Numerical ordering ability mediates the relation between number-sense and arithmetic competence

New measures of number line estimation performance reveal children's ordinal understanding of numbers

Children's performance on the number line estimation task, often measured by the percentage of absolute error, predicts their later mathematics achievement. This task may also reveal (a) children's ordinal understanding of the target numbers in relation to each other and the benchmarks (e.g., endpoints, midpoint) and (b) the ordinal skills that are a necessary precursor to children's ability to understand the interval nature of a number line as measured by percentage of absolute error. Using data from 104 U.S. kindergartners, we measured whether children's estimates were correctly sequenced across trials and correctly positioned relative to given benchmarks within trials at two time points. For both time points, we found that each ordinal error measure revealed a distinct pattern of data distribution, providing opportunities to tap into different aspects of children's ordinal understanding. Furthermore, children who made fewer ordinal errors scored higher on the Test of Early Mathematics Ability and showed greater improvement on their interval understanding of numbers as reflected by a larger reduction of percentage of absolute error from Time 1 to Time 2. The findings suggest that our number line measures reveal individual differences in children's ordinal understanding of numbers, and that such understanding may be a precursor to their interval understanding and later mathematics performance.

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.

(Dis)similarities between non-symbolic and symbolic number representations: Insights from vector space models

Empirical evidence in support of a shared system for non-symbolic and symbolic number processing has been inconclusive. The current study aims to address this question in a novel way, specifically by testing whether the efficient coding principle based on co-occurrence of number symbols in natural language holds for both non-symbolic and symbolic number processing. The efficient coding principle postulates that perception is optimized when stimuli frequently co-occur in a natural environment. We hypothesized that both numerical ratios and co-occurrence frequencies of symbolic numbers would significantly influence participants' performance on a non-symbolic and symbolic number comparison task. To test this hypothesis, we employed latent semantic analysis on a TASA corpus to quantify number co-occurrence in natural language and calculate language similarity estimates. We engaged 73 native English speakers (mean age = 19.36, standard deviation = 1.83) with normal or corrected vision and no learning disorders in a number comparison task involving non-symbolic (dot arrays) and symbolic stimuli (Arabic numerals and English number words). Results showed that numerical ratios significantly predicted participants' performances across all number formats (ps < 0.001). Language similarity estimates derived from everyday language also predicted performance on the non-symbolic task and the symbolic task involving number words (ps < 0.007). Our results highlight the complex nature of numerical processing, pointing to the co-occurrence of number symbols in natural language as an auxiliary factor in understanding the shared characteristics between non-symbolic and symbolic number representations. Given that our study focused on a limited number range (5 to 16) and a specific task type, future studies should explore a wider range of tasks and numbers to further test the role of the efficient coding principle in number processing.

Benefits of Playing at School: Filler Board Games Improve Visuospatial Memory and Mathematical Skills

The aim of the study was to test the effectiveness of cognitive interventions based on modern board games in school settings to improve memory outcomes and math skills. A parallel, quasi-experimental study was carried out with children (n = 234) into third and fourth grades (8-10 years old). School centres were allocated into a general domain intervention (playing memory board games), a specific domain intervention (playing mathematical board games) or a control group (regular classes without playing). Teachers carried out bi-weekly sessions during the last 30 min of mathematical lessons (8 weeks, 15 sessions). Before and after intervention, we individually measured verbal and visuospatial memory outcomes (short-term memory and working memory updating) and mathematical skills (number operations, number ranking, number production and problem solving). The results showed significant transfer effects of both memory and math trainings. In third grade, we found that playing math games showed medium-large effect sizes in visuospatial short-term memory and updating memory, number operations and number ranking compared to the control group. In fourth grade, we found that playing memory games showed significant small effect sizes in problem solving compared to the control group. Playing board games could be a methodology that enhances cognitive and mathematical development in children.

