problem solving concepts and theories

Problem-solving concepts and theories

Affiliation.

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

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

Publication types

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

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21 Problem Solving

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Problem Representation: The Gestalt Legacy

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

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

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

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

A solution to the nine-dot problem.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Two Legacies Converge

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

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

Relevance of the Gestalt Ideas to the Solution of Search Problems

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

Problem Isomorphs

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

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

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

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

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

Expertise and Its Development

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

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

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

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

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

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

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

Relevance of Search to Insight Solutions

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

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

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

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

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

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

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

Effects of Perception and Knowledge in Problem Solving in Academic Disciplines

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

The Role of Visual Perception

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

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

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

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

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

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

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

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

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

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

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

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

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

The Role of Background Knowledge

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

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

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

Conclusions and Future Directions

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

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

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

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Problem-Solving Theory: The Task-Centred Model

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

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

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

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

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

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THEORIES AND PRINCIPLES OF PROBLEM SOLVING IN MATHEMATICS

2023, Article

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

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18.12: Chapter 14- Problem Solving, Categories and Concepts

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This page is a draft and under active development. Please forward any questions, comments, and/or feedback to the ASCCC OERI ( [email protected] ).

• Kenneth A. Koenigshofer
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Learning Objectives

• Define problem types
• Describe problem solving strategies
• Define algorithm and heuristic
• Describe the role of insight in problem solving
• Explain some common roadblocks to effective problem solving
• What is meant by a search problem
• Describe means-ends analysis
• How do analogies and restructuring contribute to problem solving
• Explain how experts solve problems and what gives them an advantage over non-experts
• Describe the brain mechanisms in problem solving

In this section we examine problem-solving strategies. People face problems every day—usually, multiple problems throughout the day. Sometimes these problems are straightforward: To double a recipe for pizza dough, for example, all that is required is that each ingredient in the recipe be doubled. Sometimes, however, the problems we encounter are more complex. For example, say you have a work deadline, and you must mail a printed copy of a report to your supervisor by the end of the business day. The report is time-sensitive and must be sent overnight. You finished the report last night, but your printer will not work today. What should you do? First, you need to identify the problem and then apply a strategy, usually a set of steps, for solving the problem.

Defining Problems

We begin this module on Problem Solving by giving a short description of what psychologists regard as a problem. Afterwards we are going to present different approaches towards problem solving, starting with gestalt psychologists and ending with modern search strategies connected to artificial intelligence. In addition we will also consider how experts do solve problems and finally we will have a closer look at two topics: The neurophysiological background on the one hand and the question what kind of role can be assigned to evolution regarding problem solving on the other.

The most basic definition is “A problem is any given situation that differs from a desired goal”. This definition is very useful for discussing problem solving in terms of evolutionary adaptation, as it allows to understand every aspect of (human or animal) life as a problem. This includes issues like finding food in harsh winters, remembering where you left your provisions, making decisions about which way to go, repeating and varying all kinds of complex movements by learning, and so on. Though all these problems were of crucial importance during the evolutionary process that created us the way we are, they are by no means solved exclusively by humans. We find a most amazing variety of different solutions for these problems of adaptation in animals as well (just consider, e.g., by which means a bat hunts its prey, compared to a spider ).

However, for this module, we will mainly focus on abstract problems that humans may encounter (e.g. playing chess or doing an assignment in college). Furthermore, we will not consider those situations as abstract problems that have an obvious solution: Imagine a college student, let's call him Knut. Knut decides to take a sip of coffee from the mug next to his right hand. He does not even have to think about how to do this. This is not because the situation itself is trivial (a robot capable of recognizing the mug, deciding whether it is full, then grabbing it and moving it to Knut’s mouth would be a highly complex machine) but because in the context of all possible situations it is so trivial that it no longer is a problem our consciousness needs to be bothered with. The problems we will discuss in the following all need some conscious effort, though some seem to be solved without us being able to say how exactly we got to the solution. Still we will find that often the strategies we use to solve these problems are applicable to more basic problems, as well as the more abstract ones such as completing a reading or writing assignment for a college class.

Non-trivial, abstract problems can be divided into two groups:

Well-defined Problems

For many abstract problems it is possible to find an algorithmic solution. We call all those problems well-defined that can be properly formalized, which comes along with the following properties:

• The problem has a clearly defined given state. This might be the line-up of a chess game, a given formula you have to solve, or the set-up of the towers of Hanoi game (which we will discuss later ).
• There is a finite set of operators, that is, of rules you may apply to the given state. For the chess game, e.g., these would be the rules that tell you which piece you may move to which position.
• Finally, the problem has a clear goal state: The equations is resolved to x, all discs are moved to the right stack, or the other player is in checkmate.

Not surprisingly, a problem that fulfills these requirements can be implemented algorithmically (also see convergent thinking ). Therefore many well-defined problems can be very effectively solved by computers, like playing chess.

Ill-defined Problems

Though many problems can be properly formalized (sometimes only if we accept an enormous complexity) there are still others where this is not the case. Good examples for this are all kinds of tasks that involve creativity , and, generally speaking, all problems for which it is not possible to clearly define a given state and a goal state: Formalizing a problem of the kind “Please paint a beautiful picture” may be impossible. Still this is a problem most people would be able to access in one way or the other, even if the result may be totally different from person to person. And while Knut might judge that picture X is gorgeous, you might completely disagree.

Nevertheless ill-defined problems often involve sub-problems that can be totally well-defined. On the other hand, many every-day problems that seem to be completely well-defined involve a great deal of creativity and many ambiguities. For example, suppose Knut has to read some technical material and then write an essay about it.

If we think of Knut's fairly ill-defined task of writing an essay, he will not be able to complete this task without first understanding the text he has to write about. This step is the first sub-goal Knut has to solve. Interestingly, ill-defined problems often involve subproblems that are well-defined.

Knut’s situation could be explained as a classical example of problem solving: He needs to get from his present state – an unfinished assignment – to a goal state - a completed assignment - and has certain operators to achieve that goal. Both Knut’s short and long term memory are active. He needs his short term memory to integrate what he is reading with the information from earlier passages of the paper. His long term memory helps him remember what he learned in the lectures he took and what he read in other books. And of course Knut’s ability to comprehend language enables him to make sense of the letters printed on the paper and to relate the sentences in a proper way.

Same place, different day. Knut is sitting at his desk again, staring at a blank paper in front of him, while nervously playing with a pen in his right hand. Just a few hours left to hand in his essay and he has not written a word. All of a sudden he smashes his fist on the table and cries out: "I need a plan!

How is a problem represented in the mind?

Generally speaking, problem representations are models of the situation as experienced by the agent. Representing a problem means to analyze it and split it into separate components:

• objects, predicates
• state space
• selection criteria

Therefore the efficiency of Problem Solving depends on the underlying representations in a person’s mind. Analyzing the problem domain according to different dimensions, i.e., changing from one representation to another, results in arriving at a new understanding of a problem. This is basically what is described as restructuring.

There are two very different ways of approaching a goal-oriented situation . In one case an organism readily reproduces the response to the given problem from past experience. This is called reproductive thinking .

The second way requires something new and different to achieve the goal, prior learning is of little help here. Such productive thinking is (sometimes) argued to involve insight . Gestalt psychologists even state that insight problems are a separate category of problems in their own right.

Tasks that might involve insight usually have certain features – they require something new and non-obvious to be done and in most cases they are difficult enough to predict that the initial solution attempt will be unsuccessful. When you solve a problem of this kind you often have a so called "AHA-experience" – the solution pops up all of a sudden. At one time you do not have any ideas of the answer to the problem, you do not even feel to make any progress trying out different ideas, but in the next second the problem is solved.

Sometimes, previous experience or familiarity can even make problem solving more difficult. This is the case whenever habitual directions get in the way of finding new directions – an effect called fixation .

Functional fixedness

Functional fixedness concerns the solution of object-use problems . The basic idea is that when the usual way of using an object is emphasised, it will be far more difficult for a person to use that object in a novel manner.

An example is the two-string problem : Knut is left in a room with a chair and a pair of pliers given the task to bind two strings together that are hanging from the ceiling. The problem he faces is that he can never reach both strings at a time because they are just too far away from each other. What can Knut do?

Figure \(\PageIndex{1}\): Put the two strings together by tying the pliers to one of the strings and then swing it toward the other one.

Mental fixedness

Functional fixedness as involved in the examples above illustrates a mental set – a person’s tendency to respond to a given task in a manner based on past experience. Because Knut maps an object to a particular function he has difficulties to vary the way of use (pliers as pendulum's weight).

Problem-Solving Strategies

When you are presented with a problem—whether it is a complex mathematical problem or a broken printer, how do you solve it? Before finding a solution to the problem, the problem must first be clearly identified. After that, one of many problem solving strategies can be applied, hopefully resulting in a solution. Regardless of strategy, you will likely be guided, consciously or unconsciously, by your knowledge of cause-effect relations among the elements of the problem and the similarity of the problem to previous problems you have solved before. As discussed in earlier sections of this chapter, innate dispositions of the brain to look for and represent causal and similarity relations are key components of general intelligence (Koenigshofer, 2017).

A problem-solving strategy is a plan of action used to find a solution. Different strategies have different action plans associated with them. For example, a well-known strategy is trial and error. The old adage, “If at first you don’t succeed, try, try again” describes trial and error. In terms of your broken printer, you could try checking the ink levels, and if that doesn’t work, you could check to make sure the paper tray isn’t jammed. Or maybe the printer isn’t actually connected to your laptop. When using trial and error, you would continue to try different solutions until you solved your problem. Although trial and error is not typically one of the most time-efficient strategies, it is a commonly used one.

Another type of strategy is an algorithm. An algorithm is a problem-solving formula that provides you with step-by-step instructions used to achieve a desired outcome (Kahneman, 2011). You can think of an algorithm as a recipe with highly detailed instructions that produce the same result every time they are performed. Algorithms are used frequently in our everyday lives, especially in computer science. When you run a search on the Internet, search engines like Google use algorithms to decide which entries will appear first in your list of results. Facebook also uses algorithms to decide which posts to display on your newsfeed. Can you identify other situations in which algorithms are used?

A heuristic is another type of problem solving strategy. While an algorithm must be followed exactly to produce a correct result, a heuristic is a general problem-solving framework (Tversky & Kahneman, 1974). You can think of these as mental shortcuts that are used to solve problems. A “rule of thumb” is an example of a heuristic. Such a rule saves the person time and energy when making a decision, but despite its time-saving characteristics, it is not always the best method for making a rational decision. Different types of heuristics are used in different types of situations, but the impulse to use a heuristic occurs when one of five conditions is met (Pratkanis, 1989):

• When one is faced with too much information
• When the time to make a decision is limited
• When the decision to be made is unimportant
• When there is access to very little information to use in making the decision
• When an appropriate heuristic happens to come to mind in the same moment

Working backwards is a useful heuristic in which you begin solving the problem by focusing on the end result. Consider this example: You live in Washington, D.C. and have been invited to a wedding at 4 PM on Saturday in Philadelphia. Knowing that Interstate 95 tends to back up any day of the week, you need to plan your route and time your departure accordingly. If you want to be at the wedding service by 3:30 PM, and it takes 2.5 hours to get to Philadelphia without traffic, what time should you leave your house? You use the working backwards heuristic to plan the events of your day on a regular basis, probably without even thinking about it.

Another useful heuristic is the practice of accomplishing a large goal or task by breaking it into a series of smaller steps. Students often use this common method to complete a large research project or long essay for school. For example, students typically brainstorm, develop a thesis or main topic, research the chosen topic, organize their information into an outline, write a rough draft, revise and edit the rough draft, develop a final draft, organize the references list, and proofread their work before turning in the project. The large task becomes less overwhelming when it is broken down into a series of small steps.

Problem Solving as a Search Problem

The idea of regarding problem solving as a search problem originated from Alan Newell and Herbert Simon while trying to design computer programs which could solve certain problems. This led them to develop a program called General Problem Solver which was able to solve any well-defined problem by creating heuristics on the basis of the user's input. This input consisted of objects and operations that could be done on them.

As we already know, every problem is composed of an initial state, intermediate states and a goal state (also: desired or final state), while the initial and goal states characterise the situations before and after solving the problem. The intermediate states describe any possible situation between initial and goal state. The set of operators builds up the transitions between the states. A solution is defined as the sequence of operators which leads from the initial state across intermediate states to the goal state.

The simplest method to solve a problem, defined in these terms, is to search for a solution by just trying one possibility after another (also called trial and error ).

As already mentioned above, an organised search, following a specific strategy, might not be helpful for finding a solution to some ill-defined problem, since it is impossible to formalise such problems in a way that a search algorithm can find a solution.

As an example we could just take Knut and his essay: he has to find out about his own opinion and formulate it and he has to make sure he understands the sources texts. But there are no predefined operators he can use, there is no panacea how to get to an opinion and even not how to write it down.

Means-End Analysis

In Means-End Analysis you try to reduce the difference between initial state and goal state by creating sub-goals until a sub-goal can be reached directly (in computer science, what is called recursion works on this basis).

An example of a problem that can be solved by Means-End Analysis is the " Towers of Hanoi "

Figure \(\PageIndex{2}\): Towers of Hanoi with 8 discs – A well defined problem (image from Wikimedia Commons; https://commons.wikimedia.org/wiki/F..._of_Hanoi.jpeg , by User:Evanherk .licensed under the Creative Commons Attribution-Share Alike 3.0 Unported license).

The initial state of this problem is described by the different sized discs being stacked in order of size on the first of three pegs (the “start-peg“). The goal state is described by these discs being stacked on the third pegs (the “end-peg“) in exactly the same order.

Figure \(\PageIndex{3}\): This animation shows the solution of the game "Tower of Hanoi" with four discs. (image from Wikimedia Commons; https://commons.wikimedia.org/wiki/F...of_Hanoi_4.gif ; by André Karwath aka Aka ; licensed under the Creative Commons Attribution-Share Alike 2.5 Generic license).

There are three operators:

• You are allowed to move one single disc from one peg to another one
• You are only able to move a disc if it is on top of one stack
• A disc cannot be put onto a smaller one.

In order to use Means-End Analysis we have to create sub-goals. One possible way of doing this is described in the picture:

1. Moving the discs lying on the biggest one onto the second peg.

2. Shifting the biggest disc to the third peg.

3. Moving the other ones onto the third peg, too

You can apply this strategy again and again in order to reduce the problem to the case where you only have to move a single disc – which is then something you are allowed to do.

Strategies of this kind can easily be formulated for a computer; the respective algorithm for the Towers of Hanoi would look like this:

1. move n-1 discs from A to B

2. move disc #n from A to C

3. move n-1 discs from B to C

where n is the total number of discs, A is the first peg, B the second, C the third one. Now the problem is reduced by one with each recursive loop.

Means-end analysis is important to solve everyday-problems – like getting the right train connection: You have to figure out where you catch the first train and where you want to arrive, first of all. Then you have to look for possible changes just in case you do not get a direct connection. Third, you have to figure out what are the best times of departure and arrival, on which platforms you leave and arrive and make it all fit together.

Analogies describe similar structures and interconnect them to clarify and explain certain relations. In a recent study, for example, a song that got stuck in your head is compared to an itching of the brain that can only be scratched by repeating the song over and over again. Useful analogies appears to be based on a psychological mapping of relations between two very disparate types of problems that have abstract relations in common. Applied to STEM problems, Gray and Holyoak (2021) state: "Analogy is a powerful tool for fostering conceptual understanding and transfer in STEM and other fields. Well-constructed analogical comparisons focus attention on the causal-relational structure of STEM concepts, and provide a powerful capability to draw inferences based on a well-understood source domain that can be applied to a novel target domain." Note that similarity between problems of different types in their abstract relations, such as causation, is a key feature of reasoning, problem-solving and inference when forming and using analogies. Recall the discussion of general intelligence in module 14.2. There, similarity relations, causal relations, and predictive relations between events were identified as key components of general intelligence, along with ability to visualize in imagination possible future actions and their probable outcomes prior to commiting to actual behavior in the physical world (Koenigshofer, 2017).

