The approximate number system and its relation to early math achievement: evidence from the preschool years

Neural Basis of Number Sense in Larval Zebrafish

Number sense, the ability to discriminate the quantity of objects, is crucial for survival. To understand how neurons work together and develop to mediate number sense, we used two-photon fluorescence light sheet microscopy to capture the activity of individual neurons throughout the brain of larval Danio rerio, while displaying a visual number stimulus to the animal. We identified number-selective neurons as early as 3 days post-fertilization and found a proportional increase of neurons tuned to larger quantities after 3 days. We used machine learning to predict the stimulus from the neuronal activity and observed that the prediction accuracy improves with age. We further tested ethanol's effect on number sense and found a decrease in number-selective neurons in the forebrain, suggesting cognitive impairment. These findings are a significant step towards understanding neural circuits devoted to discrete magnitudes and our methodology to track single-neuron activity across the whole brain is broadly applicable to other fields in neuroscience.

Number sense-arithmetic link in Grade 1 and Grade 2: A case of fluency

BACKGROUND

Recent research suggested fluent processing as an explanation on why number sense contributes to simple arithmetic tasks-'Fluency hypothesis'.

AIMS

The current study investigates whether number sense contributes to such arithmetic tasks when other cognitive factors are controlled for (including those that mediate the link); and whether this contribution varies as a function of participants' individual maths fluency levels.

SAMPLE

Four hundred and thirty-seven Chinese schoolchildren (186 females; Mage = 83.49 months) completed a range of cognitive measures in Grade 1 (no previous classroom training) and in Grade 2 (a year later).

METHODS

Number sense, arithmetic (addition and subtraction), spatial ability, visuo-spatial working memory, perception, reaction time, character reading and general intelligence were measured.

RESULTS

Our data showed that the link between number sense and arithmetic was weaker in Grade 1 (Beta = .15 for addition and .06 (ns) for subtraction) compared to Grade 2 (.23-.28), but still persisted in children with no previous maths training. Further, math's performance in Grade 1 did not affect the link between number sense and maths performance in Grade 2.

CONCLUSION

Our data extended previous findings by showing that number sense is linked with simple maths task performance even after controlling for multiple cognitive factors. Our results brought some evidence that number sense-arithmetic link is somewhat sensitive to previous formal maths education. Further research is needed, as the differences in effects between grades were quite small, and arithmetic in Grade 1 did not moderate the link at question in Grade 2.

Frontoparietal and salience network synchronizations during nonsymbolic magnitude processing predict brain age and mathematical performance in youth

The development and refinement of functional brain circuits crucial to human cognition is a continuous process that spans from childhood to adulthood. Research increasingly focuses on mapping these evolving configurations, with the aim to identify markers for functional impairments and atypical development. Among human cognitive systems, nonsymbolic magnitude representations serve as a foundational building block for future success in mathematical learning and achievement for individuals. Using task-based frontoparietal (FPN) and salience network (SN) features during nonsymbolic magnitude processing alongside machine learning algorithms, we developed a framework to construct brain age prediction models for participants aged 7-30. Our study revealed differential developmental profiles in the synchronization within and between FPN and SN networks. Specifically, we observed a linear increase in FPN connectivity, concomitant with a decline in SN connectivity across the age span. A nonlinear U-shaped trajectory in the connectivity between the FPN and SN was discerned, revealing reduced FPN-SN synchronization among adolescents compared to both pediatric and adult cohorts. Leveraging the Gradient Boosting machine learning algorithm and nested fivefold stratified cross-validation with independent training datasets, we demonstrated that functional connectivity measures of the FPN and SN nodes predict chronological age, with a correlation coefficient of .727 and a mean absolute error of 2.944 between actual and predicted ages. Notably, connectivity within the FPN emerged as the most contributing feature for age prediction. Critically, a more matured brain age estimate is associated with better arithmetic performance. Our findings shed light on the intricate developmental changes occurring in the neural networks supporting magnitude representations. We emphasize brain age estimation as a potent tool for understanding cognitive development and its relationship to mathematical abilities across the critical developmental period of youth. PRACTITIONER POINTS: This study investigated the prolonged changes in the brain's architecture across childhood, adolescence, and adulthood, with a focus on task-state frontoparietal and salience networks. Distinct developmental pathways were identified: frontoparietal synchronization strengthens consistently throughout development, while salience network connectivity diminishes with age. Furthermore, adolescents show a unique dip in connectivity between these networks. Leveraging advanced machine learning methods, we accurately predicted individuals' ages based on these brain circuits, with a more mature estimated brain age correlating with better math skills.