The inferior fronto-occipital fasciculus correlates with early precursors of mathematics and reading before the start of formal schooling

Diffusion-weighted imaging studies in preschoolers have almost exclusively been done in the field of reading. As a result, virtually nothing is known about white matter tracts associated with individual differences in mathematics at this age. Studying the preschoolers' brain is crucial because it allows us to identify individual differences in brain anatomy without influences of formal mathematics and reading instruction. To fill this gap, we investigated for the first time before the start of formal school entry the associations between white matter tracts and precursors of mathematics and reading simultaneously. We also investigated whether these associations were specific to mathematics and to reading, or not. We focused on four bilateral white matter tracts (arcuate fasciculus (direct, anterior), inferior fronto-occipital fasciculus, inferior longitudinal fasciculus), which have been previously correlated with mathematical performance in older children and with reading performance in children of a similar age as the current study. Participants were 56 5-year-old children (Mage = 67 months; SD = 1.8), none of which received formal instruction. Our results showed an association between the bilateral inferior fronto-occipital fasciculus and precursors of mathematics (numerical ordering, numeral knowledge) and reading (phonological awareness, letter knowledge). Follow-up regression analyses revealed that the associations found with the inferior fronto-occipital fasciculus were neither specific to mathematics nor specific to reading. These findings suggest that, already before the start of formal schooling, the inferior fronto-occipital fasciculus might be related to the neural overlap between mathematics and reading. This overlap potentially reflects one of their many shared mechanisms, such as the reliance on phonological codes or the processing of visual symbols, and these mechanisms should be exploited in future studies.

The longitudinal contribution of mapping to arithmetic: Do numeral knowledge, inhibition or analogical reasoning matter?

BACKGROUND

Recent studies have revealed the association between mapping and arithmetic (Ferres-Forga et al., J. Numer. Cogn., 8, 2022, 123; LeFevre et al., J. Numer. Cogn., 8, 2022, 1).

AIM

The underlying mechanism remains unclear.

MATERIALS & METHODS

The current study recruited 118 kindergarten children and followed up on them three times at 6-month intervals. They completed measures to assess mapping skills (T1), non-verbal IQ (T1), numeral knowledge (T2), inhibitory control (T2), analogical reasoning (T2) and arithmetic (T3).

RESULTS

The results showed that mapping accounted for significant variance in arithmetic ability over and above age, gender and non-verbal IQ. Furthermore, analogical reasoning played an important role in the relationship between mapping and mathematics ability.

DISCUSSION

The findings suggest the association between mapping and mathematics ability prior to formal schooling.

CONCLUSION

Analogical reasoning, rather than numeral knowledge or inhibitory control, may drive that association in young children.

How learning influences non-symbolic numerical processing: effects of feedback in the dot comparison task

It has long been debated how humans estimate the numerosity of sets of elements and what role continuous visual properties play in this process. The dot comparison task, in which the more numerous of two dot arrays must be selected, is a dominant method to investigate this phenomenon. It has been shown that the visual properties of the two dot patterns strongly influence the comparison. This influence can be systematically investigated by manipulating visual properties congruently and incongruently with numerosity. However, it remains unclear how learning and prior experience affect the influence of the visual properties. To address this question, we introduced feedback into the classical dot comparison task: during the learning phase, participants in the experimental group received feedback after each trial indicating whether their answer was correct whereas participants in the control group did not. After the learning phase, neither group received feedback. The convex hull of the dot patterns and the average dot diameter were manipulated congruently and incongruently with numerosity. Our results show that feedback had no effect on overall performance. However, when manipulated separately, dot diameter no longer affected performance in the experimental group after the learning phase, but it did in the control group. Moreover, this effect remained visible even when diameter and convex hull were manipulated simultaneously. This pattern of results is consistent with the notion of sensory integration which proposes that weights are assigned to different visual cues and that numerical judgments depend on an additive combination of these weights. We also found a correlation between performance on an arithmetic task and performance on trials in which dot size was manipulated incongruently with numerosity. However, there were no correlations between an inhibition task and performance in the dot comparison task. Taken together, the current results suggest that learning with feedback may affect some visual properties but not others. Future studies should further investigate a wider range of visual properties to examine which of them can be influenced by learning and under what conditions learning occurs.

Basic Symbolic Number Skills, but Not Formal Mathematics Performance, Longitudinally Predict Mathematics Anxiety in the First Years of Primary School

