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Psychological Bulletin

Copyright 1994 by the American Psychological Association. Inc. 0033-2909/94/S3.00

1994, Vol. 1 1 5 . No. 3. 381^(00

The Role of Content and Abstract Information in Analogical Transfer Lauretta M. Reeves and Robert W. Weisberg Analogical transfer in problem solving is one example of analogical cognition, which also includes metaphors, similes, and case-based reasoning. The dominant theories in this area posit that abstract schemata mediate transfer (K. J. Holyoak, 1984a, 1985) or that problem solving by means of analogy is accomplished through application of the formal or deep structural characteristics of one problem to another (D. Centner, 1983, 1989). More recently, exemplar-based accounts (D. L. Medin&B. H. Ross, 1989; B. H. Ross, 1987) have emphasized problem content and exemplar-specific details in the various stages of transfer. The present article reviews research on analogical transfer and analyzes the theoretical models in light of this evidence. An adequate theory of analogical transfer must account not only for the use of schematic knowledge but also for the importance of surface information in all stages of transfer (Reeves & Weisberg, 1993a). As such, it will be a hybrid of the various models presented, with exemplar-based models such as that of B. H. Ross as a base.

In many domains, the goal of learning and of teaching is to inculcate knowledge of abstract scientific or mathematical rules (Nisbett, Fong, Lehman, & Cheng, 1987). This learning is typically preceded by a stage in which specific problems, rather than the general principles instantiated by the problems, are first comprehended and used. Learning algebra provides an apt example of this process: During class, students are presented with a number of algebraic principles (such as the Pythagorean theorem) and the attending equations (a2 + b~ = c2), along with a few illustrative problem examples. They are then instructed to do exercises for homework using the information learned in class. However, although the teacher's intention is to have the students apply the abstract algebraic principles to each homework problem, the usual modus operandi of the students is to look back at the examples from class to find one that has the same structure as a homework problem (Medin & Ross, 1989; Reeves & Weisberg, 1993b; Ross, 1984) and to use that example to determine not only which formula to use but also how to apply it to the new problem. This strategy for problem solving is known as analogical transfer: the use of a familiar problem (or base analogue) to solve a novel problem of the same type (the target - problem). Some researchers have argued that analogical transfer is the main method used for solving novel problems in all domains, both tn and out of school (Polya, 1957; Rumelhart, 1989); others have adopted the stronger position that it is the only means (Moore & Newell, 1973, and Sternberg, 1982, cited in Holyoak, 1984a). Moreover, the use of one problem to solve another may help many students to comprehend the abstract

algebraic principles they were supposed to have learned in the first place. Analogical problem solving is only one way in which analogies are used in cognition. Often, inferences are made about newly encountered people or objects based on their perceived similarity to people or objects with which one is familiar. This is known as case-based reasoning and is prevalent in law and medicine (e.g., when a judge relies on an earlier, similar case to make a ruling). Analogies can serve to elucidate unfamiliar territory through comparison with the familiar. In science class, for instance, many have developed an understanding of the structure of an atom by being told of Rutherford's analogy of the atom as a miniature solar system. Likewise, comprehension of metaphors and similes (e.g.. "His legs were broomsticks") involves transportation of a quality of a familiar entity to an unfamiliar one (Gentner, 1982; Holyoak, 1982; Miller, 1979). Psychological theories about how people solve problems1 have typically appealed to formal or abstract rules that subjects induce through experience with problem-solving situations (e.g., Cheng & Holyoak, 1985; Cheng, Holyoak, Nisbett, & Oliver, 1986; Fong, Krantz, & Nisbett, 1986; Larkin, McDermott, Simon, & Simon, 1980; Nisbett et al., 1987). This tradition, with roots in Greek philosophy, has since been buttressed by the computer model of mind prevalent in cognitive theories, which makes a distinction between the abstract syntactic rules by which cognitive processes are conducted and the semantically laden representations on which these rules operate (Lachman, Lachman, & Butterfield, 1979). This has led to cognitive theories, in problem solving and other arenas, that postulate a set of "program" rules that operate independently of the content or subject matter of stored representations (Braine, 1978; Fong etal., 1986; Fong & Nisbett, 1991; Rips, 1983). The abstract-rules view, however, has come under fire recently

We are indebted to Dedre Gentner, Mary Gick, Catherine Hanson, Kathy Hirsh-Pasek, Nora Newcombe, and Brian Ross for helpful comments on earlier versions of this article. Correspondence concerning this article should be addressed to Lauretta M. Reeves, Psychology Department, Rowan College, 201 Mullica Hill Road, Glassboro, New Jersey 08028, or to Robert W. Weisberg, Department of Psychology, Temple University, Weiss Hall, Philadelphia, Pennsylvania 19122. Electronic mail may be sent to [email protected].

1 Although deductive reasoning could be considered one form of problem solving, this research has typically been conducted as an independent venture. Thus, research in deductive reasoning is not covered here, except in cases in which an experiment tested transfer of one deductive reasoning problem to an analogous problem.

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as a result of findings that problem content greatly affects people's performance in both problem solving (e.g., Holyoak & Koh, 1987; Keane, 1987; Reeves & Weisberg, 1993a; Ross, 1987, 1989) and deductive reasoning (e.g., Griggs& Cox, 1982; Johnson-Laird, Legrenzi, & Legrenzi, 1972; Roberge, 1977). New theories have therefore been proposed in which content plays an integral role in cognitive processes, sometimes at the expense of abstract principles. Such theories appeal instead to mental models (Johnson-Laird, 1983), domain-specific principles (Griggs & Cox, 1982), or content-laden problem exemplars (Medin & Ross, 1989; Reed & Bolstad, 1991; Ross, 1987). Contemporary theories of analogical transfer can thus be distinguished by their commitment to the relative importance of abstract schemata versus specific problem content in the various stages of problem solving. In this article, we review theories that purport to explain how transfer is accomplished, and we analyze these theories in light of the empirical findings. Stages of Transfer Four major stages of analogical transfer have been proposed, the first three having been generally agreed on (e.g., Gick & Holyoak, 1983; Holyoak, 1984a; Holyoak & Koh, 1987; Keane, 1987; Reeves & Weisberg, 1990, 1993b; Ross, 1987, 1989): (a) encoding of the base and target analogues, (b) retrieval of a base analogue on presentation of the target (sometimes separated into activation of multiple base analogues and selection of a single analogue), and (c) application (or mapping) of the base to the target problem. Application can also include a substage of adapting a solution principle to fit the target problem (Holyoak & Koh, 1987; Novick & Holyoak, 1991; Reed, Ackinclose, & Voss, 1990). The fourth proposed stage, which has been gaining favor recently, is that of schema induction, which arises from the use of the base analogue to solve a target problem (e.g., Gentner, 1989; Holland, Holyoak. Nisbett, & Thagard, 1986; Holyoak, 1985: Holyoak & Thagard, 1989b; Ross & Kennedy, 1990), if such induction had not already taken place at the time of initial encoding of the base analogues. Abstract Principles, Problem Content, and Experimental Context

nonformal domains, is often referred to as the schema (Gick & Holyoak, 1980, 1983; Holyoak, 1984a, 1985) or deep structure (Centner, 1983, 1989). Schematic information is also thought to entail information about how problems should be classified (Cooper & Sweller, 1987; Gick & Holyoak, 1983; Holyoak, 1984b) as in "constant acceleration problem." Similarity between two problems can exist on any level, although true analogies are considered to be those problems that share a similar deep structure but not necessarily specific content (e.g., the analogy of the atom as a solar system).3 Problems can also be more or less similar in regard to the context in which they appear (i.e., the setting, the experimenter, and the description under which the various parts of an experiment take place [e.g., reading comprehension task vs. problem-solving task]). Critical to the psychological study of problem solving is how subjects represent problems, that is, what details are stored, and how the relations between surface elements are comprehended. Experiments in analogical transfer have used problems in domains as diverse as algebra and physics, as well as less formal ones, such as Duncker's (1945) "tumor" problem: Suppose you are a doctor faced with a patient who has a malignant tumor in his stomach. It is impossible to operate on the patient, but unless the tumor is destroyed the patient will die. There is a kind of ray that can be used to destroy the tumor. If the rays reach the tumor all at once at a sufficiently high intensity, the tumor will tie destroyed. Unfortunately, at this intensity the healthy tissue that the rays pass through on the way to the tumor will also be destroyed. At lower intensities the rays are harmless to healthy tissue, but they will not affect the tumor either. What type of procedure might be used to destroy the tumor with the rays, and at the same time avoid destroying the healthy tissue? (cited in Holyoak, 1984a, p. 205)

The tumor problem can be solved efficiently according to a "simultaneous convergence" principle: Reduce the strength of the rays and send them toward the tumor from several directions, so that they simultaneously converge in full force only on the tumor. This simultaneous convergence principle is also exemplified in "the general" story (see Appendix A), which describes a general who, to avoid setting off land mines on several roads leading to an enemy fort, divides his army so that the 2

The structure of a problem (and potentially of a subject's mental representation of a problem) may be characterized as a hierarchical organization ranging from the concrete details of a problem to the abstract description of its solution. The conjunction of the semantic domain and the surface elements of a problem constitutes the problem content; semantic domain is defined as the superordinate classification of a problem topic, often based on the surface elements that appear in it. Thus, a problem with the specific surface objects2 of carpenter, saw, and wooden beam would be categorized as a "carpentry problem." The abstract or structural details of a problem would include its solution principle, stated either as a formula (as in algebra or logic) or as a proposition (for nonformal problems) that need not be tied to a specific content; for instance, the formula for constant acceleration in physics is applicable to problems of different subject matter (e.g., trains vs. falling objects). The solution principle in more ill-defined problems, such as those in

The term features—as in "surface features" and "structural features"—is prevalent in the literature on analogical transfer. However, because of this term's implications in categorization research, we have tried to avoid it by referring instead to surface "elements" or "objects" and structural "principles" or "details." Please forgive any deviations from this system. 3 Centner and Toupin (1986, Footnote 1, p. 279) distinguished three categories of similarity among problems. Those that overlap in surface elements but not in underlying solution principle are known as "mereappearance matches." Problems that are "literally similar" share both surface and structural information. Those that overlap only in schematic structure are known as "analogies." Hence, Gentner and Toupin (1986) reserved the label of analogy for a very limited set of corresponding problems. Within the study of analogical transfer, a base analogue can be any previous problem that is retrieved for use in a problemsolving situation, whether that analogue overlaps with a target problem in deep structure or not (indeed, subjects often attempt to use mereappearance matches as base analogues). Herein, we use the less restricted definition of analogy.

