MATHEMATICAL MODELING OF KNOWLEDGE ACCUMULATION IN MATHEMATICS LEARNING DURING STANDARDIZED TEST PREPARATION

Abstract

Standardized examinations — the SAT, the GRE, and their analogues — test something harder to measure than content recall: the capacity to reason quantitatively under pressure, to interpret unfamiliar data, to hold several logical constraints in mind simultaneously. Preparing students for this kind of assessment demands more than a syllabus of practice problems. It requires a coherent theory of how mathematical knowledge actually develops.

This study proposes and analyzes a mathematical model describing the growth of mathematical competence over time. The model treats knowledge acquisition as a nonlinear process that asymptotically approaches a theoretical ceiling — a pattern consistent with decades of empirical work on skill learning. To test its applicability, we conducted a ten-week pedagogical experiment with sixty undergraduate students divided into control and experimental groups. The experimental group followed a structured preparation program combining diagnostic assessment, guided problem-solving, and continuous instructor feedback.

The results were clear-cut. Students in the experimental group outperformed their peers on every assessment — final scores of 74.2 versus 63.5 — with the gap widening as the course progressed. Model parameters, fitted from empirical data, revealed a substantially higher learning rate in the structured group (α = 0.106) compared with the conventional format (α = 0.082). These findings suggest that mathematical modeling is not merely a descriptive convenience: it can serve as a practical instrument for designing and evaluating instructional programs.