Complexities in Identifying and Defining Mathematics Learning Disability in the Primary School-Age Years

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

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

Examining the role of the visuospatial sketchpad in children's math calculation skills using Baddeley and Hitch's model of working memory

Math difficulties (MDs) occur in about 3-7 % of children and have been associated with academic, health, and occupational challenges. To date, findings about the role of working memory in MDs have been conflicting. The Automated Working Memory Assessment Battery (AWMA), which assesses all components of Baddeley and Hitch's model of working memory, was used to investigate which component of the model was most related to math calculation skills in elementary-school children. Participants were 94 (52 male) children (M age = 9 years 1 month; Range = 6 years 0 months to 11 years 8 months). As hypothesized, math calculation scores were correlated with all four working memory components (phonological loop, visuospatial sketchpad, verbal and visuospatial central executive). After accounting for age, phonological processing, and attention, the visuospatial sketchpad was the only memory component that contributed to the prediction of math calculation scores, explaining an additional 10.2 % of unique variance. Short-term visuospatial memory should be assessed in children having difficulty with math and children could benefit from interventions that include attention to the development of both visuospatial memory and math calculation skills. This study did not use a longitudinal design and so we cannot conclude that weak visuospatial memory impedes the development of math calculation skills. Future research should use longitudinal designs and investigate other types of math skills.

Developmental Dyscalculia in Relation to Individual Differences in Mathematical Abilities

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

Parent ratings on the MEMRY questionnaire predict children's academic performance

Given the high rates of learning challenges in children with medical conditions, efficient and reliable screening methods are crucial. This study examined whether parent report of daily learning and memory predicts academic achievement in youth. Parents of 213 youth (aged 6-18) with varied medical diagnoses completed the Multidimensional Everyday Memory Ratings for Youth (MEMRY), and youth completed subtests from the Wechsler Individual Achievement Test-Third Edition (WIAT-III) as part of a comprehensive assessment. All scales of the MEMRY (Learning, Daily Memory, Executive/Working Memory) correlated significantly with WIAT-III Spelling, Word Reading, and Numerical Operations, while only the MEMRY Learning subscale correlated significantly with WIAT-III Pseudoword Decoding. Regression analyses indicated that MEMRY Learning significantly predicted WIAT-III Word Reading and Spelling, while both the MEMRY Learning and MEMRY Daily Memory scales significantly predicted WIAT-III Numerical Operations. When Full Scale IQ was entered into the models first, the MEMRY Learning subscale accounted for an additional 4% of variance in WIAT-III Word Reading and 7% of variance in WIAT-III Spelling, but did not account for additional variance in WIAT-III Numerical Operations or Pseudoword Decoding. Analyses in a subset of children with broadly normal intellectual functioning demonstrated very similar results, with even higher variance in academic testing accounted for by the MEMRY. In sum, the MEMRY questionnaire may serve as an efficient screen to identify children at risk for reading, spelling, and math deficits.

Emergent math difficulties among English learners: can the odds be reduced?

Solving word problems in mathematics presents difficulties for many English learners (ELs), including those who use Spanish as a home language. In the early stage of learning to solve mathematics word problems, some children, including ELs, experience difficulties. By English status, EL refers to those children whose home language was Spanish while in the process of developing English at school. Math difficulties (MD) refers to those children with low normative mathematical problem-solving scores in both English and Spanish. The purpose of this study was to determine those measures that increase the odds of identifying EL children with emergent MD. Elementary school children (grades 1, 2, and 3) were administered a battery of math, vocabulary, reading, and cognitive measures (short-term memory [STM], inhibition, working memory [WM]) in both Spanish (L1) and English (L2) in Year 1 and again one year later. Multilevel growth modeling compared MD children identified one year later who manifested MD to children who were average math achievers or poor math achievers across the two testing waves (year 1 and year 2). The results indicated that significant growth parameters (i.e., measures of estimation, WM) increased the odds of identifying emergent MD relative to children with Persistent math deficits and average achievers. The results were discussed in terms of a multidimensional model that taps domain-specific skills and general cognitive processes that increase the odds of identifying later math difficulties.

