Preschool deficits in cardinal knowledge and executive function contribute to longer-term mathematical learning disability.

In a preschool through first grade longitudinal study, we identified groups of children with persistently low mathematics achievement (n = 14) and children with low achievement in preschool but average achievement in first grade (n = 23). The preschool quantitative developments of these respective groups of children with mathematical learning disability (MLD) and recovered children and a group of typically achieving peers (n = 35) were contrasted, as were their intelligence, executive function, and parental education levels. The core characteristics of the children with MLD were poor executive function and delayed understanding of the cardinal value of number words throughout preschool. These compounded into even more substantive deficits in number and arithmetic at the beginning of first grade. The recovered group had poor executive function and cardinal knowledge during the first year of preschool but showed significant gains during the second year. Despite these gains and average mathematics achievement, the recovered children had subtle deficits with accessing magnitudes associated with numerals and addition combinations (e.g., 5 + 6 = ?) in first grade. The study provides unique insight into domain-general and quantitative deficits in preschool that increase risk for long-term mathematical difficulties.

Predicting Age of Becoming a Cardinal Principle Knower.

Children's first mathematics concept is their understanding of the quantities represented by number words (cardinal value), and the age at which they achieve this insight predicts their readiness for mathematics learning in school. We provide the first exploration of the factors that influence the age of becoming a cardinal principle knower (CPK), with a longitudinal study of 197 (94 boys) children from the beginning to the end of two years of preschool. Core symbolic and non-symbolic quantitative competencies at the beginning of preschool, as well as measures of intelligence, executive function, preliteracy skills, and parental education were used to predict timing of CPK status. Children who achieved early CPK status had higher IQ scores, knew more count words and numerals, and had a better intuitive understanding of relative quantity than their peers. Children who were delayed CPKs, in contrast, had deficits in executive function and poor preliteracy skills. The results add to our understanding of children's conceptual development in mathematics and have implications for the identification of at-risk children and design of interventions for them.

Early Conceptual Understanding of Cardinality Predicts Superior School-Entry Number-System Knowledge.

We demonstrate a link between preschoolers' quantitative competencies and their school-entry knowledge of the relations among numbers (number-system knowledge). The quantitative competencies of 141 children (69 boys) were assessed at the beginning of preschool and throughout the next 2 years of preschool, as was their mathematics and reading achievement at the end of kindergarten and their number-system knowledge at the beginning of first grade. A combination of Bayes analyses and standard regressions revealed that the age at which the children had the conceptual insight that number words represent specific quantities (cardinal value) was strongly related to their later number-system knowledge and was more consistently related to broader mathematics than to reading achievement, controlling for intelligence, executive function, and parental education levels. The key implication is that it is not simply knowledge of cardinal value but the age of acquisition of this principle that is central to later mathematical development and school readiness.

Attaching meaning to the number words: contributions of the object tracking and approximate number systems.

Children's understanding of the quantities represented by number words (i.e., cardinality) is a surprisingly protracted but foundational step in their learning of formal mathematics. The development of cardinal knowledge is related to one or two core, inherent systems - the approximate number system (ANS) and the object tracking system (OTS) - but whether these systems act alone, in concert, or antagonistically is debated. Longitudinal assessments of 198 preschool children on OTS, ANS, and cardinality tasks enabled testing of two single-mechanism (ANS-only and OTS-only) and two dual-mechanism models, controlling for intelligence, executive functions, preliteracy skills, and demographic factors. Measures of both OTS and ANS predicted cardinal knowledge in concert early in the school year, inconsistent with single-mechanism models. The ANS but not the OTS predicted cardinal knowledge later in the school year as well the acquisition of the cardinal principle, a critical shift in cardinal understanding. The results support a Merge model, whereby both systems initially contribute to children's early mapping of number words to cardinal value, but the role of the OTS diminishes over time while that of the ANS continues to support cardinal knowledge as children come to understand the counting principles.

Children's early understanding of number predicts their later problem-solving sophistication in addition.

Previous studies suggest that the sophistication of the strategies children use to solve arithmetic problems is related to a more basic understanding of number, but they have not examined the relation between number knowledge in preschool and strategy choices at school entry. Accordingly, the symbolic and nonsymbolic quantitative knowledge of 134 children (65 boys) was assessed at the beginning of preschool and in kindergarten, and the sophistication of the strategies they used to solve addition problems was assessed at the beginning of first grade. Using a combination of Bayes and standard regression models, we found that children's understanding of the cardinal value of number words at the beginning of preschool predicted the sophistication of their strategy choices 3 years later, controlling for other factors. The relation between children's early understanding of cardinality and their strategy choices was mediated by their symbolic and nonsymbolic quantitative knowledge in kindergarten. The results suggest that sophisticated strategy choices emerge from children's developing understanding of the relations among numbers, in keeping with the overlapping waves model.

