An Inquiry into the Spontaneous Transfer of Problem-Solving Skill

CONTEMPORARY EDUCATIONAL PSYCHOLOGY ARTICLE NO.

22, 472–494 (1997)

EP970948

An Inquiry into the Spontaneous Transfer of Problem-Solving Skill Edward A. Price and Marcy P. Driscoll Florida State University Problem solving, by definition, involves achieving new understanding in unfamiliar contexts, and is critical to all aspects of life, especially in the educational and scientific arenas. Students learn from many experiences to develop a repertoire of abilities, including the use of logic, which enable them to spontaneously transfer their problemsolving skill to unfamiliar situations. The purpose of this study is to explore the minimum conditions necessary to facilitate the spontaneous transfer of problem solving skill in an unfamiliar context. One hundred and seventy-five subjects were presented with logically identical problems based on the Wason selection task, which differed only in the degree to which a familiar schema could be invoked to help solve the problem. In the pretest stage, only 10.5% of subjects could solve the selection problem in an unfamiliar context, whereas 57.3% could solve it in one that was familiar. The effect of three interventions, prior exposure to a familiar scenario, repeat opportunities on like problems, and process-oriented feedback, on selection task performance in an unfamiliar context was assessed in a posttest stage. Overall, none of the interventions were effective, indicating that the minimum threshold for spontaneous transfer may be above the level of intervention included in this study. Schema theory, implications for instruction, and directions for future research are discussed. q 1997 Academic Press

INTRODUCTION

Seventh-grade students have difficulty solving word problems, even when they can successfully perform the mathematical calculations required by the problem (e.g., Schoenfeld, 1988). Educated adults, competent with everyday mathematical computations involved in shopping, managing money, and loading trucks correctly, cannot do the same tasks when they are attempted in abstract form in the laboratory (Lave, Murtaugh, & De La Rocha, 1984, cited in Singley and Anderson, 1989). Subjects faced with a problem about radiating a tumor fail to see the relevance of the solution provided in an analogous problem until prompted with a hint by the experimenters (Gick & Holyoak, 1983). These examples all point to the significant effects that content and context can have on problem solving even when tasks are logically identical. Contrary to the classical theory of reasoning which holds that the rules of

Address correspondence and reprint requests to Edward A. Price, Department of Educational Research, Florida State University, Tallahassee, FL 32306-3030. 472 0361-476X/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.

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logic are independent of context or content, what the problem is about has an enormous impact on how people think about it. What a problem is about also influences the degree to which people can apply what they know in one context to solving the same or a similar problem in a new context. Problem solving, by definition, involves achieving new understanding in unfamiliar contexts. Problem solving is the process individuals engage in when three conditions are met: first, there must be a ‘‘question,’’ task, or challenge that requires an answer or solution; second, the problem solver does not, for the moment, know how to reach the answer or solution; and, third, there is a perceived need or desire for the answer. Absent these three conditions the individual does not have a problem which requires solving (Smith, 1991). As an ability, problem solving is critical to all aspects of life, but it is particularly vital in the educational arena and scientific inquiry. Over time, students learn from both successes and failures to develop a repertoire of problem-solving abilities in many areas of knowledge. As this repertoire expands, the students’ ability to approach the unfamiliar and the unknown increases. Applying knowledge or skill acquired in one context to new instances or problems is a matter of transfer, and Salomon and Perkins (1989) suggested that some kinds of transfer require mindful abstraction of a ‘‘rule, principle, label, schematic pattern, prototype, or category’’ (p. 125). They called this ‘‘high road transfer’’ and contrasted it with ‘‘low road transfer,’’ which is the unprompted use of well-learned behavior in new contexts. Low road transfer appears to occur with extensive and varied practice so that behaviors and cognitions become automatic and stimulus controlled. Given cues in a new context that are taken to be prototypical of a particular category of situations, the learned behavior is automatically applied. Driving a truck after years of driving a car is an example of low road transfer. The conditions necessary to facilitate the mindful abstraction on which high road transfer depends, however, are less clear. Salomon and Perkins (1989) explicitly raised the questions of what learning conditions promote high road transfer and whether both kinds of transfer may contribute to some performances. In problem solving some strategies may become automatic after years of practice or study in a domain, but any given problem must first be approached mindfully if all relevant knowledge and skills are to be brought to bear on it. This potential relation between automatic and mindful transfer intrigued us and caused us to wonder what it would take to provoke spontaneous high road transfer. After all, an essential condition of problem solving as a learning outcome is the absence of guidance by the teacher; students must be able to effectively solve problems in unfamiliar contexts on their own (Gagne, Briggs, & Wager, 1992). Research on the four card selection task (e.g., Wason, 1968) provided some

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insights because performance on this problem is widely discrepant depending on the context in which the problem is encountered. The selection task requires conditional reasoning (with propositions of the form, If p, then q) and is considered to be the single most investigated problem in the deductive reasoning literature (Evans, 1996). The importance attached to this problem, from an educational point of view, comes from the fact that logical deduction has long been considered a primary means used by problem solvers to close the distance between the problem and a solution. Moreover, conditional reasoning plays an important role in scientific thinking. According to Ward, Byrnes, and Overton (1990), ‘‘students need to become proficient in such conditional reasoning in order to become successful practitioners of the scientific method and to evaluate the validity of theoretical assertions in any of the scientific disciplines’’ (p. 832). The fact that the selection problem yields such discrepant performance in diverse contexts makes it a useful framework for studying transfer. The Four Card Selection Problem in Diverse Contexts Peter Wason (1968) discovered that very few people could solve the selection task which involved determining which of four cards to turn over in order to test a conditional statement (‘‘If a label has a vowel on one side, then it has an odd number on the other side’’). Although the content of Wason’s task was abstract and unfamiliar to his subjects, it was so simple that then-current psychological theory was unable to explain why only 4% of his subjects were able to do it. (See the Appendix and Fig. 2 for a full explanation of the selection task.) Since Wason’s study, a great many investigations have been undertaken to try to improve performance on the selection task and to understand why people have so much trouble doing it. Training in formal logic was no apparent help. Cheng, Holyoak, Nisbett, and Oliver (1986) found that subjects performed only 3% better following a full semester course in formal logic than those who did not receive such training. Similarly, overall educational attainment appeared to have no determinative effect on the outcome. According to Jackson and Griggs (1990), people with Ph.D. degrees performed the selection task no better than people with bachelor’s degrees. When the selection task was put into a familiar context, however, performance changed dramatically. In one of the earliest extensions, Wason and Shapiro (1971) showed that presenting the conditional with content familiar to subjects (‘‘Every time I go to Manchester I travel by train’’) enabled more than 60% of the subjects to generate the correct card selections. Control group subjects, faced with abstract content, were still mired at about 12%. JohnsonLaird, Lagrenzi, and Lagrenzi (1972) gave a number of British subjects a scenario relating to the British postal system with which they were familiar. In