Mathematical anxiety (MA) and mathematics performance typically correlate negatively in studies of adolescents and adults, but not always amongst young children, with some theorists questioning the relevance of MA to mathematics performance in this age group. Evidence is also limited in relation to the developmental origins of MA and whether MA in young children can be linked to their earlier mathematics performance. To address these questions, the current study investigated whether basic and formal mathematics skills around 4 and 5 years of age were predictive of MA around the age of 7-8. Additionally, we also examined the cross-sectional relationships between MA and mathematics performance in 7-8-year-old children. Specifically, children in our study were assessed in their first (T1; aged 4-5), second (T2; aged 5-6), and fourth years of school (T3; aged 7-8). At T1 and T2, children completed measures of basic numerical skills, IQ, and working memory, as well as curriculum-based mathematics tests. At T3, children completed two self-reported MA questionnaires, together with a curriculum-based mathematics test. The results showed that MA could be reliably measured in a sample of 7-8-year-olds and demonstrated the typical negative correlation between MA and mathematical performance (although the strength of this relationship was dependent on the specific content domain). Importantly, although early formal mathematical skills were unrelated to later MA, there was evidence of a longitudinal relationship between basic early symbolic number skills and later MA, supporting the idea that poorer basic numerical skills relate to the development of MA.

Electrophysiological Comparison of Cumulative Area and Non-Symbolic Number Judgments

Despite the importance of representing different magnitudes (i.e., number and cumulative area) for action planning and formal mathematics, there is much debate about the nature of these representations, particularly the extent to which magnitudes interact in the mind and brain. Early interaction views suggest that there are shared perceptual processes that form overlapping magnitude representations. However, late interaction views hold that representations of different magnitudes remain distinct, interacting only when preparing a motor response. The present study sheds light on this debate by examining the temporal onset of ratio and congruity effects as participants made ordinal judgments about number and cumulative area. Event-related potentials (ERPs) were recorded to identify whether the onset of such effects aligned with early versus late views. Ratio effects for both magnitudes were observed starting in the P100. Moreover, a congruity effect emerged within the P100. That interactions were observed early in processing, at the same time that initial ratio effects occurred, suggests that number and cumulative area processes interacted when magnitude representations were being formed, prior to preparing a decision response. Our findings are consistent with an early interaction view of magnitude processing, in which number and cumulative area may rely on shared perceptual mechanisms.

Learning artificial number symbols with ordinal and magnitude information

The question of how numerical symbols gain semantic meaning is a key focus of mathematical cognition research. Some have suggested that symbols gain meaning from magnitude information, by being mapped onto the approximate number system, whereas others have suggested symbols gain meaning from their ordinal relations to other symbols. Here we used an artificial symbol learning paradigm to investigate the effects of magnitude and ordinal information on number symbol learning. Across two experiments, we found that after either magnitude or ordinal training, adults successfully learned novel symbols and were able to infer their ordinal and magnitude meanings. Furthermore, adults were able to make relatively accurate judgements about, and map between, the novel symbols and non-symbolic quantities (dot arrays). Although both ordinal and magnitude training was sufficient to attach meaning to the symbols, we found beneficial effects on the ability to learn and make numerical judgements about novel symbols when combining small amounts of magnitude information for a symbol subset with ordinal information about the whole set. These results suggest that a combination of magnitude and ordinal information is a plausible account of the symbol learning process.

Implicit Processing of Numerical Order: Evidence from a Continuous Interocular Flash Suppression Study

Processing the ordered relationships between sequential items is a key element in many cognitive abilities that are important for survival. Specifically, order may play a crucial role in numerical processing. Here, we assessed the existence of a cognitive system designed to implicitly evaluate numerical order, by combining continuous flash suppression with a priming method in a numerical enumeration task. In two experiments and diverse statistical analysis, targets that required numerical enumeration were preceded by an invisibly ordered or non-ordered numerical prime sequence. The results of both experiments showed that enumeration for targets that appeared after an ordered prime was significantly faster, while the ratio of the prime sequences produced no significant effect. The findings suggest that numerical order is processed implicitly and affects a basic cognitive ability: enumeration of quantities.

Symbolic number ordering strategies and math anxiety

Math anxiety results in a drop in performance on various math-related tasks, including the symbolic number ordering task in which participants decide whether a triplet of digits is presented in order (e.g. 3-5-7) or not (e.g. 3-7-5). We investigated whether the strategy repertoire and reaction times during a symbolic ordering task were affected by math anxiety. In study 1, participants performed an untimed symbolic number ordering task and indicated the strategy they used on a trial-by-trial basis. The use of the memory retrieval strategy, based on the immediate recognition of the triplet, decreased with high math anxiety, but disappeared when controlling for general anxiety. In the study 2, participants completed a timed version of the number order task. High math-anxious participants used the decomposition strategy (e.g. 5 is larger than 3 and 7 is larger than 5 to decide whether 3-5-7 is in the correct order) more often, and were slower in responding when both memory- and other decomposition strategies were used. Altogether, both studies demonstrate that high-math anxious participants are not only slower to decide whether a number triplet is in the correct order, but also rely more on procedural strategies.

