Lack of integrated number sense among college students: evidence from rational number cross-notation comparison

Growing evidence highlights the predictive power of cross-notation magnitude comparison (e.g., 2/5 vs. 0.25) for math outcomes, but the underlying mechanisms remain unknown. Across two studies, we investigated undergraduates’ cross-notation and within-notation comparison skills given equivalent fractions, decimals, and percentages (Study 1, N=220 and Study 2, N=183). We found participants did not perceive equivalent rational numbers equivalently. Cluster analyses revealed that approximately one-quarter of undergraduates exhibited a bias to select percentages as larger in cross-notation comparisons. Compared with the other cluster of undergraduates who showed little-to-no bias, the percentages-are-larger bias cluster performed worse on fraction number line estimation and fraction arithmetic (exact and approximate), as well as reporting lower SAT/ACT scores. Hierarchical linear regression analyses demonstrated that cross-notation comparison accuracy accounted for variance in SAT/ACT beyond within-notation accuracy. Mediation analyses revealed a potential mechanism: stronger cross-notation knowledge equips individuals to evaluate the reasonableness of solutions. Together, these results suggest the importance of an integrated understanding of rational number notations, which may not be fully assessed by within-notation measures alone.