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the familiar condition 87.5% were able to do the selection card task correctly. Interestingly, they found that other British subjects or foreigners living in Britain who were not familiar with the postal system’s rules performed very poorly. D’Andrade (cited in Rumelhart, 1980; Rumelhart & Norman, 1981; and D’Andrade, 1995), in the United States, also demonstrated the tremendous impact of problem content on results. When presented within a familiar context, known as the Sears Store Scenario (‘‘If the purchase receipt is over $30.00, then it has the manager’s signature on the reverse side’’), 70% of subjects were able to solve the problem, but when it was presented in an unfamiliar context, known as the Label Factory Scenario (using Wason’s original rule in the label factory scenario), only about 13% could do it. In a related study, Griggs and Cox (1982) compared an unrealistic scenario with a realistic one. While none of their subjects could perform the task correctly with the unrealistic scenario, 73% could accomplish it when couched in terms bound to be familiar to undergraduate students (‘‘If a person is drinking beer, then the person must be over the age of 19’’). Initially, the facilitation effect of familiar content appeared to be based on subjects’ direct experiences. Griggs and Cox (1982) proposed the ‘‘memory cue’’ hypothesis suggesting that the facilitation effect occurred when the familiar problem content cued subjects’ memories. However, that view proved to be both too broad and too narrow. On the one hand, efforts to replicate the facilitation effect with familiar content yielded mixed results, indicating that memory cueing was not always sufficient to produce a facilitation effect (Griggs, 1983; Griggs & Cox, 1982; Wason, 1983). On the other hand, researchers such as Cheng and Holyoak (1985) showed that generalized scenarios with which their subjects were not directly familiar, and thus could not be cued from memory, could lead to the facilitation effect. A Schema-Based Explanation? Researchers turned to schema theory in order to explain the disparity of performance between familiar and unfamiliar scenarios (Anderson, 1977; Anderson, Spiro, and Anderson, 1979; Rumelhart, 1980; Rumelhart & Norman, 1981; D’Andrade, 1995). Schemata are the basic units of knowledge that are learned in the context of daily living. Schemata may concern objects, events, processes, or attitudes (Driscoll, 1994). R. C. Anderson (1977) offers an effective description of the concept of schema. A schema represents generic knowledge; that is, it represents what is believed to be generally true of a class of things, events, or situations. A schema is conceived to contain a slot or place holder for each component. For instance, a Face schema includes slots for a mouth, nose, eyes and ears. (p. 2)

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Some other examples of schemata are: ‘‘house,’’ ‘‘going to the restaurant,’’ ‘‘taking a multiple choice test,’’ ‘‘writing a term paper,’’ and ‘‘getting acquainted.’’ Rumelhart (1980) explained how schemata provide essential linkages between the concepts and patterns of behavior which are the substance of what we know and do: A schema theory is basically a theory about knowledge. It is a theory about how knowledge is represented and how that representation facilitates the use of the knowledge in particular ways. According to schema theories all knowledge is packaged into units. These units are the schemata. Embedded in these packets of knowledge is, in addition to the knowledge itself, information about how this knowledge is to be used. (p. 34)

Each schema is made up of related concepts, involving both declarative and procedural knowledge. For instance, the ‘‘getting acquainted’’ schema would include such processes as giving a hand shake, saying something friendly, asking a question about the other person, and so on. Rumelhart (1980) emphasized that ‘‘most of the reasoning we do apparently does not involve the application of general purpose reasoning skills. Rather, it seems that most of our reasoning ability is tied to particular schemata related to particular bodies of knowledge’’ (p. 55). As mentioned, Griggs and Cox (1982) proposed a ‘‘memory cue’’ schema explanation. If individuals have direct experience with the scenario, prior schemata are activated and problem solving is facilitated. According to this view, people rely mainly on very domain-specific knowledge, wherein specific recollections of prior experiences are used to solve problems. Singley and Anderson (1989) refer to this view as the ‘‘radical specificity position’’ (p. 235). Cheng and Holyoak (1985) offered a more general interpretation of schema theory. They argued that even if people do not appear to use formal rules of logic there are rule systems that people do use that are more general and somewhat domain independent. These rule systems, which they argued are acquired through everyday experience, are called pragmatic reasoning schemas. According to Cheng and Holyoak (1985), their version of the selection task, the drinking age scenario, involved an example of the pragmatic schema, called the ‘‘permission schema.’’ In this schema the premise is that in order to take one action, a precondition must be satisfied. They found that the permission schema enabled 60% of the subjects to perform well, even when the content was abstract, whereas only 20% could do the abstract problem when the permission schema was not included. Evans (1989) observed that the permission schema maps to the same solutions as formal logic in the case of the four card selection problem. In other words, using a permission schema produces ‘‘card choices which coincide

