Magnitude Representations and Counting Skills in Preschool Children

Abstract

When children learn to count, they map newly acquired symbolic representations of number onto preexisting nonsymbolic representations. The nature and timing of this mapping is currently unclear. Some researchers have suggested this mapping process helps children understand the cardinal principle of counting, while other evidence suggests that this mapping only occurs once children have cardinality understanding. One difficulty with the current literature is that studies have employed tasks that only indirectly assess children's nonsymbolic-symbolic mappings. We introduce a task in which preschoolers made magnitude comparisons across representation formats (e.g., dot arrays vs. verbal number), allowing a direct assessment of mapping. We gave this task to 60 children aged 2;7 -4;10, together with counting and Give-a-Number tasks. We found that some children could map between nonsymbolic quantities and the number words they understood the cardinal meaning of, even if they had yet to grasp the general cardinality principle of counting.Keywords: counting, magnitude comparison, cardinality, preschool children, number Running head: PRESCHOOL MAGNITUDE REPRESENTATIONS 3 Magnitude Representations and Counting Skills in Preschool ChildrenWe know from more than a decade's worth of research that infants, children and adults can represent and manipulate numerical information nonsymbolically, without number words or digits. These nonsymbolic representations are robust across multiple modalities and set sizes. Children and adults can compare, add and subtract small and large quantities in visual arrays (Barth, Kanwisher, & Spelke, 2003;McCrink & Wynn, 2004), auditory sequences (Barth et al., 2003;Barth, La Mont, Lipton, & Spelke, 2005) and moving displays of actions (e.g., puppet jumps) (Wood & Spelke, 2005;Wynn, 1996;Wynn, Bloom, & Chiang, 2002).The nonsymbolic representations employed in these tasks are approximate and in an analogue format. They are inherently noisy and the variance associated with them increases with the absolute size of the magnitudes represented. As a result, success on these tasks depends on the ratio (or numerical distance) between the numerosities to be compared 1 . As the quantities get closer together, discrimination becomes more effortful and less precise.Importantly, the precision of these representations varies across individuals and increases over development. Infants can discriminate numerosities with ratios as small as 2:3, whilst preschool children show a ratio-limit of 3:4, and adults, 7:8 (Barth et al., 2003;Feigenson, Dehaene, & Spelke, 2004).When children begin to count they learn to use external symbols to represent number.These symbolic representations enable exact number comparison and manipulation. There is evidence that when children acquire this symbolic system, the preexisting nonsymbolic system is not overridden; rather, nonsymbolic representations become mapped onto the newly acquired symbolic representations. The evidence for this is at least threefold. Firstly, children...

Approximate number system task performance: Associations with domain-general and domain-specific cognitive skills in young children

“…Fuhs, Nesbitt, & O'Rear 591 vanMarle, Chu, Li, and Geary (2014) used a battery of symbolic math tasks and found that it was children's cardinality knowledge that mediated the link between ANS task performance and math achievement, suggesting that children's ANS acuity influences symbolic math achievement indirectly through specific early math skills (see also Geary & vanMarle, 2016;vanMarle et al, 2018). Also, children who are still developing an understanding of how counting connects to cardinality mapped number words they know onto non-symbolic quantities, and cardinality knowledge predicted children's ability to make these mappings (Batchelor, Keeble, & Gilmore, 2015). However, even this association may be indirect.…”

Early numeracy skills in preschool-aged children: a review of neurocognitive findings and implications for assessment and intervention

“…Instead, evidence suggests that children who have acquired symbolic skills such as the cardinality principle are more likely to succeed on non-symbolic tasks (Batchelor, Keeble, & Gilmore, 2015; Slusser, Ditta, & Samecka, 2013), leading some to conclude that acquiring symbolic knowledge likely influences non-symbolic skills, rather than the other way around (Merkley & Ansari, 2016). Similarly, relations between ANS acuity and math achievement are mediated by symbolic number knowledge such as Arabic numeral naming and cardinality (Chu, van Marle, & Geary, 2015; see also Merkley & Ansari for a review).…”

Children's Acquisition of the Mappings Among the Number Representations

“…Despite the importance of knowing the symbolic system, however, few studies have directly assessed the order in which preschool children (i.e., two-to-four-year olds) make the connections among the three representations (cf. Batchelor, Keeble, & Gilmore, 2015;Benoit, Lehalle, Molina, Tijus, & Jouen, 2013;Odic, Le Corre, & Halberda, 2015;Knudsen et al, 2015). Although most existing evidence is consistent with the view that digit-quantity mapping develops last (Bialystok, 1992;Carey, 2004;Fuson, 1988;Knudsen et al, 2015;Soltësz, Szűcs, & Szűcs, 2010;Zhou & Wang, 2004), only one previous study tested children's knowledge of the mappings in both directions among all three representations.…”

“…An increasing number of studies have been conducted on children's early mapping skills (e.g., Batchelor, Keeble & Gilmore, 2015;Benoit et al, 2013;Knudsen et al, 2015;Mundy & Gilmore, 2009). Researchers have addressed multiple issues such as: a) the relation between mapping skills and early numeracy skills such as counting, cardinality (Batchelor, Keeble, & Gilmore, 2015) and later math skills such as symbolic number comparison (Mundy & Gilmore, 2009); b) whether children learn to map in one direction (e.g., number words to quantities) prior to the other (quantities to number words) (e.g., Odic, LeCorre, & Halberda, 2015); and c) the order of acquisition of the mappings among the three representations of number (e.g., Benoit et al, 2013). Results across studies have been inconsistent.…”

“…Lipton and Spelke's results showed a positive relation between verbal counting and estimation skill: children were able to provide accurate estimates of the numerosities that were within their counting range. Batchelor, Keeble, and Gilmore (2015) further investigated the relationship between verbal counting, and children's ability to map number words to non-symbolic quantities for two-to four-year-old children; importantly, the researchers also assessed the relation between word-quantity mapping and children's understanding of cardinality.…”