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soldiers will converge en masse in the center and thus capture the fortress. The general and tumor problems are thus schematically similar but different in content, with distinct semantic domains (army problem and medical problem, respectively) and disparate surface features (doctor, rays, tumor, and patient corresponding to general, soldiers, and fortress). Subjects who are given the general problem as a base analogue are more likely to solve the tumor problem than are control subjects who receive no relevant analogues (Gick & Holyoak, 1980, 1983; Keane, 1985, 1987; Reeves & Weisberg, 1990; Spencer & Weisberg, 1986), especially if given a hint that the general analogue can be used to solve the tumor problem. The dominant models of analogical transfer have posited either that abstract schemata, rather than individual problem exemplars, mediate transfer (e.g., Holyoak, 1984a, 1984b, 1985) or that when specific base exemplars are used, problem solving occurs through the transfer of deep structural characteristics from the base to the target problem (Centner, 1983, 1989; Holyoak, 1984a, 1985). Surface details are assumed to operate mainly in the retrieval of base analogues and less so in the later stages of selection and application (although surface details may have more of an influence in the later stages among the untutored; Gentner & Toupin, 1986). Content has been afforded a more prominent role in problem solving by exemplar-based theorists (e.g., Brooks, 1978; Reed et al., 1990; Reed & Bolstad, 1991; Ross, 1987, 1989), who have argued that the semantic domain of base and target problems and the rpecific objects and actors within those problems influence all stages of analogical transfer. Several key issues permeate the empirical and theoretical work on analogical transfer in problem solving. First, most researchers agree that an abstract solution principle can be derived from, and represented independently of, base exemplars. However, whether this process is automatic or strategic4 (i.e., based on passive or intentional encoding by a subject; Lewis & Anderson. 1985; Reeves & Weisberg, 1993b) is a source of debate. Automatic schema abstraction (e.g., Anderson, Kline, & Beasley, 1979; Carbonell, 1983; Michalski, 1983) assumes that when several problems similar in solution principle are presented, the cognitive system automatically tabulates the degree of overlap among them and stores the composite of overlapping features as a separate problem representation. Conversely, strategic abstraction is the claim that schema induction is based on explicit comparison of two or more analogues for likenesses (Catrambone & Hoiyoak, 1989; Reeves & Weisberg, 1990), active processing of the schematic principle of one or several exemplars (Lewis & Anderson, 1985;Needham&Begg, 1991), or the use of one problem to solve another (Medin & Ross, 1989; Novick & Holyoak, 1991; Ross & Kennedy, 1990). A second issue centers on the amount of exemplar-specific information that is maintained, even after schema induction has occurred. It is possible, for the sake of cognitive economy, that only structural information is maintained and exemplarspecific details are discarded after schema induction. This position has been called eliminative induction (Mackie, 1974). An alternative view is that of conservative induction (Medin & Ross, 1989), in which information about training examples is preserved. This leads to the question of whether schematic and surface elements of problems are stored in functionally indepen-

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dent representations (so that one could be lost without affecting the other). Connected to the issue of conservative induction, it is possible that episodic or contextual cues of the problem-solving session are maintained (Spencer & Weisberg, 1986), along with details of the problems themselves. Finally, if both abstract and exemplar-specific information is retained, the question arises as to which types of details are most influential in each of the proposed stages of transfer. These three major issues can be restated as follows: 1. Is schema induction from problem exemplars accomplished automatically or strategically? 2. Are the surface details of problem exemplars or the episodic details of the learning situation maintained after schema abstraction (conservative induction), or are they discarded (eliminative induction)? 3. What is the relative importance of surface and structuralschematic elements of base and target problems in retrieval and application processes? In the following sections, we review the major theories of analogical transfer on the basis of empirical research with human problem solvers.5 It is our conclusion that a viable theory of analogical transfer must be an amalgam of the existing models. Because so much of the recent evidence suggests that problem content plays an integral role in the analogical problem solving of novices in a field, exemplar models must form the core of any theory, predicting as they do that problem content influences retrieval, selection, and mapping processes. Exemplar theories are also able to explain any potential role of episodic memory factors in problem solving, such as context effects (Spencer & Weisberg, 1986). A comprehensive theory of transfer must also incorporate the abstraction that arises out of increasing knowledge and problem-solving skill, such as the development of expertise in a field (Larkin et al., 1980), and delineate the way in which mapping of deep structural relations predominates over mapping based on surface elements with an increase in domainspecific knowledge (Gentner, 1992; Gentner & Ratterman, 1991). The structural and pragmatic perspectives are well equipped to explain these effects.

Theoretical Perspectives There are three major classes of theories of analogical transfer, with different assumptions as to the nature of the rep4

Medin and Ross (1989) labeled abstraction that arises from mapping a base to a target problem, "nonautonomous abstraction." We have chosen the term strategic abstraction (Reeves & Weisberg, 1993b) both to make the comparison with automatic abstraction more overt and to more adequately convey that schema abstraction requires cognitive effort. Furthermore, this terminology reflects a difference between our view and that of Medin and Ross; whereas they asserted only that abstraction can occur through the use of analogies in a problem-solving endeavor, we hypothesize that explicit comparison of two examples and effortful processing of the schematic principle can lead to schema induction as well. 5 Although there are many computer models of analogical problem solving (e.g., Anderson, 1983; Carbonell, 1983; Hofstadter, 1984; Hofstadter, Mitchell, & French, 1982; Kedar-Cabelli, 1985; Michalski, 1983; Rumelhart, 1989; Winston, 1980), we discuss only those models (and occasionally their accompanying computer programs) that have been derived from research on human subjects.

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resentations used in problem solution and the relative importance of schemata versus content in transfer. Two theories— Centner's structure-mapping view and Holyoak's pragmatic schema view—fall toward the abstract end of the spectrum in that they posit that structural or schematic information predominates over surface information in the transfer of a solution from the base to the target analogue, with problem content playing an influential role primarily in the access of previously learned analogues (Centner, 1983, 1989; Holyoak, 1985; Holyoak & Thagard, 1989b; Thagard, Holyoak, Nelson, & Gochfeld, 1989). The structure-mapping and pragmatic schema theories differ in what they consider to constitute the deep structure of a problem. These two perspectives do, however, share a set of assumptions; (a) abstraction of the causal or structural factors of a solution principle from single or multiple base analogues, (b) diminution of the importance of surface features in problem solving, and (c) use of the most abstract level available to initiate and guide transfer of the base to the target problem. As Holyoak and Thagard (1989b, p. 316) have stated, both the structuremapping and pragmatic schema views provide "content-independent accounts of the mapping process." On the "concrete" end of the continuum of theories, exemplar theories posit that solution principles are linked not only to a given content domain but to specific problems and experiences within that domain. Very low-level information (e.g., previously learned exemplars) is stored and used to solve novel problems (Medin & Ross, 1989; Reed, 1989; Reed & Bolstad, 1991). Although some exemplar models deny that schema abstraction takes place at all (e.g., the artificial-intelligence model of Stanfill & Waltz, 1986), most assume the development of schematic representations, which are, however, inextricably tied to the specific exemplars from which the schema was induced6 (e.g., Hintzman, 1986: Ross & Kennedy, 1990). Furthermore, exemplar views argue that a base exemplar will take priority over an abstract principle in problem solving (Ross, 1987) or will serve to direct the application of an abstract principle to a target problem (Ross, 1989). Content is thus assumed to play a vital role in all stages of analogical transfer. We move now to a discussion of the psychological models exemplifying each viewpoint and illustrate how each would explain analogical transfer from the general problem to the tumor problem. The specific models can be described as viewing the process of analogical problem solving as structure driven (structural theories), goal driven (pragmatic theories), or content driven (exemplar theories), respectively.

Structure-Mapping Model The dominant structural model of analogical transfer is the structure-mapping theory of Centner (1983, 1989; Clement & Centner, 1991; Centner & Centner. 1983; Centner & Toupin, 1986), which is largely concerned with the selection and application of base to target problems. According to Centner, analogical transfer involves a structure mapping whereby the relations between elements within a base analogue are retrieved and then applied to the target to solve a problem. The content of base and target problems, including their semantic domain and the objects that instantiate the relations, is of lesser impor-

tance: "Analogy is a way of focusing on relational commonalities [between analogues] independently of the objects in which those relations are embedded" (Centner, 1989, p. 201). Because of the emphasis on the formal structure being transferred during problem solving, Centner's theory has sometimes been labeled syntactic (e.g., Holyoak, 1985), although this is misleading.7 Centner differentiated surface and structural elements in the following way: Surface elements include both (a) entities, which represent individuals and objects (represented as x and y in the following examples) such as the general or the fortress, and (b) one-place predicates, termed attributes, which are propositionally satisfied by specifying a single variable such as RED (x) or, in the case of the general problem, BRICK (fortress) or TALL (general). Problem structure includes both (a) first-order relations such as GREATER THAN (x, y), which require two objects as arguments to form a complete proposition [e.g., CONVERGE (army, fortress)], and (b) higher order relations, which use propositions themselves as arguments, such as CAUSE [CONVERGE (army, fortress), CONQUER (army, fortress)] (i.e., "That the army converges on the fortress causes the army to conquer the fortress"). Higher order predicates are typically expressions of relations such as "causes," "implies," or "depends on" (e.g., "the attraction of bodies DEPENDS ON distance" when considering the attraction of the planets to the sun). As exemplified in the "structure-mapping engine" program of Falkenhainer, Forbus, and Centner (1986), Centner has proposed that initial matches between base and target analogues can be made at any level: surface features, predicates, or multiplace predicates. Among all potential matches, those that yield similarities at the highest level will be chosen for mapping to a target problem. According to the structure-mapping theory, then, both content and pragmatic goals can play a role in retrieval but should yield to structural relations in final selection and application processes for accurate problem solving (Gentner, 1989). This selection principle, in which systems of higher order relations are preferred over isolated predicates (e.g., attributes), is known as "systematicity." Systematicity entails that true analogies, in which base and target problems share deep 6 Related to exemplar theories is the "memory-cuing" view of Griggs and Cox (1982; see also Jackson & Griggs, 1990), based on Manktelow and Evans (1979), which postulates that principles induced from problem exemplars are domain specific (i.e., linked to the content of that problem or experience) and thus are not abstract in nature. The memory-cuing model thus predicts that cross-domain transfer should be negligible, because any learned principles should not readily be applicable to problems in other content domains. Because this view has largely been developed from work on deductive reasoning, it is beyond our scope here, although it could clearly be adapted as a theory of analogical transfer within the realm of exemplar theories. 7 Previously, such theories have been labeled syntactic and grouped with those that posit that formulaic representations are used in reasoning and problem solving (e.g., Braine, 1978; Rips, 1983). This, however, is deceptive because there are fundamental differences between the two. Structural theories emphasize abstract elements of problems as those that guide transfer, but these abstract elements are semantic in that they are relations between objects or labels in semantic networks (JohnsonLaird. 1989, Note 2).

