The Contributions of Domain-General and Numerical Factors to Third-Grade Arithmetic Skills and Mathematical Learning Disability

Verbalized arithmetic principles correlate with mathematics achievement

BACKGROUND

When mathematical knowledge is expressed in general language, it is called verbalized mathematics. Previous studies on verbalized mathematics typically paid attention to mathematical vocabulary or educational practice. However, these studies did not exclude the role of symbolic mathematics ability, and almost no research has focused on verbalized mathematical principles.

AIMS

This study is aimed to investigate whether verbalized mathematics ability independently predicts mathematics achievement. The current study hypothesized that verbalized mathematics ability supports mathematics achievement independent of general language, related cognitive abilities and even symbolic mathematical ability.

SAMPLE

A sample of 241 undergraduates (136 males, 105 females, mean age = 21.95, SD = 2.38) in Beijing, China.

METHODS

A total of 12 tests were used, including a verbalized arithmetic principle test, a mathematics achievement test, and tests on general language (sentence completion test), symbolic mathematical ability (including symbolic arithmetic principles test, simple arithmetic computation and complex arithmetic computation), approximate number sense ability (numerosity comparison test) and several related cognitive covariates (including the non-verbal matrix reasoning, the syllogism reasoning, mental rotation, figure matching and choice reaction time).

RESULTS

Results showed that the processing of verbalized arithmetic principles displayed a significant role in mathematics achievement after controlling for general language, related cognitive abilities, approximate number sense ability and symbolic mathematics ability.

CONCLUSIONS

The results suggest that verbalized mathematics ability was an independent predictor and provided empirical evidence supporting the verbalized mathematics role on achievement as an independent component in three-component mathematics model.

Higher level domain specific skills in mathematics; The relationship between algebra, geometry, executive function skills and mathematics achievement

Algebra and geometry are important components of mathematics that are often considered gatekeepers for future success. However, most studies that have researched the cognitive skills required for success in mathematics have only considered the domain of arithmetic. We extended models of mathematical skills to consider how executive function skills play both a direct role in secondary-school-level mathematical achievement as well as an indirect role via algebra and geometry, alongside arithmetic. We found that verbal and visuospatial working memory were indirectly associated with mathematical achievement via number fact knowledge, calculation skills, algebra and geometry. Inhibition was also indirectly associated with mathematical achievement via number fact knowledge and calculation skills. These findings highlight that there are multiple mechanisms by which executive function skills may be involved in mathematics outcomes. Therefore, using specific measures of mathematical processes as well as context-rich assessments of mathematical achievement is important to understand these mechanisms.

Influences of cognitive control on number processing: New evidence from switching between two numerical tasks

A growing body of research suggests that basic numerical abilities such as number magnitude and number parity processing are influenced by cognitive control. So far, however, evidence for number processing being influenced by cognitive control came primarily from observed adaptations to stimulus set characteristics (e.g., ratio or order of specific stimulus types) and switches between a numerical and non-numerical task. Complementing this previous research, the present study employed a task switching paradigm exclusively involving numerical tasks (i.e., magnitude comparisons and parity judgements) to examine how cognitive control processes influence number processing. Participants were presented with a single-digit number and had to either judge its parity or compare its magnitude with a standard of 5, depending on a preceding cue. Based on previous results, we expected the numerical distance effect and the spatial-numerical association of response codes (SNARC) effect to be modulated in switch trials requiring the exertion of cognitive control. Partly in line with our expectations, the numerical distance effect was reduced in switch trials. However, no modulation of the SNARC effect was observed. The results pattern suggests that number processing is influenced by cognitive control, depending on task requirements and the type of numerical information (i.e., numerical magnitude vs spatial association of numbers) that is processed. To reconcile the present and previous results, we propose an information prioritisation account, suggesting that cognitive control primarily influences the processing of the information type that requires the most explicit processing.

[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.

The importance of domain-specific number abilities and domain-general cognitive abilities for early arithmetic achievement and development

BACKGROUND

Children's numerical and arithmetic skills differ greatly already at an early age. Although research focusing on accounting for these large individual differences clearly demonstrates that mathematical performance draws upon several cognitive abilities, our knowledge concerning key abilities underlying mathematical skill development is still limited.

AIMS

First, to identify key cognitive abilities contributing to children's development of early arithmetic skills. Second, to examine the extent to which early arithmetic performance and early arithmetic development rely on different or similar constellations of domain-specific number abilities and domain-general cognitive abilities.

SAMPLE

In all, 134 Swedish children (Mage  = 6 years and 4 months, SD = 3 months, 74 boys) participated in this study.

METHOD

Verbal and non-verbal logical reasoning, non-symbolic number comparison, counting knowledge, spatial processing, verbal working memory and arithmetic were assessed. Twelve months later, arithmetic skills were reassessed. A latent change score model was computed to determine whether any of the abilities accounted for variations in arithmetic development.

