Children’s Mapping between Non-Symbolic and Symbolic Numerical Magnitudes and Its Association with Timed and Untimed Tests of Mathematics Achievement

The ability to map between non-symbolic numerical magnitudes and Arabic numerals has been put forward as a key factor in children’s mathematical development. This mapping ability has been mainly examined indirectly by looking at children’s performance on a symbolic magnitude comparison task. The present study investigated mapping in a more direct way by using a task in which children had to choose which of two choice quantities (Arabic digits or dot arrays) matched the target quantity (dot array or Arabic digit), thereby focusing on small quantities ranging from 1 to 9. We aimed to determine the development of mapping over time and its relation to mathematics achievement. Participants were 36 first graders (M = 6 years 8 months) and 46 third graders (M = 8 years 8 months) who all completed mapping tasks, symbolic and non-symbolic magnitude comparison tasks and standardized timed and untimed tests of mathematics achievement. Findings revealed that children are able to map between non-symbolic and symbolic representations and that this mapping ability develops over time. Moreover, we found that children’s mapping ability is related to timed and untimed measures of mathematics achievement, over and above the variance accounted for by their numerical magnitude comparison skills.     

The Processing of Symbolic and Nonsymbolic Ratios in School-Age Children

This study tested the processing of ratios of natural numbers in school-age children. Nine- and eleven-year-olds were presented collections made up of orange and grey dots (i.e., nonsymbolic format) and fractions (i.e., symbolic format). They were asked to estimate ratios between the number of orange dots and the total number of dots and fractions by producing an equivalent ratio of surface areas (filling up a virtual glass). First, we tested whether symbolic notation of ratios affects their processing by directly comparing performance on fractions with that on dot sets. Second, we investigated whether children’s estimates of nonsymbolic ratios of natural numbers relied at least in part on ratios of surface areas by contrasting a condition in which the ratio of surface areas occupied by dots covaried with the ratio of natural numbers and a condition in which this ratio of surface areas was kept constant across ratios of natural numbers. The results showed that symbolic notation did not really have a negative impact on performance among 9-year-olds, while it led to more accurate estimates in 11-year-olds. Furthermore, in dot conditions, children’s estimates increased consistently with ratios between the number of orange dots and the total number of dots even when the ratio of surface areas was kept constant but were less accurate in that condition than when the ratio of surface areas covaried with the ratio of natural numbers. In summary, these results indicate that mental magnitude representation is more accurate when it is activated from symbolic ratios in children as young as 11 years old and that school-age children rely at least in part on ratios of surface areas to process nonsymbolic ratios of natural numbers when given the opportunity to do so.     

The Mediating Relation between Symbolic and Nonsymbolic Foundations of Math Competence

This study investigated the relation between symbolic and nonsymbolic magnitude processing abilities with 2 standardized measures of math competence (WRAT Arithmetic and KeyMath Numeration) in 150 3^rd- grade children (mean age 9.01 years). Participants compared sets of dots and pairs of Arabic digits with numerosities 1–9 for relative numerical magnitude. In line with previous studies, performance on both symbolic and nonsymbolic magnitude processing was related to math ability. Performance metrics combining reaction and accuracy, as well as weber fractions, were entered into mediation models with standardized math test scores. Results showed that symbolic magnitude processing ability fully mediates the relation between nonsymbolic magnitude processing and math ability, regardless of the performance metric or standardized test.     

Symbolic Numerical Magnitude Processing Is as Important to Arithmetic as Phonological Awareness Is to Reading

In this article, we tested, using a 1-year longitudinal design, whether symbolic numerical magnitude processing or children’s numerical representation of Arabic digits, is as important to arithmetic as phonological awareness is to reading. Children completed measures of symbolic comparison, phonological awareness, arithmetic, reading at the start of third grade and the latter two were retested at the start of fourth grade. Cross-sectional and longitudinal correlations indicated that symbolic comparison was a powerful domain-specific predictor of arithmetic and that phonological awareness was a unique predictor of reading. Crucially, the strength of these independent associations was not significantly different. This indicates that symbolic numerical magnitude processing is as important to arithmetic development as phonological awareness is to reading and suggests that symbolic numerical magnitude processing is a good candidate for screening children at risk for developing mathematical difficulties.     

