Individual differences in children's approximations of area correlate with competence in basic geometry

Predicting individual differences in preschoolers' numeracy and geometry knowledge: The role of understanding abstract relations between objects and quantities

Preschoolers' mathematics knowledge develops early and varies substantially. The current study focused on two ontogenetically early emerging cognitive skills that may be important predictors of later math skills (i.e., geometry and numeracy): children's understanding of abstract relations between objects and quantities as evidenced by their patterning skills and the approximate number system (ANS). Children's patterning skills, the ANS, numeracy, geometry, nonverbal intelligence (IQ), and executive functioning (EF) skills were assessed at age 4 years, and their numeracy and geometry knowledge was assessed again a year later at age 5 (N = 113). Above and beyond children's initial knowledge in numeracy and geometry, as well as IQ and EF, patterning skills and the ANS at age 4 uniquely predicted children's geometry knowledge at age 5, but only age 4 patterning uniquely predicted age 5 numeracy. Thus, although patterning and the ANS are related, they differentially explain variation in later geometry and numeracy knowledge. Results are discussed in terms of implications for early mathematics theory and research.

Geometry in the brain optimized for sign language - A unique role of the anterior superior parietal lobule in deaf signers

Geometry has been identified as a cognitive domain where deaf individuals exhibit relative strength, yet the neural mechanisms underlying geometry processing in this population remain poorly understood. This fMRI study aimed to investigate the neural correlates of geometry processing in deaf and hearing individuals. Twenty-two adult deaf signers and 25 hearing non-signers completed a geometry decision task. We found no group differences in performance, while there were some differences in parietal activation. As expected, the posterior superior parietal lobule (SPL) was recruited for both groups. The anterior SPL was significantly more activated in the deaf group, and the inferior parietal lobule was significantly more deactivated in the hearing group. In conclusion, despite similar performance across groups, there were differences in the recruitment of parietal regions. These differences may reflect inherent differences in brain organization due to different early sensory and linguistic experiences.

Visual perception and linguistic abilities, not quantitative knowledge, count in geometric knowledge of kindergarten children

Geometric knowledge is one of the important mathematical skills acquired by children at a young age and is a major area of future mathematical learning; however, there is no direct research on the factors influencing kindergarteners' early geometric knowledge. The pathways model to mathematics was modified to examine the cognitive mechanisms underlying geometric knowledge in Chinese kindergarten children aged 5-7 (n = 99). Quantitative knowledge, visual-spatial processing, and linguistic abilities were stepped into hierarchical multiple regression models. The results revealed that after age, sex, and nonverbal intelligence were statistically controlled, visual perception, phonological awareness, and rapid automatized naming in linguistic abilities significantly predicted the variation in geometric knowledge. For quantitative knowledge, neither dot comparison nor number comparison test could be a significant precursor of geometry skills. The findings indicate that visual perception and linguistic abilities, not quantitative knowledge, account for the geometric knowledge of kindergarten children.

Examining the relations between spatial skills and mathematical performance: A meta-analysis

Much recent research has focused on the relation between spatial skills and mathematical skills, which has resulted in widely reported links between these two skill sets. However, the magnitude of this relation is unclear. Furthermore, it is of interest whether this relation differs in size based on key demographic variables, such as gender and grade-level, and the extent to which this relation can be accounted for by shared domain-general reasoning skills across the two domains. Here we present the results of two meta-analytic studies synthesizing the findings from 45 articles to identify the magnitude of the relation, as well as potential moderators and mediators. The first meta-analysis employed correlated and hierarchical effects meta-regression models to examine the magnitude of the relation between spatial and mathematical skills, and to understand the effect of gender and grade-level on the association. The second meta-analysis employed meta-analytic structural equation modeling to determine how domain-general reasoning skills, specifically fluid reasoning and verbal skills, influence the relationship. Results revealed a positive moderate association between spatial and mathematical skills (r = .36, robust standard error = 0.035, τ² = 0.039). However, no significant effect of gender or grade-level on the association was found. Additionally, we found that fluid reasoning and verbal skills mediated the relationship between spatial skills and mathematical skills, but a unique relation between the spatial and mathematical skills remained. Implications of these findings include advancing our understanding for how to leverage and bolster students' spatial skills as a mechanism for improving mathematical outcomes.

Training nonsymbolic proportional reasoning in children and its effects on their symbolic math abilities

Our understanding of proportions can be both symbolic, as when doing calculations in school mathematics, or intuitive, as when folding a bed sheet in half. While an understanding of symbolic proportions is crucial for school mathematics, the cognitive foundations of this ability remain unclear. Here we implemented a computerized training game to test a causal link from intuitive (nonsymbolic) to symbolic proportional reasoning and other math abilities in 4th grade children. An experimental group was trained in nonsymbolic proportional reasoning (PR) with continuous extents, and an active control group was trained on a remarkably similar nonsymbolic magnitude comparison. We found that the experimental group improved at nonsymbolic PR across training sessions, showed near transfer to a paper-and-pencil nonsymbolic PR test, transfer to symbolic proportions, and far transfer to geometry. The active control group showed only a predicted far transfer to geometry. In a second experiment, these results were replicated with an independent cohort of children. Overall this study extends previous correlational evidence, suggesting a functional link between nonsymbolic PR on one hand and symbolic PR and geometry on the other.

