CONTEMPORARY EDUCATIONAL PSYCHOLOGY 7, 371-383 (1982)

Acquisition of Inductive Biconditional Reasoning Skills: Training of Simultaneous and Sequential Processing

SEONG-SOO LEE University of British Columbia

One hundred forty-four lOth-grade students received training on one of three processing methods: coding-mapping (simultaneous), coding only, or decision tree (sequential). Then they learned a biconditional rule under one of eight transfer test conditions based on a 2 (paradigm: rule vs complete learning) x 2 (memory aids: 0 vs 4) x 2 (focus instance: presence vs absence) design. Although the coding-map- ping students processed concept instances in much the same way as the coding- only students, they acquired the target rule more frequently, and they processed instances more quickly and more consistently than the decision-tree students. The observed ordinality of the responses of four truth-table classes was found to be more consistent with the simultaneity than with the sequentiality hypothesis. As expected, training interacted with paradigm and also with memory aids and focus instance. The induced simultaneous processing strategy apparently works opti- mally under rule learning, while the sequential strategy is difficult to induce and/or not optimal for rule-learning operations.

The present study investigated the transfer effects of strategy training on simultaneous vs sequential processing in the inductive learning of a biconditional rule. Two alternative strategy trainings were formulated and implemented according to two plausible models of logicoconceptual rule learning. The two models are (a) attribute coding and rule formation (hereafter referred to as coding-mapping) and (b) decision-tree structure (hereafter referred to as decision-tree) model, which can be characterized as being based on simultaneous and sequential processing, respectively.

Because bidimensional logical rule learning obviously involves the combination of two relevant dimensions of stimuli by a certain logical connective, an interesting question arises: Do learners treat (or process) both dimensional attributes simultaneously or one at a time, sequentially?

The study reported herein was funded by the NRC (Grant No. APA 7456) and also facilitated by the University of British Columbia grants-in-aid. The author extends his thanks to the IOth-grade students, teachers, and principal of Kitsilano High School, who participated in this project with sustained interest, and finally the Vancouver School Board Authority for their support. The author is also grateful to Florence L. Gerrard for her careful execution of the experiment, and S. F. Foster and N. S. Suzuki for their helpful comments on an early draft of this paper. Requests for reprints should be sent to Seong-Soo Lee, Department of Educational Psychology, Faculty of Education, University of British Colum- bia, Vancouver, B.C. V6T 125, Canada.

371 0361-476x182/040371-13$02.00/O Copyright @ 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.

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One way to find the answer to this question would be to train learners to process dimensional attributes according to either one of the models and then see if they could easily learn the transfer conceptual rule. If they do, the trained model or processing can be taken as being compatible with the learners’ mode of cognitive operations, and thus as being a plausible underlying process of logical rule learning (cf. Glaser & Resnick, 1972; Pellegrino & Glaser, Note 1).

In general, the processing of dimensional stimuli takes place at percep- tual (peripheral) as well as conceptual (central) levels (Neisser, 1966; Lockhead, 1972; Garner, 1970, 1974, 1976; Paivio, 1971). Such processing can be simultaneous just as synchronous, parallel operations may be per- formed on more than one dimensional information. Or processing can be sequential just as temporal, serial operations may be performed on one dimension after the other, the processing of the latter dimension depend- ing on that of the former (cf. Paivio, 1971, p. 34).’ It should be noted that typical concept-learning trials with instances successively presented in- volves the serial integration of trial outcomes into the current trial of hypothesis testing (Newell & Simon, 1972, p. 796). The sequentially de- pendent processing of interest here needs to be distinguished from such serial integration processes.

Both modes of processing at the conceptual level can be effectively promoted with perceptually separable stimulus dimensions. Obviously, adequate parallel or serial processing presupposes two or more dimen- sions of stimuli which are first definable and then separable, rather than being integral or unitary (Garner, 1970, 1974; Lockhead, 1972). As pointed out by Garner, the issue of parallel vs serial processing can be meaningless unless the conditions mentioned above are met. In the pres- ent study, however, the multidimensional geometric designs of unitary representation were employed, since logical concept learners deal with unitary stimulus dimensions more often than with separable ones. Thus, the present study was intended to show that perceptually less separable dimensions could be transformed into conceptually separable dimensions by processing training. In view of the conceptual rules, usually stated in terms of dimensions, relevant dimensions should be separable not only from the irrelevant ones, but also from one another within a set of relevant ones (Garner, 1976). In the following, the two models of rule learning presumably underlying the induction of the biconditional rule are formu- lated for experimental validation. The paradigm of rule learning viewed as one of two aspects of concept learning should prove to be useful as a basis

1 The concepts of simultaneous and sequential processing should not be confused with the similar labels exclusively discussed in terms of certain psychometric tests by Das, Kirby, and Jarman (1979).