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structure, will be favored over mere-appearance matches (i.e., base analogues that overlap in content, but not deep structure, with the target problem) and that mapping is more dependent on the relational commonalities between problems than on the similarity of objects. If available, surface elements may be used to aid application, because mapping will be easier when the surface objects and relational features of two problems correspond (Centner & Toupin, 1986), although the system is geared toward use of structural relations to determine mapping. Application of one analogue to another proceeds according to a one-to-one mapping of corresponding objects from the base to target analogue and is dependent on matching predicate relations between the two. Attributes of objects (i.e., one-place predicates) are assumed to be irrelevant to relations among objects and, thus, are dropped from this mapping process (Gentner & Toupin, 1986). For instance, suppose that the general problem were to mention that the fortress is brick or the general tall (see Appendix A); these attributes play no causal role in the solution and, thus, would not be transferred to the target problem. Predicate relations dictate object correspondences between the base and the target problem. Thus, in mapping the general problem to the tumor problem, higher order relations such as CAUSE [CONVERGE (remedy, problem), CONQUER (problem)] must match identically between base and target to accomplish transfer. One strength of Centner's structure-mapping theory is that it can explain problem solving from a single analogue and can indicate how mapping from the base to the target problem should occur to yield optimum problem-solving performance (Reed, 1987). Largely concerned with selection of base analogues and with mapping to target problems, the structuremapping theory provides no mechanism for schema induction, although Gentner (1989) acknowledged that it does occur. One logical critique of this view is that an unsolved target problem must be matched to a base analogue on the basis of structural features, which may not be known in sufficient detail for the target problem (if the structural details of the target were known, presumably it could be solved without resort to an analogue). Thus, the structure-mapping model may require either that problem solvers be explicitly led to the analogy (e.g., by being told that "The atom is like the solar system") or that they ~be able to initiate selection and mapping of the base analogue on the basis of partial structural information known about the target problem. Application could then proceed in a step-bystep mapping of the solution principle, initially arising out of only a partial match between the two problems. In such a case, cross-domain transfer will be negligible without some a priori realization of the structural relations between objects in the target problem. Pragmatic Schema Model The pragmatic schema theory of analogical transfer, developed by Holyoak and colleagues (Holland et al., 1986; Holyoak, 1985;Holyoak&Thagard, 1989a, 1989b;ThagardetaI., 1989), is similar to Centner's structure-mapping view in that it postulates that the selection and mapping of base analogues is dependent on relatively abstract, high-level information. However, it differs both in how it defines those abstract elements and in its

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greater emphasis on schema induction as a means of facilitating problem solving. The theory's advocates claim that there are semantic, pragmatic, and structural constraints that guide problem solving, with different constraints operating at each stage (Holyoak & Thagard, 1989b) and with an emphasis on goal-oriented features (Holland et al., 1986; Holyoak, 1985). Pragmatic schemata are abstract representations that contain causal information directly related to the accomplishment of problem solving (hence "pragmatic"). Recognition of the causal or goal-directed elements shared by two problems guides retrieval and selection of a relevant base analogue, as well as its application to a target problem: "The perceived structure of an analogy is heavily influenced by the pragmatic context of its use" (Holyoak, 1985, p. 76). Holyoak (1984a) distinguished between problem solving based on a schema and problem solving based on analogy (see also Winston, 1980). The first is the application of an abstract principle to a concrete problem; the second is the application of one problem to another at the same level, usually concrete to concrete or abstract to abstract (e.g., wave models of light and sound). Gick and Holyoak's (1980, 1983) pioneering work on the use of analogies in problem solving resulted in the conclusion that, although problem solving could be accomplished through reasoning from a single analogue, development of an adequate schema was the key determinant of successful transfer of base information to a target analogue. Schema induction, under the pragmatic schema view, is assumed to occur strategically, either through mapping of one analogue onto another during problem solving (Holyoak, 1984a, 1985; Holyoak & Thagard, 1989b) or through comparison of two analogous problems (Catrambone & Holyoak, 1989). One analogue is not considered sufficient for schema induction, because there is no opportunity for mapping one problem to another (Gick & Holyoak, 1983). Schema abstraction involves the decomposition of a set of analogues into their similarities and differences (Hesse, 1966; Tversky, 1977) by means of an "abstraction operator." Schema abstraction can also be based on causal analysis of the problem solution (Kedar-Cabelli. 1985) or on instruction, as from a teacher (Holyoak & Thagard, I989b). Details with causal relevance to the solution will be encoded within the schema, whereas noncausal elements will not. This results in the separate storage of the core meaning of the schematic principle shared by two (or more) analogues and the residual meaning specific to each particular analogue (Holyoak, 1984a). During this abstraction process, some transformation of the corresponding concepts between two analogues into more superordinate concepts may be needed. For example,. in the general and tumor problems, the conjoint of the two analogues would be represented as OVERCOME(TARGET); CAPTURE(fortress) and DESTROY(tumor) and would become CAPTURE'(OVERCOME) and DESTROY'(OVERCOME), respectively, within the residual representations. An illustration of how pragmatic schemata differ from the deep structure of a problem, as defined by Gentner (1983, 1989; Gentner & Toupin, 1986), is that even one-place predicates (or attributes, which are not mapped according to Gentner's theory) can be causally relevant for a problem solution and thus "deep" in the pragmatic view. For example, in an experiment conducted by Holyoak, Junn, and Billman (1984), chil-

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dren were tested on their ability to transfer the solution principle from a story about Miss Piggy rolling her carpet into a tube to transfer jewels from her jewel box to a safe to a target problem involving transfer of marbles from one bowl to another. In this case, the capacity of the carpet (and its corresponding object— paper—in the target problem) to be rolled is causal to problem solution and thus schematic according to Holyoak even though it is not structural under Centner's criteria. According to the tenets of the pragmatic schema view, the base analogue is used to solve the target problem because it shares an abstract goal and a number of constraints with the marble problem. Under Holyoak's view, retrieval of a base analogue or schema can occur through the overlap of surface features between problems (Thagard et al., 1989), by a matching of goals (Holyoak & Thagard, 1989b), or both (Holyoak & Koh, 1987). Reliance on similar semantic content, however, will be useful only for problems from the same domain (Holyoak, 1984a, 1984b; Holyoak & Thagard, 1989b); interdomain transfer, in which two problem exemplars share few or no surface elements, will be dependent on access of a base analogue through the "detection of an abstract similarity between corresponding goals, constraints, object descriptions, or operators" (i.e., through overlap of pragmatic elements of the problems; Holyoak, 1985, pp. 71-72). Holyoak (1984a, 1984b) argued that schema induction facilitates analogical problem solving because the independent schematic representation will be more similar to the target problem than will a base problem with different details (Tversky, 1977). In such a case, "a similarity-based retrieval mechanism will be more likely to access a schema than a specific analogue" (Holyoak, 1984b, p. 213). Although the pragmatic schema model does not necessarily predict any role of context effects in transfer, there is recognition that contextual overlap may facilitate the retrieval of base information (Catrambone & Holyoak, 1989; Holyoak, 1984a). If a number of analogues are activated, only one will be chosen and mapped. Selection will be based on the relative strength of the base analogue representations (determined by past usefulness), the degree of activation (based on degree of overlap with the input, i.e., the target problem), and their direct relevance to accomplishing the problem-solving goal. The selection process is facilitated by feedback from attempted mappings. For instance, presentation of the tumor problem may cue the schemata of several analogues (e.g., the general problem or other medical problems), but the general problem will be selected because it'shares the goals of "Defeat a central obstacle" and "Preserve surrounding objects." Appendix B shows a breakdown of the goals, constraints, and facts of the general and tumor problems, numbered according to their corresponding ideas. The two also share the constraint "Full force cannot be used without damage." It is thus presumed that subjects encode exemplars in terms of the goal of their solution principle and that the matching of goals between problems serves to cue or activate relevant base analogues during problem solving. Aside from aiding the retrieval of cross-domain analogues, development of an abstract schema also facilitates mapping of the solution principle to problems of different content from those from which the schema was first induced. During mapping, only elements causally related to the solution in the base problem will be applied to the target proble n (Holyoak, 1985).8

The pragmatic schema view is instantiated in several computer programs: the Processes of Induction program (PI; Holland et al., 1986; Holyoak & Thagard, 1989a), the Analog Retrieval by Constraint Satisfaction program (Thagard et al., 1989), and the Analogical Constraint Mapping Engine program (Holyoak & Thagard, 1989b; see these sources for more details). The pragmatic schema model, as supplemented by the PI program, provides the most thorough account of schema induction among the models of analogical transfer and postulates the storage of both individual exemplars and abstracted schematic principles (in the form of problem-solving goals). Both the pragmatic schema and structure-mapping models agree that content or surface features are influential mainly in the initial retrieval of base analogues (Holyoak & Thagard, 1989a, 1989b; Thagard et al., 1989) but that subsequent selection and mapping follows a search for "structure-preserving" base analogues (i.e., those similar in solution principle but not necessarily in subject matter; Holyoak, 1984a). Holyoak et al.'s theory, however, suffers from the same problem as Centner's (1983, 1989) structure-mapping view: If base-target problem matches are to be made at a level of abstraction above that of surface elements (i.e., either at the level of goals or deep structure), subjects must induce the causal elements from a target analogue to accomplish a match, thereby achieving at least a partial solution and rendering less need for a base analogue. The other alternative is for subjects to institute a remind-and-map strategy (Bassok & Holyoak, 1989), in which retrieval is based on partial overlap of schematic features, and then proceed to full mapping of the solution principle from the base to the target analogue. As Centner (1989) has stated, determining which are surface and which are schematic details on the basis of their relevance to solution attainment means that the system must decide what is relevant before it can operate. This may result in a bootstrapping problem for pragmatic models. The structure-mapping and pragmatic accounts of analogical transfer often make similar predictions about what should occur in problem-solving experiments. Another set of theories— exemplar models—diverges from these two accounts by emphasizing the importance of content over abstract features of base and target problems. -V'

Exemplar Theories Multiple-trace model. Initially posited to model empirical findings in memory and categorization research, Hintzman's (1986, 1988) multiple-trace memory model is relevant to analogical transfer because of his treatment of schema induction and retrieval of items from memory. Hintzman's (1986) MINERVA 2 program posits that each experience or exemplar creates its own episodic memory trace. Both surface and structural details are encoded and are accorded equal weight (although individuals may give greater weight to salient features in an ex8

Centner (1989) limited pragmatic information to the retrieval of base analogues, whereas Holyoak and Thagard (1989a, 1989b) claimed that pragmatic goals operate during both retrieval and mapping and, thus, determine which elements are bootstrapped from the base to the target problem.

ANALOGICAL TRANSFER IN PROBLEM SOLVING

emplar, as a particular task dictates).9 Contextual information about the setting and the time of encoding is also registered, consistent with an episodic memory trace. Abstraction, when it takes place, occurs in Hintzman's model in one of two ways. First, schema induction largely occurs automatically, as a byproduct of the retrieval of numerous stored exemplars. A probe (e.g., the target problem) will activate in parallel multiple exemplar traces, with the level of activation determined by their degree of similarity (in number of features) to the probe. The resulting representation (called the "echo") can match the target problem on purely surface elements (including contextual information), schematic information, or both, depending on the elements that predominate (see also Anderson, 1983; McClelland &Rumelhart, 1985). For example, when confronted with the tumor problem, a person would activate all medical exemplars based on overlap in surface details with that problem (e.g., old episodes of medical shows on TV) or problems that shared certain structural traits (e.g., a central obstacle that needs to be overcome). Depending on how the person had initially represented the base exemplars, the summed activation of these problems could yield an echo in which the dominant strain suggested the breakup of a force that was to be applied in parts. A second, although less prevalent, way in which schema abstraction can occur is through deliberate encoding of schematic features of an exemplar or exemplars (Hintzman, 1986). The schema resulting from this intentional induction is stored as a new memory trace, and, because of the association of the schematic and exemplar traces, activation of one would be expected to spread to the other. Either an individual problem exemplar or the echoic schematic representation can then be applied to the target problem to attempt problem solution. Accordingly, Hintzman's model can account for analogical transfer based on mere-appearance matches that are dependent on surface element overlap or on abstract similarities between problems. Three important points deserve to be made about Hintzman's multiple-trace model. First, induction is conservative rather than eliminative, because schemata can be induced and stored in a separate representation while still preserving the details specific to the base exemplars. The model thereby permits both schema abstraction and problem solving based on appeal to'single exemplars. Second, because episodic elements of the context can be encoded along with the content of base exemplars, the multiple-trace model can accommodate any effects of overlapping context between training and testing sessions. Third, the multiple-trace model posits mechanisms for both automatic and strategic abstraction of schemata during massed retrieval and deliberate schema encoding, respectively. Although Hintzman's multiple-trace model is limited to an exposition of schema abstraction, a second exemplar-based model does provide details on how mapping of a base analogue to a target problem might be accomplished. Reminding! theory. Ross (1984, 1987, 1989; Ross & Kennedy, 1990; Ross & Sofka, 1986) has developed a theory of problem solving that emphasizes the application of previously learned exemplars to a target problem while also permitting the induction of more general solution principles from the use of one problem to solve another (Ross & Kennedy, 1990). Ross

