Typical and atypical development of basic numerical skills in elementary school

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.

A transdiagnostic examination of cognitive heterogeneity in children and adolescents with neurodevelopmental disorders

Children and adolescents with neurodevelopmental disorders demonstrate extensive cognitive heterogeneity that is not adequately captured by traditional diagnostic systems, emphasizing the need for alternative assessment and classification techniques. Using a transdiagnostic approach, a retrospective cohort study of cognitive functioning was conducted using a large heterogenous sample (n = 1529) of children and adolescents 7 to 18 years of age with neurodevelopmental disorders. Measures of short-term memory, verbal ability, and reasoning were administered to participants with attention-deficit/hyperactivity disorder (ADHD), autism spectrum disorder (ASD), comorbid ADHD/ASD, and participants without neurodevelopmental disorders (non-NDD) using a 12-task, web-based neurocognitive testing battery. Unsupervised machine learning techniques were used to create a self-organizing map, an artificial neural network, in conjunction with k-means clustering to identify data-driven subgroups. The study aims were to: 1) identify cognitive profiles in the sample using a data-driven approach, and 2) determine their correspondence with traditional diagnostic statuses. Six clusters representing different cognitive profiles were identified, including participants with varying forms of cognitive impairment. Diagnostic status did not correspond with cluster-membership, providing evidence for the application of transdiagnostic approaches to understanding cognitive heterogeneity in children and adolescents with neurodevelopmental disorders. Additionally, the findings suggest that many typically developing participants may have undiagnosed learning difficulties, emphasizing the need for accessible cognitive assessment tools in school-based settings.

Children's comparison of different-length numbers: Managing different attributes in multidigit number processing

In everyday life the comparison of numbers usually occurs between numbers with different numbers of digits. However, experimental research here is scarce. Recent research has shown that adults respond faster to congruent pairs (the initial digit in the number with more digits is larger, e.g., 2384 vs. 107) than to incongruent pairs (the initial digit is larger in the number with fewer digits, e.g., 2675 vs. 398). This has been interpreted as support for the processing of multiple attributes in parallel and against serial accounts. The current research asked whether there is a change in the relevance of these attributes as school grades increase. School-age children from the second to sixth grades (N = 206) were presented with pairs of numbers that had either the same number of digits (3 vs. 3 or 4 vs. 4) or a different number of digits (3 vs. 4). In this latter condition, the stimuli, matched by distance, could be either length/digit congruent (e.g., 2384 vs. 107) or length/digit incongruent (e.g., 2675 vs. 398). Linear mixed models showed a length/digit congruity effect from second graders. Interestingly, in the response time measure, congruity interacted with school grade and the side in which the larger number of the pair was presented. Whereas these results support a model that considers number comparison as a process that weighs different attributes in parallel, it is also argued that developmental changes are associated with differences in the level of automatization of the componential skills involved in the comparison.

Delayed development of basic numerical skills in children with developmental dyscalculia

Research suggests that children with developmental dyscalculia (DD) have deficits in basic numerical skills. However, there is conflicting evidence on whether basic numerical skills in children with DD are qualitatively different from those in typically developing children (TD) or whether basic numerical skills development in children with DD is simply delayed. In addition, there are also competing hypotheses about deficits in basic numerical skills, assuming (1) a general deficit in representing numerosities (Approximate Number System, ANS), (2) specific deficits in an object-based attentional system (Object Tracking System, OTS), or (3) deficits in accessing numerosities from symbols (Access Deficit, AD). Hence, the purpose of this study was to investigate whether deficits in basic numerical skills in children with DD are more indicative of a developmental delay or a dyscalculia-specific qualitative deviation and whether these deficits result from (selective) impairment of core cognitive systems involved in numerical processing. To address this, we tested 480 children (68 DD and 412 TD) in the 2nd, 3rd, and 4th grades with different paradigms for basic numerical skills (subitizing, counting, magnitude comparison tasks, number sets, and number line estimation tasks). The results revealed that DD children's impairments did not indicate qualitatively different basic numerical skills but instead pointed to a specific developmental delay, with the exception of dot enumeration. This result was corroborated when comparing mathematical profiles of DD children in 4th grade and TD children in 2nd grade, suggesting that DD children were developmentally delayed and not qualitatively different. In addition, specific deficits in core markers of numeracy in children with DD supported the ANS deficit rather than the AD and OTS deficit hypothesis.

A neural network model of mathematics anxiety: The role of attention

Anxiety about performing numerical calculations is becoming an increasingly important issue. Termed mathematics anxiety, this condition negatively impacts performance in numerical tasks which can affect education outcomes and future employment. The disruption account proposes poor performance is due to anxiety disrupting limited attentional and inhibitory resources leaving fewer cognitive resources for the current task. This study provides the first neural network model of math anxiety. The model simulates performance in two commonly-used tasks related to math anxiety: the numerical Stroop and symbolic number comparison. Different model modifications were used to simulate high and low math-anxious conditions by modifying attentional processes and learning; these model modifications address different theories of math anxiety. The model simulations suggest that math anxiety is associated with reduced attention to numerical stimuli. These results are consistent with the disruption account and the attentional control theory where anxiety decreases goal-directed attention and increases stimulus-driven attention.

