researches on problem solving

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Problem solving articles from across Nature Portfolio

Problem solving is the mental process of analyzing a situation, learning what options are available, and then choosing the alternative that will result in the desired outcome or some other selected goal.

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21 Problem Solving

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Problem Representation: The Gestalt Legacy

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

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

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

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

A solution to the nine-dot problem.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Two Legacies Converge

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

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

Relevance of the Gestalt Ideas to the Solution of Search Problems

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

Problem Isomorphs

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

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

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

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

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

Expertise and Its Development

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

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

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

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

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

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

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

Relevance of Search to Insight Solutions

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

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

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

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

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

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

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

Effects of Perception and Knowledge in Problem Solving in Academic Disciplines

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

The Role of Visual Perception

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

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

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

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

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

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

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

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

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

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

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

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

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

The Role of Background Knowledge

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

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

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

Conclusions and Future Directions

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

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

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

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Cognitive Predictors of Everyday Problem Solving across the Lifespan

Center for Vital Longevity, School of Behavioral and Brain Sciences, University of Texas at Dallas

Christopher Hertzog

School of Psychology, Georgia Institute of Technology

Denise C. Park

An important aspect of successful aging is maintaining the ability to solve everyday problems encountered in daily life. The limited evidence today suggests that everyday problem solving ability increases from young adulthood to middle age, but decreases in older age.

The present study examined age differences in the relative contributions of fluid and crystallized abilities to solving problems on the Everyday Problems Test (EPT; [ 1 ]). We hypothesized that due to diminishing fluid resources available with advanced age, crystallized knowledge would become increasingly important in predicting everyday problem solving with greater age.

Two hundred and twenty-one healthy adults from the Dallas Lifespan Brain Study, aged 24–93 years, completed a cognitive battery that included measures of fluid ability (i.e., processing speed, working memory, inductive reasoning) and crystallized ability (i.e., multiple measures of vocabulary). These measures were used to predict performance on the Everyday Problems Test.

Everyday problem solving showed an increase in performance from young to early middle age, with performance beginning to decrease at about age of fifty. As hypothesized, fluid ability was the primary predictor of performance on everyday problem solving for young adults, but with increasing age, crystallized ability became the dominant predictor.

This study provides evidence that everyday problem solving ability differs with age, and, more importantly, that the processes underlying it differ with age as well. The findings indicate that older adults increasingly rely on knowledge to support everyday problem solving, whereas young adults rely almost exclusively on fluid intelligence.

An important aspect of successful aging is maintaining the ability to solve everyday problems encountered in daily life. Instrumental Activities of Daily Living (IADLs) represent one important domain of these problems. IADLs are complex behaviors required for independent management of one’s life, including adherence to complex medical regimens, ability to use increasingly complex communication devices, and management of financial resources [ 2 ]. Other everyday problems involve situations where a conflict is present or a goal cannot be reached without some inferential reasoning [ 3 ]. Cross-sectional data show that the practical ability to solve everyday problems increases from young adulthood until middle age [ 4 – 6 ], but that older age is characterized by diminishing performance[ 5 – 8 ].

One reason for peak performance during middle adulthood in everyday problem solving may be that middle-aged adults have the ideal balance of fluid and crystallized resources needed for everyday problem solving. Crystallized ability represents accumulated experience and knowledge of the world, and is typically measured by vocabulary and general knowledge. It does not decline, and may even grow, well into late adulthood [ 9 , 10 ]. In contrast, fluid ability – the ability to abstract and perform efficient mental operations – shows consistent age-related decline beginning in the 20s [ 10 ], but nevertheless, performance is still relatively high in middle-aged adults [ 11 ]. Fluid ability is best measured by different types of inductive and deductive reasoning tasks, and is closely related to the construct of processing resources [ 12 ] as operationalized by working memory [ 13 ].

Previous studies have found fluid ability to be an important predictor of everyday problem solving in healthy older adults [ 7 , 8 , 14 – 18 ]. Gross et al. [ 16 ] found that although memory, reasoning and processing speed were all significant predictors for everyday functioning and everyday problem solving, inductive reasoning (measured by Letter Series, Word Series, and Letter Sets tasks) independently accounted for the most variance in everyday functioning (measured by Everyday Problems Test, the Observed Tasks of Daily Living, and the Timed Instrumental Activities of Daily Living test). Willis et al. [ 19 ] also showed that older adults who underwent reasoning training showed less functional decline in IADLs than an untrained control group, indicating the importance of reasoning for everyday problem solving.

Everyday problem solving is also related to other aspects of fluid ability that decline with age, especially working memory and processing speed. Importantly, age-related decreases in working memory, using traditional measures that include Reading Span, Computation Span, and Operation Span tasks, have been strongly associated with lower performance on everyday problem solving tasks [ 14 , 20 ]. Age-related slowing in processing speed has also been associated with decreased everyday problem solving [ 8 , 21 ]. Rebok and colleagues [ 22 ] reported evidence that older adults who had extensive training on processing speed in the ACTIVE trial reported less difficulty in performing IADL’s ten years after training, suggesting that such an intervention confers protection in later life. In sum, there is little doubt that fluid ability plays an important role in everyday problem solving.

What is less certain is the role that crystallized ability and knowledge play in everyday problem solving. There have been a few studies that examined the joint contributions of both fluid ability and crystallized ability to everyday problem solving and all suggest an important role of fluid ability [ 7 , 14 , 15 , 17 , 21 , 23 , 24 ]. However, the importance of crystallized ability in everyday problem solving seems to be different depending on the age range of the sample included in the study. Three studies in older adults all found that both fluid and crystallized ability played very important roles in everyday problem solving. Diehl and colleagues [ 15 ] used structural equation modeling and found that both fluid and crystallized abilities had significant paths to everyday problem solving, measured by Observed Tasks of Daily Living (OTDL). In addition, the effects of memory and speed on OTDL were mediated by crystallized ability, indexed by vocabulary. Burton [ 21 ] used hierarchical regression and found that verbal ability, measured by verbal fluency and vocabulary tasks, predicted performance in Everyday Problems Test beyond the effect of fluid ability and demographic variables (e.g., age, education). Allaire and Marsiske [ 14 ] also found relationship between vocabulary and some domains of everyday problem solving, measured by Everyday Cognition Battery. However, studies including middle-aged adults yielded somewhat different conclusions on the role of crystallized ability in the relation to everyday problem solving. Kimbler [ 7 ] studied healthy middle-aged and older adults (age 50 to 92) and found no relationship between performance in vocabulary and Everyday Problems Test. Thornton and colleagues [ 24 ] reported that, although in a sample of healthy adults and chronic disease patients, crystallized ability (measured by ETS vocabulary) mediated age effects on performance in Everyday Problems Test, the relationship was not significant when the analysis was limited only to healthy adults.

These findings suggest that there is a discrepancy in the role of crystallized ability in predicting everyday problem solving across the adulthood lifespan. A potential explaination is that there may be an age-related shift in the contribution of fluid versus crystallized abilities in solving everyday problems. This shift can only be detected by using a lifespan sample with broader age range. We are aware of only two adult lifespan studies on the cognitive predictors of performance in everyday problem solving [ 6 , 23 ]. In both studies, the correlation of fluid and crystallized cognitive predictors to everyday problem solving (practical problem solving in [ 6 ]) was significant. However, when the effects of age and education on everyday problem solving were controlled, neither predictor was significant [ 6 ], limiting the understanding of their respective contribution. Moreover, we were unable to find any studies that include young, middle-age and older adults that examined how age affects the contribution of cognitive predictors to everyday problem solving. Therefore, the present study focuses on two important and unresolved issues. First, what are the strength of the contributions of fluid and crystallized abilities to everyday problem solving? And second, do these contributions shift in importance as a function of age?