PROCESSING STRATEGIES IN CONCEPT LEARNING 373

Sequential DECISION -TREE STRUCTURE model Operations

Prefroininp TCXLS Tranrfer Tel,

FIG. 1. Five decision-tree structures and coding-mapping schemes for pretraining and transfer tasks.

of a viable instructional strategy.* Rule learning here refers to the task of inducing a rule that lawfully combines a pair of dimensions designated as relevant; attribute identification refers to that of identifying relevant attri- butes, given a rule instructed to learners.

The Decision-Tree Model

Given two dimensions designated as relevant, learners are assumed to raise a series of two unidimensional questions and answers, one after the other (sequentially), as illustrated in the left margin of Fig. 1. Having worked with each rule instance, learners must arrive at an endpoint to find its response category, which is to be confirmed by the experimenter’s feedback. Eventually, learners should acquire a sequential pattern of test outcomes. For a conditional decision tree, it consists of (YES-YES = +), (YES-NO = -), (NO-0 = +) with a self-terminating endpoint deci- sion; for a biconditional one, it consists of (YES-YES = +), (YES-NO = -), (NO-YES = -), and (NO-NO = +) with exhaustive endpoint decisions. These decision trees result from processing trials in a top-down, left-right sequential order.

It can be readily seen that the conceptual difficulty of adapting the model is inherent in both concept learning (Hunt, Martin, & Stone, 1966)

2 Rosch’s (1973) approach to the definition of natural concepts may well be used for assessing students’ level of understanding of the nonlogical concepts not definable in or- thogonal attributes, but it has little to do with logical concept instruction problems.

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and attribute-identification paradigms (Trabasso, Rollins, & Shaughnessy, 1971). Some equivocal results of Trabasso et al. with respect to process- ing mode, sequential or simultaneous, may well be due to such difficulty as involved in instructing learners on a complex rule.

Decision-tree construction training was given on each of the four sim- pler rules. These trees are diagrammed in the first half of Fig. 1 (details are illustrated under Procedure). Given a decision tree constructed as was expected, each instance of a biconditional transfer rule task was hypothesized to be processed sequentially. If processing is sequential, instances corresponding to the left-to-right endpoints should be subjected to an increasing error tendency and a longer latency for correct process- ing. This prediction is based on the assumed primacy of the negation ofp over that of q in real time (cf. similar reasoning by Trabasso et al., 1971). Thus, instances with the early logical negation ofp (not red: yellow and blue) need to be kept longer through the nodes of the decision tree than instances with the later negation of q (not circle: square and triangle). In other words, the instance ofp requires temporary holding of the negated attributes until its endpoint is reached. Thus,pq instances take more time and more information to hold in processing than pq instances, but less than p4 instances. Accordingly, the processing error and latency of the instances of four exhaustive classes at the endpoints were predicted to be in an ascending order, pq < pq < pq < @j.

The Coding -Mapping Model

The model is based on the two-component-process notion of rule learning (Lee, Note 2). Rule learning is conceived to be the integration of two components: attribute coding and rule formation. The former refers to a nonspecific process of coding many rule instances into a set of four truth-table classes (a few classes for a nonlogical rule). The latter refers to a rule-specific process of mapping the four classes (or multiplicatively combined coded attributes) onto two category responses (cf. four-to-three mapping used by Siegler & Atlas, 1976).