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(1984) claimed that most problem solving is accomplished through "remindings," which are the "unintended retrieval of . earlier specific episodes" (p. 372; see also Schank, 1980a, I980b, 1982). Retrieval is based on the degree of similarity (in either surface or structural details) between base and target problems. The reminding view assumes that, during both initial consideration of a problem and solution attempts, exemplar-specific information will be favored over abstract, content-free information (Medin & Ross, 1989). Whatever abstract information has been induced from a base analogue is bound to specific information about that analogue, and abstract information is understood and used only insofar as it is illustrated within a problem exemplar. Ross (1987) contrasted this position—the "exemplar-analogy" view—with the "principle-cuing" view. The latter predicts that, when faced with a novel problem, subjects will recall relevant abstract information, such as a formula or solution principle, from a previously learned example. Content may aid in retrieving the principle through remembrance of the exemplar, according to the principle-cuing view, but it is the abstract principle that the problem solver seeks and uses. The example-analogy view, on the other hand, asserts that content should affect all stages of analogical transfer, including the application of a base to a target problem. Ross has proposed that general principles can be induced from the use of base problems to solve novel problems (Ross & Kennedy, 1990). In drawing object correspondences between two problems during mapping, problem solvers acquire a generalization that contains structural or schematic information about the problem type that is then stored and used to influence later selection and solution of problems. For example, a student of probability theory could derive the general schematic principle or formula for permutation problems after mapping one permutation problem to another. Abstraction of problem exemplars thus occurs strategically, through the explicit use of analogues in problem-solving activity (Medin & Ross, 1989; Ross & Kennedy, 1990), even "when no separate comparison task is required" (Ross & Kennedy, 1990, p. 51). Generalized knowledge about abstract solution principles can aid noticing and retrieval processes by helping problem solvers to accurately classify problems by type (e.g., logic problems involving permutation or Newton's second law problems in physics) and can thereby lead to more appropriate selection of a base analogue. Induction of general solution principles within a domain is thought to underlie development of expertise in the domain. For example, if the relationship between the general problem and the tumor problem is pointed out to a subject, using the former to solve the latter should lead to the development of an abstract simultaneous convergence schema that could then be used to solve other similar problems. The general analogue 9 Because Hintzman's is a memory model based on categorization of natural and artificial concepts, the issue of how structural features are represented is left open. The problem is one of instantiation (how such features are to be instituted in a computer) rather than theoretical integrity, and, because we are concerned only with the explanatory power and predictions of the model, we do not attempt here to explain how Hintzman's computer program can be adapted to represent the structural elements of problems.

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should not initially be seen to be relevant to the tumor problem, because the two problems share no content, and research has indicated that spontaneous noticing is rare under such circumstances (Gick & Holyoak, 1983; Reeves & Weisberg, 1990; Spencer & Weisberg, 1986). Even after schema abstraction, however, the simultaneous convergence solution will be retrieved more readily in the presence of army or medical problems than problems from other domains. Ross (1989) bifurcated the content of a problem into the specific objects or elements mentioned within an exemplar and the overall semantic domain of a problem (e.g., medical problem), and he argued that these two aspects of content can differentially affect the various stages of transfer. "Noticing" the relevance of a previous exemplar to a target problem is based on similarity of semantic domain, which is used as a cue to the category membership of a problem (e.g., medical problem) and is thus most influential during the activation and retrieval of base analogues. Once accessed, specific surface elements of the problems are used to set up correspondences between source and target exemplars; thus, specific objects and the functional roles they play within a problem are most important in guiding application processes. The premise that remindings are episodic in nature means that they contain not only details about the exemplars learned but also autobiographical and temporal information (Ross, 1984). Contextual features are thus likely to be contained in representations of events or exemplars used in remindings, and remindings of previous problem-solving instances will be most probable when base information and target information are presented in the same setting. Ross's remindings theory strikes a balance between the induction of schematic principles from problem exemplars and the use of surface details in all stages of transfer. Although he has postulated a mechanism for schema induction, his account of this process is not as detailed as that of the pragmatic schema model or the multiple-trace model. At the same time, Ross's view is better equipped to explain the role of content in mapping than any of the other theories. Conclusion Although all models of analogical transfer attribute an important role to the content of a problem during retrieval of base analogues and .permit content to influence subsequent processes, only the exemplar-based views stress the role of such details during mapping. Exemplar accounts posit that the application of a schematic principle is integrally bound with the exemplars from which that principle was induced. Table 1 depicts the theoretical questions of interest to the study of analogical transfer and how each of the models answers those questions. These issues are next used to guide the discussion of the empirical evidence.

Empirical Evidence Automatic Versus Strategic Schema Abstraction Automatic abstraction assumes that schematic details are discerned without explicit comparison of two or more analogues.

The alternative—strategic abstraction—can come about in several ways. Schema induction can be accomplished through intentional encoding of the deep structure of two problems, either by comparison instructions or postencoding tasks that force this abstraction (Catrambone & Holyoak, 1989; Reeves & Weisberg, 1990) or through the use of one problem to solve another, with mapping leading to abstraction of a general principle (Holyoak & Thagard, 1989b; Novick & Holyoak, 1991; Ross & Kennedy, 1990). A third method of schema development is through the explicit provision of a schematic statement (Clement, 1985; Fong et al., 1986; Gick & Holyoak, 1983) or an explanation of a problem's solution principle (Ann, Brewer, & Mooney, 1992). The automatic abstraction hypothesis predicts that mere presentation of several analogous problems should be sufficient for schema induction to occur, whereas the strategic abstraction thesis asserts that intentional encoding or use of the schematic principle embodied by the problems is necessary to accomplish schema abstraction. It may be difficult to gauge when schema generalization is truly automatic because within problem-solving experiments, base exemplars are usually presented under intentional encoding instructions or semi-intentional instructions in which subjects are made aware that their memory for the base exemplars will be tested later. Subjects may tabulate the similarities between base analogues as a way to better remember them, even though such subjects would be classified as having, encoded the schema of the stories in an incidental fashion. Two measures are used to assess subjects' degree of schema induction: (a) the quality of schema synopses written by subjects after they encode the base analogues (as rated by the experimenter), which serve as a direct measure of schema abstraction, or (b) target problem solution, which serves as an indirect measure. Furthermore, two indicators of transfer are often collected: spontaneous (or prehint) transfer, in which subjects solve a target problem without being informed of the relevance of the earlier-encoded base analogue(s), and overall transfer, which includes the subjects who spontaneously solve the target problem plus the number who solve it after a hint to use the base. Because numerous studies have shown that the adequate induction of a schematic principle from relevant base analogues strongly predicts transfer to a target problem (e.g., Catrambone & Holyoak, 1989; Chi, Bassok, Lewis, Reimann, & Glaser, 1989; Crisafi & Brown, 1986; Gick & Holyoak, 1983; Holyoak & Koh, 1987; Novick & Holyoak, 1991; Reeves & Weisberg, 1990; Spencer & Weisberg, 1986), successful transfer can be used to gauge schema abstraction in the absence of more direct measures. Most experimenters who have considered the issue of schema abstraction have asserted that a single analogue does not permit schema induction in problem solving (Gick & Holyoak, 1983; Holyoak, 1984a), although this claim is not universally accepted (Ann et al., 1992; see also Elio & Anderson, 1981). That is, similarity-based approaches claim that abstraction of a schematic principle is dependent on the provision of at least two base analogues (Gick & Holyoak, 1983) or on a single base analogue being mapped onto a target problem (Holyoak, 1984a; Ross & Kennedy, 1990). In the problem-solving literature, it has been determined that, although not impossible, schema abstraction from a single

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Table 1 How Each of the Various Models of Analogical Transfer Answers the Critical Theoretical Questions

Theoretical issue

Structure mapping (Centner)

Is schema abstraction automatic or strategic?

Pragmatic schema (Holyoak)

Automatic Strategic discernment of structure Is schema induction conservative Probably Conservative (i.e., surface details of conservative problems are maintained) or eliminative? Does conservative induction Post hoc explanation No include preservation of (Catrambone & episodic or contextual cues? Holyoak, 1989; Holyoak, I984a) Are surface or structural features more important in: Both Both Retrieval? Pragmatic structure Application? Structural

problem exemplar is rare and does not occur automatically (Gick & Holyoak, 1983). We (Reeves & Weisberg, 1990, Experiment 1) found that although subjects who received a single base analogue were able to synopsize the solution principle, they tended to couch it in terms of the surface features within the analogue (i.e., in non-abstract form). Provision of multiple base analogues improved schema abstraction; there was a significant linear trend in the quality of schema synopses when subjects encoded two, three, or four base analogues. One exception to the claim that schema induction requires two or more analogues is the research of Ahn et al. (1992), who determined that subjects can acquire a schema from a single example by generalizing its explanation on the basis of either their prior knowledge or background information provided by the experimenter. As predicted by strategic abstraction, the effect of background knowledge was most evident with participatory encoding (i.e., when subjects were asked to explain the exemplar passage in detail [Experiment Ib] rather than simply to read the information: see also Ross, Perkins, & Tenpenny, 1-990). Other researchers have shown that actively generating solution attempts to a-base "problem increases subjects' rates of transfer to a target problem relative to those who simply hear the solution to the base analogue without ever attempting it themselves. Needham and Begg (1991) found that subjects who attempted to solve training problems, even unsuccessfully, before hearing the correct answers were superior at transferring their knowledge to later analogues relative to those who studied the base analogues under instructions that their memory would be tested for the problems later (Experiments 1-3; see also Lockhart, Lamon, & Gick, 1988). This advantage was reduced, however, if subjects did not receive experimenter-supplied feedback after erroneous solution attempts on the training problems (Experiment 4). Gick and McGarry (1992) argued that erroneous solutions—whether a subject generates the solutions or merely copies someone else's mistakes—are incorporated into

