Munez, David

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.

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.

The nature of the arithmetic word problems found in maths textbooks influences the way students develop their ability to solve them, as teachers use the books in their classes quite frequently. Given that students are better able to reason through and solve authentic problems that are contextualized in situations familiar to them, and that different international assessments have shown the students in Singapore to be more skilled at problem-solving than their Spanish counterparts, this study compares the level of authenticity of the problems included in the primary school textbooks from the most frequently used publishers in both countries. The results show that the books from Singapore contain problems with a higher degree of authenticity than the Spanish textbooks at all school levels, thus providing students with better opportunities to learn how to solve problems through reasoning.

Existing research has mainly examined the role of cognitive correlates of early reading and mathematics from a stationary perspective that does not consider how these skills unfold and interact over time. This approach constraints the interpretation of cross-domain associations and the specificity of domain-specific covariates. In this study, we disentangle the role of these predictors and investigate cross-domain associations between reading, math, and two related domain-specific predictors (phonological awareness and fluency with number sets) over the kindergarten years (n = 512, Mage = 54 months, SDage = 3.5, 52% females). Results reveal that the overlap between reading and math skills changes over development. Reciprocal associations between reading and math abilities are observed at earlier stages; then, reading abilities become the lead force. Findings also show that phonological awareness and fluency with number sets are domain-specific predictors that do not contribute to cross-domain gains in academic skills. Indeed, there is a trend for domain-specific skills to be more strongly related to achievement at the beginning of formal education than at the beginning of kindergarten, which suggests an increasing differentiation of domains over the kindergarten years. Such findings have implications for the timing and nature of interventions that aim to support children's reading and mathematical development.

We explored the challenges, limitations, and potential effectiveness of a large-scale computerized working memory and numeracy intervention in the classroom with children at risk of mathematical learning disabilities (n = 428, Mage = 83.85 months, 41% female). Children were assigned to four different treatment protocols (working memory [WM], working memory plus numeracy [NWM], numeracy [NUM], and active control [AC]) that were implemented as part of normally scheduled class activities for 1 year. Wide variability in training exposure highlighted the challenges of implementing an ecologically valid large-scale classroom intervention. The NUM and NWM intervention contributed to improvements in various early numeracy skills as well as math achievement after accounting for training exposure. Some of these effects emerged once the intervention concluded. However, the intervention failed to improve WM, which was likely due to insufficient training dosage in the practical setting. Findings suggest that combining both working memory and numerical skills training is worth further investigation. The study also provides evidence of challenges related to the implementation of training programs in real-life learning environments.

In this study (n = 1000, M_{age at K1entry} = 53.4 months, SD = 3.4; 53% females), we investigated the contributions of the family socioeconomic status (SES; maternal education and an income-related measure) and number and age of siblings to the development of children's math, reading, and working memory (WM) updating skills over the kindergarten years. Results from a multivariate multilevel growth curve model showed that children from more disadvantaged SES backgrounds already had a multifaceted developmental lag at kindergarten entry. Maternal education was the aspect of SES that more clearly affected the child's cognitive development; the mother's education predicted children's math, reading, and WM-updating skills at kindergarten entry as well as the rate of development of reading skills over the kindergarten years. Independently of SES status, children with more siblings also showed poorer reading and math skills than those in one-child families at kindergarten entry. We also found that both older and younger siblings affected, negatively, children's reading skills before they attended kindergarten—which suggests that the development of reading skills is more responsive to environmental factors during the first years than other aspects of the child development. The findings underscore the independent role of siblings upon entry to kindergarten, and the enduring role of maternal education even after children are exposed to formal schooling.

Recent years have witnessed an increase in research on how numeral ordering skills relate to children’s and adults’ mathematics achievement both cross-sectionally and longitudinally. Nonetheless, it remains unknown which core competency numeral ordering tasks measure, which cognitive mechanisms underlie performance on these tasks, and why numeral ordering skills relate to arithmetic and math achievement. In the current study, we focused on the processes underlying decision-making in the numeral order judgement task with triplets to investigate these questions. A drift-diffusion model for two-choice decisions was fit to data from 97 undergraduates. Findings aligned with the hypothesis that numeral ordering skills reflected the operationalization of the numerical system, where small numbers provide more evidence of an ordered response than large numbers. Furthermore, the pattern of findings suggested that arithmetic achievement was associated with the accuracy of the ordinal representations of numbers.

The acquisition of cardinal numbers represents a crucial milestone in the development of early numerical skills and more advanced math abilities. However, relatively few studies have investigated how children's grasping of the cardinality principle can be supported. It has been suggested that the richness of number inputs children receive influences the acquisition of cardinal numbers. The present study was designed to investigate whether canonical finger patterns representing numbers may contribute to this acquisition. Fifty-one 3-year-olds were randomly assigned to 1 of 2 training conditions: (a) a condition that involved counting and labeling, which has shown efficacy to support the acquisition of cardinality, and (b) a condition in which counting and labeling were enriched with finger patterns. Crucially, we aimed at providing evidence of both training programs in a real-life learning environment where teachers incorporated the training as a group-based activity into their regular schedule of daily activities. Children assigned to the finger-based condition outperformed those who received the counting-and-label training. Findings suggest that finger patterns may have a role in children's cardinality understanding. Furthermore, our study shows that instructional approaches for improving cardinality understanding can be easily and successfully implemented into real-life learning settings.

The success or failure of education systems in promoting student problem-solving skills depends on attitudinal, political, and pedagogical variables. Among these variables, the design of mathematics textbooks is thought to partially explain why students from high-achieving countries show better problem-solving ability in international assessments. In the current study, we delved into this question and compared the frequency and characteristics of arithmetic word problems (AWPs) contained in primary school math textbooks in two countries with different levels of performance in international assessments—Singapore and Spain. In our analyses, we focused on (1) the quantity of arithmetic word problems, (2) the variety of problems in terms of their additive or multiplicative structures and semantic-mathematical substructures, and (3) the quantity and nature of illustrations that were presented together with arithmetic word problems. Although a larger proportion of AWP activities was found in Singaporean textbooks, the results showed a similar variety of AWPs in both Singaporean and Spanish math textbooks. Furthermore, in both countries, math textbooks emphasized the structures classified as (additive) combine 1 and (multiplication) simple rate in AWPs. Notably, the Singaporean textbook contained a larger percentage of illustrations that reflected the semantic-mathematical structures of the problems and helped students learn how to solve AWPs (e.g., bar models). The findings are discussed in light of theories that posit that textbooks constitute a fundamental part of the teaching–learning process in the classroom.