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AbstractVery 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.

AbstractIn the present study we examined whether children with Developmental Dyscalculia (DD) exhibit a deficit in the so‐called 'Approximate Number System' (ANS). To do so, we examined a group of elementary school children who demonstrated persistent low math achievement over 4 years and compared them to typically developing (TD), aged‐matched controls. The integrity of the ANS was measured using the Panamath (www.panamath.org) non‐symbolic numerical discrimination test. Children with DD demonstrated imprecise ANS acuity indexed by larger Weber fraction (w) compared to TD controls. Given recent findings showing that non‐symbolic numerical discrimination is affected by visual parameters, we went further and investigated whether children performed differently on trials on which number of dots and their overall area were either congruent or incongruent with each other. This analysis revealed that differences in w were only found between DD and TD children on the incongruent trials. In addition, visuo‐spatial working memory strongly predicts individual differences in ANS acuity (w) during the incongruent trials. Thus the purported ANS deficit in DD can be explained by a difficulty in extracting number from an array of dots when area is anti‐correlated with number. These data highlight the role of visuo‐spatial working memory during the extraction process, and demonstrate that close attention needs to be paid to perceptual processes invoked by tasks thought to represent measures of the ANS.

AbstractResearch demonstrating that infants discriminate between small (e.g., 1 vs. 3 dots) and large numerosities (e.g., 8 vs. 16 dots) is central to theories concerning the origins of human numerical abilities. To date, there has been no quantitative meta‐analysis of the infant numerical competency data. Here, we quantitatively synthesize the evidential value of the available literature on infant numerosity discrimination using a meta‐analytic tool called p‐curve. In p‐curve the distribution of available p‐values is analyzed to determine whether the published literature examining particular hypotheses contains evidential value. p‐curves demonstrated evidential value for the hypotheses that infants can discriminate between both small and large unimodal and cross‐modal numerosities. However, the analyses also revealed that the published data on infants' ability to discriminate between large numerosities is less robust and statistically powered than the data on their ability to discriminate small numerosities. We argue there is a need for adequately powered replication studies to enable stronger inferences in order to use infant data to ground theories concerning the ontogenesis of numerical cognition.

Abstract The numerical distance effect (inverse relationship between numerical distance and reaction time in relative number comparison tasks) has frequently been used to characterize the mental representation of number. The size of the distance effect decreases over developmental time. However, it is unclear whether this reduction simply reflects developmental changes in domain‐general speed of processing and whether it is specific to numerical compared with non‐numerical magnitude. To examine these open questions, we conducted a cross‐sectional study with 6‐, 7‐, and 8‐year‐old children as well as adult college students. Participants performed comparisons on Arabic numerals, arrays of squares, squares of varying luminance and bars of varying height. To control for general age‐related changes in reaction time, a measure of speed of processing was used as a covariate in the analysis. A significant developmental decrease in the distance effect was found across numerical and non‐numerical comparison tasks over and above general changes in processing speed. However, this change was not found to differ as a function of format. These data suggest that developmental changes in the distance effect are reflective of changes in a domain‐general comparison process, rather than domain‐specific developmental changes in number representations. However, analysis of overall reaction times revealed significantly greater developmental changes for numerical relative to non‐numerical comparison tasks. These findings highlight the importance of taking multiple measures into account when characterizing developmental changes in numerical magnitude processing. Implications for theories of numerical cognition and its development are discussed.

Abstract
How are different formats of magnitudes represented in the human brain? We used functional magnetic resonance imaging adaptation to isolate representations of symbols, quantities, and physical size in 45 adults. Results indicate that the neural correlates supporting the passive processing of number symbols are largely dissociable from those supporting quantities and physical size, anatomically and representationally. Anatomically, passive processing of quantities and size correlate with activation in the right intraparietal sulcus, whereas symbolic number processing, compared with quantity processing, correlates with activation in the left inferior parietal lobule. Representationally, neural patterns of activation supporting symbols are dissimilar from neural activation patterns supporting quantity and size in the bilateral parietal lobes. These findings challenge the longstanding notion that the culturally acquired ability to conceptualize symbolic numbers is represented using entirely the same brain systems that support the evolutionarily ancient system used to process quantities. Moreover, these data reveal that regions that support numerical magnitude processing are also important for the processing of non-numerical magnitudes. This discovery compels future investigations of the neural consequences of acquiring knowledge of symbolic numbers.

AbstractWhile there is evidence for an association between the development of reading and arithmetic, the precise locus of this relationship remains to be determined. Findings from cognitive neuroscience research that point to shared neural correlates for phonological processing and arithmetic as well as recent behavioral evidence led to the present hypothesis that there exists a highly specific association between phonological awareness and single‐digit arithmetic with relatively small problem sizes. The present study examined this association in 37 typically developing fourth and fifth grade children. Regression analyses revealed that phonological awareness was specifically and uniquely related to arithmetic problems with a small but not large problem size. Further analysis indicated that problems with a high probability of being solved by retrieval, but not those typically associated with procedural problem‐solving strategies, are correlated with phonological awareness. The specific association between phonological awareness and arithmetic problems with a small problem size and those for which a retrieval strategy is most common was maintained even after controlling for general reading ability and phonological short‐term memory. The present findings indicate that the quality of children's long‐term phonological representations mediates individual differences in single‐digit arithmetic, suggesting that more distinct long‐term phonological representations are related to more efficient arithmetic fact retrieval.