Restructuring by Using Analogies

One special kind of restructuring, the way already mentioned during the discussion of the Gestalt approach, is analogical problem solving. Here, to find a solution to one problem – the so called target problem, an analogous solution to another problem – the source problem, is presented.

An example for this kind of strategy is the radiation problem posed by K. Duncker in 1945:

As a doctor you have to treat a patient with a malignant, inoperable tumour, buried deep inside the body. There exists a special kind of ray, which is perfectly harmless at a low intensity, but at the sufficient high intensity is able to destroy the tumour – as well as the healthy tissue on his way to it. What can be done to avoid the latter?

When this question was asked to participants in an experiment, most of them couldn't come up with the appropriate answer to the problem. Then they were told a story that went something like this:

A General wanted to capture his enemy's fortress. He gathered a large army to launch a full-scale direct attack, but then learned, that all the roads leading directly towards the fortress were blocked by mines. These roadblocks were designed in such a way, that it was possible for small groups of the fortress-owner's men to pass them safely, but every large group of men would initially set them off. Now the General figured out the following plan: He divided his troops into several smaller groups and made each of them march down a different road, timed in such a way, that the entire army would reunite exactly when reaching the fortress and could hit with full strength.

Here, the story about the General is the source problem, and the radiation problem is the target problem. The fortress is analogous to the tumour and the big army corresponds to the highly intensive ray. Consequently a small group of soldiers represents a ray at low intensity. The solution to the problem is to split the ray up, as the general did with his army, and send the now harmless rays towards the tumour from different angles in such a way that they all meet when reaching it. No healthy tissue is damaged but the tumour itself gets destroyed by the ray at its full intensity.

M. Gick and K. Holyoak presented Duncker's radiation problem to a group of participants in 1980 and 1983. Only 10 percent of them were able to solve the problem right away, 30 percent could solve it when they read the story of the general before. After given an additional hint – to use the story as help – 75 percent of them solved the problem.

With this results, Gick and Holyoak concluded, that analogical problem solving depends on three steps:

1. Noticing that an analogical connection exists between the source and the target problem. 2. Mapping corresponding parts of the two problems onto each other (fortress → tumour, army → ray, etc.) 3. Applying the mapping to generate a parallel solution to the target problem (using little groups of soldiers approaching from different directions → sending several weaker rays from different directions)

The concept that links the target problem with the analogy (the “source problem“) is called problem schema. Gick and Holyoak obtained the activation of a schema on their participants by giving them two stories and asking them to compare and summarize them. This activation of problem schemata is called “schema induction“.

The two presented texts were picked out of six stories which describe analogical problems and their solution. One of these stories was "The General."

After solving the task the participants were asked to solve the radiation problem. The experiment showed that in order to solve the target problem reading of two stories with analogical problems is more helpful than reading only one story: After reading two stories 52% of the participants were able to solve the radiation problem (only 30% were able to solve it after reading only one story, namely: “The General“).

The process of using a schema or analogy, i.e. applying it to a novel situation, is called transduction . One can use a common strategy to solve problems of a new kind.

To create a good schema and finally get to a solution using the schema is a problem-solving skill that requires practice and some background knowledge.

How Do Experts Solve Problems?

With the term expert we describe someone who devotes large amounts of his or her time and energy to one specific field of interest in which he, subsequently, reaches a certain level of mastery. It should not be of surprise that experts tend to be better in solving problems in their field than novices (people who are beginners or not as well trained in a field as experts) are. They are faster in coming up with solutions and have a higher success rate of right solutions. But what is the difference between the way experts and non-experts solve problems? Research on the nature of expertise has come up with the following conclusions:

When it comes to problems that are situated outside the experts' field, their performance often does not differ from that of novices.

Knowledge: An experiment by Chase and Simon (1973a, b) dealt with the question how well experts and novices are able to reproduce positions of chess pieces on chessboards when these are presented to them only briefly. The results showed that experts were far better in reproducing actual game positions, but that their performance was comparable with that of novices when the chess pieces were arranged randomly on the board. Chase and Simon concluded that the superior performance on actual game positions was due to the ability to recognize familiar patterns: A chess expert has up to 50,000 patterns stored in his memory. In comparison, a good player might know about 1,000 patterns by heart and a novice only few to none at all. This very detailed knowledge is of crucial help when an expert is confronted with a new problem in his field. Still, it is not pure size of knowledge that makes an expert more successful. Experts also organise their knowledge quite differently from novices.

Organization: In 1982 M. Chi and her co-workers took a set of 24 physics problems and presented them to a group of physics professors as well as to a group of students with only one semester of physics. The task was to group the problems based on their similarities. As it turned out the students tended to group the problems based on their surface structure (similarities of objects used in the problem, e.g. on sketches illustrating the problem), whereas the professors used their deep structure (the general physical principles that underlay the problems) as criteria. By recognizing the actual structure of a problem experts are able to connect the given task to the relevant knowledge they already have (e.g. another problem they solved earlier which required the same strategy).

Analysis: Experts often spend more time analyzing a problem before actually trying to solve it. This way of approaching a problem may often result in what appears to be a slow start, but in the long run this strategy is much more effective. A novice, on the other hand, might start working on the problem right away, but often has to realise that he reaches dead ends as he chose a wrong path in the very beginning.

Creative Cognition

Divergent thinking.

The term divergent thinking describes a way of thinking that does not lead to one goal, but is open-ended. Problems that are solved this way can have a large number of potential 'solutions' of which none is exactly 'right' or 'wrong', though some might be more suitable than others.

Solving a problem like this involves indirect and productive thinking and is mostly very helpful when somebody faces an ill-defined problem , i.e. when either initial state or goal state cannot be stated clearly and operators are either insufficient or not given at all.

The process of divergent thinking is often associated with creativity, and it undoubtedly leads to many creative ideas. Nevertheless, researches have shown that there is only modest correlation between performance on divergent thinking tasks and other measures of creativity. Additionally it was found that in processes resulting in original and practical inventions things like searching for solutions, being aware of structures and looking for analogies are heavily involved, too.

Figure \(\PageIndex{4}\): functional MRI images of the brains of musicians playing improvised jazz revealed that a large brain region involved in monitoring one's performance shuts down during creative improvisation, while a small region involved in organizing self-initiated thoughts and behaviors is highly activated (Image and modified caption from Wikimedia Commons. File:Creative Improvisation (24130148711).jpg; https://commons.wikimedia.org/wiki/F...130148711).jpg ; by NIH Image Gallery ; As a work of the U.S. federal government , the image is in the public domain .

Convergent Thinking

Convergent thinking patterns are problem solving techniques that unite different ideas or fields to find a solution. The focus of this mindset is speed, logic and accuracy, also identification of facts, reapplying existing techniques, gathering information. The most important factor of this mindset is: there is only one correct answer. You only think of two answers, namely right or wrong. This type of thinking is associated with certain science or standard procedures. People with this type of thinking have logical thinking, are able to memorize patterns, solve problems and work on scientific tests. Most school subjects sharpen this type of thinking ability.

Research shows that the creative process involves both types of thought processes.

Brain Mechanisms in Problem Solving

Presenting Neurophysiology in its entirety would be enough to fill several books. Instead, let's focus only on the aspects that are especially relevant to problem solving. Still, this topic is quite complex and problem solving cannot be attributed to one single brain area. Rather there are systems or networks of several brain areas working together to perform a specific problem solving task. This is best shown by an example, playing chess:

Table 2: Brain areas involved in a complex cognitive task.

One of the key tasks, namely planning and executing strategies , is performed by the prefrontal cortex (PFC) , which also plays an important role for several other tasks correlated with problem solving. This can be made clear from the effects of damage to the PFC on ability to solve problems.

Patients with a lesion in this brain area have difficulty switching from one behavioral pattern to another. A well known example is the wisconsin card-sorting task . A patient with a PFC lesion who is told to separate all blue cards from a deck, would continue sorting out the blue ones, even if the experimenter next told him to sort out all brown cards. Transferred to a more complex problem, this person would most likely fail, because he is not flexible enough to change his strategy after running into a dead end or when the problem changes.

Another example comes from a young homemaker, who had a tumour in the frontal lobe. Even though she was able to cook individual dishes, preparing a whole family meal was an impossible task for her.

Mushiake et al. (2009) note that to achieve a goal in a complex environment, such as problem‐solving situations like those above, we must plan multiple steps of action. When planning a series of actions, we have to anticipate future outcomes that will occur as a result of each action, and, in addition, we must mentally organize the temporal sequence of events in order to achieve the goal. These researchers investigated the role of lateral prefrontal cortex (PFC) in problem solving in monkeys. They found that "PFC neurons reflected final goals and immediate goals during the preparatory period. [They] also found some PFC neurons reflected each of all the forthcoming steps of actions during the preparatory period and they increased their [neural] activity step by step during the execution period. [Furthermore, they] found that the transient increase in synchronous activity of PFC neurons was involved in goal subgoal transformations. [They concluded] that the PFC is involved primarily in the dynamic representation of multiple future events that occur as a consequence of behavioral actions in problem‐solving situations" (Mushiake et al., 2009, p. 1). In other words, the prefrontal cortex represents in our imagination the sequence of events following each step that we take in solving a particular problem, guiding us step by step to the solution.

As the examples above illustrate, the structure of our brain is of great importance regarding problem solving, i.e. cognitive life. But how was our cognitive apparatus designed? How did perception-action integration as a central species-specific property of humans come about? The answer, as argued extensively in earlier sections of this book, is, of course, natural selection and other forces of genetic evolution. Clearly, animals and humans with genes facilitating brain organization that led to good problem solving skills would be favored by natural selection over genes responsible for brain organization less adept at solving problems. We became equipped with brains organized for effective problem solving because flexible abilities to solve a wide range of problems presented by the environment enhanced ability to survive, to compete for resources, to escape predators, and to reproduce (see chapter on Evolution and Genetics in this text).

In short, good problem solving mechanisms in brains designed for the real world gave a competitive advantage and increased biological fitness. Consequently, humans (and many other animals to a lesser degree) have "innate ability to problem-solve in the real world. Solving real world problems in real time given constraints posed by one's environment is crucial for survival . . . Real world problem solving (RWPS) is different from those that occur in a classroom or in a laboratory during an experiment. They are often dynamic and discontinuous, accompanied by many starts and stops . . . Real world problems are typically ill-defined, and even when they are well-defined, often have open-ended solutions . . . RWPS is quite messy and involves a tight interplay between problem solving, creativity, and insight . . . In psychology and neuroscience, problem-solving broadly refers to the inferential steps taken by an agent [human, animal, or computer] that leads from a given state of affairs to a desired goal state" (Sarathy, 2018, p. 261-2). According to Sarathy (2018), the initial stage of RWPS requires defining the problem and generating a representation of it in working memory. This stage involves activation of parts of the " prefrontal cortex (PFC) , default mode network (DMN) , and the dorsal anterior cingulate cortex (dACC) ." The DMN includes the medial prefrontal cortex , posterior cingulate cortex , and the inferior parietal lobule . Other structures sometimes considered part of the network are the lateral temporal cortex , hippocampal formation , and the precuneus . This network of structures is called "default mode" because these structures show increased activity when one is not engaged in focused, attentive, goal-directed actions, but rather a "resting state" (a baseline default state) and show decreased neural activity when one is focused and attentive to a particular goal-directed behavior (Raichle, et al., 2001).

Moral Reasoning

Jeurissen, et al., (2014) examined a special type of reasoning, moral reasoning, using TMS (Transcranial Magnetic Stimulation). The dorsolateral prefrontal cortex (DLPFC) and temporal-parietal junction (TPJ) have both been shown to be involved in moral judgments, but this study by Jeurissen, et al., (2014) uses TMS to tease out the different roles these brain areas play in different scenarios involving moral dilemmas.

Moral dilemmas have been categorized by researchers as moral-impersonal (e.g., trolley or switch dilemma-- save the lives of five workmen at the expense of the life of one by switching train to another track) and moral-personal dilemmas (e.g., footbridge dilemma-- push a strange r in front of a train to save the lives of five others). In the first scenario, the person just pulls a switch resulting in death of one person to save five, but in the second, the person pushes the victim to their death to save five others.

Dual-process theory proposes that moral decision-making involves two components: an automatic emotional response and a voluntary application of a utilitarian decision-rule (in this case, one death to save five people is worth it). The thought of being responsible for the death of another person elicits an aversive emotional response, but at the same time, cognitive reasoning favors the utilitarian option. Decision making and social cognition are often associated with the DLPFC. Neurons in the prefrontal cortex have been found to be involved in cost-benefit analysis and categorize stimuli based on the predicted consequences.

Theory-of-mind (TOM) is a cognitive mechanism which is used when one tries to understand and explain the knowledge, beliefs, and intention of others. TOM and empathy are often associated with TPJ functioning .

In the article by Jeurissen, et al., (2014), brain activity is measured by BOLD. BOLD refers to Blood-oxygen-level-dependent imaging , or BOLD-contrast imaging, which is a way to measure neural activity in different brain areas in MRI images .

Greene et al., 2001 (Links to an external site.) , 2004 (Links to an external site.) reported that activity in the prefrontal cortex is thought to be important for the cognitive reasoning process , which can counteract the automatic emotional response that occurs in moral dilemmas like the one in Jeurissen, et al., (2014). Greene et al. (2001) (Links to an external site.) found that the medial portions of the medial frontal gyrus, the posterior cingulate gyrus, and the bilateral angular gyrus showed a higher BOLD response in the moral-personal condition than the moral-impersonal condition. The right middle frontal gyrus and the bilateral parietal lobes showed a lower BOLD response in the moral-personal condition than in the moral impersonal. Furthermore, Greene et al. (2004) (Links to an external site.) showed an increased BOLD response for the bilateral amygdale in personal compared to the impersonal dilemmas.

Given the role of the prefrontal cortex in moral decision-making, Jeurissen, et al., (2014) hypothesized that when magnetically stimulating prefrontal cortex , they will selectively influence the decision process of the moral personal dilemmas because the cognitive reasoning for which the DLPFC is important is disrupted , thereby releasing the emotional component making it more influential in the resolution of the dilemma. Because the activity in the TPJ is related to emotional processing and theory of mind ( Saxe and Kanwisher, 2003 (Links to an external site.) ; Young et al., 2010 (Links to an external site.) ), Jeurissen, et al., (2014) hypothesized that when magnetically stimulating this area, the TPJ, during a moral decision, this will selectively influence the decision process of moral-impersonal dilemmas.

Results of this study by Jeurissen, et al., (2014) showed an important role of the TPJ in moral judgment . Experiments using fMRI ( Greene et al., 2004 (Links to an external site.) ), have found the cingulate cortex to be involved in moral judgment . In earlier studies, the cingulate cortex was found to be involved in the emotional response. Since the moral-personal dilemmas are more emotional ly salient, the higher activity observed for TPJ in the moral-personal condition (more emotional) is consistent with this view. Another area that is hypothesized to be associated with the emotional response is the temporal cortex . In this study by Jeurissen, et al., (2014) , magnetic stimulation of the right DLPFC and right TPJ shows roles for right DLPFC (reasoning and utilitarian) and right TPJ (emotion) in moral impersonal and moral personal dilemmas respectively. TMS over the right DLPFC (disrupting neural activity here) leads to behavior changes consistent with less cognitive control over emotion . After right DLPFC stimulation, participants show less feelings of regret than after magnetic stimulation of the right TPJ. This last finding indicates that the right DLPFC is involved in evaluating the outcome of the decision process. In summary, this experiment by Jeurissen, et al., (2014) adds to evidence of a critical role of right DLPFC and right TPJ in moral decision-making and supports that hypothesis that the former is involved in judgments based on cognitive reasoning and anticipation of outcomes, whereas the latter is involved in emotional processing related to moral dilemmas.