Adolescents’ Functional Numeracy Is Predicted by Their School Entry Number System Knowledge

One in five adults in the United States is functionally innumerate; they do not possess the mathematical competencies needed for many modern jobs. We administered functional numeracy measures used in studies of young adults’ employability and wages to 180 thirteen-year-olds. The adolescents began the study in kindergarten and participated in multiple assessments of intelligence, working memory, mathematical cognition, achievement, and in-class attentive behavior. Their number system knowledge at the beginning of first grade was defined by measures that assessed knowledge of the systematic relations among Arabic numerals and skill at using this knowledge to solve arithmetic problems. Early number system knowledge predicted functional numeracy more than six years later (ß = 0.195, p = .0014) controlling for intelligence, working memory, in-class attentive behavior, mathematical achievement, demographic and other factors, but skill at using counting procedures to solve arithmetic problems did not. In all, we identified specific beginning of schooling numerical knowledge that contributes to individual differences in adolescents’ functional numeracy and demonstrated that performance on mathematical achievement tests underestimates the importance of this early knowledge.     

Processing Demands Impact 3-Year-Olds’ Performance in a Spontaneous-Response Task: New Evidence for the Processing-Load Account of Early False-Belief Understanding

Prior to age four, children succeed in non-elicited-response false-belief tasks but fail elicited-response false-belief tasks. To explain this discrepancy, the processing-load account argues that the capacity to represent beliefs emerges in infancy, as indicated by early success on non-elicited-response tasks, but that children’s ability to demonstrate this capacity depends on the processing demands of the task and children’s processing skills. When processing demands exceed young children’s processing abilities, such as in standard elicited-response tasks, children fail despite their capacity to represent beliefs. Support for this account comes from recent evidence that reducing processing demands improves young children’s performance: when demands are sufficiently reduced, 2.5-year-olds succeed in elicited-response tasks. Here we sought complementary evidence for the processing-load account by examining whether increasing processing demands impeded children’s performance in a non-elicited-response task. 3-year-olds were tested in a preferential-looking task in which they heard a change-of-location false-belief story accompanied by a picture book; across children, we manipulated the amount of linguistic ambiguity in the story. The final page of the book showed two images: one that was consistent with the main character’s false belief and one that was consistent with reality. When the story was relatively unambiguous, children looked reliably longer at the false-belief-consistent image, successfully demonstrating their false-belief understanding. When the story was ambiguous, however, this undermined children’s performance: looking times to the belief-consistent image were correlated with verbal ability, and only children with verbal skills in the upper quartile of the sample demonstrated a significant preference for the belief-consistent image. These results support the processing-load account by demonstrating that regardless of whether a task involves an elicited response, children’s performance depends on the processing demands of the task and their processing skills. These findings also have implications for alternative, deflationary accounts of early false-belief understanding.     

School Performance: A Matter of Health or Socio-Economic Background? Findings from the PIAMA Birth Cohort Study

Background

Performance in primary school is a determinant of children’s educational attainment and their socio-economic position and health inequalities in adulthood. We examined the relationship between five common childhood health conditions (asthma symptoms, eczema, general health, frequent respiratory infections, and overweight), health related school absence and family socio-economic status on children’s school performance.

Methods

We used data from 1,865 children in the Dutch PIAMA birth cohort study. School performance was measured as the teacher’s assessment of a suitable secondary school level for the child, and the child’s score on a standardized achievement test (Cito Test). Both school performance indicators were standardised using Z-scores. Childhood health was indicated by eczema, asthma symptoms, general health, frequent respiratory infections, overweight, and health related school absence. Children’s health conditions were reported repeatedly between the age of one to eleven. School absenteeism was reported at age eleven. Highest attained educational level of the mother and father indicated family socio-economic status. We used linear regression models with heteroskedasticity-robust standard errors for our analyses with adjustment for sex of the child.

