Commutativity–the fact that changing the order of operands in an operation does not change the result–is a fundamental principle of mathematics. How does its understanding develop in children? While this question has mostly been addressed for addition, here we focus on multiplication. We ask whether children’s formal learning of multiplication commutativity relies on pre-existing non-symbolic intuitions, or whether it is first learnt symbolically and only later elaborated as a general principle that extends to concrete situations. 2nd and 3rd graders received an intervention on commutative multiplication, before and after which they played a number comparison game testing their understanding of commutativity in symbolic and non-symbolic contexts. Our results suggest that the commutative principle of multiplication may not be available to intuition before formal teaching: perceiving that 2 groups of 3 dots and 3 groups of 2 dots contain the same number of dots requires mastering commutativity symbolically first.

Among primates, humans are special in their ability to create and manipulate highly elaborate structures of language, mathematics or music. We show that this sensitivity is present in a much simpler domain: the visual perception of geometric shapes. We asked human subjects to detect an intruder shape among six quadrilaterals. Although the intruder was defined by an identical amount of displacement of a single vertex, the results revealed a geometric regularity effect: detection was considerably easier with most regular shapes. This effect was replicated in several tasks and in both uneducated adults and preschoolers. Baboons, however, showed no such geometric regularity effect even after extensive training. Baboon behavior was captured by convolutional neural networks (CNNs) but a symbolic model was needed to fit human behavior. Our results indicate that the human propensity for symbolic abstraction permeates even elementary shape perception and they suggest a new putative signature of human singularity.