Many different strategies exist for solving problems. Typical strategies include trial and error, applying algorithms, and using heuristics. To solve a large, complicated problem, it often helps to break the problem into smaller steps that can be accomplished individually, leading to an overall solution. The brain mechanisms involved in problem solving vary to some degree depending upon the sensory modalities involved in the problem and its solution, however, the prefrontal cortex is one brain region that appears to be centrally involved in all problem-solving. The prefrontal cortex is required for flexible shifts in attention, for representing the problem in working memory, and for holding steps in problem solving in working memory along with representations of future consequences of those actions permitting planning and execution of plans. Also implicated is the Default Mode Network (DMN) including medial prefrontal cortex, posterior cingulate cortex, and the inferior parietal module, and sometimes the lateral temporal cortex, hippocampus, and the precuneus. Moral reasoning involves a different set of brain areas, primarily the dorsolateral prefrontal cortex (DLPFC) and temporal-parietal junction (TPJ).

Review Questions

• an algorithm
• a heuristic
• a mental set
• trial and error

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Mushiake, H., Sakamoto, K., Saito, N., Inui, T., Aihara, K., & Tanji, J. (2009). Involvement of the prefrontal cortex in problem solving. International review of neurobiology , 85 , 1-11.

Jeurissen, D., Sack, A. T., Roebroeck, A., Russ, B. E., & Pascual-Leone, A. (2014). TMS affects moral judgment, showing the role of DLPFC and TPJ in cognitive and emotional processing. Frontiers in neuroscience , 8 , 18.

Kahneman, D. (2011). Thinking, fast and slow. New York: Farrar, Straus, and Giroux.

Koenigshofer, K. A. (2017). General Intelligence: Adaptation to Evolutionarily Familiar Abstract Relational Invariants, Not to Environmental or Evolutionary Novelty. The Journal of Mind and Behavior , 119-153.

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Mushiake, H., Sakamoto, K., Saito, N., Inui, T., Aihara, K., & Tanji, J. (2009). Involvement of the prefrontal cortex in problem solving. International review of neurobiology , 85 , 1-11. https://www.sciencedirect.com/scienc...74774209850010

Attributions

"Overview," "Problem Solving Strategies," adapted from Problem Solving by OpenStax Colleg licensed CC BY-NC 4.0 via OER Commons

"Defining Problems," "Problem Solving as a Search Problem," "Creative Cognition," "Brain Mechanisms in Problem-Solving" adapted by Kenneth A. Koenigshofer, Ph.D., from 2.1, 2.2, 2.3, 2.4, 2.5, 2.6 in Cognitive Psychology and Cognitive Neuroscience (Wikibooks) https://en.wikibooks.org/wiki/Cognit...e_Neuroscience ; unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0 . Legal ; the LibreTexts libraries are Powered by MindTouch

Moral Reasoning was written by Kenneth A. Koenigshofer, Ph.D, Chaffey College.

Categories and Concepts

People form mental concepts of categories of objects, which permit them to respond appropriately to new objects they encounter. Most concepts cannot be strictly defined but are organized around the “best” examples or prototypes, which have the properties most common in the category. Objects fall into many different categories, but there is usually a most salient one, called the basic-level category, which is at an intermediate level of specificity (e.g., chairs, rather than furniture or desk chairs). Concepts are closely related to our knowledge of the world, and people can more easily learn concepts that are consistent with their knowledge. Theories of concepts argue either that people learn a summary description of a whole category or else that they learn exemplars of the category. Recent research suggests that there are different ways to learn and represent concepts and that they are accomplished by different neural systems.

• Understand the problems with attempting to define categories.
• Understand typicality and fuzzy category boundaries.
• Learn about theories of the mental representation of concepts.
• Learn how knowledge may influence concept learning.

Introduction

Consider the following set of objects: some dust, papers, a computer monitor, two pens, a cup, and an orange. What do these things have in common? Only that they all happen to be on my desk as I write this. This set of things can be considered a category , a set of objects that can be treated as equivalent in some way. But, most of our categories seem much more informative—they share many properties. For example, consider the following categories: trucks, wireless devices, weddings, psychopaths, and trout. Although the objects in a given category are different from one another, they have many commonalities. When you know something is a truck, you know quite a bit about it. The psychology of categories concerns how people learn, remember, and use informative categories such as trucks or psychopaths.

The mental representations we form of categories are called concepts . There is a category of trucks in the actual physical world, and I also have a concept of trucks in my head. We assume that people’s concepts correspond more or less closely to the actual category, but it can be useful to distinguish the two, as when someone’s concept is not really correct.

Concepts are at the core of intelligent behavior . We expect people to be able to know what to do in new situations and when confronting new objects. If you go into a new classroom and see chairs, a blackboard, a projector, and a screen, you know what these things are and how they will be used. You’ll sit on one of the chairs and expect the instructor to write on the blackboard or project something onto the screen. You do this even if you have never seen any of these particular objects before , because you have concepts of classrooms, chairs, projectors, and so forth, that tell you what they are and what you’re supposed to do with them. Furthermore, if someone tells you a new fact about the projector—for example, that it has a halogen bulb—you are likely to extend this fact to other projectors you encounter. In short, concepts allow you to extend what you have learned about a limited number of objects to a potentially infinite set of entities (i.e. generalization ). Notice how categories and concepts arise from similarity, one of the abstract features of the world that has been genetically internalized into the brain during evolution , creating an innate disposition of brains to search for and to represent groupings of similar things, forming one component of general intelligence. One property of the human brain that distinguishes us from other animals is the high degrees of abstraction in similarity relations that the human brain is capable of encoding compared to the brains of non-human animals (Koenigshofer, 2017).

Simpler organisms, such as animals and human infants, also have concepts ( Mareschal, Quinn, & Lea, 2010 ). Squirrels may have a concept of predators, for example, that is specific to their own lives and experiences. However, animals likely have many fewer concepts and cannot understand complex concepts such as mortgages or musical instruments.

You know thousands of categories, most of which you have learned without careful study or instruction. Although this accomplishment may seem simple, we know that it isn’t, because it is difficult to program computers to solve such intellectual tasks. If you teach a learning program that a robin, a swallow, and a duck are all birds, it may not recognize a cardinal or peacock as a bird. However, this shortcoming in computers may be at least partially overcome when the type of processing used is parallel distributed processing as employed in artificial neural networks (Koenigshofer, 2017), discussed in this chapter. As we’ll shortly see, the problem for computers is that objects in categories are often surprisingly diverse.

Nature of Categories

Traditionally, it has been assumed that categories are well-defined . This means that you can give a definition that specifies what is in and out of the category. Such a definition has two parts. First, it provides the necessary features for category membership: What must objects have in order to be in it? Second, those features must be jointly sufficient for membership: If an object has those features, then it is in the category. For example, if I defined a dog as a four-legged animal that barks, this would mean that every dog is four-legged, an animal, and barks, and also that anything that has all those properties is a dog.

Unfortunately, it has not been possible to find definitions for many familiar categories. Definitions are neat and clear-cut; the world is messy and often unclear. For example, consider our definition of dogs. In reality, not all dogs have four legs; not all dogs bark. I knew a dog that lost her bark with age (this was an improvement); no one doubted that she was still a dog. It is often possible to find some necessary features (e.g., all dogs have blood and breathe), but these features are generally not sufficient to determine category membership (you also have blood and breathe but are not a dog).

Even in domains where one might expect to find clear-cut definitions, such as science and law, there are often problems. For example, many people were upset when Pluto was downgraded from its status as a planet to a dwarf planet in 2006. Upset turned to outrage when they discovered that there was no hard-and-fast definition of planethood: “Aren’t these astronomers scientists? Can’t they make a simple definition?” In fact, they couldn’t. After an astronomical organization tried to make a definition for planets, a number of astronomers complained that it might not include accepted planets such as Neptune and refused to use it. If everything looked like our Earth, our moon, and our sun, it would be easy to give definitions of planets, moons, and stars, but the universe has not conformed to this ideal.

Fuzzy Categories

Borderline items.

Experiments also showed that the psychological assumptions of well-defined categories were not correct. Hampton ( 1979 ) asked subjects to judge whether a number of items were in different categories. He did not find that items were either clear members or clear nonmembers. Instead, he found many items that were just barely considered category members and others that were just barely not members, with much disagreement among subjects. Sinks were barely considered as members of the kitchen utensil category, and sponges were barely excluded. People just included seaweed as a vegetable and just barely excluded tomatoes and gourds. Hampton found that members and nonmembers formed a continuum, with no obvious break in people’s membership judgments. If categories were well defined, such examples should be very rare. Many studies since then have found such borderline members that are not clearly in or clearly out of the category.

McCloskey and Glucksberg ( 1978 ) found further evidence for borderline membership by asking people to judge category membership twice, separated by two weeks. They found that when people made repeated category judgments such as “Is an olive a fruit?” or “Is a sponge a kitchen utensil?” they changed their minds about borderline items—up to 22 percent of the time. So, not only do people disagree with one another about borderline items, they disagree with themselves! As a result, researchers often say that categories are fuzzy , that is, they have unclear boundaries that can shift over time.

A related finding that turns out to be most important is that even among items that clearly are in a category, some seem to be “better” members than others ( Rosch, 1973 ). Among birds, for example, robins and sparrows are very typical . In contrast, ostriches and penguins are very atypical (meaning not typical). If someone says, “There’s a bird in my yard,” the image you have will be of a smallish passerine bird such as a robin, not an eagle or hummingbird or turkey.

You can find out which category members are typical merely by asking people. Table 1 shows a list of category members in order of their rated typicality. Typicality is perhaps the most important variable in predicting how people interact with categories. The following text box is a partial list of what typicality influences.

We can understand the two phenomena of borderline members and typicality as two sides of the same coin. Think of the most typical category member: This is often called the category prototype . Items that are less and less similar to the prototype become less and less typical. At some point, these less typical items become so atypical that you start to doubt whether they are in the category at all. Is a rug really an example of furniture? It’s in the home like chairs and tables, but it’s also different from most furniture in its structure and use. From day to day, you might change your mind as to whether this atypical example is in or out of the category. So, changes in typicality ultimately lead to borderline members.

Source of Typicality

Intuitively, it is not surprising that robins are better examples of birds than penguins are, or that a table is a more typical kind of furniture than is a rug. But given that robins and penguins are known to be birds, why should one be more typical than the other? One possible answer is the frequency with which we encounter the object: We see a lot more robins than penguins, so they must be more typical. Frequency does have some effect, but it is actually not the most important variable ( Rosch, Simpson, & Miller, 1976 ). For example, I see both rugs and tables every single day, but one of them is much more typical as furniture than the other.

The best account of what makes something typical comes from Rosch and Mervis’s ( 1975 ) family resemblance theory . They proposed that items are likely to be typical if they (a) have the features that are frequent in the category and (b) do not have features frequent in other categories. Let’s compare two extremes, robins and penguins. Robins are small flying birds that sing, live in nests in trees, migrate in winter, hop around on your lawn, and so on. Most of these properties are found in many other birds. In contrast, penguins do not fly, do not sing, do not live in nests or in trees, do not hop around on your lawn. Furthermore, they have properties that are common in other categories, such as swimming expertly and having wings that look and act like fins. These properties are more often found in fish than in birds.

According to Rosch and Mervis, then, it is not because a robin is a very common bird that makes it typical. Rather, it is because the robin has the shape, size, body parts, and behaviors that are very common (i.e. most similar) among birds—and not common among fish, mammals, bugs, and so forth.

In a classic experiment, Rosch and Mervis ( 1975 ) made up two new categories, with arbitrary features. Subjects viewed example after example and had to learn which example was in which category. Rosch and Mervis constructed some items that had features that were common in the category and other items that had features less common in the category. The subjects learned the first type of item before they learned the second type. Furthermore, they then rated the items with common features as more typical. In another experiment, Rosch and Mervis constructed items that differed in how many features were shared with a different category. The more features were shared, the longer it took subjects to learn which category the item was in. These experiments, and many later studies, support both parts of the family resemblance theory.

Category Hierarchies

Many important categories fall into hierarchies , in which more concrete categories are nested inside larger, abstract categories. For example, consider the categories: brown bear, bear, mammal, vertebrate, animal, entity. Clearly, all brown bears are bears; all bears are mammals; all mammals are vertebrates; and so on. Any given object typically does not fall into just one category—it could be in a dozen different categories, some of which are structured in this hierarchical manner. Examples of biological categories come to mind most easily, but within the realm of human artifacts, hierarchical structures can readily be found: desk chair, chair, furniture, artifact, object.

Brown ( 1958 ), a child language researcher, was perhaps the first to note that there seems to be a preference for which category we use to label things. If your office desk chair is in the way, you’ll probably say, “Move that chair,” rather than “Move that desk chair” or “piece of furniture.” Brown thought that the use of a single, consistent name probably helped children to learn the name for things. And, indeed, children’s first labels for categories tend to be exactly those names that adults prefer to use ( Anglin, 1977 ).

This preference is referred to as a preference for the basic level of categorization , and it was first studied in detail by Eleanor Rosch and her students ( Rosch, Mervis, Gray, Johnson, & Boyes-Braem, 1976 ). The basic level represents a kind of Goldilocks effect, in which the category used for something is not too small (northern brown bear) and not too big (animal), but is just right (bear). The simplest way to identify an object’s basic-level category is to discover how it would be labeled in a neutral situation. Rosch et al. ( 1976 ) showed subjects pictures and asked them to provide the first name that came to mind. They found that 1,595 names were at the basic level, with 14 more specific names ( subordinates ) used. Only once did anyone use a more general name ( superordinate ). Furthermore, in printed text, basic-level labels are much more frequent than most subordinate or superordinate labels (e.g., Wisniewski & Murphy, 1989 ).

The preference for the basic level is not merely a matter of labeling. Basic-level categories are usually easier to learn. As Brown noted, children use these categories first in language learning, and superordinates are especially difficult for children to fully acquire. [1] People are faster at identifying objects as members of basic-level categories ( Rosch et al., 1976 ).

Rosch et al. ( 1976 ) initially proposed that basic-level categories cut the world at its joints, that is, merely reflect the big differences between categories like chairs and tables or between cats and mice that exist in the world. However, it turns out that which level is basic is not universal. North Americans are likely to use names like tree, fish , and bird to label natural objects. But people in less industrialized societies seldom use these labels and instead use more specific words, equivalent to elm, trout, and finch ( Berlin, 1992 ). Because Americans and many other people living in industrialized societies know so much less than our ancestors did about the natural world, our basic level has “moved up” to what would have been the superordinate level a century ago. Furthermore, experts in a domain often have a preferred level that is more specific than that of non-experts. Birdwatchers see sparrows rather than just birds, and carpenters see roofing hammers rather than just hammers ( Tanaka & Taylor, 1991 ). This all suggests that the preferred level is not (only) based on how different categories are in the world, but that people’s knowledge and interest in the categories has an important effect.

One explanation of the basic-level preference is that basic-level categories are more differentiated: The category members are similar to one another, but they are different from members of other categories ( Murphy & Brownell, 1985 ; Rosch et al., 1976 ). (The alert reader will note a similarity to the explanation of typicality I gave above. However, here we’re talking about the entire category and not individual members.) Chairs are pretty similar to one another, sharing a lot of features (legs, a seat, a back, similar size and shape); they also don’t share that many features with other furniture. Superordinate categories are not as useful because their members are not very similar to one another. What features are common to most furniture? There are very few. Subordinate categories are not as useful, because they’re very similar to other categories: Desk chairs are quite similar to dining room chairs and easy chairs. As a result, it can be difficult to decide which subordinate category an object is in ( Murphy & Brownell, 1985 ). Experts can differ from novices in which categories are the most differentiated, because they know different things about the categories, therefore changing how similar the categories are.

[1] This is a controversial claim, as some say that infants learn superordinates before anything else (Mandler, 2004). However, if true, then it is very puzzling that older children have great difficulty learning the correct meaning of words for superordinates, as well as in learning artificial superordinate categories (Horton & Markman, 1980; Mervis, 1987). However, it seems fair to say that the answer to this question is not yet fully known.