[Formula: see text] Domain-general and domain-specific cognitive correlates of developmental dyscalculia: a systematic review of the last two decades' literature

Developmental dyscalculia is a neurodevelopmental disorder, influencing the learning of mathematics in developing children. In the last two decades, continuous growth of research has helped in the advancement of the state of knowledge of dyscalculia. This upsurge in the number of studies makes it relevant to conduct a systematic review, covering all the empirical evidence, but there is a dearth of review studies synthesizing findings of the studies in the recent past. Therefore, the current study aims to systematically review studies investigating the underlying cognitive causal factors associated with developmental dyscalculia in the last two decades. To investigate the underlying cognitive factors associated with dyscalculia, two prominent approaches have been used: domain-general and domain-specific. While the domain-general approach argues for the deficit in general cognitive abilities, the domain-specific approach argues for the deficit in core numerical abilities. In the present review, the PRISMA method is followed. Articles were searched using two methods: firstly, through database sources of Google Scholar, Web of Science, and ScienceDirect, 1738 abstracts were screened, of which 46 articles met the specific inclusion criteria; and secondly, through recently published systematic reviews and meta-analyses, 29 studies were included. A total of 75 studies, 48 studies from domain-general and 27 studies from domain-specific approaches, have been selected. This review discusses domain-general and domain-specific approaches of developmental dyscalculia, along with specific theories associated with both approaches. Based on the discussed findings, visuospatial working memory and symbolic number processing abilities emerged as the best predictor of math ability in children with dyscalculia.

Arabic digit processing in adults with mathematical learning disability

The processing of Arabic digits is a core difficulty of children suffering from mathematical learning disability (MLD). Dominant accounts assume a semantic impairment affecting either the magnitude representation per se or its access from numerical symbols. But recent data have raised the hypothesis that the impaired processing of Arabic digits may be explained by a selective deficit of digit visual recognition (i.e., recognising a symbol as one of the digits, no matter its identity or numerical meaning). This study aims at testing whether the difficulty to process Arabic digits remains prevalent in adults with MLD and whether it is effectively associated with a digit visual recognition deficit. To do so, we compared 19 adults with MLD to 19 matched controls in an Arabic digit comparison task that required to identify the largest of two digits, and in an Arabic digit lexical decision task that required to decide whether a visual stimulus is a digit or not. The results showed that MLD participants took more time than control participants to perform the comparison task. In contrast, their performance in the digit lexical decision task was within the range of the control participants. Overall, this finding indicates that adults with MLD continue to experience difficulties to process the magnitude of Arabic digits efficiently, and this cannot be explained by a visual recognition deficit for Arabic digits. We conclude that their difficulties are best explained by an impaired representation of number magnitude or by an impaired access to this representation.

Does Conceptual Transparency in Manipulatives Afford Place-Value Understanding in Children at Risk for Mathematics Learning Disabilities?

We investigated the effect of conceptual transparency in the physical structure of manipulatives on place-value understanding in typically developing children and those at risk for mathematics learning disabilities. Second graders were randomly assigned to one of three manipulatives conditions: (a) attachable beads that did not make the denominations or ones in the denominations transparent, (b) pipe cleaners that made only the denominations transparent, and (c) string beads that made both the denominations and the ones in the denominations transparent. Participants used the manipulatives to represent double- and triple-digit numerals. Statistical analyses indicated that the transparency of the denominations, but not the transparency of the ones in the denominations, is responsible for children's number representation and place-value understanding. Descriptive analyses of their responses revealed that the at-risk children were at a greater disadvantage than their typically developing peers with the attachable beads, failing to use place-value concepts to interpret their representations.

Inconsistencies between Subjective Reports of Cognitive Difficulties and Performance on Cognitive Tests are Associated with Elevated Internalising and Externalising Symptoms in Children with Learning-related Problems

Children with learning difficulties are commonly assumed to have underlying cognitive deficits by health and educational professionals. However, not all children referred for psycho-educational assessment will be found to have deficits when their abilities are measured by performance on cognitive tasks. The primary aim of this study was to estimate the prevalence of this inconsistent cognitive profile (ICP) in a transdiagnostic sample of children referred by health and education service providers for problems related to attention, learning and memory (N = 715). A second aim was to explore whether elevated mental health problems were associated with ICPs. Findings suggest that approximately half of this sample could be characterised as having an ICP. Cognitive difficulties, whether identified by parent ratings or task performance, were associated with elevated internalising and externalising difficulties. Crucially, a larger discrepancy between a parent's actual ratings of a child's cognitive difficulties and the ratings that would be predicted based on the child's performance on cognitive tasks was associated greater internalising and externalising difficulties for measures of working memory, and greater externalising difficulties for measures of attention. These findings suggest that subjective cognitive difficulties occurring in the absence of any task-based performance deficits may be a functional problem arising from mental health problems.

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.

What Skills Could Distinguish Developmental Dyscalculia and Typically Developing Children: Evidence From a 2-Year Longitudinal Screening

Developmental dyscalculia (DD) is a mathematics learning disorder that affects approximately 5% to 7% of the population. This study aimed to detect the underlying domain-specific and domain-general differences between DD and typically developing (TD) children. We recruited 9-year-old primary school children to form the DD group via a 2-year longitudinal screening process. In total, 75 DD children were screened from 1,657 children after the one-time screening, and 13 DD children were screened from 1,317 children through a consecutive 2-year longitudinal screening. In total, 13 experimental tasks were administered to assess their cognitive abilities to test the domain-specific magnitude representation hypothesis (including symbolic and nonsymbolic magnitude comparisons) and four alternative domain-general hypotheses (including working memory, executive function, attention, and visuospatial processing). The DD group had worse performance than the TD group on the number sense task, finger sense task, shifting task, and one-back task after both one-time and two-time screening. Logistic regressions further indicated the differences on the shifting task and the nonsymbolic magnitude comparison task could distinguish DD and TD children. Our findings suggest that domain-specific nonsymbolic magnitude representation and domain-general executive function both contribute to DD. Thus, both domain-specific and domain-general abilities will be necessary to investigate and to intervene in DD groups in the future.