The Cognitive Ability of Chinese Students with Dyslexia and Mathematical Learning Disabilities

This study aims to investigate the core cognitive factors that affect reading and math performance of children of the grades 1-6 in Xi'an, Shaanxi Province, China, as well as the differences between children with dyslexia and mathematical disabilities (MD). Therefore, this study mainly evaluated the Cattell Horn Carroll (CHC) cognitive factors for 427 Chinese children and explored the core cognitive factors that affect Chinese children's reading and math performance. Students with dyslexia (n = 34), students with mathematics learning disabilities (n = 34), and 34 normal children were randomly selected as the control group. In order to explore the differences in cognitive development, we analyzed the differences among the three groups (Dyslexia, mathematical learning disabilities (MD), and normal children). The results revealed the following: (1) almost all cognitive ability factors in this study are significantly related to students' reading and mathematical achievements. (2) the core cognitive factors for predicting Chinese dyslexia students are crystallized intelligence, auditory processing and working memory. Executive function, spatial relationship and working memory are the core cognitive factors to predict Chinese children's mathematical achievements. (3) in addition, there are differences in cognitive deficits between disabled Chinese children in reading and math, among which those with reading deficits have extensive auditory processing deficits; while children with mathematic deficits have worse executive function. Recommendations were made based on these findings.

A Neuropsychological Profile of Developmental Dyscalculia: The Role of Comorbidity

Developmental dyscalculia (DD) has long been thought to be determined by multiple components. Dyscalculia has high comorbidity with other learning and developmental disabilities, including reading and writing disorders, attention deficits, and problems in visual/spatial skills, short memory, and working memory. This study aims to assess prevalence rates for isolated as well as comorbid DD in a sample of Italian-speaking children. In addition, we studied the neuropsychological profile of children with isolated or combined dyscalculia. We tested 380 children (176 males and 204 females) between the ages of 8.17 and 9.33 years using an extensive battery to determine the neuropsychological profile. The assessment included an arithmetic battery and nonverbal intelligence, short-term memory, reading, and writing tests. The results indicated that children with DD more frequently have a reading disorder and writing disorder. They also have a lower nonverbal intelligence quotient (IQ) and obtain significantly lower scores in short-term memory tests and on a visuospatial skills questionnaire. They also had significantly higher scores (indicative of greater attentional difficulties) in the Conners subscale for attentional problems. Children with DD present different cognitive and neuropsychological profiles.

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.

Discerning Developmental Dyscalculia and Neurodevelopmental Models of Numerical Cognition in a Disadvantaged Educational Context

Developmental Dyscalculia (DD) signifies a failure in representing quantities, which impairs the performance of basic math operations and schooling achievement during childhood. The lack of specificity in assessment measures and respective cut-offs are the most challenging factors to identify children with DD, particularly in disadvantaged educational contexts. This research is focused on a numerical cognition battery for children, designed to diagnose DD through 12 subtests. The aims of the present study were twofold: to examine the prevalence of DD in a country with generally low educational attainment, by comparing z-scores and percentiles, and to test three neurodevelopmental models of numerical cognition based on performance in this battery. Participants were 304 Brazilian school children aged 7–12 years of both sexes (143 girls), assessed by the Zareki-R. Performances on subtests and the total score increase with age without gender differences. The prevalence of DD was 4.6% using the fifth percentile and increased to 7.4% via z-score (in total 22 out of 304 children were diagnosed with DD). We suggest that a minus 1.5 standard deviation in the total score of the Zareki-R is a useful criterion in the clinical or educational context. Nevertheless, a percentile ≤ 5 seems more suitable for research purposes, especially in developing countries because the socioeconomic environment or/and educational background are strong confounder factors to diagnosis. The four-factor structure, based on von Aster and Shalev’s model of numerical cognition (Number Sense, Number Comprehension, Number Production and Calculation), was the best model, with significant correlations ranging from 0.89 to 0.97 at the 0.001 level.

Predicting Risk for Comorbid Reading and Mathematics Disability Using Fluency-Based Screening Assessments

The first purpose of this study was to examine the prevalence of risk for comorbid reading and mathematics disabilities (RMD) at start of first grade, when measured in a representative sample of 3,062 students with first-grade fluency measures (word reading; computation). The second purpose was to examine the utility of these measures for predicting RMD status within a sample of 577 students when RMD status was assessed at the end of second grade in terms of reading and math accuracy. When set at or below the 16^th percentile, first-grade risk for RMD was two times more common than chance; at or below the 7^th percentile, it was five times more common. Logistic regression showed that the two first-grade fluency measures accurately distinguished students with and without RMD in second grade; however, when cut scores were set to capture 85% of students with RMD, false positives were high. Overall, the results provide support for the use of fluency measures as an initial gating procedure in first grade, but additional gating steps appear necessary in the screening process to reduce false positives.