Development of number line estsimation in Chinese preschoolers: a comparison between numerical and non-numerical symbols

To examine the level of number line estimation (NLE) in Chinese children with respect to representations of both numerical (Arabic numerals) and non-numerical symbols (dots), a total of 192 Chinese preschoolers aged between 4 and 5 years participated in four different NLE tasks. These tasks were paired to evaluate the accuracy and patterns of children's estimations in both numerical and non-numerical symbol contexts. Our findings indicate that, for Chinese preschoolers, relatively precise numerical symbol representations begin to emerge as early as 4 years of age. The accuracy of number line estimates for both 4- and 5-year-old children gradually increases in tasks involving both numerical and non-numerical symbols. Additionally, the development and patterns observed in the number line estimates of 4- and 5-year-old Chinese preschoolers are similar in both numerical symbol and non-numerical symbol tasks. These results indicate that the initiation of relatively precise numerical symbol representation and the turning point in the developmental trajectory, where the relatively precise representation for numerical symbols surpasses that of non-numerical ones, occur earlier in Chinese children than in their Western counterparts.

Developmental Dyscalculia in Relation to Individual Differences in Mathematical Abilities

There is still much debate about the exact nature and frequency of developmental dyscalculia, and about how it should be defined. This article examines several key questions in turn: Is developmental dyscalculia a distinct disorder, or should it be seen as the lower end of a continuum-or possibly more than one continuum-of numerical ability? Do individuals with developmental dyscalculia show atypical brain structure or function? Does the study of acquired dyscalculia have anything to teach us about developmental dyscalculia? In studying dyscalculia, should we look less at arithmetical ability as a single entity, and more at separable components of arithmetical ability? How heterogeneous is developmental dyscalculia, and how important is it to study individual profiles? To what extent is developmental dyscalculia influenced by domain-specific versus domain-general abilities? The conclusion is that, though a significant amount has been discovered through existing research, and though this has some important implications for screening and diagnosis of dyscalculia, there is much more research that still needs to be conducted if we are to answer all of these questions fully. In particular, the study of developmental dyscalculia must be more integrated with the study of individual differences in mathematics in the population as a whole.

Heterogeneity in children at risk of math learning difficulties

This study recruited 428 Singaporean children at risk of math learning difficulties (MLD; M_age  = 83.9 months, SD_age  = 4.35 months; 41% female). Using a factor mixture model that considered both quantitative and qualitative differences in math ability, two qualitatively different groups were identified: one with generalized difficulties across different math skills and the other with more focal difficulties in arithmetic fluency. Reading, working memory capacity, and numeracy (number line estimation skills and numerical discrimination) uniquely explained group membership. Children within each group differed in the extent of difficulties they exhibited, with numeracy variables differentially contributing to math ability in each group. Findings speak against a dimensional view of MLD and underscore the conceptual limitations of using basic numeracy performance to profile learning difficulties.

How many seconds was that? Teaching children about time does not refine their ability to track durations

Over development, children acquire symbols to represent abstract concepts such as time and number. Despite the importance of quantity symbols, it is unknown how acquiring these symbols impacts one's ability to perceive quantities (i.e., nonsymbolic representations). While it has been proposed that learning symbols shapes nonsymbolic quantitative abilities (i.e., the refinement hypothesis), this hypothesis has been understudied, especially in the domain of time. Moreover, the majority of research in support of this hypothesis has been correlational in nature, and thus, experimental manipulations are critical for determining whether this relation is causal. In the present study, kindergarteners and first graders (N = 154) who have yet to learn about temporal symbols in school completed a temporal estimation task during which they were either (1) trained on temporal symbols and effective timing strategies ("2 s" and counting on the beat), (2) trained on temporal symbols only ("2 s"), or (3) participated in a control training. Children's nonsymbolic and symbolic timing abilities were assessed before and after training. Results revealed a correlation between children's nonsymbolic and symbolic timing abilities at pre-test (when controlling for age), indicating this relation exists prior to formal classroom instruction on temporal symbols. Notably, we found no support for the refinement hypothesis, as learning temporal symbols did not impact children's nonsymbolic timing abilities. Implications and future directions are discussed.

Omit needless words: Sentence length perception

Short sentences improve readability. Short sentences also promote social justice through accessibility and inclusiveness. Despite this, much remains unknown about sentence length perception-an important factor in producing readable writing. Accordingly, we conducted a psychophysical study using procedures from Signal Detection Theory to examine sentence length perception in naive adults. Participants viewed real-world full-page text samples and judged whether a bolded target sentence contained more or fewer than 17 words. The experiment yielded four findings. First, naïve adults perceived sentence length in real-world text samples quickly (median = 300-400 ms) and precisely (median = ~90% correct). Second, flipping real-world text samples upside-down generated no reaction-time cost and nearly no loss in the precision of sentence length perception. This differs from the large inversion effects that characterize other highly practiced, real-world perceptual tasks involving canonically oriented stimuli, most notably face perception and reading. Third, participants significantly underestimated the length of mirror-reversed sentences-but not upside-down, nor standard sentences. This finding parallels participants' familiarity with commonly occurring left-justified right-ragged text, and suggests a novel demonstration of left-lateralized anchoring in scene syntax. Fourth, error patterns demonstrated that participants achieved their high speed, high precision sentence-length judgments by heuristically counting text lines, not by explicitly counting words. This suggests practical advice for writing instructors to offer students. When copy editing, students can quickly and precisely identify their long sentences via a line-counting heuristic, e.g., "a 17-word sentence spans about 1.5 text lines". Students can subsequently improve a long sentence's readability and inclusiveness by omitting needless words.