Infants' Understanding of Preferences When Agents Make Inconsistent Choices.

This study showed that 8.5-month-old infants seemed to consider the consistency of an agent's choices in attributing preferences to her. When the agent consistently chose one object over another, three or four times consecutively, infants acted as if they had interpreted her actions as evidence for her preference. In contrast, when the agent inconsistently chose between the two objects, at the ratio of 1:3, infants did not seem to interpret her actions as suggesting her preference. Converging evidence was obtained from infants' responses across a looking-time task and an action task. The results are discussed in terms of how infants might use frequencies of agents' actions directed toward different objects to understand agents' preferences.

Controlling for continuous variables is not futile: What we can learn about number representation despite imperfect control.

Leibovich et al. argue that because it is impossible to isolate numerosity in a stimulus set, attempts to show that number is processed independently of continuous magnitudes are necessarily in vain. I propose that through clever design and manipulation of confounding variables, we can gain deep insight into number representation and its relationship to the representation of other magnitudes.

Kindergartners' fluent processing of symbolic numerical magnitude is predicted by their cardinal knowledge and implicit understanding of arithmetic 2years earlier.

Fluency in first graders' processing of the magnitudes associated with Arabic numerals, collections of objects, and mixtures of objects and numerals predicts current and future mathematics achievement. The quantitative competencies that support the development of fluent processing of magnitude, however, are not fully understood. At the beginning and end of preschool (M=3years 9months at first assessment, range=3years 3months to 4years 3months), 112 children (51 boys) completed tasks measuring numeral recognition and comparison, acuity of the approximate number system, and knowledge of counting principles, cardinality, and implicit arithmetic and also completed a magnitude processing task (number sets test) in kindergarten. Use of Bayesian and linear regression techniques revealed that two measures of preschoolers' cardinal knowledge and their competence at implicit arithmetic predicted later fluency of magnitude processing, controlling domain-general factors, preliteracy skills, and parental education. The results help to narrow the search for the early foundation of children's emerging competence with symbolic mathematics and provide direction for early interventions.

Early numerical foundations of young children's mathematical development.

This study focused on the relative contributions of the acuity of the approximate number system (ANS) and knowledge of quantitative symbols to young children's early mathematical learning. At the beginning of preschool, 191 children (Mage=46 months) were administered tasks that assessed ANS acuity and explicit knowledge of the cardinal values represented by number words, and their mathematics achievement was assessed at the end of the school year. Children's executive functions, intelligence, and preliteracy skills and their parents' educational levels were also assessed and served as covariates. Both the ANS and cardinality tasks were significant predictors of end-of-year mathematics achievement with and without control of the covariates. As simultaneous predictors and with control of the covariates, cardinality remained significantly related to mathematics achievement, but ANS acuity did not. Mediation analyses revealed that the relation between ANS acuity and mathematics achievement was fully mediated by cardinality, suggesting that the ANS may facilitate children's explicit understanding of cardinal value and in this way may indirectly influence early mathematical learning.

Infants use different mechanisms to make small and large number ordinal judgments.

Previous research has shown indirectly that infants may use two different mechanisms--an object tracking system and an analog magnitude mechanism--to represent small (<4) and large (≥4) numbers of objects, respectively. The current study directly tested this hypothesis in an ordinal choice task by presenting 10- to 12-month-olds with a choice between different numbers of hidden food items. Infants reliably chose the larger amount when choosing between two exclusively small (1 vs. 2) or large (4 vs. 8) sets, but they performed at chance when one set was small and the other was large (2 vs. 4) even when the ratio between the sets was very favorable (2 vs. 8). The current findings support the two-mechanism hypothesis and, furthermore, suggest that the representations from the object tracking system and the analog magnitude mechanism are incommensurable.

Enumeration takes time: Accuracy improves even after stimuli disappear.

Numerical cognition is widespread among animal species and present from birth in humans. Laboratory studies show that longer stimulus duration improves numerical discrimination accuracy (Inglis & Gilmore, 2013; Wood & Spelke, 2005). Inglis and Gilmore (2013) suggested that longer durations allow subjects to resample the stimulus image multiple times, resulting in a more accurate final estimate. The current study tested this "multiple sampling" model alongside two competing models - "serial foveal accumulator" (Cheyette & Piantadosi, 2019) and "longer processing time" - to determine how stimulus duration relates to accuracy. Adult subjects completed a fully within-subject, 2AFC task in which they judged which of two arrays had more dots; accuracy was the dependent measure. Experiment 1 revealed higher accuracy in the 500 ms stimulus duration condition compared to the 100 ms condition, replicating previous results. Experiments 2 and 4 extended this finding. When stimulus duration was held constant at 100 ms, adding a 400 ms delay between stimulus offset and mask onset improved accuracy. Accuracy was similar in the 500 ms stimulus duration condition (E1) and the 100 ms duration + 400 ms mask delay conditions (E2 and E4), indicating that post-stimulus processing time improves accuracy, rather than stimulus duration per se. This contrasts with both the multiple sampling model and the serial foveal model, although we cannot exclude the possibility that sampling continues in iconic memory. Numerosity was a significant factor in all four experiments, suggesting a serial component in the enumeration process. These findings shed light on the cognitive processes underlying nonverbal number representation in humans.

