Arithmetic Fluency Research Articles

Symbolic Number Ordering and its Underlying Strategies Examined Through Self-Reports.

Symbolic number ordering has been related to arithmetic fluency; however, the nature of this relation remains unclear. Here we investigate whether the implementation of strategies can explain the relation between number ordering and arithmetic fluency. In the first study, participants (N = 16) performed a symbolic number ordering task (i.e., “is a triplet of digits presented in order or not?”) and verbally reported the strategy they used after each trial. The analysis of the verbal responses led to the identification of three main strategies: memory retrieval, triplet decomposition, and arithmetic operation. All the remaining strategies were grouped in the fourth category “other”. In the second study, participants were presented with a description of the four strategies. Afterwards, they (N = 61) judged the order of triplets of digits as fast and as accurately as possible and, after each trial, they indicated the implemented strategy by selecting one of the four pre-determined strategies. Participants also completed a standardized test to assess their arithmetic fluency. Memory retrieval strategy was used more often for ordered trials than for non-ordered trials and more for consecutive than non-consecutive triplets. Reaction times on trials solved by memory retrieval were related to the participants’ arithmetic fluency score. For the first time, we provide evidence that the relation between symbolic number ordering and arithmetic fluency is related to faster execution of memory retrieval strategies.

The Higher Prevalence of Learning Disabilities in Children With Idiopathic Epilepsy in China

Background: Although school-age children with epilepsy are at a high risk of developing comorbid learning disabilities, there is a lack of empirical research to confirm the prevalence of learning disabilities among children with idiopathic epilepsy in China. Further, previous research has not confirmed whether children with epilepsy have a higher prevalence rate of learning disabilities than other school-age children. Methods: The current study aims to examine the prevalence and gender ratios of dyscalculia, dyslexia, and comorbidity among children with idiopathic epilepsy and other school-age children in China. Specific screening tests in arithmetic fluency (dyscalculia), reading fluency (dyslexia), and their co-occurrence were administered in a large population-based sample of children with epilepsy (N = 2,282) and other school-age children (N = 2,371). The prevalence of dyscalculia, dyslexia and comorbidity were calculated based on three cutoff criteria. Findings: The results showed that children with idiopathic epilepsy, when compared to other school-age children, have significantly higher prevalence rate of dyscalculia (-1 SD cutoff: 7·6% vs. 3·8%, -1·5 SD cutoff: 4·1% vs. 2·4%), dyslexia (-1 SD cutoff: 9·0% vs. 3·7%, -1·5 SD cutoff: 3·9% vs. 2·0%) and comorbidity (-1 SD cutoff: 5·1% vs. 1·4%, -1·5 SD cutoff: 1·5% vs. 0·4%, -2 SD cutoff: 0·4% vs. 0·1%). Moreover, gender differences in the prevalence rate of dyslexia only appeared among other school-age children. Interpretation: The results suggest that children with idiopathic epilepsy are more vulnerable to dyscalculia and dyslexia compared to other school-age children. Risk factors in the idiopathic epilepsy population should be identified to formulate effective prevention strategies. Moreover, gender differences on the prevalence of dyscalculia, dyslexia and comorbidity only exist among typical school-age children, which might be explained by clinical factors related to idiopathic epilepsy, therefore, future studies could make further examination. Funding: Funding Natural Science Foundation of China (Grant No. 31700977, 31671151), Natural Science Foundation of Beijing (Grant No. 5212004), The 111 Project (Grant No. BP0719032), Advanced Innovation Center for Future Education (Grant No. 27900-110631111), Directional Guidance Fund for the Capital Institute of Pediatrics (Grant No. FX-2019-03). Declaration of Interest: We declare no competing interests. Ethical Approval: This study was approved by the Human Research Ethics Committee at the Capital Institute of Pediatrics (approval no.: SHERLL 2019001). We also obtained oral consent from participants and informed consent signed by their guardians.