Different impacts of long-term abacus training on symbolic and non-symbolic numerical magnitude processing in children

Abacus-based mental calculation (AMC) has been shown to be effective in promoting math ability in children. Given that AMC relies on a visuospatial strategy to perform rapid and precise arithmetic, previous studies mostly focused on the promotion of AMC training on arithmetic ability and mathematical visual-spatial ability, as well as its transfer of advanced cognitive ability. However, little attention has been given to its impact on basic numerical comparison ability. Here, we aim to examine whether and how long-term AMC training impacts symbolic and non-symbolic numerical comparisons. The distance effect (DE) was utilized as a marker, indicating that the comparison between two numbers becomes faster as their numerical distance enlarges. In the current study, forty-one children matched for age and sex were recruited at primary school entry and randomly assigned to the AMC group and the control group. After three years of training, the event-related potential (ERP) recording technique was used to explore the temporal dynamics of number comparison, of which tasks were given in symbolic (Arabic number) or non-symbolic (dot array) format. In the symbolic task, the children in the AMC group showed a smaller DE than those in the control group. Two ERP components, N1 and P2p, located in parietal areas (PO7, PO8) were selected as neural markers of numerical processing. Both groups showed DE in the P2p component in both tasks, but only the children in the AMC group showed DE in the N1 component in the non-symbolic task. In addition, the DE size calculated from reaction times and ERP amplitudes was correlated with higher cognitive capacities, such as coding ability. Taken together, the present results provide evidence that long-term AMC training may be beneficial for numerical processing in children, which may be associated with neurocognitive indices of parietal brain regions.

The association between non-symbolic number comparison and mathematical abilities depends on fluency

Numerous studies have explored the correlation between non-symbolic number comparison and mathematical abilities in children, but the results have been inconsistent. The underlying mental processing featuring fluency may affect the correlation. The current study tested the fluency hypothesis that non-symbolic number comparison is associated with mathematical fluency in the development of mathematical ability. Non-symbolic number comparison, arithmetic computation, mathematical reasoning, non-symbolic number estimation, symbolic number comparison, and a series of basic cognitive processing tasks, including mental rotation, non-verbal matrix reasoning, and choice reaction time, were administered to 1072 first- to fourth-grade children. The results show that non-symbolic number comparison (measured via numerosity comparison) was the only independent predictor of arithmetic computation in higher grades, even after controlled for age, gender, basic cognitive processing, non-symbolic number estimation (measured via numerosity estimation), and symbolic number comparison (measured via digit comparison). However, it did not correlate with mathematical reasoning in any grade. These findings support the fluency hypothesis for developmental correlation between non-symbolic number comparison and mathematical abilities. That is, non-symbolic number comparison correlates with mathematical ability featuring fluency.

Can training in the approximate number system improve the informal mathematics ability of preschoolers?

Recent studies show that comparison or arithmetic training in the approximate number system (ANS) can improve the early mathematics ability of preschool children. However, no studies have compared the training effects of ANS comparison training with those of ANS arithmetic training on the early mathematics ability of preschool children. The current study pseudorandomly assigned 87 children aged 4-5 years to one of three training groups (the ANS comparison, ANS arithmetic, and control groups) for 4 weeks of training. The results showed that compared with the control group, the ANS comparison training and ANS arithmetic training equally improved the ANS acuity and informal mathematics ability of preschool children. In addition, the study found that there may be a bidirectional causal relationship between ANS and mathematics in preschoolers, but this relationship needs to be further investigated using longitudinal studies. Taken together, these findings emphasize the importance of ANS-based training in improving preschoolers' ANS acuity and informal mathematics ability before formal school enrollment.

Visual Perception Supports Adults in Numerosity Processing and Arithmetical Performance

Previous studies have found a correlation between numerosity processing and arithmetical performance. Visual perception has already been indicated as the shared cognitive mechanism between these two; however, these studies mostly focused on children. It is not clear whether the association between numerosity processing and arithmetical performance still existed following the development of individual arithmetical performance. Consequently, the underlying role of visual perception in numerosity processing and arithmetical performance has not been sufficiently studied in adults. For this study, researchers selected a total of 205 adult participants with an average age of 22years. The adults were administered arithmetic tests, numerosity comparison, and visual figure matching. Mental rotation, choice reaction time, and nonverbal intelligence were used as cognitive covariates. Results showed that numerosity comparison of adults correlated with their arithmetical performance, even after controlling for age and gender differences as well as general cognitive processing. However, after controlled for visual figure matching, the well-established association between numerosity comparison and arithmetic performance disappeared. These results supported the visual perception hypothesis, that visual perception measured by visual figure matching can account for the correlation between numerosity comparison and arithmetic performance. This indicated that even for adult populations, visual perceptual ability was the underlying component of numerosity processing and arithmetic performance.