RESULTS

Arithmetic performance was supported by counting knowledge, verbal and non-verbal logical reasoning and spatial processing. Arithmetic skill development was only supported by spatial processing.

CONCLUSIONS

Results show that young children's early arithmetic performance and arithmetic development are supported by different cognitive processes. The findings regarding performance supported Fuchs et al.'s model (Dev Psychol, 46, 2010b, 1731) but the developmental findings did not. The developmental findings align partially to Geary et al.'s (J Educ Psychol, 109, 2017, 680) hypothesis stating that young children's early arithmetic development is more dependent on general cognitive abilities than number abilities.

A potential dissociation between perception and production version for bounded but not unbounded number line estimation

BACKGROUND

What, exactly, do number line estimation (NLE) tasks measure? Different versions of the task were observed to have different effects on performance.

METHOD

We investigated associations between the production (indicating the location) and perception version (indicating the number) of the bounded and unbounded NLE task and their relationship to arithmetic.

RESULTS

A stronger correlation was observed between the production and perception version of the unbounded than the bounded NLE task, indicating that both versions of the unbounded-but not the bounded-NLE task measure the same construct. Moreover, overall low but significant associations between NLE performance and arithmetic were only observed for the production version of the bounded NLE task.

CONCLUSION

These results substantiate that the production version of bounded NLE seems to rely on proportion judgment strategies, whereas both unbounded versions and the perception version of the bounded NLE task may rely more on magnitude estimation.

Longitudinal relationship between number line estimation and other mathematical abilities in Chinese preschool children

Many cross-sectional studies have shown that number line estimation is associated with other mathematical skills, but there has been limited longitudinal research. To systematically examine such associations longitudinally at the earliest stage of mathematical learning, the current study tested 40 Chinese preschoolers (mean age = 4.97 years, SD = 0.18) and followed them up 8 months later. For both waves of data collection, children were administered six tasks: number line estimation, dot counting, comparison of two dot arrays, comparison of triple dot arrays, symbolic number comparison, and simple addition. Results of two-wave cross-lagged panel analysis showed that (a) dot counting and non-symbolic numerical comparison at Time 1 had significant longitudinal associations with number line estimation at Time 2, (b) number line estimation had bidirectional associations with symbolic number comparison, and (c) number line estimation at Time 1 had a marginally significant longitudinal association with simple addition at Time 2. These results extend the small but accumulating literature on the longitudinal relations between number line estimation and other mathematical skills and specify the important role of number line estimation in the early development of mathematical skills.

Subtypes of mathematical learning disability and their antecedents: A cognitive diagnostic approach

Using cognitive diagnostic modeling (CDM), this study identified subtypes of mathematics learning disability (MLD) based on children's numerical skills and examined the language and spatial precursors of these subtypes. Participants were 99 MLD children and 420 low achievers identified from 1839 Finnish children (966 boys) who were followed from preschool (age 6) to fourth grade (2007-2011). Five subtypes were identified: the arithmetic fluency deficit only subtype, the counting deficit subtype, the pervasive deficit subtype, the symbolic deficit subtype, and the counting and concept deficit subtype. Different subtypes depended on different constellations of language and spatial deficits. Findings highlight the effectiveness of CDM in identifying MLD subtypes and underscore the importance of understanding the specific deficits and antecedents of the subtypes.

Applying an Individual Word-Problem Intervention to a Small-Group Setting: A Pilot Study's Evidence of Improved Word-Problem Performance for Students Experiencing Mathematics Difficulty

The purpose of this pilot study was to determine whether positive results from a word-problem intervention implemented one-to-one contributed to similar outcomes when implemented in small groups of three to four students. Third-grade students experiencing mathematics difficulty (N = 76) were randomly assigned to word-problem intervention (n = 56) or business-as-usual comparison (n = 20). Intervention occurred for 13 weeks, 3 times per week, 30 min per session. Multilevel models revealed the intervention condition significantly outperformed the BaU on a proximal word-problem outcome, corroborating results from our prior individual intervention. When comparing student performance in the individual versus small-group intervention, findings suggest students received added benefit from the individual intervention. The word-problem intervention successfully translated to a small-group setting, which holds important implications for educators working with students in supplemental, targeted, or Tier 2 mathematics intervention settings.

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.

The Cognitive Profile of Math Difficulties: A Meta-Analysis Based on Clinical Criteria

Math difficulties (MD) manifest across various domain-specific and domain-general abilities. However, the existing cognitive profile of MD is incomplete and thus not applicable in typical settings such as schools or clinics. So far, no review has applied inclusion criteria according to DSM or ICD, summarized domain-specific abilities or examined the validity of response time scores for MD identification. Based upon stringent clinical criteria, the current meta-analysis included 34 studies which compared cognitive performances of a group with MD (n = 680) and a group without MD (n = 1565). Criteria according to DSM and ICD were applied to identify MD (percentile rank ≤ 16, age range 8–12 years, no comorbidities/low IQ). Effect sizes for 22 abilities were estimated and separated by their level and type of scoring (AC = accuracy, RT = response time). A cognitive profile of MD was identified, characterized by distinct weaknesses in: (a) computation (calculation [AC], fact retrieval [AC]), (b) number sense (quantity processing [AC], quantity-number linking [RT], numerical relations [AC]), and (c) visual-spatial short-term storage [AC]. No particular strength was found. Severity of MD, group differences in reading performance and IQ did not significantly moderate the results. Further analyses revealed that (a) effects are larger when dealing with numbers or number words than with quantities, (b) MD is not accompanied by any weakness in abilities typically assigned to reading, and (c) weaknesses in visual-spatial short-term storage emphasize the notion that number and space are interlinked. The need for high-quality studies investigating domain-general abilities is discussed.

