Mastery of the counting principles in toddlers: A crucial step in the development of budding arithmetic abilities?

Do children need counting principle knowledge to count on their fingers?

It has been established that young children who use their fingers to solve arithmetic problems outperform those who do not. However, it remains unclear whether finger counting itself enhances arithmetic performance or if children with already advanced numerical abilities are more inclined to use this strategy. In the current study, to shed light on this matter, we observed the behavior of 189 4- and 5-year-old children in an addition task and a task assessing their knowledge of the three "how-to-count" principles (i.e., stable order, one-to-one correspondence, and cardinality principles). Of these children, 169 were reassessed 1 year later (the second testing point). At the first testing point, our results revealed that finger users better know the counting principles than non-finger users. Nevertheless, some children use their fingers without knowing the principles, but in this case they present low performance in the addition task. Moreover, we found that knowing the counting principles does not naturally prompt finger use. Finally, we did not find evidence supporting the idea that finger use has a specific role in the development of counting principles, which questions the idea that finger counting has a functional role in the construction of the number concept. All in all, our results tend to show that children need to know the counting principles to be efficient finger users. Therefore, finger counting seems to be a useful tool when used by children who already possess advanced numerical knowledge.

What aspects of counting help infants attend to numerosity?

Recent work shows that 18-month old infants understand that counting is numerically relevant-infants who see objects counted are more likely to represent the approximate number of objects in the array than infants who see the objects labeled but not counted. Which aspects of counting signal infants to attend to numerosity in this way? Here we asked whether infants rely on familiarity with the count words in their native language, or on procedures instantiated by the counting routine, independent of specific tokens. In three experiments (N = 48), we found that 18-month old infants from English-speaking households successfully distinguished four hidden objects from two when the objects were counted correctly, regardless of their familiarity with the count words (i.e., when objects were counted in familiar English and in unfamiliar German). However, when the objects were counted using familiar English count words in ways that violated basic counting principles, infants no longer represented the arrays, failing to distinguish four hidden objects from two. Together with previous findings, these results suggest that children may link the procedure of counting with numerosity years before they learn the meanings of the count words.

Counting many as one: Young children can understand sets as units except when counting

Young children frequently make a peculiar counting mistake. When asked to count units that are sets of multiple items, such as the number of families at a party, they often count discrete items (i.e., individual people) rather than the number of sets (i.e., families). One explanation concerns children's incomplete understanding of what constitutes a unit, resulting in a preference for discrete items. Here we demonstrate that children's incomplete understanding of counting also plays a role. In an experiment with 4- and 5-year-old children (N = 43), we found that even if children are able to name sets, group items into sets, and create one-to-one correspondences with sets, many children are nevertheless unable to count sets as units. We conclude that a nascent understanding of the abstraction principle of counting is also a cause of some children's counting errors.

Verbal count sequence knowledge underpins numeral order processing in children

Recent research has suggested that numeral order processing - the speed and accuracy with which individuals can determine whether a set of digits is in numerical order or not - is related to arithmetic and mathematics outcomes. It has therefore been proposed that ordinal relations are a fundamental property of symbolic numeral representations. However, order information is also inherent in the verbal count sequence, and thus verbal count sequence knowledge may instead explain the relationship between performance on numeral order tasks and arithmetic. We explored this question with 62 children aged 6- to 8-years-old. We found that performance on a verbal count sequence knowledge task explained the relationship between numeral order processing and arithmetic. Moreover many children appeared to explicitly base their judgments of numerical order on count sequence information. This suggests that insufficient attention may have been paid to verbal number knowledge in understanding the sources of information that give meaning to numbers.