Park has argued that older adults maintain performance on many cognitive tasks by increasingly relying on knowledge and experience to compensate for declines in fluid abilities [ 25 ]. Congruent with this perspective, Baltes also suggested that crystallized ability can compensate, to some extent, for age-related declines in processing efficiency with advanced age [ 26 ]. In support of this theorizing, Hedden et al. reported that performance on a verbal memory task was mediated by fluid abilities in young and middle-aged adults, but that older adults relied more on vocabulary (an index of crystallized ability) for optimal performance [ 27 ]. In the present study, we determine whether such an age-related shift occurs for everyday problem solving in an adult lifespan sample. We predicted that young adults who are rich in cognitive resources such as speed, working memory and reasoning would rely on fluid processing for success; however, as age increased, crystallized ability would play an increasingly important role in everyday problem solving.

Participants

A total of 221 healthy adults from the Dallas Lifespan Brain Study (DLBS; 148 women, 73 men, age range: 24–93 years, Mini-Mental Status Examination scores ≥ 26, mean = 28.37) were recruited locally from the community. All participants were right-handed with normal or corrected to normal vision. Participants with any of following conditions were excluded: history of major psychiatric or neurological disorder, history of prescription drug abuse/illegal drug use, and/or any head trauma. Participants were compensated fifteen dollars per hour for their participation. They completed two 2.5-hour sessions that are described below.

Each participant completed a battery of cognitive tests as well as the Everyday Problems Test (EPT) [ 1 ]. This comprehensive battery included both paper-and-pencil and computerized tasks. The cognitive constructs assessed and the tasks associated with each construct included the following:

Processing speed was measured by Digit Comparison [ 28 ], WAIS-III Digit Symbol [ 29 ] and Pattern Comparison task taken from NIH Toolbox Cognition Battery (NIHTB-CB) [ 30 ].

Working memory was measured by the Spatial Working Memory (SWM) task of Cambridge Neuropsychological Test Automated Battery (CANTAB) [ 31 ], WAIS-III Letter-Number Sequencing [ 29 ] and NIHTB-CB List Sorting [ 30 ].

Inductive reasoning was measured by Educational Testing Service (ETS) Letter Sets [ 32 ], Raven’s Progressive Matrices [ 33 ], and Stockings of Cambridge (SOC) of CANTAB [ 31 ].

Crystallized ability was measured by NIHTB-CB Picture Vocabulary [ 30 ], NIHTB-CB Oral Reading Recognition [ 30 ] and the ETS Advanced Vocabulary Scale [ 32 ]. Although the ETS Vocabulary task was timed, we made sure that no participants failed to complete the task because of the time limit so the performance on this task was not affected by their speed.

Everyday problem solving ability was measured by the Everyday Problems Test (EPT) [ 1 ]. It is a paper-pencil task that has 42 questions, which assesses the ability to solve tasks that are important to live independently in our society. The EPT is comprised of seven scales that include problems from domains of Health/Medications, Meal Preparation/Nutrition, Phone Usage, Consumer/Shopping, Financial Management, Household Management, and Transportation. For each of these seven scales, participants are presented with three sample stimuli (e.g., prescription drug label, bus schedule, catalog order form) and two questions about each stimulus. Figure 1 is an example of one EPT stimulus with two questions based on the stimulus. The performance on this task is measured as the total number of correct answers to the 42 questions. Compared to other neuropsychological tasks that assessed traditional problem solving ability, everyday problems test (EPT) was designed to be a better indicator of problem solving performance in real-life. Schmitter-Edgecombe and colleagues [ 34 ] found that EPT was strongly associated with directly observed everyday functioning performance in real world, and therefore considered to be a valid and useful measure for assessing everyday functioning in cognitively healthy older population.

a . Age-related differences in fluid ability. Fluid ability is comprised of the measures on processing speed, working memory and inductive reasoning. b . Age-related differences in crystallized ability. Crystallized ability is comprised of ETS Advanced Vocabulary Scale, NIHTB-CB Picture Vocabulary and NIHTB-CB Oral Reading Recognition. Figure 3c . Age-related differences in everyday problem solving. Everyday problem solving is measured by number of correct answers on the Everyday Problems Test (EPT).

Data Analyses

Altogether, there were 13 tasks subjected to analyses: three measures each for processing speed, working memory, inductive reasoning, crystallized ability, and a single measure of everyday problem solving. We created standard scores for the 12 cognitive measures that were used for further analyses. A confirmatory factor analyses (CFA) validated the expected factor structure of cognitive measures, Χ 2 (60) = 147.941, p < .001, CFI = .953, RMSEA = .081, SRMR = .076 (see Figure 2 ). The standardized scores for each crystallized and fluid test were averaged to produce composite crystallized and fluid scores in a standard-score (z-score) metric in the aggregate cross-sectional sample.

Confirmatory Factor Analysis of cognitive tasks, after controlling for age. Χ 2 (60)= 147.941, p < .001, CFI = .953, RMSEA = .081, SRMR = .076.

To test the hypothesis that crystallized intelligence would be a more potent predictor of everyday problem solving for older adults, relative to earlier ages, we conducted a hierarchical moderated regression analysis with age, fluid ability, and crystallized ability as predictors, using product variables to capture interactions. Prior to evaluating the interaction effects, we introduced quadratic age effects to test for possible curvilinearity in the relation of age and the variables to the EPT score. This approach was taken because curvilinear age relations were expected in abilities [ 35 ] and everyday problem solving performance, and because methodological studies have shown that failing to account for curvilinear relations of predictors to dependent variables in the context of moderated regression can create spurious product variable effects that are an artifact of curvilinear relations of both predictors to the dependent variable [ 36 ]. To foreshadow our results, we did detect curvilinear relations of age and abilities to EPS tests, requiring that moderated regression tests for age × ability interaction effects include quadratic terms for each predictor variable.

Linear predictor terms were first centered at the sample mean, and then squared predictors were computed to reduce collinearity issues in the multiple regression. Significant moderated regression effects were decomposed by computing simple slopes at the mean and at ± 1 SD of the predictor variables.

To further understand age differences in the predictive utility of fluid and crystallized abilities for everyday problem solving, we used bootstrapping to examine the regression coefficients for each of the three age groups (young, middle-aged, older). Finally, to assess the stability of the observed effects across individual problem solving domains, we examined the contributions of fluid and crystallized abilitites for each of the seven domains in everyday problem solving for young, middle-aged, and older adults.

Demographic Data and Age-related Differences

Demographic data are presented in Table 1 , broken down by 3 age groups (Young: 24–49 years old; Middle: 50–69 years old; Old: 70–93 years old). The three age groups differed on years of formal education, F (2, 218) = 6.16, p =.002, with young adults having somewhat higher levels than the other two age groups. Means and standard deviations of cognitive measures and EPT scores are also presented in Table 1 . For descriptive purposes, we presented age effects associated with fluid ability, crystallized ability and everyday problem solving score in scatter plots ( Figure 3 ). Figure 3a portrays a significant linear age-related decrease in fluid ability, R 2 = .626, R 2 adjusted = 625, F(1, 212) = 355.312, p < .001, and the quadratic relationship was also statistically significant, R 2 = .64, R 2 adjusted = .637, F(2, 211) = 187.72, p < .001. In contrast, crystallized ability ( Figure 3b ) did not have a significant linear relationship with age, p = .628. However, there was a significant quadratic relationship between crystallized ability and age, R 2 = .038, R 2 adjusted = .029, F(2, 217) = 4.258, p = .015, with increasing performance until about age 59. We also examined both linear and quadratic relationships between everyday problem solving ability and age. While the simple linear relationship showed significance, R 2 = .237, R 2 adjusted = .234, F(1, 219) = 68.091, p < .001, adding age 2 significantly improved the model, ΔR 2 = .105, ΔF = 34.810, p <.001, suggesting a quadratic relation with age was a better fit for everyday problem solving ability ( Figure 3c ), R 2 = .342, R 2 adjusted = .336, F (2, 218) = 56.707, p < .001, with the peak performance at 47.2 years of age.

