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.

Executive function and mathematics in preschool children: Training and transfer effects.

Identifying the underpinnings of mathematics proficiency is relevant for all societies. A growing literature supports a relation between executive function (EF) and mathematics across a wide age range, but causal links are not well understood. In the current study, typically developing preschool children (N = 104) were randomly assigned to one of four training conditions: EF, Number, EF + Number, or an active Control. They participated in three brief training sessions and pretest and posttest sessions measuring EF and mathematics skills. EF training improved EF skills on a task similar to the training but did not extend to an untrained EF task. In addition, the EF training improved number skills but not general mathematics skills. The EF + Number training improved number and general mathematics skills but not EF skills. The EF + Number training did not yield significantly greater benefits for EF and mathematics beyond other training conditions. Finally, differential training effects emerged, such that children with lower pretest EF skills had greater EF benefits on only the trained EF skill. In addition, children from lower versus higher socioeconomic households had greater gains in numerical skills following EF training. No training condition improved verbal knowledge, suggesting that results were specific to the targeted skills. These results extend prior findings on the effectiveness of improving EF and mathematical skills through short-term trainings during early childhood.

Relational language influences young children's number relation skills.

Relational language is thought to influence mathematical skills. This study examines the association between relational language and number relation skills-knowledge of cardinal, ordinal, and spatial principles-among 104 U.S. kindergartners (5.9 years; 44% boys; 37% White, 25% Black, 14% Asian, 24% other) in the 2017-2018 academic year. Controlling for general verbal knowledge, executive function, and counting and number identification skills, relational language predicted later number relation skills, specifically number line estimation, ß = .30. Relational language did not differentially predict number line estimation performance in children with low or high number relation skills, likely due to the restricted ranges of data within subgroups. Number relation skills, specifically number line estimation and number ordering, may be a pathway between relational language and mathematical skills.

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.

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.

Competing features influence children's attention to number.

Spontaneous focus on numerosity (SFON), an attentional process that some consider distinct from number knowledge, predicts later mathematical skills. Here we assessed the "spontaneity" and malleability of SFON using a picture-matching task. We asked children to view a target picture and to choose which of four other pictures matched the target. We tested whether attention to number (defined as number-based matches) was affected by (a) age, (b) the presence of very noticeable (or salient) features among alternative match choices, and (c) the examiner's use of motor actions to emphasize numerosity. Although adults attended to number more frequently than did preschoolers, the salience of competing features affected responses to number in both age groups. Specifically, number-based matches were more likely when alternative choices matched the target on features of low versus high salience (e.g., the relative location within a picture frame vs. color). In addition, adults' attention to number was more frequent if their first exposure to number-based matches occurred with alternative choices that matched the target on low salience features. This order by salience interaction was not observed among children. Simply observing motor actions that emphasized number (i.e., tapping stimuli) did not enhance children's attention to number. The results extend previous findings on SFON and provide evidence for the contextual influences on, and malleability of, attention to number.

From classroom environment to mathematics achievement: The mediating role of self-perceived ability and subject interest.

Drawing on Bandura's triadic reciprocal causation model, perceived classroom environment and three intrapersonal factors (mathematics self-efficacy, maths interest and academic self-concept) were considered as predictors of test performance in two correlated mathematics assessments: a public examination (GCSE) and an on-line test, both taken by UK pupils at age 16 (n = 6689). Intrapersonal factors were significantly associated with both test scores, even when the alternative score was taken into account. Classroom environment did not correlate with mathematics achievement once intrapersonal factors and alternative test performance were included in the model, but was associated with subject interest and academic self-concept. Perceptions of classroom environment may exercise an indirect influence on achievement by boosting interest and self-concept. In turn, these intrapersonal factors have direct relationships with achievement and were found to mediate the relationship between perceived classroom environment and maths performance. Findings and their implications for mathematics education are discussed.

Is Math Anxiety Always Bad for Math Learning? The Role of Math Motivation.

