MODELING MOTIVATIONAL DYNAMICS IN UNIVERSITY ENGLISH LEARNING USING THE ARCS FRAMEWORK AND RUNGE-KUTTA METHODS 

Abstract

The ARCS model (Attention, Relevance, Confidence, Satisfaction) provides a robust framework for fostering motivation in educational settings, yet its dynamic evolution remains underexplored. This study introduces a computational approach to model motivational dynamics in university English classrooms by integrating the ARCS framework with the fourth-order Runge-Kutta method (RK4). A system of ordinary differential equations (ODEs) is developed to capture the temporal interactions among ARCS components, reflecting the impact of instructional strategies such as interactive listening tasks, culturally relevant readings, and gamified speaking activities. Applied to 120 university English learners, the model simulates how interventions influence motivation over a semester. RK4 simulations reveal nonlinear motivational trajectories, identifying optimal intervention timings to enhance engagement and language proficiency. Validated against empirical data, the model offers educators a quantitative tool to design motivation-driven English curricula. This interdisciplinary study bridges educational psychology and numerical analysis, with implications for personalized learning design.