Demographic and descriptive data.

Cognitive Predictors Across the Lifespan

We used hierarchical multiple regression to examine the role that fluid and crystallized abilities play in solving everyday problems. In the first model, we included years of education and linear and quadratic components for age. Then in the second model, we added fluid ability and crystallized ability as cognitive predictors. In the third model, we included quadratic components (crystallized 2 and fluid 2 ) to examine if there was a curvilinear relationship between cognitive predictors and everyday problem solving. In the fourth model, we added interactions among fluid ability, crystallized ability and age. Each of aforementioned steps improved the fit of the overall model significantly ( Table 2 ). We also examined a further model that included interactions between cognitive ability and age 2 , and found that it did not improve the model significantly. Therefore, the fourth model was chosen as the final model depicting the relationship between cognitive predictors and everyday problem solving across the lifespan. As shown in Table 2 , Model 4 explained a substantial amount of variance in everyday problem solving, R 2 = .683, R 2 Adjusted = 666. There was a main effect of age, age 2 , fluid ability, and crystallized ability on everyday problem solving. Although the quadratic terms of fluid ability and crystallized ability were not each statistically significant in the final model, adding quadratic terms of these predictors significantly improved the fit of the model. The partial residual plots of crystallized ability ( Figure 4a ) and fluid ability ( Figure 4b ) showed that these two predictors both evidenced a similar curvilinear pattern to everyday problem solving. Curvilinearity occurred because for lower ability participants compared to those of higher ability, cognitive ability had a stronger relationship to everyday problem solving.

a . Partial residual plot of crystallized ability. b . Partial residual plot of fluid ability. For both cognitive predictors, the effect of crystallized and fluid ability follows a similar curvilinear pattern regardless of age and the other cognitive level: for people who have lower cognitive ability, the level of cognitive ability has a strong effect on everyday problem solving, while for people who have high cognitive ability, higher cognitive ability does not affect everyday problem solving as much.

Hierarchical Multiple Regression.

Critically, we also found a significant Age × Crystallized ability interaction, b = 0.046, SEb = 0.016, t (201) = 2.943, β = .152, p = .004, 95% CI = [0.015, 0.076], indicating the relationship between crystallized ability and everyday problem solving differed across the lifespan. In order to better interpret the significant interaction, simple slopes (displayed in Figure 5 ) for the relationship between crystallized ability and everyday problem solving were tested for younger age (−1 SD below the mean), middle age (mean), and older age (+1 SD above the mean). Simple slope tests showed that the relationship of crystallized ability to everyday problem solving at a younger age was not significant, b = 0.708, SEb = 0.433, t (201) = 1.636, β = .125, p = .103, 95% CI = [−0.146, 1.562]. However, both the middle age model, b = 1.576, SEb = 0.292, t (201) = 5.391, β = .279, p < .001, 95% CI = [0.999, 2.152], and the older age model, b = 2.44, SEb = 0.397, t (201) = 6.141, β = .432, p < .001, 95% CI = [1.656, 3.223], revealed a significant positive association between crystallized ability and everyday problem solving. We then tested the difference between regression coefficients across models, and found that the effect of crystallized ability was stronger for both old ( z = −3.027, p = .001) and middle age ( z = −1.719, p = .043) compared to young, and that the effect was even stronger for the old age compared to middle, ( z = −1.753, p = .04), suggesting that crystallized ability played a continuously increasingly important role in solving everyday problems as age increased. Note that the interaction between fluid and crystallized ability was not significant ( p = .351), suggesting that the contribution of crystallized ability did not change across people with different fluid ability, after age-related effects taken into account.

Simple slopes of Age × Crystallized ability. Simple slope was not significantly different from 0 at Age = 40 (1SD below mean), but was significant at Age = 59 (mean age) and Age = 78 (1SD above mean). Based on comparison using z-tests, the effect of crystallized ability was stronger at older age ( z = −3.027, p = .001) and middle age ( z = −1.719, p = .043), than at a younger age, and the effect was even stronger at a older age than middle, ( z = −1.753, p = .04).

Comparing Cognitive Predictors in Three Age-groups

To further examine which cognitive predictor – fluid or crystallized ability – was more important for everyday problem solving at different stages of the lifespan, we generated bootstrapped standard errors for regression coefficients in three age subgroups: younger adults (24–49 years old), middle-aged adults (50–69 years old), and older adults (70–93 years old). In each multiple regression, the predictor variables were age, fluid ability, crystallized ability, fluid 2 , crystallized 2 and the fluid × crystallized interaction. This model was derived from Model 4 used for the whole sample with first order age-related effects removed since this analysis was on each age group. We generated 95% confidence intervals (CI) using bias-corrected and accelerated (BCa) bootstrap (with 1000 iterations in each group) as presented in Table 3 . We then compared the BCa CI using a conservative rule by examining the overlap of confidence intervals [ 37 ]. Put simply, the rule assesses whether the 95% confidence intervals have less than 50% proportion overlap, expressed as a proportion of average margin of error. If the result is affirmative, the two estimates are significantly different ( p < .05). As shown in Figure 6 , for the young group, the lower end of 95% CI of the crystallized ability parameter was below zero, confirming its non-significance and that only the fluid ability value was predictive, as we found in simple slope analysis. For the middle age, the 95% CIs of fluid and crystallized abilities overlapped more than 50%, suggesting that both were predictive but not significantly different in middle-aged adults. Finally, for the older group, the predictive utility of crystallized ability was significantly larger than fluid ability, with the proportion overlap = 42.8%, p < .05. Hence, in middle-aged and older adults, everyday problem solving was associated with both fluid and crystallized abilities. Importantly for older adults, crystallized ability was a significantly stronger predictor compared to fluid ability (see Figure 6 ).

95% BCa CI for fluid and crystallized regression coefficients. In older adults, everyday problem solving was predicted more by crystallized ability than fluid ability, proportion overlap = 42.8%, p <.05.

Regression coefficient estimates and 95% BCa CI in three age groups.

We also note that we found no evidence for a Fluid × Crystallized interaction within any age group. The absence of the interaction suggests that fluid and crystallized ability made independent contributions to everyday problem solving, regardless of level of performance on either ability.

In a final analysis, we assessed the stability of the effects of fluid and crystallized ability for each of the seven problem-solving domains, within each age group, using the same bootstrapping approach. The main finding was that for older adults, crystallized ability played an important role for all EPT domains except meal preparation , which was marginally significant. In addition, fluid ability was significant for shopping, finance and meal preparation in older adults (see Table 4 ). Table 4 also shows that for young adults, fluid ability was significant for finance, household and transportation , and for finance, medication and transportation in middle-aged adults. Crystallized ability played no significant role for young adults, and significantly predicted only shopping in middle age.

Regression coefficient estimates and 95% BCa CI for seven EPT domains.

The main goal of this study was to understand how fluid and crystallized ability differ across the lifespan in predicting everyday problem solving. We hypothesized that due to diminished fluid resources with age, crystallized knowledge would become increasingly important in predicting everyday problem solving as a function of age. Congruent with this hypothesis, crystallized ability (measured by verbal knowledge in this study) played a more important role in predicting everyday problem solving as age increased. In contrast, fluid ability (measured by speed, working memory, and inductive reasoning) consistently explained variance for all age groups. This pattern of findings suggests that older adults are relying more on crystallized knowledge to solve everyday problems, whereas young adults rely more heavily on the efficiency of basic cognitive-mechanisms (e.g., processing speed, working memory, inductive reasoning) that comprise fluid ability.