Cognitive mediators of US-China differences in early symbolic arithmetic

Chinese children routinely outperform American peers in standardized tests of mathematics knowledge. To examine mediators of this effect, 95 Chinese and US 5-year-olds completed a test of overall symbolic arithmetic, an IQ subtest, and three tests each of symbolic and non-symbolic numerical magnitude knowledge (magnitude comparison, approximate addition, and number-line estimation). Overall Chinese children performed better in symbolic arithmetic than US children, and all measures of IQ and number knowledge predicted overall symbolic arithmetic. Chinese children were more accurate than US peers in symbolic numerical magnitude comparison, symbolic approximate addition, and both symbolic and non-symbolic number-line estimation; Chinese and U.S. children did not differ in IQ and non-symbolic magnitude comparison and approximate addition. A substantial amount of the nationality difference in overall symbolic arithmetic was mediated by performance on the symbolic and number-line tests.

Relations among spatial skills, number line estimation, and exact and approximate calculation in young children

Decades of research have established that spatial skills correlate with numerical skills. However, because both spatial and numerical skills are multidimensional, we sought to determine how specific spatial skills relate to specific numeracy skills. We used a cohort-sequential design, assessing a large diverse sample of students (N = 612, initially in pre-kindergarten [pre-K]-3rd grade, 4-9 years of age) at four time points spanning 2 years. We examined how initial levels of five spatial skills (visuospatial working memory [VSWM], mental transformation, mental rotation, proportional reasoning, and analog magnitude system [AMS] acuity) related to initial levels and growth rates in exact and approximate calculation skills, and we further investigated number line estimation as a potential mediator. We found unique patterns of relations between spatial skills and numeracy. Initial levels of mental rotation, proportional reasoning, and AMS acuity related to initial levels of exact calculation skill; initial levels of AMS acuity related to initial levels of approximate calculation; and initial levels of proportional reasoning related to initial levels of number line estimation. VSWM and mental transformation did not relate to numeracy skills after controlling for other spatial skills. Initial levels of number line estimation related to both exact and approximate calculation after controlling for spatial skills. Notably, neither spatial skills nor number line estimation predicted growth in exact or approximate calculation skills. These results indicate that there is specificity in the time-invariant relations between spatial skills and numeracy, and they suggest that researchers and educators should treat spatial skills and numeracy as multidimensional constructs with complex and unique interrelations.

Approximate arithmetic training does not improve symbolic math in third and fourth grade children

BACKGROUND

Prior studies reported that practice playing an approximate arithmetic game improved symbolic math performance relative to active control groups in adults and preschool children (e.g. Park & Brannon, 2013, 2014; Park et al., 2016; Szkudlarek & Brannon, 2018). However, Szkudlarek, Park and Brannon (2021) recently failed to replicate those findings in adults. Here we test whether approximate arithmetic training yields benefits in elementary school children who have intermediate knowledge of arithmetic.

METHOD

We conducted a randomized controlled trial with a pre and post-test design to compare the effects of approximate arithmetic training and visuo-spatial working memory training on standardized math performance in third and fourth grade children.

RESULTS

We found that approximate arithmetic training did not yield any significant gains on standardized measures of symbolic math performance.

CONCLUSION

A Bayesian analysis supports the conclusion that approximate arithmetic provides no benefits for symbolic math performance.

Neural association between non-verbal number sense and arithmetic fluency

Non‐verbal number sense has been shown to significantly correlate with arithmetic fluency. Accumulated behavioral evidence indicates that the cognitive mechanism relies on visual perception. However, few studies have investigated the neural mechanism underlying this association. Following the visual perception account, we hypothesized that there would be a neural association in occipital areas of the brain between non‐verbal number sense, arithmetic fluency, and visual perception. We analyzed event‐related potentials that are sensitive to neural responses while participants performed five cognitive tasks: simple addition, simple subtraction, numerosity comparison, figure matching, and character rhyming. The single‐trial ERP‐behavior correlation approach was used to enhance the statistical power. The results showed that the N1 component significantly correlated with reaction time at occipital electrodes on all tasks except for character rhyming. The N1 component for arithmetic fluency (simple addition and subtraction) and character rhyming correlated with the reaction time for numerosity comparison and figure matching. The results suggest that there are neural associations between arithmetic fluency, non‐verbal number sense, and visual perception in the occipital cortex, and that visual perception is the shared mechanism for both non‐verbal number sense and arithmetic fluency.