Patterns of symbolic numerical magnitude processing and working memory as predictors of early mathematics performance

Although the roles of symbolic numerical magnitude processing (SNMP) and working memory (WM) in mathematics performance are well acknowledged, studies examining their joint effects are few. Here, we investigated the profiles of SNMP (1- and 2-digit comparison) and WM (verbal, visual and central executive) among Norwegian first graders (N = 256), and how these predict performance in counting, arithmetic facts and word problem–solving. Using latent class cluster analysis, four groups were identified: (1) weak SNMP (33.6%), (2) strong SNMP (25.8%), (3) weak SNMP and WM (23.4%) and (4) strong WM (17.2%). Group differences in mathematics performance were significant with explained variance ranging from 7 to 16%, even after controlling for relevant demographics and domain-general cognitive skills. Our findings suggest that children may display relative strengths in SNMP and WM, and that they both have a unique, even compensatory role in mathematics performance.

Nonsymbolic and Symbolic Numerical Magnitude Processing in the Brazilian Children with Mathematics Difficulties

It is still debated if the main deficit in mathematical difficulties (MD) is nonsymbolic or symbolic numerical magnitude processing.

Sharpening, focusing, and developing: A study of change in nonsymbolic number comparison skills and math achievement in 1st grade

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

Reactive and proactive cognitive control as underlying processes of number processing in children

Cognitive control is crucial to resolve conflict in tasks such as the flanker task. Reactive control is used when conflict is rare, whereas proactive control is more efficient in situations where conflict is frequent. Macizo and Herrera (Psychological Research, 2013, Vol. 77, pp. 651-658) found that these two control processes can also underlie two-digit number comparison in adults. Specifically, they observed that the unit-decade compatibility effect decreased in a block containing many conflict trials as compared with a block containing few conflict trials (i.e., a list-wide proportion congruency effect). In the current study, we assessed whether this finding also applies to children (7-, 9-, and 11-year-olds). Participants performed a flanker task and a two-digit number comparison task. In both tasks, the proportion of conflict was manipulated (80% vs. 20%). Results from the flanker task showed a typical list-wide proportion congruency effect in reaction times in all participating age groups. In the number comparison task, we observed list-wide proportion congruency effects in both reaction times and error rates, which did not interact with age. Our findings support the assumption that children as young as 7 years can effectively use proactive and reactive control strategies. We showed that this effect is not limited to standardized artificial laboratory tasks, such as the flanker task, but also underlies more daily life tasks, such as the processing of Arabic numbers.

Higher-order dimensions of psychopathology in a neurodevelopmental transdiagnostic sample

Hierarchical dimensional models of psychopathology derived for adult and child community populations offer more informative and efficient methods for assessing and treating symptoms of mental ill health than traditional diagnostic approaches. It is not yet clear how many dimensions should be included in models for youth with neurodevelopmental conditions. The aim of this study was to delineate the hierarchical dimensional structure of psychopathology in a transdiagnostic sample of children and adolescents with learning-related problems, and to test the concurrent predictive value of the model for clinically, socially, and educationally relevant outcomes. A sample of N = 403 participants from the Centre for Attention Learning and Memory (CALM) cohort were included. Hierarchical factor analysis delineated dimensions of psychopathology from ratings on the Conner's Parent Rating Short Form, the Revised Children's Anxiety and Depression Scale, and the Strengths and Difficulties Questionnaire. A hierarchical structure with a general p factor at the apex, broad internalizing and broad externalizing spectra below, and three more specific factors (specific internalizing, social maladjustment, and neurodevelopmental) emerged. The p factor predicted all concurrently measured social, clinical, and educational outcomes, but the other dimensions provided incremental predictive value. The neurodevelopmental dimension, which captured symptoms of inattention, hyperactivity, and executive function and emerged from the higher-order externalizing factor, was the strongest predictor of learning. This suggests that in struggling learners, cognitive and affective behaviors may interact to influence learning outcomes. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Cognitive Dimensions of Learning in Children With Problems in Attention, Learning, and Memory

A data-driven, transdiagnostic approach was used to identify the cognitive dimensions linked with learning in a mixed group of 805 children aged 5 to 18 years recognised as having problems in attention, learning and memory by a health or education practitioner. Assessments included phonological processing, information processing speed, short-term and working memory, and executive functions, and attainments in word reading, spelling, and maths. Data reduction methods identified three dimensions of phonological processing, processing speed and executive function for the sample as a whole. This model was comparable for children with and without ADHD. The severity of learning difficulties in literacy was linked with phonological processing skills, and in maths with executive control. Associations between cognition and learning were similar across younger and older children and individuals with and without ADHD, although stronger links between learning-related problems and both executive skills and processing speed were observed in children with ADHD. The results establish clear domain-specific cognitive pathways to learning that distinguish individuals in the heterogeneous population of children struggling to learn.

Developmental dyscalculia: the progress of cognitive modeling in the field of numerical cognition deficits for children

The study of dyscalculia requires an analysis of the current developed hypotheses which describe the cognitive mechanisms involved in this neurodevelopmental disorder. The objective of our review is to determine any progress in modeling developmental dyscalculia. The first hypothesis suggests that dyscalculia is the consequence of a specific deficit level number on the precise number system and the approximate system. Then, the second hypothesis states that developmental dyscalculia is linked to a failure to process non-symbolic representations of numbers. On the other hand, the third suggests that dyscalculia is caused by a lack of access to numerical quantities from symbols. However, the last hypothesis asserts that developmental dyscalculia is linked to general deficits. All these hypotheses are compatible with recent neuroimaging results and raise new horizons for experimentation, which will allow the development of precise diagnostic tools and the improvement of intervention strategies and the remediation of developmental dyscalculia.