Learners may well start with unidimensional coding (i.e., affirma- tion-p or q-and negation-p or 4). After a few nonreinforced trials, however, they are assumed to do the simultaneous, multiplicative coding of two dimensions (e.g., pq, p?j, pq, and p4, referred to as “base associations” by Inhelder & Piaget, 1958), as is shown in the bottom of Fig. 1. At the same time, the learners must map the four classes onto the positive and negative categories. This mapping actually involves a three- dimensional mental construction of an interdimensional relationship be- tween the two relevant dimensions, dissociated from one another, and a response dimension. Thus, a mentally constructed scheme for a bicondi- tional rule requires a mapping pattern of (pq -+ +), (~4 + -), (pq + -)

PROCESSING STRATEGIES IN CONCEPT LEARNING 375

and (PS + +), givenp standing for “one” and q for “large.” Four other schemes are shown in Fig. 1.

Coding-mapping training consisted of (a) attribute-coding practice via sorting rule instances into a 2 x 2 spatial matrix and (b) mapping practices for each of the four simpler rules, as shown in each cell of the 2 x 2 tables in Fig. 1 (details are illustrated under Procedure). Given a three- dimensional mental matrix constructed as was expected, learners were hypothesized to carry out two interdependent processes through the si- multaneous processing of two relevant dimensions on each instance. If processing is simultaneous, rule-learning proficiency or difficulty should be a joint function of two separate indices of coding and mapping diflicul- ties.

The coding difficulty can be determined by the direct product of two difficulty vectors, one indexing combinatory operations (i.e., given a tri- nary dimensional task, negation: 0, 2, 2, 4 and combination: 1, 1, 1, 1 for pq, ~4, pq, and p4, respectively, their direct product is 0, 2, 2, 4). The mapping difficulty can be derived from the inference model proposed by Salatas and Bourne (1974), which is in essence a model of conjunction- biased operations. Given the four truth-table classes, it was specified to include four operations-(a) all pq + +, (b) allpq -+ -, (c) allpq andpq -+ FG’s category, and (d) pq andpq -+ different categories. (The findings of Dominowski & Wetherick, 1976, appear to be relevant to the validity of these assumptions, but are not based on a reliable test.) Many encounters violating (a) and (b) lead to assigningpq to a negative andpq to a positive category, respectively. The operations (a), (b), and (c) were assumed to be equally difficult, and so weighted with 1; but if (d) was violated, a factor of 2 was applied to each of the three. With the assumption of the difficulty metric, the violation of operations (b), (c), and (d), which are necessary for a biconditional rule, determines the vector of the mapping difficulty: 0, 2, 2, 2, for pq, ~4, pq, pq, respectively.

Accordingly, overall rule-learning difficulty was expected to result in a vector of 0, 4, 4, 8 based on a direct product of the coding and mapping difficulties. According to the above vector, the pattern of latencies and error tendencies should be in an order of pq < pq = pq < pg. Further, learners given the coding-mapping training should process instances faster with more consistency than those given the decision-tree training because simultaneous processing would take less real time than sequential processing.

Design and Subjects

METHOD

A factorial design of 24 conditions was used, which resulted from combining three training and eight transfer test conditions. The training included coding-mapping, coding-only, and

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decision-tree conditions.3 Each learner was trained on one of the three and then given a biconditional task under one of the eight test conditions. These eight were the combinations of three variables: (a) paradigm (rule vs complete learning), (b) memory aids (0 vs 4 feedback instances), and (c) a focus instance (presence vs absence).

The three variables were included to assess the generalizability of the transfer of process- ing training with respect to those variables known to be relevant. Either simultaneous or sequential processing may not work under a complete-learning condition as effectively as under rule learning. Given the decision-tree training, memory aids should be conducive to sequential processing because they could serve as template instances, especially when a focus card as a nontransitory structural anchor is available. Given the coding-mapping training, however, the focus as an infallible pivotal class for coding and mapping should be conducive to simultaneous processing, but its pivotal role would be attenuated in the pres- ence of memory aids.

One hundred forty-four lOth-grade students (72 boys and 72 girls ranging from 15.2 to 16.1 years of age with a mean age of 15.6 years) were randomly assigned to 24 conditions, six to each condition.