Results

The health indicators used in our study were not associated with children’s school performance, independently from parental educational level, with the exception of asthma symptoms (-0.03 z-score / -0.04 z-score with Cito Test score after adjusting for respectively maternal and paternal education) and missing more than 5 schooldays due to illness (-0.18 z-score with Cito Test score and -0.17 z-score with school level assessment after adjustment for paternal education). The effect estimates for these health indicators were much smaller though than the effect estimates for parental education, which was strongly associated with children’s school performance.

Conclusion

Children’s school performance was affected only slightly by a number of common childhood health problems, but was strongly associated with parental education.     

The Influence of Texting Language on Grammar and Executive Functions in Primary School Children

When sending text messages on their mobile phone to friends, children often use a special type of register, which is called textese. This register allows the omission of words and the use of textisms: instances of non-standard written language such as 4ever (forever). Previous studies have shown that textese has a positive effect on children’s literacy abilities. In addition, it is possible that children’s grammar system is affected by textese as well, as grammar rules are often transgressed in this register. Therefore, the main aim of this study was to investigate whether the use of textese influences children’s grammar performance, and whether this effect is specific to grammar or language in general. Additionally, studies have not yet investigated the influence of textese on children’s cognitive abilities. Consequently, the secondary aim of this study was to find out whether textese affects children’s executive functions. To investigate this, 55 children between 10 and 13 years old were tested on a receptive vocabulary and grammar performance (sentence repetition) task and various tasks measuring executive functioning. In addition, text messages were elicited and the number of omissions and textisms in children’s messages were calculated. Regression analyses showed that omissions were a significant predictor of children’s grammar performance after various other variables were controlled for: the more words children omitted in their text messages, the better their performance on the grammar task. Although textisms correlated (marginally) significantly with vocabulary, grammar and selective attention scores and omissions marginally significantly with vocabulary scores, no other significant effects were obtained for measures of textese in the regression analyses: neither for the language outcomes, nor for the executive function tasks. Hence, our results show that textese is positively related to children’s grammar performance. On the other hand, use of textese does not affect—positively nor negatively—children’s executive functions.     

Lord’s Paradox in a Continuous Setting and a Regression Artifact in Numerical Cognition Research

In this paper we review, and elaborate on, the literature on a regression artifact related to Lord’s paradox in a continuous setting. Specifically, the question is whether a continuous property of individuals predicts improvement from training between a pretest and a posttest. If the pretest score is included as a covariate, regression to the mean will lead to biased results if two critical conditions are satisfied: (1) the property is correlated with pretest scores and (2) pretest scores include random errors. We discuss how these conditions apply to the analysis in a published experimental study, the authors of which concluded that linearity of children’s estimations of numerical magnitudes predicts arithmetic learning from a training program. However, the two critical conditions were clearly met in that study. In a reanalysis we find that the bias in the method can fully account for the effect found in the original study. In other words, data are consistent with the null hypothesis that numerical magnitude estimations are unrelated to arithmetic learning.     

Levers for Language Growth: Characteristics and Predictors of Language Trajectories between 4 and 7 Years

Background

Evidence is required as to when and where to focus resources to achieve the greatest gains for children’s language development. Key to these decisions is the understanding of individual differences in children’s language trajectories and the predictors of those differences. To determine optimal timing we must understand if and when children’s relative language abilities become fixed. To determine where to focus effort we must identify mutable factors, that is those with the potential to be changed through interventions, which are associated with significant differences in children’s language scores and rate of progress.

Methods

Uniquely this study examined individual differences in language growth trajectories in a population sample of children between 4 and 7 years using the multilevel model for change. The influence of predictors, grouped with respect to their mutability and their proximity to the child (least-mutable, mutable-distal, mutable-proximal), were estimated.

Results

A significant degree of variability in rate of progress between 4 and 7 years was evident, much of which was systematically associated with mutable-proximal factors, that is, those factors with evidence that they are modifiable through interventions with the child or family, such as shared book reading, TV viewing and number of books in the home. Mutable-distal factors, such as family income, family literacy and neighbourhood disadvantage, hypothesised to be modifiable through social policy, were important predictors of language abilities at 4 years.