The linear relations between math anxiety and math cognition have been frequently studied. However, the relations between anxiety and performance on complex cognitive tasks have been repeatedly demonstrated to follow a curvilinear fashion. In the current studies, we aimed to address the lack of attention given to the possibility of such complex interplay between emotion and cognition in the math-learning literature by exploring the relations among math anxiety, math motivation, and math cognition. In two samples-young adolescent twins and adult college students-results showed inverted-U relations between math anxiety and math performance in participants with high intrinsic math motivation and modest negative associations between math anxiety and math performance in participants with low intrinsic math motivation. However, this pattern was not observed in tasks assessing participants' nonsymbolic and symbolic number-estimation ability. These findings may help advance the understanding of mathematics-learning processes and provide important insights for treatment programs that target improving mathematics-learning experiences and mathematical skills.

Why mental arithmetic counts: brain activation during single digit arithmetic predicts high school math scores.

Do individual differences in the brain mechanisms for arithmetic underlie variability in high school mathematical competence? Using functional magnetic resonance imaging, we correlated brain responses to single digit calculation with standard scores on the Preliminary Scholastic Aptitude Test (PSAT) math subtest in high school seniors. PSAT math scores, while controlling for PSAT Critical Reading scores, correlated positively with calculation activation in the left supramarginal gyrus and bilateral anterior cingulate cortex, brain regions known to be engaged during arithmetic fact retrieval. At the same time, greater activation in the right intraparietal sulcus during calculation, a region established to be involved in numerical quantity processing, was related to lower PSAT math scores. These data reveal that the relative engagement of brain mechanisms associated with procedural versus memory-based calculation of single-digit arithmetic problems is related to high school level mathematical competence, highlighting the fundamental role that mental arithmetic fluency plays in the acquisition of higher-level mathematical competence.

Individual differences in non-verbal number acuity correlate with maths achievement.

Human mathematical competence emerges from two representational systems. Competence in some domains of mathematics, such as calculus, relies on symbolic representations that are unique to humans who have undergone explicit teaching. More basic numerical intuitions are supported by an evolutionarily ancient approximate number system that is shared by adults, infants and non-human animals-these groups can all represent the approximate number of items in visual or auditory arrays without verbally counting, and use this capacity to guide everyday behaviour such as foraging. Despite the widespread nature of the approximate number system both across species and across development, it is not known whether some individuals have a more precise non-verbal 'number sense' than others. Furthermore, the extent to which this system interfaces with the formal, symbolic maths abilities that humans acquire by explicit instruction remains unknown. Here we show that there are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual differences in the present correlate with children's past scores on standardized maths achievement tests, extending all the way back to kindergarten. Moreover, this correlation remains significant when controlling for individual differences in other cognitive and performance factors. Our results show that individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense. Further research will determine whether early differences in number sense acuity affect later maths learning, whether maths education enhances number sense acuity, and the extent to which tertiary factors can affect both.

Individual differences in left parietal white matter predict math scores on the Preliminary Scholastic Aptitude Test.

Mathematical skills are of critical importance, both academically and in everyday life. Neuroimaging research has primarily focused on the relationship between mathematical skills and functional brain activity. Comparatively few studies have examined which white matter regions support mathematical abilities. The current study uses diffusion tensor imaging (DTI) to test whether individual differences in white matter predict performance on the math subtest of the Preliminary Scholastic Aptitude Test (PSAT). Grades 10 and 11 PSAT scores were obtained from 30 young adults (ages 17-18) with wide-ranging math achievement levels. Tract based spatial statistics was used to examine the correlation between PSAT math scores, fractional anisotropy (FA), radial diffusivity (RD) and axial diffusivity (AD). FA in left parietal white matter was positively correlated with math PSAT scores (specifically in the left superior longitudinal fasciculus, left superior corona radiata, and left corticospinal tract) after controlling for chronological age and same grade PSAT critical reading scores. Furthermore, RD, but not AD, was correlated with PSAT math scores in these white matter microstructures. The negative correlation with RD further suggests that participants with higher PSAT math scores have greater white matter integrity in this region. Individual differences in FA and RD may reflect variability in experience dependent plasticity over the course of learning and development. These results are the first to demonstrate that individual differences in white matter are associated with mathematical abilities on a nationally administered scholastic aptitude measure.