Examining the impact of a tutoring program implemented with community support on math proficiency and growth

The current study evaluated the impact of a math tutoring program delivered in 20 schools to students in 4th through 8th grades by community members over one academic year. Students were randomly assigned to treatment and control groups. Multi-level linear and generalized linear mixed models were used to evaluate group differences in post-test scores and the probability of attaining the spring proficiency benchmark on two increasingly distal measures of math achievement. Intent-to-treat analyses identified higher achievement scores among students assigned to treatment on a measure of fact fluency and a computer adaptive measure of overall math achievement. Students assigned to treatment also had a higher probability of reaching grade-level benchmarks on the computer adaptive test. No statistically significant effects were observed on a state proficiency test. Implications for significant and null findings are discussed within the context of intervention content and delivery.

Kölner Fragebogen zur Sprachlosigkeit

Hintergrund

Mit dem Kölner Fragebogen zur Sprachlosigkeit (KFS) liegt erstmals ein validiertes Erhebungsinstrument zur Erfassung von Sprachlosigkeit vor. Die empirische Prüfung des KFS hinsichtlich seiner Eignung zur validen Unterscheidung von Personen mit gering und hoch ausgeprägter Sprachlosigkeit ist von zentraler Bedeutung für den klinischen Einsatz des Instruments.

Material und Methoden

Die Stichprobe umfasste die KFS-Daten von insgesamt 205 teilnehmenden Personen einer Onlineerhebung. Mithilfe mehrerer inferenzstatistischer Verfahren wurden die prädiktiven Eigenschaften des KFS-Item 12 analysiert, um es für die Bestimmung eines Schwellenwerts beim Einsatz des KFS nutzbar zu machen. Basierend auf dem 75. Perzentil des KFS-Item 12 erfolgte eine Dichotomisierung der Stichprobe mit anschließender „Receiver-operating-characteristic“(ROC)-Analyse zur KFS-Gesamtskala.

Ergebnisse

Die Ergebnisse der ROC-Analyse („area under the curve“ [AUC]: 0,863; p < 0,001) ergaben bei Verwendung des Youden-Index den optimalen Schwellenwert von >29, mit einer Sensitivität von 76 % und einer Spezifität von 79 % für die Gesamtskala des KFS.

Diskussion

Der Schwellenwert der KFS-Gesamtskala ist zur Differenzierung von Personen mit gering oder stark ausgeprägter Sprachlosigkeit geeignet. Das Item 12 des KFS kann zudem als Prädiktor einer potenziellen Sprachlosigkeit herangezogen werden.

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.

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

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

No evidence for a core deficit in developmental dyscalculia or mathematical learning disabilities

BACKGROUND

Two hypotheses were tested regarding the characteristics of children with mathematical learning disabilities (MLD): (a) that children with MLD would have a 'core deficit' in basic number processing skills; and (b) that children with MLD would be at the end of a developmental continuum and have impairments in many cognitive skills.

METHODS

From a large sample (N = 1,303) of typically developing children, we selected a group definable as having MLD. The children were given measures of basic number processing and domain-general constructs. Differences between the observed sample and a simulated population were estimated using Cohen's d and Bayes factors. Receiver operating characteristic curves were plotted, and the area under the curve was computed to ascertain the diagnostic power of measures.

RESULTS

Results suggest that the differences between the MLD and control group can be defined along with general characteristics of the population rather than assuming single or multiple 'core deficits'. None of the measures of interest exceeded the diagnostic power that could be derived via simulation from the dimensional characteristics of the general population.

CONCLUSIONS

There is no evidence for core deficit(s) in MLD. We suggest that future research should focus on representative samples of typical populations and on carefully tested clinical samples confirming to the criteria of international diagnostic manuals. Clinical diagnoses require that MLD is persistent and resistant to intervention, so studies would deliver results less exposed to measurement fluctuations. Uniform diagnostic criteria would also allow for the easy cross-study comparison of samples overcoming a serious limitation of the current literature.

How do phonological awareness, rapid automatized naming, and vocabulary contribute to early numeracy and print knowledge of Filipino children?

