Arithmetic Fluency Research Articles

Bridging the Connection between Fluency in Reading and Arithmetic.

This study examines the contribution of early executive functions (EFs) in the association between fluency in reading and arithmetic. Kindergarten children (N = 1185) were assessed on executive functions skills and on reading and arithmetic fluency in Grade 1 and Grade 3. The analysis revealed that beyond the connection within each domain there is a unidirectional effect between fluency measures, with Grade 1 reading fluency significantly influencing the development of arithmetic fluency in Grade 3. Furthermore, the findings indicate that kindergarten EFs significantly contribute to arithmetic fluency at both time points and to reading fluency in the first grade. Early EF skills also emerged as significant contributors to the associations between fluency performance in reading and arithmetic, suggesting that the influence of EFs extends beyond individual academic domains. These findings have implications for understanding the cognitive mechanisms that underlie the relations between these academic skills.

Affective and cognitive factors associated with Chinese and Italian children’s arithmetic performance

BackgroundThis study aimed to investigate the cognitive and affective factors associated with cross-cultural differences in arithmetic tasks.MethodsA total of 404 third- and fourth- graders were recruited from China and Italy to complete exact arithmetic, arithmetic estimation and cognitive tasks (i.e., short-term memory, executive functions, and fluid reasoning). Their mathematical anxiety was also measured.ResultsThe results showed that Chinese children performed better than Italian children in both arithmetic tasks and in shifting task. Italian children performed better in visuospatial updating task and reported higher levels of mathematical anxiety than their Chinese peers. Multi-group path analyses showed that the patterns of relations among cognitive factors (i.e., short-term memory, inhibition and shifting), mathematical anxiety, and arithmetic performance were similar across groups. The only exception was that visuospatial updating uniquely predicted arithmetic estimation for Chinese but not for Italian children.ConclusionsChinese children outperformed their Italian peers in the exact arithmetic task, likely due to the greater emphasis on arithmetic fluency in Chinese mathematics education, both in schools and at home. They also had a slight advantage than Italian peers in the arithmetic estimation task. The unique link between updating and arithmetic estimation found in Chinese children but not Italian children suggests that, although arithmetic estimation is not emphasized in the curricula of either country, instruction and practice in exact arithmetic may enhance Chinese children’s efficiency in solving arithmetic estimation problems.

The interconnection of orthographic, phonetic, and semantic skills with arithmetic fluency.

Arithmetic fluency is considered considers highly rely on language processing, encompassing essential skills. However, the independent predictive power of phonetic, semantic, or orthographic skills in relation to arithmetic fluency remains an unresolved query. This study introduces the common component hypothesis to elucidate the inconsistent findings in previous research. The hypothesis posits that significant correlations between language and mathematics hinge on whether the language and mathematics utilized in a given task share a common component. According to this hypothesis, processing skills for each of the three fundamental language elements (i.e., phonetic, semantic, orthographic) should correlate with arithmetic fluency, as these elements are also integral to simple arithmetic processing. A cohort of one hundred and ninety-eight primary school students participated in the study, undertaking a battery of tests assessing general cognitive abilities, psycholinguistic elements, and arithmetic fluency. The results showed that orthographic, phonetic, and semantic abilities independently predicted arithmetic fluency, even after accounting for all other cognitive predictors. These findings substantiate the common component hypothesis, providing empirical support for explaining the association between language and mathematics. This evidence contributes to addressing the interplay between language and mathematics in educational contexts.

Arithmetic fluency and number processing skills in identifying students with mathematical learning disabilities