Multiple trace (Hintzman) Both

Reminding view (Ross) Strategic

Conservative Conservative

Yes

Yes

Both —

Surface Largely surface

subjects' schematic representation of the base analogue. They then serve as useful retrieval cues for the base when subjects recognize that a target problem has the same constraints. Likewise, Lewis and Anderson (1985) determined that actively generating hypotheses about a piece of missing information necessary to solve base geometry problems and receiving experimenter feedback about these hypotheses led to an enhanced ability to select the correct solution strategy for a set of target problems. They concluded that although automatic abstraction appears to work during concept-formation tasks (e.g., Brooks, 1978), strategic processing is necessary for schema abstraction in problem solving. Experiments that have manipulated the instructions under which subjects encode multiple base analogues usually confirm the strategic abstraction hypothesis, with one caveat. We (Reeves & Weisberg, 1990, Experiment 2) asked subjects to encode either two or four base exemplars under instructions to explicitly compare the analogues for commonalities (compare condition) or to rate each analogue on its likelihood of occurrence in real life (no-compare condition). Both groups were told that their memory for the stories might be tested later. Because only the compare condition subjects were asked to write schema synopses, spontaneous (i.e., prehint) transfer was used as a measure of schema induction. Subjects in the two-analogue, nocompare condition evinced the least transfer, and those in the four-analogue, compare condition exhibited the highest rates of transfer. However, the latter did not significantly differ from the two-analogue, compare or the four-analogue, no-compare subjects, suggesting that schema synopsis may be accomplished either strategically (when attention is drawn to the schematic principle through comparison instructions; see also Catrambone & Holyoak, 1985) or automatically (through provision of numerous analogues [e.g., four]). Automatic schema induction appears to be possible only when large numbers of base problems are supplied. However, this conclusion must be tentative because, even in the no-compare group, subjects may have ab-

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stracted the schema after consciously noticing the similarities between problems, thus rendering schema abstraction strategic rather than automatic. Underscoring the solution principle of a set of base analogues within an experimental design can further enhance schema induction. Catrambone and Holyoak (1989, Experiment 1) found that subjects who first summarized and then synopsized the similarities of two analogous base stories had higher rates of spontaneous transfer than those who only summarized the analogues. However, there was a significant difference in spontaneous transfer (when schema effects are most in evidence), with the summarize-and-synopsize group yielding a higher rate of problem solution than the summarize-only group. Alterations in base analogues themselves may also highlight critical features of a schematic solution principle. Gick and Paterson (1992) supplied subjects with two simultaneous convergence principle analogues to the tumor problem (used as the target) plus either a third, unrelated problem or a contrasting example in which a force (e.g., the army in a modified version of the general problem) was divided but did not converge simultaneously on the problematic entity (e.g., the fortress). The "near miss" aspect of the contrasting example emphasized the need for simultaneity of convergence in the solution principle: Quality of schema synopses and spontaneous transfer after a delay were higher in the contrasting example condition. The possibility that schema induction can occur through the use of one analogue to solve another problem (also a tenet of strategic abstraction) was explored by Ross and Kennedy (1990). Using a within-subjects design, they first provided subjects with four study examples, each illustrating one of four probability principles (the training problems). Subjects were then requested to attempt to solve a set of four test problems exemplifying each of the study principles (the first-test problems). Two of the four first-test problems were cued as to which study example would be useful in solving them, whereas the other two were uncued. In the final phase of the experiment, subjects were asked to solve a second set of analogous problems with no benefit of cuing. The relevant formula was accessed more frequently (Experiments 1 and 3), and performance was better (Experiment 2) on the second test problems in the cued than the noncued conditions. Ross and Kennedy's (1990) and others' (Novick & Holyoak, 1991) research demonstrates that schema induction can result from mapping one problem onto another, as-well as from comparison of two analogous problems. Reed (1989) was less sanguine about the ability of problem solvers to induce a schema from mapping one algebra problem to another and suggested that instruction on the general principles within a domain may be necessary for schema abstraction. In fact, numerous experiments have found that providing a schematic statement or diagram along with several problem exemplars leads to both better schema induction and higher rates of transfer (Clement, 1985; Fong et al., 1986; Gick & Holyoak, 1983) than supplying base analogues alone, thereby supporting Reed's contention. In total, the results of many problem-solving experiments suggest that subjects are inefficient at inducing a schematic principle from a single exemplar (Clement, 1985; Gick & Holyoak, 1983, Experiments 1-3; Reeves & Weisberg, 1990) unless they have sufficient prior knowledge about a topic (Ahn

et al., 1992). It may be that automatic abstraction does not occur until provision of four or more exemplars, as we have suggested (Reeves & Weisberg, 1990; and it might not have been automatic even then, if subjects had noticed similarities among the problems and thus engaged in strategic abstraction). In conclusion, in almost all cases, subjects must either work at schema induction by comparing the similarity between base analogues (Catrambone & Holyoak, 1985; Reeves & Weisberg, 1990), mapping one analogue onto another (Ross & Kennedy, 1990), or being explicitly provided with schematic principles that accompany the base analogues (Fong et al., 1986; Gick & Holyoak, 1983).

Maintenance of Surface Details Experiments show that, during analogical transfer, problem content influences retrieval and application of base to target problems (Holyoak & Koh, 1987; Keane, 1987; Ross, 1987, 1989), suggesting that some surface details are maintained in memory even after schema abstraction, as predicted by the conservative induction hypothesis. Furthermore, schematic and exemplar-specific details may be stored in separate cognitive representations; research has revealed that these types of information are differentially susceptible to loss over time (Fong & Nisbett, 1991, Experiments 2 and 3) and may independently contribute to analogical transfer (Novick & Holyoak, 1991). Many researchers have claimed that failure to retrieve previously learned information that is relevant to a novel problem is the biggest impediment to problem solving (Adams et al., 1988; Perfetto, Bransford, & Franks, 1983; Reed, Ernst, & Banerji, 1974; Spencer & Weisberg, 1986; Stein, Way, Benningfield, & Hedgecough, 1986). The strongest evidence for the claim that exemplar-specific information is retained comes from studies of analogical transfer showing that overlap in surface details, semantic domain (i.e., when two problems share a superordinate label [e.g., medical problem]), or both is often highly conducive to access and use of a base analogue in solving a target problem. For example, Keane (1987, Experiment 1) introduced both a change of context and up to a 3-day delay between experimental sessions to examine noticing and retrieval processes in transfer. Subjects were scored on whether they mentioned the relevant base analogue in their verbal protocol while solving the target tumor problem. A single analogue from the same semantic domain (medical problem) that also shared surface elements (e.g., doctor, tumor [albeit in the brain], and X rays) with the target problem was retrieved significantly more often than a remote analogue from a different semantic domain (such as the general problem). Ross (1989) concurred that similar story lines enhanced retrieval processes in analogical transfer. Keane (1987, Experiment 2) also found that retrieval was aided by the similarity of surface elements alone (e.g., rays), even when the semantic domain of the base and target problems was different (army and medical problems, respectively; see also Holyoak & Koh, 1987, Experiment 1). However, retrieval rates (58% in Experiment 2) were not as high as those in a literal analogue condition (88% in Experiment 1) in which both semantic domain and surface elements overlapped, suggesting that there may be an additive effect of semantic domain and

ANALOGICAL TRANSFER IN PROBLEM SOLVING

surface details in the probability of retrieval of base analogues, furthermore, corresponding surface elements did not need to be identical—only similar—to facilitate retrieval: The mention of rays in the target tumor problem (Keane, 1987) elicited equal rates of retrieval of a different-domain analogue that included either rays (identical element) or lasers (similar element; 58% vs. 53%, respectively). Such findings provide support for the conservative induction thesis, because problem content is preserved and used even after a delay of several days between training and testing sessions. This is not to argue that surface details are always maintained; superficial information that is not seen as relevant to the solution principle may not be incorporated into subjects' representations of the base or target problems and, thus, may not facilitate later processes of transfer such as retrieval (Gick & McGarry, 1992, Experiments 1-3). However, it is generally agreed that one reason cross-domain problem solving is rarer than same-domain transfer is the lack of surface similarity between base and target problems in cross-domain transfer, which deters retrieval of the base information (Gick & Holyoak, 1983; Holyoak, 1984a; Keane, 1987; Ross, 1987). For this to be the case, surface details of the problems must be maintained.

Contextual Factors in Conservative Induction Induction may also be conservative through the encoding and preservation of features of the learning context itself. That is, if it can be shown that, on presentation of a target problem, retrieval of potentially relevant base analogues is facilitated by these analogues being in the same context as when one first learned them, it seems evident that episodic information has been stored along with details of the problems themselves. Evidence from memory experiments suggests that the recall of information is optimum when subjects are tested in the same setting in which they originally learned the material (Godden & Baddeley, 1975: Smith, Glenberg, & Bjork, 1978). In problem-solving experiments, there are typically three major aspects of the context in which training exemplars are learned: the setting in which the experiment is conducted, the experimenter who conducts it, and the description under which the task is presented (e.g., psychology experiment or, more specifically, story comprehension experiment; Holyoak, 1984a). Changing one or more aspects of the context between presentation of base and target, problems has been found to have a deleterious effect on spontaneous transfer (once a hint is provided, however, subjects are often able to retrieve information about the base exemplars). In the first experiment to test the effect of context in analogical transfer, Spencer and Weisberg (1986, Experiment 2) varied both the descriptions under which the two tasks were presented (from psychology experiment to in-class demonstration) and the person who conducted each part of the transfer task (outside experimenter vs. class instructor). The change in context between base and target analogue presentations almost completely eliminated spontaneous transfer, even for subjects who had produced adequate schema synopses. Transfer rates in both the same- and different-context conditions were improved by a hint as to the relevance of the base analogues (Spencer & Weisberg, 1986, Experiment 1), suggesting that contextual cues served as

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a basis for the retrieval of base information in the same-context group (see also Catrambone & Holyoak, 1989). When contextual features are not available, as in differentcontext conditions, a direct hint about the relevance of previously learned base information (Spencer & Weisberg, 1986, Experiment 1) must be provided, or other details maintained during conservative induction must cue retrieval (e.g., shared surface and schematic components of the base and target problems [Catrambone & Holyoak, 1989; Holyoak & Koh, 1987; Keane, 1987, Experiment 2] or overlap of semantic domain plus surface elements [Keane, 1987, Experiment 1]). Temporal delays can interact with contextual changes to deleteriously affect transfer (Catrambone & Holyoak, 1989, Experiment 2; Spencer & Weisberg, 1986), although they appear to have no effect alone (Reeves & Weisberg, 1990). The evidence thus upholds the hypothesis that schema induction is conservative in preserving not only surface details of base and target problems but low-level features of the learning situation.

Relative Importance of Structural and Surface Details of Problems in Retrieval and Application The importance of surface details and context in eliciting activation of base information that may be relevant to a current problem does not mean that schematic details play no part in retrieval (Bassok & Holyoak, 1989; Holyoak, 1984a; Holyoak & Koh, 1987). If no same-content base exemplars are available, if same-content exemplars are ineffective at accomplishing target problem solution, or if one has acquired expertise within a field, base analogues overlapping in structural or pragmatic details may be used. Bassok and Holyoak (1989) noted that there are three cues used to assess when a base analogue is relevant to a novel problem: surface cues, underlying structure, and context. The relative importance of the first two is discussed in the next section.10 Retrieval and selection of base analogues. At this point, it may be useful to bifurcate retrieval itself into stages, each of which may differentially rely on surface and structural problem elements. Some researchers have distinguished "noticing" (Ross, 1987) or activation of base analogues (Rumelhart^ 1989) and selection of a base to use in an attempted solution of the target (Clement & Gentner, 1991; Gentner, Ratterman, & Forbus, 1993; Reed et al., 1990). Thus, retrieval requires, first, that one or more base analogues be activated and, second, that the one (or ones) thought to be most relevant to solving the target problem be selected for attempted application.. Retrieval in transfer may operate on the basis of implicit rather than explicit memory processes (Needham & Begg, 1991; Roediger, 1990), because the target problem acts as a probe for previously learned analogues (Weisberg, 1980). Although most researchers agree on the importance of sur10 Although it is recognized that the segregation of surface and schematic details within problems may not always be clear cut. most researchers (e.g., Gentner, 1989; Gentner, Ratterman, & Forbus, 1993; Holyoak & Thagard, 1989a; Reeves & Weisberg, 1993b) agree that such a division may be made and the relative contribution of surface and schematic information to retrieval and application in transfer examined.