The current study investigated the contributions of phonological awareness, rapid automatized naming (RAN), and vocabulary to early numeracy and print knowledge developmental trajectories. A total of 128 young Filipino children were tracked three times at mean ages of 4.5, 5.0, and 5.5 years. The initial level (the intercept) and the growth rate (the slope) of early numeracy and print knowledge were estimated. Results showed that phonological awareness, vocabulary, and age significantly predicted the initial level of early numeracy. RAN and vocabulary explained significant variance in the growth rate of early numeracy. Phonological awareness, RAN, and vocabulary accounted for unique variance in the initial level of print knowledge. Results highlight the differential roles of phonological awareness, RAN, and vocabulary knowledge in the development of early numeracy and print knowledge among Filipino children.

Hand preference and Mathematical Learning Difficulties: New data from Greece, the United Kingdom, and Germany and two meta-analyses of the literature

Increased rates of atypical handedness are observed in neurotypical individuals who are low-performing in mathematical tasks as well as in individuals with special educational needs, such as dyslexia. This is the first investigation of handedness in individuals with Mathematical Learning Difficulties (MLD). We report three new studies (N = 134; N = 1,893; N = 153) and two sets of meta-analyses (22 studies; N = 3,667). No difference in atypical hand preference between MLD and Typically Achieving (TA) individuals was found when handedness was assessed with self-report questionnaires, but weak evidence of a difference was found when writing hand was the handedness criterion in Study 1 (p = .049). Similarly, when combining data meta-analytically, no hand preference differences were detected. We suggest that: (i) potential handedness effects require larger samples, (ii) direction of hand preference is not a sensitive enough measure of handedness in this context, or that (iii) increased rates of atypical hand preference are not associated with MLD. The latter scenario would suggest that handedness is specifically linked to language-related conditions rather than conditions related to cognitive abilities at large. Future studies need to consider hand skill and degree of hand preference in MLD.

The importance of visuospatial abilities for verbal number skills in preschool: Adding spatial language to the equation

Children's verbal number skills set the foundation for mathematical development. Therefore, it is central to understand their cognitive origins. Evidence suggests that preschool children rely on visuospatial abilities when solving counting and number naming tasks despite their predominantly verbal nature. We aimed to replicate these findings when controlling for verbal abilities and sociodemographic factors. Moreover, we further characterized the relation between visuospatial abilities and verbal number skills by examining the role of spatial language. Because spatial language encompasses the verbalization of spatial thinking, it is a key candidate supporting the interplay between visuospatial and verbal processes. Regression analysis indicated that both visuospatial and verbal abilities, as assessed by spatial perception and phonological awareness, respectively, uniquely predicted verbal number skills when controlling for their respective influences, age, gender, and socioeconomic status. This confirms the spatial grounding of verbal number skills. Interestingly, adding spatial language to the model abolished the predictive effects of visuospatial and verbal abilities, whose influences were completely mediated by spatial language. Verbal number skills thus concurrently depend on specifically those visuospatial and verbal processes jointly indexed through spatial language. The knowledge of spatial terms might promote verbal number skills by advancing the understanding of the spatial relations between numerical magnitudes on the mental number line. Promoting spatial language in preschool thus might be a successful avenue for stimulating mathematical development prior to formal schooling. Moreover, measures of spatial language could become an additional promising tool to screen preschool children for potential upcoming difficulties with mathematical learning.

Dyscalculia and Typical Math Achievement Are Associated With Individual Differences in Number-Specific Executive Function

Deficits in numerical magnitude perception characterize the mathematics learning disability developmental dyscalculia (DD), but recent studies suggest the relation stems from inhibitory control demands from incongruent visual cues in the nonsymbolic number comparison task. This study investigated the relation among magnitude perception during differing congruency conditions, executive function, and mathematics achievement measured longitudinally in children (n = 448) from ages 4 to 13. This relation was investigated across achievement groups and as it related to mathematics across the full range of achievement. Only performance on incongruent trials related to achievement. Findings indicate that executive function in a numerical context, beyond magnitude perception or executive function in a non-numerical context, relates to DD and mathematics across a wide range of achievement.

