Applying Time-Inhomogeneous Markov Chains to Math Performance Rating

In this paper, we present a case study in collaboration with mathematics education and probability theory, with one providing the use case and the other providing tests and data. The application deals with the reason and execution of automation of difficulty classification of mathematical tasks and their users’ skills based on the Elo rating system. The basic method is to be extended to achieve numerically fast converging ranks as opposed to the usual weak convergence of Elo numbers. The advantage over comparable state-of-the-art ranking methods is demonstrated in this paper by rendering the system an inhomogeneous Markov Chain. The usual Elo ranking system, which for equal skills (Chess, Math, ...) defines an asymptotically stationary time-inhomogeneous Markov process with a weakly convergent probability law. Our main objective is to modify this process by using an optimally decreasing learning rate by experiment to achieve fast and reliable numerical convergence. The time scale on which these ranking numbers converge then may serve as the basis for enabling digital applicability of established theories of learning psychology such as spiral principal and Cognitive Load Theory. We argue that the so further developed and tested algorithm shall lay the foundation for easier and better digital assignment of tasks to the individual students and how it is to be researched and tested in more detail in future.