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face details in eliciting noticing of base analogues (Gentner, 1989; Holyoak, 1985; Keane, 1987), similarity in the deep structure of problems can also affect retrieval processes, especially selection (Gentner et al., 1993). Holyoak and Koh (1987, Experiment 2) used Duncker's tumor problem as a target and devised several variations of an analogous light bulb problem in which the bulb filament is in need of repair. Surface similarity to the X rays in the tumor problem was varied by using either lasers (similar) or ultrasound waves (dissimilar) to mend the filament. The similarity of structural features to the target problem was manipulated by problem constraints, either that a force of too high an intensity would break the bulb glass (structurally similar) or that the laser/ultrasound machines could not independently generate sufficient force to repair the filament (structurally dissimilar). Structural and surface dissimilarity and similarity were factorially combined to yield four versions of the light bulb problem, which served as the base analogue. Holyoak and Koh (1987) found an additive effect of the two types of similarity in spontaneous transfer, with the similar surface/similar structure group evincing the highest prehint solution rate (69%). However, structural similarity was the main factor that determined successful overall solutions (prehint plus posthint). This makes intuitive sense, because the structurally dissimilar problems have constraints that prohibit accurate solution of the target problem (see also Ross, 1987, Experiment 2). These results have led some theorists to suggest an additive effect of surface and structural similarity in retrieval of base analogues that may be explained in one of two ways. Holyoak and Koh (1987) invoked a "summed-activation" theory to explain how all shared details can contribute to the likelihood of a base analogue being activated. According to this position— similar to Himzman's (1986) retrieval mechanism—surface and structural features are treated equally during the noticing and retrieval stages of transfer. Alternatively, an "access-thenselection view" (Gentner et al., 1993) advocates that access is predominantly affected by the surface similarity of base and target problems, whereas selection is dependent on overlap in structural details. The empirical findings of Holyoak and Koh (1987) cannot discriminate between the summed-activation and access-thenselection views because they tested only spontaneous transfer, which does not allow separation of the activation, selection, and use of base analogues. However, recent research conducted by Gentner et al. (1993) addressed the differential effect of surface and structural similarity in noticing and selection processes, respectively. Subjects studied 32 stories (base exemplars) that were surface matches, structurally similar analogies, or first-order-relations matches (i.e., first-order predicates matched, but not higher level predicates) to 18 cue stories. One week later, subjects were provided with the cue stories and asked to write down any stories they were reminded of from the previous experimental session. Surface-similar matches were more likely to be recalled and to be recalled in greater detail than were structurally similar stories. However, when asked to judge which of the training stories were most sound as analogies to the cue problems (soundness being defined as follows: "Two situations match enough for a strong argument, so that one can infer things about the second story from the first"), subjects most often chose the structurally similar analogues. Hence, activation

of a base analogue was dependent on its surface similarity to a target probe, but selection was based on the degree of relational similarity. Further insight into selection processes is provided by categorization tasks, which require subjects to sort problems by "type." Chi, Feltovich, and Glaser (1981) determined that physics experts based similarity judgments of physics problems on formulaic or structural overlap (e.g., problems that could be solved by application of Newton's third law), whereas novices used surface aspects of problems (e.g., mention of an inclined plane; see also Adelson, 1981; Hinsley, Hayes, & Simon, 1977). Silver (1979) determined that, among novices, good problem solvers were more likely to sort math problems by their deep structure, whereas poor problem solvers were more influenced by story topic (which could be one reason why they were poor problem solvers). Also, with further training in a subject (such as math), novices are able to categorize problems schematically (Hinsley et al., 1977; Mayer, 1981; Reed, 1987; Schoenfeld & Hermann, 1982). Collectively, these findings suggest that as one's knowledge of a domain increases, there is a transition from categorization based on exemplar-specific information to categorization based on abstract knowledge. Novices should be more taken in by "fool's gold," or mere-appearance matches, whereas experts should be better able to avoid such mistakes, rendering them superior at solving novel problems as well. If one has a number of base analogues available, then those with too much structural information (e.g., additional steps in a math formula) should be preferred over those with too little, because the latter may not provide sufficient information to adequately solve a target problem. This is known as the principle of inclnsiveness (Gentner, 1980). Reed et al. (1990) tested subjects' ability to choose either an inclusive or an isomorphic (i.e., formulaically equivalent) base algebra word problem over a surface-similar alternative when presented with a target problem. Subjects were most likely to use surface similarity as the basis for problem selection (Experiment 1); only when there was no surface similarity did subjects rely on structural principles to guide selection (Experiment 2). There was, however, a difference in the perceived and actual usefulness of selected problems; the use of a more inclusive base exemplar best predicted accurate solution of the algebra target problems. Math majors (Experiment 4) and those who saw the solutions to the base problems (Experiment 3) were able to choose isomorphic problems over surface matches on a consistent basis but did not select a more inclusive over a less inclusive problem. Surface cues, however, may not affect retrieval and selection processes in all domains to the same extent; it is possible that some domains are represented as less "content bound" than others. Bassok and Holyoak (1989) trained high school subjects on either algebra or physics problems that were solved using arithmetic progressions (structurally isomorphic to constant acceleration problems in physics). The following day, those trained in algebra took a physics test, and vice versa. Nonsymmetrical transfer was shown: Algebra subjects readily transferred their knowledge of arithmetic progressions to physics problems, but physics subjects showed negligible transfer to algebra questions (Experiment 1). This was true even when all of the algebra training problems were derived from the same content domain (e.g., finance); thus, relevance to other domains

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was not suggested by the training materials (Experiment 2). The researchers claimed that students realize that algebra formulas are meant to be abstract and not limited in their applicability to a single topic area and that they then represent algebra principles accordingly (Bassok & Holyoak, 1989). Physics principles, on the other hand, tend to be embedded in the content in which they were learned, and it is not until they are "disembedded" during training that cross-domain transfer to algebra is seen (Experiment 3). This difference in how abstractly the two domains are represented could be based on students' greater familiarity with mathematics and algebra. The classification and selection studies suggest that although novices are not incapable of selection or categorization based on structural aspects of problems (Chi et al., 1981; Hinsley et al, 1977; Mayer, 1981), for them, surface elements often act as more salient cues (Reed et al., 1990). Structural similarity is used either in the absence of overlapping surface elements (Reed et al., 1990) or as one acquires expertise in a field (Chi et al., 1981; Hinsley etal., 1977; Reed etal., 1990; Schoenfeld& Hermann, 1982). Centner and Ratterman (1991) attributed the ability to use structural information to the development of domain-specific knowledge, which would explain the oft-found developmental shift from reliance on surface to structural similarity (Gentner & Toupin, 1986) as well as the novice-to-expert shift (as illustrated by the studies just discussed). This domainspecific knowledge may include knowing whether solution principles should be abstracted beyond the content of the problems in which they were first learned (Bassok & Holyoak, 1989). Mapping of base analogue to target. Because true analogies are defined as those that share a schematic structure, analogical transfer will be most successful when a structurally similar exemplar is selected and applied to a target problem, and transfer will be most efficient when the analogy emanates from overlap at the highest structural level. The preference for mapping connected systems of relations over isolated predicates is known as systematicity (Clement & Gentner, 1991; Gentner, 1989; Gentner & Gentner, 1983; Gentner & Toupin, 1986), and strict use of this strategy precludes that mapping be influenced by surface features that violate structural constraints." Thus, systematicity predicts that a fact from a base analogue is more likely to be transported to a target domain when that fact is embedded within a system of relations that are consistent with propositions contained in the target problem than when it is an isolated proposition unrelated to any of the other facts in either the base or the target. Clement and Gentner (1991) asked subjects to read two analogues, one complete (which served as the base analogue) and one with missing information (which served as the target analogue). They were requested to draw correspondences between the two stories and to explain the ways in which they were and were not similar. Subjects then judged which of two facts about the target story (both of which were analogous to facts in the base story) could more likely be inferred about the target when one fact statement was systematically related to other information contained in the target story and the other statement was not.12 Consistent with the principle of systematicity, subjects more often chose the shared-system fact over the different-system fact and rated it higher on its importance to the target passage (Experiment 1). When subjects were allowed to make their

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own inferences about the target story, their predictions were governed by systematicity (Experiment 2), and they often cited causal relations between propositions as justification for their responses (Experiments 1 and 2). Subjects thus prefer to import inferences from a familar to an unfamiliar domain when those inferences are embedded in a complex structure of relations rather than those that stand alone. An alternative strategy to systematicity is transparency (Gentner & Toupin, 1986): the tendency to rely on the similarity of surface elements, rather than structural details, to determine the mapping of base analogue to target problem. One way to test the relative importance of transparency versus systematicity is to observe subjects' mappings when they are provided with base and target problems containing similar objects that carry out corresponding functions (thereby making object correspondences easier) and when they are provided with base and target problems with reversed correspondences (i.e., similar objects carry out different functions in each problem, potentially leading to incorrect mapping of the schematic principle). To the extent that subjects rely on systematicity to guide application processes in analogical transfer, they should perform equally in the same-correspondence and reversed-correspondence conditions. Use of transparency as a mapping strategy predicts a decrement in correct application in the case of reversed correspondences. Gentner and Toupin (1986) examined the degree of children's reliance on either transparency or systematicity in guiding application processes in analogical transfer. In the systematic condition, in which the structural features of stories wrere made evident by providing a reason for the solution in the base analogues (e.g., a cat steals a wagon out of his jealousy over his friend, the walrus, playing with someone else), the researchers determined that there was a developmental change in mapping strategy from dependence on transparency to the use of systematicity. Younger children (5-7 years old) relied on the transparency of object correspondences between base and target problem to guide mapping in both the systematic and nonsystematic conditions, even when such mapping violated the solution principle. Older children (8-10 years old), on the other hand, were able to ignore surface similarities between problems and to map according to a structure-preserving strategy when provided with a reason for the story action (systematic condition). Gent" This is not to say that when surface-similar elements share corresponding relational roles between base and target, transfer is not made easier (Gentner & Toupin. 1986; Holyoak &Thagard, I989b). 12 One base story, for example, involved fictional creatures called "tarns" who consume minerals in rocks through their underbellies. When the minerals are exhausted in one place, the tarns stop using their underbelly. Also, the tarns' underbellies become specialized for one kind of rock and, thus, cannot function on new rocks. In the corresponding target story, when subjects were told that robots use their probes to gather data and can exhaust data collection in one place, they were more likely to infer that "The robots sometimes stop using their probes" than "The probes cannot function on new planets." This occurred even though both facts had appeared in the base analogue and had been deemed equally plausible by a control group not exposed to the base analogue. According to Clement and Gentner (1991), "The robots sometimes stop using their probes" was causally related to a system of facts, whereas the other statement was not.