Learning Materials

Typical geometric design cards were used for training and transfer tasks. They varied on six trinary dimensions: color (red, yellow, blue), shape (triangle, square, circle), texture (hatched, crosshatched, solid), outline (thin, medium, thick), number (1, 2, 3), and size (small, medium, large). Each card measuring 6.35 x 8.89 cm bore a combination of six dimensional variations. A training task was based on two relevant (number and size) and two irrelevant dimensions (outline and texture) with color and shape kept constant. A deck of 36 cards contained nine (all possible combinations of two irrelevant dimensions) replications of four truth-table class instances, which resulted from the two trinary-relevant dimensions.

A warm-up task contained 18 instances, which varied on one relevant (color) and two irrelevant (outline and texture) dimensions. The two category response labels used were “A” and “B.” A biconditional transfer task was constructed by using two relevant (color and shape) and two irrelevant (outline and texture) dimensions with number and size kept constant. The task contained a deck of 36 cards which represented equal numbers of four truth-table class instances (9 x 4) and equal numbers of positive and negative instances (2 x 18). Three duplicates of each deck were made, each with one of three values (1, 2, 3) kept constant with large figures only. No two successive cards in sequencing were allowed to be in the same truth-table class, so as to avoid any obvious response patterns.

Apparatus Stimulus cards were presented one at a time through windows fixed at learners’ eyesight

level by two concept-learning devices. One was designed for straightforward successive presentations of cards, and the other for successive presentations with the feature of holding four memory aids of feedback instances for the learners’ view. Two stop watches were used, one for taking response latency measures and the other for timing procedural operations.

3 Because of the research logistic constraints imposed, a control condition with no specific training (i.e., coding, mapping, or any other truth-table strategy) was not included in the present study. The findings by Lee (Note 2) seemed to make it redundant to show that the coding-only training is superior to the no-training control condition again in the present study, which has been consistently demonstrated in the literature.

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Procedure

As each student reported to a small experimental room in the school, s/he received standard instructions for category concept learning, along with three typical cards and a warm-up task. Each learner was asked to pace the presentation of each card, to study its attributes, and then to determine its response category. S/he was further told about the importance of feedback, the speed as well as accuracy of responses, the maximum limit of 15 set for each response, and the mastery criterion of 16 consecutive correct responses. Upon completion of the warm-up task, each learner engaged in one of the following three training treatments.

Coding -mapping. Each learner first practiced sorting a deck of 36 cards into 2 x 2 matrix bins by “number and size.” S/he was given the experimenter’s corrective feedback for each and continued until s/he got 36 consecutive correct placements. Immediately after the completion of practice, the mapping practice began. It consisted of placing the four sorted classes of cards into either the positive or negative category. Before such placement, s/he detected and counted the dimensional differences between each sorted card and a pq pivotal class instance in terms of the number of attributes. At the same time, s/he was told to detect and verbalize an interdimensional relationship between two response categories and a pattern of the dimensional differences counted, using a set of four truth-table class instances.

A conjunctive rule pattern consisted of the mapping: 0 difference goes to “+” and 1, 2 differences to “-.” Three other patterns used were joint denial (0,) -+ “+“; 2 + “-“) and conditional (0,l on number, 2 -+ “ + “; 1 on size + “-“) rules. This type of mapping was evidently used by college students who were reported to be very efficient in rule induction (Shepard, Hovland, & Jenkins, 1961). The counting was intended to ensure the separability of the two relevant dimensions. Garner (1976) has argued that separability is a necessary condition for either simultaneous or sequential processing to occur.

Coding only. Each learner in this training engaged in exactly the same coding phase as involved in the coding-mapping training and then in a filler activity of familiarization. The activity amounted to labeling each bin with one of four letters, a, b, c, and d in four different random patterns. It lasted for the amount of time yoked by the time spent on the mapping training phase by the coding-mapping subject from a block of three subjects.

recision tree. At least four sets of four truth-table class instances were used to induce four decision-tree structures, as diagrammed in Fig. 1. The first two sets were used to indicate to the learner the preliminary task requirements. Each learner sorted the first set into either the positive or negative category, and then the second set with “number” and “size” designated as relevant according to a decision-tree structure and the experimenter’s feedback. The last two sets were used for the guided establishment of a sequential routine for mental operations. Given the third set, the experimenter presented each card and told the learner its response category. S/he was then asked to raise two unidimensional questions sequentially, while trying to figure out the relationship between each answer and its as- sociated response category. Given the fourth set, s/he was asked to follow the same sequen- tial question-answer series with no prompting but feedback from the experimenter.