Domain-General Cognitive Skills in Children with Mathematical Difficulties and Dyscalculia: A Systematic Review of the Literature

Mathematical performance implies a series of numerical and mathematical skills (both innate and derived from formal training) as well as certain general cognitive abilities that, if inadequate, can have a cascading effect on mathematics learning. These latter skills were the focus of the present systematic review. Method: The reviewing process was conducted according to the PRISMA statement. We included 46 studies comparing school-aged children’s performance with and without math difficulties in the following cognitive domains: processing speed, phonological awareness, short- and long-term memory, executive functions, and attention. Results: The results showed that some general cognitive domains were compromised in children with mathematical difficulties (i.e., executive functions, attention, and processing speed). Conclusions: These cognitive functions should be evaluated during the diagnostic process in order to better understand the child’s profile and propose individually tailored interventions. However, further studies should investigate the role of skills that have been poorly investigated to date (e.g., long-term memory and phonological awareness).

Cognitive control in number processing: new evidence from number compatibility effects in task-switching

A growing body of research suggests that basic numerical abilities such as number magnitude processing are influenced by cognitive control processes. So far, evidence for number processing being affected by cognitive control processes stems primarily from observed adaptations of numerical effects to stimulus set characteristics (e.g. order or ratio of specific stimulus types). Complementing previous research on adaptation to stimulus set characteristics as an index of influences of cognitive control, the present study employed a task-switching paradigm to examine how cognitive control processes influence number processing. Participants were presented with a two-digit number and had to either judge its parity or compare its magnitude to a standard depending on a preceding cue. We expected numerical congruency effects (i.e. the unit-decade compatibility effect for magnitude comparisons and the parity congruity effect for parity judgements) to be larger in switch trials, as persisting activation of the task set of the preceding trial should increase interference. In contrast to our expectations, both numerical congruity effects were reduced following task switches as compared to repetitions. This interaction of task-switching with numerical congruency effects suggests an influence of cognitive control on basic number processing in form of persisting inhibition of previously abandoned task sets, so that these exert less influence on current number processing demands.

Comorbidity between persistent reading and mathematics disabilities: The nature of comorbidity

The current study aimed at investigating the comorbidity between reading disability (RD) and mathematical disability (MD) in a non-alphabetic language context. Over 1,900 Chinese first graders were screened on their reading and mathematics achievement twice. Children who scored consistently below the 10^th percentile in reading and/or mathematics were identified as RD and/or MD respectively. A subsample of these children, together with a group of typically-achieving children, were further assessed on their cognitive capacities. Results suggested that while there were cognitive deficits that were specifically found in RD (shifting) versus MD (spatial working memory, inhibition, processing speed, visual attention) groups, deficits in naming speed was found in both RD and MD groups. The cognitive profile of the comorbid group was an additive combination of those of the two single LD groups. The findings suggest that RD and MD are two dissociable learning disabilities with distinct cognitive profiles. Effective screening and intervention can be developed based on the cognitive profiles of different disability groups.

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.

Examining the relevance of basic numerical skills for mathematical achievement in secondary school using a within-task assessment approach

Previous research repeatedly found basic numerical abilities (e.g., magnitude understanding, arithmetic fact knowledge, etc.) to predict young students' current and later arithmetic achievement as assessed by achievement tests - even when controlling for the influence of domain-general abilities (e.g., intelligence, working memory). However, to the best of our knowledge, previous studies hardly addressed this issue in secondary school students. Additionally, they primarily assessed basic numerical abilities in a between-task approach (i.e., using different tasks for different abilities). Finally, their relevance for real-life academic outcomes such as mathematics grades has only rarely been investigated. The present study therefore pursued an approach using one and the same task (i.e., a within-task approach) to reduce confounding effects driven by between-task differences. In particular, we evaluated the relevance of i) number magnitude understanding, ii) arithmetic fact knowledge, and iii) conceptual and procedural knowledge for the mathematics grades of 81 students aged between ten and thirteen (i.e., in Grades 5 and 6) employing the number bisection task. Results indicated that number magnitude understanding, arithmetic fact knowledge, and conceptual and procedural knowledge contributed to explaining mathematics grades even when controlling for domain-general cognitive abilities. Methodological and practical implications of the results are discussed.