Past studies have been inconclusive about the relative roles of crystallized versus fluid abilities in everyday problem solving at different ages, because none that have examined this issue have included a lifespan sample. The inclusion of the entire adult lifespan was an important feature of the present study, as it allowed us to begin to clarify when in the lifespan crystallized knowledge assumes importance in everyday problem solving. We began to observe a small contribution of crystallized ability to everyday problem solving in middle age, with a large contribution at older ages. The present findings provide clear evidence for the importance of including middle-aged samples in studies.

We also note that the present findings replicate a pattern reported by Hedden et al. [ 27 ] for a very different task—a verbal cued recall task that required participants to memorize associations between paired cues and target words. Hedden et al [ 27 ] used crystallized and fluid ability to predict performance on the verbal recall task. Just as reported in the present study, they found that crystallized ability (vocabulary scores) explained more variance for older compared to middle-aged and young adults. The similarity of the findings for these two very different tasks suggests that increasing reliance on crystallized ability may be a general characteristic of aging. Buttressing this conclusion, was the finding that crystallized ability accounted for significant variance in older adults in six of the seven EPT domains, suggesting that the breadth of the effect was reliable across domains. Moreover, the crystallized ability effect was nearly absent in the young and middle-aged adults, with only one significant effect for shopping in the middle-aged.

The notion that age differentially affects the type of cognitive ability drawn upon to perform everyday cognitive tasks has not received much attention in the literature. The present findings suggest that crystallized knowledge may help older adults maintain cognitive function in the face of declining fluid ability. Other studies of problem-solving support this interpretation. For example, older adults actually showed better problem-solving abilities than young and middle-aged adults when they were presented with problems associated with social conflict and interpersonal conflict. The solution to these types of problems rely more on wisdom and a broad range of social experiences rather than fluid ability [ 38 ]. Similarly, there is evidence that older adults develop adaptive, experience-based heuristics for solving everyday problems and make decisions that minimize the need to rely on fluid reasoning [ 39 ]. Conversely, there are also domains where crystallized ability makes a scant contribution, even for older adults. We suggest that these would be domains that require extensive on-line processing, such as constantly switching and updating information of different ingredients and procedures when cooking.

It is also important to recognize that everyday problem solving ability is a crucial skill that greatly affects older adults’ life quality, but few studies have examined the predictive utility of respondent-based, laboratory problem solivng tasks (such as the EPT) in the real world. In support of the use of such laboratory measures, there is a small body of evidence suggesting that the EPT explains substantial variance in every day functioning [ 17 , 34 , 40 ]; but much more research is needed. Moreover, the EPT consists of sets of questions that address well defined, but relatively narrow everyday problems. Real world problems are typically more complex, are more open-ended (ill-defined), and are comprised of many smaller interrelated problems that require different aspects of knowledge, skills and abilities. Thus, the EPT may not adequately mirror the complexity of real world problems. Additional investigation of ability predictors of everyday problem solving tasks would help to address this concern.

A limitation of this study is that crystallized ability was measured by vocabulary tasks, which have been traditionally considered as a proxy of knowledge and experience in cognitive psychology studies and everyday problem solving research. However, we acknowledge that a broader assessment of crystallized ability would incorporate experience and other types of world knowledge. Future research with more comprehensive assessment of knowledge and experience beyond measures of vocabulary may help to understand the individual differences in people’s utilization of cognition in solving everyday problems. One option might be to assess expertise and familiarity participants have in each problem solving domain in an effort to understand how life experiences asset problem solving. Similar strategies could be adapted to different problem solving paradigms.

We also recognize that it would be ideal to have longitudinal data on both cognitive and everyday problem solving so that the changing relationship between cognitive measures and everyday performance could be assessed as people grow and age. Cross-sectional designs are vulnerable to cohort differences and age × selection confounds. Finally, the compensatory role of crystallized ability may be maximized in high-functioning samples of older adults. Participants in this study were well-educated (mean years of education = 16.6); individuals with lower levels of educational attainment may not show the same degree of compensatory benefit (although we found no evidence of fluid × crystallized interactions in predicting EPS performance). It would therefore be useful to evaluate these relationships in a more representative sample of the population that included low-education individuals.

In conclusion, the present study suggests that young adults may solve everyday problems based on cognitive resources and mechanisms that are traditionally associated with effective problem solving. However, crystallized knowledge becomes a more predominant influence on everyday problem solving in older adults.

Example questions of the Everyday Problems Test.

Acknowledgments

This work was supported by National Institute on Aging at the National Institutes of Health (grant number 5R37AG006265-29 to D. C. P.).

Contributor Information

Xi Chen, Center for Vital Longevity, School of Behavioral and Brain Sciences, University of Texas at Dallas.

Christopher Hertzog, School of Psychology, Georgia Institute of Technology.

Denise C. Park, Center for Vital Longevity, School of Behavioral and Brain Sciences, University of Texas at Dallas.

• Open access
• Published: 05 February 2018

The role of problem solving ability on innovative behavior and opportunity recognition in university students

• Ji Young Kim 1 ,
• Dae Soo Choi 1 ,
• Chang-Soo Sung 1 &
• Joo Y. Park 2  

Journal of Open Innovation: Technology, Market, and Complexity volume  4 , Article number:  4 ( 2018 ) Cite this article

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Universities engage in entrepreneurship education to increase social value creation, through students’ new opportunities recognition. However, there are not enough of empirical researches on whether the current entrepreneurship education can be differentiated from other curriculum to improve the opportunity recognition process. This study argues that it is very important for cognitive abilities to be manifested as behavior when students in university are new opportunities recognition. For this purpose, the relationship between problem solving ability, innovation behavior, and opportunity perception was verified empirically. This study was conducted on 203 students who took entrepreneurship education courses at Korean universities. The results of this study showed that problem solving ability positively influenced innovation behavior and opportunity perception. Innovation behavior was identified as a key parameter that partially mediated the relationship between problem solving ability and innovation behavior. The implication of this study is to prove the relationship between individual ‘s problem - solving ability considering the characteristics of education in Korea and the opportunity through innovative behavior and various learning strategies to help entrepreneurship education to design better courses for the future It has important implications for strategic pedagogy that can enhance behavioral elements in development.

It is the new opportunity recognition that all firms focus on for a new economic paradigm (Ancona and Caldwell, 1992 ). Recognizing high opportunities can significantly improve profit, growth, and / or competitive positioning. And this new opportunity leads to innovation. From a conceptual point of view, research is continuing on the question of ‘what is opportunity’ and ‘where is opportunity’ (Gartner and Carter, 2003 ; Venkataraman & Sarasvathy, 2001 ). Research on the discovery and realization of new opportunities is a very important research area that suggests how to discover and utilize creative opportunities that create new value and profit for pre-service workers, and is the ultimate goal of entrepreneurship education. (Kim et al., 2016 ). Particularly, there is a lot of debate about the relationship between opportunity perception and personal characteristics. Despite many arguments, however, research on individual characteristics and opportunity perceptions is still insufficient, and a unified opinion has not been created due to differences between cognitive and behavioral theories (Ko & Butler, 2003 ). In particular, there is much controversy over the relationship between opportunity recognition and personal traits, and research has been continuing to demonstrate that organizational learning in organizations can influence opportunity recognition (Shane & Venkataraman, 2000 ). In particular, learning enhances cognitive ability, which is an opportunity that leads to opportunity recognition through the manifestation of behavior (Lumpkin and Dess, 2004 ). Many studies have also demonstrated the difference in behavior that successful entrepreneurs see as contributing to their ability to recognize opportunities and create innovative business ideas (Dyer et al., 2008 ; Kim et al., 2017 ). For example, Alvarez and Barney ( 2005 ) argue for mountain climbing and mountain building to understand the implications of entrepreneurial behavior in relation to these theories. In other words, a new opportunity for entrepreneurs is not a passive case that is generally found and climbed by climbers such as mountains, but rather by the actions of entrepreneurs, creating competition for the market, creating another market, Is the same. Therefore, in order for a person’s cognitive ability to recognize a new opportunity, it must focus on manifesting an action that can realize an innovative idea. In this regard, Kanter ( 1988 ) proved the relationship between new opportunity recognition and those with innovative tendencies and regarded this new opportunity recognition as innovation activity through organizational education. Scott and Bruce ( 1994 ) have integrated a number of research flows into innovation pioneers to develop and test individual innovative behavioral models. In particular, they argued that individual problem-solving styles are very important to induce innovative behavior. Although there are a number of studies on problem solving ability, innovation behavior, and new opportunities, most of the opportunistic researches have been conducted in organizational units of companies. Is still insufficient. Furthermore, unified opinions were not created due to differences between cognitive theory and behavioral theory (Ko & Butler, 2003 ). It is also true that the effects of entrepreneurship education in university have not been studied empirically because they are mainly focused on promoting cognitive ability and applied to various kinds of teaching methods.