Evaluating the rapid automatized naming and arithmetical fluency relationship in Chilean first grade students

Several studies have shown that Rapid Automatised Naming (RAN) is an important reading predictor. However, it has also been found that RAN is a valid predictor of mathematical performance. This study explores the relationship between RAN, counting and arithmetic fluency in Chilean first graders. A total of 102 participants were evaluated with RAN letters, phonological awareness, counting, arithmetical fluency, inhibition and verbal working memory. Results show moderate correlations between RAN and counting with arithmetical fluency. After controlling for the effect of all the variables, RAN still explains a significant variance of arithmetical fluency. Although the nature of the relationship between RAN and arithmetical fluency is multi-componential, our results only confirm the role of working memory when simultaneous counting is considered in the model. Therefore, RAN is not only an important cognitive variable related to reading processes but also a key one for mathematical skills, especially when they involve a fluency component.

Comorbidity Between Math and Reading Problems: Is Phonological Processing a Mutual Factor?

There is a relationship between reading and math skills, as well as comorbidity between reading and math disorders. A mutual foundation for this comorbidity could be that the quality of phonological representations is important for both early reading and arithmetic. In this study, we examine this hypothesis in a sample traced longitudinally from preschool to first grade (N = 259). The results show that phonological awareness does not explain development in arithmetic, but that there is an indirect effect between phoneme awareness in preschool and arithmetic in first grade via phoneme awareness in first grade. This effect is, however, weak and restricted to verbal arithmetic and not arithmetic fluency. This finding is only partly in line with other studies, and a reason could be that this study more strongly controls for confounders and previous skills than other studies.

Supplemental Material for Individual Differences in the Development of Children’s Arithmetic Fluency From Grades 2 to 3

Supplemental Material for Individual Differences in the Development of Children’s Arithmetic Fluency From Grades 2 to 3

Prevalence of Learning Disabilities Among Chinese Children, Adolescents, and Young Adults: Comorbidity, Age Trend, and Gender Differences

Background: Early identification and prevalence estimates of learning disabilities (LDs) is critical to future intervention. However, in China, most studies were conducted in a single area and they only investigated children of a specific age group. There still lacks a large-sample study on the prevalence of LDs and their comorbidities in Chinese children, adolescents, and young adults. Methods: For the first time, this study examined the prevalence of LDs and their comorbidity rates and age trend and gender difference in 9236 Chinese children (6–11 years old, mean age = 9.55, standard deviation (SD) = 1.17), adolescents (12–17 years old, mean age = 14.03, SD = 1.65), and young adults (18–35 years old, mean age = 20.63, SD = 2.72). Computerized screening tests addressing arithmetic fluency (dyscalculia), reading fluency (dyslexia), and their comorbidity were administered in a large population-based sample of children (n = 3541, 1833 males and 1708 females), adolescents (n = 3641, 1928 males and 1713 females), and young adults (n = 2054, 1045 males and 1009 females). The prevalence of dyscalculia, dyslexia, and comorbidity were analyzed based on two cutoff criteria. Findings: The results showed that with -1 SD cutoff criteria, the prevalence rates of dyscalculia and dyslexia were 10.8% and 12.3%, respectively. The comorbidity rate of dyscalculia with reading disabilities was 34.8% and the comorbidity rate of dyslexia with arithmetic disabilities was 29.6%. Using -1.5 SD cutoff criteria, the prevalence rates of dyscalculia and dyslexia were 5.8% and 6.3%, respectively. The comorbidity rate of dyscalculia with reading disabilities was 24.3% and the comorbidity rate of dyslexia with arithmetic disabilities was 22.3%. When more stringent cutoff criteria were applied, the comorbidity rates of dyscalculia and dyslexia decreased. Among the three age groups, children showed the highest prevalence rate of dyscalculia and dyslexia. In addition, we observed a preponderance of males with dyslexia, whereas the gender ratio for dyscalculia was balanced among the three age groups. Interpretation: These findings suggested that the prevalence rates of dyscalculia and dyslexia in China are similar to that of other counties. The comorbidity rates of dyscalculia and dyslexia decreased as cutoff criteria became more stringent, suggesting that dyslexia and dyscalculia may have a common biological basis. Under the two cutoff criteria among the three age groups, the prevalence rate of dyslexia was higher than dyscalculia, which suggested that dyslexia affected a wider population than dyscalculia. Children were more vulnerable to dyscalculia and dyslexia compared to adolescents and young adults. Among the three age groups, gender differences existed in the prevalence rate of dyslexia; males were more likely to have dyslexia than females. Funding: This study is funded by the Natural Science Foundation of China (Grant No. 31700977, 31671151), Natural Science Foundation of Beijing (Grant No. 5212004), The 111 Project (Grant No. BP0719032), Advanced Innovation Center for Future Education (Grant No. 27900-110631111). Declaration of Interest: We declare no competing interests. Ethical Approval: The study design was approved by the State Key Laboratory of Cognitive Neuroscience and Learning at Beijing Normal University and the school principals.