The prevalence of specific learning disorder in mathematics and comorbidity with other developmental disorders in primary school-age children

Mathematics difficulties are common in both children and adults, and they can have a great impact on people's lives. A specific learning disorder in mathematics (SLDM or developmental dyscalculia) is a special case of persistent mathematics difficulties, where the problems with maths cannot be attributed to environmental factors, intellectual disability, or mental, neurological or physical disorders. The aim of the current study was to estimate the prevalence rate of SLDM, any gender differences in SLDM, and the most common comorbid conditions. The DSM-5 provides details regarding these only for specific learning disorders in general, but not specifically for SLDM. We also compared the prevalence rates obtained on the basis of the DSM-IV and DSM-5 criteria. We investigated the performance of 2,421 primary school children on standardized tests of mathematics, English, and IQ, and several demographic factors over the primary school years. We applied the DSM-5 diagnostic criteria to identify children with a potential diagnosis of SLDM. Six per cent of our sample had persistent, severe difficulties with mathematics, and, after applying the exclusion criteria, 5.7% were identified as having an SLDM profile. Both persistent maths difficulties and consistently exceptionally high performance in maths were equally common in males and females. About half of the children with an SLDM profile had some form of language or communication difficulty. Some of these children also had a diagnosis of autism, social, emotional, and behavioural difficulties or attention deficit and hyperactivity disorder. Our findings have important implications for research and intervention purposes, which we discuss in the study.

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.

Are WISC IQ scores in children with mathematical learning disabilities underestimated? The influence of a specialized intervention on test performance

BACKGROUND

Intelligence measures play a pivotal role in the diagnosis of mathematical learning disabilities (MLD). Probably as a result of math-related material in IQ tests, children with MLD often display reduced IQ scores. However, it remains unclear whether the effects of math remediation extend to IQ scores.

AIMS

The present study investigated the impact of a special remediation program compared to a control group receiving private tutoring (PT) on the WISC IQ scores of children with MLD.

METHODS

We included N=45 MLD children (7-12 years) in a study with a pre- and post-test control group design. Children received remediation for two years on average.

RESULTS

The analyses revealed significantly greater improvements in the experimental group on the Full-Scale IQ, and the Verbal Comprehension, Perceptual Reasoning, and Working Memory indices, but not Processing Speed, compared to the PT group. Children in the experimental group showed an average WISC IQ gain of more than ten points.

CONCLUSION

Results indicate that the WISC IQ scores of MLD children might be underestimated and that an effective math intervention can improve WISC IQ test performance. Taking limitations into account, we discuss the use of IQ measures more generally for defining MLD in research and practice.

TEMA and Dot Enumeration Profiles Predict Mental Addition Problem Solving Speed Longitudinally

Different math indices can be used to assess math potential at school entry. We evaluated whether standardized math achievement (TEMA-2 performance), core number abilities (dot enumeration, symbolic magnitude comparison), non-verbal intelligence (NVIQ) and visuo-spatial working memory (VSWM), in combination or separately, predicted mental addition problem solving speed over time. We assessed 267 children’s TEMA-2, magnitude comparison, dot enumeration, and VSWM abilities at school entry (5 years) and NVIQ at 8 years. Mental addition problem solving speed was assessed at 6, 8, and 10 years. Longitudinal path analysis supported a model in which dot enumeration performance ability profiles and previous mental addition speed predicted future mental addition speed on all occasions, supporting a componential account of math ability. Standardized math achievement and NVIQ predicted mental addition speed at specific time points, while VSWM and symbolic magnitude comparison did not contribute unique variance to the model. The implications of using standardized math achievement and dot enumeration ability to index math learning potential at school entry are discussed.

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.

Error Patterns with Fraction Calculations at Fourth Grade as a Function of Students' Mathematics Achievement Status

The goal of the present study was to describe fraction-calculation errors among 4^th-grade students and determine whether error patterns differed as a function of problem type (addition vs. subtraction; like vs. unlike denominators), orientation (horizontal vs. vertical), or mathematics-achievement status (low- vs. average- vs. high-achieving). We specifically addressed whether mathematics-achievement status was related to students' tendency to operate with whole number bias. We extended this focus by comparing low-performing students' errors in two instructional settings that focused on two different types of fraction understandings: core instruction that focused on part-whole understanding vs. small-group tutoring that focused on magnitude understanding. Results showed students across the sample were more likely to operate with whole number bias on problems with unlike denominators. Students with low or average achievement (who only participated in core instruction) were more likely to operate with whole number bias than students with low achievement who participated in small-group tutoring. We suggest instruction should emphasize magnitude understanding to sufficiently increase fraction understanding for all students in the upper elementary grades.