Effects of Visual Training of Approximate Number Sense on Auditory Number Sense and School Math Ability

Research with children and adults suggests that people's math performance is predicted by individual differences in an evolutionarily ancient ability to estimate and compare numerical quantities without counting (the approximate number system or ANS). However, previous work has almost exclusively used visual stimuli to measure ANS precision, leaving open the possibility that the observed link might be driven by aspects of visuospatial competence, rather than the amodal ANS. We addressed this possibility in an ANS training study. Sixty-eight 6-year-old children participated in a 5-week study that either trained their visual ANS ability or their phonological awareness (an active control group). Immediately before and after training, we assessed children's visual and auditory ANS precision, as well as their symbolic math ability and phonological awareness. We found that, prior to training, children's precision in a visual ANS task related to their math performance - replicating recent studies. Importantly, precision in an auditory ANS task also related to math performance. Furthermore, we found that children who completed visual ANS training showed greater improvements in auditory ANS precision than children who completed phonological awareness training. Finally, children in the ANS training group showed significant improvements in math ability but not phonological awareness. These results suggest that the link between ANS precision and school math ability goes beyond visuospatial abilities and that the modality-independent ANS is causally linked to math ability in early childhood.

Understanding the unique contributions of home numeracy, inhibitory control, the approximate number system, and spontaneous focusing on number for children's math abilities

Math abilities are important predictors of both children's academic achievement and their outcomes in adulthood such as full-time employment and income. Previous work indicates that parenting factors (i.e., parental education, parent math ability, frequency of math activities) relate to children's math performance. Further, research demonstrates that both domain-general (i.e., language skills, inhibitory control) and domain-specific (i.e., approximate number system acuity, tendency to spontaneously focus on number) cognitive predictors are related to math during early childhood. However, work to date has not examined all of these factors together to identify their unique contributions for young children's math abilities. Thus, in the present study we examine whether parent-level and child-level factors uniquely explain children's math abilities. To this end, 112 four-year-old children and one of their parents completed a battery of assessments and questionnaires. Results indicate that children's math performance is uniquely predicted by the frequency of home math activities reported by the parents, as well as children's own inhibitory control, approximate number system acuity, and tendency to spontaneously focus on number. These parent- and child-level factors provide independent targets for future interventions aimed at improving early math performance.

Subitizing and counting impairments in children with developmental coordination disorder

Developmental coordination disorder (DCD) interferes with academic achievement and daily life, and is associated with persistent academic difficulties, in particular within mathematical learning. In the present study, we aimed to study numerical cognition using an approach that taps very basic numerical processes such as subitizing and counting abilities in DCD. We used a counting task and a subitizing task in forty 7-10 years-old children with or without DCD. In both tasks, children were presented with arrays of one to eight dots and asked to name aloud the number of dots as accurately and quickly as possible. In the subitizing task, dots were presented during 250 ms whereas in the counting task they stayed on the screen until the participants gave a verbal response. The results showed that children with DCD were less accurate and slower in the two enumeration tasks (with and without a time limit), providing evidence that DCD impairs both counting and subitizing. These impairments might have a deleterious impact on the ability to improve the acuity of the Approximate Number System through counting, and thus could play a role in the underachievement of children with DCD in mathematics.

Do the Processes Engaged During Mathematical Word-Problem Solving Differ Along the Distribution of Word-Problem Competence?

Conventional research methods for understanding sources of individual differences in word-problem solving (WPS) only permit estimation of average relations between component processes and outcomes. The purpose of the present study was instead to examine whether and if so how the component processes engaged in WPS differ along the spectrum of WPS performance. Second graders (N = 1,130) from 126 classrooms in 17 schools were assessed on component processes (reasoning, in-class attentive behavior, working memory, language comprehension, calculation fluency, word reading) and WPS. Multilevel, unconditional quantile multiple regression indicated that 3 component processes, calculation fluency, language comprehension, and working memory, are engaged in WPS differentially depending on students' overall word-problem skill. The role of calculation fluency and language comprehension was stronger with more competent word-problem solving ability. By contrast, the role of working memory was stronger with intermediate-level than for strong problem solving. Results deepen insight into the role of these processes in WPS and provide the basis for hypothesizing how instructional strategies may be differentiated depending on students' overall level of WPS competence.

Numerical Estimation and Mathematical Learning Methodology in Preschoolers

One of the means for representing quantities/magnitudes is the mental number line. It is still a nonsolved question as to whether the method of learning mathematics in the early years could improve this type of estimating. A total of 233 students, aged four and five years, who learned mathematics with a new method called Open Algorithm Based on Number or the more traditional Closed Based on Ciphers approach, were evaluated with a reliable estimation on the number line test. Results revealed significant differences in participants' estimation functions based on the learning method used. Students who learned mathematics through the Open Algorithm Based on Number method used a linear representation more efficiently than those who were taught with the Closed Based on Ciphers methodology. This group exhibited a logarithmic function in their approach. We discussed whether these differences can be attributed to a recurrent practice in estimation tasks at school, characterized by the Open Algorithm Based on Number methodology.