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with those prescribed by the logical analysis of the problem’’ (p. 85). When subjects interpret the card selection problem as one concerning permission, the permission schema is evoked and enables them to make the correct selections. The idea that subjects might be using a relevant schema to help them solve the selection problem is supported by results of studies investigating analogical reasoning as well as arithmetic problem solving. Gick and Holyoak (1983), for example, conducted a series of studies in which subjects first read one or more stories illustrating a problem and then attempted to solve a different but analogous problem. Their results indicated that subjects who induced a problem schema from the story had greater success in solving the analogous problem than subjects who did not induce a problem schema. In addition, Sweller (1989; Cooper & Sweller, 1987) argued that students solve arithmetic problems rapidly and with relative ease when they access schemata of problem types. Does schema theory explain the disparity of performance among logically identical situations that is seen in the investigations of the selection problem? Can schema theory offer guidance in generating the conditions that will induce people to make the transfer between identical problems in different contexts? When we considered the array of prior research on the selection problem, we noted that all of the studies exposed subjects to either an unfamiliar or a familiar scenario, but never both and never the same scenario a second time. Thus, the prior studies do not speak to the issue of potentially beneficial effects of sequential or repeated exposure to the selection task, nor to the effect of prior exposure to the familiar version on solving the one that is unfamiliar. If schemata are the operating mechanisms or knowledge structures that are enabling or inhibiting performance when subjects encounter the unfamiliar version of the selection task, and if possession of an applicable schema is believed to be the reason why so many more subjects are able to perform well in the familiar version, then it is possible that prior exposure to a familiar scenario should facilitate problem solving when an unfamiliar scenario is encountered right afterward. If all other conditions, such as wording, formatting, and problem structure, are held constant, we should be able to detect the effect, thus supporting a schema-based explanation. Mindful of the fact that performance was so much higher in the familiar condition, we reasoned that prior exposure to a familiar scenario would be one of the minimum conditions necessary to facilitate spontaneous transfer of problem solving in the unfamiliar condition. Less likely, but also evidence of a schema-based explanation, would be improved performance on the selection problem when encountered repeatedly in the unfamiliar scenario. Given these supports, it may be possible for subjects to abstract a problem schema and transfer it to the unfamiliar scenario.

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Recognizing that people find the unfamiliar version of the four card selection problem difficult, however, we considered that prior exposure to the familiar scenario and repeat exposures might still not be enough to facilitate spontaneous transfer. Then the issue of additional factors leading to schema building and schema extension becomes a primary concern. Thus, we hypothesized that problem solving in the unfamiliar scenario could be enhanced by a feedback intervention that induced schema restructuring or schema tuning, two of the three ways schemata are assumed to be modified. (As noted below, the third means of modifying schemata, accretion, does not apply to this study.) Our three research questions were generated, therefore, from our interpretation of schema theory and by our concern to know how much (or how little) assistance problem solvers would require in order to spontaneously generate their own correct solutions to problems of logic: 1. If a subject is exposed to the selection task in a familiar context before immediately going on to one that is unfamiliar, does performance in the unfamiliar context improve? 2. If subjects encounter the selection problem multiple times, does performance improve? 3. If feedback is provided between attempts, does performance improve and, if so, what kind of feedback is most effective? Design and Rationale for the Study Figure 1 portrays the three-stage structure of the study. Stage 1 replicates the conditions of the D’Andrade study and establishes the existence of a schema in our subjects. This enabled us to place the subsequent stages in the same experimental context as the D’Andrade study. Stages 2 and 3 provide our extensions of the original design. (A full description of the selection task is provided in the appendix to this paper. See also Fig. 2.) In stage 1, half the subjects began with the unfamiliar scenario in which they were told to imagine themselves as label checkers in a label factory. They were given the rule—‘‘If one side of the label has a vowel on it, then the other side of the label has an odd number’’—and told to turn over only those labels required to verify if the rule is being followed or not. The remaining subjects began with the familiar scenario. They were told to imagine themselves as clerks in a Sears store who were required to check receipts. They were given the rule—‘‘If the store receipt is more than $30.00, then it has the manager’s signature on the other side’’—and provided the same instructions as the subjects in the unfamiliar scenario. In both scenarios, the rules were carefully worded to exclude words such as ‘‘must’’ or ‘‘should’’ in order to avoid inadvertently triggering one of the

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Fig. 1. Design of the study.

pragmatic schemas—the ‘‘schema of obligation’’ that Manktelow and Over (1992) discussed. Understanding the rules in terms of obligation tends to change the nature of subjects’ reasoning in the selection task. Instead of approaching the problem using logical deduction, they select cards based on expected utility of conforming or not conforming to the rule. We wanted to assure a clear test of schema-based logical reasoning and avoid introducing an extraneous variable. In accord with D’Andrade’s results, we expected that subjects given the familiar scenario would outperform those given the unfamiliar scenario. Stage 2 provides a test of the hypothesis that prior exposure to a familiar scenario would assist problem solving by virtue of evoking an existing schema. The question is whether apprehending the problem schema in the familiar Sears story context will enable subjects to abstract a general problem schema that they can then apply to the problem in the unfamiliar label factory context. In stage 2, all subjects were presented with an unfamiliar scenario (as label checkers) and the rule, ‘‘If one side of the label has the letter ‘P’ on it, then it has an even number on the other side.’’ The subjects exposed first to a familiar scenario were expected to perform better on the problem in the unfamiliar scenario of stage 2 than those exposed first to an unfamiliar scenario. No feedback was provided between stages 1 and 2 for either group. Stage 2 also answers the question of whether repeat exposures to the same problem in the same context is enough for subjects to abstract a useful problem schema. Should performance on the problem in the unfamiliar scenario im-

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prove from stage 1 to stage 2, independent of the context seen before, then we would conclude that multiple exposures are sufficient to induce spontaneous transfer of problem solving. Stage 3 enables us to examine the impact of an intervention—rule-based feedback—that was designed to facilitate problem-solving transfer by modifying subjects’ problem schemata. The selection of feedback types was guided by schema theory. According to theory, schemata are extended in three ways: accretion, restructuring, and tuning (Rumelhart, 1980; Rumelhart & Norman, 1978). Accretion refers to the encoding of new information in terms of existing schemata. Since no new information was being provided to subjects in this study, accretion was not applicable. Restructuring occurs when information does not fit into an individual’s existing collection of schemata and a new schema must be created. Restructuring most commonly occurs by modeling the new schema on an existing schema and modifying it slightly, i.e., reasoning by analogy (Rumelhart & Norman, 1978, 1981). To support restructuring in stage 3 we used analogical feedback in one of the two feedback conditions to be compared with no feedback. Any analogy requires two elements for comparison. Thus no feedback could be given following stage 1 but was provided between stages 2 and 3. The analogical feedback for the Sears scenario was presented as follows: In the first scenario the rule you were asked to follow as a Sears Clerk was ‘‘If a receipt is over $30.00, the manager’s signature appears on the back.’’ In the second scenario the rule you were asked to follow as a Label Checker was ‘‘If there is a letter P on one side of the label, it has an even number on the other side.’’ Please take note of the fact that having a receipt that is over $30.00 in the first scenario is like having a letter P on one side of the label in the second scenario. Likewise, having a receipt with the manager’s signature in the first scenario is like having an even number on one side of the label in the second scenario. When you perform this task one last time, please keep this in mind.