The Multifactorial Nature of Early Numeracy and Its Stability

Early numeracy is a robust predictor of later mathematical abilities. So far, early numeracy has typically been presented as a unitary or two-factorial construct. Nevertheless, there is recent evidence suggesting that it may also be reflected by more basic numerical competences. However, the structure and stability of such a multifactorial model of early numeracy over time has not been investigated yet. In the present study, we used data from a large, longitudinal sample (N = 1292) in the United States with assessments of math ability in prekindergarten and kindergarten to evaluate both the factorial structure of early numeracy and its stability over time. Confirmatory factor analysis identified four distinct basic numerical competences making up early numeracy in prekindergarten: patterning/geometry, number sense, arithmetic, and data analysis/statistics. Stability as tested by means of measurement invariance indicated configural invariance of these four factors from prekindergarten to kindergarten. This reflected that early numeracy in kindergarten was made up by the same four basic numerical competences as in prekindergarten and thus seemed rather stable over the course of preschool. These findings may not only have implications for research on numerical cognition but particularly for diagnostic processes or the development of interventions in educational practice.

Cognitive control in number processing: new evidence from task switching

Recently, it was demonstrated that even basic numerical cognition such as the processing of number magnitude is under cognitive control. However, evidence so far primarily came from adaptation effects to stimulus characteristics (e.g., relative frequency of specific stimulus categories). Expanding this approach, we evaluated a possible influence of more active exertion of cognitive control on basic number processing in task switching. Participants had to perform a magnitude comparison task while we manipulated the order of compatible and incompatible input–output modalities (i.e., auditory/vocal input–visual/manual output vs. auditory/visual input–manual/vocal output, respectively) on the trial level, differentiating repeat vs. switch trials. Results indicated that the numerical distance effect but not the problem size effect was increased after a switch in input–output modality compatibility. In sum, these findings substantiate that basic number processing is under cognitive control by providing first evidence that it is influenced by the active exertion of cognitive control as required in task switching.

Working memory growth predicts mathematical problem-solving growth among emergent bilingual children

An area of mathematics found to be difficult for emergent bilingual children whose first language (L1) is Spanish in the United States is solving mathematical word problems. The purpose of this study was to determine the relationship between growth in the executive component of working memory (WM) and growth in mathematical word-problem solving in children whose L1 is Spanish. Elementary school children (Grades 1, 2, and 3) were administered a battery of mathematical, vocabulary, reading, and cognitive measures (short-term memory [STM], inhibition, and WM) in both Spanish (L1) and English (second language [L2]) in Year 1 and again 1 year later. Multilevel growth modeling showed that growth in WM significantly predicted growth in L1 and L2 mathematical word-problem solving. Furthermore, the contributions of WM to mathematical word-problem-solving growth in both L1 and L2 were independent of language skills in vocabulary, reading, estimation, naming speed, inhibition, STM, and calculation. Overall, the results suggest that the mental activities that underlie WM play a significant role in predictions of L1 and L2 mathematical word-problem-solving accuracy.

Relation of attention deficit hyperactivity disorder (ADHD) to basic number skills and arithmetic fact retrieval in children

Although research has provided evidence for a clear association of core executive function (CEF) to math performance, fewer studies have been carried out on arithmetic fact retrieval in relation to the symptoms of attention deficit hyperactivity disorder (ADHD). The present study assessed mathematical achievement on a basic number skill test and on an arithmetic fact retrieval task. Besides math achievement, self-ratings of ADHD symptoms (attention deficits, hyperactivity, impulsivity), core executive functions (inhibition, cognitive flexibility, working memory capacity, global index) and state anxiety in 646 fourth and fifth grade students (48.1 % girls) were measured. CEF was evaluated by means of a tablet-based test. Regression analysis showed different predictors for both math abilities. While all CEF measures, inattention and impulsivity self-ratings predicted basic number skills, only inhibition, WMC and inattention self-rating were predictors of arithmetic fact retrieval. On the basis of a cluster analysis of ADHD self-ratings and CEF performance, three different groups were identified: 1: low ADHD, average CEF; 2: high ADHD, low CEF; 3: moderate ADHD, high CEF. Cluster 2 showed scores below the mean in both math scores, whereas cluster 3 exhibited performances above the mean in both math scores. No math differences were found between cluster 1 and 3. The data yield evidence that the association between CEF and math achievement depends on the specific math skill and test type. The results underline the importance of differentiating between CEF components and specific math abilities. Implications for future research on the associations between mathematics, ADHD and CEF are discussed.