Typical Crossmodal Numerosity Perception in Preterm Newborns

Premature birth is associated with a high risk of damage in the parietal cortex, a key area for numerical and non-numerical magnitude perception and mathematical reasoning. Children born preterm have higher rates of learning difficulties for school mathematics. In this study, we investigated how preterm newborns (born at 28-34 weeks of gestation age) and full-term newborns respond to visual numerosity after habituation to auditory stimuli of different numerosities. The results show that the two groups have a similar preferential looking response to visual numerosity, both preferring the incongruent set after crossmodal habituation. These results suggest that the numerosity system is resistant to prematurity.

Different neural activations for an approaching friend versus stranger: Linking personal space to numerical cognition

INTRODUCTION

Typically, humans place themselves at a preferred distance from others. This distance is known to characterize human spatial behavior. Here, we focused on neurocognitive conditions that may affect interpersonal distances. The current study investigated whether neurocognitive deficiencies in numerical and spatial knowledge may affect social perception and modulate personal space.

METHOD

In an event-related potential (ERP) study, university students with developmental dyscalculia (DD) and typically developing control participants were given a computerized version of the comfortable interpersonal distance task, in which participants were instructed to press the spacebar when they began to feel uncomfortable by the approach of a virtual protagonist.

RESULTS

Results showed that students with deficiencies in numerical and spatial skills (i.e., DD) demonstrated reduced variability in their preferred distance from an approaching friend. Importantly, DD showed decreased amplitude of the N1 wave in the friend condition.

CONCLUSION

These results suggest that people coping with deficiencies in spatial cognition have a less efficient allocation of spatial attention in the service of processing personal distances. Accordingly, the study highlights the fundamental role of spatial neurocognition in organizing social space.

Predicting individual differences in reading, spelling and maths in a sample of typically developing children: A study in the perspective of comorbidity

We examined reading, spelling, and mathematical skills in an unselected group of 129 Italian fifth graders by testing various cognitive predictors for each behaviour. As dependent variables, we measured performance in behaviours with a clear functional value in everyday life, such as reading a text, spelling under dictation and doing mental and written computations. As predictors, we selected cognitive dimensions having an explicit relation with the target behaviour (called proximal predictors), and prepared various tests in order to select which task had the best predictive power on each behaviour. The aim was to develop a model of proximal predictors of reading (speed and accuracy), spelling (accuracy) and maths (speed and accuracy) characterized by efficacy also in comparison to the prediction based on general cognitive factors (i.e., short-term memory, phonemic verbal fluency, visual perceptual speed, and non-verbal intelligence) and parsimony, pinpointing the role of both common and unique predictors as envisaged in the general perspective of co-morbidity. With one exception (reading accuracy), the proximal predictors models (based on communality analyses) explained a sizeable amount of variance, ranging from 27.5% in the case of calculation (accuracy) to 48.7% of reading (fluency). Models based on general cognitive factors also accounted for some variance (ranging from 6.5% in the case of spelling to 19.5% in the case of reading fluency) but this was appreciably less than that explained by models based on the hypothesized proximal predictors. In general, results confirmed the efficacy of proximal models in predicting reading, spelling and maths although they offered only limited support for common predictors across different learning skills; namely, performance in the Orthographic Decision test entered as a predictor of both reading and spelling indicating that a single orthographic lexicon may account for performance in reading and spelling. Possible lines of research to expand on this approach are illustrated.

Numerical processing profiles in children with varying degrees of arithmetical achievement

Recent studies show basic cognitive abilities such as the rapid and precise apprehension of small numerosities in object sets ("subitizing"), verbal counting and numerical magnitude comparison significantly influence the acquisition of arithmetic and continues to modulate more advanced stages of mathematical cognition. Additionally, children with low arithmetic achievement (LAA) and Developmental Dyscalculia (DD) exhibit significant deficits in these cognitive processes. Nevertheless, the different cognitive profiles of children with varying degrees of numerical and arithmetic processing deficits have not been sufficiently characterized, despite its potential relevance to the stimulation of numerical cognition and the design of appropriate intervention strategies. Here, the cognitive profiles of groups of typically developing children, children with low arithmetical achievement and DD, exhibiting typical and atypical subitizing ability were contrasted. The results suggest that relatively independent neurocognitive mechanisms may produce distinct profiles at the behavioral level and suggest children with low arithmetic performance exhibiting atypical subitizing abilities are not only significantly slower, but rely on compensatory mechanisms and strategies compared to typical subitizers. The role of subitizing as a correlate of arithmetic fluency is revised in the light of the present findings.

Assessing Mathematical School Readiness

Early math skills matter for later formal mathematical performances, academic and professional success. Accordingly, it is important to accurately assess mathematical school readiness (MSR) at the beginning of elementary school. This would help identifying children who are at risk of encountering difficulties in math and then stimulate their acquisition of mathematical skills as soon as possible. In the present study, we present a new test that allows professionals working with children (e.g., teachers, school psychologists, speech therapists, and school doctors) to assess children’s MSR when they enter formal schooling in a simple, rapid and efficient manner. 346 children were assessed at the beginning of 1st Grade (6-to-7-year-olds) with a collective test assessing early mathematical abilities (T1). In addition, children’s math skills were evaluated with classical curriculum math tests at T1 and a year later, in 2nd Grade (T2, 7-to-8-year-olds). After assessing internal consistency, three tasks were retained for the final version of the MSR test. Test performance confirmed to be essentially unidimensional and systematically related to the scores children obtained in classical tests in 1st and 2nd Grade. By using the present MSR test, it is possible to identify pupils at risk of developing low math skills right from the start of formal schooling in 1st Grade. Such a tool is needed, as children’s level in math at school beginning (or school readiness) is known to be foundational for their future academic and professional carrier.