To help facilitate the creation of a problem schema for subjects who did not spontaneously invoke it, the analogical feedback made the schema explicit and linked it by analogy in the two scenarios. Tuning, the third way by which schemata are modified, is a process of refinement that adds and deletes characteristics to an existing schema. Tuning leads to increased accuracy of a schema (Rumelhart & Norman, 1978). In this study, declarative feedback (Dempsey, Driscoll, & Swindell, 1993) was used in the second feedback condition of stage 3 to facilitate tuning. Declarative feedback is rule-based and informative in nature. Declarative feedback elaborates on the instructions given in the first stage. Briefly recall

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that, in effect, the instructions were structured in this way: ‘‘The rule is if A is true, then B is true. Using the items given, identify those which need to be checked.’’ Chapman (1993) noted that the most common error that people make on the selection involves affirming the consequent. In effect, affirming the consequent logically reverses the rule into ‘‘If B is true, then A is true.’’ Chapman reported that people may not affirm the consequent when they are provided direct information pointing away from that fallacy, hence our decision to include this form of declarative, rule-based feedback. We anticipated that if subjects knew which error to avoid, they might then replace the wrong item with a correct one. Specifically, declarative feedback provided in the unfamiliar scenario was as follows (feedback provided in the Sears scenario was nearly identical): . . . the rule you were asked to follow as a Label Checker was ‘‘If a label has a vowel on one side, it has an odd number on the other side.’’ Please take note of the fact that the rule was NOT ‘‘If a label has an odd number on one side, it has a vowel on the other side . . .’’ When you perform this task one last time, please keep this in mind.

According to schema theory, this kind of feedback is most similar to tuning because it helps the subject increase the clarity and accuracy of an existing schema. As noted, and consistent with Gick and Holyoak (1983), subjects were invited to use the information provided in the feedback when they attempt the selection task an additional time. All subgroups in stage 3 performed the same task, that is, to examine the set of labels provided in an unfamiliar scenario as follows: ‘‘If there is a letter P on one side of the label, then there is an even number on the other side.’’ The problem in stage 3 is identical to the problem in stage 2 in order to avoid possible confounding caused by a problem change. The letters and numbers used were changed, however, to prevent memorization of responses. The hypothesis of this portion of the study was that, in keeping with schema theory, analogical feedback should be more effective than declarative feedback, which in turn should be more effective than no feedback at all. A corollary of our earlier hypothesis is that the respective feedback effects were expected to be greater among the subjects working within a familiar schema than with those with the unfamiliar schema. Finally, if multiple exposures to the problem have any effect, then the performance of all subjects should be better in stage 3 than in the prior two stages. Since our interest in this study was in discovering the minimum conditions necessary to promote spontaneous transfer, we did not provide subjects with correct response feedback, nor did we elaborate on how to solve the selection

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task. We wanted to test first for the effects of rule-based feedback. In future research, we intend to add both instructional and correct response feedback. METHOD

Subjects One hundred eighty (180) subjects were recruited for the study. Of these, 169 were recruited from students in the College of Education at the Florida State University, Tallahassee, Florida. Eleven were recruited from a local church in Tallahassee, Florida. Five packets were discarded from the data set because they were not filled out (n Å 4) or were filled out incorrectly (n Å 1). Thus, the final sample size was 175. Subjects were divided into six subgroups (see Fig. 1) that were equal in size initially. Subgroup sizes were 28, 29, 29, 30, 30, and 29, respectively.

Materials All materials were in print format. The first page described the task and what the subjects were supposed to do and included a consent form. Subjects were told that their names were not required. The second page contained stage 1. On that page a brief description of the label checking or Sears store scenario was provided along with the problem-solving task, depending on the individual’s group assignment. The third page (stage 2) presented another label-checking scenario with the same instructions as before. At the fourth page (stage 3), subjects’ paths diverged. The no-feedback group encountered their third and final exposure to the selection problem. The declarative and analogical feedback groups were presented with feedback as previously described, followed by their third and final exposure to the problem.

Procedure The study was administered to small groups in the classroom. The researcher explained that this was a study in human problem solving and that the task would take between 10 and 15 minutes, but they would be given all the time they required. Subjects were asked to note their start and end times at the top of the first page. As explained above, the subjects were divided into six categories, thus there were six different packets. Packets were placed in sequential order and inserted into standard office envelopes ahead of time to assure that assignment to the six subgroups would be random. The researcher did not know at the time the study was being administered which subjects were being assigned to which groups. Packets were then collected and scored.

Measures Scoring of the selection task is straightforward. Subjects must select both affirming the antecedent and denying the consequent. All other combinations are incorrect. (See the Appendix and Fig. 2 for details.)

RESULTS

A summary of results indicating the percent of subjects responding correctly in each stage of the study is presented in Table 1. Statistical significance is also indicated using the x2 technique.

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Fig. 2. The four possibilities of the four-card selection task.

Is There a Schema Effect in Stage 1? In stage 1, we replicated the findings of the Wason and D’Andrade studies. As expected, subjects performed significantly better in the familiar scenario than they did in the unfamiliar scenario. In the familiar Sears scenario, 57.3% of subjects were able to solve the problem correctly, whereas only 10.5% of subjects were able to do so in the unfamiliar label-checker scenario. Therefore, subjects were about 5.5 times more likely to solve the problem when it was

TABLE 1 Percent of Subjects with Correct Responses Group One: Unfamiliar

Group Two: Familiar

Statistical significance (x2, a Å 0.05)

Stage One

10.5%

57.3%

Stage Two Stage Three No feedback provided Declarative feedback provided . . . Analogical feedback provided . . .