Development of early domain-specific and domain-general cognitive precursors of high and low math achievers in grade 6

This study investigated from a longitudinal retrospective perspective what characterizes and predicts 6th graders (M_age = 12.95, SD = 0.27) with low (LMA) or high (HMA) math achievement concerning the development of early domain-specific and domain-general cognitive abilities. They were examined and compared to average achievers (n = 88) at four-time points from kindergarten (M_age = 6.58, SD = 0.36) to third grade (M_age = 9.53, SD = 0.33). The LMA (n = 27) or HMA (n = 41) children exhibited persistent multi-weakness and multi-strength profiles, respectively, present already prior to formal schooling. The cognitive profiles of the two groups, and their development, were mostly qualitatively similar, but there were also important qualitative differences. Logistic regression analyzes showed that superior verbal arithmetic, logical reasoning, and executive functions are vital for developing superior mathematical skills while inferior verbal arithmetic, logical reasoning, and spatial processing ability constitute unique potential risk factors for low mathematical skills.

Mathematical Difficulties vs. High Achievement: An Analysis of Arithmetical Cognition in Elementary School

This study analyzed the contribution of cognitive processes (planning, attention, simultaneous and successive processing) and domain-specific skills (counting, number processing and conceptual comprehension) to the arithmetic performance achieved in the last three grades (4th, 5th, and 6th) of elementary school. Three groups of students with a different arithmetic achievement level were characterized. The predictive value of the cognitive processes and the math specific skills are explored through diverse covariance and discriminant analyses. Participants were 110 students (M = 10.5 years, SD = 1.17) classified in three groups: mathematical difficulties (MD; n = 26), high achieving (HA; n = 26), and typical achieving (TA; n = 58). Cognitive processes and domain-specific skills were evaluated in two individual sessions at the end of the school year. Nonverbal intelligence was assessed in a final collective session with each class. The mathematical difficulties group's achievement was deficient in simultaneous and successive processing, number processing, and conceptual comprehension compared to the typical achievement group. High achievement children obtained significantly better results than the typical achievement children in simultaneous processing, counting, number processing, and conceptual comprehension. Number processing and conceptual comprehension were the most consistent classifiers, although successive and simultaneous processing, respectively, also contributed to identifying students with mathematical difficulties and high achievement. These findings have practical implications for preventive and intervention proposals linked to the observed profiles.

Numerical Estimation and Mathematical Learning Methodology in Preschoolers

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

Logical Reasoning, Spatial Processing, and Verbal Working Memory: Longitudinal Predictors of Physics Achievement at Age 12-13 Years

To date, few studies have tried to pinpoint the mechanisms supporting children’s skills in science. This study investigated to what extent logical reasoning, spatial processing, and working memory, tapped at age 9–10 years, are predictive of physics skills at age 12–13 years. The study used a sample of 81 children (37 girls). Measures of arithmetic calculation and reading comprehension were also included in the study. The multiple regression model accounted for 24% of the variation in physics achievement. The model showed that spatial processing (4.6%) and verbal working memory (4.5%) accounted for a similar amount of unique variance, while logical reasoning accounted for 5.7% variance. The measures of arithmetic calculation and reading comprehension did not account for any unique variance. Nine percent of the accounted variance was shared variance. The results demonstrate that physics is a multivariate discipline that draws upon numerous cognitive resources. Logical reasoning ability is a key component in order for children to learn about abstract physics facts, concepts, theories, and applying complex scientific methods. Spatial processing is important as it may sub-serve the assembly of diverse sources of visual-spatial information into a spatial-schematic image. The working memory system provides a flexible and efficient mental workspace that can supervise, coordinate, and execute processes involved in physics problem-solving.

Executive Function: Association with Multiple Reading Skills

Executive function (EF) is related to reading. However, there is a lack of clarity around (a) the relative contribution of different components of EF to different reading components (word reading, fluency, comprehension), and (b) how EF operates in the context of known strong language predictors (e.g., components of the Simple View of Reading or SVR), and other skills theoretically related to reading (e.g., vocabulary, processing speed) and/or to EF (e.g., short-term memory, motor function). In a large sample of 3rd to 5th graders oversampled for struggling readers, this paper evaluates the impact of EF derived from a bifactor model (Cirino, Ahmed, Miciak, Taylor, Gerst, & Barnes, 2018) in the context of well-known covariates and demographics. Beyond common EF, five specific factors (two related to working memory, and factors of fluency, self-regulated learning, and behavioral inattention/metacognition) were addressed. EF consistently showed a unique contribution to already-strong predictive models for all reading outcomes; for reading comprehension, EF interacted with SVR indices (word reading and listening comprehension). The findings extend and refine our understanding of the contribution of EF to reading skill.

Early Cognitive Precursors of Children's Mathematics Learning Disability and Persistent Low Achievement: A 5-Year Longitudinal Study

Mathematical difficulties have been distinguished as mathematics learning disability (MLD) and persistent low achievement (LA). Based on 1,880 Finnish children who were followed from kindergarten (age 6) to fourth grade, this study examined the early risk factors for MLD and LA. Distinct groups of MLD (6.0% of the sample) and LA (25.7%) children were identified on the basis of their mathematics performance between first and fourth grades with latent class growth modeling. Impairment in the same set of cognitive skills, including language, spatial, and counting skills, was found to underlie MLD and LA. The finding highlights the importance of monitoring mathematical development across the early grades and identifying early cognitive precursors of MLD and LA for screening and intervention efforts.

