Numerical Magnitude Comparison Research Articles

Effects of online parent coaching on Filipino children's skills

Technology today can possibly help make early childhood interventions more accessible to young children in need. In the present study, we conducted all assessment and training of both parents and children entirely online to examine the effectiveness of an online parental coaching programme in facilitating young Filipino children's language, literacy, and numeracy skills. Ninety-four families with 3-to 5-year-old Cebuano-speaking children were randomly assigned into either the intervention group or a waitlist control group. The 13-week long online intervention group showed significant improvement in their receptive vocabulary, syllable matching, letter name knowledge, numeral identification, object counting, rote counting, missing number identification, numerical magnitude comparison and addition skills but not in their expressive vocabulary and syllable deletion relative to the waitlist control group. No moderation effect of socioeconomic status (SES) on the online intervention outcome was found. For the intervention group, a main effect for parental involvement on rote counting, missing number identification, and addition was found, with higher levels of participation in the coaching programme translating to greater gains. The present study underscores the validity and feasibility of an online format of intervention and assessment procedure, in a low-to-middle-income country context (LMIC). The current findings demonstrate the possible added benefits of an online intervention format and the importance of an integrated cross-domains intervention format.

How does working memory matter in young children's arithmetic skills: The mediating role of basic number processing.

The current study investigated whether and how each component of the working memory model was associated with kindergarten children’s arithmetic performance. A total of 103 Chinese kindergarten children were administered tests of the visuospatial sketchpad, the phonological loop, the central executive, and basic number processing (i.e., number line estimation, nonverbal numerosity estimation and numerical magnitude comparison). The results showed that among the three working memory components, the central executive accounted for a significant proportion of the variance in young children’s arithmetic performance. In terms of basic number processing, number line estimation and numerical magnitude comparison had significant influences on young children’s arithmetic performance. Furthermore, numerical magnitude comparison played a mediating role between the visuospatial sketchpad and early arithmetic skills. These findings highlight the importance of working memory and basic number processing in early arithmetic skills and reveal different pathways through which the three working memory components influence young children’s arithmetic performance.

Understanding additive composition is important for symbolic numerical magnitude processing

The present study examined additive composition as a longitudinal predictor for symbolic numerical magnitude comparison. Two hundred and twenty-four Chinese kindergarteners (Mage= 63.76 months; 103 boys, 121 girls) participated in this study across three waves of cognitive assessments. Cross-lagged panel analyses showed that additive composition predicted symbolic numerical magnitude comparison longitudinally but not vice versa. The cross-lagged effects remained significant beyond the influence of general intelligence, working memory, nonsymbolic magnitude comparison, and counting abilities. Theoretically, this study suggests that additive composition is a key factor underlying the exact representational system of numbers. It also entails educational implications that understanding the quantitative relations between numbers should be the focus of learning and teaching in early childhood mathematics education.

Cognitive mediators of US-China differences in early symbolic arithmetic.

Chinese children routinely outperform American peers in standardized tests of mathematics knowledge. To examine mediators of this effect, 95 Chinese and US 5-year-olds completed a test of overall symbolic arithmetic, an IQ subtest, and three tests each of symbolic and non-symbolic numerical magnitude knowledge (magnitude comparison, approximate addition, and number-line estimation). Overall Chinese children performed better in symbolic arithmetic than US children, and all measures of IQ and number knowledge predicted overall symbolic arithmetic. Chinese children were more accurate than US peers in symbolic numerical magnitude comparison, symbolic approximate addition, and both symbolic and non-symbolic number-line estimation; Chinese and U.S. children did not differ in IQ and non-symbolic magnitude comparison and approximate addition. A substantial amount of the nationality difference in overall symbolic arithmetic was mediated by performance on the symbolic and number-line tests.

