Formulation and application of a theory of educational undercurrent

Abstract

Mathematics instruction at the pre-college level often focuses on teaching a collection of skills, such as computational algorithms. Although mathematical procedures can be learned and practiced in this piecemeal fashion, the conceptualization of mathematics as a collection of isolated rules and procedures seems unlikely to foster the sorts of knowledge representations that best facilitate future application.

The goals of this research were to: (1) analyze representative examples of highly-successful instructional materials from a fifteen-year software development effort, (2) determine some key considerations in the design of the materials, (3) formulate the theory of instructional design for mathematical experiences, and (4) test essential aspects of this theory in a realistic classroom context.

The materials analyzed in this study are characterized by careful attention to aspects of the student's experience beyond the evident content goals. Analysis of the materials distilled some themes of this undercurrent of educational experiences.

An experiment was designed to evaluate the effect of restructuring a unit on trigonometric identities around microcomputer activities based on the identified themes of educational undercurrent. Two experimental conditions were tested, in appropriate curriculum context, with 30 high school trigonometry students. One treatment focused on trigonometric identities and related graphing activities; the other treatment approached the same content with careful attention to the themes of educational undercurrent.

Analysis of covariance indicated significantly higher posttest performance for the undercurrent-enhanced treatment on relating trigonometric functions to their graphical representations, with no significant difference in posttest performance on the standard content of proving trigonometric identities. The results of this study provide evidence that designing instruction around the identified themes of educational undercurrent can produce a richer learning experience without significant detrimental effect on the mastery of standard content.