This study examined how book features influence talk during shared book reading. We used data from a study in which parent-child dyads (n = 157; child's M-age = 43.99 months; 88 girls, 69 boys; 91.72% of parents self-reported as white) were randomly assigned to read two number books. The focus was comparison talk (i.e., talk in which dyads count a set and also label its total), as this type of talk has been shown to promote children's understanding of cardinality. Replicating previous findings, dyads produced relatively low levels of comparison talk. However, book features influenced the talk. Books containing a greater number of numerical representations (e.g., number word, numeral, and non-symbolic set) and a greater word count elicited more comparison talk.

Counting books are a potential source of input for children's learning of early mathematics concepts. However, little is known about the factors that affect the counting books parents choose for their children. Parents (N = 696) of preschoolers (ages 2;6–4;11) were surveyed about their preferences for two specific counting book features, tactility and narrative quality. These two features were studied both covertly, by experimentally manipulating the types of books parents saw and asking parents why they would choose particular books, and explicitly, by asking parents to rate the importance of various factors when choosing counting books for their children. The a priori hypotheses were that parents would prefer tactile over non-tactile counting books for boys and narrative over non-narrative counting books for girls and that education level would be positively associated with counting book reading. Results did not support these hypotheses. Instead, parents’ preferences for the features depended on their education level. Higher education levels were generally associated with decreased preference for tactility and increased preference for narrative quality. Results raise the question of whether the books parents choose for their children may be one way parent education shapes children's early learning environments. © 2018 Elsevier Inc.

A common measure of number word understanding is the give-N task. Traditionally, to receive credit for understanding a number, N, children must understand that N does not apply to other set sizes (e.g. a child who gives three when asked for 'three' but also when asked for 'four' would not be credited with knowing 'three'). However, it is possible that children who correctly provide the set size directly above their knower level but also provide that number for other number words ('N + 1 givers') may be in a partial, transitional knowledge state. In an integrative analysis including 191 preschoolers, subset knowers who correctly gave N + 1 at pretest performed better at posttest than did those who did not correctly give N + 1. This performance was not reflective of 'full' knowledge of N + 1, as N + 1 givers performed worse than traditionally coded knowers of that set size on separate measures of number word understanding within a given timepoint. Results support the idea of graded representations (Munakata, Trends in Cognitive Sciences, 5, 309-315, 2001.) in number word development and suggest traditional approaches to coding the give-N task may not completely capture children's knowledge.

How does improving children's ability to label set sizes without counting affect the development of understanding of the cardinality principle? It may accelerate development by facilitating subsequent alignment and comparison of the cardinal label for a given set and the last word counted when counting that set (Mix et al., 2012). Alternatively, it may delay development by decreasing the need for a comprehensive abstract principle to understand and label exact numerosities (Piantadosi et al., 2012). In this study, preschoolers (N = 106, M-age( )= 4;8) were randomly assigned to one of three conditions: (a) count-and-label, wherein children spent 6 weeks both counting and labeling sets arranged in canonical patterns like pips on a die; (b) label-first,wherein children spent the first 3 weeks learning to label the set sizes without counting before spending 3 weeks identical to the count-and-label condition; (c) print referencing control. Both counting conditions improved understanding of cardinality through increases in children's ability to label set sizes without counting. In addition to this indirect effect, there was a direct effect of the count-and-label condition on progress toward understanding of cardinality. Results highlight the roles of set labeling and equifinality in the development of children's understanding of number concepts.