Rethinking the implications of numerical ratio effects for understanding the development of representational precision and numerical processing across formats.

Numerical ratio effects are a hallmark of numerical comparison tasks. Moreover, ratio effects have been used to draw strong conclusions about the nature of numerical representations, how these representations develop, and the degree to which they generalize across stimulus formats. Here, we compute ratio effects for 1,719 children from Grades K-6 for each individual separately by computing not just the average ratio effect for each person, but also the variability and statistical magnitude (effect-size) of their ratio effect. We find that individuals' ratio effect-sizes in fact increase over development, calling into question the view that decreasing ratio effects over development indicate increasing representational precision. Our data also strongly caution against the use of ratio effects in inferring the nature of symbolic number representation. While 75% of children showed a statistically significant ratio effect for nonsymbolic comparisons, only 30% did so for symbolic comparisons. Furthermore, whether a child's nonsymbolic ratio effect was significant did not predict whether the same was true of their symbolic ratio effect. These results undercut the notions (a) that individuals' ratio effects are indicative of representational precision in symbolic numbers, and (b) that a common process generates ratio effects in symbolic and nonsymbolic formats. Finally, for both formats, it was the variability of an individual child's ratio effect (not its slope or even effect-size) that correlated with arithmetic ability. Taken together, these results call into question many of the long-held tenets regarding the interpretation of ratio effects-especially with respect to symbolic numbers.