The plural counts: Inconsistent grammatical number hinders numerical development in preschoolers - A cross-linguistic study

The role of grammar in numerical development, and particularly the role of grammatical number inflection, has already been well-documented in toddlerhood. It is unclear, however, whether the influence of grammatical language structure further extends to more complex later stages of numerical development. Here, we addressed this question by exploiting differences between Polish, which has a complex grammatical number paradigm, leading to a partially inconsistent mapping between numerical quantities and grammatical number, and German, which has a comparatively easy verbal paradigm: 151 Polish-speaking and 123 German-speaking kindergarten children were tested using a symbolic numerical comparison task. Additionally, counting skills (Give-a-Number and count-list), and mapping between non-symbolic (dot sets) and symbolic representations of numbers, as well as working memory (Corsi blocks and Digit span) were assessed. Based on the Give-a-Number and mapping tasks, the children were divided into subset-knowers, CP-knowers-non-mappers, and CP-knowers-mappers. Linguistic background was related to performance in several ways: Polish-speaking children expectedly progressed to the CP-knowers stage later than German children, despite comparable non-numerical capabilities, and even after this stage was achieved, they fared worse in the numerical comparison task. There were also meaningful differences in spatial-numerical mapping between the Polish and German groups. Our findings are in line with the theory that grammatical number paradigms influence. the development of representations and processing of numbers, not only at the stage of acquiring the meaning of the first number-words but at later stages as well, when dealing with symbolic numbers.

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.

Nonparametric cognitive diagnosis of profiles of mathematical knowledge of teacher education candidates

Traditionally, the selection process of teacher candidates has emphasized the assessment of subject matter and pedagogical knowledge using psychometric methodologies, which simply organize candidates in continuous scales and require a large number of samples. However, these methods do not allow for the identification of candidates' knowledge profiles and learning paths, which would help develop programs tailored to support students in their training process. In this study, an evaluation instrument was developed by using the nonparametric approach to model diagnostic classifications and was then validated on a sample of 119 participants. This instrument allows for disaggregating candidates' initial knowledge and establishing relationships between its components. The results showed that candidates present a variety of profiles, which may consider more than one attribute. Not only does it provide a score that can be used for selection processes, it also provides useful information for initial teacher training methods.

Reduced choice-confidence in negative numerals

Negative numbers are central in math. However, they are abstract, hard to learn, and manipulated slower than positive numbers regardless of math ability. It suggests that confidence, namely the post-decision estimate of being correct, should be lower than positives. We asked participants to pick the larger single-digit numeral in a pair and collected their implicit confidence with button pressure (button pressure was validated with three empirical signatures of confidence). We also modeled their choices with a drift-diffusion decision model to compute the post-decision estimate of being correct. We found that participants had relatively low confidence with negative numerals. Given that participants compared with high accuracy the basic base-10 symbols (0–9), reduced confidence may be a general feature of manipulating abstract negative numerals as they produce more uncertainty than positive numerals per unit of time.

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.

Mathematics and Numerosity but Not Visuo-Spatial Working Memory Correlate with Mathematical Anxiety in Adults

Many individuals, when faced with mathematical tasks or situations requiring arithmetic skills, experience exaggerated levels of anxiety. Mathematical anxiety (MA), in addition to causing discomfort, can lead to avoidance behaviors and then to underachievement. However, the factors inducing MA and how MA deploys its detrimental effects are still largely debated. There is evidence suggesting that MA affects working memory capacity by further diminishing its limited processing resources. An alternative account postulates that MA originates from a coarse early numerical cognition capacity, the perception of numerosity. In the current study, we measured MA, math abilities, numerosity perception and visuo-spatial working memory (VSWM) in a sample of neurotypical adults. Correlational analyses confirmed previous studies showing that high MA was associated with lower math scores and worse numerosity estimation precision. Conversely, MA turned out to be unrelated to VSWM capacities. Finally, partial correlations revealed that MA fully accounted for the relationship between numerosity estimation precision and math abilities, suggesting a key role for MA as a mediating factor between these two domains.