Development of reading and arithmetic skills across Grades 1 to 4 in two groups of children receiving part-time special education

This study investigated why some Finnish students receive part-time special education in Grade 1, duration of that support, and its relation to student reading fluency, reading comprehension, and arithmetic fluency development. The participants included two groups from Grade 1 receiving part-time special education (1–2 years vs. 3–4 years) and a control group comprising the remaining participants from the study. Teachers identified reading and expressive language difficulties as the main reasons for part-time special education in Grade 1. By Grade 4, students who received support until Grade 1 or 2 caught up the level of the control group especially in reading fluency. Students who had received support until Grade 3 or 4 exhibited more persistent and overlapping difficulties with attention, receptive language, memory, and motivation. Additionally, by Grade 4, differences from the control group remained statistically significant for all three skills. Implications for enhancing special education interventions for students with persistent and overlapping learning difficulties are discussed.

Failure to replicate the benefit of approximate arithmetic training for symbolic arithmetic fluency in adults

Previous research reported that college students' symbolic addition and subtraction fluency improved after training with non-symbolic, approximate addition and subtraction. These findings were widely interpreted as strong support for the hypothesis that the Approximate Number System (ANS) plays a causal role in symbolic mathematics, and that this relation holds into adulthood. Here we report four experiments that fail to find evidence for this causal relation. Experiment 1 examined whether the approximate arithmetic training effect exists within a shorter training period than originally reported (2 vs 6 days of training). Experiment 2 attempted to replicate and compare the approximate arithmetic training effect to a control training condition matched in working memory load. Experiments 3 and 4 replicated the original approximate arithmetic training experiments with a larger sample size. Across all four experiments (N = 318) approximate arithmetic training was no more effective at improving the arithmetic fluency of adults than training with control tasks. Results call into question any causal relationship between approximate, non-symbolic arithmetic and precise symbolic arithmetic.

Longitudinal Effects of the Home Learning Environment and Parental Difficulties on Reading and Math Development Across Grades 1-9.

This study focuses on parental reading and mathematical difficulties, the home literacy environment, and the home numeracy environment as well as their predictive role in Finnish children’s reading and mathematical development through Grades 1–9. We examined if parental reading and mathematical difficulties directly predict children’s academic performance and/or if they are mediated by the home learning environment. Mothers (n = 1590) and fathers (n = 1507) reported on their reading and mathematical difficulties as well as on the home environment (shared reading, teaching literacy, and numeracy) when their children were in kindergarten. Tests for reading fluency, reading comprehension, and arithmetic fluency were administered to children in Grades 1, 2, 3, 4, 7, and 9. Parental reading difficulties predicted children’s reading fluency, whereas parental mathematical difficulties predicted their reading comprehension and arithmetic fluency. Familial risk was associated with neither formal nor informal home environment factors, whereas maternal education had a significant relationship with both, with higher levels of education among mothers predicting less time spent on teaching activities and more time spent on shared reading. In addition, shared reading was significantly associated with the development of reading comprehension up to Grades 3 and 4, whereas other components of the home learning environment were not associated with any assessed skills. Our study highlights that taken together, familial risk, parental education, and the home learning environment form a complex pattern of associations with children’s mathematical and reading skills.

Neural association between non-verbal number sense and arithmetic fluency.