BackgroundStudents with mathematical learning disabilities (MLD) struggle with number processing skills (e.g., enumeration and number comparison) and arithmetic fluency. Traditionally, MLD is identified based on arithmetic fluency. However, number processing skills are suggested to differentiate low achievement (LA) from MLD. AimsThis study investigated the accuracy of number processing skills in identifying students with MLD and LA, based on arithmetic fluency, and whether the classification ability of number processing skills varied as a function of grade level. Methods and proceduresThe participants were 18,405 students (girls = 9080) from Grades 3–9 (ages 9–15). Students’ basic numerical skills were assessed with an online dyscalculia screener (Functional Numeracy Assessment –Dyscalculia Battery, FUNA-DB), which included number processing and arithmetic fluency as two factors. Outcomes and resultsConfirmatory factor analyses supported a two-factor structure of FUNA-DB. The two-factor structure was invariant across language groups, gender, and grade levels. Receiver operating characteristics curve analyses indicated that number processing skills are a fair classifier of MLD and LA status across grade levels. The classification accuracy of number processing skills was better when predicting MLD (cut-off < 5 %) compared to LA (cut-off < 25 %). Conclusions and implicationsResults highlight the need to measure both number processing and arithmetic fluency when identifying students with MLD.

Short-term number sense training recapitulates long-term neurodevelopmental changes from childhood to adolescence.

Number sense is fundamental to the development of numerical problem-solving skills. In early childhood, children establish associations between non-symbolic (e.g., a set of dots) and symbolic (e.g., Arabic numerals) representations of quantity. The developmental estrangement theory proposes that the relationship between non-symbolic and symbolic representations of quantity evolves with age, with increased dissociation across development. Consistent with this theory, recent research suggests that cross-format neural representational similarity (NRS) between non-symbolic and symbolic quantities is correlated with arithmetic fluency in children but not in adolescents. However, it is not known if short-term training (STT) can induce similar changes as long-term development. In this study, children aged 7-10 years underwent a theoretically motivated 4-week number sense training. Using multivariate neural pattern analysis, we investigated whether short-term learning could modify the relation between cross-format NRS and arithmetic skills. Our results revealed a significant correlation between cross-format NRS and arithmetic fluency in distributed brain regions, including the parietal and prefrontal cortices, prior to training. However, this association was no longer observed after training, and multivariate predictive models confirmed these findings. Our findings provide evidence that intensive STT during early childhood can promote behavioral improvements and neural plasticity that resemble and recapitulate long-term neurodevelopmental changes that occur from childhood to adolescence. More generally, our study contributes to our understanding of the malleability of number sense and highlights the potential for targeted interventions to shape neurodevelopmental trajectories in early childhood. RESEARCH HIGHLIGHTS: We tested the hypothesis that short-term number sense training induces the dissociation of symbolic numbers from non-symbolic representations of quantity in children. We leveraged a theoretically motivated intervention and multivariate pattern analysis to determine training-induced neurocognitive changes in the relation between number sense and arithmetic problem-solving skills. Neural representational similarity between non-symbolic and symbolic quantity representations was correlated with arithmetic skills before training but not after training. Short-term training recapitulates long-term neurodevelopmental changes associated with numerical problem-solving from childhood to adolescence.

Longitudinal predictors of reading and arithmetic at different attainment levels

Reading and arithmetic are distinct academic skills that share similarities in skill acquisition and use. Previous research investigated the cognitive basis of associations and dissociations between reading and arithmetic by using either subtyping or dimensional approaches. In the current study, we aim to bridge the gap between these two approaches by investigating common and distinct predictors of reading and arithmetic at different performance levels with quantile regression models. This allowed us to look more closely at the lower tail of the ability distributions, and to test whether predictions for children with low reading and arithmetic fluency differed from the typical performance range. We analyzed longitudinal data of 357 children speaking English or German. Outcome variables were reading and arithmetic fluency assessed at the end of Grade 1, 2, and 3. Predictors were assessed in Grade 1. Results confirmed nonverbal IQ and working memory as domain-general predictors of reading and arithmetic. The association of reading and arithmetic was mainly explained by nonverbal IQ, phonological awareness, RAN and multi-digit transcoding. Across grades and performance levels, phonological awareness and RAN made a specific contribution to reading. Magnitude processing and multi-digit transcoding were specific predictors of arithmetic. Counting also made a specific prediction to arithmetic in Grade 3, but only in the low performance range. Our findings indicate partly distinct underlying cognitive mechanisms for reading and arithmetic. Shared predictors are involved in retrieval efficiency, language processing and cross-format integration. These results have important implications, as they suggest that most predictors are equally relevant for children with low, typical or even excellent reading and arithmetic fluency.