Limited knowledge of fraction representations differentiates middle school students with mathematics learning disability (dyscalculia) versus low mathematics achievement.

Fractions pose significant challenges for many children, but for some children those challenges persist into high school. Here we administered a fractions magnitude comparison test to 122 children, from Grades 4 to 8, to test whether their knowledge of fractions typically learned early in the sequence of formal math instruction (e.g., fractions equivalent to one-half, fraction pairs with common denominators) differentiates those with mathematics learning disability (MLD) versus low achievement (LA) or typical achievement (TA) in mathematics and whether long-term learning trajectories of this knowledge also differentiate these groups. We confirmed that although fourth graders with TA (n=93) were more accurate in evaluating "one-half" fractions than in evaluating "non-half" fractions (until they reached ceiling performance levels on both types of fractions), children with MLD (n=11) did not show a one-half advantage until Grade 7 and did not reach ceiling performance even by Grade 8. Both the MLD and LA groups had early difficulties with fractions, but by Grade 5 the LA group approached performance levels of the TA group and deviated from the MLD group. All groups showed a visual model advantage over Arabic number representation of fractions, but this advantage was short-lived for the TA group (because ceiling level was achieved across formats), whereas it was slightly more persistent for the LA group and persisted through Grade 8 for children with MLD. Thus, difficulties with fractions persist through Grade 8 for many students, but the nature and trajectories of those difficulties vary across children with math difficulties (MLD or LA).

Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia).

Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. The current study hypothesizes that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. In this study of 71 ninth graders, it is shown that students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low, typically, and high achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relation persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low-achieving from typically achieving students, suggesting an ANS deficit that is specific to MLD.

Correction: Variation in early number skills and mathematics achievement: Implications from cognitive profiles of children with or without Turner syndrome.

[This corrects the article DOI: 10.1371/journal.pone.0239224.].

Variation in early number skills and mathematics achievement: Implications from cognitive profiles of children with or without Turner syndrome.

Individuals with Mathematics Learning Disabilities have persistent mathematics underperformance but vary with respect to their cognitive profiles. The present study examined mathematics ability and achievement, and associated mathematics-specific numerical skills and domain-general cognitive abilities, in young children with Turner syndrome compared to their matched peers. We utilized two independent peer groups so that group comparisons would account for verbal skills, a hypothesized strength of girls with Turner syndrome, and nonsymbolic magnitude comparison skills, a hypothesized difference of girls with Turner syndrome. This individual matching approach afforded characterization of mathematics profiles of girls with Turner syndrome and girls without Turner syndrome that share potential key features of the Turner syndrome phenotype. Results indicated differences in mathematics ability and nonsymbolic magnitude comparison tasks between girls with Turner syndrome and peers with similar levels of verbal skill. Mathematics ability and mathematics achievement scores of girls with Turner syndrome did not differ significantly from their peers with similar levels of accuracy on a nonsymbolic magnitude comparison task. Cognitive correlates of mathematics outcomes showed disparate patterns across groups. These quantitative and qualitative differences across profiles enhance our understanding of variation in mathematics ability in early childhood and inform how mathematics skills develop in young children with or without Turner syndrome.

Effects of treatment with oxandrolone for 4 years on the frequency of severe arithmetic learning disability in girls with Turner syndrome.

OBJECTIVES: To study androgen treatment effects on arithmetic performance in girls with Turner syndrome. STUDY DESIGN: Forty-four girls, ages 10 to 14 years at baseline, completed 4 years of treatment with oxandrolone (Ox) or placebo (Pl). All received growth hormone and estrogen replacement therapy. We assessed the number of girls with severe learning disability (LD, standard score

The nature of the association between number line and mathematical performance: An international twin study.