The importance of visuospatial abilities for verbal number skills in preschool: Adding spatial language to the equation

Children's verbal number skills set the foundation for mathematical development. Therefore, it is central to understand their cognitive origins. Evidence suggests that preschool children rely on visuospatial abilities when solving counting and number naming tasks despite their predominantly verbal nature. We aimed to replicate these findings when controlling for verbal abilities and sociodemographic factors. Moreover, we further characterized the relation between visuospatial abilities and verbal number skills by examining the role of spatial language. Because spatial language encompasses the verbalization of spatial thinking, it is a key candidate supporting the interplay between visuospatial and verbal processes. Regression analysis indicated that both visuospatial and verbal abilities, as assessed by spatial perception and phonological awareness, respectively, uniquely predicted verbal number skills when controlling for their respective influences, age, gender, and socioeconomic status. This confirms the spatial grounding of verbal number skills. Interestingly, adding spatial language to the model abolished the predictive effects of visuospatial and verbal abilities, whose influences were completely mediated by spatial language. Verbal number skills thus concurrently depend on specifically those visuospatial and verbal processes jointly indexed through spatial language. The knowledge of spatial terms might promote verbal number skills by advancing the understanding of the spatial relations between numerical magnitudes on the mental number line. Promoting spatial language in preschool thus might be a successful avenue for stimulating mathematical development prior to formal schooling. Moreover, measures of spatial language could become an additional promising tool to screen preschool children for potential upcoming difficulties with mathematical learning.

The Implicit Contribution of Fine Motor Skills to Mathematical Insight in Early Childhood

Understanding number magnitude is an important prerequisite for children's mathematical development. One early experience that contributes to this understanding is the common practice of finger counting. Recent research suggested that through repeated finger counting, children internalize their fingers as representations of number magnitude. Furthermore, finger counting habits have been proposed to predict concurrent and future mathematical performance. However, little is known about how finger-based number representations are formed and by which processes they could influence mathematical development. Regarding the emergence of finger-based number representations, it is likely that they result from repeated practice of finger counting. Accordingly, children need sufficient fine motor skills (FMS) to successfully count on their fingers. However, the role that different types of FMS (such as dexterity and graphomotor skills) might play in the development of finger-based number representations is still unknown. In the current study, we investigated (a) whether children's FMS (dexterity and graphomotor skills) are associated with their emerging finger-based number representations (ordinal and cardinal), (b) whether FMS explain variance in children's finger-based number representations beyond the influence of general cognitive skills, and (c) whether the association between FMS and numerical skills is mediated by finger-based representations. We tested associations between preschool children's (N = 80) FMS (dexterity and graphomotor skills), finger-based number representations, and numerical skills. Furthermore, visuo-spatial working memory and nonverbal intelligence were controlled for. Dexterity was related to children's finger-based number representations as well as numerical skills after controlling for chronological age, but not after also controlling for cognitive skills. Moreover, the relationship between dexterity and numerical skills was mediated by finger-based number representations. No such associations were observed for graphomotor skills. These results suggest that dexterity plays a role in children's development of finger-based number representations, which in turn contribute to their numerical skills. Possible explanations are discussed.

A latent growth curve analysis of precursor cognitive abilities and academic achievement

BACKGROUND

Language-based cognitive fitness describes the ability to carry out cognitive tasks with vigour and alertness, to learn, and to adapt.

AIMS

The purpose of this research was to test the efficacy of a brain fitness training programme to better understand the role played by three precursor abilities for language-based cognitive development-verbal reasoning, visual synthesis, and active analysis-and their impact on academic achievement.

SAMPLE

The sample included 92 children from a private school designed to deliver a prescriptive educational model for the spectrum of challenged, average, and gifted students.

METHOD

Students completed cognitive tests at entry to the program and at the end of each academic year. Students completed daily brain-based exercises. Associative latent variable growth curve modelling using Mplus was used to test growth in all variables. Bayesian estimation analysis was used for missing data imputation. Age and baseline cognitive ability were included as covariates.

RESULTS

Model fit was excellent, χ² (104) = 115.98, p = .20, (χ² /df) = 1.11, CFI = 0.98, TLI = 0.98, RMSEA = 0.04, SRMR = 0.05. Students who started with the lowest ability levels initially grew faster than those students with higher initial two-wave performance levels consistent with a closing of an ability gap. This had a lagged effect on achievement that accelerated in growth after the first 2 years.

CONCLUSIONS

Results provide evidence for the importance of interventions that focus on strengthening precursor cognitive abilities directly involved with language and mathematics achievement.