The effect of visual parameters on neural activation during nonsymbolic number comparison and its relation to math competency

Nonsymbolic numerical comparison task performance (whereby a participant judges which of two groups of objects is numerically larger) is thought to index the efficiency of neural systems supporting numerical magnitude perception, and performance on such tasks has been related to individual differences in math competency. However, a growing body of research suggests task performance is heavily influenced by visual parameters of the stimuli (e.g. surface area and dot size of object sets) such that the correlation with math is driven by performance on trials in which number is incongruent with visual cues. Almost nothing is currently known about whether the neural correlates of nonsymbolic magnitude comparison are also affected by visual congruency. To investigate this issue, we used functional magnetic resonance imaging (fMRI) to analyze neural activity during a nonsymbolic comparison task as a function of visual congruency in a sample of typically developing high school students (n = 36). Further, we investigated the relation to math competency as measured by the preliminary scholastic aptitude test (PSAT) in 10th grade. Our results indicate that neural activity was modulated by the ratio of the dot sets being compared in brain regions previously shown to exhibit an effect of ratio (i.e. left anterior cingulate, left precentral gyrus, left intraparietal sulcus, and right superior parietal lobe) when calculated from the average of congruent and incongruent trials, as it is in most studies, and that the effect of ratio within those regions did not differ as a function of congruency condition. However, there were significant differences in other regions in overall task-related activation, as opposed to the neural ratio effect, when congruent and incongruent conditions were contrasted at the whole-brain level. Math competency negatively correlated with ratio-dependent neural response in the left insula across congruency conditions and showed distinct correlations when split across conditions. There was a positive correlation between math competency in the right supramarginal gyrus during congruent trials and a negative correlation in the left angular gyrus during incongruent trials. Together, these findings support the idea that performance on the nonsymbolic comparison task relates to math competency and ratio-dependent neural activity does not differ by congruency condition. With regards to math competency, congruent and incongruent trials showed distinct relations between math competency and individual differences in ratio-dependent neural activity.

Using assessment to individualize early mathematics instruction

Accumulating evidence suggests that assessment-informed personalized instruction, tailored to students' individual skills and abilities, is more effective than more one-size-fits-all approaches. In this study, we evaluate the efficacy of Individualizing Student Instruction in Mathematics (ISI-Math) compared to Reading (ISI-Reading) where classrooms were randomly assigned to ISI-Math or ISI-Reading. The literature on child characteristics X instruction or skill X treatment interaction effects point to the complexities of tailoring instruction for individual students who present with constellations of skills. Second graders received mathematics instruction in small flexible learning groups based on their assessed learning needs. Results of the study (n=32 teachers, 370 students) revealed significant treatment effects on standardized mathematics assessments. With effect sizes (d) of 0.41-0.60, we show that we can significantly improve 2nd graders' mathematics achievement, including for children living in poverty, by using assessment data to individualize the mathematics instruction they receive. The instructional regime, ISI-Math, was implemented by regular classroom teachers and it led to about a 4-month achievement advantage on standardized mathematics tests when compared to students in control classrooms. These results were realized within one school year. Moreover, treatment effects were the same regardless of school-level poverty and students' gender, initial mathematics or vocabulary scores.

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.

Children's visuospatial memory predicts mathematics achievement through early adolescence

A previous study showed that gains in visuospatial memory from first to fifth grade predicted end-of-fifth grade mathematics but not reading achievement, controlling other factors. In this follow up study, these relations were assessed from sixth to ninth grade, inclusive (n = 145). The results showed that growth in visuospatial memory across the elementary school years was related to growth in mathematics achievement after fifth grade, controlling intelligence, the central executive and phonological memory components of working memory, in-class attentive behavior, parental education, and fifth grade mathematics achievement. As found for fifth grade, this relation was not found for reading achievement after fifth grade. In total, the results suggest that visuospatial memory has a unique influence on ease of learning some types of mathematics and that this influence becomes more important across successive grades.

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.

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.

