A Reciprocal Model of Practice and Skill: Navigating Between Dropout and Expertise

Understanding the reciprocal relationship between practice and skill, or learning success, is essential for designing effective interventions to prevent dropout. Classical learning curves model the effect of practice on skill in the case of enforced learning, where dropout is not an option. However in many learning contexts, such as learning a new hobby or taking an online course, learners can, and often do, drop out. For this case, previous research in educational psychology has suggested a reciprocal relationship between motivation and practice, on the one hand, and achievement and skill, on the other. We provide a formal model of this relationship by integrating classical models of learning and forgetting into a model of the practice-success cycle. We show that in this Reciprocal-Practice-Success (RPS) model of learning, long-term learning outcomes are particularly sensitive to the shape of the learning curve - the risk of dropping out is much higher when the learning curves are sigmoid than when they are concave. Through bifurcation analysis of a simplified formulation of the model, we demonstrate how modifying the minimum practice rate and success sensitivity can mitigate attrition in cases where learning curves are sigmoid. In addition, we explore various mechanisms to shape the learning curve in a concave form, further reducing the risk of dropout. Finally, we present extensions that incorporate spacing effects and temporal discounting to describe learning dynamics in more realistic contexts.