Theories of Concept Representation

Now that we know these facts about the psychology of concepts, the question arises of how concepts are mentally represented. There have been two main answers. The first, somewhat confusingly called the prototype theory suggests that people have a summary representation of the category, a mental description that is meant to apply to the category as a whole. (The significance of summary will become apparent when the next theory is described.) This description can be represented as a set of weighted features ( Smith & Medin, 1981 ). The features are weighted by their frequency in the category. For the category of birds, having wings and feathers would have a very high weight; eating worms would have a lower weight; living in Antarctica would have a lower weight still, but not zero, as some birds do live there.

The idea behind prototype theory is that when you learn a category, you learn a general description that applies to the category as a whole: Birds have wings and usually fly; some eat worms; some swim underwater to catch fish. People can state these generalizations, and sometimes we learn about categories by reading or hearing such statements (“The kimodo dragon can grow to be 10 feet long”).

When you try to classify an item, you see how well it matches that weighted list of features. For example, if you saw something with wings and feathers fly onto your front lawn and eat a worm, you could (unconsciously) consult your concepts and see which ones contained the features you observed. This example possesses many of the highly weighted bird features, and so it should be easy to identify as a bird.

This theory readily explains the phenomena we discussed earlier. Typical category members have more, higher-weighted features. Therefore, it is easier to match them to your conceptual representation. Less typical items have fewer or lower-weighted features (and they may have features of other concepts). Therefore, they don’t match your representation as well (less similarity). This makes people less certain in classifying such items. Borderline items may have features in common with multiple categories or not be very close to any of them. For example, edible seaweed does not have many of the common features of vegetables but also is not close to any other food concept (meat, fish, fruit, etc.), making it hard to know what kind of food it is.

A very different account of concept representation is the exemplar theory ( exemplar being a fancy name for an example; Medin & Schaffer, 1978 ). This theory denies that there is a summary representation. Instead, the theory claims that your concept of vegetables is remembered examples of vegetables you have seen. This could of course be hundreds or thousands of exemplars over the course of your life, though we don’t know for sure how many exemplars you actually remember.

How does this theory explain classification? When you see an object, you (unconsciously) compare it to the exemplars in your memory, and you judge how similar it is to exemplars in different categories. For example, if you see some object on your plate and want to identify it, it will probably activate memories of vegetables, meats, fruit, and so on. In order to categorize this object, you calculate how similar it is to each exemplar in your memory. These similarity scores are added up for each category. Perhaps the object is very similar to a large number of vegetable exemplars, moderately similar to a few fruit, and only minimally similar to some exemplars of meat you remember. These similarity scores are compared, and the category with the highest score is chosen . [2]

Why would someone propose such a theory of concepts? One answer is that in many experiments studying concepts, people learn concepts by seeing exemplars over and over again until they learn to classify them correctly. Under such conditions, it seems likely that people eventually memorize the exemplars ( Smith & Minda, 1998 ). There is also evidence that close similarity to well-remembered objects has a large effect on classification . Allen and Brooks ( 1991 ) taught people to classify items by following a rule. However, they also had their subjects study the items, which were richly detailed. In a later test, the experimenters gave people new items that were very similar to one of the old items but were in a different category. That is, they changed one property so that the item no longer followed the rule. They discovered that people were often fooled by such items. Rather than following the category rule they had been taught, they seemed to recognize the new item as being very similar to an old one and so put it, incorrectly, into the same category.

Many experiments have been done to compare the prototype and exemplar theories. Overall, the exemplar theory seems to have won most of these comparisons . However, the experiments are somewhat limited in that they usually involve a small number of exemplars that people view over and over again. It is not so clear that exemplar theory can explain real-world classification in which people do not spend much time learning individual items (how much time do you spend studying squirrels? or chairs?). Also, given that some part of our knowledge of categories is learned through general statements we read or hear, it seems that there must be room for a summary description separate from exemplar memory.

Many researchers would now acknowledge that concepts are represented through multiple cognitive systems. For example, your knowledge of dogs may be in part through general descriptions such as “dogs have four legs.” But you probably also have strong memories of some exemplars (your family dog, Lassie) that influence your categorization. Furthermore, some categories also involve rules (e.g., a strike in baseball). How these systems work together is the subject of current study.

[2] Actually, the decision of which category is chosen is more complex than this, but the details are beyond this discussion.

The final topic has to do with how concepts fit with our broader knowledge of the world. We have been talking very generally about people learning the features of concepts. For example, they see a number of birds and then learn that birds generally have wings, or perhaps they remember bird exemplars. From this perspective, it makes no difference what those exemplars or features are—people just learn them. But consider two possible concepts of buildings and their features in Table 2.

Imagine you had to learn these two concepts by seeing exemplars of them, each exemplar having some of the features listed for the concept (as well as some idiosyncratic features). Learning the donker concept would be pretty easy. It seems to be a kind of underwater building, perhaps for deep-sea explorers. Its features seem to go together. In contrast, the blegdav doesn’t really make sense. If it’s in the desert, how can you get there by submarine, and why do they have polar bears as pets? Why would farmers live in the desert or use submarines? What good would steel windows do in such a building? This concept seems peculiar. In fact, if people are asked to learn new concepts that make sense, such as donkers, they learn them quite a bit faster than concepts such as blegdavs that don’t make sense ( Murphy & Allopenna, 1994 ). Furthermore, the features that seem connected to one another (such as being underwater and getting there by submarine) are learned better than features that don’t seem related to the others (such as being red).

Such effects demonstrate that when we learn new concepts, we try to connect them to the knowledge we already have about the world. If you were to learn about a new animal that doesn’t seem to eat or reproduce, you would be very puzzled and think that you must have gotten something wrong. By themselves, the prototype and exemplar theories don’t predict this. They simply say that you learn descriptions or exemplars, and they don’t put any constraints on what those descriptions or exemplars are. However, the knowledge approach to concepts emphasizes that concepts are meant to tell us about real things in the world, and so our knowledge of the world is used in learning and thinking about concepts.

We can see this effect of knowledge when we learn about new pieces of technology. For example, most people could easily learn about tablet computers (such as iPads) when they were first introduced by drawing on their knowledge of laptops, cell phones, and related technology. Of course, this reliance on past knowledge can also lead to errors, as when people don’t learn about features of their new tablet that weren’t present in their cell phone or expect the tablet to be able to do something it can’t.

One important aspect of people’s knowledge about categories is called psychological essentialism ( Gelman, 2003 ; Medin & Ortony, 1989 ). People tend to believe that some categories—most notably natural kinds such as animals, plants, or minerals—have an underlying property that is found only in that category and that causes its other features. Most categories don’t actually have essences, but this is sometimes a firmly held belief. For example, many people will state that there is something about dogs, perhaps some specific gene or set of genes, that all dogs have and that makes them bark, have fur, and look the way they do. Therefore, decisions about whether something is a dog do not depend only on features that you can easily see but also on the assumed presence of this cause.

Belief in an essence can be revealed through experiments describing fictional objects. Keil ( 1989 ) described to adults and children a fiendish operation in which someone took a raccoon, dyed its hair black with a white stripe down the middle, and implanted a “sac of super-smelly yucky stuff” under its tail. The subjects were shown a picture of a skunk and told that this is now what the animal looks like. What is it? Adults and children over the age of 4 all agreed that the animal is still a raccoon. It may look and even act like a skunk, but a raccoon cannot change its stripes (or whatever!)—it will always be a raccoon.

Importantly, the same effect was not found when Keil described a coffeepot that was operated on to look like and function as a bird feeder. Subjects agreed that it was now a bird feeder. Artifacts don’t have an essence.

Signs of essentialism include (a) objects are believed to be either in or out of the category, with no in-between; (b) resistance to change of category membership or of properties connected to the essence; and (c) for living things, the essence is passed on to progeny.

Essentialism is probably helpful in dealing with much of the natural world, but it may be less helpful when it is applied to humans. Considerable evidence suggests that people think of gender, racial, and ethnic groups as having essences, which serves to emphasize the difference between groups and even justify discrimination ( Hirschfeld, 1996 ). Historically, group differences were described by inheriting the blood of one’s family or group. “Bad blood” was not just an expression but a belief that negative properties were inherited and could not be changed. After all, if it is in the nature of “those people” to be dishonest (or clannish or athletic ...), then that could hardly be changed, any more than a raccoon can change into a skunk.

Research on categories of people is an exciting ongoing enterprise, and we still do not know as much as we would like to about how concepts of different kinds of people are learned in childhood and how they may (or may not) change in adulthood. Essentialism doesn’t apply only to person categories, but it is one important factor in how we think of groups.

Concepts are central to our everyday thought. When we are planning for the future or thinking about our past, we think about specific events and objects in terms of their categories. If you’re visiting a friend with a new baby, you have some expectations about what the baby will do, what gifts would be appropriate, how you should behave toward it, and so on. Knowing about the category of babies helps you to effectively plan and behave when you encounter this child you’ve never seen before. Such inferences from knowledge about a category are highly adaptive and an important component of thinking and intelligence.

Learning about those categories is a complex process that involves seeing exemplars (babies), hearing or reading general descriptions (“Babies like black-and-white pictures”), general knowledge (babies have kidneys), and learning the occasional rule (all babies have a rooting reflex). Current research is focusing on how these different processes take place in the brain. It seems likely that these different aspects of concepts are accomplished by different neural structures ( Maddox & Ashby, 2004 ). However, it is clear that the brain is genetically predisposed to seek out similarities in the environment and to represent groupings of things forming categories that can be used to make inferences about new instances of the category which have never been encountered before. In this way knowledge is organized and expectations from this knowledge allow improved adaptation to newly encountered environmental objects and situations by virtue of their similarity to a known category previously formed (Koenigshofer, 2017).

Another interesting topic is how concepts differ across cultures. As different cultures have different interests and different kinds of interactions with the world, it seems clear that their concepts will somehow reflect those differences. On the other hand, the structure of the physical world also imposes a strong constraint on what kinds of categories are actually useful. The interplay of culture, the environment, and basic cognitive processes in establishing concepts has yet to be fully investigated.

Discussion Questions

• Pick a couple of familiar categories and try to come up with definitions for them. When you evaluate each proposal (a) is it in fact accurate as a definition, and (b) is it a definition that people might actually use in identifying category members?
• For the same categories, can you identify members that seem to be “better” and “worse” members? What about these items makes them typical and atypical?
• Going around the room, point to some common objects (including things people are wearing or brought with them) and identify what the basic-level category is for that item. What are superordinate and subordinate categories for the same items?
• List some features of a common category such as tables. The knowledge view suggests that you know reasons for why these particular features occur together. Can you articulate some of those reasons? Do the same thing for an animal category.
• Choose three common categories: a natural kind, a human artifact, and a social event. Discuss with class members from other countries or cultures whether the corresponding categories in their cultures differ. Can you make a hypothesis about when such categories are likely to differ and when they are not?
• Allen, S. W., & Brooks, L. R. (1991). Specializing the operation of an explicit rule. Journal of Experimental Psychology: General, 120 , 3–19.
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• Hirschfeld, L. A. (1996). Race in the making: Cognition, culture, and the child's construction of human kinds . Cambridge, MA: MIT Press.
• Horton, M. S., & Markman, E. M. (1980). Developmental differences in the acquisition of basic and superordinate categories. Child Development , 51, 708–719.
• Keil, F. C. (1989). Concepts, kinds, and cognitive development . Cambridge, MA: MIT Press.
• Koenigshofer, K. A. (2017). General Intelligence: Adaptation to Evolutionarily Familiar Abstract Relational Invariants, Not to Environmental or Evolutionary Novelty. The Journal of Mind and Behavior , 38(2):119-153.
• Maddox, W. T., & Ashby, F. G. (2004). Dissociating explicit and procedural-based systems of perceptual category learning. Behavioural Processes, 66 , 309–332.
• Mandler, J. M. (2004). The foundations of mind: Origins of conceptual thought . Oxford, UK: Oxford University Press.
• Mareschal, D., Quinn, P. C., & Lea, S. E. G. (Eds.) (2010). The making of human concepts . Oxford, UK: Oxford University Press.
• McCloskey, M. E., & Glucksberg, S. (1978). Natural categories: Well defined or fuzzy sets? Memory & Cognition, 6 , 462–472.
• Medin, D. L., & Ortony, A. (1989). Psychological essentialism. In S. Vosniadou & A. Ortony (Eds.), Similarity and analogical reasoning (pp. 179–195). Cambridge, UK: Cambridge University Press.
• Medin, D. L., & Schaffer, M. M. (1978). Context theory of classification learning. Psychological Review , 85, 207–238.
• Mervis, C. B. (1987). Child-basic object categories and early lexical development. In U. Neisser (Ed.), Concepts and conceptual development: Ecological and intellectual factors in categorization (pp. 201–233). Cambridge, UK: Cambridge University Press.
• Murphy, G. L., & Allopenna, P. D. (1994). The locus of knowledge effects in concept learning. Journal of Experimental Psychology: Learning, Memory, and Cognition, 20 , 904–919.
• Murphy, G. L., & Brownell, H. H. (1985). Category differentiation in object recognition: Typicality constraints on the basic category advantage. Journal of Experimental Psychology: Learning, Memory, and Cognition, 11 , 70–84.
• Norenzayan, A., Smith, E. E., Kim, B. J., & Nisbett, R. E. (2002). Cultural preferences for formal versus intuitive reasoning. Cognitive Science, 26 , 653–684.
• Rosch, E., & Mervis, C. B. (1975). Family resemblance: Studies in the internal structure of categories. Cognitive Psychology , 7, 573–605.
• Rosch, E., Mervis, C. B., Gray, W., Johnson, D., & Boyes-Braem, P. (1976). Basic objects in natural categories. Cognitive Psychology, 8 , 382–439.
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• Rosch, E. H. (1973). On the internal structure of perceptual and semantic categories. In T. E. Moore (Ed.), Cognitive development and the acquisition of language (pp. 111–144). New York, NY: Academic Press.
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• Smith, J. D., & Minda, J. P. (1998). Prototypes in the mist: The early epochs of category learning. Journal of Experimental Psychology: Learning, Memory, and Cognition, 24 , 1411–1436.
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• Gregory Murphy is Professor of Psychology at New York University. He previously taught at the University of Illinois and Brown University. His research focuses on concepts and reasoning, and he is the author of The Big Book of Concepts (MIT Press, 2002).

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Adapted by Kenneth Koenigshofer, PhD, from Categories and Concepts by Gregory Murphy , licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License . Permissions beyond the scope of this license may be available in our Licensing Agreement .

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Murphy, G. (2021). Categories and concepts. In R. Biswas-Diener & E. Diener (Eds), Noba textbook series: Psychology. Champaign, IL: DEF publishers. Retrieved from http://noba.to/6vu4cpkt

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Categories, Concepts, and Conceptual Development

Vladimir m. sloutsky.

a Department of Psychology, The Ohio State University

Wei (Sophia) Deng

b Department of Psychology, University of Macau

Concepts (i.e., lexicalized classes of real or fictitious entities) play a central role in many human intellectual activities, including planning, thinking, reasoning, problem solving, and decision making. How do people acquire concepts in the course of development and learning and use them in their thinking about the world? In this article, we attempt to provide an overview of conceptual development. We suggest that concepts can originate (1) in interactions with the world and get lexicalized later or (2) in the language and get grounded later. The first route is from category learning to a concept, and we discuss this route by focusing on the mechanisms of category learning and developmental changes in these mechanisms. The second route is from a word to a concept, and we discuss this route by focusing on inferring word meanings without visual referents. We then consider proposals of how concepts get organized into networks and hierarchies.

INTRODUCTION

There are many abilities that reflect the remarkable intelligence of humans: people make inferences, develop and use scientific theories, make laws, preserve knowledge and pass it onto new generations, write fiction, reason about past and future, and make counterfactual arguments.