This study argues that it is very important for cognitive abilities to be manifested as behavior that. “Through” courses, In other words, it is very important to induce students to act through ‘learning through process’ learning through behavioral learning by providing students with some (virtual or real) business to start doing some of the actions of the entrepreneur. When students in university are new opportunity recognition. Especially, entrepreneurship education, which ultimately focuses on whether it is a new opportunity, is very important to induce behavior through behavior learning beyond the cognitive ability as the general education curriculum. Particularly, innovative behaviors that create and realize innovative ideas are very important for new opportunity recognition (Paine & Organ, 2000 ).In order to achieve this, various kinds of teaching methods are being pursued in the university, but studies on the effectiveness of behavioral learning have not been studied yet. In this study, we are based on team-based learning among various teaching methods for behavior learning that leads to innovative behaviors. Team learning instructional activity sequence designed by Michaelsen and Sweet ( 2008 ), the most well known team-based learning in entrepreneurship education as in class-primarily group work and outside class-primarily individual work. In this way, we demonstrate empirically the relationship between individual problem solving ability and opportunity through innovative behavior, and develop a variety of learning strategies that help entrepreneurship education to design better courses for the future. I would like to point out some implications for strategic pedagogy to increase the element.

The paper proceeds as follows: Initially we present the theory of innovative behavior with individual problem-solving ability, innovative behavior and opportunity recognition. We develop hypotheses to confirm its basic predictions in the student context. Finally, we link the findings with the wider social effect of entrepreneurship literature and highlight the theoretical contributions and practical implications.

Theoretical background

‘opportunity recognition’ as entrepreneurship education unit of analysis.

A commonly focused analysis in entrepreneurship research over the last 30 years has been the ‘opportunity’, most simply defined as any situation in which new products or services can be development of production (Casson, 1982 ; Shane & Venkataraman, 2000 ; Venkataraman, 1997 ). The definition of opportunity recognition is defined in many ways, but opportunity is defined as a perceived means of generating economic value (ie, profit) that has not been exploited previously and is not currently exploited by others. If opportunity is defined in this way, opportunity recognition can be defined as a cognitive process (or process) that concludes that an individual has identified an opportunity (Baron and Ensley, 2006 ). Kirzner ( 1997 ) pointed out that the distribution of information in society affects the discovery of entrepreneurial opportunities and that only a few individuals can identify and recognize specific opportunities in the market. The process of finding opportunities also depends on the individual’s ability and discovery (Stevenson & Gumpert, 1985 ). For example, people may miss opportunities due to a lack of cognitive ability to change external environments (Stevenson & Gumpert, 1985 ). Only those who recognize and value the existence of opportunity can benefit from new opportunities (Ardichvili et al., 2003a , b ; Shane & Venkataraman, 2000 ). Opportunity recognition is an early step in transforming value into a business concept that creates value and generates revenue and distinguishes it from the aggressive stages of detailed assessment and development of recognized opportunities and potential economic value. The focus of the new venture business is also an innovative opportunity to create new opportunities rather than merely expanding or repeating existing business models (Gaglio & Katz, 2001 ). As a result, universities need to make use of a variety of initiatives to educate students to recognize innovative opportunities. Therefore, entrepreneurship education aimed at a new opportunity recognition should be able to provide learning opportunities based on various theories of favorable conditions for new business creation and the types of traits required for new ventures (Garavan & O’Cinne’ide, 1994 ).

Based on these considerations, we also define opportunity recognition as the formation of beliefs that can be translated into actions in order to understand the signals of change (new information on new conditions) and respond to these changes.

Problem-solving ability and innovative behavior of education for students

Problem-solving abilities have been proven to be one of the key factors for success in organizations and personal careers (Anderson & Anderson 1995 ). Through decades of research data, organizations and schools have studied factors that affect improvement. Problem-solving abilities are defined in a number of prior studies, and problem-solving abilities in a volatile and sophisticated knowledge- and technology-based industry are an important ability to drive innovation and sustainable growth and development in the industry. Table  1 show the concept of problem solving ability defined in previous research.

There have been a number of previous studies, emphasis has been placed on the importance and meaning of rational problem-solving processes in order to improve problem-solving abilities, and research has focused on individual problem solving styles (Woodman et al., 1993 ; Scott & Bruce, 1994 ). According to the personal innovation behavior model of Scott and Bruce ( 1994 ), climate has shown individual innovative behavior as a result of individuals signaling the organization’s expectations of behavior and the potential consequences of action. Innovative organizations are, last but not least, equipment, facilities and time, including the direction of creativity and innovative change (Kanter, 1983 ; Siegel & Kaemmerer, 1978 ) Proper supply of such resources is important to innovation (Amabile, 1988 ; Van de Ven & Angle, 1989 ; Dubickis & Gaile-Sarkane, 2017 ). Based on a study of Koestler’s ( 1964 ) creative thinking, Jabri conceptualized a problem-solving style consisting of two independent thinking styles. He uses a structured problem-solving styles that is based on associative thinking, follows a set of rules, resolves reasonably logically, and uses an intuitive problem-solving ability that focuses on problem-solving, not tied to existing rules with multiple ideas. Intuitive problem solving styles tend to process information from different paradigms simultaneously. It is therefore more likely to create new problem solutions as possible (Isaksen, 1987 ; Kirton, 1976 ). However, style assessment is not desirable because the style of problem solving affects style differently depending on the individual problem-solving situations (Scott & Bruce, 1994 ). We are proposing a role for the University to encourage innovative behavior based on the individuality of our students in order to recognize new opportunities through education about Scott and Bruce’s innovative behavioral models and diverse entrepreneurship education approaches. And involvement of resources, such as entrepreneurship awareness programs, ultimately leads to the identification of individual characteristics and innovation. In addition, current Korean entrepreneurship education is mainly focused on cognitive learning to improve problem solving ability, and one aspect of cognitive learning plays an important role in learning process of new venture firms. This study has a more direct focus on behavior learning such as team-based learning.

Hypothesis development

Problem-solving ability and innovative behavior.

Problem solving is to discover knowledge and skills that reach the target country by interfering with a set of processes and goals where the solution is unknown, unfamiliar, or reaching a new state of goal (Jonassen, 2004 ; Inkinen, 2015 ). There are various approaches to solve this problem. To solve problems and improve problem solving with a successful solution experience, you should adopt the method that best suits your problem solution. You need to select the appropriate inputs for the solution elements and a flexible process structure. Problem solving ability has been recognized as a key element of innovative behavior in responding to rapid changes with the ability to find various alternatives and predict outcomes from these alternatives to maximize positive results, minimize negative consequences, and select solutions to problems (Barron & Harrington, 1981 ; Jabri, 1991 ; Kirton, 1976 ). We pose the following hypotheses:

Hypothesis 1: Individual problem-solving ability has an effect on the innovative behavior of students.

