A Bayesian Ordinal Analysis of Students’ Progression Across Van Hiele Levels

Abstract

Understanding students’ progression across hierarchical levels of geometric reasoning requires analytical approaches that respect ordinal structure and quantify uncertainty. The purpose of this study is to model students’ progression across Van Hiele levels probabilistically and to estimate level-specific transition probabilities under uncertainty using paired pretest–posttest data from Grade 8 and Grade 9 students. The method used in this study is Bayesian ordinal regression with a cumulative logit specification estimated via MCMC sampling. Van Hiele levels were modeled as ordered categorical outcomes, and posterior summaries, transition matrices, and posterior predictive checks were used to characterize progression and assess model adequacy. The results indicate that progression is predominantly upward in both grades, with negligible posterior support for regression. Grade-dependent differences are evident: Grade 8 shows broader, more heterogeneous transitions from a uniformly low baseline, whereas Grade 9 exhibits more constrained, incremental progression from a higher, more dispersed initial distribution. Posterior predictive checks confirm that the model adequately reproduces the observed posttest patterns, supporting the validity of the Bayesian ordinal specification. Pedagogically, these findings imply that students at lower baseline levels tend to undergo broader conceptual shifts, whereas those at higher levels require sustained, targeted instructional support to advance further. These findings indicate that baseline ordinal structure shapes progression dynamics and that Bayesian ordinal modeling offers a coherent alternative to significance-based approaches for analyzing hierarchical learning outcomes. Educationally, this underscores the need to align instruction with students’ initial Van Hiele levels to support optimal conceptual advancement.