Discrimination of Small Forms in a Deviant-Detection Paradigm by 10-month-old Infants

Using eye tracking, we investigated if 10-month-old infants could discriminate between members of a set of small forms based on geometric properties in a deviant-detection paradigm, as suggested by the idea of a core cognitive system for Euclidian geometry. We also investigated the precision of infants' ability to discriminate as well as how the discrimination process unfolds over time. Our results show that infants can discriminate between small forms based on geometrical properties, but only when the difference is sufficiently large. Furthermore, our results also show that it takes infants, on average, <3.5 s to detect a deviant form. Our findings extend previous research in three ways: by showing that infants can make similar discriminative judgments as children and adults with respect to geometric properties; by providing a first crude estimate on the limit of the discriminative abilities in infants, and finally; by providing a first demonstration of how the discrimination process unfolds over time.

Number, time, and space are not singularly represented: Evidence against a common magnitude system beyond early childhood

Our ability to represent temporal, spatial, and numerical information is critical for understanding the world around us. Given the prominence of quantitative representations in the natural world, numerous cognitive, neurobiological, and developmental models have been proposed as a means of describing how we track quantity. One prominent theory posits that time, space, and number are represented by a common magnitude system, or a common neural locus (i.e., Bonn & Cantlon in Cognitive Neuropsychology, 29(1/2), 149-173, 2012; Cantlon, Platt, & Brannon in Trends in Cognitive Sciences, 13(2), 83-91, 2009; Meck & Church in Animal Behavior Processes, 9(3), 320, 1983; Walsh in Trends in Cognitive Sciences, 7(11), 483-488, 2003). Despite numerous similarities in representations of time, space, and number, an increasing body of literature reveals striking dissociations in how each quantity is processed, particularly later in development. These findings have led many researchers to consider the possibility that separate systems may be responsible for processing each quantity. This review will analyze evidence in favor of a common magnitude system, particularly in infancy, which will be tempered by counter evidence, the majority of which comes from experiments with children and adult participants. After reviewing the current data, we argue that although the common magnitude system may account for quantity representations in infancy, the data do not provide support for this system throughout the life span. We also identify future directions for the field and discuss the likelihood of the developmental divergence model of quantity representation, like that of Newcombe (Ecological Psychology, 2, 147-157, 2014), as a more plausible account of quantity development.

Development of early numerical abilities of Spanish-speaking Mexican preschoolers: A new assessment tool

This article presents a tool for assessing the early numerical abilities of Spanish-speaking Mexican preschoolers. The Numerical Abilities Test, from the Evaluación Neuropsicológica Infantil-Preescolar (ENI-P), evaluates four core abilities of number development: magnitude comparison, counting, subitizing, and basic calculation. We evaluated 307 Spanish-speaking Mexican children aged 2 years 6 months to 4 years 11 months. Appropriate internal consistency and test-retest reliability were demonstrated. We also investigated the effect of age, children's school attendance, maternal education, and sex on children's numerical scores. The results showed that the four subtests captured development across ages. Critically, maternal education had an impact on children's performance in three out of the four subtests, but there was no effect associated with children's school attendance or sex. These results suggest that the Numerical Abilities Test is a reliable instrument for Spanish-speaking preschoolers. We discuss the implications of our outcomes for numerical development.

Representations of numerical and non-numerical magnitude both contribute to mathematical competence in children

A growing body of evidence suggests that non-symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non-numerical magnitudes not only share analog format but also form part of a general magnitude system raises questions about whether the non-symbolic basis of mathematical thinking is unique to numerical magnitude. Here we examined this issue in 5- and 6-year-old children using comparison tasks of non-symbolic number arrays and cumulative area as well as standardized tests of math competence. One set of findings revealed that scores on both magnitude comparison tasks were modulated by ratio, consistent with shared analog format. Moreover, scores on these tasks were moderately correlated, suggesting overlap in the precision of numerical and non-numerical magnitudes, as expected under a general magnitude system. Another set of findings revealed that the precision of both types of magnitude contributed shared and unique variance to the same math measures (e.g. calculation and geometry), after accounting for age and verbal competence. These findings argue against an exclusive role for non-symbolic number in supporting early mathematical understanding. Moreover, they suggest that mathematical understanding may be rooted in a general system of magnitude representation that is not specific to numerical magnitude but that also encompasses non-numerical magnitude.

Children and Adults Use Physical Size and Numerical Alliances in Third-Party Judgments of Dominance

Humans and other social animals interact regularly with conspecifics as part of affiliative groups. Many of these interactions are cooperative, but many others involve competition for resources. Competitive exchanges are often resolved on the basis of dominance relationships, with higher-ranking individuals receiving priority access to desired goods. Although no single cue can establish permanent dominance relationships, there are some cues that predict dominance fairly reliably across context. In the present study, we focused on two such cues relevant to competing groups: (i) the physical sizes of individual members, and (ii) their relative number. Using a social competition task, we examined whether, and how, preschool-aged children and adults used differences in physical size and numerical alliances to judge which of two groups should prevail in a competitive exchange for a desired object. These judgments were made when either physical size or number differed between groups (Experiment 1), and when both were available but pitted against each other (Experiments 1 and 2). Our findings revealed that by 3 years of age, humans use multiple perceptible cues in third-party judgments of dominance. Our findings also revealed that 3-year-olds, like adults, weighted these cues flexibly according to the additional factor of overall group size, with the physical sizes of individuals determining dominance in smaller groups (e.g., 2 vs. 4 characters) and the relative number of individuals determining dominance in larger groups (e.g., 15 vs. 30 characters). Taken together, our findings suggest that a basic formula for determining dominance in competitive exchanges, which weights physical size of individuals and numerical alliances as a function of overall group size, is available to young children and appears fairly stable through to adulthood.