THE EFFECTS OF PROBLEM EXEMPLAR VARIATIONS ON FRACTION IDENTIFICATION IN ELEMENTARY SCHOOL CHILDREN

Abstract

A major purpose of this study was to investigate the relationship between fraction-identification rule abstraction and training exemplars. Fractions can be identified with at least two different rules. One of these, the denominator rule, is general in that it yields correct responses across a wide variety of fraction identification problems. The second, the one-element rule, is appropriate only when the number of elements in the set and the denominator-specified number of subsets are equivalent. Because these rules are not equally serviceable, a question of major importance is what factors determine which of the two fraction identification rules a child will learn during training. The main hypothesis within this study specified that the nature of the fraction identification rule abstracted by a learner would be influenced by the nature of the examples used in training. It was further hypothesized that mastery of the denominator rule would positively affect performance on one-element problems, and that denominator-rule problem errors consistent with the one-element rule would occur significantly more frequently than would be expected by chance. The study addressed two additional questions. These related to recent work in the area of information processing and concerned both changes in learner behavior across training/posttesting sessions and consistency between verbal reports of thinking processes and the fraction rules hypothesized to control fraction identification. A pretest was used to determine eligibility to participate in the study. Eighty-two children incapable of set/subset fraction identification participated in the study. Two additional children were involved in an exploratory phase. The children ranged in age from six to ten and in grade level from one to four. The participants were mainly from middle and lower-class Anglo and Mexican-American homes. Children were randomly assigned to one of two training groups. In both training groups learners were provided, through symbolic (verbal) modeling, the general denominator rule for fraction identification. Children in one training group were also provided examples of fraction identification requiring the denominator rule. In the second training group children were provided simple examples in keeping with the denominator rule stated as part of instruction and yet also in keeping with the unstated one-element rule. A modified path analysis procedure was used to assess the effects of training group assignment on fraction identification performance. Results of this analysis suggest a training group main effect. That is, children's performance on fraction identification posttest problems was in keeping with rules associated with the training examples they had been provided. The results suggest that the strongest effects of training were related to performance on denominator rule problems in that the odds of passing the denominator rule posttest were 11.52 times greater for children taught with denominator-rule exemplars than for children taught with the one-element exemplars. The findings also suggest that performance on either of the fraction identification tasks influenced performance on the other. A further finding was that training with the one-element exemplars was associated with performance congruent with inappropriate use of the one-element rule. Recall that the one-element rule was never stated and that the exemplars, while compatible with the one-element rule were equally compatible with the stated denominator rule. Protocols from the children in the exploratory portion of the study suggest that the child taught with the ambiguous exemplars did abstract the one-element rule while the child taught with the denominator rule exemplars abstracted the denominator rule. The protocols also suggest that the child taught with the denominator rule made changes in his thinking as training and posttesting progress progressed.