Abstract
Introduction
Student learning is a dynamic process that depends on both current engagement and accumulated learning experience. Conventional statistical and machine learning approaches often overlook the temporal memory effects that influence learning progression. This study develops a fractional differential equation model to represent student learning dynamics using anonymized learning analytics data.
Methods
The proposed framework uses the publicly available Open University Learning Analytics Dataset (OULAD) to construct continuous, time-dependent variables from assessment outcomes and interactions in the virtual learning environment. A Caputo fractional differential equation is formulated to incorporate memory-dependent learning behavior. Model parameters are estimated from empirical data, and the numerical solution is obtained using the L1 finite difference scheme. Model performance is evaluated through prediction accuracy, residual analysis, and sensitivity analysis.
Results
The simulated learning state evolves from initial values of approximately 0.52 to 0.58 to a steady-state range of 0.66 to 0.78, depending on the student group. The proposed model achieves a mean squared error of approximately 0.004 and a coefficient of determination of about 0.88, indicating strong agreement between predicted and observed learning trajectories. Residuals remain small and unbiased, while sensitivity analysis identifies the fractional order as the most influential parameter governing long-term learning behavior.
Discussion
The results demonstrate that fractional differential equations provide an accurate and interpretable framework for modeling memory-dependent learning processes. The proposed approach offers practical insights for learning analytics by supporting the identification of students who may benefit from timely instructional interventions based on their engagement patterns and memory-related learning characteristics.