Non‐verbal number sense has been shown to significantly correlate with arithmetic fluency. Accumulated behavioral evidence indicates that the cognitive mechanism relies on visual perception. However, few studies have investigated the neural mechanism underlying this association. Following the visual perception account, we hypothesized that there would be a neural association in occipital areas of the brain between non‐verbal number sense, arithmetic fluency, and visual perception. We analyzed event‐related potentials that are sensitive to neural responses while participants performed five cognitive tasks: simple addition, simple subtraction, numerosity comparison, figure matching, and character rhyming. The single‐trial ERP‐behavior correlation approach was used to enhance the statistical power. The results showed that the N1 component significantly correlated with reaction time at occipital electrodes on all tasks except for character rhyming. The N1 component for arithmetic fluency (simple addition and subtraction) and character rhyming correlated with the reaction time for numerosity comparison and figure matching. The results suggest that there are neural associations between arithmetic fluency, non‐verbal number sense, and visual perception in the occipital cortex, and that visual perception is the shared mechanism for both non‐verbal number sense and arithmetic fluency.

Cognitive predictors of arithmetic, reading, and spelling in Brazilian Portuguese-speaking children

We examined the contribution of a major predictor of basic literacy ability—phoneme awareness—to individual differences in two arithmetic computation tasks: a speeded task consisting of simple computation problems (arithmetic fluency) and an untimed, more complex computation task involving multi-digit operands (numerical operations). The contribution of phoneme awareness to word-reading fluency, untimed word-reading accuracy, and untimed spelling accuracy also was evaluated. Participants were 134 Brazilian Portuguese-speaking 4th and 5th grade students aged 8.67 to 11.75 years (M = 10.13 years; SD = 0.58). Individual differences in phoneme awareness accounted for significant and unique variance in performance on the arithmetic fluency task even after taking into account individual differences in grade in school, verbal ability, nonverbal ability, rapid naming, and verbal working memory. In contrast, phoneme awareness did not contribute uniquely to variance in performance on the more complex, untimed numerical operations task. In keeping with De Smedt’s (2018) argument regarding the role of phoneme awareness in arithmetic ability, our findings suggest that the contribution of phoneme awareness is primarily restricted to the ability to solve very simple arithmetic problems for which the solution is likely to be retrieved directly and automatically from long-term memory. As expected, phoneme awareness and rapid naming were the two strongest predictors of individual differences in all literacy measures.

The interplay between spatial ordinal knowledge, linearity of number-space mapping, and arithmetic skills

The ability to map numbers onto space has been widely investigated with the number line (NL) task. Accurate (linear) placement is typically preceded by a developmental phase in which children assign more space to small numbers and compress large numbers to the right part of the line, thereby resembling a biased log-like mapping. Here, we exploited a task that separately assesses direction, order and accuracy of spatial mapping (DOS task) to investigate the origin of the biased NL mapping in a sample of first and second graders. Children with reduced ordinal knowledge in the DOS task showed a more biased NL mapping, which was formally assessed using the leading mathematical models of the NL task. However, the log-like biased mapping did not emerge in the DOS task, thereby showing a lack of generalisation across different number-space mapping tasks. Both ordinal knowledge in the DOS task and linearity in the NL task related to arithmetic fluency beyond domain-general cognitive factors. We conclude that the NL task is a meter of children’s arithmetic skills rather than an expression of the mental representation of numbers.

Early Engagement of Parietal Cortex for Subtraction Solving Predicts Longitudinal Gains in Behavioral Fluency in Children.

