Modelling Progress in Educational Interventions for Dyscalculia Using Controlled Markov Chains

This work introduces a controlled Markov chain approach to describe and optimize educational interventions for students with dyscalculia. The model represents the progression of a student through a finite collection of educational states, beginning with diagnosis and ending in one of two possible outcomes: successful recovery or discontinuation of the intervention. By treating these final outcomes as absorbing states, the framework makes it possible to derive relevant quantitative measures, including the fundamental matrix, the expected time spent in the intervention process, and the probabilities associated with recovery and dropout. To incorporate the influence of educational decisions, transition probabilities are allowed to vary according to a control variable representing alternative intervention strategies. This leads naturally to a controlled Markov chain formulation in which the evolution of the student depends not only on the current educational state but also on the intervention selected at each stage. Within this setting, several theoretical properties are established, including the well-posedness of the controlled stochastic process, the existence of the corresponding fundamental matrix, and the existence of an optimal intervention policy. The resulting mathematical framework provides a basis for comparing educational strategies and identifying policies that balance three main objectives: increasing the likelihood of recovery, reducing the expected duration of the intervention, and limiting the probability of educational intervention discontinuation. By bringing together absorbing Markov chains, stochastic modelling, and optimal control, the proposed approach offers quantitative tools for studying the dynamics and effectiveness of educational interventions for dyscalculia. The same methodology may also be adapted to other intervention settings characterized by discrete stochastic evolution and decision-dependent transitions.