Innovative behavior and opportunity recognition

Innovation involves introducing ideas from outside the organization, through creative processes, and linking these ideas to products or processes. Many scholars studying innovation recognize that designing ideas is only one step in the innovation process (Kanter, 1988 ). Innovation is changing at the organizational or individual level. Kanter, Scott and Bruce defined personal innovation. In other words, an innovation act starts with recognition of a problem, adoption of a new idea, or creation of a solution, and an individual with an innovative tendency wants to create a realistically realizable group with the sympathy of such an idea. Innovative individuals create prototypes for innovations that enable ideas to be realized specifically with goods or services and become productive use and social day merchandising. According to previous studies, opportunity perception can be seen as an individual’s corporate strategy that focuses on the perception and exploitation of individuals about potential business ideas and opportunities and finds resources to create innovative outcomes (Manev et al., 2005 ). New Venture Ideas (NVI) are imaginary combinations of product/service offerings; potential markets or users, and means of bringing these offerings into existence (Davidsson, 2015 ). From the viewpoint of a potential entrepreneur like a university student, entrepreneurship starts with an idea. This process continues with a range of practices including attractiveness and feasibility of an idea, gathering information to minimize value-related uncertainty and possibility and perhaps the main idea’s conformity ratio in terms of newly discovered needs (Hayton & Cholakova, 2012 ). Earlier we proposed that the program as a whole increases the students’ innovative behavior and that innovative performance is the new venture ideas. Since it is logical to assume that the relationship between innovative behavior and opportunity recognition. We pose the following hypotheses:

Hypothesis 2: Innovative behavior will be a more potent inducer of opportunity recognition.

Problem-solving ability and opportunity recognition

Among the many factors influencing opportunity perception, the problems that arise in the fourth industry, the knowledge-based industry of the twenty-first century, are unpredictable and unstructured; they cannot be solved with existing solutions and require creative problem-solving skills. In order to determine how to solve problem situations that are different from the current situation and have unknown results, problems are solved through the process of adjusting previous experience, knowledge, and intuition (Charles & Lester, 1982 ). Experience, knowledge, and intuition are applied simultaneously to a single problem, not individually or collectively, and the intellectual and creative results that can be quickly and effectively solved in problem solving are seen as problem solving abilities (Ardichvili et al., 2003a , b ). Empirical studies of problem-solving abilities and opportunity perceptions have provided strong evidence that there is a positive relationship between theoretical integrative processes and corporate opportunity recognition (Ucbasaran et al., 2009 ). Therefore, we hypothesized that:

Hypothesis 3: Problem solving ability has an effect on the opportunity recognition.

The respondents for this study were randomly selected from three universities in Korea. Most of the respondents in this study were Korean university students who experienced team-based learning during behavioral learning through entrepreneurship education. Since then, we have been guided by two main criteria when choosing these universities. First, students who take entrepreneurship courses are critical to their innovation behavior. This led us to realize that innovative behavior is an important factor in an individual’s survival and growth. The second is that the parallel process of theoretical and behavioral learning is highly satisfied. A pilot study was conducted to verify the reliability and validity of the research measurements with 28 students at a university. The results of the pilot study showed high clarity and reliability (Cronbach ‘s alphas were all above 0.70) ​​of the research measurements. The sample of the pilot study was not incorporated in the present study.

This study was conducted in a four - year undergraduate course (various majors) that took entrepreneurship courses in Korea university programs. Students in this course have a mix of students who have previously experienced entrepreneurship and those who have not. During the course, students were taught the theoretical lessons for 8 weeks and the team for the 8 weeks. The questionnaire was administered during the last week of the course.

The data were analyzed from 203 participants, out of a total of 209, of which 7 were not appropriate. Of the 203 participants, 27% were female and 73% were male and the grade distribution was 3% for freshmen, 12% for grade 2, 26% for grade 2, and 59% for grade 2. The main distribution is 26% in social science, 16% in business and economics, 39% in engineering, 11% in music and athletics and 7% in others (see Table  2 ).

Measurement

The structure of the model was measured by questionnaires (problem-solving ability, innovation behavior and opportunity recognition questionnaire) consisting of the scale taken from questionnaires verified in previous studies. Tool selection was performed on two criteria. First, the selected tool should measure the same structure (ie, the original measured structure had to be conceptually identical to the way the structure was defined in this study model). Secondly, the psychometric qualities of the instrument for the student had to be high.

Assessment of the factors was carried out through principal component analyses (varimax rotation with eigenvalues of 1.0 or above) of the scales connected to the same level of the model to confirm the uniqueness of the scales with respect to each other. This was supplemented by the computation of the internal consistency reliability of the scales (Cronbach’s α). These analyses were executed using the individual participants’ responses (Nunnally & Bernstein, 1994 ).

Problem- solving ability was measured on a 7-point Likert-scale (1 = ‘completely disagree’; 7 = ‘completely agree’). Jabri ( 1991 ) used a measurement tool to measure individual problem solving ability.

Innovative behavior was measured on a 7-point Likert-scale (1 = ‘completely disagree’; 7 = ‘completely agree’). In order to measure innovation behavior, we modified the questionnaire items to fit the intention of this study among the questionnaire items used by Scott and Bruce ( 1994 ) and Kim and Rho ( 2010 ).

Opportunity recognition was measured on a 7-point Likert-scale (1 = ‘completely disagree’; 7 = ‘completely agree’). In order to measure opportunity recognition, we modified the questionnaire items to fit the intention of this study among the questionnaire items used by Kim and Rho ( 2010 ).

Methods of analysis

The first two parts of the analysis were primarily based on (multiple) regression analyses. The last part of the analysis was informed through the path analyses. The adequacy of the models was assessed by AMOS 18(Arbuckle & Wothke, 2003 ). Models were all tested with standardized coefficients obtained from the Principal Component Analysis. To ascertain the model fit, we analyzed the comparative fit index (CFI), the normed fit index (NFI), the Root Mean Square Err of Approximation (RMSEA), the standardized root mean square residual (SRMR) and the chi-square test statistic.

Reliability and validity are essential psychometrics to be reported. The first step to evaluate those aspects was to use the Cronbach’s alpha and the composite reliability to test reliability of the proposed scales. The usual threshold level is 0.7 for newly developed measures (Fornell and Larcker, 1981 ). Values range from 0.69 to 0.79 in the case of Cronbach’s alpha, and from 0.85 to 0.92 in the case of composite reliability (see Table  3 ). Therefore, these scales may be considered as reliable. Next, we estimated the research model, displayed in Fig.  1 , using structural equation modeling (SEM) and AMOS 18 (Arbuckle & Wothke, 2003 ). Our analysis revealed an adequate measurement model with high factor loadings for all the items on the expected factors and communalities of each item exceeding 0.50. We discuss three fit indices that are generally considered as important (Hu & Bentler, 1998 ). First, the CFI-value represents the overall difference between observed and predicted correlations. A value of 0.04 which is situated well below the cut-off value of 0.08, suggests that the hypothesized model resembles the actual correlations. Secondly, Bentler’s CFI (comparative fit index) greater than 0.90 and 0.95 which is above the cut-off of 0.90 (Schumacker & Lomax, 1996 ). Thirdly, NFI greater greater than 0.90 and 0.95 which is above the cut-off of 0.90 (Schumacker & Lomax, 1996 ). Fourthly, the standardized root mean square residual (SRMR) value of 0.0392 which is situated well below the cut-off value of 0.05(Hu & Bentler, 1998 ), and the chi-square value of 3581.622 which is situated well below the cut-off value of 0.0005. Finally, the RMSEA (root mean square error of approximation) equals 0.04 with a 90% confidence interval between 0.03 and 0.05.

Analysis of mediation effect

The value and confidence interval are situated over but below the cut-off value of 0.1 which suggests not a great but a good fit. Factor analysis was verified by factor analysis using principal component analysis and only factors with an eigenvalue of 1 or more by orthogonal rotation method were selected. Factor loading was considered to be significant at 0.5 or more (Hair et al., 2006a , b ). As a result of the analysis, cumulative explanation for 72.4% of the total variance. Confirmatory factor analysis thus supported the differentiation of the three components Also we tested the confirmatory validity of the construct by testing whether the structural linkage of each square is greater than the mean variance extraction (AVE) of each structure. The AVE ranged from 0.52 to 0.53, reaching the recommended level of .50 for both Fornell and Larcker ( 1981 ). Therefore, all constructs showed sufficient convergent validity (see Table 3 ).