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ner and Ratterman (1991; see also Centner, 1992) made a compelling case, based on considerable evidence (e.g., Brown & Kane, 1988; Crisafi & Brown, 1986; Centner, 1988; Kotovsky & Centner, 1990), that the developmental shift from reliance on surface to structural features is due to increases in domainspecific knowledge (but see Halford, 1992, for an alternative view). Ross (1989) used probability problems to determine the degree to which statistics novices used base analogues to solve target problems, even when they were explicitly provided with the correct solution formula. If transfer is purely structurally mediated, the formula alone should be sufficient to induce accurate problem solving, without requiring the use of the base problems. In the cases in which the base exemplars were used to facilitate problem solution, Ross was also interested in the degree to which transparency guided mapping of base analogues to target problems. Subjects received analogous study and test examples (i.e., embodying the same solution formula, such as permutation problems), in which the content of each problem pair was the same (e.g., IBM departments) or different and in which the correspondences between objects were either the same (e.g., mechanics chose cars to work on) or reversed (e.g., in the study example, mechanics chose cars to work on; in the test example, car owners chose which mechanic would work on their cars). Although the formula was provided with each target problem, subjects used the specific details in the training examples to determine application of the formula to the target problem. The transparency of object correspondences between base and target problems guided subjects' mapping, even when such correspondences led to incorrect application of the solution (Experiment 1). Correct solutions were rarer in the reversed-correspondence condition, as predicted by use of transparency as a mapping structure. Ross's (1989) novices thus tended to operate much like the young children in Centner and Toupin's( 1986) study. Not only the surface elements themselves but also similarity between how corresponding elements in base and target analogues are presented can affect analogical problem solving. Bassok (1990, Experiment 2) determined that spontaneous transfer was greater when two problems shared rates involving two entities (e.g., miles per hour or words typed per minute) than when one of them was presented in less of a ratelike form, with only one entity (e.g., amount of money earned) and the time spread over-a longer period (e.g., per year).13 Both retrieval and application processes were affected by two problems having similar units, as evidenced in verbal protocols provided by subjects. These findings indicate that surface details of problems play a critical role not only in retrieval processes but in the assignment of object correspondences during analogical transfer, even in experiments in which an abstract solution principle is explicitly provided (Ross, 1989). This is not to suggest that surface features always overrule systematicity; there appears to be a developmental trend from mapping based on transparency toward mapping according to structural constraints (Centner & Toupin, 1986), and this developmental change may parallel the novice-to-expert shift within a given domain. Further evidence of the importance of content in the mapping stage of transfer comes from Fong and Nisbett (1991), who

trained subjects on the law of large numbers and provided several illustrative training problems from a single domain (e.g., sports examples). Subjects then attempted to solve test problems that were from either the same domain as or a different domain from the training problems. There were no effects of a change in problem domain among those who received both training and test problems in the same experimental session. However, after a 2-week delay, higher rates of transfer were attained by subjects in the same-domain condition (even though a hint was provided that subjects should solve the problems by using the law of large numbers), indicating a difference in the ability to apply the abstract principle. Thus, at least after a delay, receiving target problems from the same domain in which a general principle was learned facilitates problem solving. Recent research by Novick and Holyoak (1991) also demonstrated that application of a solution from base to target is not exclusively based on structural features of the solution and that low-level details are often important in guiding mapping procedures. They found that subjects who received mapping cues about which specific numbers from a problem should be fitted into each variable slot in a mathematical equation outperformed both subjects who received cues about which concepts should be mapped onto which variables and those who received nondirective clues to simply use a base problem to solve a target problem. A second way to test the relative importance of surface versus structural features in application is to compare the overall transfer rates of subjects who receive a principle only, an exemplar only, or both an exemplar and principle during the training session. If surface features aid in application, as claimed in exemplar-based accounts (e.g., Ross, 1987, 1989), learning an abstract principle alone should not be as effective in producing successful transfer as an example plus a principle (and perhaps not as effective as an example itself) because a principle alone provides no clues for its application.14 Cheng et al. (1986, Experiment 1) found that neither rules training nor examples training alone was sufficient for accurate solution of deductive reasoning problems, and the best performance was shown by the exemplar plus rules group. These results, confirmed by Fong et al. (1986, Experiment 1), show that optimum transfer is produced when training on an abstract principle or set of 13 Bassok (1990), following Kaput (1986), distinguished between extensive quantities, which involve only one entity (e.g., dollars earned), and intensive or ratelike quantities, which involve two entities (e.g., miles per hour). Kaput (1986) also stated that extensive quantities are additive—the amount earned in the previous year can be added to the amount earned in the current year for a total dollar amount—whereas intensive quantities are not additive. For example, if one travels at 45 miles per hour (72 km/hr) for half an hour and 55 miles per hour (89 km/hr) for another half hour, one's total number of miles per hour is not 100(161 km/hr). 14 Proponents of an abstract-rules view of transfer (Fong & Nisbett, 1991) have argued that the benefit of problem examples being provided to supplement a principle derives from the development of "coding rules" that help to map an abstract principle onto concrete target problems. These coding rules are not exemplar specific but are thought to be relatively abstract rules detailing how to map a solution principle to a base analogue (but see Reeves & Weisberg, 1993a, for a critique of this argument).

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problem-solving rules within a domain is supplemented with examples that illustrate the principle(s). In conclusion, problem content affects application of a base analogue to a target problem, as evidenced by the superiority of same- over different-domain transfer and the frequent use of transparency as a mapping strategy. There is, however, a limit on the role of problem content in application; research has shown that structural details are more likely than surface details to guide mapping of a base to a target problem as one's comprehension of structural information increases (Centner & Toupin, 1986).

Conclusion and Analysis We return now to our original questions regarding the theoretical issues pertaining to analogical transfer. The available evidence clearly indicates that schema induction from analogues takes place, often to the great advantage of problem solving. This induction is largely strategic; that is, people must work at developing and comprehending the abstract principles inherent in problem exemplars. The small number of base analogues provided in most analogical transfer studies is insufficient to promote schema induction unless (a) there are explicit comparison instructions (Catrambone & Holyoak, 1985; Reeves & Weisberg, 1990); (b) encoding tasks that emphasize schematic details are used (Catrambone & Holyoak, 1989); (c) a base analogue is mapped onto a target problem to aid in its solution, from which a more general solution principle may be abstracted (Ross & Kennedy, 1990); or (d) sufficient explanation of the schema is provided (Ahn et al., 1992). Increasing the number of base analogues (e.g., from two to four) may promote development of a schematic representation (Reeves & Weisberg, 1990) even under incidental encoding instructions, suggesting the possibility of automatic abstraction with a large enough number of base analogues. Although subjects may be able to memorize an abstract solution principle without benefit of an accompanying example, optimum benefit is gained by provision of base exemplars plus an explicitly stated rule, which has been found to lead to better schema induction (Gick & Holyoak, 1983) and higher rates of transfer (Cheng et al., 1986; Fong et al., 1986; Gick & Holyoak, 1983) than does either provision of base exemplars alone or " statement of a solution principle alone. These findings, too, support the case for strategic abstraction (Holyoak, 1985; Reeves & Weisberg, 1993b; Ross & Kennedy, 1990), because schema induction appears to require both an explicit synopsis of the schematic principle and at least two base analogues illustrating that principle (Gick & Holyoak, 1983). Single analogues are effective in schema induction only when there is active processing of an explanation of the solution principle (Ahn et al., 1992). In fact, strategic processing of a base exemplar, either through subject-generated solution attempts and experimenter feedback (Needham & Begg, 1991) or through hypothesis generation (Lewis & Anderson, 1985), has been found to facilitate learning of a solution principle and thus transfer. Abstraction of the solution principle from a set of analogues does not compromise the ability of the subject to maintain details of those analogues. It is thus likely that people maintain multiple levels of representation of a set of problems, because

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schematic and surface details have been found to be differentially lost in memory tasks (Fong & Nisbett, 1991). Evidence for the conservative induction thesis comes from studies revealing that problem content—both semantic domain and surface elements—can influence the retrieval of base analogues (Holyoak & Koh, 1987; Keane, 1987) and the application of a solution principle to a target problem (Ross, 1989). This effect is especially pronounced after a delay between the presentation of base and target analogues (Fong & Nisbett, 1991). However, surface features will not be influential in transfer unless subjects have incorporated them into their mental representations (Gick & McGarry, 1992). Episodic features of the learning situation are also maintained, because a change in context jeopardizes the retrieval of relevant base analogues (Catrambone & Holyoak, 1989; Gick & Paterson, 1992; Spencer & Weisberg, 1986). Some researchers have claimed that both surface and structural details can influence retrieval (Holyoak & Koh, 1987), but it is probably best to bifurcate retrieval into stages of noticing (or access or activation) and selection (Clement & Centner, 1991; Gentner et al., 1993; Reed et al., 1990; Reeves & Weisberg, 1993b; Ross, 1987, 1989), with the former influenced largely by surface elements or problem content and the latter influenced by schematic or structural elements. This bifurcation is consistent with empirical findings that access to base analogues is influenced by overlap in content (Gentner et al., 1993; Keane, 1985, 1987; Ross, 1984, 1987) and that structural features may guide selection only in the absence of overlapping surface details (Reed et al, 1990). Furthermore, classification of problems as similar becomes increasingly based on structural criteria (a) as one's expertise in a field grows (Chi et al., 1981; Gentner et al., 1993; Hinsley et al., 1977; Schoenfeld & Hermann, 1982) and (b) among better problem solvers (Silver, 1979). There may also be developmental differences in reliance on surface versus structural information in base problem selection (Gentner & Ratterman, 1991; Gentner & Toupin, 1986), with increasing attention paid to problem structure. Theoretically, selection is often thought to be influenced by feedback from attempted mapping of the base to the target problem (Gentner. 1989; Holyoak & Thagard, 1989a), which may explain the greater emphasis on schematic details during this stage; attempted mappings of base problems similar only in content will not succeed in solving target problems adequately, and, thus, alternative base analogues may be used. During the application stage, mapping often depends on the surface details of the base analogues from which the solution principle was induced (Bassok, 1990; Ross, 1989). Cues about how to apply a solution to a target problem facilitate accurate mapping (Novick & Holyoak, 1991). Furthermore, provision of several examples illustrating an explicitly stated solution principle is more useful than either the principle statement or base exemplars alone (Cheng et al., 1986; Fong et al., 1986; Gick & Holyoak, 1983) in aiding target problem solution. All of the theories of analogical transfer discussed herein receive partial support from the empirical evidence, although Ross's exemplar theory is the only one that (a) advocates strategic abstraction, (b) predicts conservative induction of both surface details of base and target problems and episodic features of the learning situation, and (c) emphasizes the potential impor-