The distance effect on discrimination ability and response bias during magnitude comparison in a go/no-go task

The distance effect is the change in the performance during numerical magnitude comparison, depending on the numerical distance between the compared numbers (Moyer & Landauer, Nature, 215[5109], 1519-1520, 1967). This effect is generally accepted as evidence for the mental number line (MNL) hypothesis, which proposes that the mental representation of the numbers align in an increasing linear (or monotone) order. The majority of studies investigating the distance effect are focused on the reaction time (RT) findings, which show slower responses for closer numbers. In the present study, we examined the distance effect by applying signal detection theory (SDT) to a magnitude comparison task. We aimed to reveal whether discrimination ability and the response bias measures were affected by the location of numbers on the MNL. To accomplish this, we developed a magnitude comparison task using a go/no-go procedure in which participants performed a magnitude comparison based on a reference number (i.e., 5). Results revealed a substantial distance effect in both sensitivity and response bias measures-a better discrimination performance for far numbers, and a larger response bias for close numbers. In addition, an RT distribution analysis revealed that the distance effect seems to originate mainly from slower responses. Based on the current data, we suggest that sensitivity and response bias measures could offer comprehensive information in the understanding of number-based decisions.

Nonsymbolic and Symbolic Numerical Magnitude Processing in the Brazilian Children with Mathematics Difficulties.

ABSTRACTIt is still debated if the main deficit in mathematical difficulties (MD) is nonsymbolic or symbolic numerical magnitude processing.Objectives:In the present study, our main goal was to investigate nonsymbolic and symbolic numerical magnitude processing in MD and the relationship between these abilities and arithmetic.Methods:The Brazilian school-age children with MD completed a nonsymbolic and a symbolic numerical magnitude comparison task and an arithmetic task. We compared their performance with a group of children with typical achievement (TA) and investigated the association between numerical magnitude processing and arithmetic with a series of regression analyses.Results:Results indicated that children with MD had low performance in the nonsymbolic numerical magnitude comparison task. Performance in both nonsymbolic and symbolic numerical magnitude comparison tasks predicted arithmetic abilities in children with TA, but not in children with MD.Conclusions:These results indicate that children with MD have difficulties in nonsymbolic numerical magnitude processing, and do not engage basic numerical magnitude representations to solve arithmetic.

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.

Numerical processing profiles in children with varying degrees of arithmetical achievement

Recent studies show basic cognitive abilities such as the rapid and precise apprehension of small numerosities in object sets (“subitizing”), verbal counting and numerical magnitude comparison significantly influence the acquisition of arithmetic and continues to modulate more advanced stages of mathematical cognition. Additionally, children with low arithmetic achievement (LAA) and Developmental Dyscalculia (DD) exhibit significant deficits in these cognitive processes. Nevertheless, the different cognitive profiles of children with varying degrees of numerical and arithmetic processing deficits have not been sufficiently characterized, despite its potential relevance to the stimulation of numerical cognition and the design of appropriate intervention strategies. Here, the cognitive profiles of groups of typically developing children, children with low arithmetical achievement and DD, exhibiting typical and atypical subitizing ability were contrasted. The results suggest that relatively independent neurocognitive mechanisms may produce distinct profiles at the behavioral level and suggest children with low arithmetic performance exhibiting atypical subitizing abilities are not only significantly slower, but rely on compensatory mechanisms and strategies compared to typical subitizers. The role of subitizing as a correlate of arithmetic fluency is revised in the light of the present findings.

Benefits of Playing Numerical Card Games on Head Start Children’s Mathematical Skills

Low-income preschoolers have lower average performance on measures of early numerical skills than middle-income children. The present study examined the effectiveness of numerical card games in improving children’s numerical and executive functioning skills. Low-income preschoolers (N = 76) were randomly assigned to play a numerical magnitude comparison card game, a numerical memory and matching card game, or a shape and color matching card game across four 15-minute sessions. Children who played either of the numerical games improved their numeral identification skills, while only children who played the numerical magnitude comparison game improved their symbolic magnitude comparison skills. These improvements were maintained eight weeks later. The results suggest that a brief, low-cost intervention can successfully improve the numerical skills of low-income children.

Differences in Counting Skills Between Chinese and German Children Are Accompanied by Differences in Processing of Approximate Numerical Magnitude Information.

