Creating AI-driven adaptive systems in calculus

Education through dynamical modeling

Abstract

This research presents a mathematical framework for AI-driven adaptive learning systems (ALS) in calculus education, combining nonlinear dynamical systems, reinforcement learning (RL), and fairness-aware optimization. The author derives a delay differential equations (DDEs) system to model knowledge retention and prove global stability via Lyapunov functions. The ALS employs a partially observable Markov decision process (POMDP) to optimize instructional policies, with a fairness penalty term minimizing demographic disparities. Empirical validation involves a year-long study (N=450 students), showing a 28% increase in mastery rates (p < 0.001, alpha=0.01) and a 63% reduction in equity gaps. Theoretical contributions include a bifurcation analysis of the DDE system and a proof of regret bounds for the RL algorithm. The work advances the integration of control theory and AI in mathematics education.