Different Subcomponents of Executive Functioning Predict Different Growth Parameters in Mathematics: Evidence From a 4-Year Longitudinal Study With Chinese Children

Executive functioning (EF), an umbrella term used to represent cognitive skills engaged in goal-directed behaviors, has been found to be a unique predictor of mathematics performance. However, very few studies have examined how the three core EF subcomponents (inhibition, shifting, and working memory) predict the growth parameters (intercept and slope) in mathematics skills and even fewer studies have been conducted in a non-Western country. Thus, the purpose of this study was to examine how inhibition, shifting, and working memory predict the growth parameters in arithmetic accuracy and fluency in a group of Chinese children (n = 179) followed from Grade 2 (mean age = 97.89 months) to Grade 5 (mean age = 133.43 months). In Grade 2, children were assessed on measures of nonverbal IQ, number sense, speed of processing, inhibition, shifting, and working memory. In addition, in Grades 2–5, they were assessed on arithmetic accuracy and fluency. Results of structural equation modeling showed that nonverbal IQ, speed of processing, and number sense predicted the intercept in arithmetic accuracy, while working memory was the only EF subcomponent to predict the slope (rate of growth) in arithmetic accuracy. In turn, number sense, speed of processing, inhibition, and shifting were significant predictors of the intercept in arithmetic fluency. None of the EF subcomponents predicted the slope in arithmetic fluency. Our findings reinforce those of previous studies in North America and Europe showing that EF contributes to mathematics performance over and above other key predictors of mathematics, and suggest that different EF subcomponents may predict different growth parameters in mathematics.

Working memory components that predict word problem solving: Is it merely a function of reading, calculation, and fluid intelligence?

The purpose of this study was to assess whether the differential effects of working memory (WM) components (the central executive, phonological loop, and visual-spatial sketchpad) on math word problem-solving accuracy in children (N = 413, ages 6-10) are completely mediated by reading, calculation, and fluid intelligence. The results indicated that all three WM components predicted word problem solving in the nonmediated model, but only the storage component of WM yielded a significant direct path to word problem-solving accuracy in the fully mediated model. Fluid intelligence was found to moderate the relationship between WM and word problem solving, whereas reading, calculation, and related skills (naming speed, domain-specific knowledge) completely mediated the influence of the executive system on problem-solving accuracy. Our results are consistent with findings suggesting that storage eliminates the predictive contribution of executive WM to various measures Colom, Rebollo, Abad, & Shih (Memory & Cognition, 34: 158-171, 2006). The findings suggest that the storage component of WM, rather than the executive component, has a direct path to higher-order processing in children.

Assessment and conceptual remediation of basic calculation skills in elementary school students

The specific domain model for math disabilities postulates a core number deficit which presents a prime target for remedial interventions. This longitudinal study identified two groups of Grade 3 students based on their basic calculation abilities: students with persistent difficulties through Grade 4 (PD group) and students whose performance improved into the average range (IP group). Baseline data revealed a distinct cognitive profile for students in the PD group featuring predominant deficits in symbolic number processing. A conceptual intervention based on explicit teaching of basic arithmetic procedures was implemented when students attended Grade 5 or 6. Students in the PD group benefited more from the programme, especially in performing written calculations and in multiplication speed. Statement of contribution What is already known on this subject? Most interventions focus on young students' basic arithmetical skills to prevent serious math problems in future. Few interventions target older students who often face persistent math difficulties. These interventions are usually procedural and focus on age-appropriate math skills. What does this study add? A conceptual intervention was implemented to remediate basic calculation deficits at the end of elementary school. The aim was to help students compensate for their gaps in knowledge and motivate them to engage in math activities. Neuropsychological testing of arithmetic abilities revealed difficulties in symbolic number processing.

A Systematic Review of Longitudinal Studies of Mathematics Difficulty

Some students may be diagnosed with a learning disability in mathematics or dyscalculia, whereas other students may demonstrate below-grade-level mathematics performance without a disability diagnosis. In the literature, researchers often identify students in both groups as experiencing math difficulty. To understand the performance of students with math difficulty, we examined 35 studies that reported longitudinal results of mathematics achievement (i.e., mathematics performance measured across at least a 12-month span). Our primary goal was to conduct a systematic review of these studies and to understand whether the growth of students with math difficulty was comparable or stagnant when compared with that of students without math difficulty. We also analyzed whether identification of math difficulty was predictive of mathematics achievement in later grades and whether a diagnosis of math difficulty was stable across grade levels. Results indicate that students with math difficulty demonstrate growth on mathematics measures, but this growth still leads to lower performance than that of students without math difficulty. Identification of math difficulty is strongly related to math performance in subsequent grades, and this diagnosis is often stable. Collectively, this literature indicates that students with math difficulty continue to struggle with mathematics in later grades.