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tance of problem content to the later stages of transfer, such as selection and application. Centner's structure-mapping theory correctly predicts that selection of a base analogue on the basis of structural similarity is the best predictor of a correct solution within analogical transfer (Reed, 1987; Reed et al., 1990), and she has provided an explanation for the accurate application of a base to a target problem. This view also supplies a viable account of how analogies can be used to elucidate unfamiliar problems during instruction (i.e., learning by analogy) and how one analogue is applied to another after a hint as to the base analogue's relevance. However, the structure-mapping theory has difficulty accounting for the spontaneous retrieval of base analogues that are taken from a domain different from that of the target problem. Initial selection of base analogues may be sensitive to overlap in goal structures, as predicted by the pragmatic view (Holyoak, 1985; Holyoak & Thagard, 1989a, 1989b), which could account for how initial matches are made. Mapping could then proceed one concept at a time (Bassok & Holyoak, 1989) rather than through the preferable channel of classifying problems as being of the same "type" and subsequently initiating application of base to target. Although Centner's model permits surface similarities to influence mapping in young children or novices, or both, it appears that reliance on surface elements to guide selection (Reed et al., 1990) and application of base to target problems (Ross, 1989) is more prevalent than predicted by the structure-mapping view. Holyoak and colleagues' pragmatic schema model of problem solving (Holland et al., 1986; Holyoak, 1985; Holyoak & Thagard, 1989a, 1989b; Thagard et al., 1989) provides a thorough account of schema induction. Such a mechanism is necessary for a comprehensive theory of analogical transfer, given extensive findings that schema abstraction strongly predicts transfer (Catrambone & Holyoak, 1989; Gick & Holyoak, 1983; Reeves &Weisberg, 1990; Spencer & Weisberg, 1986) and that principles learned in one domain can be useful in solving problems from others (e.g., Bassok & Holyoak, 1989; Cheng et al., 1986;Fongetal., 1986;Fong&Nisbett, 1991). However, the pragmatic position, too, underestimates the role of content and surface elements in the later processes of analogical transfer (e.g., selection and application). The pragmatic schema theory has the potential to explain how surface elements of particular problem exemplars can influence application, even when a schema has been induced, because it assumes maintenance of both the schematic representation and exemplar-specific information. This, however, has not been discussed in detail by the view's proponents, nor does this view adequately address context effects in problem solving (although such effects are not inconsistent with the model). Whether pragmatic goals, rather than structural details, are the basis of selection and mapping of base analogues requires further research. Hintzman's (1986) multiple-trace model concentrates on explaining schema abstraction and retrieval of base exemplars. Its predictions are largely consistent with evidence from the problem-solving literature on these points, except that it is not clear that automatic abstraction of a solution principle is as prevalent in problem solving as in concept formation. Because Hintzman's theory is based on the development of semantic memory from episodic memory traces, it is well equipped to explain

both context effects in retrieval of base analogues and the maintenance of surface details from specific problems, which may prove useful in retrieval and application. The multiple-trace model, however, is limited to showing how the early stages of analogical transfer may be carried out, and it may thus need to be combined with other exemplar accounts, such as Ross's, to fully explain how later stages (e.g., mapping) are accomplished. The theory that best accounts for the majority of data on analogical transfer is that of Ross (1987, 1989; Medin & Ross, 1989; Ross & Kennedy, 1990; Ross & Sofka, 1986), which we have characterized as an exemplar view. The reminding perspective postulates that specific exemplars will often be used in problem solving but that more general solution principles can be learned from mapping a base to a target problem (Ross & Kennedy, 1990). Furthermore, in this exemplar view, induction is conservative (Medin & Ross, 1989), and the remindings theory can thus explain the role of problem content in all stages of transfer: retrieval (Holyoak & Koh. 1987; Keane, 1985, 1987; Ross, 1987), selection (Reed et al., 1990), and application (e.g., Ross, 1987, 1989). Ross's exemplar model of analogical problem solving considers transfer to be, in part, a product of episodic memory, and thus it is intrinsically equipped to account for context effects (Catrambone & Holyoak, 1989; Spencer & Weisberg, 1986). Such effects are predicted by the other models only on a post hoc basis. Although Ross's reminding position is valuable as a core for a comprehensive theory of problem solving, it must be accom-,/ panied by elements of the structure-mapping and pragmatic schema theories to fully explain the available data. For example, Holyoak and colleagues' pragmatic model provides a detailed account of schema induction that is missing in the reminding view. Furthermore, although the evidence supports the exemplar theories' contention that surface and episodic cues play a role in retrieval, selection, and mapping of base analogues to target problems, these processes may also be influenced by similarities in pragmatic goals, as predicted by Holyoak and Thagard (1989a, 1989b). Centner's structure-mapping model furnishes an exposition of the types of mappings most likely to lead to problem-solving success and predicts that application based on structural features should increase with domain-specific knowledge, such as that found among experts in a field. Thus, to explain all of the available data, a hybrid of the structural, pragmatic, and exemplar views is necessary. The past decade has seen much theoretical and experimental progress in elucidating the cognitive processes and knowledge representations underlying the use of analogies in problem solving. Since the emergence of interest in analogical transfer fostered by Gick and Holyoak's (1980, 1983) research, the field has burgeoned into a healthy subdiscipline in cognitive science and has incorporated the work of investigators from a multitude of areas, addressing as it does issues of memory, knowledge representation, the development of expertise, similarity and categorization, language, and so forth. Yet psychologists' (and computer scientists') research in this area is not exhausted. More work needs to be done to determine what types of details of base problems are maintained in memory; for example, are surface details dealing with causal actions (e.g., that an army, rather than the marines, carry out the actions in the general problem) better remembered than noncausal surface details (e.g., what

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city the raid was carried out in)? Most of the studies showing how surface elements can influence application processes have been conducted in domains using formal principles, such as statistics (Ross, 1989) and mathematics (Reed et al., 1990), and it may be worthwhile to carry out such experiments using less formally specified problems. Also, although the novice-to-expert shift from reliance on surface to structural details as a basis for classification of problems is well documented (Chi et al., 1981;Hinsleyetal., 1977; Schoenfeld& Hermann, 1982; Silver, 1979), less is known about the actual change in knowledge structures responsible for this shift. As noted in our initial example about the algebra student who comprehends mathematical principles and succeeds in solving homework problems through appeal to familiar examples, research on analogical transfer has practical as well as theoretical implications. Psychologists' recognition of the potential educational benefits to be gained by the use of analogies in problem solving is reflected both in increased interest in the use of analogies in learning a skill (Reed, 1989; Reed &Bolstad, 1991) and in developmental studies that focus on children's acquisition of the skills underlying analogical transfer (Brown & Kane, 1988; Brown, Kane, &Echols, 1986; Crisafi& Brown, 1986; Centner & Toupin, 1986; Holyoak et al., 1984). Furthermore, transfer fits into a wider spectrum of ways in which analogies may be used in cognitive activities, such as analogical reasoning, casebased reasoning, and the comprehension of similes and metaphors. The recognized importance of work on the use of analogies in transfer guarantees that the progress of the last decade will continue into the next. References Adams. L. T.. Kasserman. J. E.. Yearwood, A. A., Perfetto. G. A., Bransford, J. D., & Franks, J. J. (1988). Memory access: The effects of fact-oriented versus problem-oriented acquisition. Memory & Cognition: 16, 167-175. Adelson. B. (1981). Problem solving and the development of abstract categories in programming languages. Memory & Cognition. 9, 422433. Ahn, W., Brewer, W. F, & Mooney, R. J. (1992). Schema acquisition from a single example. Journal of Experimental Psychology: Learning. Memory, and Cognition. 18, 391-412. Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: • Harvard University Press. Anderson, J. R.. Kline, P. G., & Beasley, C. M. (1979). A general learning theory and its application to schema abstraction. In G. H. Bower (Ed.), The psychology of learning and motivation (Vol. 13, pp. 277318). San Diego, CA: Academic Press. Bassok, M. (1990). Transfer of domain-specific problem-solving procedures. Journal of Experimental Psychology: Learning, Memory, and Cognition. 16. 522-533. Bassok, M., & Holyoak, K. J. (1989). Interdomain transfer between isomorphic topics in algebra and physics. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15, 153-166. Braine, M. D. S. (1978). On the relation between the natural logic of reasoning and standard logic. Psychological Review, 85, 1-21. Brooks, L. R. (1978). Nonanalytic concept formation and memory for instances. In E. Rosen & B. B. Lloyd (Eds.), Cognition and categorization (pp. 169-215). Hillsdale, NJ: Erlbaum. Brown, A, L.. & Kane, M. J. (1988). Preschool children can learn to transfer: Learning to learn and learning from example. Cognitive Psychology, 20, 493-523.

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(Appendixes follow on next page)

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Appendix A The General Problem A small country fell under the iron rule of a dictator. The dictator ruled the country from a strong, [brick] fortress. The fortress was situated in the middle of the country, surrounded by farms and villages. Many roads radiated outward from the fortress like spokes on a wheel. A great general, [tall, and resplendent in his navy blue uniform], arose who raised a large army at the border and vowed to capture the fortress and free the country of the dictator. The general knew that if his entire army could attack the fortress at once it could be captured. His troops were poised at the head of one of the roads leading to the fortress, ready to attack. However, a spy brought the general a disturbing report. The ruthless dictator had planted mines on each of the roads. The mines were set so that small bodies of men could pass over them safely, since the dictator needed to be able to move troops and workers to and from the fortress. However, any large forces would detonate the mines. Not

only would this blow up the road and render it impassable, but the dictator would then destroy many villages in retaliation. A full scale direct attack on the fortress therefore appeared impossible. The general, however, was undaunted. He divided his army up into small groups and dispatched each group to the head of a different road. When all was ready he gave the signal, and each group charged down a different road. All of the small groups passed safely over the mines, and the army then attacked the fortress in full strength. In this way, the general was able to capture the fortress and overthrow the dictator. (The information included in brackets was added to elucidate theoretical points made in the text. From "Schema Induction and Analogical Transfer" by M. L. Gick and K. J. Holyoak, 1983, Cognitive Psychology. 15, pp. 35-36. Copyright 1983 by Academic Press. Reprinted by permission.)

Appendix B Breakdown of the General and Tumor Problems The General 1. 2. 3. 4.

FACT: A fortress is located in the center of a country. FACT: Several roads emanate from the fortress. GOAL: A general desires to capture the fortress with an army. GOAL: The general desires to prevent mines in the roads from destroying the army and surrounding villages. 4.A. CONSTRAINT: The mines will explode when a large number of soldiers passes over them, which prevents an attack by the entire army going down one road to the fortress. 5. FACT: The force of the entire army is required to capture the fortress. 5.A. CONSTRAINT: The need for the entire army prevents the general from attacking the fortress with a small group which would not detonate the mines. 5.B. SOLUTION: The general's desire to prevent the mines from destroying the army and villages results in the dividing of the army into small groups. 5.C. SOLUTION: The fact that there are several roads radiating from the fortress allows the general to position a small group at the head of each road. 5.D. SOLUTION: This enables the small groups to simultaneously converge on the fortress. 5.E. SOLUTION: This allows the army to capture the fortress.

The Tumor 1. 2. 3. 4.

FACT: A tumor is located in the interior of a patient's body. FACT: The tumor is surrounded by healthy tissue. GOAL: A doctor desires to destroy the tumor with rays. GOAL: The doctor desires to prevent the rays from destroying healthy tissue. 4.A. CONSTRAINT: The susceptibility of healthy tissue to the high-intensity rays prevents the doctor from applying one highintensity beam of rays to the tumor. 5. FACT: A ray of high intensity is required to destroy the tumor. 5.A. CONSTRAINT: The need for a high intensity of rays makes it impossible to apply a low-intensity beam of rays, which would not destroy healthy tissue, to the tumor. 5.B. SOLUTION: The doctor's desire to prevent the rays from destroying healthy tissue results in the doctor's dividing the rays into beams of low-intensity rays. 5.C. SOLUTION: This enables the doctor to position the beams at multiple locations around the tumor. 5.D. SOLUTION: This enables the low-intensity rays to simultaneously converge on the tumor. 5.E. SOLUTION: This results in destruction of the tumor.

Received September 11, 1992 Revision received August 23, 1993 Accepted August 24, 1993 •