Conclusions

Potential levers for language interventions lie in the child’s home learning environment from birth to age 4. However, the role of a family’s material and cultural capital must not be ignored, nor should the potential for growth into the school years. Early Years services should acknowledge the effects of multiple, cascading and cumulative risks and seek to promote child language development through the aggregation of marginal gains in the pre-school years and beyond.     

Impact of high mathematics education on the number sense

In adult number processing two mechanisms are commonly used: approximate estimation of quantity and exact calculation. While the former relies on the approximate number sense (ANS) which we share with animals and preverbal infants, the latter has been proposed to rely on an exact number system (ENS) which develops later in life following the acquisition of symbolic number knowledge. The current study investigated the influence of high level math education on the ANS and the ENS. Our results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education. However, performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-symbolic quantities was significantly better the higher mathematics achievement. Our findings suggest that high level math education in adults shows little influence on their ANS, but it seems to be associated with a better anchored ENS and better mapping abilities between ENS and ANS.     

Early Social Experience Predicts Referential Communicative Adjustments in Five-Year-Old Children

A large body of work has focused on children’s ability to attribute mental states to other people, and whether these abilities are influenced by the extent and nature of children’s social interactions. However, it remains largely unknown which developmental factors shape children’s ability to influence the mental states of others. Building on the suggestion that collaborative experiences early in life might be crucial for the emergence of mental coordination abilities, here we assess the relative contribution of social exposure to familial and non-familial agents on children’s communicative adjustments to their mental model of an addressee (‘audience design’). During an online interactive game, five-year-olds spontaneously organized their non-verbal communicative behaviors according to their beliefs about an interlocutor. The magnitude of these communicative adjustments was predicted by the time spent at daycare, from birth until four years of age, over and above effects of familial social environment. These results suggest that the degree of non-familial social interaction early in life modulates the influence that children’s beliefs have on their referential communicative behavior.     

Preschoolers' Dot Enumeration Abilities Are Markers of Their Arithmetic Competence

The abilities to enumerate small sets of items (e.g., dots) and to compare magnitudes are claimed to be indexes of core numerical competences that scaffold early math development. Insofar as this is correct, these abilities may be diagnostic markers of math competence in preschoolers. However, unlike magnitude comparison abilities, little research has examined preschoolers'' ability to enumerate small sets, or its significance for emerging math abilities; which is surprising since dot enumeration is a marker of school-aged children''s math competence. It is nevertheless possible that general cognitive functions (working memory, response inhibition in particular) are associated with preschoolers'' math abilities and underlie nascent dot enumeration abilities. We investigated whether preschoolers'' dot enumeration abilities predict their non-verbal arithmetic ability, over and above the influence of working memory and response inhibition. Two measures of dot enumeration ability were examined—inverse efficiency and paradigm specific (response time profiles) measures—to determine which has the better diagnostic utility as a marker of math competence. Seventy-eight 42-to-57 month-olds completed dot enumeration, working memory, response inhibition, and non-verbal addition and subtraction tasks. Dot enumeration efficiency predicted arithmetic ability over and above the influence of general cognitive functions. While dot enumeration efficiency was a better predictor of arithmetic ability than paradigm specific response time profiles; the response time profile displaying the smallest subitizing range and steepest subitizing slope, also displayed poor addition abilities, suggesting a weak subitizing profile may have diagnostic significance in preschoolers. Overall, the findings support the claim that dot enumeration abilities and general cognitive functions are markers of preschoolers'' math ability.     

