Mathematical Modeling of Student Learning Outcomes using E-learning-Based Remedial Programs

Research on mathematical modeling of learning outcomes remains limited, despite its potential to evaluate educational processes and inform placement decisions in schools, classes, learning resources, and remedial programs. This quantitative study aims to construct a mathematical model in the form of an ordinary differential equation (ODE) to represent the dynamics of students’ learning in a remedial program. The model is developed using empirical data obtained from multiple-choice diagnostic tests designed to identify common learning difficulties and a series of three remedial assessments conducted after e-learning-based interventions. The dataset includes students’ assessment scores and time records during the remedial learning process. The Nelder–Mead method was used to estimate the model parameters, followed by a stability analysis and an RMSE-based evaluation of the model’s accuracy. The model captures changes in student understanding over time and reveals that students with lower initial scores tend to show greater improvements through remedial programs. However, as the duration of the remedial program increases, the rate of score improvement decreases—suggesting a decline in student focus and learning efficiency over time. These findings highlight the nonlinear nature of learning progress in remedial program. The model provides for predicting student outcomes and analyzing the effectiveness of remedial programs. It offers practical implications for optimizing the structure and timing of remedial programs and can support the development of adaptive learning systems tailored to student needs. This research demonstrates the potential of mathematical modeling for decision-making in education.