Starting small: exploring the origins of successor function knowledge

Abstract

Although most U.S. children can accurately count sets by 4 years of age, many fail to understand the structural analogy between counting and numberthat adding 1 to a set corresponds to counting up 1 word in the count list. While children are theorized to establish this Structure Mapping coincident with learning how counting is used to generate sets, they initially have an item-based understanding of this relationship, and can infer that, e.g, adding 1 to "five" is "six", while failing to infer that, e.g., adding 1 to "twenty-five" is "twenty-six" despite being able to recite these numbers when counting aloud. The item-specific nature of children's successes in reasoning about the relationship between changes in cardinality and the count list raises the possibility that such a Structure Mapping emerges later in development, and that this ability does not initially depend on learning to count. We test this hypothesis in two experiments and find evidence that children can perform item-based addition operations before they become competent counters. Even after children learn to count, we find that their ability to perform addition operations remains item-based and restricted to very small numbers, rather than drawing on generalized knowledge of how the count list represents number. We discuss how these early item-based associations between number words and sets might play a role in constructing a generalized Structure Mapping between counting and quantity.

Acquiring the cardinal knowledge of number words: A conceptual replication

“…One plausible account is that some children use a mix of exact number processing strategies combining subitizing small subsets and addition. Several pieces of evidence indeed showed that young children start being able to perform small nonverbal addition problems with no counting possibility (as items are occluded) around the age of 4 years onwards (Huttenlocher et al, 1994;Sarnecka & Carey, 2008;Schneider, Pankonin, Schachner & Barner, 2021, see also Secada et al, 1983 for counting on ability with large numbers in 6 year-old children). This evidence leaves open the possibility that children could have subitized a first subset of items and then silently counted on from the first subset to reach the requested number with no apparent sign of one-by-one processing (i.e., a condition for a trial to be classified as counting in the present experiment).…”

Features in children's counting books that lead dyads to both count and label sets during shared book reading

“…Noticing this relation among small sets allows children to infer that this pattern holds for larger set sizes (Gentner & Christie, 2010). Recent work has suggested that children may not need to have an explicit understanding of the successor function to understand the cardinality principle (e.g., Baroody & Lai, 2022; Cheung et al, 2017; Schneider et al, 2021; Spaepen et al, 2018). Instead, children may use structure mapping to form an analogy drawing on the commonality between the last word of the count and the labeled total, understanding that counting can be used to represent the total number of items in any set (Mix et al, 2012).…”

Children’s early understanding of the successor function

“…Only after months or even years of having known the cardinality principle do children appear to fully understand that S(n) = n + 1 (Davidson et al, 2012;Schneider et al, 2021a;Schneider et al, 2021b;Spaepen et al, 2018) One source of evidence is the inconsistent performance of cardinal principle knowers on the unit-task. In the unit-task, children are presented with an occluded set of objects, told the cardinality of the set is n, shown the addition of one object and then asked if the cardinality of the set is now n + 1 or n + 2 (Sarnecka & Gelman, 2004;Sarnecka & Carey, 2008;.…”

“…These findings have been taken to contradict Carey's account, in which becoming a cardinal principle knower represents a moment of conceptual induction and discovery of the structure of the number system. Instead, it has been proposed that becoming a cardinal principle-knower may simply equip children with a procedure and practice that allows them to eventually discover the structure of the number system (Barner, 2017;Davidson et al, 2012;Schneider et al, 2021a;Spaepen et al, 2018). Potentially, only once children start using counting to produce and reason about sets are they able to notice how adding one item to a set changes the cardinality of the set not only to a different number but to always exactly the next number in the number sequence (Spaepen et al, 2018).…”