Adaptive Instructional Strategies for Teaching Rules in Mathematics

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  • Adaptive Instructional Strategies for Teaching Rules in Mathematics Steven M, Ross Ernest A, Rakow

    Steven M. Ross and Ernest A. Rakow are associate professors in the Department of Foundations of Educa- tion at Memphis State University, Memphis, TN 38152. The authors acknowledge the contribution of Duncan N. Hansen, who directed the research project involving performance of the present studies and par- ticipated in the writing of earlier drafts. Requests for reprints should be sent to Steven M. Ross, Foun- dations of Education, Memphis State University.

    These researchers explored ways of adapting instruction to individual students. In one experiment, as an incentive for focusing on more difficult learning tasks, they assigned more points to tasks predicted to be difficult for the student. In a second experiment, they varied both incentives and number of examples offered to demonstrate the tasks, again on the basis of pretest scores. Included is a discussion of how the f indings-that both techniques are benef ic ia l -can be applied in classroom instruction.

    ECTJ, VOL. 30, NO. 2, PAGES 67-74 ISSN 0148-5806

    Research on individualized instruction, as exemplified by studies of aptitude- treatment interactions (ATI), has tended to focus on group variables in attempting to adapt instruction to individuals (for re- views, see Berliner & Cahen, 1973; Cron- bach & Snow, 1977). Since there usually is considerable variation within groups, the potential of such orientations for optimizing learning for individuals appears limited. In addition, the majority of previous studies have based adaptations primarily on trait (pretask) variables, representing general- ized aptitudes or predispositions for learn- ing, to the exclusion of state variables, rep- resenting students' needs and interests at the time of instruction (Tennyson & Rothen, 1977; Tobias, 1976). Consequently, these applications have achieved relatively limited empirical success and offer the prac- titioner few suggestions for individualizing instruction in applied contexts.

    The objective of the research described in this article was to test an adaptation ap- proach designed to extend the ATI concept by applying instructional variations to indi- viduals as opposed to groups. The context for the experimentation was an introduc- tory mathematics lesson adapted from the beginning module in an undergraduate statistics course using the Keller (1968) PSI Plan (see Kulik, Kulik, & Cohen, 1979; Rob- in, 1976). Compared with lecture-based instruction, PSI (personalized system of in- struction) represents a condition of high in-

  • 68 ECTJ SUMMER 1982

    structional support by emphasizing the availability of supplementary study mate- rial and teacher assistance for those who need them. The problem, however, is that individuals may lack the sophistication or motivation to use these resources appropri- ately on their own, given that learning ob- jectives and study materials are stan- dardized for an entire class, the only adap- tation has to do with the pace at which stu- dents complete the lesson. With regard to the "sufficiency" of learning, high achiev- ers may tend to receive more support than they need, and low achievers too little.

    On the basis of this rationale, we under- took a preliminary study to explore ways of adapting the quality of instructional support given individuals (Hansen, Ross, & Rakow, 1977). Subjects learned a series of 10 math rules under full (individual) adaptation, partial (group-based) adaptation, and sev- eral forms of standard instruction. For the full adaptive strategy, scores on a battery of entry measures provided the basis for as- signing subjects to different learning style groups via cluster analysis and then deriv- ing predictions of individual performance, within groups, via multiple regression techniques. Prescriptions specifying the number of supporting examples to be pre- sented for each rule were matched to the predicted scores and refined during instruc- tion on the basis of on-task performance. For the partial adaptation strategy, pre- scriptions were matched to group predic- tions only. Results on a cumulative posttest favored full adaptation over partial adapta- tion and both adaptive treatments over standard instruction.