Human beings are supposed to possess an approximate number system (ANS) dedicated to extracting and representing approximate numerical magnitude information as well as an object tracking system (OTS) for the rapid and accurate enumeration of small sets. It is assumed that the OTS and the ANS independently contribute to the acquisition of more elaborate numerical concepts. Chinese children have been shown to exhibit more elaborate numerical concepts than their non-Chinese peers, but it is still an open question whether similar cross-national differences exist with regard to the underlying systems, namely the ANS and the OTS. In the present study, we investigated this question by comparing Chinese and German preschool children with regard to their performance in a non-symbolic numerical magnitude comparison task (assessing the ANS) and in an enumeration task (assessing the OTS). In addition, we compared children’s counting skills. To ensure that possible between-group differences could not be explained by differences in more general performance factors, we also assessed children’s reasoning ability and processing speed. Chinese children showed a better counting performance and a more accurate performance in the non-symbolic numerical magnitude comparison task. These differences in performance could not be ascribed to differences in reasoning abilities and processing speed. In contrast, Chinese and German children did not differ significantly in the enumeration of small sets. The superior counting performance of Chinese children was thus found to be reflected in the ANS but not in the OTS.

It’s more than just fun and games: Play-based mathematics activities for Head Start families

Discrepancies in early mathematical knowledge between children from different socioeconomic backgrounds have been found before the start of kindergarten. The early home environment is one context that can address these discrepancies. This study examined whether an informal mathematical activity that has been successful at promoting children’s numerical knowledge could be translated into a home activity for families from lower-income backgrounds. Families from Head Start programs (n = 39) were randomly assigned to play either a numerical magnitude comparison game or a shape and color matching game. Results showed that playing the numerical magnitude comparison game did not improve children’s numerical knowledge, although playing the shape and color matching game did improve children’s shape knowledge. However, parental reports of the frequency of game playing at home related to children’s learning from both games. Analyses of audio recordings of the families playing the games at home revealed there was wide variability in how parents assisted the children during the card game play. Results are discussed in terms of the benefits and challenges of mathematical interventions targeting the home context.

The influence of visual–spatial skills on the association between processing of nonsymbolic numerical magnitude and number word sequence skills

Nonsymbolic numerical magnitude processing skills are assumed to be fundamental to mathematical learning. Recent findings suggest that visual–spatial skills account for associations between children’s performance in visually presented nonsymbolic numerical magnitude comparison tasks and their performance in visually presented arithmetic tasks. The aim of the current study was to examine whether associations between children’s performance in visually presented tasks assessing nonsymbolic numerical magnitude processing skills and their performance in tasks assessing early mathematical skills, which do not involve visual stimulation, may also be mediated by visual–spatial skills. This line of reasoning is based on the assumption that children make use of mental visualization processes when working on tasks assessing early mathematical skills, such as knowledge of the sequence of number words, even when these tasks do not involve visual stimulation. We assessed 4- to 6-year-old children’s performance in a nonsymbolic numerical magnitude comparison task, in tasks concerning knowledge of the sequence of number words, and in a developmental test to assess visual–spatial skills. Children’s nonsymbolic numerical magnitude processing skills were found to be associated with their number word sequence skills. This association was fully mediated by interindividual differences in visual–spatial skills. The effect size of this mediation effect was small. We assume that the ability to construct mental visualizations constitutes the key factor underlying this mediation effect.

Narrowing the Early Mathematics Gap: A Play-Based Intervention to Promote Low-Income Preschoolers' Number Skills.

Preschoolers from low-income households lag behind preschoolers from middle-income households on numerical skills that underlie later mathematics achievement. However, it is unknown whether these gaps exist on parallel measures of symbolic and non-symbolic numerical skills. Experiment 1 indicated preschoolers from low-income backgrounds were less accurate than peers from middle-income backgrounds on a measure of symbolic magnitude comparison, but they performed equivalently on a measure of non-symbolic magnitude comparison. This suggests activities linking non-symbolic and symbolic number representations may be used to support children's numerical knowledge. Experiment 2 randomly assigned low-income preschoolers (M age = 4.7 years) to play either a numerical magnitude comparison or a numerical matching card game across four 15 min sessions over a 3-week period. The magnitude comparison card game led to significant improvements in participants' symbolic magnitude comparison skills in an immediate posttest assessment. Following the intervention, low-income participants performed equivalently to an age- and gender-matched sample of middle-income preschoolers in symbolic magnitude comparison. These results suggest a brief intervention that combines non-symbolic and symbolic magnitude representations can support low-income preschoolers' early numerical knowledge.