For example, upon learning that all animals are heterotrophs (i.e., they use other organisms as a source of energy) one may conclude that cats are heterotrophs as well. Furthermore, this conclusion follows with logical necessity from knowledge of taxonomy (i.e., cat is a proper subset of animal ) and of the logic of classes (i.e., if class X is properly included in class Y, then whatever is attributable to Y is attributable to X). Therefore, all properties of class Y (i.e., animals) is shared by class X (i.e., cats). One can also entertain a counterfactual (which in this case follows with logical necessity): if cats were not heterotrophs, they would not have been animals .

All these abilities are based on concepts, and, in fact, it is difficult to imagine reasoning (or any intellectual activity) without concepts. Therefore, one of the most interesting challenges in the study of human cognitive development is to understand how people acquire concepts in the course of development and learning and use them in their thinking about the world.

In this article, we attempt to provide a brief overview of conceptual development. Because concepts are classes (rather than individuals) they are more general or abstract . We use these terms interchangeably: abstract or general versus concrete or specific is a dimension reflecting how inclusive the class is. More inclusive classes are also more abstract and general. For example, cats is a more abstract class than an individual (e.g., my cat Fluffy ), but less abstract than a more inclusive class of mammals or living things . More abstract concepts can be formed by the process of abstraction or generalization , whereby some (presumably more important) properties are preserved, whereas other (presumably less important) properties are dropped. For example, it is easy to see that triangle is more abstract than equilateral triangle, whereas polygon (a shape with any number of edges and vertices) is more abstract than triangle .

To distinguish concepts from other classes and sets (such as categories), we define concepts as lexicalized classes of real or fictitious entities. We also suggest that concepts can originate (1) in interactions with the world and get lexicalized later or (2) in the language and get grounded later. For example, it is conceivable that a toddler has enough encounters with dogs to form a category before learning the word for it. In contrast, the concept germ cannot originate in experience and has to originate in language. Therefore, concepts can be acquired in a bottom-up manner (i.e., originating in experience) or in a top-down manner (i.e., originating in language).

Because bottom-up concepts (i.e., those acquired in a bottom-up manner) are tightly linked to experience, there are a number of constraints as to what these concepts might be. In general, many of these are “embodiment” constraints on the concepts (cf., Yu & Smith, 2012 , for ideas on the role of embodiment in the early word learning). Among these constraints are (1) these concepts should primarily include perceptible (and possibly actionable) entities, (2) these concepts should be mostly based on objects, (3) these objects should predominate in a particular child’s experience, (4) these objects’ sizes should be within a certain range, and (5) there should be common and frequently used labels within the child’s native language to denote these objects.

In contrast, the top-down concepts (those acquired in a top-down manner) have a different set of constraints. Because these concepts originate in language, the primary constraints are (1) parents’ education and vocabulary, (2) topics of conversations with and around the child, (3) access to books and media and types of available books and media, and (4) access to formal education. Second language learning may offer an interesting illustration of these latter constraints, as both authors of this paper learned English as a second language. For example, because of the literature we read while learning English, we learned words like pride , prejudice , vanity , expectation , and curiosity long before learning words for eggshell , faucet , fingernail , eyelash , or tire .

In what follows, we will overview concepts and conceptual behaviors, introduce category learning as an important step in conceptual development, discuss both ways of acquiring concepts (i.e., the bottom-up and the top-down), and consider how concepts get organized into coherent networks that promote understanding of the world as well as thinking and reasoning about it.

What are Concepts?

In the simplest possible way, concepts can be defined as lexicalized categories , or equivalence classes . What is an equivalence class? In his chapter focusing on concepts (Chapter XII of the Principles of Psychology ), William James (1983/1890) wrote: “Our principle only lays it down that the mind makes continual use of the notion of sameness, and, if deprived of it, would have a different structure from what it has.” In other words, the mind can treat different things as if they were equivalent in some way. Once equivalence is established, it could be marked by lexicalization: by calling two different dogs d 1 and d 2 a Dog , we can express that as d 1 ≡ d 2 . When such an equivalence class (or category) is lexicalized, it becomes a concept . Lexicalization allows (a) accumulation of new (often unobservable) information about categories and (b) communicating and sharing this information with others. Examples of concepts vary from chairs (obviously, chairs are non-identical, but merely equivalent in some way) to odd numbers to extremely abstract concepts, such as cause or effect . If concepts are lexicalized categories, then development may proceed either from forming a category first to lexicalization or from acquiring a lexical item first to the formation of a category. As we discuss below, both types of progression can be observed in individual development.

We also suggest that regularities in both the world and in language are sources of conceptual development. In the world, members of many categories (e.g., cats or birds) share multiple observable features and therefore these categories can be learned without language. On the other hand, language also provides important input to conceptual development: many related categories (e.g., wolfs and dolphins) share some observable and many unobservable features and knowledge of these unobservable features comes from language. In what follows, we discuss both developments. We first review some principles of category learning – principles that may apply to learning of pre-linguistic categories. We then review how language can be a rich source of conceptual development by contributing to the formation of conceptual hierarchies.

Perceptual Groupings, Categories, Concepts, and Conceptual Networks

If concepts are lexicalized categories (some of which can be learned prior to language acquisition and some cannot be learned without language), then, like to categories to which they refer, conceptual behaviors may vary substantially in levels of complexity ranging from simple perceptual groupings of arbitrary categories, to full blown lexicalized concepts that are linked to other concepts which thereby form conceptual networks. The study of each type of conceptual behavior requires somewhat different research paradigms.

First, people can learn perceptual groupings or equivalence classes that are based on purely perceptual properties. Such groupings may include imposing categorical boundaries on sensory continua (known as categorical perception, e.g., Eimas, 1994 ), learning dot patterns coming from a single prototype and generalizing learning to distortions from the studied prototype, or forming a category based on image properties (see Bhatt & Quinn, 2010 , for a review). Perceptual groupings are the simplest form of categorization because they allow extending category membership on the basis of global familiarity. Therefore, if members of category A share some features, a novel item would be judged as a member of A to the extent that it has these features.

A more complicated variant of conceptual behavior requires one to learn two or more mutually exclusive categories (e.g., cats vs. dogs) at the same time. The categories are mutually exclusive because there are no members common to A and B (i.e., A∩B=⊘). This problem is more difficult than simple perceptual groupings because a decision of whether a novel item belongs to A or to B cannot be made on the basis of global familiarity (i.e., both A and B are equally familiar). The studied categories can be based on multiple correlated features (birds have wings, feathers, and beaks, whereas fish have scales, fins, and gills), few features (e.g., squirrels have a long, fluffy tail, whereas hamsters have a small tail), or relations among features (e.g., rectangles can be grouped into tall if the aspect ratio is less than 1, and wide if the aspect ratio is more than 1). The categories may be also deterministic (such that there is a subset of features that is sufficient to predict category membership with a 100% accuracy) or probabilistic (such that any feature or a combination of features predicts category membership only with a degree of probability). Therefore, to make a categorization decision, at the very minimum, some processing of two category structures is required. This task has been used in some studies with infants and animals, and in many category-learning studies with children and adults.

An even more complicated variant of conceptual behavior is the ability to lexicalize categories and use them in reasoning, inference, prediction, or judgment. Such lexicalized categories can be defined as concepts proper. Lexicalization is critical as it enables acquiring knowledge that may not be directly observable in a given situation (e.g., dogs are friendly pets, they like meat, and are taken to a vet for a physical exam). In other words, having a word for a category allows accumulation of knowledge from sources that are not based on direct observation of category members. These sources include conversations with others, books and other media sources, and formal education. Such concepts proper can be studied in a variety of tasks, including grouping of items, property listing, picture naming, and category judgment among others. A grouping task may require participants to put together items of the same kind (e.g., toys versus animals), whereas an attribute listing task may require a participant to list properties of categories (e.g., of cats, birds, or animals).

Finally, a conceptual network involves not only knowledge of concepts, but also of relations among these concepts. Take, for example, Newton’s second law (F = ma) that acceleration of a body is directly proportional to the net force acting on the body and inversely proportional to the mass of the body. Here, the concepts of mass , force , and acceleration are linked and are part of a broader conceptual network. For example, force is linked to work and power , whereas acceleration is linked to time and space . In other words, all these concepts are linked and each can be derived from others.

Such networks can be organized in a variety of ways; for example, networks of naturally occurring categories often have hierarchical, or taxonomical, organization (e.g., grey hound → dog → mammal → animal → living thing). One way of detecting such hierarchies is a classification task in which a diverse set of items is partitioned into N mutually exclusive and exhaustive sub-sets. These subsets can then be further partitioned into smaller groups or combined into larger groups. Although it has been argued that classification tasks may underestimate children’s concepts (the fact that a child may put together a dog and a bone does not mean that the child considers the two to be the same thing, see Fodor, 1972 ), classification tasks are useful in that they may reveal a limit on the kinds of concepts children may form.

In short, humans exhibit multiplicity of conceptual behaviors, some are universal and shared with other animals, whereas others are uniquely human. Overall, human conceptual repertoire ranges from perceptual groupings (something that can be also achieved by certain non-mammalian species) to conceptual networks that are likely to be unique to humans.

EARLY CONCEPTUAL DEVELOPMENT: FROM CATEGORIES TO EARLY CONCEPTS

Conceptual behaviors come in various forms: they range from more simple, universal, and early emerging forms (i.e., establishing equivalence between non-identical percepts) to rather complex, uniquely human, and late emerging forms (i.e., forming a conceptual network in a knowledge domain).

Category Structure and Category Learning

Are all categories the same? Perhaps not: although there is little doubt that categories differ in content, the most interesting distinctions pertain to category structure. Structural differences identified by researchers include syntactic differences (nouns versus verbs; e.g., Gentner, 1981 ), ontological differences (natural kinds versus artifacts; e.g., Barton & Komatzu, 1989 ), taxonomic differences (i.e., basic-level versus superordinate-level; e.g., Rosch & Mervis, 1975 ), differences in organizational principle (entity categories versus relational categories; e.g., Gentner & Kurtz, 2005 ), differences in concreteness (concrete versus abstract categories; e.g., Barsalou, 1999 ), differences in category coherence and confusability (e.g., Homa et al., 1979 ; Smith & Minda, 2000 ; Rouder & Ratcliff, 2004 ), and some other distinctions (for a review, see Medin, Lynch, & Solomon, 2000 ).

Kloos and Sloutsky (2008) proposed another structural distinction, one that could form the basis for many of the above distinctions. They proposed the idea of statistical density, that is a measure of category structure that (a) can (in principle) be measured independently rather than be inferred from participants’ patterns of response and (b) provides a continuous measure rather than a dichotomous one (which makes it well suited for capturing the graded nature of differences between categories). Conceptually, statistical density is a ratio of variance relevant for category membership to the total variance across members and non-members of the category. Intuitively, statistical density is a measure of how members of a category are separated from non-members (see Kloos & Sloutsky, 2008 , for a detailed discussion). For example, a category of small racing cars is dense (even when contrasted with other categories of vehicles) because there are multiple correlated features that distinguish this category. In contrast, a category of red things is sparse as there is a single feature common to the category members and distinguishing this category from any contrasting category.

The idea of statistical density has important implications for the development of category learning. One possibility is that category learning progresses from spontaneous learning of highly dense categories (i.e., when multiple dimensions are correlated within a category) to less spontaneous (and more guided or supervised) learning of sparser categories (i.e., only few dimensions are relevant; for example, members of a category are all red, but vary on multiple dimensions, such as shape, color, texture, and size).

Category Learning: What is the Mechanism and What Develops?

Category learning is the process by which one or more equivalence classes of discriminable entities are formed. How do people form these classes? One of the first ideas was that category learning is a variant of stimulus generalization: if a new item is sufficiently similar (i.e., exceeds some threshold value) to a member or members of an identified category, it would be included in this category. However, this simple and compelling idea may have difficulty explaining learning of categories based on a single dimension. For example, Shepard, Hovland, & Jenkins (1961) demonstrated that people easily learn single-dimension categories (e.g., black shapes vs. white shapes), even though within-category similarity (measured as stimulus confusability) may be small. Shepard et al. (1961) concluded that categories can be learned by selectively attending to a relevant dimension. These ideas have been captured in several influential models of categorization (e.g., Kruschke, 1992 ; Nosofsky, 1986 ; Love, Medin, & Gureckis, 2004 ).

Similar to Shepard et al. (1961) , these models suggested that categories of the same structure can be learned either by allocating attention to few relevant dimensions or by distributing attention across many dimensions. The way attention is allocated is consequential for how quickly the category will be learned and how the dimensions will be represented in memory. In particular, the former way of category learning is fast and efficient, but it may result in inattention to (and consequently, relatively poor memory for) irrelevant dimensions as these dimensions are ignored. In contrast, the latter way of learning may be slower and less efficient, but may result in attention allocated to all dimensions (and consequently, memory for both relevant and irrelevant dimensions).

Learning of similarity-based categories is a developmental default.

It is hardly controversial that selective attention undergoes protractive development (see Hanania & Smith, 2010; Lane & Pearson, 1982 ; Plude, Enns, & Brodeur, 1994 ; for reviews), with infants and young children tending to distribute attention ( Best, Yim, & Sloutsky, 2013 ; Plebanek & Sloutsky, 2017 ; Deng & Sloutsky, 2015a ). If infants and young children tend to distribute attention rather than to attend selectively, how do they learn categories? One idea is that first categories that infants and young children learn are categories that do not require selective attention – these are sufficiently perceptually distinct and have enough within-category structure to be learned without selective attention ( Sloutsky, 2010 ; Sloutsky & Fisher, 2004a ).

In an attempt to examine the mechanism of category learning and its potential change with development, Deng and Sloutsky (2015b) presented 4-year-olds, 6-year-olds, and adults with a category learning task, in which participants learned two categories. The categories had a rule-plus-similarity structure, such that there was a single deterministic (or rule) feature and multiple probabilistic features (see Figure 1 , for examples of stimuli). Therefore, this category structure allows examining what participants learn spontaneously: given the same set of stimuli, participants could learn either rule-based or similarity-based categories.

Examples of stimuli used in Deng and Sloutsky (2015b) . There were two family resemblance categories, with each training item including a single deterministic feature D (which perfectly distinguished between the two categories) and multiple probabilistic features P (with each providing imperfect probabilistic information about category membership). The body mark (introduced as a body button) was the D feature, and all the other features—the head, body, hands, feet, antennae, and tail—were the P features. Each row depicts items within a category, whereas each column identified an item role (e.g., switch item) and item type (e.g., P jalet D flurp ). The High-Match items were used in training and testing. The switch items, new-D, one-new-P, and all-new-P items were used only in testing. Neither prototype was shown in training or testing.

To establish what specifically was learned by participants, the authors presented them with categorization and memory testing (various test items are presented in Figure 1 ). Categorization trials included High-Match items (these were the items used in training), Switch items (these were the items that had the rule feature from one category and probabilistic features from another category), and All-new-P (these items had an old deterministic feature and all new probabilistic features). The goal of High-Match items was to test whether participants learned the category. The goal of Switch items was to examine what they learned about the category (i.e., whether they learned a rule-based or similarity-based category). Finally, the goal of All-new-P items was to examine whether participants could generalize on the basis of the rule features. There were also memory tests examining memory for the rule feature (i.e., New-D items) and each probabilistic feature (i.e., One-new-P items).

Results of categorization testing (see Figure 2 ) indicated that whereas participants of all age groups ably learned the categories (as evidenced by high performance on High-Match items), only older participants relied in their categorization on rule features (as evidenced by the above-chance performance on Switch items). In contrast, younger participants relied on the overall similarity (as evidenced by the below-chance performance on Switch items). In addition, younger participants were at chance for the All-new-P items, whereas older participants were reliably above chance. These results suggested that whereas 4-year-olds learned similarity-based categories, 6-year-olds and adults learned rule-based categories.