The role of mathematical vocabulary in the development of mathematical skills for Spanish-speaking students

Does mathematical vocabulary predict the change in students’ performance on mathematical tasks from one academic year to the next? Chilean Spanish-speaking students (N = 87) completed measures of mathematical vocabulary, mathematical skills (i.e., arithmetic fluency, calculation, and applied problems), receptive vocabulary, and working memory in Grade 2 (T1, Mage = 7:11 years:months, SD = 0:5, 46% girls). One year later (T2) they completed the same mathematical measures. Concurrent relations were found between mathematical vocabulary and the three mathematical skills at both time points. Together, general and mathematical vocabulary at T1 explained significant unique variance in the change in applied problems and calculation from T1 to T2. For calculation however, only mathematical vocabulary predicted significant unique variance in the change from T1 to T2. Change in arithmetic fluency was only predicted by working memory. These results address the roles of general and mathematical vocabulary in students’ mathematical development in elementary school.

Phonological awareness and RAN contribute to Chinese reading and arithmetic for different reasons.

The present study investigated how phonological awareness and rapid automatized naming (hereafter, RAN), simultaneously contributed to Chinese reading and arithmetic fluency. Specifically, we proposed a new hypothesized mechanism that processing speed would mediate the relations of RAN with Chinese reading and arithmetic fluency. One hundred and forty-five Chinese children at the fifth grade were administered with a battery of measures, including three phonological processing measures, character reading, and whole number computation, as well as nonverbal IQ, and vocabulary knowledge. Path analyses revealed that phonological awareness and RAN were uniquely related to character reading and arithmetic fluency, while phonological memory was not significantly correlated to either character reading or arithmetic fluency, after controlling for age, nonverbal IQ, and vocabulary knowledge. Further analysis indicated that processing speed demonstrated a mediating effect on the importance of RAN in character reading, rather than in arithmetic fluency. Results underscore the potential importance of phonological awareness and RAN in character reading and arithmetic fluency, and the mediating role of processing speed in RAN to promote Chinese character reading fluency.

Electrophysiological correlates of symbolic numerical order processing.

Determining if a sequence of numbers is ordered or not is one of the fundamental aspects of numerical processing linked to concurrent and future arithmetic skills. While some studies have explored the neural underpinnings of order processing using functional magnetic resonance imaging, our understanding of electrophysiological correlates is comparatively limited. To address this gap, we used a three-item symbolic numerical order verification task (with Arabic numerals from 1 to 9) to study event-related potentials (ERPs) in 73 adult participants in an exploratory approach. We presented three-item sequences and manipulated their order (ordered vs. unordered) as well as their inter-item numerical distance (one vs. two). Participants had to determine if a presented sequence was ordered or not. They also completed a speeded arithmetic fluency test, which measured their arithmetic skills. Our results revealed a significant mean amplitude difference in the grand average ERP waveform between ordered and unordered sequences in a time window of 500-750 ms at left anterior-frontal, left parietal, and central electrodes. We also identified distance-related amplitude differences for both ordered and unordered sequences. While unordered sequences showed an effect in the time window of 500-750 ms at electrode clusters around anterior-frontal and right-frontal regions, ordered sequences differed in an earlier time window (190-275 ms) in frontal and right parieto-occipital regions. Only the mean amplitude difference between ordered and unordered sequences showed an association with arithmetic fluency at the left anterior-frontal electrode. While the earlier time window for ordered sequences is consistent with a more automated and efficient processing of ordered sequential items, distance-related differences in unordered sequences occur later in time.

Cross-notation knowledge of rational numbers predicts fraction arithmetic.