Differences in arithmetic performance between Chinese and German adults are accompanied by differences in processing of non-symbolic numerical magnitude.

Human beings are assumed to possess an approximate number system (ANS) dedicated to extracting and representing approximate numerical magnitude information. The ANS is assumed to be fundamental to arithmetic learning and has been shown to be associated with arithmetic performance. It is, however, still a matter of debate whether better arithmetic skills are reflected in the ANS. To address this issue, Chinese and German adults were compared regarding their performance in simple arithmetic tasks and in a non-symbolic numerical magnitude comparison task. Chinese participants showed a better performance in solving simple arithmetic tasks and faster reaction times in the non-symbolic numerical magnitude comparison task without making more errors than their German peers. These differences in performance could not be ascribed to differences in general cognitive abilities. Better arithmetic skills were thus found to be accompanied by a higher speed of retrieving non-symbolic numerical magnitude knowledge but not by a higher precision of non-symbolic numerical magnitude representations. The group difference in the speed of retrieving non-symbolic numerical magnitude knowledge was fully mediated by the performance in arithmetic tasks, suggesting that arithmetic skills shape non-symbolic numerical magnitude processing skills.

Eye-movement patterns during nonsymbolic and symbolic numerical magnitude comparison and their relation to math calculation skills

A growing body of research suggests that the processing of nonsymbolic (e.g. sets of dots) and symbolic (e.g. Arabic digits) numerical magnitudes serves as a foundation for the development of math competence. Performance on magnitude comparison tasks is thought to reflect the precision of a shared cognitive representation, as evidence by the presence of a numerical ratio effect for both formats. However, little is known regarding how visuo-perceptual processes are related to the numerical ratio effect, whether they are shared across numerical formats, and whether they relate to math competence independently of performance outcomes. The present study investigates these questions in a sample of typically developing adults. Our results reveal a pattern of associations between eye-movement measures, but not their ratio effects, across formats. This suggests that ratio-specific visuo-perceptual processing during magnitude processing is different across nonsymbolic and symbolic formats. Furthermore, eye movements are related to math performance only during symbolic comparison, supporting a growing body of literature suggesting symbolic number processing is more strongly related to math outcomes than nonsymbolic magnitude processing. Finally, eye-movement patterns, specifically fixation dwell time, continue to be negatively related to math performance after controlling for task performance (i.e. error rate and reaction time) and domain general cognitive abilities (IQ), suggesting that fluent visual processing of Arabic digits plays a unique and important role in linking symbolic number processing to formal math abilities.

The Relationship Between Accuracy of Numerical Magnitude Comparisons and Children's Arithmetic Ability: A Study in Iranian Primary School Children.

The relationship between children’s accuracy during numerical magnitude comparisons and arithmetic ability has been investigated by many researchers. Contradictory results have been reported from these studies due to the use of many different tasks and indices to determine the accuracy of numerical magnitude comparisons. In the light of this inconsistency among measurement techniques, the present study aimed to investigate this relationship among Iranian second grade children (n = 113) using a pre-established test (known as the Numeracy Screener) to measure numerical magnitude comparison accuracy. The results revealed that both the symbolic and non-symbolic items of the Numeracy Screener significantly correlated with arithmetic ability. However, after controlling for the effect of working memory, processing speed, and long-term memory, only performance on symbolic items accounted for the unique variances in children’s arithmetic ability. Furthermore, while working memory uniquely contributed to arithmetic ability in one-and two-digit arithmetic problem solving, processing speed uniquely explained only the variance in single-digit arithmetic skills and long-term memory did not contribute to any significant additional variance for one-digit or two-digit arithmetic problem solving.

Numerical magnitude processing impairments in genetic syndromes: a cross-syndrome comparison of Turner and 22q11.2 deletion syndromes.