Categorization Performance: Proportion of rule-based responses by trial type and training condition for 4-year-old children, 6-year-old children and adults (After Deng & Sloutsky, 2015b , Experiment 1). Error bars represent standard error of the mean.

Researchers also tested participants’ memory for features. Memory results are important because they reflected what participants had learned about the categories. Whereas older children and adults remembered the rule features better than the probabilistic features, 4-year-olds remembered all the features equally well. Furthermore, follow-up studies indicated that 4-year-olds’ memory for probabilistic features was (at least numerically) better than that of older participants (cf. Sloutsky & Fisher, 2004b ). Taken together, these results indicated that whereas older children and adults spontaneously learned rule-based categories (if the structure supported such learning), young children spontaneously learned similarity-based categories.

To further examine mechanisms of early category learning, Deng & Sloutsky (2015b) attracted their attention to the deterministic feature by pointing to this feature on every training trial and commenting on the importance of this feature. Although in this condition 4-year-olds appeared to have learned a rule-based category (as evidenced by their categorization and generalization responses), they exhibited equivalently good memory for all features. Therefore, their categorization decisions pointed to selective attention, whereas their pattern of memory did not. These findings suggest that category representations and category decisions may be decoupled early in development, but they become coupled in the course of development.

The role of category labels in category learning.

Often times, children learn the category and the label, lexicalizing this category, concurrently. This happens, for example, when a new object is shown to the child and is labeled (e.g., “look, a dax”). In this case, the child needs to figure out what are the other objects that are daxes , or learn the category of dax . In the lab studies, this type of learning is referred to as category learning by classification. First examples of categories are introduced and labeled. Then the participant needs to predict which of the novel items also belong to this category (i.e., have the same category label). Given that participants learn the category and the linguistic label at the same time, it is reasonable to ask: How does the label affect learning? And does this role changes with development?

Several ideas have been proposed. Some have argued that from early in development, the category label is a category marker—an indicator that the items belong to the same category—and guides or supervises category learning ( Gelman & Markman, 1986 ; Gelman, 2003 ; Waxman & Markow, 1995 , Waxman & Gelman, 2009 ; Welder & Graham, 2001 ; Westermann & Mareschal, 2014 ). In contrast, others ( Deng & Sloutsky, 2012 ; 2013 ; 2015a ; Sloutsky & Lo, 1999 ; Sloutsky & Fisher, 2004a ; Sloutsky, Lo, & Fisher, 2001 ) have argued that, at least early in development, labels are akin to other features of items, but their role may change in the course of development. How could these positions be tested and contrasted?

In an attempt to distinguish between labels being features and category markers, Yamauchi and Markman (1998 , 2000 ) developed a paradigm potentially capable of settling the issue. The paradigm is based on the following idea. Imagine two categories A (labeled “A”) and B (labeled “B”), each having five binary dimensions (e.g., Size: large vs. small, Color: black vs. white, Shape: square vs. circle, Luminance: bright vs. dark, and Texture: smooth vs. rough). The prototype of Category A has all values denoted by “1” (i.e., “A”, 1, 1, 1, 1, 1) and the prototype of Category B has all values denoted by “0” (i.e., “B”, 0, 0, 0, 0, 0). There are two inter-related generalization tasks – classification and induction. The goal of the classification task is to infer category membership (and hence the label) on the basis of presented features. For example, participants are presented with all the values for an item (e.g., ?, 0, 1, 1, 1, 1) and have to predict category label “A” or “B”. In contrast, the goal of the induction task is to infer a feature on the basis of category label and other presented features. For example, given an item (e.g., “A”, 1, ?, 1, 0, 1), participants have to predict the value of the missing feature. A critical manipulation that could illuminate the role of labels is the “low-match” condition. For low-match induction, participants were presented with an item “A”, ?, 0, 1, 0, 0 (which had the label of A, but more features in common with the prototype of Category B) and asked to infer the missing feature. For low-match classification, participants were presented with an item “?”, 1, 0, 1, 0, 0 (which again had more features in common with the prototype of Category B) and asked to infer the missing label.

These researchers argued that if the label is just a feature then performance on the classification and the induction tasks should be symmetrical. However, if the label is more than a feature and serve as a category marker, then inferring a label when features are provided (i.e., a classification task) should elicit different performance from a task of inferring a feature when the label is provided (i.e., an induction task). Specifically, category-consistent responding should be more likely in induction tasks (where participants could rely on the category label) than in classification tasks (where participants had to infer the category label). This asymmetry should be particularly evident in the critical low-match condition: in the low-match classification task (when they predict the label and thus cannot rely on it), participants would be likely to identify low-match items (e.g., “?”, 1, 0, 1, 0, 0) as belonging to category B (because these items have more features in common with prototype B), whereas in the low-match induction (when they can rely on the label), participants would be likely to identify low-match items (e.g., “A”, ?, 0, 1, 0, 0) as belonging to category A.

Upon finding predicted asymmetries between the two conditions, these researchers concluded that category labels differed from other features in that adult participants were more likely to treat labels as category markers rather than as features. These findings have been replicated in a series of follow-up studies ( Yamauchi, Kohn, & Yu, 2007 ; Yamauchi & Yu, 2008 ; see also Markman & Ross, 2003 , for a review). However, when Deng and Sloutsky (2013) extended this paradigm to children, they found symmetric performance: regardless of the condition (i.e., classification or induction), young children relied on multiple features rather than on the label (see Figure 3 ). It was concluded, therefore, that early in development category labels may function as features of objects, but they become more than features in the course of development.

Proportion of category-consistent responses by feature match and testing condition (After Deng & Sloutsky, 2013 , Experiment 1). Note. Error bars represent standard error of the mean.

This shift in the role of labels is related to the protractive development of selective attention discussed above: For a label to be used as a category marker, people should be able to selectively attend to relevant information and ignore irrelevant information. They should also have enough experience to realize that labels have higher cue validity than other features: even considering homonyms (which reduce cue validity), the probability that X belongs to category K, given that it has label “K” is very high”.

This transition in the role of label is important because once a label denotes a category, much information about the category can be accumulated through conversations, reading, media, and education. This information can also be merged with what is accumulated through observation. As a result, linguistic labels may become “knowledge hubs” that afford non-trivial inferences that are impossible through observation, such as “cows and dolphins are mammals” or “plants and animals are alive.” Having these “knowledge hubs” in place is critical for semantic development – learning information about categories and using this information for linking the concepts together and forming conceptual networks and hierarchies. We focus on these issues in the next section.

SEMANTIC DEVELOPMENT: FROM EARLY CONCEPTS TO CONCEPTUAL NETWORKS AND TAXONOMIES

Language is not a necessary aspect of category learning: nonhuman animals and preverbal human infants can learn categories ( Lazareva & Wasserman, 2008 ; Smith, et al., 2012 , 2015 ; Eimas & Quinn, 1994 ; Madole & Oakes, 1999 ; Younger & Cohen, 1985 ). However, lexicalization of categories, or learning words for categories, is a critical step in integrating knowledge about objects, people, and events, and this integrated knowledge—semantic memory—is central to our ability to use concepts for planning, prediction, explanation, reasoning, and decision making. Furthermore, as discussed above, words may be a starting point for learning many new categories. For example, unobservable categories (such as germs , heat , or energy ) can be only learned that way.

Importantly, regardless of how a concept is learned (i.e., from a category to lexicalization or from a lexical entry to a category), words for categories eventually become part of category representation and they help connecting what is known about a given category and integrating it with newly learned information. For example, when learning a category (e.g., cat ), some information can be acquired from observing the cats (e.g., shape, texture, and the pattern of locomotion), whereas other information cannot (e.g., the fact that cats have hearts, brains, and other internal organs). This latter information requires a verbal description and lexicalization is critical here: the sentence cats have brains can only be expressed if one knows words for the cat and the brain. For the same reason, words are also important for forming conceptual hierarchies, such as grey hound → dog → mammal → animal → living thing → thing. All this is a product of development and in this section, we discuss how this development may occur.

From Words to Categories: Learning Words from Context

Between birth and adulthood, a typical English-speaking child learns on average about 8–10 words per day: learning new words starts slowly, but accelerates dramatically during the second year of life ( Bloom, 1973 ). Obviously, some of these words are learned by ostension (i.e., someone points to a putative referent and explicitly labels it), but many (if not most) are learned from context, including conversations and reading (see Nagy, Herman, & Anderson, 1985 ; Goodman, McDonough, & Brown, 1998 ), sometimes without a referent being present. When this happens, it is important to ask: How do people infer meanings of words, without having the referent present?

A number of ideas have been proposed to answer this question. Some have argued that the context provides the learner with syntactic, semantic, or social cues to meaning. Another possibility is that the context provides the learner with a rich network of associations that may help figuring out the meaning of the new word. Note that these ideas are not mutually exclusive and different sources may mutually strengthen each other.

One idea is that children use syntactic cues to disambiguate the meaning of a novel word. The idea goes back to Roger Brown (1957) who elegantly demonstrated that when presented with a novel word, 3–4 year-olds used the syntactic frame (e.g., “this is a sib” vs. “this one is sibbing”) to determine whether the word referred to an object, action, or property. These ideas generated much empirical support (see Bloom, 2000 , for an extensive review). For example, Soja (1992) found that participants could use count noun/mass noun syntax to guide their learning of novel nouns. Syntactic cues have also been shown to facilitate learning of new verbs, the process known as syntactic bootstrapping (e.g., Fisher, 1996 ; Gleitman, 1990 ; Landau & Gleitman, 1985 ; Naigles, 1990 ). Another idea is that semantic cues can assist word learning. For example, Goodman et al. (1998) demonstrated that 2-year-olds could learn novel words when given sufficient semantic cues, such as familiar verbs (e.g., “Mommy feeds the ferret”) suggesting possible meanings of a novel noun. However, it is easy to see that syntactic and semantic cues provide only a rough guide as to what the meaning of the new word might be.

The third idea is that word learning could be assisted by the “theory of mind.” In particular, solving a social problem of what the speaker is calling attention to may guide word learning (e.g., Akhtar & Tomasello, 2000 ). This social problem could be solved by focusing on the speaker’s gaze direction, facial expression ( Baldwin, 1991 ; 1993 ), or on some relevant aspects of a social situation ( Akhtar, 2002 ). For example, Akhtar (2002) presented 2-year-olds with a word learning task. In one condition, participants’ attention was attracted to the texture of objects (“this is a smooth one and this is a fuzzy one”), and in another condition their attention was attracted to the shape of the objects. Participants were then shown a triad of novel objects, and one of the novel objects was labeled (“this is a dacky one”). The remaining objects matched the labeled object either in shape or in texture. The results indicated that the context affected inferred meaning of the new adjective – participants were more likely to infer that “dacky” was a shape word in the shape-relevant context. However, while social problem solving may offer some assistance in figuring out the meaning, its assistance is rather limited when entities have multiple feature dimensions. For example, what would be the set of relevant features to communicate that the word cat refers to all cats and only to cats and how could these features be communicated?

Finally, a more recent proposal ( Sloutsky, Yim, Yao, & Dennis, 2017 ) suggests that the context in which words are presented provides associative cues that trigger a candidate meaning of a novel word. Two types of associative cues are of particular importance – syntagmatic and paradigmatic ( Brown & Berko, 1960 ; Dennis, 2005 ; Ervin-Tripp, 1970 ; Nelson, 1977 ).

Syntagmatic associations refer to words that co-occur in close temporal proximity (e.g., The dog was barking at the car). Words associated syntagmatically tend to be thematically related. Paradigmatic associations refer to words playing the same role in sentences and appearing in similar sentential contexts (e.g., “He went home to feed the cat” and “She drove home to feed the dog” ). Words associated paradigmatically also tend to be taxonomically related.

While syntagmatic associations exhibit an early onset, paradigmatic associations tend to emerge later in development appearing around 6-years of age ( McNeill, 1963 ; Nelson, 1977 ). Under this construal, early in development (i.e., before paradigmatic associations come online), only syntagmatic associations provide cues to the meaning of a novel word. At the same time, later in development, both syntagmatic and paradigmatic associations provide cues to novel words. If this is the case, then early in development syntagmatic associates should be better cues than paradigmatic ones.

To illustrate, imagine that the learner is presented by a sequence “..., furry, dax” or by a sequence “..., cat, dax”. When syntagmatic associations predominate (which is believed to be the case for young children), the word furry is associated with the word animal (strengths of such associations can be measured by a free association task ( Nelson, McEvoy, & Schreiber, 2004 ) allowing to calculate the probability that the word animal will be recalled in a free association task, given the word furry ). In this case, furry will activate the word animal , and this may result in dax being interpreted as a kind of animal. At the same time, if paradigmatic associations are not formed yet, the word cat will not be associated with the word animal . As a result, the word furry will be a better cue to the meaning of dax (i.e., suggesting that dax is an animal) than the word cat .

In the case of emergence of paradigmatic associations (which we believe are a product of development), the word furry remains associated with the word animal syntagmatically, whereas the word cat become associated with the word animal paradigmatically. As a result, both words may activate the word animal : furry via the syntagmatic route, whereas cat via the paradigmatic route. Under this construal, the probability that the word dax would be considered as referring to some sort of an animal, would be proportional to the forward associative strength between the words accompanying dax and the word animal (or, perhaps, features of animacy or the set of animals). The initial meaning of a word learned from context is broad and imprecise and additional information would be needed to zero in on a precise meaning of the word. This initially broad meaning may become more precise with additional experience with the word.

If syntagmatic and paradigmatic associations play a role in inferring meanings of novel words, young children whose associative repertoire is limited to syntagmatic associations ( Nelson, 1977 ) should learn words from contexts that involve syntagmatic associations. At the same time, adults, whose repertoire includes both syntagmatic and paradigmatic associations should learn words from contexts that have associations of either type. These predictions have been supported in new research coming out of our lab ( Sloutsky, et al., 2017 ). Specifically, in a series of experiments, Sloutsky et al. presented 4-year-olds and adults with sets of words that included a single nonsense word (e.g., dax ) and asked them to indicate whether the nonsense word was an animal or an artifact. Across experiments, adults reliably identified the appropriate category of the nonsense word when lists contained associatively (all members of the list were semantic associates of the word animal , e.g., zoo , farm , furry , creature , giraffe , hamster , bear , and feeding ) or taxonomically (all members of the list were animals referred to by count nouns, e.g., cat , dog , fish , bird , horse , squirrel , cow , and rabbit ) related items, whereas children could only identify the appropriate category when lists contained associatively related items. Furthermore, a computational model developed by these researchers indicated that only the syntagmatic network initially affected the model performance, which was sufficient to account for the child data. In contrast, to capture the adult data, additional learning in the paradigmatic network was needed, which developed later in training. These results suggest a syntagmatic to paradigmatic shift in development and provide a mechanistic account for the shift: while word co-occurrence (which appears in linguistic environment early in development) gives rise to syntagmatic associations, experience with language (which accumulates with development) gives rise to paradigmatic associations.

Linking Words and Categories: The Development of Semantic Knowledge

With the dramatic growth in language, children start to learn words for both known and unknown categories. There is much evidence that these emerging concepts (as well as fully developed ones) get organized into some form of semantic (or conceptual) network. This evidence includes studies of semantic priming, the development and decline of sematic memory, as well as of property and picture verification tasks (see Rogers & McClelland, 2004 , for a review). In short, language allows the development of a conceptual network (also referred to as semantic knowledge or semantic memory) that represents one’s knowledge about the world.

In subsequent studies, researchers attempted to examine whether priming effects in adults stem from thematic relations (or syntagmatic associations) or from taxonomic relations (or paradigmatic associations). These researchers found evidence for priming effects due taxonomic relations in the absence of thematic relations (e.g., Ferrand & New, 2003 ; Thompson-Schill, Kurtz, & Gabrieli, 1998 ) and for priming effects due thematic relations in the absence of taxonomic relations (e.g., Ferrand & New, 2003 ). However, priming effects were more reliable for words that were both taxonomically and thematically related (e.g., dog and cat) than for words that were related only taxonomically (e.g., dog and cow) or only thematically (e.g., dog and bone) ( McRae & Boisvert, 1998 ; Moss, Ostrin, Tyler, & Marslen-Wilson, 1995 ; Perea & Rosa, 2002 ).