Continuity in social cognition from infancy to childhood.

Research examining the development of social cognition has largely been divided into two areas: infant perception of intentional agents, and preschoolers' understanding of others' mental states and beliefs (theory of mind). Many researchers have suggested that there is continuity in social cognitive development such that the abilities observed in infancy are related to later preschool ability, yet little empirical evidence exists for this claim. Here, we present preliminary evidence that capacities specific to the social domain contribute to performance in social cognition tasks both during infancy and in early childhood. Specifically, looking time patterns in an infant social cognition task correlated with preschool theory of mind; however, no such relationship was found for infants in a nonsocial cognition task.

Growth of symbolic number knowledge accelerates after children understand cardinality.

Children who achieve an early understanding of the cardinal value of number words (cardinal knowledge) have a superior understanding of the relations among numerals at school entry, controlling other factors (e.g., intelligence). We tested the hypothesis that this pattern emerges because an understanding of cardinal value jump starts children's learning of the relations among numerals. Across two years of preschool, the cardinal knowledge of 179 children (85 boys) was assessed four times, as was their understanding of the relative quantity of Arabic numerals and competence at discriminating nonsymbolic quantities. Children were more accurate on nonsymbolic than numeral comparisons before they understood cardinality, but showed more rapid growth for numeral than nonsymbolic comparisons once they understood cardinality. Moreover, and with the possible exception of very small numerals (<5), before they understood cardinality children were no better than chance in their numeral comparisons, but greatly exceeded chance once they understood cardinality. These patterns were independent of the age at which children became cardinal principle knowers and independent of intelligence, executive function, and preliteracy skills. More broadly, the results provide a developmental bridge between cardinal knowledge and school-entry number knowledge.

How capuchin monkeys (Cebus apella) quantify objects and substances.

Humans and nonhuman animals appear to share a capacity for nonverbal quantity representations. But what are the limits of these abilities? Results of previous research with human infants suggest that the ontological status of an entity as an object or a substance affects infants' ability to quantify it. We ask whether the same is true for another primate species-the New World monkey Cebus apella. We tested capuchin monkeys' ability to select the greater of two quantities of either discrete objects or a nonsolid substance. Participants performed above chance with both objects (Experiment 1) and substances (Experiment 2); in both cases, the observed performance was ratio dependent. This finding suggests that capuchins quantify objects and substances similarly and do so via analog magnitude representations.

Young children's core symbolic and nonsymbolic quantitative knowledge in the prediction of later mathematics achievement.

At the beginning of preschool (M = 46 months of age), 197 (94 boys) children were administered tasks that assessed a suite of nonsymbolic and symbolic quantitative competencies as well as their executive functions, verbal and nonverbal intelligence, preliteracy skills, and their parents' education level. The children's mathematics achievement was assessed at the end of preschool (M = 64 months). We used a series of Bayesian and standard regression analyses to winnow this broad set of competencies down to the core subset of quantitative skills that predict later mathematics achievement, controlling other factors. This knowledge included children's fluency in reciting the counting string, their understanding of the cardinal value of number words, and recognition of Arabic numerals, as well as their sensitivity to the relative quantity of 2 collections of objects. The results inform theoretical models of the foundations of children's early quantitative development and have practical implications for the design of early interventions for children at risk for poor long-term mathematics achievement. (PsycINFO Database Record

Predicting Children's Reading and Mathematics Achievement from Early Quantitative Knowledge and Domain-General Cognitive Abilities.