Strategic Development for Middle School Students Struggling With Fractions: Assessment and Intervention

Research has suggested that different strategies used when solving fraction problems are highly correlated with students' problem-solving accuracy. This study (a) utilized latent profile modeling to classify students into three different strategic developmental levels in solving fraction comparison problems and (b) accordingly provided differentiated strategic training for students starting from two different strategic developmental levels. In Study 1 we assessed 49 middle school students' performance on fraction comparison problems and categorized students into three clusters of strategic developmental clusters: a cross-multiplication cluster with the highest accuracy, a representation strategy cluster with medium accuracy, and a whole-number strategy cluster with the lowest accuracy. Based on the strategic developmental levels identified in Study 1, in Study 2 we selected three students from the whole-number strategy cluster and another three students from the representation strategy cluster and implemented a differentiated strategic training intervention within a multiple-baseline design. Results showed that both groups of students transitioned from less advanced to more advanced strategies and improved their problem-solving accuracy during the posttest, the maintenance test, and the generalization test.

Multidimensionality in the Measurement of Math-Specific Anxiety and its Relationship with Mathematical Performance

Traditionally, mathematical anxiety has been utilized as a unidimensional construct. However, math-specific anxiety may have distinguishable factors, and taking these factors into account may better illuminate the relationship between anxiety and mathematics performance. Drawing from the Western Reserve Reading and Math Project (N = 244 children, mean age = 12.28 years), the present study examined math-specific anxiety and mathematics problem evaluation, utilizing a structural equation modeling approach with an item-level measurement model structure. Results suggested math-specific anxiety tapped into three factors: anxiety about performing mathematical calculations, anxiety about math in classroom situations, and anxiety about math tests. Among the three math anxiety factors, only calculation anxiety was significantly and negatively related to math performance while holding other anxiety factors constant. Implications for the measurement of math-specific anxiety are discussed.

Kindergartners' fluent processing of symbolic numerical magnitude is predicted by their cardinal knowledge and implicit understanding of arithmetic 2years earlier

Fluency in first graders' processing of the magnitudes associated with Arabic numerals, collections of objects, and mixtures of objects and numerals predicts current and future mathematics achievement. The quantitative competencies that support the development of fluent processing of magnitude, however, are not fully understood. At the beginning and end of preschool (M=3years 9months at first assessment, range=3years 3months to 4years 3months), 112 children (51 boys) completed tasks measuring numeral recognition and comparison, acuity of the approximate number system, and knowledge of counting principles, cardinality, and implicit arithmetic and also completed a magnitude processing task (number sets test) in kindergarten. Use of Bayesian and linear regression techniques revealed that two measures of preschoolers' cardinal knowledge and their competence at implicit arithmetic predicted later fluency of magnitude processing, controlling domain-general factors, preliteracy skills, and parental education. The results help to narrow the search for the early foundation of children's emerging competence with symbolic mathematics and provide direction for early interventions.

Who Is At Risk for Persistent Mathematics Difficulties in the United States?

We analyzed two nationally representative, longitudinal data sets of U.S. children to identify risk factors for persistent mathematics difficulties (PMD). Results indicated that children from low socioeconomic households are at elevated risk of PMD at 48 and 60 months of age, as are children with cognitive delays, identified developmental delays or disabilities, and vocabulary difficulties. In contrast, children attending preschool either in Head Start or non-Head Start classrooms are at initially lower risk of PMD. Kindergarten-aged children experiencing either low socioeconomic status or mathematics difficulties are at greatest risk for PMD across third, fifth, and eighth grades. Also at risk for PMD between third and eighth grades are children displaying reading difficulties or inattention and other learning-related behaviors problems, children with identified disabilities, and those who are retained. Educationally relevant and potentially malleable factors for decreasing young children's risk for PMD may include increasing children's access to preschool, decreasing their risk of experiencing vocabulary or reading difficulties, and avoiding use of grade retention.

Cognitive Profiles of Mathematical Problem Solving Learning Disability for Different Definitions of Disability

Three cohorts of third-grade students (N= 813) were evaluated on achievement, cognitive abilities, and behavioral attention according to contrasting research traditions in defining math learning disability (LD) status: low achievement versus extremely low achievement and IQ-achievement discrepant versus strictly low-achieving LD. We use methods from these two traditions to form math problem solving LD groups. To evaluate group differences, we used MANOVA-based profile and canonical analyses to control for relations among the outcomes and regression to control for group definition variables. Results suggest that basic arithmetic is the key distinguishing characteristic that separates low-achieving problem solvers (including LD, regardless of definition) from typically achieving students. Word problem solving is the key distinguishing characteristic that separates IQ-achievement-discrepant from strictly low-achieving LD students, favoring the IQ-achievement-discrepant students.