Limits to intuition in number compositionality: the case of multiplicative commutativity

Commutativity—the principle that operand order does not affect the result of an operation—is a core feature of two foundational operations of arithmetic: addition and multiplication. Recent theories propose that such compositional structures emerge from an innate “language of thought” that enables abstract principles to apply flexibly across symbolic and non-symbolic formats. While children possess early intuitions about additive commutativity, less is known about their grasp of the commutativity of multiplication. Here, we ask whether children’s understanding of the commutative principle of multiplication stems from pre-existing intuitions that the order in which sets are grouped does not matter, or whether it instead emerges from reasoning about structural relations among numerical symbols. We tested 8- to 9-year-old children in a number comparison game, probing their understanding of multiplicative commutativity in both symbolic expressions (3x2 = 2x3) and in non-symbolic sets of grouped dots. On average, children performed better with symbolic expressions than with non-symbolic arrays. Moreover, their performance on the arrays of grouped dots was directly linked to their mastery of commutativity in symbolic expressions, and independent of their intuitive, non-symbolic numerosity estimation skills. To probe children’s developing mastery of the commutativity of multiplication, a subset of the children were given brief training on commutativity and played the same game in a second session, 2 to 14 days later. Children who, in the first session, did not master the commutative principle with symbolic stimuli showed improvement on symbolic trials, while those with high initial symbolic commutativity mastery improved on non-symbolic trials. These findings suggest that symbolic learning acts as a gateway for children to apply the principle of commutativity of multiplication to concrete contexts. Moreover, they provide evidence for format- and operation-specific limits to children’s mastery of the compositional structure of the natural number system.