Optimal allocation of instructional effort to interrelated learning strands

The problem of allocating instructional effort to two interrelated blocks of learning material is studied. In many learning environments, the amount of material that has been mastered in one area of study affects the learning rate in another distinct but related area—for example, the curriculum subjects of mathematics and engineering. A model is developed that describes this phenomenon, and the Pontryagin Maximum Principle of control theory is applied to determine optimal instructional policies based on the model. The nature of this optimal solution is to allocate instructional effort so that the learner follows a maximal average learning rate “turnpike” path until near the end of the study period and then concentrates on only one strand. This strategy, when applied to a more realistic stochastic model, defines a closed-loop feedback controller that determines daily instructional allocation based on the best current estimate of how much the student has learned. This estimate is calculated by a multistage linear filter based on the Kalman filtering technique.