Why set-comparison is vital in early number learning

Cardinal numbers serve two logically complementary functions. They tell us how many things are within a set, and they tell us whether two sets are equivalent or not. Current modelling of counting focuses on the representation of number sufficient for the within-set function; however, such representations are necessary but not sufficient for the equivalence function. We propose that there needs to be some consideration of how the link between counting and set-comparison is achieved during formative years of numeracy. We work through the implications to identify how this crucial change in numerical understanding occurs.