Symbolic Magnitude Comparison Research Articles

Number and domain both affect the relation between executive function and mathematics achievement: A study of children's executive function with and without numbers.

Magnitude processing and executive functions (EFs) have emerged as robust predictors of mathematics achievement. However, the nature of these associations is still unclear. For example, it is uncertain if EFs applied in the context of domain-specific mathematical cognition (i.e., EFs applied while processing numbers) are more closely related to mathematics achievement than EFs applied in nonnumerical, domain-general contexts. Also, how distinct EF domains-that is, working memory, inhibitory control, and cognitive flexibility-and contents-that is, numerical versus nonnumerical-moderate the association between magnitude processing and mathematics achievement has not been fully understood. To address these issues, we investigated how magnitude processing, EFs applied to nonnumerical and numerical task stimuli, and their interactions were associated with mathematics achievement. Three hundred fifty-nine Brazilian third- to fifth-grade (8-10 years old) students completed measures of working memory, inhibitory control, and cognitive flexibility with numerical and nonnumerical task versions, nonsymbolic and symbolic magnitude comparison, and mathematics achievement. A series of regression models indicated that nonsymbolic and symbolic magnitude processing are consistently associated with mathematics achievement, even when controlling for working memory, inhibitory control, and cognitive flexibility measured with both numerical and nonnumerical contents. All EF measures were associated with mathematics achievement. However, cognitive flexibility measured with numerical content showed the strongest association. Results support the hypothesis that magnitude processing and EFs are uniquely associated with mathematics achievement. Furthermore, EFs measured with nonnumerical and numerical contents related differently to mathematics achievement, even when controlling for symbolic and nonsymbolic magnitude processing, suggesting they encompass somewhat distinct cognitive processes. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Spontaneous grip force fluctuations mirror semantic numerical magnitude processing: Larger numbers elicit larger forces

This study investigated the relationship between semantic numerical magnitudes and motor magnitudes. We asked whether the processing of numbers can affect motor behavior such as the size of numbers affecting the size of motor actions. For this, we recorded continuous grip force fluctuations from 43 healthy adults during a symbolic magnitude comparison task. We found that numbers induced spontaneous grip force fluctuations during number processing. Smaller numbers induced lower grip forces, whereas larger numbers induced larger forces. This result constitutes strong behavioral support for a generalized magnitude processing by continuously quantifying the response that challenges binary accounts of cross-domain interactions.

Interrelations Between Acuity of the Approximate Number System and Symbolic Skills in Preschool Children

ABSTRACT This study investigates how the approximate number system (ANS) and young children’s symbolic skills jointly develop and interact. Specifically, the study aims at disentangling the directionality of the association between ANS acuity and a wide range of symbolic skills that reflect 4- to 5-year-olds’ symbolic quantitative knowledge (enumeration skills, knowledge of the verbal count sequence, symbolic comparison skills, and single-digit arithmetic). After accounting for individual differences in several domain-general skills (visuospatial working memory, non-verbal reasoning, and phonological processing), path models on longitudinal data collected from 4-year-old childen in Spain (N = 62) over one year revealed that earlier single-digit arithmetic and symbolic magnitude comparison skills predicted changes in ANS acuity over time. No contribution from earlier ANS to improvements in symbolic skills was found. Notably, the strength of the effect of visuospatial working memory on improvements in ANS acuity over time was like that of the auto-regressor – the correlation between measures of ANS acuity across time points. Implications for extant theories on the nature of the associations between ANS and young children’s symbolic skills are drawn.

A-B-3-Associations and dissociations of reading and arithmetic: Is domain-specific prediction outdated?