There is debate in the literature regarding how single-digit arithmetic fluency is achieved over development. While the Fact-retrieval hypothesis suggests that with practice, children shift from quantity-based procedures to verbally retrieving arithmetic problems from long-term memory, the Schema-based hypothesis claims that problems are solved through quantity-based procedures and that practice leads to these procedures becoming more automatic. To test these hypotheses, a sample of 46 typically developing children underwent functional magnetic resonance imaging (fMRI) when they were 11 years old (time 1), and 2 years later (time 2). We independently defined regions of interest (ROIs) involved in verbal and quantity processing using rhyming and numerosity judgment localizer tasks, respectively. The verbal ROIs consisted of left middle/superior temporal gyri (MTG/STG) and left inferior frontal gyrus (IFG), whereas the quantity ROIs consisted of bilateral inferior/superior parietal lobules (IPL/SPL) and bilateral middle frontal gyri (MFG)/right IFG. Participants also solved a single-digit subtraction task in the scanner. We defined the extent to which children relied on verbal vs. quantity mechanisms by selecting the 100 voxels showing maximal activation at time 1 from each ROI, separately for small and large subtractions. We studied the brain mechanisms at time 1 that predicted gains in subtraction fluency and how these mechanisms changed over time with improvement. When looking at brain activation at time 1, we found that improvers showed a larger neural problem size effect in bilateral parietal cortex, whereas no effects were found in verbal regions. Results also revealed that children who showed improvement in behavioral fluency for large subtraction problems showed decreased activation over time for large subtractions in both parietal and frontal regions implicated in quantity, whereas non-improvers maintained similar levels of activation. All children, regardless of improvement, showed decreased activation over time for large subtraction problems in verbal regions. The greater parietal problem size effect at time 1 and the reduction in activation over time for the improvers in parietal and frontal regions implicated in quantity processing is consistent with the Schema-based hypothesis arguing for more automatic procedures with increasing skill. The lack of a problem size effect at time 1 and the overall decrease in verbal regions, regardless of improvement, is inconsistent with the Fact-retrieval hypothesis.

Impact of Children’s math self-concept, math self-efficacy, math anxiety, and teacher competencies on math development

We examined to what extent children’s development of arithmetic fluency and mathematical problem-solving was influenced by their math self-concept, math self-efficacy, and math anxiety but also teacher competence, specifically: actual teaching behavior, self-efficacy, and mathematical teaching knowledge. Participants were 610 children and 31 teachers of grade four. Multi-level analyses showed children’s math self-concept to be a positive predictor of arithmetic fluency and actual teaching behavior to be a negative predictor. The development of mathematical problem-solving was predicted: positively by mathematical teaching knowledge; negatively by actual teaching behavior and teachers’ self-efficacy; and not at all by the child factors of math self-concept, math self-efficacy, or math anxiety. Promoting the self-confidence of young children is essential for their mathematical development. More research into the relationship between teaching behaviors and children’s math development is needed.

The semantic control network mediates the relationship between symbolic numerical order processing and arithmetic performance in children

Behavioural and neuroimaging studies have recently demonstrated that symbolic numerical order processing (i.e., deciding whether numbers are in an increasing/decreasing sequence or not) and symbolic numerical magnitude processing (e.g., deciding which of two numerals is larger) engage different cognitive mechanisms and brain regions. Because of this dissociation, growing interest has emerged to better understand the neurocognitive mechanisms of symbolic numerical order processing and their relationship to individual differences in arithmetic performance. In the present functional imaging work, we further investigated this link in a group of thirty children (7.2–10.25 years) from elementary school, who completed a symbolic numerical order verification (are the numbers going up? e.g., 1-2-3) and a symbolic numerical magnitude comparison task (which is the larger number? e.g., 5–7) inside the scanner, as well as an arithmetic fluency test outside the scanner. Behavioural results demonstrated the unique role of numerical order to predict children's arithmetic skills and confirmed its mediating power to explain the association between numerical magnitude and arithmetic performance. Imaging results showed a significant association between numerical order and arithmetic in the intersection of the right inferior frontal gyrus and insula, as well as the posterior middle temporal gyrus. An age-dependent change in brain activity was found in the left intraparietal sulcus. These findings solidify the developmental importance of symbolic numerical order processing in children and provide new evidence that the semantic control network mediates the relationship with arithmetic performance.

Exploring mechanisms of rapid automatized naming to arithmetic skills in Chinese primary schoolers

AbstractRapid automatized naming (RAN) is associated with children's arithmetic performance, which is multifactorial by nature and consists of arithmetic fluency, arithmetic procedure, and word problems. Yet, it is unclear whether RAN can predict all key aspects of arithmetic skills independently and what cognitive mechanisms may explain this relation. Using a sample of Chinese primary school students (N = 170), results showed that while RAN was associated with all key aspects of arithmetic skills, the relations of RAN–arithmetic procedure and RAN–word problems may be dependent on that of RAN–arithmetic fluency. Supporting the processing speed hypothesis, results further suggested that processing speed may underlie the relation between RAN and arithmetic performance. Findings made unique contribution to understanding the effect of RAN on arithmetic performance and cognitive mechanisms that may underpin this association in the context of Chinese culture.