The fourth set was repeated either until the learner understood the routine of sequential operations or for one-quarter of time spent by the coding-mapping counterpart in a block of three subjects. The majority of the decision-tree subjects completed before the time was up. The remaining three-quarters of the time was used for three other decision-tree trainings, one for each rule. Finally, the learner’s verbalization of each decision tree was coached into each self-terminating or exhaustive decision-tree form, as shown in Fig. 1.

Transfer. Upon the completion of the training, each learner received the biconditional task as rule learning with “color” and “shape” designated as relevant or as complete learning with the mere indication of two out of three (color, texture, shape) being relevant.

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TABLE 1 MEAN VECTORS OF ERROR RATE AND INTRASLJBJECT MEAN AND STANDARD DEVIATION

LATENCY/SECOND FOR CORRECT PROCESSING RESPONSES BY FOUR CLASSES UNDER

SIX TRAINING-PARADIGM CONDITIONS

Training condition

Four trnth- table

classes

Rule learning Complete learning

Errorb Mean’ SD Error Mean SD rate latency latency rate latency latency

Coding-mapping RC(+Y’ ,185 5.74 2.23 .222 7.15 2.94 EC(-) ,306 6.71 2.25 ,453 8.21 2.86 EC(-) ,295 6.55 2.21 ,433 7.65 2.64 RC(+) .461 7.44 2.62 ,479 8.04 2.87

Coding only RC(+) ,185 6.95 2.86 ,241 7.87 2.72 act-1 ,375 7.55 2.82 ,446 8.70 2.71 RC(-1 ,318 7.55 2.73 ,405 8.87 2.83 RC(+) ,414 7.95 2.89 ,532 9.37 2.74

Decision tree KC+) ,177 6.31 2.97 ,311 7.86 3.20 EC(-) ,313 7.51 3.30 .458 8.90 3.10 &CC-) ,340 7.19 3.21 ,388 8.95 3.22 RC(+) ,477 1.19 3.21 ,456 9.56 3.30

Note-N = 144. ’ R, R, C, and C stand for “red,” “not red,” “circle,” and “not circle,” respectively. b The error rate was obtained by dividing the number of errors by a total of 36 instances within each

troth-table class. ’ Mean and SD latency scores obtained only on correct responses were determined for each subject

within each class.

Also, on each and every trial, s/he was provided with either no focus or a focus card of apq instance (red circle), and either four feedback instances as running memory aids or none. A pair of response labels were “positive” and “negative.” S/he was instructed to apply to the transfer task whatever concept or strategy s/he learned from the training. Learners per- formed either to the mastery criterion of 18 consecutive correct responses or to the maximum limit of 60 min.

RESULTS AND DISCUSSION

Mean error rates and mean latencies along with SD latency scores by six treatments are shown in Table 1. The intrasubject means and SD’s were determined for correct response latency within each truth-table class. The trimmed mean suggested by Wainer (1977) was not used since the extent of trimming is arbitrary. And the SD latency was taken to have a logical basis, as an index of the consistency of processing strategy, whether simultaneous or sequential. Four repeated measures by four - - truth-table classes (i.e., RC, RC, RC, and RC) were transformed to yield four between-class effect scores: Overall, RC-RC, RC-RC, and -- RC-RC, by using four contrast vectors, (1, 1, 1, l), (1, - 1, 0, O), (0, 1, - 1, 0), and (0, 0, 1, - l), respectively. The last three between-class scores were used for testing simultaneity and sequentiality hypothesis. A mul- tivariate analysis of variance by l-d! linear contrasts was performed on the 3 x 2 x 2 x 2 x 4 (repeated) design data.

PROCESSING STRATEGIES IN CONCEPT LEARNING 379

Overall Effectiveness of Coding -Mapping and Decision-Tree Training

The students given coding-mapping training solved the target problem more often than those given coding only (.438 vs .333), z = 1.97, one- tailed p < .024. Both training groups are very similar to each other in terms of the process-related measures (error rate, mean, and SD laten- ties), multivariate F,(3,118) < 1.0, in terms of the three overall scores. Coding-mapping students arrived at the solution of the problem more often than decision-tree students (.438 vs .375), z = 1.70, one-tailed p < .045, and in addition, they processed concept instances faster and more consistently than the others, F,(3,118) = 4.21, p < .007. The latter process-related effects were especially noticeable in the consistency mea- sure, F(1,120) = 12.30, p < .0007.