Recent research showed that cross-notation magnitude knowledge of fractions and decimals was related to better performance in fraction arithmetic, but it remains unclear whether it made an independent contribution to fraction arithmetic longitudinally when other cognitive variables are considered. To examine the extent to which children's earlier knowledge of cross-notation magnitude predicted subsequent performance in fraction addition and subtraction as well as fraction multiplication and division longitudinally. Three hundred and fifty-four Chinese children (Mage = 112.1 months). During the first wave of assessment, a range of cognitive abilities of children were measured, including within-notation fraction and decimal magnitude comparisons, whole-number arithmetic fluency, non-verbal intelligence, attentive behaviours, counting recall, word-level reading, and phonological awareness. Twelve months later, the same children were assessed again with two tasks of fraction arithmetic: fraction addition and subtraction as well as fraction multiplication and division. Multiple linear regressions showed that within-notation fraction and decimal magnitude knowledge predicted fraction addition and subtraction longitudinally, after the effects of working memory, nonverbal intelligence, language skills, attentive behaviour, and whole-number arithmetic were controlled. Cross-notation magnitude knowledge made independent contributions to fraction addition and subtraction longitudinally beyond the influence of within-notation fraction and decimal magnitude knowledge and other covariates. However, within-notation fraction and decimal magnitude knowledge were not associated with fraction multiplication and division, whereas cross-notation magnitude knowledge remained a unique predictor. These findings suggest that it may be useful to incorporate cross-notation knowledge in the assessments of children's mathematics abilities and teaching.

Development and validation of a semi-automated, scalable response to intervention framework in mathematics

Response to Intervention (RTI) has been shown to be better at identifying and addressing learning difficulties than other frameworks. However, only a few educational systems have implemented it, and it can be difficult for schools to implement RTI on their own without policies in place. One solution is to create semi-automated digital tools that can assist with RTI without requiring a lot of extra time from the teacher. This study describes the development and implementation of such tools in schools that were not previously familiar with RTI. The effectiveness of the tools was tested by comparing 5 schools (149 students) that used RTI with 8 schools (269 students) that did not. The results showed that the extra practice was effective and that students in the intervention group made significantly more progress in arithmetic fluency than their peers in the control group. These students were also more likely to leave the “low achievement” and “at risk of Mathematics Learning Difficulties” zones than students in the control group. This suggests that it is possible to develop a scalable RTI framework and implement it in educational systems without RTI-based policies.

Developmental profiles of arithmetic fluency skills from grades 1 to 9 and their early identification.

The aim of the present study was to examine the kinds of developmental profiles of arithmetic fluency skills that can be identified across Grades 1-9 (ages 7-16) in a large Finnish sample (n = 2,518). The study also examined whether membership in the developmental profiles could be predicted using a comprehensive set of kindergarten-age factors, including information on cognitive skills; motivational, parental, and home environment factors; and gender. Four profiles of arithmetic fluency skills development were identified using a factor mixture model: persistent arithmetic difficulties (12.23%), precocious onset (50.24%), delayed onset (36.96%), and precocious onset with a Grade 7 drop (.06%). The Cholesky models predicting membership in the three largest profiles suggested that overall, the strongest kindergarten-age predictors were cognitive skills (especially counting, number concepts, spatial relations, rapid automatized naming [RAN], phonological awareness, and letter knowledge), but motivational, parental, and home environment factors were also significant. Membership in the profile with precocious onset was predicted by most of the kindergarten-age measures, suggesting that the strengths in early skills, as well as motivational, parental, and home environment factors, are reflected in the advanced start in arithmetic development at school. The profiles with delayed onset and persistent difficulties were similar in most kindergarten-age measures but differed in task avoidance and four cognitive skills (letter knowledge, counting, number concepts, and RAN), suggesting that these factors predict differential development over the longer term. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Interventions to improve equational reasoning: replication and extension of the Cuisenaire-Gattegno curriculum effect.