Numerical magnitude in the human parietal lobe; tests of representational generality and domain specificity

Behavioral evidence suggests that human adults have a single system for representing the numerical magnitude of both symbolic numbers (e.g., Arabic digits) and nonsymbolic number stimuli (e.g., dot arrays). Brain imaging studies have implicated a specific parietal region in symbolic number processing, leading to the influential hypothesis that this region is the locus of a dedicated, domain-specific number system. Here we evaluated a prediction of this hypothesis, that this region should be activated not only by symbolic but also nonsymbolic number processing. Using nonsymbolic stimuli, we tested for higher parietal activations for number than for nonnumber comparison tasks (experiment 1), fMRI adaptation for numerosity repetition (experiment 2), and greater fMRI increases with increasing task difficulty for number than nonnumber tasks (experiment 3). None of these predictions were supported by the data, posing a serious challenge to the hypothesis that a single, domain-specific parietal region underlies both symbolic and nonsymbolic number representation.     

Multi-representation of symbolic and nonsymbolic numerical magnitude in Chinese number processing

Numerical information can be conveyed by either symbolic or nonsymbolic representation. Some symbolic numerals can also be identified as nonsymbolic quantities defined by the number of lines (e.g., I, II, III in Roman and -, =, ≡ in Japanese Kanji and Chinese). Here we report that such multi-representation of magnitude can facilitate the processing of these numerals under certain circumstances. In a magnitude comparison task judging 1 to 9 (except 5) Chinese and Arabic numerals presented at the foveal (at the center) or parafoveal (3° left or right of the center) location, multi-representational small-value Chinese numerals showed a processing advantage over single-representational Arabic numerals and large-value Chinese numerals only in the parafoveal condition, demonstrated by lower error rates and faster reaction times. Further event-related potential (ERP) analysis showed that such a processing advantage was not reflected by traditional ERP components identified in previous studies of number processing, such as N1 or P2p. Instead, the difference was found much later in a N400 component between 300-550 msec over parietal regions, suggesting that those behavioral differences may not be due to early processing of visual identification, but later processing of subitizing or accessing mental number line when lacking attentional resources. These results suggest that there could be three stages of number processing represented separately by the N1, P2p and N400 ERP components. In addition, numerical information can be represented simultaneously by both symbolic and nonsymbolic systems, which will facilitate number processing in certain situations.     

Personalized brain stimulation for effective neurointervention across participants

Accumulating evidence from human-based research has highlighted that the prevalent one-size-fits-all approach for neural and behavioral interventions is inefficient. This approach can benefit one individual, but be ineffective or even detrimental for another. Studying the efficacy of the large range of different parameters for different individuals is costly, time-consuming and requires a large sample size that makes such research impractical and hinders effective interventions. Here an active machine learning technique is presented across participants—personalized Bayesian optimization (pBO)—that searches available parameter combinations to optimize an intervention as a function of an individual’s ability. This novel technique was utilized to identify transcranial alternating current stimulation (tACS) frequency and current strength combinations most likely to improve arithmetic performance, based on a subject’s baseline arithmetic abilities. The pBO was performed across all subjects tested, building a model of subject performance, capable of recommending parameters for future subjects based on their baseline arithmetic ability. pBO successfully searches, learns, and recommends parameters for an effective neurointervention as supported by behavioral, simulation, and neural data. The application of pBO in human-based research opens up new avenues for personalized and more effective interventions, as well as discoveries of protocols for treatment and translation to other clinical and non-clinical domains.     

Distracted by the Unthought – Suppression and Reappraisal of Mind Wandering under Stereotype Threat

Previous research has found that subtle reminders of negative stereotypes about one’s group can lead individuals to underperform on stereotype-relevant tests (e.g., women in math, ethnic minorities on intelligence tests). This so called stereotype threat effect can contribute to systematic group differences in performance that can obscure the true abilities of certain social groups and thereby sustain social inequalities. In the present study, we examined processes underlying stereotype threat effects on women’s math performance, specifically focusing on the role of suppression of mind wandering (i.e., task-irrelevant thinking) in stereotype threat (ST) and no threat (NT) situations. Based on a process model of stereotype threat effects on performance, we hypothesized that women under stereotype threat spontaneously suppress mind wandering, and that this suppression impairs performance. An alternative regulation strategy that prevents suppression (i.e., reappraising task-irrelevant thoughts as normal) was predicted to prevent stereotype threat effects on performance. We manipulated stereotype threat (ST vs. NT) and cognitive regulation strategy (suppression, reappraisal, or no strategy) and measured women’s performance on a math and a concentration task (N = 113). We expected three groups to perform relatively more poorly: Those in ST with either no strategy or suppression and those in NT with a suppression strategy. We tested the performance of these groups against the remaining three groups hypothesized to perform relatively better: those in NT with no strategy or reappraisal and those in ST with reappraisal. The results showed the expected pattern for participants’ math performance, but not for concentration achievement. This pattern suggests that ineffective self-regulation by suppressing mind wandering can at least partly explain stereotype threat effects on performance, whereas a reappraisal strategy can prevent this impairment. We discuss implications for the understanding of processes underlying stereotype threat effects and the benefits of reappraising subjective experience under threat.     