Adolescents with Developmental Dyscalculia Do Not Have a Generalized Magnitude Deficit - Processing of Discrete and Continuous Magnitudes

The link between number and space has been discussed in the literature for some time, resulting in the theory that number, space and time might be part of a generalized magnitude system. To date, several behavioral and neuroimaging findings support the notion of a generalized magnitude system, although contradictory results showing a partial overlap or separate magnitude systems are also found. The possible existence of a generalized magnitude processing area leads to the question how individuals with developmental dyscalculia (DD), known for deficits in numerical-arithmetical abilities, process magnitudes. By means of neuropsychological tests and functional magnetic resonance imaging (fMRI) we aimed to examine the relationship between number and space in typical and atypical development. Participants were 16 adolescents with DD (14.1 years) and 14 typically developing (TD) peers (13.8 years). In the fMRI paradigm participants had to perform discrete (arrays of dots) and continuous magnitude (angles) comparisons as well as a mental rotation task. In the neuropsychological tests, adolescents with dyscalculia performed significantly worse in numerical and complex visuo-spatial tasks. However, they showed similar results to TD peers when making discrete and continuous magnitude decisions during the neuropsychological tests and the fMRI paradigm. A conjunction analysis of the fMRI data revealed commonly activated higher order visual (inferior and middle occipital gyrus) and parietal (inferior and superior parietal lobe) magnitude areas for the discrete and continuous magnitude tasks. Moreover, no differences were found when contrasting both magnitude processing conditions, favoring the possibility of a generalized magnitude system. Group comparisons further revealed that dyscalculic subjects showed increased activation in domain general regions, whilst TD peers activate domain specific areas to a greater extent. In conclusion, our results point to the existence of a generalized magnitude system in the occipito-parietal stream in typical development. The detailed investigation of spatial and numerical magnitude abilities in DD reveals that the deficits in number processing and arithmetic cannot be explained with a general magnitude deficiency. Our results further indicate that multiple neuro-cognitive components might contribute to the explanation of DD.

Examining the relationship between rapid automatized naming and arithmetic fluency in Chinese kindergarten children

Rapid automatized naming (RAN) has been found to predict mathematics. However, the nature of their relationship remains unclear. Thus, the purpose of this study was twofold: (a) to examine how RAN (numeric and non-numeric) predicts a subdomain of mathematics (arithmetic fluency) and (b) to examine what processing skills may account for the RAN-arithmetic fluency relationship. A total of 160 third-year kindergarten Chinese children (83 boys and 77 girls, mean age=5.11years) were assessed on RAN (colors, objects, digits, and dice), nonverbal IQ, visual-verbal paired associate learning, phonological awareness, short-term memory, speed of processing, approximate number system acuity, and arithmetic fluency (addition and subtraction). The results indicated first that RAN was a significant correlate of arithmetic fluency and the correlations did not vary as a function of type of RAN or arithmetic fluency tasks. In addition, RAN continued to predict addition and subtraction fluency even after controlling for all other processing skills. Taken together, these findings challenge the existing theoretical accounts of the RAN-arithmetic fluency relationship and suggest that, similar to reading fluency, multiple processes underlie the RAN-arithmetic fluency relationship.

Limited Near and Far Transfer Effects of Jungle Memory Working Memory Training on Learning Mathematics in Children with Attentional and Mathematical Difficulties

The goal of this randomized controlled trial was to investigate whether Jungle Memory working memory training (JM) affects performance on working memory tasks, performance in mathematics and gains made on a mathematics training (MT) in school aged children between 9-12 years old (N = 64) with both difficulties in mathematics, as well as attention and working memory. Children were randomly assigned to three groups and were trained in two periods: (1) JM first, followed by MT, (2) MT first, followed by JM, and (3) a control group that received MT only. Bayesian analyses showed possible short term effects of JM on near transfer measures of verbal working memory, but none on visual working memory. Furthermore, support was found for the hypothesis that children that received JM first, performed better after MT than children who did not follow JM first or did not train with JM at all. However, these effects could be explained at least partly by frequency of training effects, possibly due to motivational issues, and training-specific factors. Furthermore, it remains unclear whether the effects found on improving mathematics were actually mediated by gains in working memory. It is argued that JM might not train the components of working memory involved in mathematics sufficiently. Another possible explanation can be found in the training's lack of adaptivity, therefore failing to provide the children with tailored instruction and feedback. Finally, it was hypothesized that, since effect sizes are generally small, training effects are bound to a critical period in development.

A longitudinal study on predictors of early calculation development among young children at risk for learning difficulties

The purpose of this study was to explore domain-general cognitive skills, domain-specific academic skills, and demographic characteristics that are associated with calculation development from first grade to third grade among young children with learning difficulties. Participants were 176 children identified with reading and mathematics difficulties at the beginning of first grade. Data were collected on working memory, language, nonverbal reasoning, processing speed, decoding, numerical competence, incoming calculations, socioeconomic status, and gender at the beginning of first grade and on calculation performance at four time points: the beginning of first grade, the end of first grade, the end of second grade, and the end of third grade. Latent growth modeling analysis showed that numerical competence, incoming calculation, processing speed, and decoding skills significantly explained the variance in calculation performance at the beginning of first grade. Numerical competence and processing speed significantly explained the variance in calculation performance at the end of third grade. However, numerical competence was the only significant predictor of calculation development from the beginning of first grade to the end of third grade. Implications of these findings for early calculation instructions among young at-risk children are discussed.

