Catastrophes in learning: Dynamical system models for the reciprocal relation between practice and success

We propose a dynamic system model for the investigation of the reciprocal relation between practice and success in learning under conditions of free practice, where practice leads to success and success reinforces practice. In free practice, one may quit, in contrast to forced practice, a case that has been extensively studied in mathematical learning theory. The forced ‘law of practice’ models studied in mathematical learning theory are the main building blocks of our model. It is shown that the equilibrium behavior of the reciprocal practice-success (RPS) model depends mainly on the choice of the ‘law of practice’ function. For concave practice functions, the resulting dynamics can be characterized as a fold catastrophe. For S-shaped practice functions, the behavior is governed by a cusp catastrophe, in which sudden transitions between optimal and deprived learning states occur. As such, the model offers new explanations for drop-out, the Matthew effect, and the development of expertise. The psychological interpretation of this model, its practical implications, and limitations are discussed.