Extending ideas of numerical order beyond the count-list from kindergarten to first grade

Ordinal processing plays a fundamental role in both the representation and manipulation of symbolic numbers. As such, it is important to understand how children come to develop a sense of ordinality in the first place. The current study examines the role of the count-list in the development of ordinal knowledge through the investigation of two research questions: (1) Do K-1 children struggle to extend the notion of numerical order beyond the count-list, and if so (2) does this extension develop incrementally or manifest as a qualitative re-organization of how children recognize the ordinality of numerical sequences. Overall, we observed that although young children reliably identified adjacent ordered sequences (i.e., those that match the count-list; '2-3-4') as being in the correct ascending order, they performed significantly below chance on non-adjacent ordered trials (i.e., those that do not match the count-list but are in the correct order; '2-4-6') from the beginning of kindergarten to the end of first grade. Further, both qualitative and quantitative analyses supported the conclusion that the ability to extend notions of ordinality beyond the count-list emerged as a conceptual shift in ordinal understanding rather than through incremental improvements. These findings are the first to suggest that the ability to extend notions of ordinality beyond the count-list to include non-adjacent numbers is non-trivial and reflects a significant developmental hurdle that most children must overcome in order to develop a mature sense of ordinality.

The relationship between numerosity perception and mathematics ability in adults: the moderating role of dots number

Background

It has been proposed that numerosity perception is the cognitive underpinning of mathematics ability. However, the existence of the association between numerosity perception and mathematics ability is still under debate, especially in adults. The present study examined the relationship between numerosity perception and mathematics ability and the moderating role of dots number (i.e., the numerosity of items in dot set) in adults.

Methods

Sixty-four adult participants from Anshun University completed behavioral measures that tested numerosity perception of small numbers and large numbers, mathematics ability, inhibition ability, visual-spatial memory, and set-switching ability.

Results

We found that numerosity perception of small numbers correlated significantly with mathematics ability after controlling the influence of inhibition ability, visual-spatial memory, and set-switching ability, but numerosity perception of large numbers was not related to mathematics ability in adults.

Conclusions

These findings suggest that the dots number moderates the relationship between numerosity perception and mathematics ability in adults and may contribute to explaining the contradictory findings in the previous literature about the link between numerosity perception and mathematics ability.

When A is greater than B: Interactions between magnitude and serial order

Processing ordinal information is an important aspect of cognitive ability, yet the nature of such ordinal representations remains largely unclear. Previously, it has been suggested that ordinal position is coded as magnitude, but this claim has not yet received direct empirical support. This study examined the nature of ordinal representations using a Stroop-like letter order judgment task. If ordinal position is coded as magnitude, then letter ordering and font size should interact. Experiments 1 and 2 identified a significant interaction between letter size and ordering. Specifically, a facilitation effect was observed for alphabetically ordered sequences with decreasing font size (e.g., B C D). This suggests an overlap in the mechanisms for order and magnitude processing. The finding also suggests that earlier ranks may be represented as "more" in such a magnitude-based code, and vice versa for later ranks.

A rational explanation for links between the ANS and math

The proposal by Clarke and Beck offers a new explanation for the association between the approximate number system (ANS) and math. Previous explanations have largely relied on developmental arguments, an underspecified notion of the ANS as an "error detection mechanism," or affective factors. The proposal that the ANS represents rational numbers suggests that it may directly support a broader range of math skills.

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.

Common and distinct predictors of non-symbolic and symbolic ordinal number processing across the early primary school years

What are the cognitive mechanisms supporting non-symbolic and symbolic order processing? Preliminary evidence suggests that non-symbolic and symbolic order processing are partly distinct constructs. The precise mechanisms supporting these skills, however, are still unclear. Moreover, predictive patterns may undergo dynamic developmental changes during the first years of formal schooling. This study investigates the contribution of theoretically relevant constructs (non-symbolic and symbolic magnitude comparison, counting and storage and manipulation components of verbal and visuo-spatial working memory) to performance and developmental change in non-symbolic and symbolic numerical order processing. We followed 157 children longitudinally from Grade 1 to 3. In the order judgement tasks, children decided whether or not triplets of dots or digits were arranged in numerically ascending order. Non-symbolic magnitude comparison and visuo-spatial manipulation were significant predictors of initial performance in both non-symbolic and symbolic ordering. In line with our expectations, counting skills contributed additional variance to the prediction of symbolic, but not of non-symbolic ordering. Developmental change in ordering performance from Grade 1 to 2 was predicted by symbolic comparison skills and visuo-spatial manipulation. None of the predictors explained variance in developmental change from Grade 2 to 3. Taken together, the present results provide robust evidence for a general involvement of pair-wise magnitude comparison and visuo-spatial manipulation in numerical ordering, irrespective of the number format. Importantly, counting-based mechanisms appear to be a unique predictor of symbolic ordering. We thus conclude that there is only a partial overlap of the cognitive mechanisms underlying non-symbolic and symbolic order processing.

