Modeling master’s students’ competence development based on the variational principle

Context and relevance. Despite the widespread adoption of the competency-based approach in higher education, a gap remains between the understanding of competence as a dynamic process and the tools available for its design and management. Dominant practices of learning outcomes assessment rely on static “snapshots,” which limits the possibilities for forecasting and purposeful development of competence. In this context, there is a growing need for formal modeling of competence development trajectories based on principles of optimality derived from the mathematical theories of the calculus of variations and optimal control. Objective. To develop and empirically test a model of master’s students’ competence development based on the variational principle, which conceptualizes the educational process as a problem of optimal control over the trajectory of key competence formation. Methods and materials.The study involved 24 first-year master’s students enrolled in the program School Leadership and Educational Policy (83% women). Competence was assessed using seven indicators reflecting the ability to establish causal relationships between norms and practices (scale 0–3; total score range 0–21). Data were collected across three measurement points during the completion of an analytical learning task. The theoretical model was constructed using tools of the calculus of variations (Euler–Lagrange equations, transversality conditions), and the empirical validation was conducted through multilevel regression analysis with random effects models. Results. The analytical solution of the extremal problem demonstrated that the optimal trajectory of competence development is linear in nature and corresponds to the principle of minimizing the “length” of the educational path, interpreted as the minimization of cognitive and motivational overload. Empirical data confirmed a predominantly linear pattern of competence growth for most students. The multilevel regression model revealed a statistically significant increase in performance across measurement points, as well as a negative association between initial competence level and growth rate, indicating a compensatory pattern of development. The high value of the conditional coefficient of determination highlights the crucial role of individual trajectories in competence formation. Conclusions. The findings suggest that competence development should be conceptualized as a non-ergodic process, which requires a shift from the analysis of interindividual differences to the modeling of intraindividual dynamics. The variational approach provides a foundation for the normative design of educational trajectories and enables the integration of mathematical models of optimality into educational analytics and learning support systems. The proposed model is recommended as a tool for forecasting and planning pedagogical interventions aimed at fostering sustainable and resource-efficient development of students’ key competencies.