As shown in Table  4 , the AVE value of each variable has a higher value than that of other factors. Therefore, the discriminant validity of the proposed model can be judged as appropriate.

Means, standard deviations, and correlations among the study variables are shown in Table  5 .

The mean scores for the conceptual model were as follows for problem-solving ability (MD. 5.20, SD.1.08), innovative behavior (MD.5.20, SD.1.03), and opportunity recognition (MD. 5.14, SD. 1.06) conditions. The means of problem-solving ability, innovative behavior, and opportunity recognition were high. Furthermore, those variables correlated positively with each other.

Figure  1 showed that all paths and their significance levels are presented in Table  6 . The path between the latent variables problem-solving ability and innovative behavior was significant (p, 0.001), consistent with Hypotheses 1. In addition, there was innovative behavior and opportunity recognition (p, 0.01), this result provide empirical support for Hypothesis 2.

H3 proposed that Problem-solving ability is positively related to opportunity recognition. The results of the correlation analysis: The coefficient of problem solving and opportunity perception weakened from .717 to .444, but it is still partly mediated because it is still significant (C. R  = 7.604 ***). This supports H3 (see Table 6 ).

In order to verify the significance of the indirect effect, the bootstrapping must be performed in AMOS, and the actual significance test should be identified using two-tailed significance. As a result, the significance of indirect effect is 0.04 ( p  < 0.05), which is statistically significant (see Table  7 ).

Discussion and conclusion

We have tried to demonstrate the effects of behavior and its significance by differentiating from the general curriculum emphasizing cognitive effects as a model of problem solving ability emerging as innovative behavior through opportunity of university entrepreneurship education.. This supports the premise that entrepreneurship education can improve opportunities or processes through behavioral learning. The results of this study support the role of entrepreneurship education in creating opportunities for innovative behavior and problem solving abilities. Entrepreneurship education should provide different types of learning for new opportunities and focus on what is manifested in behavior.

In addition, based on previous research, we propose whether the following contents are well followed and whether it is effective. First, the emergence of innovative behavior in problem-solving abilities increases as the cognitive diversity of students with diverse majors and diverse backgrounds increases. Second, the more entrepreneurial learning experiences, the greater the chance of new opportunities. Third, it is necessary to investigate students’ problem solving style and problem-solving ability first, and then a teaching strategy based on this combination of systematic and effective theory and practice is needed. Of course, as demonstrated by many studies, it may be easier to enhance the effectiveness of opportunity recognition through cognitive learning. This is because it emphasizes the achievement of knowledge and understanding with acquiring skills and competence. This process, however, is not enough for entrepreneurship education. However, we do not support full team-based behavioral learning in the class designed by Michaelsen and Sweet ( 2008 ). As with the results of this study, problem solving ability is positively related to opportunity perception directly. As previously demonstrated in previous studies, problem solving ability can be enhanced by cognitive learning (Anderson et al., 2001 ; Charles & Lester, 1982 ).

Therefore, it has been demonstrated that it is more efficient to balance a certain level of cognitive learning and behavior learning in consideration of the level of students in a course. Also this study satisfies the need for empirical research by Lumpkin and Lichtenstein ( 2005 ) and Robinson et al. ( 2016 ) and others. This will help to improve understanding of how entrepreneurship training is linked to various learning models and their effectiveness and to design better courses for the future. Finally, this study sought to provide an awareness of entrepreneurship education as the best curriculum for solutions that evolved into innovative behaviors that create new values and ultimately represent new opportunities. This study shows that it can positively influence the social effect of creating new value, that is, not only the cognitive effect of general pedagogy, but also the innovation behavior. By providing this awareness, we have laid the groundwork for empirical research on entrepreneurship education in order to create more opportunities for prospective students in education through education and to expand their capabilities.

Limitation and future research

Indeed, the concepts presented here and the limitations of this study have important implications that can fruitfully be addressed in future research. First, we selected a sample of college students taking entrepreneurship training. However, since it is not the whole of Korean university students, it is difficult to extend the research results to all college students in Korea. Second, there is no precedent research on the role of innovation behavior as intermedia in college students. Therefore, we were forced to proceed as an exploratory study.

The ability to recognize opportunities can provide significant benefits that can remain firm and competitive in an ever-changing environment. Future research should therefore expand these insights and try to empirically test more ways in which entrepreneurship pedagogy teaches how learning methods can be integrated into venture creation and growth processes to help new process opportunities. By providing this study, we will help entrepreneurship education in the university to create more opportunities and expand the capacity of prospective members.

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Kim, J.Y., Choi, D.S., Sung, CS. et al. The role of problem solving ability on innovative behavior and opportunity recognition in university students. J. open innov. 4 , 4 (2018). https://doi.org/10.1186/s40852-018-0085-4

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PROBLEM SOLVING AS ACTION RESEARCH

This chapter excerpt describes two problem solving models and describes how to use these as the basis for an action research project.

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Mathematical Problem-Posing Research: Thirty Years of Advances Building on the Publication of “On Mathematical Problem Posing”

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• Jinfa Cai 6 ,
• Stephen Hwang 6 &
• Matthew Melville 7  

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In 1994, Ed Silver published a seminal paper entitled “On Mathematical Problem Posing.” Silver both helped to lay a foundation for problem-posing research and pointed out key directions that problem-posing research could explore. This chapter provides a brief review of the problem-posing literature in the past three decades, showing that there have been marked advances in problem-posing research. We not only provide a review of the advances in problem-posing research, but also of the impact of Silver’s seminal paper on problem-posing research. The chapter ends with a discussion of three specific areas of research on mathematical problem posing (one of these areas is Problem-Posing-Based Learning [P-PBL]) that are ripe for progress and could significantly move the entire field forward.

During the preparation of this chapter, the authors were supported by a grant from the National Science Foundation (DRL 2101552). Any opinions expressed herein are those of the authors and do not necessarily represent the views of the National Science Foundation.

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Cai, J., Hwang, S., Melville, M. (2023). Mathematical Problem-Posing Research: Thirty Years of Advances Building on the Publication of “On Mathematical Problem Posing”. In: Cai, J., Stylianides, G.J., Kenney, P.A. (eds) Research Studies on Learning and Teaching of Mathematics. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-35459-5_1

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The Robots That Will Change the World Are Already Among Us

Climate change more like ecosocial crisis, openmind books, scientific anniversaries, kary mullis, the genius of a scientist, the eccentricity of a celebrity, featured author, latest book, solving problems visually.

What’s the best way to approach (and successfully solve) a mathematical problem statement? Perhaps by drawing a picture? That, at least, is the conclusion of a striking new study by researchers at the Universities of Geneva and Burgundy.

It’s not a trivial assumption. It is thought that when we face a mathematical problem that contains both mathematical information (numbers and arithmetic operations) and non-mathematical information (the context of the problem and the characteristics of the entities involved), our brains process this combination of verbal and numerical information and convert it into a mental representation in order to identify the best strategy for solving it. On the other hand, more and more studies suggest that the schematic drawings that are usually made to solve this type of problem are a reflection of these mental representations.

Game 1: Not a game, an experiment

In the study, participants were asked to solve 12 simple arithmetic problems in as few steps as possible and to draw a picture that would help them understand and solve the problem.

Here are two of these problems, and we invite you to solve them in the same way: in as few steps as possible, and with a drawing to help you understand the problem.