Numeral order and the operationalization of the numerical system

Recent years have witnessed an increase in research on how numeral ordering skills relate to children's and adults' mathematics achievement both cross-sectionally and longitudinally. Nonetheless, it remains unknown which core competency numeral ordering tasks measure, which cognitive mechanisms underlie performance on these tasks, and why numeral ordering skills relate to arithmetic and math achievement. In the current study, we focused on the processes underlying decision-making in the numeral order judgement task with triplets to investigate these questions. A drift-diffusion model for two-choice decisions was fit to data from 97 undergraduates. Findings aligned with the hypothesis that numeral ordering skills reflected the operationalization of the numerical system, where small numbers provide more evidence of an ordered response than large numbers. Furthermore, the pattern of findings suggested that arithmetic achievement was associated with the accuracy of the ordinal representations of numbers.

Order processing of number symbols is influenced by direction, but not format

This study probed the cognitive mechanisms that underlie order processing for number symbols, specifically the extent to which the direction and format in which number symbols are presented influence the processing of numerical order, as well as the extent to which the relationship between numerical order processing and mathematical achievement is specific to Arabic numerals or generalisable to other notational formats. Seventy adults who were bilingual in English and Chinese completed a Numerical Ordinality Task, using number sequences of various directional conditions (i.e., ascending, descending, mixed) and notational formats (i.e., Arabic numerals, English number words, and Chinese number words). Order processing was found to occur for ascending and descending number sequences (i.e., ordered but not non-ordered trials), with the overall pattern of data supporting the theoretical perspective that the strength and closeness of associations between items in the number sequence could underlie numerical order processing. However, order processing was found to be independent of the notational format in which the numerical stimuli were presented, suggesting that the psychological representations and processes associated with numerical order are abstract across different formats of number symbols. In addition, a relationship between the processing speed for numerical order judgements and mathematical achievement was observed for Arabic numerals and Chinese number words, and to a weaker extent, English number words. Together, our findings have started to uncover the cognitive mechanisms that could underlie order processing for different formats of number symbols, and raise new questions about the generalisability of these findings to other notational formats.

First-year students' math anxiety predicts STEM avoidance and underperformance throughout university, independently of math ability

Math anxiety is widely considered a potential barrier to success in STEM. Current thinking holds that math anxiety is directly linked to avoidance of and underperformance in STEM domains. However, past evidence supporting these claims is limited in important ways. Perhaps most crucially, it is possible that math anxiety predicts STEM outcomes merely as a proxy for poor math skills. Here, we tested the link between math anxiety and subsequent STEM outcomes by measuring math anxiety, math ability, and several covariates in 183 first-semester university students. We then tracked students' STEM avoidance and achievement through four years at university via official academic transcripts. Results showed that math anxiety predicted both a reduction in how many STEM courses students took and, separately (i.e., controlling for one another), lower STEM grades. Crucially, these associations held after controlling for math ability (and other covariates). That math anxiety predicts math-related academic achievement independently of Math Ability suggests that, contrary to current thinking, math anxiety's effects on academic performance likely operate via mechanisms other than negatively affecting math ability. Beyond this, we show evidence that math anxiety can account for associations between math ability and STEM outcomes, suggesting that past links between math ability and real-world outcomes may, in fact, be at least partially explainable by attitudes toward math. These findings provide clear impetus for developing and testing interventions that target math anxiety specifically and suggest that focusing on math ability without additional attention to math anxiety may fail to optimally boost STEM outcomes.

Verbal count sequence knowledge underpins numeral order processing in children

Recent research has suggested that numeral order processing - the speed and accuracy with which individuals can determine whether a set of digits is in numerical order or not - is related to arithmetic and mathematics outcomes. It has therefore been proposed that ordinal relations are a fundamental property of symbolic numeral representations. However, order information is also inherent in the verbal count sequence, and thus verbal count sequence knowledge may instead explain the relationship between performance on numeral order tasks and arithmetic. We explored this question with 62 children aged 6- to 8-years-old. We found that performance on a verbal count sequence knowledge task explained the relationship between numeral order processing and arithmetic. Moreover many children appeared to explicitly base their judgments of numerical order on count sequence information. This suggests that insufficient attention may have been paid to verbal number knowledge in understanding the sources of information that give meaning to numbers.