One hundred children (44 boys) participated in a 3-year longitudinal study of the development of basic quantitative competencies and the relation between these competencies and later mathematics and reading achievement. The children's preliteracy knowledge, intelligence, executive functions, and parental educational background were also assessed. The quantitative tasks assessed a broad range of symbolic and nonsymbolic knowledge and were administered four times across 2 years of preschool. Mathematics achievement was assessed at the end of each of 2 years of preschool, and mathematics and word reading achievement were assessed at the end of kindergarten. Our goals were to determine how domain-general abilities contribute to growth in children's quantitative knowledge and to determine how domain-general and domain-specific abilities contribute to children's preschool mathematics achievement and kindergarten mathematics and reading achievement. We first identified four core quantitative competencies (e.g., knowledge of the cardinal value of number words) that predict later mathematics achievement. The domain-general abilities were then used to predict growth in these competencies across 2 years of preschool, and the combination of domain-general abilities, preliteracy skills, and core quantitative competencies were used to predict mathematics achievement across preschool and mathematics and word reading achievement at the end of kindergarten. Both intelligence and executive functions predicted growth in the four quantitative competencies, especially across the first year of preschool. A combination of domain-general and domain-specific competencies predicted preschoolers' mathematics achievement, with a trend for domain-specific skills to be more strongly related to achievement at the beginning of preschool than at the end of preschool. Preschool preliteracy skills, sensitivity to the relative quantities of collections of objects, and cardinal knowledge predicted reading and mathematics achievement at the end of kindergarten. Preliteracy skills were more strongly related to word reading, whereas sensitivity to relative quantity was more strongly related to mathematics achievement. The overall results indicate that a combination of domain-general and domain-specific abilities contribute to development of children's early mathematics and reading achievement.

Analog Magnitudes Support Large Number Ordinal Judgments in Infancy.

Few studies have explored the source of infants' ordinal knowledge, and those that have are equivocal regarding the underlying representational system. The present study sought clear evidence that the approximate number system, which underlies children's cardinal knowledge, may also support ordinal knowledge in infancy; 10 - to 12-month-old infants' were tested with large sets (>3) in an ordinal choice task in which they were asked to choose between two hidden sets of food items. The difficulty of the comparison varied as a function of the ratio between the sets. Infants reliably chose the greater quantity when the sets differed by a 2:3 ratio (4v6 and 6v9), but not when they differed by a 3:4 ratio (6v8) or a 7:8 ratio (7v8). This discrimination function is consistent with previous studies testing the precision of number and time representations in infants of roughly this same age, thus providing evidence that the approximate number system can support ordinal judgments in infancy. The findings are discussed in light of recent proposals that different mechanisms underlie infants' reasoning about small and large numbers.

Quantitative deficits of preschool children at risk for mathematical learning disability.

The study tested the hypothesis that acuity of the potentially inherent approximate number system (ANS) contributes to risk of mathematical learning disability (MLD). Sixty-eight (35 boys) preschoolers at risk for school failure were assessed on a battery of quantitative tasks, and on intelligence, executive control, preliteracy skills, and parental education. Mathematics achievement scores at the end of 1 year of preschool indicated that 34 of these children were at high risk for MLD. Relative to the 34 typically achieving children, the at risk children were less accurate on the ANS task, and a one standard deviation deficit on this task resulted in a 2.4-fold increase in the odds of MLD status. The at risk children also had a poor understanding of ordinal relations, and had slower learning of Arabic numerals, number words, and their cardinal values. Poor performance on these tasks resulted in 3.6- to 4.5-fold increases in the odds of MLD status. The results provide some support for the ANS hypothesis but also suggest these deficits are not the primary source of poor mathematics learning.

Physics for infants: characterizing the origins of knowledge about objects, substances, and number.

Adults possess a great deal of knowledge about how objects behave and interact in our every day environment, yet several puzzles remain unsolved regarding how we manage this ubiquitous skill. The notion of intuitive physics has been a central focus of research on cognitive development in infancy. This article focuses on the origins of knowledge about objects, substances, and number concepts in infancy. The article reviews common themes of solidity, continuity, cohesion, and property changes as they have been studied with regard to infants' knowledge about objects and more recently with regard to infants' knowledge about substances. In addition, we review how object and substance knowledge interfaces with number knowledge systems. The evidence supports the view that certain core principles about these domains are present as early as we can test for them and the nature of the underlying representation is best characterized as primitive initial concepts that are elaborated and refined through learning and experience. WIREs Cogn Sci 2012, 3:19-27. doi: 10.1002/wcs.157 For further resources related to this article, please visit the WIREs website.

Infants' auditory enumeration: evidence for analog magnitudes in the small number range.

Vigorous debate surrounds the issue of whether infants use different representational mechanisms to discriminate small and large numbers. We report evidence for ratio-dependent performance in infants' discrimination of small numbers of auditory events, suggesting that infants can use analog magnitudes to represent small values, at least in the auditory domain. Seven-month-old infants in the present study reliably discriminated two from four tones (a 1:2 ratio) in Experiment 1, when melodic and continuous temporal properties of the sequences were controlled, but failed to discriminate two from three tones (a 2:3 ratio) under the same conditions in Experiment 2. A third experiment ruled out the possibility that infants in Experiment 1 were responding to greater melodic variety in the four-tone sequences. The discrimination function obtained here is the same as that found for infants' discrimination of large numbers of visual and auditory items at a similar age, as well as for that obtained for similar-aged infants' duration discriminations, and thus adds to a growing body of evidence suggesting that human infants may share with adults and nonhuman animals a mechanism for representing quantities as "noisy" mental magnitudes.