Do Young Children Understand Relative Value Comparisons?

Many forms of judgments, such as those used in economic games or measures of social comparison, require understanding relative value, as well as the more complex ability to make comparisons between relative values. To examine whether young children can accurately compare relative values, we presented children 4 to 7 years with simple judgments of relative value in two scenarios. Children then were asked to compare the relative values in the two scenarios. Results show that even the youngest children downgraded evaluations of a reward when another has a larger amount, indicating the ability to make relative value judgments. When asked to compare relative values however, only the oldest children were able to make these comparisons consistently. We then extended this analysis to economic game performance. Specifically, previous results using economic games suggest that younger children are more generous than older ones. We replicate this result, and then show that a simple change in procedure, based on the initial study, is sufficient to change young children’s choices. Our results strongly suggest that conclusions regarding young children’s pro-social motives based on relative value comparisons should be viewed cautiously.     

“Vision for Action” in Young Children Aligning Multi-Featured Objects: Development and Comparison with Nonhuman Primates

Effective vision for action and effective management of concurrent spatial relations underlie skillful manipulation of objects, including hand tools, in humans. Children’s performance in object insertion tasks (fitting tasks) provides one index of the striking changes in the development of vision for action in early life. Fitting tasks also tap children’s ability to work with more than one feature of an object concurrently. We examine young children’s performance on fitting tasks in two and three dimensions and compare their performance with the previously reported performance of adult individuals of two species of nonhuman primates on similar tasks. Two, three, and four year-old children routinely aligned a bar-shaped stick and a cross-shaped stick but had difficulty aligning a tomahawk-shaped stick to a matching cut-out. Two year-olds were especially challenged by the tomahawk. Three and four year-olds occasionally held the stick several inches above the surface, comparing the stick to the surface visually, while trying to align it. The findings suggest asynchronous development in the ability to use vision to achieve alignment and to work with two and three spatial features concurrently. Using vision to align objects precisely to other objects and managing more than one spatial relation between an object and a surface are already more elaborated in two year-old humans than in other primates. The human advantage in using hand tools derives in part from this fundamental difference in the relation between vision and action between humans and other primates.     

Comparing Different Models of the Development of Verb Inflection in Early Child Spanish

How children acquire knowledge of verb inflection is a long-standing question in language acquisition research. In the present study, we test the predictions of some current constructivist and generativist accounts of the development of verb inflection by focusing on data from two Spanish-speaking children between the ages of 2;0 and 2;6. The constructivist claim that children’s early knowledge of verb inflection is only partially productive is tested by comparing the average number of different inflections per verb in matched samples of child and adult speech. The generativist claim that children’s early use of verb inflection is essentially error-free is tested by investigating the rate at which the children made subject-verb agreement errors in different parts of the present tense paradigm. Our results show: 1) that, although even adults’ use of verb inflection in Spanish tends to look somewhat lexically restricted, both children’s use of verb inflection was significantly less flexible than that of their caregivers, and 2) that, although the rate at which the two children produced subject-verb agreement errors in their speech was very low, this overall error rate hid a consistent pattern of error in which error rates were substantially higher in low frequency than in high frequency contexts, and substantially higher for low frequency than for high frequency verbs. These results undermine the claim that children’s use of verb inflection is fully productive from the earliest observable stages, and are consistent with the constructivist claim that knowledge of verb inflection develops only gradually.