Understanding of exact equality emerges after and builds on symbolic number knowledge

Establishing whether two sets of objects have the same number of objects turns out to be surprisingly challenging for children even if they have some basic number word and counting knowledge. Here we study the relationship between understanding of exact equality of sets and symbolic number knowledge in preschool children (N=208, Age Range=2.89–5.09 years) at various stages of symbolic number word acquisition. To do this, we gave children two classic verbal symbolic number word knowledge tasks (Give-N, How Many?) and two comparable but non-verbal set-matching tasks in which they were asked to produce a set of objects that numerically matched a target set. We find strong evidence that symbolic number knowledge is related to set-matching for exact equality, both replicating and extending recent findings. Specifically, set-matching accuracy was better for children who understood symbolic number cardinality (i.e., the cardinally principle) compared to those who did not, even after accounting for age and other general cognitive abilities (e.g., executive functions). Furthermore, this effect was seen across the range of set sizes tested (1-8), including smaller set sizes (1-4) typically thought to be within the cognitive limits to be compared for exact equality non-verbally. Finally, set-matching performance was below ceiling even at set sizes corresponding to individual children’s specific level of symbolic number knowledge (N) as well as the preceding quantity (N-1). Together these results suggest that understanding of exact equality comes after and builds on symbolic number knowledge. More broadly, our results support the emerging view that understanding symbolic number cardinality is only a single, early step towards understanding the symbolic number system.