UC San Diego

Although numerate humans’ experience with symbolic number is finite, we nevertheless are capable of abstract numerical thought extending well beyond these limits. Such abstract thought is argued to depend on extracting two properties of exact number available within symbolic counting systems: the successor function and exact equality. On standard theories of number acquisition, children are proposed to simultaneously acquire both principles upon mastering their symbolic counting system by establishing an analogical mapping between the count list and cardinality. In this dissertation, I argue that this analogical mapping is not established until well after children master counting and that, consequently, implicit knowledge of both the successor function and exact equality emerge years after children are capable counters. In Chapter 1, I show that children who have just learned to deploy counting do not understand that adding one item to a set requires counting up one number in the count list—an analogical mapping compatible with implicit successor function. However, if implicit successor knowledge is not acquired in conjunction with counting, how does it emerge? In Chapter 2, I explore one hypothesized mechanism—mastery of counting’s recursive structure—in five languages which vary in the transparency of these recursive structures (English, Cantonese, Slovenian, Hindi, and Gujarati). Across language groups, I show that implicit successor knowledge is strongly predicted by recursive counting mastery. In Chapter 3, I test another hypothesized mechanism—arithmetic instruction. I show that although arithmetic instruction is correlated with successor knowledge, it is not causally related; instead, compatible with Chapter 2, I show that recursive counting is once again predictive of implicit successor knowledge. Finally, in Chapter 4, I turn to exact equality—another property of exact number—and show that although learning to deploy counting marks a substantial shift in children’s understanding of exact equality, counting on its own is insufficient to guarantee a full understanding of this logical property. Together, these studies suggest that, instead of marking a conceptual inflection point in children’s understanding of number, mastering a symbolic counting system is only a first step in a gradual process of extracting conceptual knowledge from procedures.