Linear-linear piecewise growth mixture models with unknown random knots: A primer for school psychology

Studying change over time requires rigorous and sometimes novel statistical methods that can support increasingly complex applied research questions. In this article, we provide an overview of the potential of piecewise growth mixture models. This type of longitudinal model can be used to advance our understanding of group and individual growth that may follow a segmented, or disjointed, pattern of change, and where the data come from a mixture of two or more latent classes. We then demonstrate the practical utility of piecewise growth mixture models by applying it to a subsample of students from the Early Childhood Longitudinal Study - Kindergarten Cohort of 1998 (ECLS-K) to ascertain whether mathematics achievement is characterized by one or two latent classes akin to students with and without mathematics difficulties. We discuss the applicability for school psychological research and provide supplementary online files that include an instructional sample dataset and corresponding R routine with explanatory annotations to assist in understanding the R routine before applying this approach in novel applications (https://doi.org/10.1016/j.jsp.2019.03.004).

Understanding comorbidity of learning disorders: task-dependent estimates of prevalence

BACKGROUND

Reading disorder (RD) and mathematics disorder (MD) frequently co-occur. However, the exact comorbidity rates differ largely between studies. Given that MD is characterised by high heterogeneity on the symptom level, differences in comorbidity rates may result from different mathematical subskills used to define MD. Comorbidity rates with RD are likely to be higher when MD is measured by mathematical subskills that do not only build on number processing, but also require language (i.e. arithmetic fluency), than when measured by magnitude processing skills.

METHODS

The association between literacy, arithmetic fluency and magnitude processing as well as the overlap between deficits in these domains were assessed in a representative sample of 1,454 third Graders.

RESULTS

Associations were significantly higher between literacy and arithmetic, than between literacy and magnitude processing. This was also reflected in comorbidity rates: comorbidity rates between literacy and arithmetic deficits were four times higher than expected by chance, whereas comorbidity rates between literacy and magnitude processing deficits did not exceed chance rate. Deficits in the two mathematical subskills showed some overlap, but also revealed dissociations, corroborating the high heterogeneity of MD. Results are interpreted within a multiple-deficit framework and implications for diagnosis and intervention are discussed.

CONCLUSIONS

The overlap between RD and MD depends on the subskills used to define MD. Due to shared domain-general factors mathematical subskills that draw on language skills are more strongly associated with literacy than those that do not require language. The findings further indicate that the same symptom, such as deficits in arithmetic, can be associated with different cognitive deficits, a deficit in language skills or a deficit in number processing.

Automatic and intentional processing of numerical order and its relationship to arithmetic performance

Recent findings have demonstrated that numerical order processing (i.e., the application of knowledge that numbers are organized in a sequence) constitutes a unique and reliable predictor of arithmetic performance. The present work investigated two central questions to further our understanding of numerical order processing and its relationship to arithmetic. First, are numerical order sequences processed without conscious monitoring (i.e., automatically)? Second, are automatic and intentional ordinal processing differentially related to arithmetic performance? In the first experiment, adults completed a novel ordinal congruity task. Participants had to evaluate whether number triplets were arranged in a correct (e.g., ) physical order or not (e.g., ). Results of this experiment showed that participants were faster to decide that the physical size of ascending numbers was in-order when the physical and numerical values were congruent compared to when they were incongruent (i.e., congruency effect). In the second experiment, a new group of participants was asked to complete an ordinal congruity task, an ordinal verification task (i.e., are the number triplets in a correct order or not) and an arithmetic fluency test. Results of this experiment revealed that the automatic processing of ascending numerical order is influenced by the numerical distance of the numbers. Correlation analysis further showed that only reaction time measures of the intentional ordinal verification task were associated with arithmetic performance. While the findings of the present work suggest that ascending numerical order is processed automatically, the relationship between numerical order processing and arithmetic appears to be limited to the intentional manipulation of numbers. The present findings show that the mental engagement of verifying the order of numbers is a crucial factor for explaining the link between numerical order processing and arithmetic performance.

Place-value computation in children with mathematics difficulties

Recent research has provided initial evidence that children with math difficulties (MD) experience problems in processing place-value information in basic numerical tasks. However, it remains unclear whether these problems generalize to basic arithmetic operations. For instance, multi-digit addition problems with carryover specifically require the computation of place-value information. Yet little is known about the carry effect in children with MD. Therefore, the current study investigated whether problems in processing place-value information among third-grade children with MD (n = 29 9-year-olds) compared with an age-matched control group (n = 50) generalize to two-digit addition. The results indicate an increased carry effect for response latencies and error rates in children with MD. These findings suggest that deficits in processing place-value information among children with MD generalize to place-value computations in multi-digit arithmetic. Potential contributions of strategy use and working memory for difficulties in processing place-value information are discussed.