Automatic non-symbolic numerosity processing in preschoolers

There has recently been an increasing focus on the development of automatic processing of numerical magnitude. However, little effort has been made to explore automatic access to non-symbolic numerical magnitude in preschool children. In experiment 1, we used a non-symbolic physical size comparison task in 3- to 6-year-olds to examine developmental changes and the effect of ratio and counting principle knowledge. Results showed that the existence of automatic non-symbolic numerical processing began at age 3–4 years and size congruity effects tended to reduce with increasing age from 4 years old. The study also found that non-counting-principle knowers had a larger congruity effect, and in low ratio conditions the size congruity effect was more easily found. In addition, symbolic number comparison ability was negatively related to size congruity effect. In experiment 2, we explored the relationship between inhibition skill and size congruity effects, as well as interference and facilitatory components in children aged 4 years old. Results showed no correlation between inhibition skills and the size congruity effect and only interference effects were found. We also found a larger interference effect in low ratio conditions than in high ratio conditions.

How Parents Read Counting Books and Non-numerical Books to Their Preverbal Infants: An Observational Study

Studies have stressed the importance of counting with children to promote formal numeracy abilities; however, little work has investigated when parents begin to engage in this behavior with their young children. In the current study, we investigated whether parents elaborated on numerical information when reading a counting book to their preverbal infants and whether developmental differences in numerical input exist even in the 1st year of life. Parents and their 5–10 months old infants were asked to read, as they would at home, two books to their infants: a counting book and another book that did not have numerical content. Parents’ spontaneous statements rarely focused on number and those that did consisted primarily of counting, with little emphasis on labeling the cardinality of the set. However, developmental differences were observed even in this age range, such that parents were more likely to make numerical utterances when reading to older infants. Together, results are the first to characterize naturalistic reading behaviors between parents and their preverbal infants in the context of counting books, suggesting that although counting books promote numerical language in parents, infants still receive very little in the way of numerical input before the end of the 1st year of life. While little is known regarding the impact of number talk on the cognitive development of young infants, the current results may guide future work in this area by providing the first assessment of the characteristics of parental numerical input to preverbal infants.

Early language and executive skills predict variations in number and arithmetic skills in children at family-risk of dyslexia and typically developing controls

Two important foundations for learning are language and executive skills. Data from a longitudinal study tracking the development of 93 children at family-risk of dyslexia and 76 controls was used to investigate the influence of these skills on the development of arithmetic. A two-group longitudinal path model assessed the relationships between language and executive skills at 3-4 years, verbal number skills (counting and number knowledge) and phonological processing skills at 4-5 years, and written arithmetic in primary school. The same cognitive processes accounted for variability in arithmetic skills in both groups. Early language and executive skills predicted variations in preschool verbal number skills, which in turn, predicted arithmetic skills in school. In contrast, phonological awareness was not a predictor of later arithmetic skills. These results suggest that verbal and executive processes provide the foundation for verbal number skills, which in turn influence the development of formal arithmetic skills. Problems in early language development may explain the comorbidity between reading and mathematics disorder.

Cognitive and mathematical profiles for different forms of learning difficulties

The purpose of this study was to compare subgroups of students with various forms of learning difficulties (< 25th percentile) on cognitive and mathematics characteristics. Students with mathematics difficulty (MD, n = 105), reading difficulty (RD, n = 65), both (MDRD, n = 87), or neither (NoLD, n = 403) were evaluated on an array of cognitive measures (e.g., working memory and language) and on mathematics measures of foundational numerical competencies, computation, and problem solving. Results revealed expected level differences among groups in both domains: NoLD outperformed RD, and MD outperformed MDRD. Profile differences were noted among pairs of subgroups on cognitive measures. On mathematics measures, profile differences were noted between RD and other subgroups, but not between MD and MDRD subgroups. The most discriminating cognitive measures were processing speed and language; the most discriminating mathematics measures depended on the subgroups being compared. Results were further evaluated according to more severe (< 10th percentile) criteria for MD and RD, which generally affected level differences more than the profile patterns. Results have implications for understanding comorbid MD and RD and for conceptualizing core deficits in MD.