Abstract Previous studies have suggested that typically developing 6‐month‐old infants are able to discriminate between small and large numerosities. However, discrimination between small numerosities in young infants is only possible when variables continuous with number (e.g. area or circumference) are confounded. In contrast, large number discrimination is successful even when variables continuous with number are systematically controlled for. These findings suggest the existence of different systems underlying small and large number processing in infancy. How do these develop in atypical syndromes? Williams syndrome (WS) is a rare neurocognitive developmental disorder in which numerical cognition has been found to be impaired in older children and adults. Do impairments of number processing have their origins in infancy? Here this question is investigated by testing the small and large number discrimination abilities of infants and toddlers with WS. While infants with WS were able to discriminate between 2 and 3 elements when total area was confounded with numerosity, the same infants did not discriminate between 8 and 16 elements, when number was not confounded with continuous variables. These findings suggest that a system for tracking the features of small numbers of object (object‐file representation) may be functional in WS, while large number discrimination is impaired from an early age onwards. Finally, we argue that individual differences in large number processing in infancy are more likely than small number processing to be predictive of later development of numerical cognition.

AbstractMath relies on mastery and integration of a wide range of simpler numerical processes and concepts. Recent work has identified several numerical competencies that predict variation in math ability. We examined the unique relations between eight basic numerical skills and early arithmetic ability in a large sample (N = 1391) of children across grades 1–6. In grades 1–2, children's ability to judge the relative magnitude of numerical symbols was most predictive of early arithmetic skills. The unique contribution of children's ability to assess ordinality in numerical symbols steadily increased across grades, overtaking all other predictors by grade 6. We found no evidence that children's ability to judge the relative magnitude of approximate, nonsymbolic numbers was uniquely predictive of arithmetic ability at any grade. Overall, symbolic number processing was more predictive of arithmetic ability than nonsymbolic number processing, though the relative importance of symbolic number ability appears to shift from cardinal to ordinal processing.

AbstractWhich dimension of a set of objects is more salient to young children: number or size? The 'Build‐A‐Train' task was developed and used to examine whether children spontaneously use a number or physical size approach on an un‐cued matching task. In the Build‐A‐Train task, an experimenter assembles a train using one to five blocks of a particular length and asks the child to build the same train. The child's blocks differ in length from the experimenter's blocks, causing the child to build a train that matches based on either the number of blocks or length of the train, as it is not possible to match on both. One hundred and nineteen children between 2 years 2 months and 6 years 0 months of age (M = 4.05, SD = 0.84) completed the Build‐A‐Train task, and the Give‐a‐Number task, a classic task used to assess children's conceptual knowledge of verbal number words. Across train lengths and verbal number knowledge levels, children used a number approach more than a size approach on the Build‐A‐Train task. However, children were especially likely to use a number approach over a size approach when they knew the verbal number word that corresponded to the quantity of blocks in the train, particularly for quantities smaller than four. Therefore, children's attention to number relates to their knowledge of verbal number words. The Build‐A‐Train task and findings from the current study set a foundation for future longitudinal research to investigate the causal relationship between children's acquisition of symbolic mathematical concepts and attention to number.

AbstractA long‐standing debate in the field of numerical cognition concerns the degree to which symbolic and non‐symbolic processing are related over the course of development. Of particular interest is the possibility that this link depends on the range of quantities in question. Behavioral and neuroimaging research with adults suggests that symbolic and non‐symbolic quantities may be processed more similarly within, relative to outside of, the subitizing range. However, it remains unclear whether this unique link exists in young children at the outset of formal education. Further, no study has yet taken numerical size into account when investigating the longitudinal influence of these skills. To address these questions, we investigated the relation between symbolic and non‐symbolic processing inside versus outside the subitizing range, both cross‐sectionally and longitudinally, in 540 kindergarteners. Cross‐sectionally, we found a consistently stronger relation between symbolic and non‐symbolic number processing within versus outside the subitizing range at both the beginning and end of kindergarten. We also show evidence for a bidirectional relation over the course of kindergarten between formats within the subitizing range, and a unidirectional relation (symbolic → non‐symbolic) for quantities outside of the subitizing range. These findings extend current theories on symbolic and non‐symbolic magnitude development by suggesting that non‐symbolic processing may in fact play a role in the development of symbolic number abilities, but that this influence may be limited to quantities within the subitizing range.

AbstractChildren's ability to discriminate nonsymbolic number (e.g., the number of items in a set) is a commonly studied predictor of later math skills. Number discrimination improves throughout development, but what drives this improvement is unclear. Competing theories suggest that it may be due to a sharpening numerical representation or an improved ability to pay attention to number and filter out non‐numerical information. We investigate this issue by studying change in children's performance (N = 65) on a nonsymbolic number comparison task, where children decide which of two dot arrays has more dots, from the middle to the end of 1st grade (mean age at time 1 = 6.85 years old). In this task, visual properties of the dot arrays such as surface area are either congruent (the more numerous array has more surface area) or incongruent. Children rely more on executive functions during incongruent trials, so improvements in each congruency condition provide information about the underlying cognitive mechanisms. We found that accuracy rates increased similarly for both conditions, indicating a sharpening sense of numerical magnitude, not simply improved attention to the numerical task dimension. Symbolic number skills predicted change in congruent trials, but executive function did not predict change in either condition. No factor predicted change in math achievement. Together, these findings suggest that nonsymbolic number processing undergoes development related to existing symbolic number skills, development that appears not to be driving math gains during this period.

Children's ability to discriminate nonsymbolic number improves throughout development. Competing theories suggest improvement due to sharpening magnitude representations or changes in attention and inhibition.
The current study investigates change in nonsymbolic number comparison performance during first grade and whether symbolic number skills, math skills, or executive function predict change.
Children's performance increased across visual control conditions (i.e., congruent or incongruent with number) suggesting an overall sharpening of number processing.
Symbolic number skills predicted change in nonsymbolic number comparison performance.