Simultaneous vs Sequential Processing Hypothesis

The simultaneity and sequentiality hypotheses predicted error rates and mean latencies in the two different orders of the four truth-table classes, -- RC < RC = RC < RC and RC < RC < RC < RC, respectively. In other words, an interaction was expected to appear between coding-mapping vs decision tree and RC vs RC contrasts, the latter contrast being indexed by RC-RC effect score. A strong and weak test can be performed, the former by testing a three-way interaction including a rule- vs complete- learning contrast, and the latter by a two-way interaction without the last contrast included. No such two-way interaction effect was obtained, F,(6,115) = .83, p < 55, even in terms of univariate F’s. But the three- way interaction was significant, F,(6,115) = 3.01, p < .009, the effects of which were saturated only in the RC-RC between-class effects of error rate and latency, precisely as expected, F’s( 1,120) = 3.34 and 3.11, p < .07 and .08, for error rate and mean latency, respectively; step-down F’s(1,118 and 117) = 7.97 and 3.52,~ < .006 and .06 after removing the RC -RC between-class effects in error rate and mean latency. The signifi- cant three-way interaction stems from the negligible RC-RC between- class effect scores of coding-mapping students’ error rate and mean la- tency under rule learning, as predicted, and a measurable reverse of the effect score of mean latency under complete learning. In contrast, decision-tree students’ effect scores of mean latency under rule learning and that of error rate under complete learning are observed in a direction contrary to the prediction, as can be seen in Table 1.

Neither the significant two-way interaction nor the significant three- way interaction effects were observed in terms of the RC-RC and -- RC-RC between-class effects scores of error rate and mean latency, as predicted. Accordingly, a more parsimonious test was performed by pooling across the three training conditions and two paradigms. The test

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TABLE 2 OVERALL MEANS OF ERROR RATES, MEAN LATENCY, AND SD LATENCY FOR

12 TRAINING-TRANSFER TEST TREATMENTS

Four memory aids No memory aid

Error Mean SD Error Mean SD Training Focus rate latency latency rate latency latency

Coding- Presence .386 8.39 2.93 ,283 6.07 2.15 mapping Absence ,331 8.60 2.87 .409 5.69 2.37

Coding only Presence ,373 7.92 2.72 ,323 6.79 2.74 Absence ,349 9.72 2.96 ,416 7.99 2.73

Decision tree Presence .327 9.10 3.42 ,350 6.53 3.11 Absence ,385 8.71 3.60 .398 7.71 2.63

Note. N = 144.

of the RC-RC effect scores of error rate and mean latency was highly significant, F’s( 1,120) = 125.32 and 77.90, p’s < .OOOl, respectively; and -- so were the RC-RC effect scores, F’s(1,120) = 37.30 and 26.37, p’s < .OOOl, respectively, for error rate and mean latency. The results of these tests are entirely consistent with the predicted order of the four classes by the simultaneity hypothesis given the rule-learning task, but only partially given the complete-learning task. The results obtained from decision-tree students’ performance are mixed at best and inconsistent with the se- quentiality hypothesis under both the rule-learning and complete-learning tasks.

Training Effects as Affected by Four Memory Aids and a Focus Card

Mean error rates and mean and SD latencies by 12 treatments are shown in Table 2. Since the roles of the memory aids and the focus card were not specified with respect to the between-class effects of the four truth-table classes, only the overall scores of the three response measures are used for testing a three-way interaction. As expected, no significant interaction effect was observed between the contrast of coding-mapping vs coding only and the memory aids and focus factors. However, the contrast of coding- mapping vs decision-tree training appeared to interact with the two factors significantly, F,(3,118) = 3.64,~ < .015; F’s(1,20) = 3.90, 1.04, and 1.83, p’s < .05, .31, and .18, for the overall error rate, mean latency, and SD latency, respectively. The pattern of mean error rates, as can be seen in Table 2, indicates that coding-mapping students made fewer errors when given either the memory aids or focus card than when given neither or both, while decision-tree students made the least number of errors when given both. In general, this interaction occurred in

PROCESSING STRATEGIES IN CONCEPT LEARNING 381

a trade-off manner between accuracy and response latency, but coding-mapping students processed more consistently.