Although considerable progress has been made in characterizing the structure of adults’ semantic knowledge, origins of this knowledge and how it develops remain unclear. Findings from studies of word associations in children are controversial. Some researchers argue that the ability to represent semantic relations transpires very early in development, with infants as young as 24 months of age exhibiting evidence of semantic priming. Others argue for protracted development, presenting evidence of semantic (or taxonomic) sensitivity emerging during elementary school years.

For example, Arias-Trejo and Plunkett (2009) used an intermodal preferential looking paradigm to examine 18- and 21-month-old infants’ responses to related prime-target pairs such as cat-dog, compared with unrelated pairs such as plate-dog. The related pairs were strongly associated according to adult associative norms. Infants first heard a phrase such as “I saw a cat”, followed by a target word (dog). They then concurrently saw two images, one related (e.g., a dog) and one unrelated (e.g., a door). The critical manipulation was whether the initial phrase contained a prime word related to the target word and picture (e.g., “I saw a cat…dog”) or an unrelated word (e.g., “I saw a swing…dog”). The researchers found that 18-month-olds looked significantly longer to the picture named by the target word, regardless of whether it followed a related or an unrelated prime phrase. In contrast, 21-month-olds looked significantly longer to the named picture in the related prime–target condition (“I saw a cat…dog”) but not in the unrelated prime–target condition (“I saw a swing…dog”). The older infants’ sensitivity to the word relatedness provides evidence for lexical organization in infants by 21 months of age.

To further identify the type of lexical relatedness, Arias-Trejo and Plunkett (2013) used the same paradigm with 21- and 24-month-olds, with prime-target word pairs that were either thematically or taxonomically related or were unrelated. Twenty-four-month-olds, but not 21-month-olds, exhibited a priming effect for words that were either thematically or taxonomically related, and the age-related differences suggest that semantic relationships between words developed between 21 and 24 months of age. Similarly, Willits, Wojcik, Seidenberg, & Saffran (2013) also provide convergent evidence showing that, even in the absence of visual referents, infants by 24 months of age are able to represent semantic relations between words when processing language.

However, a recent study by Unger et al. (2016) found that even preschoolers were unable to represent relations that were only taxonomic: these participants appear to recognize only links between concepts that are related along multiple dimensions (both thematic and taxonomic). Older children increasingly recognize links between concepts that are related along one dimension (either thematic or taxonomic), and starting at approximately the age of second grade, taxonomic relations are prioritized over thematic relations. This finding is in contrast to the infant literature where semantic priming is found in 24-month infants, which could be possibly due to the methodological differences between studies with infants and with older children (see also Sloutsky, Deng, Fisher, & Kloos, 2015 , for related evidence).

Although the precise structuring principles of children’s semantic network remains unclear, most researchers would agree that there is a dramatic development in children’s semantic knowledge: children learn links between words and categories and form connections among concepts. Importantly, the development of semantic knowledge is critical and fundamental to the development of conceptual networks and hierarchies.

Development of Conceptual Hierarchies

One critical step in acquiring conceptual significance is establishing a structure for concepts. Among various possible conceptual structures ( Kemp, Shafto, & Tenenbaum, 2012 ), taxonomic hierarchy is the most general and well-studied one. An example of taxonomic hierarchy is grey hound → dog → mammal → animal → living thing → thing. This kind of hierarchy is based on class-inclusion relations, with lower-level categories being exhaustive with respect to a higher-level category, and sub-classes of higher-level category being mutually exclusive, with no common members in it.

In order to form a taxonomic hierarchy of concepts, some researchers ( Inhelder & Piaget, 1964 ) argued that the ability to understand the class-inclusion relations, or the logical constraints, is of critical importance. The idea of the logic of classes is that multidimensional sets of stimuli can be divided into proper subsets focusing on one dimension at a time. As proposed by Inhelder and Piaget (1964) , the development of conceptual hierarchies is a function of the development of the logic of classes. According to this view, fundamental developmental changes occur with respect to understanding of class-inclusion relations, which is closely tied to the ability of understanding quantifiers, such as all , some , some are not , and none . Once these are mastered, a classification scheme based on these relations can be applied to any domain of knowledge. However, logic alone is not sufficient for building the taxonomic hierarchies of concepts. One also needs to know dimensions that distinguish sub-categories, to learn words denoting different level of classes, and to acquire knowledge of a domain in which a taxonomy is to be built. Therefore, most contemporary theories consider domain knowledge as a necessary component of the development of conceptual hierarchies (e.g., Carey, 1985 ; Chi, Hutchinson, & Robin, 1989 ; Inagaki & Hatano, 2002 ; Keil, 1981 ) and argue that a taxonomic hierarchy of concepts may result from knowledge of a domain. According to this view, knowledge of a domain reveals class-inclusion relations, which gives rise to one’s hierarchical representations of concepts. For example, knowledge of mammals may help a child understand that all dogs are mammals, but not all mammals are dogs. In this case, the child does not necessarily need to understand logic and apply class-inclusion relations.

An alternative view, has been proposed by Rogers and McClelland (2004) , according to which conceptual hierarchies can be formed implicitly, on the basis of shared predicates. For example, salmon and trout share most of the predicates (e.g., can swim, had gills, has scales, has skin, has bones, etc.), whereas salmon and eagle share only some of the predicates (e.g., has skin, has bones). Therefore, salmon and eagle are members of a broader class (i.e., vertebrates) than salmon and trout . Although this is a promising approach, it runs into some obvious difficulties. First, in order to bypass the problem of class inclusion, one need to have a representative sample of predicates associated with a given category. However, given vast differences in experience, there is no guarantee that this would happen. As a result, there may be more individual variability in conceptual hierarchies than is currently observed.

Another potential problem is that simply adding the predicates may result in a wrong taxonomy. For example, animals sharing the habitat (e.g., dolphin and tuna or bat and hawk ) may share more predicates with members of a broader class that with members of a narrow class (e.g., dolphin and cow ), yet people eventually learn to classify such animal correctly.

In sum, although people often behave as if they have conceptual hierarchies, it is not fully understood if , when , and how such conceptual hierarchies are developed. Evidence for the early onset of conceptual hierarchies is limited. Even if a child exhibits the ability to classify items at a superordinate level or draws inductive inferences on the basis of a superordinate class, this ability does not necessarily indicate the presence of a conceptual hierarchy—because the child can rely on similarity. Although it is clear that the ability to understand class-inclusion relations (along with the ability to use quantifiers) and the knowledge of how these relations can be applied in a particular domain can greatly contribute to the development of conceptual hierarchies, it remains controversial as to whether the former is really necessary for such development (cf. McClelland & Rogers, 2004 ).

CONCLUSIONS

Conceptual development supports many uniquely human behaviors ranging from accumulation of knowledge that is not directly observable to multiple ways of using this knowledge. Importantly, conceptual development has humble origins – it is based on the ability to form categories, the ability that humans share with many non-human animals. However, this ability is greatly amplified (if not transformed) by language: (a) lexicalization helps turning categories into knowledge hubs as well as to mark to-be-learned categories and (b) language is an important source of knowledge about the concepts. Knowledge acquired through perceptual experience coupled with knowledge acquired through language (including reading) leads to the formation of conceptual networks and hierarchies.

However, much is unknown about each constituent part of conceptual development (i.e., the development of category learning, lexicalization, and sematic organization) as well as about their interaction. Therefore, a challenge for future research is to establish precise details of conceptual development, including the way the constituent components change and interact in the course of development.

Acknowledgments

Writing of this manuscript is supported by IES grant R305A140214 and NIH grants R01HD078545 and P01HD080679 to Vladimir Sloutsky. We thank members of the Cognitive Development Lab for helpful comments.

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What Are Psychological Theories?

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk,  "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

Verywell / Colleen Tighe 

5 Major Psychological Theories

• Types of Theories

Psychological theories are fact-based ideas that describe a phenomenon of human behavior. These theories are based on a hypothesis , which is backed by evidence. Thus, the two key components of a psychological theory are:

• It must describe a behavior.
• It must make predictions about future behaviors.

The term "theory" is used with surprising frequency in everyday language. It is often used to mean a guess, hunch, or supposition. You may even hear people dismiss certain information because it is "only a theory."

But in the realm of science, a theory is not merely a guess. A theory presents a concept or idea that is testable. Scientists can test a theory through empirical research and gather evidence that supports or refutes it.

As new evidence surfaces and more research is done, a theory may be refined, modified, or even rejected if it does not fit with the latest scientific findings. The overall strength of a scientific theory hinges on its ability to explain diverse phenomena.

Some of the best-known psychological theories stem from the perspectives of various branches within psychology . There are five major types of psychological theories.

Behavioral Theories

Behavioral psychology, also known as behaviorism, is a theory of learning based on the idea that all behaviors are acquired through conditioning.

Advocated by famous psychologists such as John B. Watson and B.F. Skinner , behavioral theories dominated psychology during the early half of the twentieth century. Today, behavioral techniques are still widely used by therapists to help clients learn new skills and behaviors.

Cognitive Theories

Cognitive theories of psychology are focused on internal states, such as motivation, problem-solving, decision-making , thinking, and attention. Such theories strive to explain different mental processes including how the mind processes information and how our thoughts lead to certain emotions and behaviors.

Humanistic Theories

Humanistic psychology theories began to grow in popularity during the 1950s. Some of the major humanist theorists included Carl Rogers and Abraham Maslow .

While earlier theories often focused on abnormal behavior and psychological problems, humanist theories about behavior instead emphasized the basic goodness of human beings.

Psychodynamic Theories

Psychodynamic theories examine the unconscious concepts that shape our emotions, attitudes, and personalities. Psychodynamic approaches seek to understand the root causes of unconscious behavior.

These theories are strongly linked with Sigmund Freud and his followers. The psychodynamic approach is seen in many Freudian claims—for instance, that our adult behaviors have their roots in our childhood experiences and that the personality is made up of three parts: the ID, the ego, and the superego.

Biological Theories

Biological theories in psychology attribute human emotion and behavior to biological causes. For instance, in the nature versus nurture debate on human behavior, the biological perspective would side with nature.

Biological theories are rooted in the ideas of Charles Darwin, who is famous for theorizing about the roles that evolution and genetics play in psychology.

Someone examining a psychological issue from a biological lens might investigate whether there are bodily injuries causing a specific type of behavior or whether the behavior was inherited.

Different Types of Psychological Theories

There are many psychology theories, but most can be categorized as one of four key types.

Developmental Theories

Theories of development provide a framework for thinking about human growth, development, and learning. If you have ever wondered about what motivates human thought and behavior, understanding these theories can provide useful insight into individuals and society.

Developmental theories provide a set of guiding principles and concepts that describe and explain human development. Some developmental theories focus on the formation of a particular quality, such as Kohlberg's theory of moral development. Other developmental theories focus on growth that happens throughout the lifespan, such as  Erikson's theory of psychosocial development .

Grand Theories

Grand theories are those comprehensive ideas often proposed by major thinkers such as Sigmund Freud,  Erik Erikson , and  Jean Piaget . Grand theories of development include psychoanalytic theory,  learning theory , and  cognitive theory .

These theories seek to explain much of human behavior, but are often considered outdated and incomplete in the face of modern research. Psychologists and researchers often use grand theories as a basis for exploration, but consider smaller theories and recent research as well.

Mini-Theories

Mini-theories describe a small, very particular aspect of development. A mini-theory might explain relatively narrow behaviors, such as how self-esteem is formed or early childhood socialization. These theories are often rooted in the ideas established by grand theories, but they do not seek to describe and explain the whole of human behavior and growth.

Emergent Theories

Emergent theories are those that have been created relatively recently. They are often formed by systematically combining various mini-theories. These theories draw on research and ideas from different disciplines but are not yet as broad or far-reaching as grand theories. The  sociocultural theory  proposed by Lev Vygotsky  is a good example of an emergent theory of development.

The Purpose of Psychological Theories

You may find yourself questioning how necessary it is to learn about different psychology theories, especially those that are considered inaccurate or outdated.

However, theories provide valuable information about the history of psychology and the progression of thought on a particular topic. They also allow a deeper understanding of current theories. Each one helps contribute to our knowledge of the human mind and behavior.

By understanding how thinking has progressed, you can get a better idea not only of where psychology has been, but where it might be going in the future.

Studying scientific theories can improve your understanding of how scientific explanations for behavior and other phenomena in the natural world are formed, investigated, and accepted by the scientific community.

While debates continues to rage over hot topics, it is worthwhile to study science and the psychological theories that have emerged from such research, even when what is often revealed might come as a harsh or inconvenient truth.

As Carl Sagan once wrote, "It is far better to grasp the universe as it really is than to persist in delusion, however satisfying and reassuring."

Examples of Psychological Theories

These are a few examples of psychological theories that have maintained relevance, even today.

Maslow's Hierarchy of Needs

Maslow's hierarchy of needs theory is commonly represented by a pyramid, with five different types of human needs listed. From bottom to top, these needs are:

• Physiological : Food, water, shelter
• Safety needs : Security, resources
• Belongingness and love : Intimate relationships
• Esteem needs : Feeling accomplished
• Self-actualization : Living your full potential creatively and spiritually

According to Maslow, these needs represent what humans require to feel fulfilled and lead productive lives. However, one must satisfy these needs from the bottom up, according to Maslow.

For instance, the most basic and most immediate needs are physiological. Once those are met, you can focus on subsequent needs like relationships and self-esteem.

Piaget's Theory of Cognitive Development

Piaget's theory of cognitive development focuses on how children learn and evolve in their understanding of the world around them. According to his theory, there are four stages children go through during cognitive development:

• Sensorimotor stage : This stage lasts from birth to age two. Infants and toddlers learn about the world around them through reflexes, their five senses, and motor responses.
• Preoperational stage : This stage occurs from two to seven years old. Kids start to learn how to think symbolically, but they struggle to understand the perspectives of others.
• Concrete operational stage : This stage lasts from seven to 11 years old. Kids begin to think logically and are capable of reasoning from specific information to form a general principle.
• Formal operational stage : This stage starts at age 12 and continues from there. This is when we begin to think in abstract terms, such as contemplating moral, philosophical, and political issues.

Freud's Psychoanalytic Theory

Still widely discussed today is Freud's famous psychoanalytic theory . In his theory, Freud proposed that a human personality is made up of the id, the ego, and the superego.

The id, according to Freud, is a primal component of personality. It is unconscious and desires pleasure and immediate gratification. For instance, an infant crying because they're hungry is an example of the id at work. In order to get their needs met, they respond to hunger by crying.

The ego is responsible for managing the impulses of the id so they conform to the norms of the outside world. As you age, your ego develops.

For instance, as an adult, you know that crying doesn't get you the same type of attention and care that it did as an infant. So the ego manages the id's primal impulses, while making sure your responses are appropriate for the time and place.

The superego is made up of what we internalize to be right and wrong based on what we've been taught (our conscience is part of the superego). The superego works to make our behavior acceptable and it urges the ego to make decisions based on what's idealistic (not realistic).

A Word From Verywell

Much of what we know about human thought and behavior has emerged thanks to various psychology theories. For example, behavioral theories demonstrated how conditioning can be used to promote learning. By learning more about these theories, you can gain a deeper and richer understanding of psychology's past, present, and future.

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By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Jerome Bruner’s Theory Of Learning And Cognitive Development

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

Jerome Bruner believed that children construct knowledge and meaning through active experience with the world around them. He emphasized the role of culture and language in cognitive development, which occurs in a spiral fashion with children revisiting basic concepts at increasing levels of complexity and abstraction.

