Socioeconomic Inequality and Mathematics Achievement: A Three-Regime Nonlinear Gradient

Socioeconomic status (SES) is a powerful predictor of children’s mathematics achievement, yet its functional form is rarely modeled flexibly. Using longitudinal data from ECLS-K:2011 (N = 18,174 students; 163,566 person-wave observations), this study examines the nonlinear SES–achievement relationship. Multilevel spline models reveal a structured three-regime gradient: steep gains at low SES (constraint release), attenuated marginal gains in the middle (capability activation), and moderate re-acceleration at high SES (enrichment accumulation). These regions reflect variation in the marginal effect of SES along a continuous gradient rather than discrete categories.To assess robustness, we implement a multimethod strategy combining propensity score matching, inverse probability weighting, difference-in-differences, and sensitivity analysis. Results indicate a strong and robust association between SES and mathematics achievement, and sensitivity analysis suggests that substantial unobserved confounding would be required to overturn inference.Interpreted through a developmental resource-conversion framework, findings suggest that socioeconomic resources yield the largest marginal returns among the most disadvantaged children. The study highlights the importance of flexible modeling of inequality gradients and the potential for targeted interventions.