Effects of Paradigm, Memory Aids, and Focus Card

The predicted effects of transfer test conditions would indicate the extent to which the present findings are internally valid. The rule-learning task was found to be easier than the complete-learning task, F,(3,118) = 7.00, p < .0003, mostly in terms of the overall scores of error rate and mean latency. The students given the memory aids made fewer errors processing all class instances, but took longer with less consistency than those given no aid, F,(3,118) = 7.99, p < .OOOl. Also, they made fewer errors processing RC instances relative to RC instances than those given no aid, F(1,120) = 7.30,~ < .008. The students given the focus card made fewer errors than those given no focus card, F( 1,120) = 624, p < .04, especially fewer errors processing RC instances relative to RC instances, F(1,120) = 4.48, p c-.03. Also, as expected, they took less time process- ing RC relative to RC instances than those given no focus card, F( 1,120) = 6.79, p < .Ol. In general, when no memory aid was provided, the students given the focus card made fewer errors than those given no focus card, but when memory aids were available, the converse was observed, F(1,120) = 5.86, p < .017. All of these findings are consistent with rea- sonable predications.

The analysis of the between-class response structure and pass/fail data clearly indicates that simultaneous processing can be carried out rela- tively faster and with fewer errors, especially under rule learning. This learned processing strategy led learners to more frequent induction of a high-order biconditional rule than that strategy by decision-tree or coding-only training. The rule-mapping component seems necessary for the successful generation of the rule.

Error rate and mean and SD latency data inconsistent with the sequen- tial hypothesis may be the case when either the decision-tree training was not successful or the sequential processing strategy, even if induced, was not adopted by the learners. Considering that the training was reformu- lated and well implemented as prescribed by Hunt, Martin, and Stone (1966), the learners more likely failed to adopt the sequential strategy. One can only speculate that abstract sequential processing, relative to simultaneous processing employing a concrete 2 x 2 matrix image, could not be readily adopted for processing a new set of dimensional instances under either the rule- or complete-learning paradigm.

In any case, the major findings seem to be internally consistent with the other aspects of the present data. First, the role of a focus instance as a pivot in simultaneous processing was found to be significant. It was also found that role can be attenuated given extra memory aids which could

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functionally substitute for the pivotal role of the focus instance. Although success in the induction of sequential processing is not clear, the focus and memory aids, perhaps serving as templates when made available, appear to be beneficial to the development of a biconditional decision tree. Second, the three main effects of transfer test conditions were ob- tained as expected. Rule learning caused faster processing with fewer errors, resulting in more frequent success in the generation of the target rule than compiete learning. Given the memory aids, a typical tradeoff occurred between processing accuracy and time, since they too required time for processing. When an RC instance was given as a focus instance - 2 RC class instances were most accurately processed, and errors in the RC class were significantly reduced.

As noted earlier, the decision-tree processing mode was claimed to underlie concept learning (Hunt et al., 1966). Trabasso et al.‘s (1971) chronometric data of the four truth-table classes instances suggested that attribute-identification processes may be of sequential as well as parallel nature (cf. Lockhead, 1972). In view of the difficulty of verbally instruct- ing complex rules, the paradigm previously used might not be a good choice. The present study employed a more well-defined paradigm, rule learning. Bidimensional, logical rule learning is clearly shown to involve simultaneous processing. The involvement of sequential processing in rule or concept learning is not clear, however.

One obvious practical implication for instruction is that teachers of logic may usefully set logical rule-learning tasks for their students in such a way as to facilitate simultaneous processing of dimensional information, before engaging them on the deductive application of logical rules. In par- ticular, the use of such exercises as the 2 x 2 matrix coding combined with dissociative rule mapping used in the present study should help students having difficulty inducing the target rules. Such instructional procedures would be effective, especially when complex (or real life), concept instances, such as medical symptoms used in a study by Sowder (1973, are not as clearly dimensioned as in geometric designs.

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