The Relationship Between Non-symbolic and Symbolic Numerosity Representations in Elementary School: The Role of Intelligence

This study aimed to estimate the extent to which the development of symbolic numerosity representations relies on pre-existing non-symbolic numerosity representations that refer to the Approximate Number System. To achieve this aim, we estimated the longitudinal relationships between accuracy in the Number Line (NL) test and “blue–yellow dots” test across elementary school children. Data from a four-wave longitudinal study involving schoolchildren in grades 1–4 in Russia and Kyrgyzstan (N = 490, mean age 7.65 years in grade 1) were analyzed. We applied structural equation modeling and tested several competing models. The results revealed that at the start of schooling, the accuracy in the NL test predicted subsequent accuracy in the “blue–yellow dots” test, whereas subsequently, non-symbolic representation in grades 2 and 3 predicted subsequent symbolic representation. These results indicate that the effect of non-symbolic representation on symbolic representation emerges after a child masters the basics of symbolic number knowledge, such as counting in the range of twenty and simple arithmetic. We also examined the extent to which the relationships between non-symbolic and symbolic representations might be explained by fluid intelligence, which was measured by Raven’s Standard Progressive Matrices test. The results revealed that the effect of symbolic representation on non-symbolic representation was explained by fluid intelligence, whereas at the end of elementary school, non-symbolic representation predicted subsequent symbolic representation independently of fluid intelligence.

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.

Does 1 + 1 = 2nd? The relations between children's understanding of ordinal position and their arithmetic performance

The current study examined the relations between 5- and 6-year-olds' understanding of ordinality and their mathematical competence. We focused specifically on "positional operations," a property of ordinality not contingent on magnitude, in an effort to better understand the unique contributions of position-based ordinality to math development. Our findings revealed that two types of positional operations-the ability to execute representational movement along letter sequences and the ability to update ordinal positions after item insertion or removal-predicted children's arithmetic performance. Nevertheless, these positional operations did not mediate the relation between magnitude processing (as measured by the acuity of the approximate number system) and arithmetic performance. Taken together, these findings suggest a unique role for positional ordinality in math development. We suggest that positional ordinality may aid children in their mental organization of number symbols, which may facilitate solving arithmetic computations and may support the development of novel numerical concepts.

Preschoolers and multi-digit numbers: A path to mathematics through the symbols themselves

Numerous studies from developmental psychology have suggested that human symbolic representation of numbers is built upon the evolutionally old capacity for representing quantities that is shared with other species. Substantial research from mathematics education also supports the idea that mathematical concepts are best learned through their corresponding physical representations. We argue for an independent pathway to learning "big" multi-digit symbolic numbers that focuses on the symbol system itself. Across five experiments using both between- and within-subject designs, we asked preschoolers to identify written multi-digit numbers with their spoken names in a two-alternative-choice-test or to indicate the larger quantity between two written numbers. Results showed that preschoolers could reliably map spoken number names to written forms and compare the magnitudes of two written multi-digit numbers. Importantly, these abilities were not related to their non-symbolic representation of quantities. These findings have important implications for numerical cognition, symbolic development, teaching, and education.

Mathematics Competence Level: The Contribution of Non-symbolic and Spatial Magnitude Comparison Skills

Magnitude comparison skills have been related to mathematics competence, although results in this area vary. The current study aimed to describe the performance of 75 children (aged 4-5 years) in two comparison tasks; and examine the strength of the relationship between each of the two tasks and mathematics competence level (MCL). Participants were assessed with the Early Numeracy Test which provides a global MCL score. Magnitude comparison skills were assessed with two tasks: a non-symbolic number comparison task and a spatial comparison task. Results of the Pearson correlation analysis showed a relationship between the two tasks with better performance in the spatial comparison task. Regression analysis with the stepwise method showed that only the non-symbolic number comparison task had a significant value in the prediction of the MCL pointing to the need to take these kinds of tasks into account in the first years of school.

Metaphor processing in middle childhood and at the transition to early adolescence: the role of chronological age, mental age, and verbal intelligence

Metaphor development in conjunction with verbal intelligence and linguistic competence in middle childhood and at the transition to early adolescence was investigated. 298 individuals between seven and ten years (chronological age) who attended grades two-four (mental age) were tested for metaphor processing by the Metaphoric Triads Task, for linguistic competence (HELD), and verbal intelligence (WISC-III). Chronological age significantly predicted metaphor processing with a breakpoint of 8.2 years regarding identification and comprehension, and 10.2 years regarding preference. Fourth-graders showed highest metaphor processing scores. Verbal intelligence significantly predicted metaphor processing; this effect became stronger with increasing age. Attributional metaphors were best understood and most preferred. Chronological and mental age are associated with metaphor processing in an age span that is seemingly crucial for metaphor development. Verbal analogical reasoning, concept formation, verbal abstraction, and semantic knowledge predicted metaphor comprehension. Understanding facts, principles, and social situations, and resultant inferential verbal reasoning predicted metaphor preference.

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.