Reading and arithmetic are core domains of academic achievement with marked impact on career opportunities and socioeconomic status. While associations between reading and arithmetic are well established, evidence on underlying mechanisms is inconclusive. The main goal of this study was to reevaluate the domain-specificity of established predictors and to enhance our understanding of the (co-)development of reading and arithmetic. In a sample of 885 German-speaking children, standard domain-specific predictors of reading and arithmetic were assessed before and/or at the onset of formal schooling. Reading and arithmetic skills were measured at the beginning and end of second grade. Latent variables were extracted for all relevant constructs: Grapheme-phoneme processing (phonological awareness, letter identification), RAN (RAN-objects, RAN-digits), number system knowledge (number identification, successor knowledge), and magnitude processing (non-symbolic and symbolic magnitude comparison), as well as the criterion measures reading and arithmetic. Four structural equation models tested distinct research questions. Grapheme-phoneme processing was a specific predictor of reading, and magnitude processing explained variance specific to arithmetic. RAN explained variance in both domains, and it explained variance in reading even after controlling for arithmetic. RAN and number system knowledge further explained variance in skills shared between reading and arithmetic. Reading and arithmetic entail domain-specific cognitive components, and they both require tight networks of visual, verbal, and semantic information, as reflected by RAN. This perspective provides a useful background to explain associations and dissociations between reading and arithmetic performance.

Common and distinct predictors of non-symbolic and symbolic ordinal number processing across the early primary school years.

What are the cognitive mechanisms supporting non-symbolic and symbolic order processing? Preliminary evidence suggests that non-symbolic and symbolic order processing are partly distinct constructs. The precise mechanisms supporting these skills, however, are still unclear. Moreover, predictive patterns may undergo dynamic developmental changes during the first years of formal schooling. This study investigates the contribution of theoretically relevant constructs (non-symbolic and symbolic magnitude comparison, counting and storage and manipulation components of verbal and visuo-spatial working memory) to performance and developmental change in non-symbolic and symbolic numerical order processing. We followed 157 children longitudinally from Grade 1 to 3. In the order judgement tasks, children decided whether or not triplets of dots or digits were arranged in numerically ascending order. Non-symbolic magnitude comparison and visuo-spatial manipulation were significant predictors of initial performance in both non-symbolic and symbolic ordering. In line with our expectations, counting skills contributed additional variance to the prediction of symbolic, but not of non-symbolic ordering. Developmental change in ordering performance from Grade 1 to 2 was predicted by symbolic comparison skills and visuo-spatial manipulation. None of the predictors explained variance in developmental change from Grade 2 to 3. Taken together, the present results provide robust evidence for a general involvement of pair-wise magnitude comparison and visuo-spatial manipulation in numerical ordering, irrespective of the number format. Importantly, counting-based mechanisms appear to be a unique predictor of symbolic ordering. We thus conclude that there is only a partial overlap of the cognitive mechanisms underlying non-symbolic and symbolic order processing.

Cognitive abilities and computational thinking at age 5: Evidence for associations to sequencing and symbolic number comparison

Computational thinking (CT) is currently used as an umbrella term in education to describe the processes associated with learning and applying computer science concepts and strategies for problem-solving. Despite a growing body of literature on the subject within education and the development of a grand variety of tools designed to enhance computational thinking abilities, the relations between CT and other cognitive abilities have not been reported in young children. Understanding CT from a cognitive development perspective could be useful for creating targeted interventions tailored for young children. In the present study, we compare a previously reported computational thinking assessment with a battery of nine cognitive tests, which include fluid intelligence, working memory, planning, sequencing, mental rotation, vocabulary, and early math precursors such as numerical transcoding and symbolic magnitude comparison. Mixed-effects linear regressions were implemented with CT as a dependent variable to explore the associations between our variables. A subsample of children who participated in an educational robotics intervention were assessed in their robot programming performance, and its outcomes were compared with their CT assessments. Results suggest temporal sequencing ability and symbolic magnitude comparison are significant predictors of CT in kindergarten. Positive significant associations between robot programming and CT performance are reported.

Deficits in or preservation of basic number processing in Parkinson's disease? A registered report.

Neurodegenerative diseases such as Parkinson's disease (PD) have a huge impact on patients, caregivers, and the health-care system. To date, the diagnosis of mild cognitive impairments in PD has been established based on domain-general functions such as executive functions, attention, or working memory. However, specific numerical deficits observed in clinical practice have not yet been systematically investigated. PD-immanent deterioration of domain-general functions and domain-specific numerical areas suggests the mechanisms of both primary and secondary dyscalculia. The current study will systematically investigate basic number processing performance in PD patients for the first time, targeting domain-specific cognitive representations of numerosity and the influence of domain-general factors. The overall sample consists of patients with a diagnosis of PD, according to consensus guidelines, and healthy controls. PD patients will be stratified into patients with normal cognition or mild cognitive impairment (level I-PD-MCI based on cognitive screening). Basic number processing will be assessed using transcoding, number line estimation, and (non)symbolic number magnitude comparison tasks. Discriminant analysis will be employed to assess whether basic number processing tasks can differentiate between a healthy control group and both PD groups. All participants will be subjected to a comprehensive numerical and a neuropsychological test battery, as well as sociodemographic and clinical measures. Study results will give the first broad insight into the extent of basic numerical deficits in different PD patient groups and will help us to understand the underlying mechanisms of the numerical deficits faced by PD patients in daily life.