Identifying subtypes among children with developmental coordination disorder and mathematical learning disabilities, using model-based clustering

A relationship between motor and mathematical skills has been shown by previous research. However, the question of whether subtypes can be differentiated within developmental coordination disorder (DCD) and/or mathematical learning disability (MLD) remains unresolved. In a sample of children with and without DCD and/or MLD, a data-driven model-based clustering was used to identify subgroups of individuals with relatively homogeneous profiles on measures associated with motor and mathematical skills. One subgroup of children with motor problems was found based on motor variables. Based on mathematical variables, two clinical clusters were found: a subtype with number fact retrieval problems and a subtype with procedural calculation problems. Clustering with motor and mathematical skills revealed two clinical clusters: a cluster with number fact retrieval as well as procedural calculation problems and below average motor and visual-motor integration skills. A second cluster of children had only procedural calculation and visual-motor problems. Our results raise questions about the usefulness of placing children who have below average mathematical skills into a single diagnostic category. Furthermore, we inform ongoing debates about the overlap between DCD and MLD, as below average motor skills were found in both MLD subgroups, although a different motor profile is linked to a different mathematical profile.

Refining the quantitative pathway of the Pathways to Mathematics model

In the current study, we adopted the Pathways to Mathematics model of LeFevre et al. (2010). In this model, there are three cognitive domains--labeled as the quantitative, linguistic, and working memory pathways--that make unique contributions to children's mathematical development. We attempted to refine the quantitative pathway by combining children's (N=141 in Grades 2 and 3) subitizing, counting, and symbolic magnitude comparison skills using principal components analysis. The quantitative pathway was examined in relation to dependent numerical measures (backward counting, arithmetic fluency, calculation, and number system knowledge) and a dependent reading measure, while simultaneously accounting for linguistic and working memory skills. Analyses controlled for processing speed, parental education, and gender. We hypothesized that the quantitative, linguistic, and working memory pathways would account for unique variance in the numerical outcomes; this was the case for backward counting and arithmetic fluency. However, only the quantitative and linguistic pathways (not working memory) accounted for unique variance in calculation and number system knowledge. Not surprisingly, only the linguistic pathway accounted for unique variance in the reading measure. These findings suggest that the relative contributions of quantitative, linguistic, and working memory skills vary depending on the specific cognitive task.

Mathematical problems in children with developmental coordination disorder

Developmental coordination disorder (DCD) is a heterogeneous disorder, which is often co-morbid with learning disabilities. However, mathematical problems have rarely been studied in DCD. The aim of this study was to investigate the mathematical problems in children with various degrees of motor problems. Specifically, this study explored if the development of mathematical skills in children with DCD is delayed or deficient. Children with DCD performed significantly worse for number fact retrieval and procedural calculation in comparison with age-matched control children. Moreover, children with mild DCD differed significantly from children with severe DCD on both number fact retrieval and procedural calculation. In addition, we found a developmental delay of 1 year for number fact retrieval in children with mild DCD and a developmental delay of 2 years in children with severe DCD. No evidence for a mathematical deficit was found. Diagnostic implications are discussed.

Early numeracy in cerebral palsy: review and future research

Children with cerebral palsy (CP) often have problems with arithmetic, but the development of numerical abilities in these children has received only minor attention. In comparison, detailed accounts have been written on the arithmetic abilities of typically developing children, but a theoretical framework is still lacking. A promising perspective is the embodied cognition framework, which focuses on the influence of perception and action behaviours on cognition. We searched the literature to find the available studies on the early numeracy capacities of children with CP. We reviewed eight studies in which primary school-aged children with CP with a verbal IQ of at least 70 participated. The selected studies showed that these children are regularly delayed in performing simple arithmetic operations compared with their typically developing peers. However, owing to the limited number of studies no definite conclusions can be drawn regarding the precursors and developmental trajectories of arithmetic abilities in children with CP. We argue that the embodied cognition framework is well suited to scrutinize the arithmetic abilities of children with CP and provide future directions for research.

Development of the cardinality principle in Macedonian preschool children

Cardinality principle refers to the fact that the last number tag used in counting determines the cardinality of a set. Macedonian kindergarten children were tested with the give-a-number task for understanding of this principle. It was found that Macedonian children, unlike their western counterparts, pass through an additional stage, 5-knowers, before they master the cardinality principle. Also, the age at which they pass through the individual stages is somewhat higher than the age of children coming from western samples. Possible explanations are offered and discussed.