The effect of cognitive style on the acquisition of mathematical concepts presented through emphasis on positive and negative instances.

Abstract

Purpose. The purpose of this study was to examine the interaction between the field-independent/field-dependent cognitive styles and the use of positive-negative instances on the acquisition of "real-world" classroom concepts selected from an introductory calculus course. It was specifically directed to the following questions: (1) Are field-independent subjects superior to field-dependent subjects in the initial achievement and retention of the selected calculus concepts? (2) Does the inclusion of negative instances with positive instances aid in the initial achievement and retention of the selected calculus concepts? (3) Can significant interaction between positive instances versus positive and negative instances and the field-independent/field-dependent cognitive styles be observed.?

Analysis of covariance was used to adjust for initial between group differences in the intact classes. A test of pre-calculus algebraic skills was used as the covariate. The Hidden Figures Test (HFT) was used to assess the field-independent, mobile, and field-dependent cognitive styles. At the conclusion of the three week treatment, an initial achievement test was administered. Two weeks later, a similar test was used to assess student retention of the differentiation concepts.

While the literature has revealed mixed findings for the use of negative instances, it appears that the sequence of positive and negative instances is equally as effective as a sequence of all positive instances when teaching the introductory differentiation concepts involved in this study.

Findings. The analysis of covariance for the initial achievement data and the retention data revealed no statistically significant differences between the experimental group receiving both positive and negative instances and the control group receiving all positive instances. It was also found that field-independent subjects were not superior to the field-dependent subjects in initial achievement or retention as hypothesized, nor was there found a statistically significant interaction between the treatments and the cognitive styles.

The calculus concepts used in the experiment consisted of introductory differentiation rules for finding the derivative of the product and the quotient of two differentiable functions and the extended power rule. Each of these concepts was presented in a lecture-discussion approach, supplemented by in-class handouts and outside assignments emphasizing positive and negative instances for the experimental treatment and all positive instances for the control treatment.

Methods and Procedures. The study was conducted over a three week period in the fall semester, 1980, at the University of Oklahoma. Sixty two students, enrolled in two intact sections of introductory calculus for the business, life, and social sciences majors, were used in the experiment. One section was randomly chosen to receive the experimental treatment consisting of equal numbers of positive and negative instances. The other section was designated the control group and received an equal number of all positive instances.

Recommendations. In attempting to meet the individual needs of students with differing learning styles, the use of negative instances in conjunction with positive instances is but one important strategy that classroom practitioners may want to employ. Consequently, further research is needed to assess the impact of the use of negative instances on student learning in additional non-laboratory "real-world" concepts found in the classroom environment.