Cheung, Pierina

Arithmetic skills – the ability to add, subtract, multiply, and divide – are the building blocks of mathematics. Poor arithmetic skills can lead to poor job prospects and life outcomes. It is thus important to investigate the development of arithmetic skills. What constitute the foundations for arithmetic skills? When do they develop? Previous studies have highlighted the importance of the toddler and preschool period as providing foundations for later math learning. In this chapter, we provide an overview of key factors across domain-specific and domain-general areas that support the development of arithmetic skills. We then draw on existing data from the Singapore Kindergarten Impact Project (SKIP) and describe the performance of basic numeracy skills at entry to kindergarten that are relevant for arithmetic learning. These skills include counting, informal arithmetic, and the reading and writing of Arabic digits. Finally, we conclude with guidelines for promoting the development of early mathematical knowledge in the classroom and at home.

Mastery motivation predicts achievement, but intricacies amongst pre-schoolers are unclear. In keeping with the Specificity Principle, school-age, and adolescent research demonstrates the importance of considering the setting conditions in which mastery motivation is observed. Here, Singaporean 4-year-olds’ (N = 63) mastery-motivation-related behaviour (MMRB) (e.g. signs of persistence, focus, and pleasure) in mathematical and non-mathematical activities were observed. Relations between numeracy and MMRB during a mathematical game (outcome relevant setting) were determined, controlling for MMRB in other activities (outcome irrelevant settings). Association between MMRB during the mathematical game and receptive language (outcome irrelevant setting) was also examined. Consistent with the Specificity Principle, MMRB during the mathematical game was (i) associated with numeracy, after controlling for MMRB in other activities and (ii) did not predict language. Enhancing preschoolers’ experiences, especially when implemented in contexts related to areas targeted for improvement, may benefit outcomes. These skills acquired in early life can become important predictors of future ability.

Although it is thought that young children focus on the magnitude of the target dimension across ratio sets during binary comparison of ratios, it is unknown whether this is the default approach to ratio reasoning, or if such approach varies across representation formats (discrete entities and continuous amounts) that naturally afford different opportunities to process the dimensions in each ratio set. In the current study, 132 kindergarteners (Mage = 68 months, SD = 3.5, range = 62–75 months) performed binary comparisons of ratios with discrete and continuous representations. Results from a linear mixed model revealed that children followed an additive strategy to ratio reasoning—i.e., they focused on the magnitude of the target dimension across ratio sets as well as on the absolute magnitude of the ratio set. This approach did not vary substantially across representation formats. Results also showed an association between ratio reasoning and children’s math problem-solving abilities; children with better math abilities performed better on ratio reasoning tasks and processed additional dimensions across ratio sets. Findings are discussed in terms of the processes that underlie ratio reasoning and add to the extant debate on whether true ratio reasoning is observed in young children.

Studies on children's understanding of counting examine when and how children acquire the cardinal principle: the idea that the last word in a counted set reflects the cardinal value of the set. Using Wynn's (1990) Give-N Task, researchers classify children who can count to generate large sets as having acquired the cardinal principle (cardinal-principle-knowers) and those who cannot as lacking knowledge of it (subset-knowers). However, recent studies have provided a more nuanced view of number word acquisition. Here, we explore this view by examining the developmental progression of the counting principles with an aim to elucidate the gradual elements that lead to children successfully generating sets and being classified as CP-knowers on the Give-N Task. Specifically, we test the claim that subset-knowers lack cardinal principle knowledge by separating children's understanding of the cardinal principle from their ability to apply and implement counting procedures. We also ask when knowledge of Gelman & Gallistel's (1978) other how-to-count principles emerge in development. We analyzed how often children violated the three how-to-count principles in a secondary analysis of Give-N data (N = 86). We found that children already have knowledge of the cardinal principle prior to becoming CP-knowers, and that understanding of the stable-order and word-object correspondence principles likely emerged earlier. These results suggest that gradual development may best characterize children's acquisition of the counting principles and that learning to coordinate all three principles represents an additional step beyond learning them individually.

Children learn the first few number words one at a time, before learning the meaning of counting. Past research, however, has focused on monolingual children, with only a handful of studies on bi/multilingual children. The current study investigated whether knowledge of number words in one language predicted knowledge of those same number words in another language. We studied Mandarin-, Malay-, and Tamil-English bilingual children (N = 131, ages 2–5) growing up in Singapore, a multilingual and multicultural society, and found that their number knowledge was strongly correlated between languages. Contrary to previous studies in the U.S., we also found some evidence that meanings for small number words are transferred across languages. Our findings point to a possibility that early number words can be learned as translation equivalents (e.g., “one” in English is “yi” in Mandarin Chinese) in bilingual children and highlight the importance of taking an inclusive, global approach to studying numerical development that goes beyond studying White monolingual children.

Very large numbers words such as “hundred,” “thousand,” “million,” “billion,” and “trillion” pose a learning problem for children because they are sparse in everyday speech and children's experience with extremely large quantities is scarce. In this study, we examine when children acquire the relative ordering of very large number words as a first step toward understanding their acquisition. In Study 1, a hundred and twenty-five 5–8-year-olds participated in a verbal number comparison task involving very large number words. We found that children can judge which of two very large numbers is more as early as age 6, prior to entering first grade. In Study 2, we provided a descriptive analysis on the usage of very large number words using the CHILDES database. We found that the relative frequency of large number words does not change across the years, with “hundred” uttered more frequently than others by an order of magnitude. We also found that adults were more likely to use large number words to reference units of quantification for money, weight, and time, than for discrete, physical entities. Together, these results show that children construct a numerical scale for large number words prior to learning their precise cardinal meanings, and highlight how frequency and context may support their acquisition. Our results have pedagogical implications and highlight a need to investigate how children acquire meanings for number words that reference quantities beyond our everyday experience.

We suggest that prior to school entry, our earliest “teachers” and “learning settings” —that is, our parents, caregivers, and homes—provide signals about our environmental conditions. In turn, our brains may interpret this information as cues indicating the types of environments we will likely face and adapt accordingly. We discuss ways in which two such early-life cues—bilingual exposure and sensitive caregiving quality, influence “domain general” neurocircuitry and associated functioning (e.g., temperament and emotional reactivity, emotion regulation, relational memory, exploratory play, and executive functioning), as well as pre-academic outcomes. We conclude by discussing the need for early upstream intervention programmes, as well as the need for additional research including our upcoming “BE POSITIVE” study, designed to help bridge the gap between the community, home, and school environments.

Number books are an effective tool to increase math talk and promote children's mathematical thinking, but little is known about book genre effects. How do different types of number books impact mathematical input? This study examined whether book genre affects the amount and types of teachers' math talk. Using a within-subjects design, we asked 31 teachers of 4- to 5-year-olds to read a narrative number book and a nonnarrative number book in a counterbalanced order. The books were matched in target numerical content. We found that the nonnarrative number book elicited more math talk than the narrative number book. In a qualitative analysis, we analyzed multiple conversational turns and found that both books allowed teachers to engage in simple math discussions, but there were differences in other types of math discussions. Together, these findings show that the quantity and quality of mathematical input differ as a function of genre.