Processing of Ordinal Information in Math-Anxious Individuals

This study aimed to investigate whether the ordinal judgments of high math-anxious (HMA) and low math-anxious (LMA) individuals differ. Two groups of 20 participants with extreme scores on the Shortened Mathematics Anxiety Rating Scale (sMARS) had to decide whether a triplet of numbers was presented in ascending order. Triplets could contain one-digit or two-digit numbers and be formed by consecutive numbers (counting condition), numbers with a constant distance of two or three (balanced) or numbers with variable distances between them (neutral). All these triplets were also presented unordered: sequence order in these trials could be broken at the second (D2) or third (D3) number. A reverse distance effect (worse performance for ordered balanced than for counting trials) of equal size was found in both anxiety groups. However, HMA participants made more judgment errors than their LMA peers when they judged one-digit counting ordered triplets. This effect was related to worse performance of HMA individuals on a symmetry span test and might be related to group differences on working memory. Importantly, HMAs were less accurate than LMA participants at rejecting unordered D2 sequences. This result is interpreted in terms of worse cognitive flexibility in HMA individuals.

Separable effects of the approximate number system, symbolic number knowledge, and number ordering ability on early arithmetic development

There is evidence that early variations in the development of an approximate number system (ANS) and symbolic number understanding are both influences on the later development of formal arithmetic skills. We report a large-scale (N = 552) longitudinal study of the predictors of arithmetic spanning a critical developmental period (the first 3 years of formal education). Variations in early knowledge of symbolic representations of number and the ordinal associations between them are direct predictors of later arithmetic skills. The development of number ordering ability is in turn predicted by earlier variations in arithmetic, the ANS (numerosity judgments), and rapid automatized naming (RAN). These findings have important implications for theories of numerical and arithmetical development and potentially for the teaching of these skills.

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.

Children's Knowledge of Symbolic Number in Grades 1 and 2: Integration of Associations

How do children develop associations among number symbols? For Grade 1 children (n = 66, M = 78 months), sequence knowledge (i.e., identify missing numbers) and number comparison (i.e., choose larger number) predicted addition, both concurrently and indirectly at the end of Grade 1. Number ordering (i.e., touch numbers in order) did not predict addition but was predicted by number comparison, suggesting that magnitude associations underlie ordering performance. In contrast, for Grade 2 children (n = 80, M = 90 months), number ordering predicted addition concurrently and at the end of Grade 2; number ordering was predicted by number comparison, sequencing, and inhibitory processing. Development of symbolic number competence involves the hierarchical integration of sequence, magnitude, order, and arithmetic associations.

Longitudinally adaptive assessment and instruction increase numerical skills of preschool children

Social inequality in mathematical skill is apparent at kindergarten entry and persists during elementary school. To level the playing field, we trained teachers to assess children's numerical and spatial skills every 10 wk. Each assessment provided teachers with information about a child's growth trajectory on each skill, information designed to help them evaluate their students' progress, reflect on past instruction, and strategize for the next phase of instruction. A key constraint is that teachers have limited time to assess individual students. To maximize the information provided by an assessment, we adapted the difficulty of each assessment based on each child's age and accumulated evidence about the child's skills. Children in classrooms of 24 trained teachers scored 0.29 SD higher on numerical skills at posttest than children in 25 randomly assigned control classrooms (P = 0.005). We observed no effect on spatial skills. The intervention also positively influenced children's verbal comprehension skills (0.28 SD higher at posttest, P < 0.001), but did not affect their print-literacy skills. We consider the potential contribution of this approach, in combination with similar regimes of assessment and instruction in elementary schools, to the reduction of social inequality in numerical skill and discuss possible explanations for the absence of an effect on spatial skills.

Grouping Mechanisms in Numerosity Perception

Enumeration of a dot array is faster and easier if the items form recognizable subgroups. This phenomenon, which has been termed "groupitizing," appears in children after one year of formal education and correlates with arithmetic abilities. We formulated and tested the hypothesis that groupitizing reflects an ability to sidestep counting by using arithmetic shortcuts, for instance, using the grouping structure to add or multiply rather than just count. Three groups of students with different levels of familiarity with mathematics were asked to name the numerosity of sets of 1-15 dots in various arrangements, for instance, 9 represented as a single group of 9 items, three distinct groups of 2, 3, and 4 items (affording addition 2 + 3 + 4), or three identical groups of 3 items (affording multiplication 3 × 3). Grouping systematically improved enumeration performance, regardless of whether the items were grouped spatially or by color alone, but only when an array was divided into subgroups with the same number of items. Response times and error patterns supported the hypothesis of a multiplication process. Our results demonstrate that even a simple enumeration task involves mental arithmetic.

