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.

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.

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 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.

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.

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.

Number sense and mathematics: Which, when and how?

Individual differences in number sense correlate with mathematical ability and performance, although the presence and strength of this relationship differs across studies. Inconsistencies in the literature may stem from heterogeneity of number sense and mathematical ability constructs. Sample characteristics may also play a role as changes in the relationship between number sense and mathematics may differ across development and cultural contexts. In this study, 4,984 16-year-old students were assessed on estimation ability, one aspect of number sense. Estimation was measured using 2 different tasks: number line and dot-comparison. Using cognitive and achievement data previously collected from these students at ages 7, 9, 10, 12, and 14, the study explored for which of the measures and when in development these links are observed, and how strong these links are and how much these links are moderated by other cognitive abilities. The 2 number sense measures correlated modestly with each other (r = .22), but moderately with mathematics at age 16. Both measures were also associated with earlier mathematics; but this association was uneven across development and was moderated by other cognitive abilities. (PsycINFO Database Record

Early Executive Function and Mathematics Relations: Correlation Does Not Ensure Concordance.

In this chapter, we address one potentially overlooked component of the relation between executive function (EF) skills and early mathematics, a relation for which there is widespread empirical support. Evidence for this relation has, thus far, been largely correlational. Here we emphasize that because positive correlations do not guarantee concordance among all members of a sample or population, a small but meaningful number of children may either fare well in mathematics despite poor EF skills, or may have strong EF skills despite weak mathematics skills. We propose that attention to different profiles of discordance for EF and mathematics may help identify individualized learning needs for students at risk for mathematics difficulties and disabilities.

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.

Knowing right from wrong in mental arithmetic judgments: calibration of confidence predicts the development of accuracy.

Does knowing when mental arithmetic judgments are right--and when they are wrong--lead to more accurate judgments over time? We hypothesize that the successful detection of errors (and avoidance of false alarms) may contribute to the development of mental arithmetic performance. Insight into error detection abilities can be gained by examining the "calibration" of mental arithmetic judgments-that is, the alignment between confidence in judgments and the accuracy of those judgments. Calibration may be viewed as a measure of metacognitive monitoring ability. We conducted a developmental longitudinal investigation of the relationship between the calibration of children's mental arithmetic judgments and their performance on a mental arithmetic task. Annually between Grades 5 and 8, children completed a problem verification task in which they rapidly judged the accuracy of arithmetic expressions (e.g., 25 + 50 = 75) and rated their confidence in each judgment. Results showed that calibration was strongly related to concurrent mental arithmetic performance, that calibration continued to develop even as mental arithmetic accuracy approached ceiling, that poor calibration distinguished children with mathematics learning disability from both low and typically achieving children, and that better calibration in Grade 5 predicted larger gains in mental arithmetic accuracy between Grades 5 and 8. We propose that good calibration supports the implementation of cognitive control, leading to long-term improvement in mental arithmetic accuracy. Because mental arithmetic "fluency" is critical for higher-level mathematics competence, calibration of confidence in mental arithmetic judgments may represent a novel and important developmental predictor of future mathematics performance.

Persistent consequences of atypical early number concepts.

How does symbolic number knowledge performance help identify young children at risk for poor mathematics achievement outcomes? In research and practice, classification of mathematics learning disability (MLD, or dyscalculia) is typically based on composite scores from broad measures of mathematics achievement. These scores do predict later math achievement levels, but do not specify the nature of math difficulties likely to emerge among students at greatest risk for long-term mathematics failure. Here we report that gaps in 2nd and 3rd graders' number knowledge predict specific types of errors made on math assessments at Grade 8. Specifically, we show that early whole number misconceptions predict slower and less accurate performance, and atypical computational errors, on Grade 8 arithmetic tests. We demonstrate that basic number misconceptions can be detected by idiosyncratic responses to number knowledge items, and that when such misconceptions are evident during primary school they persist throughout the school age years, with variable manifestation throughout development. We conclude that including specific qualitative assessments of symbolic number knowledge in primary school may provide greater specificity of the types of difficulties likely to emerge among students at risk for poor mathematics outcomes.

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).

Growth in rapid automatized naming from grades K to 8 in children with math or reading disabilities.

Rapid automatized naming (RAN) is widely used to identify reading disabilities (RD) and has recently been considered a potential predictor of risk for mathematics learning disabilities (MLD). Here we longitudinally examine RAN performance from Grades K to 8, to view how growth on RAN response time differs for children with RD versus MLD. Across all participants, there was significant stability of RAN performance for all three subtests. For all three subtests, there was significant between-child variability in response times (RTs) at kindergarten and Grade 8; for select combinations of subtests, there was also between-child variability in difference scores over time. Relative to typically achieving peers, children with MLD or RD were slower on all three subtests at kindergarten, but these deficits were most pronounced for children with MLD versus RD at kindergarten and more persistent through Grade 8 for children with RD. RAN numbers and letters performance was closely related in the RD group, more so than in the MLD group, despite deficits on both subtests across groups. Our findings indicate that empirically guided modifications of RAN are needed for extending its use for evaluating risk for MLD.

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 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.