When one size does not fit all: A latent profile analysis of low-income preschoolers’ math skills

On average, preschoolers from lower-income households perform worse on symbolic numerical tasks than preschoolers from middle- and upper-income households. Although many recent studies have developed and tested mathematics interventions for low-income preschoolers, the variability within this population has received less attention. The goal of the current study was to describe the variability in low-income children’s math skills using a person-centered analysis. We conducted a latent profile analysis on six measures of preschoolers’ (N = 115, mean age = 4.6 years) numerical abilities (nonsymbolic magnitude comparison, verbal counting, object counting, cardinality, numeral identification, and symbolic magnitude comparison). The results showed different patterns of strengths and weaknesses and revealed four profiles of numerical skills: (a) poor math abilities on all numerical measures (n = 13), (b) strong math abilities on all numerical measures (n = 41), (c) moderate abilities on all numerical measures (n = 35), and (d) strong counting and numeral skills but poor magnitude skills (n = 26). Children’s age, working memory, and inhibitory control significantly predicted their profile membership. We found evidence of quantitative and qualitative differences between profiles, such that some profiles were higher performing across tasks than others, but the overall patterns of performance varied across the different numerical skills assessed.

Neural Correlate Differences in Number Sense Between Children With Low and Middle/High Socioeconomic Status.

Although some cognitive studies provided reasons that children with low socioeconomic status (SES) showed poor mathematical achievements, there was no explicit evidence to directly explain the root of lagged performance in children with low SES. Therefore, the present study explored the differences in neural correlates in the process of symbolic magnitude comparison between children with different SESs by the event-related potentials (ERPs). A total of 16 second-graders from low-SES families and 16 from middle/high-SES families participated in this study. According to the results of anterior N1 (early attention) and P2 (extraction of numerical meaning) over the frontal region, the differences among children with different SESs were manifested as differences in general neural activities in terms of attention and top-down cognitive control. In the late stage of cognitive processing, there was no significant difference in the average amplitude of the late positive component (LPC) between children with different SES, indicating that low SES did not influence the information encoding and memory updating of numerical representation, which was responsible by the parietal lobe. The educational implications of this study are mentioned in the discussion.

Fraction magnitude: Mapping between symbolic and spatial representations of proportion

Fraction notation conveys both part-whole (3/4 is 3 out of 4) and magnitude (3/4 = 0.75) information, yet evidence suggests that both children and adults find accessing magnitude information from fractions particularly difficult. Recent research suggests that using number lines to teach children about fractions can help emphasize fraction magnitude. In three experiments with adults and 9-12-year-old children, we compare the benefits of number lines and pie charts for thinking about rational numbers. In Experiment 1, we first investigate how adults spontaneously visualize symbolic fractions. Then, in two further experiments, we explore whether priming children to use pie charts vs. number lines impacts performance on a subsequent symbolic magnitude task and whether children differentially rely on a partitioning strategy to map rational numbers to number lines vs. pie charts. Our data reveal that adults very infrequently spontaneously visualize fractions along a number line and, contrary to other findings, that practice mapping rational numbers to number lines did not improve performance on a subsequent symbolic magnitude comparison task relative to practice mapping the same magnitudes to pie charts. However, children were more likely to use overt partitioning strategies when working with pie charts compared to number lines, suggesting these representations did lend themselves to different working strategies. We discuss the interpretations and implications of these findings for future research and education. All materials and data are provided as Supplementary Materials.

