A mathematical theory of instruction: Instructor/learner interaction and instruction pacing☆

Abstract In this paper, a mathematical theory of instruction applicable in the educational environment is developed from concepts of psychological learning theory. Within the framework of optimization and control theory, the dynamics of the interaction between instructor and learner are modelled, and the trade-off between instruction cost and learner achievement is formulated so that optimal instruction inputs can be determined. One important aspect of the classroom environment that is characterized by the theory is the interaction between an instructor and a group of learners with various learning abilities. A basic dynamic model that relates learner achievement and instruction cost is developed from learning theory concepts. This model, which applies to the individual learner situation, is analyzed in detail to determine instruction intensity inputs that match the learner's characteristics in order to maximize an objective that measures both achievement and cost. This basic model is used as a building block to describe how individual learner achievement depends on instruction pacing. To determine optimal instruction pacing the concept of gain, which is essentially learner achievement per unit time, is introduced. In this extended model, instruction pacing is intimately related with the concept of learner aptitude. This relationship leads immediately to the consideration of instruction pacing for a group of learners with various aptitudes and thus optimal instruction pacing is determined for nonhomogenous groups. Throughout the development of the theory, hypothetical examples are presented to demonstrate many of the implications of the theory. One of the contributions of the theory is the definition of the concepts of learner aptitude and instruction pacing within a framework that structures the empirical investigation of these concepts by means of experimental research.