14.0%

12.4%

Significant, x2 Å 42.6, df Å 4, p õ .0001 Not significant

14.3% 17.2% 3.4%

13.3% 16.7% 6.9%

Not significant Not significant Not significant

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TABLE 2 Number Who Selected the Correct Pair Stage One n Å 51 and of these c

Familiar Scenario

Unfamiliar Scenario

nÅ9 and of these c

Stage Two

Stage Three (All subgroups combined)

n Å 10 (19.6%) and of these c

n Å 8 (80.0%)

n Å 9 (100.0%) and of these c

n Å 7 (77.8%)

put in a familiar context. This difference in performance was statistically significant (x2 Å 42.6, df Å 1, p õ .001, a Å .05). Does Prior Exposure to a Familiar Scenario Facilitate Problem Solving? Because problem-solving performance in an unfamiliar scenario is always poorer than in a familiar scenario, we expected the performance to drop for those subjects exposed to the Sears scenario in stage 1 followed by a labelchecker scenario in stage 2. It did (from 57.3 to 12.4%). But we hypothesized that these subjects would outperform those who received the unfamiliar scenario in both stages 1 and 2. The difference between the two groups in stage 2 would constitute the schema facilitation effect we expected to see. However, that did not occur. Even though subjects could do the selection task more successfully in the familiar scenario, the entire performance advantage disappeared as soon as they were in the unfamiliar scenario. The difference was so dramatic that we think subjects did not even recognize that the familiar and the unfamiliar scenarios were actually the same problem. This is all the more remarkable since the materials were worded and formatted exactly alike. So far, the data we reported for stage 2 include subjects who answered correctly and incorrectly in stage 1. One might argue that the subjects in stage 1 who did not solve the selection task correctly should be excluded from the analysis because they had no effective schema to be applied to stage 2. Indeed, this is in keeping with our hypothesis that success with the familiar scenario would provide facilitation when the unfamiliar scenario is encountered. Unfortunately, when we looked only at those who answered correctly in stage 2, prior exposure to the familiar scenario still did not appear to facilitate stage 2 performance in the unfamiliar scenario. Table 2 shows the situation quite strikingly. If a subject solved the problem correctly in the familiar scenario, the chances of getting it right in the subsequent unfamiliar scenario were less than one in five (19.6%). On the other hand, although many fewer subjects answered correctly in the unfamiliar scenario of stage 1, of those who did,

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100% of them got it right when they faced another unfamiliar scenario in stage 2. Contrary to our hypothesis, prior exposure to the familiar scenario accomplished nothing. Does Multiple Exposure to the Problem Facilitate Transfer? Most subjects who failed to solve the selection task correctly in the unfamiliar scenario appeared to have no existing problem schema to work with in stage 1, and they did not apparently build a useful problem schema in stage 2. Their performance remained low regardless of their multiple exposure to the selection problem. For the subjects who initially encountered the problem in a familiar scenario, the availability of an existing schema in stage 1 provided no benefit to problem solving in stage 2. As Table 2 shows, multiple exposures to the same type problem, regardless of its context, had no facilitative effect on performance. How Does Feedback Affect the Transfer of Problem Solving? In stage 3, we wanted to determine what kind of feedback, if any, improves the transfer of problem solving. Three types of feedback conditions were provided: no feedback, declarative feedback, and analogical feedback. We hypothesized that declarative feedback and analogical feedback would improve problem-solving performance, but the data did not confirm our prediction. As Table 1 shows, the differences among subgroups in stage 3 were not significant, nor was performance in stage 3 better than performance in stage 2 for any group. Therefore, this study provided no evidence that rule-based feedback facilitates the spontaneous transfer of problem-solving skill, at least within the confines of this design. What Can Be Learned from the Errors Subjects Made? When two of our three primary hypotheses were not confirmed, we examined the error patterns made by the subjects. Although response patterns of one, three, or four selections may include the correct options of affirming the antecedent and denying the consequent, they are counted as errors because they violate the instructions given to subjects to turn over only those cards that require verification, no more and no less. While virtually all the subjects correctly chose to affirm the antecedent, the most common error pattern involved affirming the consequent. This error is tantamount to arbitrarily reversing the rule. If the rule is ‘‘if vowel, then odd,’’ affirming the consequent transforms the rule to ‘‘if odd, then vowel.’’ Stage 2 tells a striking story in comparison to stage 1 as both the familiar and the unfamiliar groups appeared nearly identical in their response pattern distribution. In stage 2 both groups displayed the same error patterns of affirming both the antecedent and the consequent (50%) and affirming the

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TABLE 3 Frequency and Percentage of Subjects Who Made a Change after Receiving Feedback Subgroup

Frequency

Unfamiliar, no feedback Familiar, no feedback Unfamiliar, declarative feedback Familiar, declarative feedback Unfamiliar, analogical feedback Familiar, analogical feedback

0 4 19 16 8 1

out out out out out out

of of of of of of

28 30 29 30 29 29

Percentage 0.0% 13.3% 65.5% 53.3% 27.6% 3.4%

antecedent only (20%). No statistical difference exists between the two groups in Stage 2. This means that many subjects who correctly selected denying the consequent in conjunction with affirming the antecedent in the familiar scenario of stage 1 abandoned denying the consequent (which was correct) for affirming the consequent (which was not correct) in stage 2 when they got to the unfamiliar scenario. Stage 3 follows the feedback treatments. We have already seen that the feedback used did not lead to a statistically significant improvement in performance in any group. Nonetheless, there was an observable effect regarding those who made a change following feedback. The change could be of any kind: from correct to incorrect, from incorrect to correct, or from one incorrect response pattern to another. As shown in Tables 3 and 4, the feedback we provided was effective only in the declarative condition and then only in eliminating one error response, affirming the consequent, which our feedback targeted. The degree of impact was about the same for all subjects who received this type of feedback, so it is safe to assume the change is caused by the feedback treatment rather than any residual schema effect based on prior exposure to the familiar scenario. TABLE 4 Frequency and Percentage of Subjects Who Eliminated ‘‘Affirming the Consequent’’ after Receiving Feedback Subgroup

Frequency

Unfamiliar, no feedback Familiar, no feedback Unfamiliar, declarative feedback Familiar, declarative feedback Unfamiliar, analogical feedback Familiar, analogical feedback

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Percentage 0.0% 0.0% 55.2% 40.0% 0.0% 3.4%

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Other than this, the number who changed to the correct pair following any of the feedback conditions is negligible. DISCUSSION