Magnitude Comparisons, Number Knowledge and Calculation in VeryPreterm Children and Children With Specific Learning Disability: A Cross-Population Study Using Eye-Tracking

Difficulties in mathematics have been described in very preterm children, but their origins are not well understood and may differ from other populations with specific learning disability. Very preterm children, children with learning disability, and typically developing children were compared in mathematics skills, using standardized tools, experimental tasks, and eye-tracker measures. We assessed symbolic and nonsymbolic magnitude comparisons, number knowledge, calculation, as well as cognitive skills of 103 Italian-speaking fourth and fifth graders. Compared to typically developing peers, very preterm children showed delays in number knowledge, slower reaction times in nonsymbolic magnitude comparisons, and an atypical gaze exploration characterized by more and shorter fixations that lacked a target preference. The profile of mathematics skills of very preterm children appeared different from that of children with learning disability. Although both populations showed mainly preserved cognitive skills and slower reaction times in nonsymbolic magnitude comparisons, children with specific learning disability showed more severe impairments in calculation and were slower in symbolic magnitude comparisons, compared to very preterm children. Including eye-tracking measures in preterm follow-up programs and planning tailored interventions are recommended.

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.

Math and Memory in Bilingual Preschoolers: The Relations Between Bilingualism, Working Memory, and Numerical Knowledge

ABSTRACTBilingual children exhibit enhanced working memory (WM) skill relative to monolingual children, which could have implications for early mathematics development. Competency in mathematics is supported by conceptual and procedural mathematical knowledge, and numerical knowledge is in turn supported by underlying cognitive processes such as WM. Building on this evidence, we investigated whether bilingual preschoolers demonstrated, relative to monolingual peers, enhanced performance on WM and greater numerical knowledge. We examined the role of WM in numerical understanding for both monolingual and bilingual children. Participants were 4- and 5-year old children (74) recruited from preschools serving families from a range of socioeconomic, racial, and ethnic backgrounds. Participants completed a nonverbal WM task and a range of numerical-knowledge measures, specifically numeral identification, addition, symbolic magnitude comparison, and nonsymbolic magnitude comparison tasks. Results revealed that bilingual children outperformed monolingual peers on WM even after controlling for age and socioeconomic status (SES). Bilingual children also demonstrated greater performance on addition and numeral identification tasks. For all tasks except numeral identification, WM predicted children’s performance on numerical knowledge measures. We discuss results in terms of the possible unique cognitive and academic advantages bilingual children may have.

The analysis of EEG coherence reflects middle childhood differences in mathematical achievement

Symbolic numerical magnitude processing is crucial to arithmetic development, and it is thought to be supported by the functional activation of several brain-interconnected structures. In this context, EEG beta oscillations have been recently associated with attention and working memory processing that underlie math achievement. Due to that EEG coherence represents a useful measure of brain functional connectivity, we aimed to contrast the EEG coherence in forty 8-to-9-year-old children with different math skill levels (High: HA, and Low achievement: LA) according to their arithmetic scores in the Fourth Edition of the Wide Range Achievement Test (WRAT-4) while performing a symbolic magnitude comparison task (i.e. determining which of two numbers is numerically larger). The analysis showed significantly greater coherence over the right hemisphere in the two groups, but with a distinctive connectivity pattern. Whereas functional connectivity in the HA group was predominant in parietal areas, especially involving beta frequencies, the LA group showed more extensive frontoparietal relationships, with higher participation of delta, theta and alpha band frequencies, along with a distinct time–frequency domain expression. The results seem to reflect that lower math achievements in children mainly associate with cognitive processing steps beyond stimulus encoding, along with the need of further attentional resources and cognitive control than their peers, suggesting a lower degree of numerical processing automation.

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.

A Diffusion Model Analysis of Magnitude Comparison in Children with and without Dyscalculia: Care of Response and Ability Are Related to Both Mathematical Achievement and Stimuli.