Cross-magnitude interactions across development: Longitudinal evidence for a general magnitude system

There is general agreement that humans represent numerical, spatial, and temporal magnitudes from early in development. However, there is disagreement about whether different magnitudes converge within a general magnitude system and whether this system supports behavioral demonstrations of cross-magnitude interactions at different developmental time points. Using a longitudinal design, we found a relation between children's cross-magnitude interactions assessed at two developmental time points with different behavioral measures. More specifically, stronger cross-magnitude interactions in infancy (M = 9.3 months) predicted a stronger cross-magnitude congruity effect at preschool age (M = 44.2 months), even when controlling for performance on measures of inhibitory control, analogical reasoning, and verbal competence at preschool age. The results suggest a common mechanism for cross-magnitude interactions at different points in development as well as stability of the underlying individual differences. We argue that this mechanism reflects a nonverbal general magnitude system that is operational early in life and that displays continuity from infancy to preschool age.

Using Hierarchical Linear Models to Examine Approximate Number System Acuity: The Role of Trial-Level and Participant-Level Characteristics

The ability to intuitively and quickly compare the number of items in collections without counting is thought to rely on the Approximate Number System (ANS). To assess individual differences in the precision of peoples' ANS representations, researchers often use non-symbolic number comparison tasks in which participants quickly choose the numerically larger of two arrays of dots. However, some researchers debate whether this task actually measures the ability to discriminate approximate numbers or instead measures the ability to discriminate other continuous magnitude dimensions that are often confounded with number (e.g., the total surface area of the dots or the convex hull of the dot arrays). In this study, we used hierarchical linear models (HLMs) to predict 132 adults' accuracy on each trial of a non-symbolic number comparison task from a comprehensive set of trial-level characteristics (including numerosity ratio, surface area, convex hull, and temporal and spatial variations in presentation format) and participant-level controls (including cognitive abilities such as visual-short term memory, working memory, and math ability) in order to gain a more nuanced understanding of how individuals complete this task. Our results indicate that certain trial-level characteristics of the dot arrays contribute to our ability to compare numerosities, yet numerosity ratio, the critical marker of the ANS, remains a highly significant predictor of accuracy above and beyond trial-level characteristics and across individuals with varying levels of math ability and domain-general cognitive abilities.

Numerical cognition is resilient to dramatic changes in early sensory experience

Humans and non-human animals can approximate large visual quantities without counting. The approximate number representations underlying this ability are noisy, with the amount of noise proportional to the quantity being represented. Numerate humans also have access to a separate system for representing exact quantities using number symbols and words; it is this second, exact system that supports most of formal mathematics. Although numerical approximation abilities and symbolic number abilities are distinct in representational format and in their phylogenetic and ontogenetic histories, they appear to be linked throughout development--individuals who can more precisely discriminate quantities without counting are better at math. The origins of this relationship are debated. On the one hand, symbolic number abilities may be directly linked to, perhaps even rooted in, numerical approximation abilities. On the other hand, the relationship between the two systems may simply reflect their independent relationships with visual abilities. To test this possibility, we asked whether approximate number and symbolic math abilities are linked in congenitally blind individuals who have never experienced visual sets or used visual strategies to learn math. Congenitally blind and blind-folded sighted participants completed an auditory numerical approximation task, as well as a symbolic arithmetic task and non-math control tasks. We found that the precision of approximate number representations was identical across congenitally blind and sighted groups, suggesting that the development of the Approximate Number System (ANS) does not depend on visual experience. Crucially, the relationship between numerical approximation and symbolic math abilities is preserved in congenitally blind individuals. These data support the idea that the Approximate Number System and symbolic number abilities are intrinsically linked, rather than indirectly linked through visual abilities.

The Developing Mental Number Line: Does Its Directionality Relate to 5- to 7-Year-Old Children's Mathematical Abilities?

Spatial representations of number, such as a left-to-right oriented mental number line, are well documented in Western culture. Yet, the functional significance of such a representation remains unclear. To test the prominent hypothesis that a mental number line may support mathematical development, we examined the relation between spatial-numerical associations (SNAs) and math proficiency in 5- to 7-year-old children. We found evidence of SNAs with two tasks: a non-symbolic magnitude comparison task, and a symbolic "Where was the number?" (WTN) task. Further, we found a significant correlation between these two tasks, demonstrating convergent validity of the directional mental number line across numerical format. Although there were no significant correlations between children's SNAs on the WTN task and math ability, children's SNAs on the magnitude comparison task were negatively correlated with their performance on a measure of cross-modal arithmetic, suggesting that children with a stronger left-to-right oriented mental number line were less competent at cross-modal arithmetic, an effect that held when controlling for age and a set of general cognitive abilities. Despite some evidence for a negative relation between SNAs and math ability in adulthood, we argue that the effect here may reflect task demands specific to the magnitude comparison task, not necessarily an impediment of the mental number line to math performance. We conclude with a discussion of the different properties that characterize a mental number line and how these different properties may relate to mathematical ability.