We began this study with the question, What does it take to provoke spontaneous high road transfer? We hypothesized that invoking an appropriate schema from a problem in a familiar context would facilitate solving the same problem in an unfamiliar context when it was encountered right after. Although our subjects demonstrated that they had an appropriate schema that assisted them in solving the problem in the familiar context (stage 1 results), they did not transfer this schema to the unfamiliar context (stage 2 results). Moreover, subjects who did not have a problem schema for solving the problem in the unfamiliar context did not build one with multiple exposures to the problem (also stage 2 results). Finally, contrary to expectation, rulebased feedback designed to facilitate schema tuning and restructuring did not improve subjects’ problem solving performance (stage 3 results). Declarative feedback, however, did have the effect of eliminating the most common error, affirming the consequent. Schema theorists, such as Anderson (1977), Rumelhart (1980), JohnsonLaird (1983), and Cheng and Holyoak (1985), have argued that subjects depend on schema building in learning how to solve problems. Salomon and Perkins (1989) argued further that the problem schema must be mindfully abstracted or decontextualized from specific instances of a problem in order for high road transfer to occur. This study adds strength to the idea that schemata exist and that they powerfully influence problem solving (from stage 1 results). However, there is no evidence that our subjects spontaneously abstracted a useful schema while trying to solve the selection problems nor did the feedback conditions appear to promote such abstraction. In one respect, our results seem surprising. Subjects in Gick and Holyoak’s (1983) research demonstrated increased transfer when they were cued as to the relevance of the first story to the solution of the problem in the second story and when two prior analogs were provided. In contrast, our subjects could not solve the selection problem in the unfamiliar context even when provided with feedback analogously linking it to the problem in the familiar context, which they could solve. Two prior analogs were of little benefit to our subjects even when subjects were cued to the relevance of the earlier scenarios and when encouraged to use the information on the third and final problem-solving attempt. So what did the subjects in this study actually do? In the unfamiliar scenario, given the absence of a relevant schema, we think they acted with what Evans (1984, 1989, 1996) and others have referred to as matching bias. The rule

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said, ‘‘If vowel, then odd,’’ so they tended to notice the items mentioned in the rule, select them, and then quickly turn the page. In the familiar scenario, possession of a relevant schema (i.e., familiarity with store receipts and getting forms signed) enabled them to score higher, but with no visible transfer of this ability to the unfamiliar context. Simply put, they saw two items in the rule, so in most cases those were the ones they selected. We doubt the thinking was any more mindful than that. In a study conducted after ours, Evans (1996) suggested that the matching bias is cued by a preconscious determination of relevance that is qualitatively different from formal logical reasoning. Using the drinking age version of the selection problem (Griggs & Cox, 1982), which has a reliable facilitation effect, Evans (1996) noted that subjects required little time to make correct selections. His study, which was computer-based, was designed to collect data on time spent considering card choices, as well as the actual selections. Evans reported that the subjects who think about the card for a reasonable amount of time—as measured by inspection times—end up selecting it. Those who do not think about the card (there are many zero times at the level of individual subjects) or inspect it only briefly do not end up selecting the card . . . . Individual differences in perceived relevance (as measured by inspection times) are also strongly correlated with selection decisions on particular cards. (p. 223)

Does this mean that subjects are deciding what cards they will pick before they think about them? Although this is counterintuitive, Evans thinks so. If, as we have seen, problem-solving ability is heavily content and context dependent, and yet, as we also have seen, not entirely dependent on direct experience, then what Evans is proposing makes sense. In everyday life individuals seldom have the time or the opportunity to search through all the possibilities of a problem space before a decision is required. Often key facts are missing, parts of a problem are difficult to understand, or sufficient time is not available to fully analyze a problem. And yet, despite these limitations, people are often able to discern a correct problem solution in situations where they have an incomplete set of relevant knowledge to draw upon. Such an ability would essentially consist of pattern recognition (Margolis, 1987). These patterns would be stored and organized as schemata, ready for use on demand when needed. Living in a complex and ever-changing environment would require that these schemata be fairly durable in the sense of being correctly tuned to appropriate contexts, so that the right schemata would be triggered in the right context. Some of them would be very specific and not very transferable. The demands of survival would also require some moderately generalizable schemata as well in order to promote learning and

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adaptation. Nesting schemata within schemata, as schema theory proposes, would certainly add to this flexibility. Given the usual constraints of both time and limited working memory, most individuals, most of the time, will not search through an entire problem space; indeed, the habit of doing so could often create an overwhelming cognitive burden (Carlson, 1997). As Perkins and Salomon (1989) noted with the example of famous chess masters, the number of combinations and permutations on a typical chess board is far too large for an individual to explore fully. The chess master does not attempt this futile task, however, for he or she has already memorized thousands of patterns that commonly occur on the chess board along with their probable strengths and weaknesses. Using these patterns as a heuristic, the chess master rapidly recognizes a few of the most promising moves and then concentrates all of his or her analytical ability on those. On this account, patterns are judged relevant or worthy of further cognitive attention as a result of the interaction between the schemata that individuals possess and the contexts in which they find themselves. The schemata people use could be based on direct experience as Griggs and Cox (1982), JohnsonLaird (1983), Evans (1984, 1989), D’Andrade (1995), and others have proposed. When these are absent, the schemata can be moderately generalized, as in the pragmatic schemas proposed by Cheng and Holyoak (1985), Manktelow and Evans (1979), Manktelow and Over (1992), Cosmides (1989), and others. What does appear clear, however, given the long history of research on the selection task and other problems, is that fully context-independent logical analysis is not the normative response for most people. Thus, it may well be that the judgment of relevance is determined heuristically and preconsciously, through pattern recognition of subtle pragmatic cues (Evans, 1984, 1996; Margolis, 1987). Evans (1996) noted: Not only is attention selectively focused on the presented information, but the retrieval of relevant prior knowledge, including rules, heuristics or schemas, also occurs the same way. The success of such conscious, analytic reasoning as does occur is very highly constrained by the ‘‘relevant’’ information on which it is focused. (p. 223)

In sum, this study reinforces the point that new schemata do not easily arise spontaneously, nor are they easily transferred. We were not able to develop evidence supporting our hypotheses that exposure to a familiar scenario would facilitate the transfer of problem-solving skill to an unfamiliar scenario, nor that practice opportunities alone enhance performance, nor that the rule-based feedback that we provided in this study by itself leads to enhanced performance. We do have evidence that declarative feedback may help eliminate errors, which is valuable to know. If these factors are to have