BACKGROUND: The number line task assesses the ability to estimate numerical magnitudes. People vary greatly in this ability, and this variability has been previously associated with mathematical skills. However, the sources of individual differences in number line estimation and its association with mathematics are not fully understood. AIMS: This large-scale genetically sensitive study uses a twin design to estimate the magnitude of the effects of genes and environments on: (1) individual variation in number line estimation and (2) the covariation of number line estimation with mathematics. SAMPLES: We used over 3,000 8- to 16-year-old twins from the United States, Canada, the United Kingdom, and Russia, and a sample of 1,456 8- to 18-year-old singleton Russian students. METHODS: Twins were assessed on: (1) estimation of numerical magnitudes using a number line task and (2) two mathematics components: fluency and problem-solving. RESULTS: Results suggest that environments largely drive individual differences in number line estimation. Both genes and environments contribute to different extents to the number line estimation and mathematics correlation, depending on the sample and mathematics component. CONCLUSIONS: Taken together, the results suggest that in more heterogeneous school settings, environments may be more important in driving variation in number line estimation and its association with mathematics, whereas in more homogeneous school settings, genetic effects drive the covariation between number line estimation and mathematics. These results are discussed in the light of development and educational settings.

Parts and 'holes': gaps in rational number sense among children with vs. without mathematical learning disabilities.

Many middle-school students struggle with decimals and fractions, even if they do not have a mathematical learning disability (MLD). In the present longitudinal study, we examined whether children with MLD have weaker rational number knowledge than children whose difficulty with rational numbers occurs in the absence of MLD. We found that children with MLD failed to accurately name decimals, to correctly rank order decimals and/or fractions, and to identify equivalent ratios (e.g. 0.5 = 1/2); they also 'identified' incorrect equivalents (e.g. 0.05 = 0.50). Children with low math achievement but no MLD accurately named decimals and identified equivalent pairs, but failed to correctly rank order decimals and fractions. Thus failure to accurately name decimals was an indicator of MLD; but accurate naming was no guarantee of rational number knowledge - most children who failed to correctly rank order fractions and decimals tests passed the naming task. Most children who failed the ranking tests at 6th grade also failed at 8th grade. Our findings suggest that a simple task involving naming and rank ordering fractions and decimals may be a useful addition to in-class assessments used to determine children's learning of rational numbers.

Arousal modulation in females with fragile X or Turner syndrome.

The present study was carried out to examine physiological arousal modulation (heart activity and skin conductance, across baseline and cognitive tasks, in females with fragile X or Turner syndrome and a comparison group of females with neither syndrome. Relative to the comparison group, for whom a greater increase in skin conductance was associated with poor arithmetic performance and less risk taking behavior, females with fragile X displayed a minimal increase in heart activity that was nevertheless associated with poor performance on mental arithmetic. In contrast, no arousal-cognitive performance relationship emerged for the group with Turner syndrome. Taken together, our findings suggest that distinct profiles of arousal modulation might be associated with cognitive deficits in these syndrome populations.

Mathematics learning disabilities in girls with fragile X or Turner syndrome during late elementary school.

The present study focuses on math and related skills among 32 girls with fragile X (n = 14) or Turner (n = 18) syndrome during late elementary school. Performance in each syndrome group was assessed relative to Full Scale IQ-matched comparison groups of girls from the general population (n = 32 and n = 89 for fragile X syndrome and Turner syndrome, respectively). Differences between girls with fragile X and their comparison group emerged on untimed arithmetic calculations, mastery of counting skills, and arithmetic problem verification accuracy. Relative to girls in the comparison group, girls with Turner syndrome did not differ on untimed arithmetic calculations or problem verification accuracy, but they had limited mastery of counting skills and longer response times to complete the problem verification task. Girls with fragile X or Turner syndrome also differed from their respective comparison groups on math-related abilities, including visual-spatial, working memory, and reading skills, and the associations between math and those related skills. Together, these findings support the notion that difficulty with math and related skills among girls with fragile X or Turner syndrome continues into late elementary school and that the profile of math and related skill difficulty distinguishes the two syndrome groups from each other.