Development of arithmetic fluency: A direct effect of reading fluency?

The present longitudinal study investigated the developmental trajectories of addition, subtraction, and multiplication fluency; their relationships to general cognitive functions; and potential interrelations between reading fluency and growth in arithmetic fluency for different operations. The central prediction was that measures of arithmetic fluency for different operations would be differentially related to reading fluency because of variations in the timing and format of arithmetic instruction. Panel data on children’s arithmetic fluency and reading fluency performance were collected twice per year in third, fourth, and fifth grades (N = 449). For each operation tested (addition, subtraction, and multiplication), a series of bivariate (reading/arithmetic) latent growth and autoregressive latent trajectory models were constructed. After controlling for background cognitive variables, significant short- and long-term relationships to reading fluency were observed only for multiplication fluency. Specifically, reading fluency in early third grade predicted both immediately subsequent multiplication fluency scores and the slope of growth in multiplication fluency across subsequent grade levels. Exploratory analyses of covariate effects showed that whole number line estimation accuracy was related to growth in multiplication and subtraction fluency, while general verbal ability had a unique relationship to subtraction fluency that was not observed for either addition or multiplication. Overall, results indicate that children’s level of reading fluency and differences among arithmetic operations should be considered both when conducting research and when making instructional decisions about how to build students’ arithmetic fluency. (PsycINFO Database Record (c) 2019 APA, all rights reserved)

Responsive home numeracy as children progress from kindergarten through Grade 1

The aim of the present study was to investigate longitudinal associations between formal home numeracy activities and children’s arithmetic fluency skills. Children were followed during the transition from the end of kindergarten (T1; Mage = 6.87 years) to the beginning of Grade 1 (T2), and again at the end of the Grade 1 (T3). Participants were Lithuanian children (n = 341) and their parents. At each time point, parents reported the frequency of the formal home numeracy activities they engaged in with their children; the children completed addition and subtraction tasks at the same three time points. Using a cross-lagged analysis, we found that parents adjusted the frequency of their home numeracy activities in response to children’s arithmetic performance. The frequency of home numeracy activities, however, did not predict changes in arithmetic skills. Finally, maternal education was found to be related to children’s arithmetic skills rather than to home numeracy activities.

The hierarchical symbol integration model of individual differences in mathematical skill

Symbolic number knowledge is strongly related to mathematical performance for both children and adults. We present a model of symbolic number relations in which increasing skill is a function of hierarchical integration of symbolic associations. We tested the model by contrasting the performance of two groups of adults. One group was educated in China (n = 71) and had substantially higher levels of mathematical skill compared to the other group who was educated in Canada (n = 68). Both groups completed a variety of symbolic number tasks, including measures of cardinal number knowledge (number comparisons), ordinal number knowledge (ordinal judgments) and arithmetic fluency, as well as other mathematical measures, including number line estimation, fraction/algebra arithmetic and word problem solving. We hypothesized that Chinese-educated individuals, whose mathematical experiences include a strong emphasis on acquiring fluent access to symbolic associations among numbers, would show more integrated number symbol knowledge compared to Canadian-educated individuals. Multi-group path analysis supported the hierarchical symbol integration hypothesis. We discuss the implications of these results for understanding why performance on simple number processing tasks is persistently related to measures of mathematical performance that also involve more complex and varied numerical skills.

Adaptive number knowledge in secondary school students: Profiles and antecedents

The present study aims to examine inter-individual differences in adaptive number knowledge in secondary school students. Adaptive number knowledge is defined as a well-connected network of knowledge of numerical characteristics and arithmetic relations. Substantial and relevant qualitative differences in the strategies and expression of adaptive number knowledge have been found in primary school students still in the process of learning arithmetic. We present a study involving 879 seventh-grade students that examines the structure of individual differences in adaptive number knowledge with students who have completed one year of algebra instruction. Results of a latent profile analysis reveal a model that is similar than was previously found in primary school students. As well, arithmetic fluency and the development of arithmetic fluency are strong predictors of adaptive number knowledge latent profile membership. These results suggest that adaptive number knowledge may be characteristic of high-level performance extending into secondary school, even after formal instruction with arithmetic concludes.