Bruner’s Ideas

• Like  Ausubel (and other cognitive psychologists), Bruner sees the learner as an active agent; emphasizing the importance of existing schemata in guiding learning.
• Bruner argues that students should discern for themselves the structure of subject content – discovering the links and relationships between different facts, concepts and theories (rather than the teacher simply telling them).
• Bruner (1966) hypothesized that the usual course of intellectual development moves through three stages: enactive, iconic, and symbolic, in that order. However, unlike Piaget’s stages, Bruner did not contend that these stages were necessarily age-dependent, or invariant.
• Piaget and, to an extent, Ausubel, contended that the child must be ready, or made ready, for the subject matter. But Bruner contends just the opposite. According to his theory, the fundamental principles of any subject can be taught at any age, provided the material is converted to a form (and stage) appropriate to the child.
• The notion of a “spiral curriculum” embodies Bruner’s ideas by “spiraling” through similar topics at every age, but consistent with the child’s stage of thought.
• His spiral curriculum revisits basic ideas repeatedly, building upon them into more complex, abstract concepts over time in a developmentally appropriate sequence.
• The aim of education should be to create autonomous learners (i.e., learning to learn).
• Cognitive growth involves an interaction between basic human capabilities and “culturally invented technologies that serve as amplifiers of these capabilities.”
• These culturally invented technologies include not just obvious things such as computers and television, but also more abstract notions such as the way a culture categorizes phenomena, and language itself.
• Bruner would likely agree with  Vygotsky  that language serves to mediate between environmental stimuli and the individual’s response.

Three Modes of Representation

Modes of representation are how information or knowledge is stored and encoded in memory.

Rather than neat age-related stages (like Piaget), the modes of representation are integrated and only loosely sequential as they “translate” into each other.

Bruner (1966) was concerned with how knowledge is represented and organized through different modes of thinking (or representation).

In his research on the cognitive development of children,  Jerome Bruner proposed three modes of representation:

• Enactive representation (action-based)
• Iconic representation (image-based)
• Symbolic representation (language-based)

Bruner’s constructivist theory suggests it is effective when faced with new material to follow a progression from enactive to iconic to symbolic representation; this holds true even for adult learners.

Bruner’s work also suggests that a learner even of a very young age is capable of learning any material so long as the instruction is organized appropriately, in sharp contrast to the beliefs of Piaget and other stage theorists.

Enactive Mode (0-1 year)

In the  enactive mode , knowledge is stored primarily in the form of motor responses. This mode is used within the first year of life (corresponding with Piaget’s sensorimotor stage ).

Thinking is based entirely on physical actions , and infants learn by doing, rather than by internal representation (or thinking).

It involves encoding physical action-based information and storing it in our memory. For example, in the form of movement as muscle memory, a baby might remember the action of shaking a rattle.

And this is not just limited to children. Many adults can perform a variety of motor tasks (typing, sewing a shirt, operating a lawn mower) that they would find difficult to describe in iconic (picture) or symbolic (word) form.

This mode continues later in many physical activities, such as learning to ride a bike.

Iconic Mode (1-6 years)

Information is stored as sensory images (icons), usually visual ones, like pictures in the mind. For some, this is conscious; others say they don’t experience it.

This may explain why, when we are learning a new subject, it is often helpful to have diagrams or illustrations to accompany the verbal information.

Thinking is also based on using other mental images (icons), such as hearing, smell or touch.

Symbolic Mode (7 years onwards)

This develops last. In the  symbolic stage , knowledge is stored primarily as language, mathematical symbols, or in other symbol systems.

This mode is acquired around six to seven years old (corresponding to Piaget’s concrete operational stage ).

In the symbolic stage, knowledge is stored primarily as words, mathematical symbols, or other symbol systems, such as music.

Symbols are flexible in that they can be manipulated, ordered, classified, etc., so the user isn’t constrained by actions or images (which have a fixed relation to that which they represent).

According to Bruner’s taxonomy, these differ from icons in that symbols are “arbitrary.” For example, the word “beauty” is an arbitrary designation for the idea of beauty in that the word itself is no more inherently beautiful than any other word.

The Importance of Language

Language is important for the increased ability to deal with abstract concepts.

Bruner argues that language can code stimuli and free an individual from the constraints of dealing only with appearances, to provide a more complex yet flexible cognition.

The use of words can aid the development of the concepts they represent and can remove the constraints of the “here & now” concept.

Bruner views the infant as an intelligent & active problem solver from birth, with intellectual abilities basically similar to those of the mature adult.

Educational Implications

Education should aim to create autonomous learners (i.e., learning to learn).

For Bruner (1961), the purpose of education is not to impart knowledge, but instead to facilitate a child’s thinking and problem-solving skills which can then be transferred to a range of situations. Specifically, education should also develop symbolic thinking in children.

In 1960 Bruner’s text, The Process of Education was published. The main premise of Bruner’s text was that students are active learners who construct their own knowledge.

Bruner (1960) opposed Piaget’s notion of readiness . He argued that schools waste time trying to match the complexity of subject material to a child’s cognitive stage of development.

This means students are held back by teachers as certain topics are deemed too difficult to understand and must be taught when the teacher believes the child has reached the appropriate stage of cognitive maturity .

The Spiral Curriculum

Bruner (1960) adopts a different view and believes a child (of any age) is capable of understanding complex information:

“We begin with the hypothesis that any subject can be taught effectively in some intellectually honest form to any child at any stage of development.” (p. 33)

Bruner (1960) explained how this was possible through the concept of the spiral curriculum. This involved information being structured so that complex ideas can be taught at a simplified level first, and then re-visited at more complex levels later on.

The underlying principle in this is that the student should review particular concepts at over and over again during their educative experience; each time building and their understanding and requiring more sophisticated cognitive strategies (and thus increase the sophistication of their understanding).

Therefore, subjects would be taught at levels of gradually increasing difficultly (hence the spiral analogy). Ideally, teaching his way should lead to children being able to solve problems by themselves.

Bruner argues that, as children age, they are capable of increasingly complex modes of representation (basically, ways of thinking) – and the spiral curriculum should be sensitive to this development;

• Initially, children learn better using an  enactive  mode of representation (i.e. they learn better through “doing things” such as physical and manual tasks) – for instance, the concept of addition might be first taught by asking the child to combine piles of beads and counting the results.
• As they grow older – and more familiar with subject content – pupils become more confident in using an  iconic  mode of representation; they are able to perform tasks by imagining concrete pictures in their heads. To continue the above example; as the child becomes more confident with addition, they should be able to imagine the beads in order to complete additions (without physically needing to manipulate the piles).
• Finally, students become capable of more abstract,  symbolic  modes of representation; without the need for either physical manipulation or mental imagery. Consequently, at this point, the student should have little problem with completing a series of written calculations; of numbers which are higher than is possible by “imagining beads”.

Discovery Learning Theory

Bruner (1960) developed the concept of Discovery Learning – arguing that students should “not be presented with the subject matter in its final form, but rather are required to organize it themselves…[requiring them] to discover for themselves relationships that exist among items of information”.

Bruner (1961) proposes that learners construct their own knowledge and do this by organizing and categorizing information using a coding system.

Bruner believed that the most effective way to develop a coding system is to discover it rather than being told by the teacher.

The concept of discovery learning implies that students construct their own knowledge for themselves (also known as a constructivist approach ).

The result is an extremely active form of learning, in which the students are always engaged in tasks, finding patterns or solving puzzles – and in which they constantly need to exercise their existing schemata , reorganizing and amending these concepts to address the challenges of the task.

The role of the teacher should not be to teach information by rote learning, but instead to facilitate the learning process. This means that a good teacher will design lessons that help students discover the relationship between bits of information.

To do this a teacher must give students the information they need, but without organizing for them. The use of the spiral curriculum can aid the process of discovery learning .

For example, in teaching a particular concept, the teacher should present the set of instances that will best help learners develop an appropriate model of the concept. The teacher should also model the inquiry process. Bruner would likely not contend that all learning should be through discovery.

For example, it seems pointless to have children “discover” the names of the U.S. Presidents, or important dates in history.

Bruner’s theory is probably clearest when illustrated with practical examples. The instinctive response of a teacher to the task of helping a primary-school child understand the concept of odd and even numbers, for instance, would be to explain the difference to them.

However, Bruner would argue that understanding of this concept would be much more genuine if the child discovered the difference for themselves; for instance, by playing a game in which they had to share various numbers of beads fairly between themselves and their friend.

Discovery is not just an instructional technique, but an important learning outcome in itself. Schools should help learners develop their own ability to find the “recurrent regularities” in their environment.

Bruner would likely not contend that all learning should be through discovery. For example, it seems pointless to have children “discover” the names of the U.S. Presidents, or important dates in history.

Scaffolding Theory

On the surface, Bruner’s emphasis on the learner discovering subject content for themselves seemingly absolves the teacher of a great deal of work.

In practice, however, his model requires the teacher to be actively involved in lessons; providing cognitive scaffolding which will facilitate learning on the part of the student.

On the one hand, this involves the selection and design of appropriate stimulus materials and activities which the student can understand and complete – however Bruner also advocates that the teacher should circulate the classroom and work with individual students, performing six core “functions” (Wood, Bruner and Ross: 1976):

• Recruitment : ensuring that the student is interested in the task, and understands what is required of them.
• Reducing degrees of freedom : helping the student make sense of the material by eliminating irrelevant directions and thus reducing the “trial and error” aspect of learning.
• Direction Maintenance : ensuring that the learner is on-task and interest is maintained – often by breaking the ultimate aim of the task into “sub-aims” which are more readily understood and achieved.
• Marking critical features : highlighting relevant concepts or processes and pointing out errors.
• Frustration Control : stopping students from “giving up” on the task.
• Demonstration : providing models for imitation or possible (partial solution).

In this context, Bruner’s model might be better described as guided discovery learning; as the teacher is vital in ensuring that the acquisition of new concepts and processes is successful.

Bruner and Vygotsky

Both Bruner and Vygotsky emphasize a child’s environment, especially the social environment, more than Piaget did. Both agree that adults should play an active role in assisting the child’s learning.

Bruner, like Vygotsky, emphasized the social nature of learning, citing that other people should help a child develop skills through the process of scaffolding.

“[Scaffolding] refers to the steps taken to reduce the degrees of freedom in carrying out some task so that the child can concentrate on the difficult skill she is in the process of acquiring” (Bruner, 1978, p. 19).

He was especially interested in the characteristics of people whom he considered to have achieved their potential as individuals.

The term scaffolding first appeared in the literature when Wood, Bruner, and Ross described how tutors” interacted with a preschooler to help them solve a block reconstruction problem (Wood et al., 1976).

The concept of scaffolding is very similar to Vygotsky’s notion of the zone of proximal development , and it’s not uncommon for the terms to be used interchangeably.

Scaffolding involves helpful, structured interaction between an adult and a child with the aim of helping the child achieve a specific goal.

The purpose of the support is to allow the child to achieve higher levels of development by:

• Simplifying the task or idea.
• Motivating and encouraging the child.
• Highlighting important task elements or errors.
• Giving models that can be imitated.

Bruner and Piaget

There are similarities between Piaget and Bruner, but a significant difference is that Bruner’s modes are not related in terms of which presuppose the one that precedes it. While sometimes one mode may dominate in usage, they coexist.

Bruner states that the level of intellectual development determines the extent to which the child has been given appropriate instruction together with practice or experience.

So – the right way of presentation and explanation will enable a child to grasp a concept usually only understood by an adult. His theory stresses the role of education and the adult.

Although Bruner proposes stages of cognitive development, he doesn’t see them as representing different separate modes of thought at different points of development (like Piaget).

Instead, he sees a gradual development of cognitive skills and techniques into more integrated “adult” cognitive techniques.

Bruner views symbolic representation as crucial for cognitive development, and since language is our primary means of symbolizing the world, he attaches great importance to language in determining cognitive development.

• Children are innately PRE-ADAPTED to learning
• Children have a NATURAL CURIOSITY
• Children’s COGNITIVE STRUCTURES develop over time
• Children are ACTIVE participants in the learning process
• Cognitive development entails the acquisition of SYMBOLS
• Social factors, particularly language, were important for cognitive growth. These underpin the concept of ‘scaffolding’.
• The development of LANGUAGE is a cause not a consequence of cognitive development
• You can SPEED-UP cognitive development. You don’t have to wait for the child to be ready
• The involvement of ADULTS and MORE KNOWLEDGEABLE PEERS makes a big difference

Bruner, J. S. (1957). Going beyond the information given. New York: Norton.

Bruner, J. S. (1960). The Process of education. Cambridge, Mass.: Harvard University Press.

Bruner, J. S. (1961). The act of discovery. Harvard Educational Review , 31, 21-32.

Bruner, J. S. (1966). Toward a theory of instruction , Cambridge, Mass.: Belkapp Press.

Bruner, J. S. (1973). The relevance of education . New York: Norton.

Bruner, J. S. (1978). The role of dialogue in language acquisition. In A. Sinclair, R., J. Jarvelle, and W. J.M. Levelt (eds.) The Child’s Concept of Language. New York: Springer-Verlag.

Wood, D. J., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychiatry and Psychology , 17(2), 89-100.

Psych 256: Cognitive Psychology, 002, SP24

Making connections between theory and reality., bilingualism, problem-solving and decision-making.

Living in a bilingual environment, I have always wondered whether speaking multiple languages contributes to the cognitive development of a child. Our cognitive system encompasses language comprehension, decision-making, problem-solving, memory, emotions, and perception. Multiple studies have shown that there is a positive correlation between bilingualism, the ability to use two languages for communication, and all areas of children’s cognitive development. Most importantly, there are numerous benefits on decision-making and problem-solving.

According to our course, decision-making involves the mental activities that occur when choosing among alternatives. Making decisions often involves dealing with uncertainty. A study conducted by Professor Boaz Keysar at the University of Chicago has demonstrated that bilingualism influences people’s decision-making and their appraisal of moral dilemmas. (Robson, 2023) In the experiment, participants who spoke Spanish as a second language were asked to consider a well-known “trolley problem” dilemma. The scenario includes a person who witnesses an oncoming train that is about to collide with five people walking on the track. The only way to save these people is to push a heavy man off the bridge to prevent the train from hitting them. In his study, Professor Boaz Keysar proved that when speaking a second language, participants were more likely to choose the utilitarian option of pushing the heavy man off the bridge to save five lives. Speaking multiple languages leads to more morally influenced decision-making and promotes more utilitarian decisions.

Additionally, bilingualism positively influences problem-solving. Problem-solving is the ability to find solutions to complex issues. The constant mental juggling that bilinguals engage in—switching between different languages, interpreting meanings, and adjusting responses according to linguistic context—contributes to improving a child’s problem-solving abilities. Besides that, research has shown that bilingual children demonstrate advantages in metalinguistic awareness, cognitive control, and conflict resolution tasks (Bialystok & Barac, 2012; Bialystok & Shapero, 2005). This set of skills can also prevent individuals from developing a termed mental set, the tendency to adopt strategies, frameworks, or procedures that have been used in the past to solve problems.  The cognitive flexibility and metalinguistic awareness enable bilinguals to approach problems from different perspectives and reflect on their experiences from multiple angles.

Overall, there are plenty of advantages of speaking multiple languages for both children and adults. Bilingualism fosters creativity, boosts critical thinking and problem-solving abilities, and encourages individuals to think outside the box. It broadens horizons and influences our perception of the world, enhances human memory, and contributes to the emotional aspects of decision-making. It’s never too late to learn a second language and embrace all the opportunities it brings along!

References:

Bialystok, E., & Shapero, D. (2005). Ambiguous benefits: The effect of bilingualism on reversing ambiguous figures. Developmental Science , 8(6), 595-604

Bialystok, E., & Barac, R. (2012). Emerging bilingualism: Dissociating advantages for metalinguistic awareness and executive control. Cognition , 122(1), 67-73.

Robson, D. (2023, September 17) ‘I couldn’t believe the data’: how thinking in a foreign language improves decision-making. https://www.theguardian.com/science/2023/sep/17/how-learning-thinking-in-a-foreign-language-improves-decision-making?CMP=Share_iOSApp_Other

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