Cross-syndrome comparisons offer an important window onto understanding heterogeneity in mathematical learning disabilities or dyscalculia. The present study therefore investigated symbolic numerical magnitude processing in two genetic syndromes that are both characterized by mathematical learning disabilities: Turner syndrome and 22q11.2 deletion syndrome (22q11DS). We further verified whether the phenotypic outcomes of these syndromes emerged from the same or different cognitive processes and therefore examined whether numerical impairments were related to working memory deficits, often observed in these syndromes. Participants were 24 girls with Turner syndrome, 25 children with 22q11DS and 48 well-matched typically developing control children. All children completed a symbolic numerical magnitude comparison task and four additional working memory tasks. Both groups of children with genetic syndromes showed similar impairments in symbolic numerical magnitude processing compared to typically developing controls. Importantly, in Turner syndrome, group differences in symbolic numerical magnitude processing disappeared when their difficulties in visual-spatial working memory were taken into account. In contrast, the difficulties in 22q11DS were not explained by poor visual-spatial working memory. These data suggest that different factors underlie the symbolic numerical magnitude processing impairments in both patient groups with mathematical learning disabilities and highlight the value of cross-syndrome comparisons for understanding different pathways to mathematical learning disabilities or dyscalculia.

Differences in Arithmetic Performance between Chinese and German Children Are Accompanied by Differences in Processing of Symbolic Numerical Magnitude.

Symbolic numerical magnitude processing skills are assumed to be fundamental to arithmetic learning. It is, however, still an open question whether better arithmetic skills are reflected in symbolic numerical magnitude processing skills. To address this issue, Chinese and German third graders were compared regarding their performance in arithmetic tasks and in a symbolic numerical magnitude comparison task. Chinese children performed better in the arithmetic tasks and were faster in deciding which one of two Arabic numbers was numerically larger. The group difference in symbolic numerical magnitude processing was fully mediated by the performance in arithmetic tasks. We assume that a higher degree of familiarity with arithmetic in Chinese compared to German children leads to a higher speed of retrieving symbolic numerical magnitude knowledge.

The Contribution of Numerical Magnitude Comparison and Phonological Processing to Individual Differences in Fourth Graders’ Multiplication Fact Ability

Although numerical magnitude processing has been related to individual differences in arithmetic, its role in children’s multiplication performance remains largely unknown. On the other hand, studies have indicated that phonological awareness is an important correlate of individual differences in children’s multiplication performance, but the involvement of phonological memory, another important phonological processing skill, has not been studied in much detail. Furthermore, knowledge about the relative contribution of above mentioned processes to the specific arithmetic operation of multiplication in children is lacking. The present study therefore investigated for the first time the unique contributions of numerical magnitude comparison and phonological processing in explaining individual differences in 63 fourth graders’ multiplication fact ability (mean age = 9.6 years, SD = .67). The results showed that children’s multiplication fact competency correlated significantly with symbolic and nonsymbolic magnitude comparison as well as with phonological short-term memory. A hierarchical regression analysis revealed that, after controlling for intellectual ability and general reaction time, both symbolic and nonsymbolic magnitude comparison and phonological short-term memory accounted for unique variance in multiplication fact performance. The ability to compare symbolic magnitudes was found to contribute the most, indicating that the access to numerical magnitudes by means of Arabic digits is a key factor in explaining individual differences in children’s multiplication fact ability.

Are Individual Differences in Arithmetic Fact Retrieval in Children Related to Inhibition?

Although it has been proposed that inhibition is related to individual differences in mathematical achievement, it is not clear how it is related to specific aspects of mathematical skills, such as arithmetic fact retrieval. The present study therefore investigated the association between inhibition and arithmetic fact retrieval and further examined the unique role of inhibition in individual differences in arithmetic fact retrieval, in addition to numerical magnitude processing. We administered measures of cognitive inhibition (i.e., numerical and non-numerical stroop tasks) and a complementary, more ecologically valid measure of children’s inhibition in the classroom (i.e., teacher questionnaire), as well as numerical magnitude processing (i.e., symbolic and non-symbolic numerical magnitude comparison) and arithmetic fact retrieval (i.e., two verification tasks) in 86 typically developing third graders. We used a correlation, a regression and a Bayesian analysis. This study failed to observe a significant association between inhibition and arithmetic fact retrieval. Consequently, our results did not reveal a unique contribution of inhibition to arithmetic fact retrieval in addition to numerical magnitude processing. On the other hand, symbolic numerical magnitude processing turned out to be a very powerful predictor of arithmetic fact retrieval, as indicated by both frequentist and Bayesian approaches.