PBL methodology. Different experiences have demonstrated that it is both possible and feasible to shift from a traditional curriculum to a PBL program, recognizing that PBL methodology is complex to plan and structure, needs a large number of human and material resources, requiring an immense teacher effort [ 28 , 31 , 94 , 127 , 128 , 129 , 130 , 131 , 132 , 133 ]. In addition, and despite its cost implication, a PBL curriculum can be successfully implemented in resource-constrained settings [ 134 , 135 ].

We conducted this scoping review to explore the effectiveness and satisfaction of PBL methodology for teaching in undergraduate medicine and, to our knowledge, it is the only study of its kind (systematic scoping review) that has been carried out in the last years. Similarly, Vernon et al. conducted a meta-analysis of articles published between 1970 and 1992 and their results generally supported the superiority of the PBL approach over more traditional methods of medical education [ 136 ]. PBL methodology is implemented in medical studies on the six continents but there is more experience (or at least more publications) from Asian countries and North America. Despite its apparent difficulties on implementation, a PBL curriculum can be successfully implemented in resource-constrained settings [ 134 , 135 ]. Although it is true that the few studies with the highest level of scientific evidence (randomized studies and meta-analysis) were carried out mainly in Asian countries (and some in North America and Europe), there were no significant differences in the main results according to geographical origin.

In this scoping review we have included a large number of publications that, despite their heterogeneity, tend to show favorable results for the usefulness of the PBL methodology in teaching and learning medicine. The results tend to be especially favorable to PBL methodology when it is compared with traditional or lecture-based teaching methods, but when compared with simulation it is not so clear. There are two studies that show neutral [ 71 ] or superior [ 80 ] results to simulation for the acquisition of specific clinical skills. It seems important to highlight that the four meta-analysis included in this review, which included a high number of participants, show results that are clearly favorable to the PBL methodology in terms of knowledge, clinical skills, problem-solving, self-learning and satisfaction [ 40 , 51 , 53 , 58 ].

Regarding the level of satisfaction described in the surveys or questionnaires, the overall satisfaction rate was higher in the PBL students when compared with traditional learning students. Students work in small groups, allowing and promoting teamwork and facilitating social and communication skills. As sessions are more attractive and dynamic than traditional classes, this could lead to a greater degree of motivation for learning.

These satisfaction results are not so favorable when tutors are asked and this may be due to different reasons; first, some studies are from the 90s, when the methodology was not yet fully implemented; second, the number of tutors included in these studies is low; and third, and perhaps most importantly, the complaints are not usually due to the methodology itself, but rather due to lack of administrative support, and/or work overload. PBL methodology implies more human and material resources. The lack of experience in guided self-learning by lecturers requires more training. Some teachers may not feel comfortable with the method and therefore do not apply it correctly.

Despite how effective and/or attractive the PBL methodology may seem, some (not many) authors are clearly detractors and have published opinion articles with fierce criticism to this methodology. Some of the arguments against are as follows: clinical problem solving is the wrong task for preclinical medical students, self-directed learning interpreted as self-teaching is not appropriate in undergraduate medical education, relegation to the role of facilitators is a misuse of the faculty, small-group experience is inherently variable and sometimes dysfunctional, etc. [ 137 ].

In light of the results found in our study, we believe that PBL is an adequate methodology for the training of future doctors and reinforces the idea that the PBL should have an important weight in the curriculum of our medical school. It is likely that training through PBL, the doctors of the future will not only have great knowledge but may also acquire greater capacity for communication, problem solving and self-learning, all of which are characteristics that are required in medical professionalism. For this purpose, Koh et al. analyzed the effect that PBL during medical school had on physician competencies after graduation, finding a positive effect mainly in social and cognitive dimensions [ 138 ].

Despite its defects and limitations, we must not abandon this methodology and, in any case, perhaps PBL should evolve, adapt, and improve to enhance its strengths and improve its weaknesses. It is likely that the new generations, trained in schools using new technologies and methodologies far from lectures, will feel more comfortable (either as students or as tutors) with methodologies more like PBL (small groups and work focused on problems or projects). It would be interesting to examine the implementation of technologies and even social media into PBL sessions, an issue that has been poorly explorer [ 139 ].

Limitations

Scoping reviews are not without limitations. Our review includes 124 articles from the 2399 initially identified and despite our efforts to be as comprehensive as possible, we may have missed some (probably few) articles. Even though this review includes many studies, their design is very heterogeneous, only a few include a large sample size and high scientific evidence methodology. Furthermore, most are single-center experiences and there are no large multi-center studies. Finally, the frequency of the PBL sessions (from once or twice a year to the whole curriculum) was not considered, in part, because most of the revised studies did not specify this information. This factor could affect the efficiency of PBL and the perceptions of students and tutors about PBL. However, the adoption of a scoping review methodology was effective in terms of summarizing the research findings, identifying limitations in studies’ methodologies and findings and provided a more rigorous vision of the international state of the art.

Conclusions

This systematic scoping review provides a broad overview of the efficacy of PBL methodology in undergraduate medicine teaching from different countries and institutions. PBL is not a new teaching method given that it has already been 50 years since it was implemented in medicine courses. It is a method that shifts the leading role from teachers to students and is based on guided self-learning. If it is applied properly, the degree of satisfaction is high, especially for students. PBL is more effective than traditional methods (based mainly on lectures) at improving social and communication skills, problem-solving and self-learning skills, and has no worse results (and in many studies better results) in relation to academic performance. Despite that, its use is not universally widespread, probably because it requires greater human resources and continuous training for its implementation. In any case, more comparative and randomized studies and/or other systematic reviews and meta-analysis are required to determine which educational strategies could be most suitable for the training of future doctors.

Abbreviations

Problem-based learning

Case-based learning

Team-based learning

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21 Problem Solving

Problem Representation: The Gestalt Legacy

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

The Two Legacies Converge

Relevance of the Gestalt Ideas to the Solution of Search Problems

Problem Isomorphs

Expertise and Its Development

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Connections between Reading Comprehension and Word-Problem Solving via Oral Language Comprehension: Implications for Comorbid Learning Disabilities

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Effectiveness of problem-based learning methodology in undergraduate medical education: a scoping review

Stage 1: Identifying the research question

Stage 2: Identifying relevant studies

Stage 3: Study selection

Stage 4: Charting the data

Stage 5: Collating, summarizing and reporting the results

Main outcome: academic performance (learning and knowledge retention)

Secondary outcomes

Student satisfaction

Tutor satisfaction

Other skills (problem solving and self-learning)

Limitations

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