    In the preliminary study, the full adaptive approach provided individualized prescrip- tions of instructional support; it increased practice opportunities on rules predicted to be most difficult to learn and economized on rules predicted to be easiest. A possible limitation of this strategy is that it provides no control over how students actually use the prescriptions; they may linger on the "restricted" prescriptions associated with easy rules and give only surface attention to "extended" prescriptions for difficult rules. Thus it would seem advantageous to invoke some type of external prompting, such as incentives, to direct students to use the ma-

    terial in the manner intended. In theorizing about the motivational influences of class- room incentives, Atkinson and Wickens (1971; also see Kribs, 1974) emphasized the advantages of varying the level of rewards within-task. Such a weighting of different parts of a lesson unevenly would provide information about which parts are more important than others. From an adaptive standpoint, a logical strategy would be to vary conditions so as to offer the highest rewards on materials the individual student is predicted to find the most difficult and the lowest rewards on those he or she is pre- dicted to find the easiest. The expected out- come would be to modify study activity so as to direct the learning emphasis to weak- nesses rather than to strengths. This idea was examined in the present experiments by determining whether adaptive varia- tions in incentives, used separately from and in combination with adaptation of in- structional support, had a more positive ef- fect on performance than weighting all les- son objectives the same. The critical feature of the adaptive treatments examined in- volved their formulation of prescriptions systematically tailored to the inferred needs of individuals.

    EXPERIMENT I: ADAPTATION OF INCENTIVES

    The purpose of the first experiment was to determine the effectiveness for learning of within-task, individualized adaptation of incentives. The main hypothesis was that, as a function of modifying study behavior in a manner consistent with individual needs, adaptive incentives would enhance overall performance relative to standard incen- tives. It was also predicted that a highly differentiated adaptive incentive schedule would provide subjects with more informa- tion and direction, and thus be more facilita- tive, than would one having less differentia- tion between values. Comparisons were made between the following five treatment variations: (a) an adaptive strategy that as- signed five different levels of incentives across rules (distributive adaptive), (b) an adaptive strategy that assigned three differ- ent levels of incentives across rules (modal adaptive), (c)a nonadaptive strategy that

  • ADAPTIVE INSTRUCTIONAL STRATEGIES 69

    assigned the five-level incentive distri- butions derived for subjects in treatment a to subjects possessing different learning characteristics (distributive nonadaptive), (d) a nonadaptive strategy misappropria- ting the three-level incentives distribution (modal nonadaptive), and (e)a control strategy that assigned a standard incentive level across all rules (standard).

    Method

    The procedure used to vary incentives in the adaptive treatments was a modification of the one developed by Hansen, Ross, and Rakow and applied in the experimentation reviewed earlier (see Hansen et al., 1977, for a complete description of the methods and statistical analyses). To provide necessary background for understanding the present treatment manipulations, some highlights of that preliminary work are reviewed be- low.

    Adaptive modeling procedure. The adaptive model developed by Hansen et al. was structured to use multiple predictors for two purposes: (a) to classify students into independent-learning-style groups, thus allowing for group-based adaptations, and (b) to generate estimates of individual per- formances, thus allowing for instructional adaptation within groups. Predictors were selected on the basis of the degree of their relation to criterion performance on the task in question (i.e., 10 instructional rules covering different mathematical opera- tions). Based on a validation sample of 315 subjects, the following predictor set was found to yield the highest multiple correla- tion (total posttest r = .776): locus of con- trol, trait anxiety, math reading com- prehension, and a pretest on the material to be learned.

    The selected predictor set was adminis- tered to students and the resultant scores were subjected to cluster analysis following the procedure of Ward (1963) and Ward and Hook (1963). In analyses conducted on three independent norming samples, simi- lar results, indicating four separate group- ings, were obtained. One group was high in aptitude and ability, and another was low in aptitude and ability, but both were close to the overall means for locus of control and

    trait anxiety. The two other groups were about average in aptitude and ability, but one was external in locus of control and high in anxiety, while the other showed the opposite personality characteristics.

    Discriminant analysis was used to derive discriminant functions for classifying new subjects into these four groups. In the vali- dation run 91.6% of the norming sample were correctly classified. The final step was generating different prediction equations within each group. This was done by re- gressing posttest performance on the four predictor measures. These regressions were done separately for each rule within each of the groups; thus we produced a unique set of multiple regression equations for each group into which the pretask scores of new students could be entered to predict their performances on each rule.