Impaired neural processing of transitive relations in children with math learning difficulty

Math learning difficulty (i.e., MLD) is common in children and can have far-reaching consequences in personal and professional life. Converging evidence suggests that MLD is associated with impairments in the intraparietal sulcus (IPS). However, the role that these impairments play in MLD remains unclear. Although it is often assumed that IPS deficits affect core numerical abilities, the IPS is also involved in several non-numerical processes that may contribute to math skills. For instance, the IPS supports transitive reasoning (i.e., the ability to integrate relations such as A > B and B > C to infer that A > C), a skill that is central to many aspects of math learning in children. Here we measured fMRI activity of 8- to 12-year-olds with MLD and typically developing (TD) peers while they listened to stories that included transitive relations. Children also answered questions evaluating whether transitive inferences were made during story comprehension. Compared to non-transitive relations (e.g., A > B and C > D), listening to transitive relations (e.g., A > B and B > C) was associated with enhanced activity in the IPS in TD children. In children with MLD, the difference in activity between transitive and non-transitive relations in the IPS was (i) non-reliable and (ii) smaller than in TD children. Finally, children with MLD were less accurate than TD peers when making transitive inferences based on transitive relations. Thus, a deficit in the online processing of transitive relations in the IPS might contribute to math difficulties in children with MLD.

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Transitive reasoning is central to mathematical thinking.

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Transitive reasoning relies on the intra-parietal sulcus (IPS) in healthy children.

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Math learning difficulty (MLD) is associated with IPS impairments.

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Transitive reasoning is impaired in children with MLD.

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Transitive reasoning does not engage the IPS in children with MLD.

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.

Symbolic magnitude processing in elementary school children: A group administered paper-and-pencil measure (SYMP Test)

The ability to compare symbolic numerical magnitudes correlates with children's concurrent and future mathematics achievement. We developed and evaluated a quick timed paper-and-pencil measure that can easily be used, for example in large-scale research, in which children have to cross out the numerically larger of two Arabic one- and two-digit numbers (SYMP Test). We investigated performance on this test in 1,588 primary school children (Grades 1-6) and examined in each grade its associations with mathematics achievement. The SYMP Test had satisfactory test-retest reliability. The SYMP Test showed significant and stable correlations with mathematics achievement for both one-digit and two-digit comparison, across all grades. This replicates the previously observed association between symbolic numerical magnitude processing and mathematics achievement, but extends it by showing that the association is observed in all grades in primary education and occurs for single- as well as multi-digit processing. Children with mathematical learning difficulties performed significantly lower on one-digit comparison and two-digit comparison in all grades. This all suggests satisfactory construct and criterion-related validity of the SYMP Test, which can be used in research, when performing large-scale (intervention) studies, and by practitioners, as screening measure to identify children at risk for mathematical difficulties or dyscalculia.

About why there is a shift from cardinal to ordinal processing in the association with arithmetic between first and second grade

Digit comparison is strongly related to individual differences in children's arithmetic ability. Why this is the case, however, remains unclear to date. Therefore, we investigated the relative contribution of three possible cognitive mechanisms in first and second graders' digit comparison performance: digit identification, digit-number word matching and digit ordering ability. Furthermore, we examined whether these components could account for the well-established relation between digit comparison performance and arithmetic. As expected, all candidate predictors were related to digit comparison in both age groups. Moreover, in first graders, digit ordering and in second graders both digit identification and digit ordering explained unique variance in digit comparison performance. However, when entering these unique predictors of digit comparison into a mediation model with digit comparison as predictor and arithmetic as outcome, we observed that whereas in second graders digit ordering was a full mediator, in first graders this was not the case. For them, the reverse was true and digit comparison fully mediated the relation between digit ordering and arithmetic. These results suggest that between first and second grade, there is a shift in the predictive value for arithmetic from cardinal processing and procedural knowledge to ordinal processing and retrieving declarative knowledge from memory; a process which is possibly due to a change in arithmetic strategies at that age. A video abstract of this article can be viewed at: https://youtu.be/dDB0IGi2Hf8.

Reliability and Validity of Nonsymbolic and Symbolic Comparison Tasks in School-Aged Children

Basic numerical processing has been regularly assessed using numerical nonsymbolic and symbolic comparison tasks. It has been assumed that these tasks index similar underlying processes. However, the evidence concerning the reliability and convergent validity across different versions of these tasks is inconclusive. We explored the reliability and convergent validity between two numerical comparison tasks (nonsymbolic vs. symbolic) in school-aged children. The relations between performance in both tasks and mental arithmetic were described and a developmental trajectories' analysis was also conducted. The influence of verbal and visuospatial working memory processes and age was controlled for in the analyses. Results show significant reliability (p < .001) between Block 1 and 2 for nonsymbolic task (global adjusted RT (adjRT): r = .78, global efficiency measures (EMs): r = .74) and, for symbolic task (adjRT: r = .86, EMs: r = .86). Also, significant convergent validity between tasks (p < .001) for both adjRT (r = .71) and EMs (r = .70) were found after controlling for working memory and age. Finally, it was found the relationship between nonsymbolic and symbolic efficiencies varies across the sample's age range. Overall, these findings suggest both tasks index the same underlying cognitive architecture and are appropriate to explore the Approximate Number System (ANS) characteristics. The evidence supports the central role of ANS in arithmetic efficiency and suggests there are differences across the age range assessed, concerning the extent to which efficiency in nonsymbolic and symbolic tasks reflects ANS acuity.