Problem 1 : Paul has five red marbles and also has some blue marbles. In total he has eleven marbles. Julie’s marbles are green and blue. Julie has as many blue marbles as Paul and also has two fewer green marbles than Paul has red marbles. How many marbles does Julie have?

Problem 2 : Lisa takes the train during the day, travels for 5 hours and arrives at her destination at 11am. Fred got on the train at the same time as Lisa and his trip took 2 hours less. What time did Fred arrive at his destination?

Independently of the above, many studies postulate that relying on drawings, diagrams or other types of graphical representations when processing information has numerous benefits: it improves our ability to learn and remember, it helps us to understand complex concepts, it reinforces critical and scientific thinking, and it fosters a transversal and interdisciplinary interpretation. And from a mathematical point of view, using these representations makes it easier to establish the relationships between different data, to visualise the information implicit in the statement and to identify the most direct and simplest solution strategy.

Use the drawings to answer these complex and hieroglyphic questions.

A recent study goes a step further by suggesting that the verbal information in the problem statement influences the type of diagram shown and also the strategy chosen to solve the problem. More specifically, the study has found that the type of diagram preferentially chosen depends on whether the statement is cardinal or ordinal in nature.

Thus, when the context alludes to the cardinal properties of the quantities involved—the number of elements in a set—a drawing based on groupings of entities (crosses, circles, etc.) that sometimes overlap (or intersect) is usually chosen. This in turn leads to a three-step arithmetic strategy. On the other hand, when the statement of the problem focuses on the ordinal properties of numbers—the position they occupy in a set—we usually opt for drawings based on axes, graduations or intervals, which lead to a more direct and simpler one-step solution strategy.

And this is observed even when the problems are analogous from a mathematical point of view: they have the same structure, the same numerical values and can be solved with the same strategy (as in the case of the two problems in Game 1).

But perhaps the most interesting reflection is that, knowing this, it is possible to guide and train the student to apply this second type of diagram, thereby facilitating the identification of the best way to solve it.

Game 3:  A high-flying challenge

Sara wants to travel from Madrid to Tokyo. To do so, she flies first to New York, from where she takes a plane to London and from there to Tokyo.

Paul also wants to go from Madrid to Tokyo, but in his case he flies directly from Madrid to London and then takes a flight to Tokyo.

If Sara flies for a total of 27hrs 15min and Paul for 14hrs 30min, and given that the flight from New York to London takes 4hrs 45min longer than the flight from Madrid to London, and the flight from London to Tokyo takes 12 hours, how long is the flight from Madrid to New York?

And if both Sara and Paul lose only one hour at each stopover, what will the local time be when they each arrive in Tokyo if they both depart Madrid at 2pm?

            M                                 NY    M-L + 4:45   L                    12:00 h                   T

                                                   M           L                                       12:00 h                    T 

14 hrs 30 min

The Madrid-London flight takes 2hrs 30min. New York to London is 2hrs 30min + 4hrs 45min = 7hrs 15min. And the Madrid to New York flight is 27hrs 15min – 12hrs – 7hrs 15min = 8 hours.

With this, and bearing in mind that each stopover only takes one hour:

If Sara leaves at 2pm from Madrid then: 2pm + 8hrs – 6hrs (time difference) + 1hr (at NY airport) + 7hrs 15min + 5hrs (time difference) + 1hr + 12hrs + 8hrs (time difference) = 2:15am on day 3.

In Paul’s case: 2pm + 2hrs 30min – 1hr (time difference) + 1hr + 12hrs + 8hrs (time difference) = 12:30pm on day 2.

Miguel Barral

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• What Purpose Do the Great Mathematical Problems Serve?
• David Hilbert: The Architect of Modern Mathematics
• Magic Squares: When Art is Squared With Mathematics

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Differentiation by Design

Blending human-centered design research and methodologies with a strategic focus on creative problem solving, dan brown helps made students seek, conceive, and create compelling solutions for the marketplace..

In the Segal Design Institute’s most recent Manufacturing and Design Engineering (MaDE) capstone, all five student teams secured provisional patents for their class projects.

But that result alone is not what brings a smile to Dan Brown ’s face.

“What’s most exciting is watching the students push themselves beyond their comfort zones and work in teams to strategically solve problems through design with distinct differentiations that create advantage,” said Brown, a clinical professor at Segal.

Brown has long championed the concept of “Differentiation by Design” – a trademarked term he has long used to describe his design process – to strategically create innovative solutions to real-world problems in a competitively advantaged way. Through their three-quarter capstone work, MaDE students put Brown’s design thinking doctrine into action.

“It’s fundamentally working smarter, not harder in a ‘designerly’ way,” Brown said.

Pursuing a competitive edge through design

Brown has lived the “Differentiation by Design” mantra over four decades in product design and entrepreneurship. A serial inventor and design engineering professional, the Chicago native boasts nearly 45 US patents as well as numerous international patents and awards for design and innovation. He’s developed celebrated, game-changing products like the Bionic Wrench and founded LoggerHead Tools in 2005.  

Brown joined the Segal faculty in 2008 after receiving a Master of Product Design and Development Management from Northwestern and subsequently pursuing a PhD at Coventry University in the UK. In the classroom, Brown leaned into his practical experiences and graduate research to educate students and demonstrate the value of strategically baking competitive advantage into the design process.

Brown defines design as “how people seek, conceive, and create new knowledge” and innovation as “how new knowledge competes with existing knowledge.” The competitive advantage, then, is innovation’s key metric – and the target Brown actively promotes with students as they pursue novel solutions.

“We’re teaching beyond the skills of research, data collection, and analysis. Our capstone focus is the application and practice of strategic critical thinking, research, and process rigor applied to real, right-now problems in a professionally appropriate way,” he said.

While promoting core elements of human-ce ntered design, such as stakeholder research and iterative testing, Brown also provides students with deep instruction on how to design with strong intellectual property in mind. Specifically, he trumpets the importance of patents and other forms of intellectual property to protecting one’s competitive edge. He also incorporates patent research as primary research to guide strategic design.

“If you’re going to invest your time, effort, and money, you need to be able to protect it and essentially design the strategy to protect your IP as you create it. Without protection, you will simply lose the control of who benefits from your efforts,” said Brown, who recently completed a three-year term on the United States Patent and Trademark Office’s Patent Public Advisory Committee. The US Secretary of Commerce appointed him to the committee to represent the voices of independent inventors.

Providing students a powerful experience

Brown’s Differentiation by Design ethos is evident in the MaDE capstone he leads alongside fellow Segal clinical professor David Gatchell, director of the MaDE undergraduate program.  

Together, Brown and Gatchell reject handing students a defined problem or a design process with strict checkpoints. Rather, they encourage exploratory research, creative critical thinking, and rigorous evidence-based reasoning and validation to propel unique design solutions to the real-world problems students identify.

Brown recalls one MaDE team designing a restaurant highchair that could be easily broken down and placed in the standard commercial dishwasher for cleaning, thereby eliminating a pain point for restaurant guests with young children.

“Both Dave and I fondly remember at one point the team, frustrated by the challenges, insisting it was physically impossible to accomplish, but we insisted they push towards a solution,” Brown said. “We often reflect and share the story of how excited the team was having proven themselves wrong.”

Brown has also watched others launch startups from products they designed at Northwestern and then gain marketplace traction given their focus on distinctive IP that allowed for investor interest.

Along the way, Brown has witnessed students’ persistence and confidence grow as they imagine new possibilities and seize compelling opportunities. Even better, they begin to see competitive advantage as an attainable goal when they marry the foundational disciplinary design knowledge of creative problem solving and critical strategic thinking alongside the technical skills and experience gained in their engineering courses.  

“In the MaDE Capstone, we want to prepare students for real-world problem solving and push them beyond what they think they can do,” Brown said. “When we embrace design as a discipline through the lens of how humans seek, conceive, and create new knowledge solutions, and focus them on ethical and responsible competitive outcomes, we are all in better positions for success in society.”

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