    Participants and design. Participants in the study were 120 undergraduate students. They were administered the test battery, assigned to cluster groups, and randomly assigned to five treatments involving differ- ent distributions of incentives. In the distributive-adaptive (five levels) treat- ment, the 10 rules were rank-ordered on the basis of the individual subject's predicted performance and assigned incentive values such that 0 points each were offered on the two rules having the highest predicted scores, 5 points were offered on the two rules having the next highest predictions, and 10, 15, and 20 points each were offered, respectively, for the remaining three pairs. In the modal-adaptive (three levels) treat- ment, the same strategy was employed, ex- cept that the incentive values applied were 15 points each for the three lowest rules, 10 points each for the four next lowest, and 5 points each for the three highest. In the distr ibutive-nonadaptive and modal- nonadaptive treatments, subjects were paired across cluster groups with counter- parts from the corresponding adaptive con- dition (distributive or modal) and adminis- tered the identical incentive distributions prepared for those counterparts. In the con- trol treatment, a standard incentive of 10 points was assigned to each of the 10 rules. The design implied a 4 (cluster group) x 5 (treatment) factorial analysis of variance.

  • 70 ECTJ SUMMER 1982

    TABLE 1 Nonadjusted Treatment Means and Standard Deviations

    Learning Pretest Posttest Gain Time Test Time

    Treatment X S,D. Y( S.D. X S.D. ~ S.D. Y( S.D.

    Standard 20.95 6.19 23.95 6.48 3.00 2.98 76.80 23.13 27.00 5.59 Modal

    Adaptive 20.10 7.99 26.05 8.08 5.95 3.77 98.75 39.00 27.25 10.44 Distributive

    Adaptive 22.85 6.83 31.45 4.62 8.60 3.44 79.55 17.77 25.50 8.19 Modal

    Nonadaptive 19.00 9.87 22.25 9.47 3.25 8.76 102.10 26.90 24.95 6.25 Distributive

    Nonadaptive 17.40 9.63 21.85 9.35 4.45 4.46 91.55 23.42 30.40 10.46

    Materials. The instructional materials were the same as those used in the Hansen et al. (1977) experiments. The content consisted of mathematical rules generally taught in introductory algebra and statistics (e.g., fractions, summation, and factorials). Each unit began with a brief rule definition and was followed by six supporting examples. The last section of each unit consisted of an immediate four-item unit posttest.

    Lesson achievement was assessed via a 40-item cumulative posttest containing 4 items per rule. The posttest was designed as a parallel form of the pretest used for entry classification.

    Procedure. The incentive distributions ad- ministered to subjects in the adaptive treatments were derived using the multiple regression model described earlier. Stu- dents' predicted rule scores were obtained, rules were rank-ordered on the basis of those scores, and incentive values totaling 100 points were determined using the ap- propriate (distributive or modal) schedule.

    For pairing subjects in nonadaptive con- ditions with adaptive treatment counter- parts, two basic criteria were used: (a) the two subjects had to be members of different cluster groups, and (b) their total predicted scores had to differ by an average of at least .125 SD. During instruction, nonadaptive subjects worked under the same incentive distribution developed for their adaptive treatment counterparts. The result, follow- ing all pairings, was two independent nonadaptive conditions, one incorporating the modal set of incentives and the other incorporating the distributive set.

    Experimental sessions were attended by small groups. Subjects were told that each rule would be assigned a certain number of points and that the amount awarded would be contingent on performance on the rule (immediate) posttest. The basic learning re- quirements consisted of the following se- quence of activities for each rule: (a) study of the introductory rule definition, (b) study of complete examples and prob- lem solving of incomplete examples, and (c) completion of the rule posttest. After a rule posttest was completed, the proctor collected the answer sheet, scored it, and returned it to the subject with an indication of the number of points awarded. At the completion of all rules, subjects took the 40-item cumulative posttest.

    Results

    A 4 x 5 analysis of covariance was per- formed on four outcome variables: (a) in- complete example score, (b) immediate posttest score, (c) cumulative posttest score, and (d) learning time. Pretest scores served as the covariate and were highly sig- nificant in all analyses (p's < .001). Thus, prior achievement was a strong correlate of task performance. Treatment means are summarized in Table 1.

    The analysis of posttest scores yielded a significant main effect due to treatments, F(4, 79)= 13.10, p < .001. Neither the cluster group main effect nor the interaction was significant. Apparently, with pret...

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