Developmental trajectories of children's symbolic numerical magnitude processing skills and associated cognitive competencies

Although symbolic numerical magnitude processing skills are key for learning arithmetic, their developmental trajectories remain unknown. Therefore, we delineated during the first 3years of primary education (5-8years of age) groups with distinguishable developmental trajectories of symbolic numerical magnitude processing skills using a model-based clustering approach. Three clusters were identified and were labeled as inaccurate, accurate but slow, and accurate and fast. The clusters did not differ in age, sex, socioeconomic status, or IQ. We also tested whether these clusters differed in domain-specific (nonsymbolic magnitude processing and digit identification) and domain-general (visuospatial short-term memory, verbal working memory, and processing speed) cognitive competencies that might contribute to children's ability to (efficiently) process the numerical meaning of Arabic numerical symbols. We observed minor differences between clusters in these cognitive competencies except for verbal working memory for which no differences were observed. Follow-up analyses further revealed that the above-mentioned cognitive competencies did not merely account for the cluster differences in children's development of symbolic numerical magnitude processing skills, suggesting that other factors account for these individual differences. On the other hand, the three trajectories of symbolic numerical magnitude processing revealed remarkable and stable differences in children's arithmetic fact retrieval, which stresses the importance of symbolic numerical magnitude processing for learning arithmetic.

The approximate number system and domain-general abilities as predictors of math ability in children with normal hearing and hearing loss

Many children with hearing loss (CHL) show a delay in mathematical achievement compared to children with normal hearing (CNH). This study examined whether there are differences in acuity of the approximate number system (ANS) between CHL and CNH, and whether ANS acuity is related to math achievement. Working memory (WM), short-term memory (STM), and inhibition were considered as mediators of any relationship between ANS acuity and math achievement. Seventy-five CHL were compared with 75 age- and gender-matched CNH. ANS acuity, mathematical reasoning, WM, and STM of CHL were significantly poorer compared to CNH. Group differences in math ability were no longer significant when ANS acuity, WM, or STM was controlled. For CNH, WM and STM fully mediated the relationship of ANS acuity to math ability; for CHL, WM and STM only partially mediated this relationship. ANS acuity, WM, and STM are significant contributors to hearing status differences in math achievement, and to individual differences within the group of CHL. Statement of contribution What is already known on this subject? Children with hearing loss often perform poorly on measures of math achievement, although there have been few studies focusing on basic numerical cognition in these children. In typically developing children, the approximate number system predicts math skills concurrently and longitudinally, although there have been some contradictory findings. Recent studies suggest that domain-general skills, such as inhibition, may account for the relationship found between the approximate number system and math achievement. What does this study adds? This is the first robust examination of the approximate number system in children with hearing loss, and the findings suggest poorer acuity of the approximate number system in these children compared to hearing children. The study addresses recent issues regarding the contradictory findings of the relationship of the approximate number system to math ability by examining how this relationship varies across children with normal hearing and hearing loss, and by examining whether this relationship is mediated by domain-general skills (working memory, short-term memory, and inhibition).

Visual Form Perception Can Be a Cognitive Correlate of Lower Level Math Categories for Teenagers

Numerous studies have assessed the cognitive correlates of performance in mathematics, but little research has been conducted to systematically examine the relations between visual perception as the starting point of visuospatial processing and typical mathematical performance. In the current study, we recruited 223 seventh graders to perform a visual form perception task (figure matching), numerosity comparison, digit comparison, exact computation, approximate computation, and curriculum-based mathematical achievement tests. Results showed that, after controlling for gender, age, and five general cognitive processes (choice reaction time, visual tracing, mental rotation, spatial working memory, and non-verbal matrices reasoning), visual form perception had unique contributions to numerosity comparison, digit comparison, and exact computation, but had no significant relation with approximate computation or curriculum-based mathematical achievement. These results suggest that visual form perception is an important independent cognitive correlate of lower level math categories, including the approximate number system, digit comparison, and exact computation.

Automatic non-symbolic numerosity processing in preschoolers

There has recently been an increasing focus on the development of automatic processing of numerical magnitude. However, little effort has been made to explore automatic access to non-symbolic numerical magnitude in preschool children. In experiment 1, we used a non-symbolic physical size comparison task in 3- to 6-year-olds to examine developmental changes and the effect of ratio and counting principle knowledge. Results showed that the existence of automatic non-symbolic numerical processing began at age 3–4 years and size congruity effects tended to reduce with increasing age from 4 years old. The study also found that non-counting-principle knowers had a larger congruity effect, and in low ratio conditions the size congruity effect was more easily found. In addition, symbolic number comparison ability was negatively related to size congruity effect. In experiment 2, we explored the relationship between inhibition skill and size congruity effects, as well as interference and facilitatory components in children aged 4 years old. Results showed no correlation between inhibition skills and the size congruity effect and only interference effects were found. We also found a larger interference effect in low ratio conditions than in high ratio conditions.

(Non-)symbolic magnitude processing in children with mathematical difficulties: A meta-analysis

Symbolic and non-symbolic magnitude representations, measured by digit or dot comparison tasks, are assumed to underlie the development of arithmetic skills. The comparison distance effect (CDE) has been suggested as a hallmark of the preciseness of mental magnitude representations. It implies that two magnitudes are harder to discriminate when the numerical distance between them is small, and may therefore differ in children with mathematical difficulties (MD), i.e. low mathematical achievement or dyscalculia. However, empirical findings on the CDE in children with MD are heterogeneous, and only few studies assess both symbolic and non-symbolic skills. This meta-analysis therefore integrates 44 symbolic and 48 non-symbolic response time (RT) outcomes reported in nineteen studies (N=1630 subjects, aged 6-14 years). Independent of age, children with MD show significantly longer mean RTs than typically achieving controls, particularly on symbolic (Hedges' g=0.75; 95% CI [0.51; 0.99]), but to a significantly lower extent also on non-symbolic (g=0.24; 95% CI [0.13; 0.36]) tasks. However, no group differences were found for the CDE. Extending recent work, these meta-analytical findings on children with MD corroborate the diagnostic importance of magnitude comparison speed in symbolic tasks. By contrast, the validity of CDE measures in assessing MD is questioned.