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any value in instruction of problem-solving skill they may well need to be combined with other kinds of instructional assistance. These reflections suggest some implications for instruction as well as directions for further research. First, instruction should focus more on schemabuilding strategies as foundational to problem-solving ability. An integral component of building these schemata for logical deduction would concentrate explicitly on increasing the problem solver’s ability to deny the consequent as well as to affirm the antecedent. Problem solvers should be guided explicitly to avoid affirming the consequent when inappropriate. Instruction should pursue these goals in the context of realistic, familiar scenarios rather than in more conventional abstract contexts. Second, instruction should facilitate schema building by providing learner feedback in the form of numerous fully worked out and explained examples or worksheets that explicitly guide learners in building their own schemata. This relates to what Cormier (1987) refers to as the principle of ‘‘encoding specificity’’—wherein the probability of retrieving a schema that has been built ‘‘is a joint function of the way in which the material was originally encoded and the cues of information available at the time of retrieval’’ (pp. 153–154). As shown in this study, feedback that was limited to a process or rule orientation was not sufficient to facilitate spontaneous problem-solving performance. Third, single exposures to problem-solving situations are unlikely to provide enough material for schema building to occur. If high road transfer is desired following instruction, multiple schema-building experiences are probably required just as they are for low road transfer. How many and what kind is still a subject for additional research. Corollary to this implication is that instructors and instructional designers should assume that problem-solving ability is cumulative not only over time but over numerous experiences. This study points in the direction of multiple exposures to problem solving scenarios, probably from differing perspectives, as the most probable way to assure that the learner actually notices that a newly encountered problem is really a previously encountered problem in a new context (Driscoll, 1994; Cheng et al., 1986). Once the learner recognizes the relevance of other schema to the current problem, he or she can bring his or her repertoire of problem-solving abilities and prior knowledge to bear upon the present situation, thus setting the stage for high road transfer to occur. Given these implications for instruction, a new focus on individuals, rather than just groups, is desirable. Research utilizing thinking aloud protocols would greatly add to this effort. We agree with Evans’ (1996) concern that ‘‘surprisingly, very few reports are available of thinking aloud protocols on the selection task’’ (p. 223).

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Finally, whatever plan instructors adopt, they must not leave schema building or knowledge transfer to chance. It does not arise spontaneously and its development cannot be assumed. APPENDIX: THE FOUR CARD SELECTION TASK

Wason (1968) introduced the selection task to test the classical view that human reasoning operates according to the rules of formal logic. Subjects are shown four cards as in Fig. 2 and told that a letter appears on one side and a number on the other side. Subjects are asked to determine which cards need to be turned over to check if a rule is being followed or not. The rule is, ‘‘If a card has a vowel on one side, then it has an odd number on the other side.’’ While subjects are instructed to turn over only those cards required to make the determination, they are not told how many cards need to be turned over. The four cards (E, F, 7, 8) represent the four logical possibilities created by the if-then conditional: affirm the antecedent, affirm the consequent, deny the antecedent and deny the consequent. Only 4% of Wason’s subjects identified the correct pair of cards, the letter ‘‘E’’ and the number ‘‘8,’’ which represent affirming the antecedent and denying the consequent, respectively. Affirming the Antecedent. Selecting the ‘‘E’’ is a called affirming the antecedent and, in formal logic, it leads to a valid conclusion. If there is an even number on the other side, the rule is being followed and the card is correct. In this case we could let the card by without looking at it. However, if there is an odd number on the other side, the rule would be violated and the card would be wrong. We cannot let a bad card through, so we have to turn the card over to make sure there is not an odd number on the other side. Denying the Antecedent. Selecting the ‘‘F’’ is a called denying the antecedent and, in formal logic, it leads to an invalid conclusion. Since the rule does not tell us to look for consonants, it does not matter if there is an odd or an even number on the other side. Therefore, we do not turn over this item. Affirming the Consequent. Selecting the ‘‘7’’ is called affirming the consequent and, in formal logic, it leads to an invalid conclusion. We do not turn this item over since the rule does not say to do anything in the case of odd numbers, rather only vowels. Thus, if there is a vowel or a consonant on the other side, it makes no difference. Seventy-nine percent of Wason’s (1968) subjects made this error even though it is logically invalid. Approximately 70% of our subjects made this error as well. Denying the Consequent. The remaining response, selecting the ‘‘8,’’ is called denying the consequent, and, in formal logic, it leads to a valid conclusion. It does not matter if there is a consonant on the other side, but if there is a vowel on the other side the rule is being violated. Thus, we must turn over this item to check. When we deny the consequent we are looking for information that is missing rather than for information which is present. Since

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the information we require is missing, more people are likely to miss this option. In fact, that is what happens. 96% of Wason’s subjects failed to combine this option with the more obvious and equally essential affirming the antecedent option. In this study, 89.5% of our subjects in the unfamiliar scenario missed this as well. D’Andrade (cited in Rumelhart, 1980; D’Andrade, 1995) modified Wason’s original design. Half of D’Andrade’s subjects repeated the original abstract and unfamiliar conditions of the Wason study. The other half were asked to imagine themselves as clerks at a Sears store with the responsibility of checking the validity of store receipts. They were told ‘‘If the receipt is over $30, then it has the manager’s signature on the back of the receipt.’’ In the Sears scenario four receipts as shown in Fig. 2 were presented: a purchase over $30, a purchase under $30, a receipt that is signed on the back, and a receipt that is not signed on the back. Logically identical to the Wason version, the correct answer is to select just the receipt over $30 and the receipt without the manager’s signature on the back, that is, to affirm the antecedent and to deny the consequent. As we can see, the logical form of the Sears problem is the same as the earlier letter and number scenario. The two scenarios differ in that one has a familiar context and the other has an unfamiliar one. One is realistic in content, the other is arbitrary. As mentioned earlier, in the familiar Sears scenario 70% of D’Andrade’s subjects selected the correct combination of responses (cited in Rumelhart, 1980). However, only 13% of D’Andrade’s subjects could perform the task when based on the abstract rule, ‘‘if vowel, then odd.’’ REFERENCES Anderson, R. C. (1977). Schema-directed processes in language comprehension. In A. Lesgold, J. Pelligrino, S. Fokkema, & R. Glaser (Eds.), Cognitive psychology and instruction. New York: Plenum. Anderson, R. C., Spiro, R. J., & Anderson, M. C. (1979). Schemata as scaffolding for the representation of information in connected discourse. American Educational Research Journal, 15(3), 433–440. Carlson, S. (1997). Algorithm of the gods. Scientific American, 276(3), 121–123. Carraher, Carraher, & Schliemann (1985). [Cited In J. A. M. Pucket & H. W. Reese (Eds.), (1993). Mechanisms of everyday cognition. Hillsdale, NJ: Lawrence Erlbaum.] Chapman, M. (1993). Everyday reasoning and the revision of belief. In J. A. M. Pucket & H. W. Reese (Eds.), Mechanisms of everyday cognition. Hillsdale, NJ: Lawrence Erlbaum. Cheng, P. W., & Holyoak, K. J. (1985). Pragmatic reasoning schemas. Cognitive Psychology, 17, 391–416. Cheng, P. W., Holyoak, K. J., Nisbett, R. E., & Oliver, L. M. (1986). Pragmatic vs syntactic approaches to training deductive reasoning. Cognitive Psychology, 18, 293–328.