Numerical Magnitude Representation in Children With Mathematical Difficulties With or Without Reading Difficulties

This study aimed to explore the spatial numerical association of response codes (SNARC), the flanker, and the numerical distance effects in children with mathematical difficulties. From a sample of 720 third, fourth, and fifth graders, 60 children were selected and divided into the following three groups: typically developing children (TD; n = 29), children with mathematical difficulties only (MD only; n = 21), and children with mathematical and reading difficulties (MD+RD; n = 10). Children were tested with a numerical Eriksen task that was built to assess SNARC, numerical distance, and flanker (first and second order congruency) effects. Children with MD only showed stronger SNARC and second order congruency effects than did TD children, whereas the numerical distance effects were similar across the three groups. Finally, the first order congruency effect was associated with reading difficulties. These results showed that children with mathematical difficulties with or without reading difficulties were globally more impaired when spatial incompatibilities were presented.

Cognitive and numerosity predictors of mathematical skills in middle school

There is a strong research base on the underlying concomitants of early developing math skills. Fewer studies have focused on later developing skills. Here, we focused on direct and indirect contributions of cognitive measures (e.g., language, spatial skills, working memory) and numerosity measures, as well as arithmetic proficiency, on key outcomes of fraction performance, proportional reasoning, and broad mathematics achievement at sixth grade (N=162) via path analysis. We expected a hierarchy of skill development, with predominantly indirect effects of cognitive factors via number and arithmetic. Results controlling for age showed that the combination of cognitive, number, and arithmetic variables cumulatively accounted for 38% to 44% of the variance in fractions, proportional reasoning, and broad mathematics. There was consistency across outcomes, with more proximal skills providing direct effects and with the effects of cognitive skills being mediated by number and by more proximal skills. Results support a hierarchical progression from domain-general cognitive processes through numerosity and arithmetic skills to proportional reasoning to broad mathematics achievement.

The effect of parents' literacy skills and children's preliteracy skills on the risk of dyslexia

The combination of investigating child and family characteristics sheds light on the constellation of risk factors that can ultimately lead to dyslexia. This family-risk study examines plausible preschool risk factors and their specificity. Participants (N = 196, 42 % girls) included familial risk (FR) children with and without dyslexia in Grade 3 and controls. First, we found impairments in phonological awareness, rapid naming, and letter knowledge in FR kindergartners with later dyslexia, and mild phonological-awareness deficits in FR kindergartners without subsequent dyslexia. These skills were better predictors of reading than arithmetic, except for rapid naming. Second, the literacy environment at home was comparable among groups. Third, having a dyslexic parent and literacy abilities of the non-dyslexic parent related to offspring risk of dyslexia. Parental literacy abilities might be viewed as indicators of offspring's liability for literacy difficulties, since parents provide offspring with genetic and environmental endowment. We propose an intergenerational multiple deficit model in which both parents confer cognitive risks.

Pathways of Number Line Development in Children

Dyscalculia, or mathematics learning disability, has received growing attention in recent years. Working memory and number sense are hypothesized to form important determinants of dyscalculia, but longitudinal assessments of number sense in children with or at-risk for dyscalculia are scarce. The current study investigated number line development in first and second grade, in addition to kindergarten predictors and mathematical proficiency as an outcome. Children (n = 396) could be divided into three latent growth classes: at-risk, catch-up, and typical, based on their number line development. Growth was predicted by kindergarten number sense and verbal working memory. According to the class to which they were assigned, children differed in mathematical proficiency at the end of grade 2. The current study makes an important contribution to the understanding of risk for dyscalculia, showing that children at-risk can be distinguished based on their number line development, and that kindergarten variables are predictive of subsequent development.

Early numerical abilities and cognitive skills in kindergarten children

In this study, a unitary path analysis model was developed to investigate the relationship between cognitive variables (derived from published studies) and early numerical abilities in children attending the last year of kindergarten. We tested 100 children starting their last year of kindergarten on the following cognitive abilities: intelligence, phonological abilities, counting, verbal and visuospatial short-term memory and working memory, processing speed, and early numerical abilities. The same children were tested again on early numerical abilities at the end of the same year. The children's early numerical abilities at the beginning of the final year of kindergarten were found to be directly related to their verbal intelligence, phonological abilities, processing speed, and working memory and to be indirectly related to their nonverbal intelligence. Early numerical abilities at the end of the same year are directly related not only to early numerical abilities assessed at the beginning of the year but also to working memory and phonological abilities as well as have an indirect relationship with verbal and nonverbal intelligence. Overall, our results showed that both general and specific abilities are related to early mathematic learning in kindergarten-age children.