Heterogeneity of Developmental Dyscalculia: Cases with Different Deficit Profiles

Developmental Dyscalculia (DD) has long been thought to be a monolithic learning disorder that can be attributed to a specific neurocognitive dysfunction. However, recent research has increasingly recognized the heterogeneity of DD, where DD can be differentiated into subtypes in which the underlying cognitive deficits and neural dysfunctions may differ. The aim was to further understand the heterogeneity of developmental dyscalculia (DD) from a cognitive psychological perspective. Utilizing four children (8–9 year-old) we administered a comprehensive cognitive test battery that shed light on the cognitive-behavioral profile of each child. The children were compared against norm groups of aged-matched peers. Performance was then contrasted against predominant hypotheses of DD, which would also give insight into candidate neurocognitive correlates. Despite showing similar mathematical deficits, these children showed remarkable interindividual variability regarding cognitive profile and deficits. Two cases were consistent with the approximate number system deficit account and also the general magnitude-processing deficit account. These cases showed indications of having domain-general deficits as well. One case had an access deficit in combination with a general cognitive deficit. One case suffered from general cognitive deficits only. The results showed that DD cannot be attributed to a single explanatory factor. These findings support a multiple deficits account of DD and suggest that some cases have multiple deficits, whereas other cases have a single deficit. We discuss a previously proposed distinction between primary DD and secondary DD, and suggest hypotheses of dysfunctional neurocognitive correlates responsible for the displayed deficits.

Defective Number Sense or Impaired Access? Differential Impairments in Different Subgroups of Children With Mathematics Difficulties

Developmental dyscalculia (DD) is a specific learning disability in mathematics that affects around 6% of the population. Currently, the core deficit of DD remains unknown. While the number sense deficit hypothesis suggests that the core deficit of DD lies in the inability to represent nonsymbolic numerosity, the access deficit hypothesis suggests that the origin of this disability lies in the inability to associate numbers with the underlying magnitude representation. The present study compared the performance of DDs with their low-achieving (LA) and normally achieving peers in nonsymbolic numerosity processing and number-magnitude mapping over 1 year (from kindergarten to 1st grade). The results demonstrated differential impairments in different subgroups of children with mathematics difficulties. While DDs showed deficits in both nonsymbolic numerosity processing and number-magnitude mapping, LAs showed deficit only in the number-magnitude mapping. Furthermore, the deficit in number-magnitude mapping among the DD group was partially explained by their number sense deficit. The number sense deficit hypothesis is supported. Theoretical and practical implications are discussed.

Cognitive factors affecting children's nonsymbolic and symbolic magnitude judgment abilities: A latent profile analysis

Early math abilities are claimed to be linked to magnitude representation ability. Some claim that nonsymbolic magnitude abilities scaffold the acquisition of symbolic (Arabic number) magnitude abilities and influence math ability. Others claim that symbolic magnitude abilities, and ipso facto math abilities, are independent of nonsymbolic abilities and instead depend on the ability to process number symbols (e.g., 2, 7). Currently, the issue of whether symbolic abilities are or are not related to nonsymbolic abilities, and the cognitive factors associated with nonsymbolic-symbolic relationships, remains unresolved. We suggest that different nonsymbolic-symbolic relationships reside within the general magnitude ability distribution and that different cognitive abilities are likely associated with these different relationships. We further suggest that the different nonsymbolic-symbolic relationships and cognitive abilities in combination differentially predict math abilities. To test these claims, we used latent profile analysis to identify nonsymbolic-symbolic judgment patterns of 124, 5- to 7-year-olds. We also assessed four cognitive factors (visuospatial working memory [VSWM], naming numbers, nonverbal IQ, and basic reaction time [RT]) and two math abilities (number transcoding and single-digit addition abilities). Four nonsymbolic-symbolic ability profiles were identified. Naming numbers, VSWM, and basic RT abilities were differentially associated with the different ability profiles and in combination differentially predicted math abilities. Findings show that different patterns of nonsymbolic-symbolic magnitude abilities can be identified and suggest that an adequate account of math development should specify the inter-relationship between cognitive factors and nonsymbolic-symbolic ability patterns.

The Contribution of Numerical Magnitude Comparison and Phonological Processing to Individual Differences in Fourth Graders' Multiplication Fact Ability

Although numerical magnitude processing has been related to individual differences in arithmetic, its role in children’s multiplication performance remains largely unknown. On the other hand, studies have indicated that phonological awareness is an important correlate of individual differences in children’s multiplication performance, but the involvement of phonological memory, another important phonological processing skill, has not been studied in much detail. Furthermore, knowledge about the relative contribution of above mentioned processes to the specific arithmetic operation of multiplication in children is lacking. The present study therefore investigated for the first time the unique contributions of numerical magnitude comparison and phonological processing in explaining individual differences in 63 fourth graders’ multiplication fact ability (mean age = 9.6 years, SD = .67). The results showed that children’s multiplication fact competency correlated significantly with symbolic and nonsymbolic magnitude comparison as well as with phonological short-term memory. A hierarchical regression analysis revealed that, after controlling for intellectual ability and general reaction time, both symbolic and nonsymbolic magnitude comparison and phonological short-term memory accounted for unique variance in multiplication fact performance. The ability to compare symbolic magnitudes was found to contribute the most, indicating that the access to numerical magnitudes by means of Arabic digits is a key factor in explaining individual differences in children’s multiplication fact ability.