Children's Non-symbolic and Symbolic Numerical Representations and Their Associations With Mathematical Ability

Most empirical evidence supports the view that non-symbolic and symbolic representations are foundations for advanced mathematical ability. However, the detailed development trajectories of these two types of representations in childhood are not very clear, nor are the different effects of non-symbolic and symbolic representations on the development of mathematical ability. We assessed 253 4- to 8-year-old children's non-symbolic and symbolic numerical representations, mapping skills, and mathematical ability, aiming to investigate the developmental trajectories and associations between these skills. Our results showed non-symbolic numerical representation emerged earlier than the symbolic one. Four-year-olds were capable of non-symbolic comparisons but not symbolic comparisons; five-year-olds performed better at non-symbolic comparisons than symbolic comparisons. This performance difference disappeared at age 6. Children at age 6 or older were able to map between symbolic and non-symbolic quantities. However, as children learn more about the symbolic representation system, their advantage in non-symbolic representation disappeared. Path analyses revealed that a direct effect of children's symbolic numerical skills on their math performance, and an indirect effect of non-symbolic numerical skills on math performance via symbolic skills. These results suggest that symbolic numerical skills are a predominant factor affecting math performance in early childhood. However, the influences of symbolic and non-symbolic numerical skills on mathematical performance both declines with age.

Approximate Arithmetic Training Improves Informal Math Performance in Low Achieving Preschoolers

Recent studies suggest that practice with approximate and non-symbolic arithmetic problems improves the math performance of adults, school aged children, and preschoolers. However, the relative effectiveness of approximate arithmetic training compared to available educational games, and the type of math skills that approximate arithmetic targets are unknown. The present study was designed to (1) compare the effectiveness of approximate arithmetic training to two commercially available numeral and letter identification tablet applications and (2) to examine the specific type of math skills that benefit from approximate arithmetic training. Preschool children (n = 158) were pseudo-randomly assigned to one of three conditions: approximate arithmetic, letter identification, or numeral identification. All children were trained for 10 short sessions and given pre and post tests of informal and formal math, executive function, short term memory, vocabulary, alphabet knowledge, and number word knowledge. We found a significant interaction between initial math performance and training condition, such that children with low pretest math performance benefited from approximate arithmetic training, and children with high pretest math performance benefited from symbol identification training. This effect was restricted to informal, and not formal, math problems. There were also effects of gender, socio-economic status, and age on post-test informal math score after intervention. A median split on pretest math ability indicated that children in the low half of math scores in the approximate arithmetic training condition performed significantly better than children in the letter identification training condition on post-test informal math problems when controlling for pretest, age, gender, and socio-economic status. Our results support the conclusion that approximate arithmetic training may be especially effective for children with low math skills, and that approximate arithmetic training improves early informal, but not formal, math skills.

Innate or Acquired? - Disentangling Number Sense and Early Number Competencies

The clinical profile termed developmental dyscalculia (DD) is a fundamental disability affecting children already prior to arithmetic schooling, but the formal diagnosis is often only made during school years. The manifold associated deficits depend on age, education, developmental stage, and task requirements. Despite a large body of studies, the underlying mechanisms remain dubious. Conflicting findings have stimulated opposing theories, each presenting enough empirical support to remain a possible alternative. A so far unresolved question concerns the debate whether a putative innate number sense is required for successful arithmetic achievement as opposed to a pure reliance on domain-general cognitive factors. Here, we outline that the controversy arises due to ambiguous conceptualizations of the number sense. It is common practice to use early number competence as a proxy for innate magnitude processing, even though it requires knowledge of the number system. Therefore, such findings reflect the degree to which quantity is successfully transferred into symbols rather than informing about quantity representation per se. To solve this issue, we propose a three-factor account and incorporate it into the partly overlapping suggestions in the literature regarding the etiology of different DD profiles. The proposed view on DD is especially beneficial because it is applicable to more complex theories identifying a conglomerate of deficits as underlying cause of DD.

Frequency of Home Numeracy Activities Is Differentially Related to Basic Number Processing and Calculation Skills in Kindergartners

Home numeracy has been shown to play an important role in children's mathematical performance. However, findings are inconsistent as to which home numeracy activities are related to which mathematical skills. The present study disentangled between various mathematical abilities that were previously masked by the use of composite scores of mathematical achievement. Our aim was to shed light on the specific associations between home numeracy and various mathematical abilities. The relationships between kindergartners' home numeracy activities, their basic number processing and calculation skills were investigated. Participants were 128 kindergartners (Mage = 5.43 years, SD = 0.29, range: 4.88-6.02 years) and their parents. The children completed non-symbolic and symbolic comparison tasks, non-symbolic and symbolic number line estimation tasks, mapping tasks (enumeration and connecting), and two calculation tasks. Their parents completed a home numeracy questionnaire. Results indicated small but significant associations between formal home numeracy activities that involved more explicit teaching efforts (i.e., identifying numerals, counting) and children's enumeration skills. There was no correlation between formal home numeracy activities and non-symbolic number processing. Informal home numeracy activities that involved more implicit teaching attempts, such as "playing games" and "using numbers in daily life," were (weakly) correlated with calculation and symbolic number line estimation, respectively. The present findings suggest that disentangling between various basic number processing and calculation skills in children might unravel specific relations with both formal and informal home numeracy activities. This might explain earlier reported contradictory findings on the association between home numeracy and mathematical abilities.