Examining aging and numerosity using an integrated diffusion model

Two experiments are presented that use tasks common in research in numerical cognition with young adults and older adults as subjects. In these tasks, one or two arrays of dots are displayed, and subjects decide whether there are more or fewer dots of one kind than another. Results show that older adults, relative to young adults, tend to rely more on the perceptual feature, area, in making numerosity judgments when area is correlated with numerosity. Also, convex hull unexpectedly shows different effects depending on the task (being either correlated with numerosity or anticorrelated). Accuracy and response time (RT) data are interpreted with the integration of the diffusion decision model with models for the representation of numerosity. One model assumes that the representation of the difference depends on the difference between the numerosities and that standard deviations (SDs) increase linearly with numerosity, and the other model assumes a log representation with constant SDs. The representational models have coefficients that are applied to differences between two numerosities to produce drift rates and SDs in drift rates in the decision process. The two tasks produce qualitatively different patterns of RTs: One model fits results from one task, but the results are mixed for the other task. The effects of age on model parameters show a modest decrease in evidence driving the decision process, an increase in the duration of processes outside the decision process (nondecision time), and an increase in the amount of evidence needed to make a decision (boundary separation). (PsycInfo Database Record (c) 2020 APA, all rights reserved).

Effect of Finger Gnosis on Young Chinese Children's Addition Skills

Evidence has revealed an association between finger gnosis and arithmetic skills in young Western children, however, it is unknown whether such an association can be generalized to Chinese children and what mechanism may underlie this relationship. This study examines whether finger gnosis is associated with addition skills in young Chinese children and, if so, what numerical skills could explain this correlation. A total of 102 Chinese children aged 5–6 years were asked to complete finger gnosis and addition tasks in Study 1. Results showed that finger gnosis was significantly associated with addition performance. However, no significant correlation was found between finger gnosis and the use of finger counting in solving addition problems. Moreover, girls’ finger gnosis was better than boys’, and children with musical training demonstrated better finger gnosis than those without. In Study 2, 16 children with high finger gnosis and 20 children with low finger gnosis were selected from the children in Study 1 and asked to perform enumeration, order judgment, number sense, and number line estimation. Children with high finger gnosis performed better in number line estimation than their counterparts with low finger gnosis. Moreover, the number line estimation fully mediated the relationship between finger gnosis and addition performance. Together, these studies provide evidence of a correlation between finger gnosis and addition skills. They also highlight the importance of number line estimation in bridging this association.

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.

Mathematical Profile Test: A Preliminary Evaluation of an Online Assessment for Mathematics Skills of Children in Grades 1-6

The domain of numerical cognition still lacks an assessment tool that is theoretically driven and that covers a wide range of key numerical processes with the aim of identifying the learning profiles of children with difficulties in mathematics (MD) or dyscalculia. This paper is the first presentation of an online collectively administered tool developed to meet these goals. The Mathematical Profile Test (MathPro Test) includes 18 subtests that assess numerical skills related to the core number domain or to the visual-spatial, memory or reasoning domains. The specific aim of this paper is to present the preliminary evaluation both of the sensitivity and the psychometric characteristics of the individual measures of the MathPro Test, which was administered to 622 primary school children (grades 1–6) in Belgium. Performance on the subtests increased across all grades and varied along the level of difficulty of the items, supporting the sensitivity of the test. The MathPro Test also showed satisfactory internal consistency and significant and stable correlation with a standardized test in mathematics across all grades. In particular, the achievement in mathematics was strongly associated with the performance on the subtests assessing the reasoning and the visuospatial domains throughout all school grades, whereas associations with the core number and memory tasks were found mainly in the younger children. MD children performed significantly lower than their peers; these differences in performance on the MathPro subtests also varied according to the school grades, informing us about the developmental changes of the weaknesses of children with MD. These results suggest that the MathPro Test is a very promising tool for conducting large scale research and for clinicians to sketch out the mathematical profile of children with MD or dyscalculia.