The respective roles of the approximate number system (ANS) and an access deficit (AD) in developmental dyscalculia (DD) are not well-known. Most studies rely on response times (RTs) or accuracy (error rates) separately. We analyzed the results of two samples of elementary school children in symbolic magnitude comparison (MC) and non-symbolic MC using a diffusion model. This approach uses the joint distribution of both RTs and accuracy in order to synthesize measures closer to ability and response caution or response conservatism. The latter can be understood in the context of the speed-accuracy tradeoff: It expresses how much a subject trades in speed for improved accuracy. We found significant effects of DD on both ability (negative) and response caution (positive) in MC tasks and a negative interaction of DD with symbolic task material on ability. These results support that DD subjects suffer from both an impaired ANS and an AD and in particular support that slower RTs of children with DD are indeed related to impaired processing of numerical information. An interaction effect of symbolic task material and DD (low mathematical ability) on response caution could not be refuted. However, in a sample more representative of the general population we found a negative association of mathematical ability and response caution in symbolic but not in non-symbolic task material. The observed differences in response behavior highlight the importance of accounting for response caution in the analysis of MC tasks. The results as a whole present a good example of the benefits of a diffusion model analysis.

Individual differences influence two-digit number processing, but not their analog magnitude processing: a large-scale online study.

Symbolic magnitude comparison is one of the most well-studied cognitive processes in research on numerical cognition. However, while the cognitive mechanisms of symbolic magnitude processing have been intensively studied, previous studies have paid less attention to individual differences influencing symbolic magnitude comparison. Employing a two-digit number comparison task in an online setting, we replicated previous effects, including the distance effect, the unit-decade compatibility effect, and the effect of cognitive control on the adaptation to filler items, in a large-scale study in 452 adults. Additionally, we observed that the most influential individual differences were participants' first language, time spent playing computer games and gender, followed by reported alcohol consumption, age and mathematical ability. Participants who used a first language with a left-to-right reading/writing direction were faster than those who read and wrote in the right-to-left direction. Reported playing time for computer games was correlated with faster reaction times. Female participants showed slower reaction times and a larger unit-decade compatibility effect than male participants. Participants who reported never consuming alcohol showed overall slower response times than others. Older participants were slower, but more accurate. Finally, higher grades in mathematics were associated with faster reaction times. We conclude that typical experiments on numerical cognition that employ a keyboard as an input device can also be run in an online setting. Moreover, while individual differences have no influence on domain-specific magnitude processing-apart from age, which increases the decade distance effect-they generally influence performance on a two-digit number comparison task.

TEMA and Dot Enumeration Profiles Predict Mental Addition Problem Solving Speed Longitudinally.

Different math indices can be used to assess math potential at school entry. We evaluated whether standardized math achievement (TEMA-2 performance), core number abilities (dot enumeration, symbolic magnitude comparison), non-verbal intelligence (NVIQ) and visuo-spatial working memory (VSWM), in combination or separately, predicted mental addition problem solving speed over time. We assessed 267 children’s TEMA-2, magnitude comparison, dot enumeration, and VSWM abilities at school entry (5 years) and NVIQ at 8 years. Mental addition problem solving speed was assessed at 6, 8, and 10 years. Longitudinal path analysis supported a model in which dot enumeration performance ability profiles and previous mental addition speed predicted future mental addition speed on all occasions, supporting a componential account of math ability. Standardized math achievement and NVIQ predicted mental addition speed at specific time points, while VSWM and symbolic magnitude comparison did not contribute unique variance to the model. The implications of using standardized math achievement and dot enumeration ability to index math learning potential at school entry are discussed.

Cognitive mechanisms underlying the relation between nonsymbolic and symbolic magnitude processing and their relation to math

Recent studies suggest that the relation between nonsymbolic magnitude processing skills and math competence is mediated by symbolic number processing. However, less is known about whether mapping between nonsymbolic and symbolic magnitude representations also mediates that relation, and whether the mediating role of symbolic number processing is explained by domain-general executive functions. Therefore, the current study examines whether symbolic comparison, mixed-format comparison, and executive function each mediate the relation between nonsymbolic magnitude processing and math. Furthermore, we investigate whether the relation between nonsymbolic and symbolic magnitude comparison is mediated by mapping between the formats and/or domain-general executive functions. Results indicate that symbolic processing mediates the relation between nonsymbolic processing and math, even after controlling for multiple components of executive function, which were also significant mediators. Cross-format comparison (i.e. mapping), on the other hand, did not mediate the relation between nonsymbolic comparison and math, but did mediate the relation between nonsymbolic and symbolic magnitude processing, even after controlling for executive function, which also mediated that relation. Taken together, our results suggest that both domain-specific and domain-general cognitive mechanisms account for the link between nonsymbolic and symbolic magnitude processing and their relation to math.

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.