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Cooper, G., & Sweller, J. (1987). The effects of schema acquisition and rule automation on mathematical problem-solving transfer. Journal of Educational Psychology, 79, 347–362. Cormier, S. M. (1987). The structural processes underlying transfer of training. In S. M. Cormier & J. D. Hagman (Eds.), Transfer of learning: Contemporary research and applications. London: Academic Press. Cosmides, L. (1989). The logic of social exchange: Has natural selection shaped how humans reason? Studies with the Wason selection task. Cognition, 31, 187–276. D’Andrade, R. (1980). [Cited in Rumelhart, D. E. (1980). Schemata: The building blocks of cognition. In R. J. Spiro, B. C. Bruce, & W. E. Brewer (Eds.), Theoretical issues in reading comprehension. Hillsdale, NJ: Lawrence Erlbaum.] D’Andrade, R. (1995). The development of cognitive anthropology. Cambridge: Cambridge University Press. Dempsey, J. V., Driscoll, M. P., & Swindell, L. K. (1993). Text-based feedback. In J. V. Dempsey & G. C. Sales (Eds.), Interactive instruction and feedback. Englewood Cliffs, NJ: Educational Technology Publications. Driscoll, M. P. (1994). Psychology of learning for instruction. Boston, MA: Allyn and Bacon. Evans, J. St. B. T. (1984). Heuristic and analytic processes in reasoning. British Journal of Psychology, 75, 451–468. Evans, J. St. B. T. (1989). Bias in human reasoning: Causes and consequences. East Sussex: Lawrence Earlbaum. Evans, J. St. B. T. (1996). Deciding before you think: Relevance and reasoning in the selection task. British Journal of Psychology, 87, 223. Gagne, R. M., Briggs, L. J., & Wager, W. W. (1992). Principles of instructional design. Orlando, FL: Harcourt Brace Jovanovich College Publishers. Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15, 1–38. Griggs, R. A. (1983). The role of problem content in the selection task and in the THOG problem. In J. St. B. T. Evans (Ed.), Thinking and reasoning: Psychological approaches. London: Routldge & Kegan Paul. Griggs, R. A., & Cox, J. R. (1982). The Elusive Thematic-Materials Effect in Wason’s Selection Task. British Journal of Psychology, 73, 407–420. Jackson, S. L., & Griggs, R. A. (1990). Education and the Selection Task. Bulletin of the Psychonomic Society, 26, 327–330. Johnson-Laird, P. N. (1983). Mental models. Cambridge, MA: Harvard University Press. Johnson-Laird, P. N., Legrenzi, P., & Lagrenzi, M. (1972). Reasoning and a sense of reality. British Journal of Psychology, 63, 392–400. Manktelow, K. I., & Evans, E. H. (1979). Facilitation of reasoning by realism: Effect or noneffect? British Journal of Psychology, 70, 477–488. Manktelow, K. I., & Over, E. D. (1992). Obligation, permission and mental models. In Y. Rogers, A. Rutherford, & P. A. Bibby (Eds.), Models in the mind: Theory, perspective and application. San Diego: Academic Press. Margolis, M. (1987). Patterns, thinking and cognition: A theory of judgment. Chicago, IL: The University of Chicago Press. Perkins, D. N., & Salomon, G. (1989). Are Cognitive Skills Context-Bound? Educational Researcher, 53, 16–25. Pucket, J. A. M., & Reese, H. W. (Eds.) (1993). Mechanisms of everyday cognition. Hillsdale, NJ: Lawrence Erlbaum. Rumelhart, D. E. (1980). Schemata: The building blocks of cognition. In R. J. Spiro, B. C. Bruce, & W. E. Brewer (Eds.), Theoretical issues in reading comprehension. Hillsdale, NJ: Lawrence Erlbaum.

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Rumelhart, D. E., & Norman, D. A. (1978). Accretion, tuning, and restructuring: Three modes of learning. In J. W. Cotton & R. L. Klatzkey (Eds.), Semantic factors in cognition. Hillsdale, NJ: Lawrence Erlbaum. Rumelhart, D. E., & Norman, D. A. (1981). Analogical processes in learning. In J. R. Anderson (Ed.), Cognitive skills and their acquisition. Hillsdale, NJ: Lawrence Erlbaum. Salomon, G., & Perkins, D. N. (1989). Rocky roads to transfer: Rethinking mechanisms of a neglected phenomenon. Educational Psychologist, 24, 113–142. Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of ‘‘welltaught’’ mathematics classes. Educational Psychologist, 23, 145–166. Singley, M. K., & Anderson, J. R. (1989). The transfer of cognitive skill. Cambridge, MA: Harvard University Press. Smith, M. U. (1991). A view from biology. In M. U. Smith (Ed.), Toward a unified theory of problem solving: Views from the content domains. Hillsdale, NJ: Lawrence Earlbaum. Sweller, J. (1989). Cognitive technology: Some procedures for facilitating learning and problem solving in mathematics and science. Journal of Educational Psychology, 81, 457–466. Ward, S. L., Byrnes, J. P., & Overton, W. F. (1990). Organization of knowledge and conditional reasoning. Journal of Educational Psychology, 82, 832–837. Wason, P. C. (1968). Reasoning about a rule. Quarterly Journal of Experimental Psychology, 20, 273–281. Wason, P. C. (1983). Realism and rationality in the selection task. In J. St. B. T. Evans (Ed.), Thinking and reasoning: Psychological approaches. London: Routldge & Kegan Paul. Wason, P. C., & Shapiro, D. (1971). Natural and contrived experience in a reasoning problem. Quarterly Journal of Experimental Psychology, 23, 63–71.

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