The ability to reason about equations in a robust and fluent way requires both instrumental knowledge of symbolic forms, syntax, and operations, as well as relational knowledge of how such formalisms map to meaningful relationships captured within mental models. A recent systematic review of studies contrasting the Cuisenaire-Gattegno (Cui) curriculum approach vs. traditional rote schooling on equational reasoning has demonstrated the positive efficacy of pedagogies that focus on integrating these two forms of knowledge. Here we seek to replicate and extend the most efficacious of these studies (Brownell) by implementing the curriculum to a high degree of fidelity, as well as capturing longitudinal changes within learners via a novel tablet-based assessment of accuracy and fluency with equational reasoning. We examined arithmetic fluency as a function of relational reasoning to equate initial performance across diverse groups and to track changes over four growth assessment points. Results showed that the intervention condition that stressed relational reasoning leads to advances in fluency for addition and subtraction with small numbers. We also showed that this intervention leads to changes in problem solving dispositions toward complex challenges, wherein students in the CUI intervention were more inclined to solve challenging problems relative to those in the control who gave up significantly earlier on multi-step problems. This shift in disposition was associated with higher accuracy on complex equational reasoning problems. A treatment by aptitude interaction emerged for both arithmetic equation reasoning and complex multi-step equational reasoning problems, both of which showed that the intervention had greatest impact for children with lower initial mathematical aptitude. Two years of intervention contrast revealed a large effect (d = 1) for improvements in equational reasoning for the experimental (CUI) group relative to control. The strong replication and extension findings substantiate the importance of embedding these teaching aides within the theory grounded curricula that gave rise to them.

We Cannot Ignore the Signs: The Development of Equivalence and Arithmetic for Students from Grades 3 to 4

ABSTRACT Students' understanding of the meaning of the equal sign develops slowly over the primary grades. In addition to updating their representations of equations to recognize that the equal sign represents an equivalence relation rather than signaling an operation, students need to move beyond full computation to efficiently solve equivalence problems. In this study, we examined the longitudinal relation between arithmetic and equivalence for students who were capable of accurately solving arithmetic problems in different formats. Chinese students (N = 612; M age = 9.0 years in Grade 3, 57% boys) completed measures of arithmetic fluency and equivalence fluency in Grade 3 and again in Grade 4. They also completed a non-verbal reasoning task in Grade 3. We tested a cross-lagged structural equation model to examine the reciprocal relations between arithmetic and equivalence fluency. We found reciprocal relations between the development of arithmetic and equivalence fluency from Grades 3 to 4, with a greater influence of arithmetic on the development of equivalence than the reverse. Furthermore, non-verbal reasoning predicted the development of equivalence, but not the development of arithmetic. Based on our findings, we conclude that for Chinese students with prior basic understanding of equivalence, flexible access to arithmetic facts supports their development of equivalence fluency.

A direct comparison of two measures of ordinal knowledge among 8-year-olds

Children’s knowledge of the ordinal relations among number symbols is related to their mathematical learning. Ordinal knowledge has been measured using judgment (i.e., decide whether a sequence of three digits is in order) and ordering tasks (i.e., order three digits from smallest to largest). However, the question remains whether performance on these two ordinal tasks tap into similar cognitive processes. Canadian children (N = 87; Age M = 8.7 years, Grade 3) completed symbolic number tasks (i.e., number comparison, ordering, and order judgment) and measures of arithmetic fluency (i.e., addition and subtraction) and working memory (i.e., digit span backward). For both ordinal tasks, there was a reverse distance effect for ordered sequences such that children responded faster to adjacent than to non-adjacent sequences (e.g., 2 3 4 vs. 4 7 9) and a canonical distance effect for unordered sequences such that children responded faster to non-adjacent than to adjacent sequences (e.g., 4 2 3 vs. 4 9 7). Working memory and number comparison each predicted unique variance in the ordinal measures (ordering, order judgment, and a latent ordinal factor based on the two measures). Furthermore, ordinal skills superseded the role of number comparison as the key predictor of arithmetic, controlling for children’s gender and working memory skills. In summary, although both ordering and order judgment tasks index ordinal knowledge, a latent factor that excludes task-specific error may be a better index than either task separately.

The factor structure of mathematics anxiety and its relation to gender and mathematics performance

AbstractMathematics anxiety (MA) refers to negative cognition and negative affect that interfere with mathematics performance. We examined the dimensionality of MA, measurement invariance across males and females, and whether the strength of relations between MA and mathematics performance varies by dimension and gender among 245 sixth‐grade students. Students were assessed on MA, arithmetic fluency, and on‐grade‐level computation skills. The confirmatory factor analysis and measurement invariance tests indicated that two distinct dimensions, cognition and affect, best represent MA across both genders. For the full sample, on‐grade‐level computation skills showed a significant negative relation with the cognitive dimension only, whereas arithmetic fluency was not significantly related to either dimension. We did not find any significant gender differences. Our findings provide support for the cognitive and affective model of MA and negative impact of MA on mathematics performance. Our findings further suggest incorporating the cognitive component in identifying and alleviating MA, and improving mathematics outcomes.

