Capturing learning persistence through fractional dynamic modeling: evidence from secondary school students.

Background: Student academic performance develops through cumulative learning experiences, yet conventional educational performance models generally assume that Educational support and learning behavior variables produce immediate effects. This study proposes a fractional dynamic framework to capture the persistence of learning and examine associations between educational support, learning behavior, and academic performance. Methods: We analyzed micro-level data from 649 Portuguese secondary school students using a fractional-order educational performance model that incorporated educational memory through a fractional formulation. The analysis examined associations between school study support, paid academic assistance, family educational support, attendance behavior, and study time and final academic performance while controlling for academic history and socioeconomic characteristics. Results: The proposed fractional model outperformed the benchmark specification by increasing the adjusted coefficient of determination from 0.849 to 0.886 and reducing the root mean square error from 1.04 to 0.87. The estimated fractional-order parameter (α = 0.83) indicated temporal dependence within the observed academic assessment structure. Family educational support and weekly study time were positively associated with academic performance, whereas student absences and previous academic failures were negatively associated with final achievement. Conclusion: The findings indicate that incorporating educational memory provides a useful representation of temporal dependence in student academic performance. The observed associations between educational support, learning behavior, and achievement provide descriptive evidence that may inform future longitudinal and intervention-based research.