Parental influences on the development of single and co-occurring difficulties in reading and arithmetic fluency

This study investigated how single and co-occurring difficulties in reading and arithmetic fluency developed among Finnish children across Grades 1–9 (N = 2151). Latent profile analysis among 391 children who had reading and/or arithmetic fluency difficulties in Grade 9 revealed profiles that followed three distinct patterns: reading difficulties (N = 121), mathematical difficulties (N = 94), and comorbid difficulties (N = 176). The profiles and typical performers were compared on parental reading and mathematical difficulties, parental education, the early home learning environment, and parental assistance with school homework across Grades 1–9. Results showed that although parents whose children had difficulties provided them with domain-specific support across all grades, the amount of support gradually declined and the performance gap between the profiles increased.

Cognitive Skills, Math-Related Emotions, and Beliefs Explaining Response to Arithmetic Fluency Intervention

We examined the associations of cognitive skills, math-related emotions and beliefs, and gender with responses to an arithmetic fluency intervention. Elementary school children with dysfluent arithmetic skills (N=69) participated in an arithmetic fluency intervention (with and without self-efficacy support) implemented in small groups in schools for 12 weeks. Hierarchical regression models including cognitive skills or math-related emotions and beliefs predicted 21% to 50% of the variation in the intervention response, i.e., improvement in arithmetic fluency. Cognitive skills were associated with the response mainly among boys, whereas math-related emotions and beliefs explained more among girls. Thus, both cognitive and non-cognitive factors and their interaction with gender should be considered when identifying non-responders who may need more individually tailored support at school.

Monolingual and bilingual children’s performance on arithmetic fluency varies by language fluency

Research has demonstrated that learning two or more languages during development (i.e., becoming bilingual) shapes children’s cognition in myriad ways. However, because such studies have largely been conducted using laboratory experiments, it is unclear how bilingualism may modulate more naturalistic cognitive skills such as arithmetic fluency. Moreover, how the relationship between speaking two (or more) languages and arithmetic varies with language fluency—specifically, the degree of bilingualism—has been understudied. Therefore, this study examined third- to fifth-grade monolingual (n = 70) and bilingual (n = 51) children’s performance on an arithmetic fluency task. Monolinguals’ and bilinguals’ performance on the arithmetic fluency task did not differ. However, individual differences in the relation between children’s arithmetic fluency and their language fluency were found, suggesting that bilingual children’s skill in their nondominant language was associated with arithmetic fluency. These findings point to the importance of examining individual differences in language fluency among bilinguals to understand how bilingualism may shape cognitive skills.

Verbal, Figural, and Arithmetic Fluency of Children with Cochlear Implants

Cochlear implantation gives children with prelingual severe hearing loss and deafness the opportunity to develop their hearing abilities, speech, language, cognitive abilities and academic skills with adequate rehabilitation. The aim of the research was to analyze verbal, figural and arithmetic fluency and their interrelationship in children with a cochlear implant (CI) and children with normal hearing (NH). A total of 46 children with CI and 110 children with NH, aged 9 to 16, participated in the research. Verbal fluency was assessed using phonemic and semantic fluency, and non-verbal fluency using figural fluency. Arithmetic fluency was assessed using simple arithmetic tasks within the number range up to 100. The results showed that children with CI achieved poorer results in phonemic fluency (z = -4.92; p < 0.001), semantic fluency (z = -3.89; p < 0.001), figural fluency (z = -3.07; p = 0.002), and arithmetic fluency (z = -4.27; p < 0.001). In both groups, a positive correlation was obtained between the measured modalities and types of fluency. In the group of children with CI, a sex difference was obtained on the phonemic fluency test, in favor of girls. The age of children with CI was correlated with arithmetic fluency. Verbal, figural and arithmetic fluency of children with CI speak in favor of the importance of early auditory and language experiences.