Nonsymbolic Number Comparison Research Articles

The importance of domain-specific number abilities and domain-general cognitive abilities for early arithmetic achievement and development.

Children's numerical and arithmetic skills differ greatly already at an early age. Although research focusing on accounting for these large individual differences clearly demonstrates that mathematical performance draws upon several cognitive abilities, our knowledge concerning key abilities underlying mathematical skill development is still limited. First, to identify key cognitive abilities contributing to children's development of early arithmetic skills. Second, to examine the extent to which early arithmetic performance and early arithmetic development rely on different or similar constellations of domain-specific number abilities and domain-general cognitive abilities. In all, 134 Swedish children (Mage = 6 years and 4 months, SD = 3 months, 74 boys) participated in this study. Verbal and non-verbal logical reasoning, non-symbolic number comparison, counting knowledge, spatial processing, verbal working memory and arithmetic were assessed. Twelve months later, arithmetic skills were reassessed. A latent change score model was computed to determine whether any of the abilities accounted for variations in arithmetic development. Arithmetic performance was supported by counting knowledge, verbal and non-verbal logical reasoning and spatial processing. Arithmetic skill development was only supported by spatial processing. Results show that young children's early arithmetic performance and arithmetic development are supported by different cognitive processes. The findings regarding performance supported Fuchs et al.'s model (Dev Psychol, 46, 2010b, 1731) but the developmental findings did not. The developmental findings align partially to Geary et al.'s (J Educ Psychol, 109, 2017, 680) hypothesis stating that young children's early arithmetic development is more dependent on general cognitive abilities than number abilities.

The association between non-symbolic number comparison and mathematical abilities depends on fluency.

Numerous studies have explored the correlation between non-symbolic number comparison and mathematical abilities in children, but the results have been inconsistent. The underlying mental processing featuring fluency may affect the correlation. The current study tested the fluency hypothesis that non-symbolic number comparison is associated with mathematical fluency in the development of mathematical ability. Non-symbolic number comparison, arithmetic computation, mathematical reasoning, non-symbolic number estimation, symbolic number comparison, and a series of basic cognitive processing tasks, including mental rotation, non-verbal matrix reasoning, and choice reaction time, were administered to 1072 first- to fourth-grade children. The results show that non-symbolic number comparison (measured via numerosity comparison) was the only independent predictor of arithmetic computation in higher grades, even after controlled for age, gender, basic cognitive processing, non-symbolic number estimation (measured via numerosity estimation), and symbolic number comparison (measured via digit comparison). However, it did not correlate with mathematical reasoning in any grade. These findings support the fluency hypothesis for developmental correlation between non-symbolic number comparison and mathematical abilities. That is, non-symbolic number comparison correlates with mathematical ability featuring fluency.

Predictors of middle school students’ growth in symbolic number comparison performance

The ability to efficiently compare number symbols, such as digits, is associated with mathematics competence across the lifespan. Performance on symbolic number comparison tasks differ across age groups; young students who are developing fluency with digits improve on symbolic number comparison, and performance is better in adults than children. However, whether this improvement continues for older students who are fluent with number symbols, and what cognitive factors relate to this improvement, is unknown. This study used a longitudinal sample of U.S. middle school students (n = 394) to examine whether symbolic number comparison performance changes over middle school (i.e., students aged 11-14), whether there are individual differences in students’ rate of change, and potential predictors of that change. Students completed measures of single-digit symbolic number comparison, nonsymbolic number comparison, executive function (EF), and mathematics competence in Grade 5 (M = 11.02 years; SD = 0.32), and double-digit symbolic number comparison in Grades 6-8. Results showed that, on average, students’ symbolic number comparison performance improved from Grades 6-8. Grade 5 Symbolic number comparison performance predicted Grade 8 symbolic number comparison and rate of change over Grades 6-8. Grade 5 nonsymbolic number comparison, EF, and mathematics competence predicted Grade 8 symbolic number comparison performance. Results suggest that numerical magnitude processing, executive functions, and mathematics competence are related to symbolic number processing well into middle school, and that students continue to refine their ability to process number symbols into adolescence.

A connectome-based neuromarker of nonverbal number acuity and arithmetic skills.

The approximate number system (ANS) is vital for survival and reproduction in animals and is crucial for constructing abstract mathematical abilities in humans. Most previous neuroimaging studies focused on identifying discrete brain regions responsible for the ANS and characterizing their functions in numerosity perception. However, a neuromarker to characterize an individual's ANS acuity is lacking, especially one based on whole-brain functional connectivity (FC). Here, based on the resting-state functional magnetic resonance imaging (rs-fMRI) data obtained from a large sample, we identified a distributed brain network (i.e. a numerosity network) using a connectome-based predictive modeling (CPM) analysis. The summed FC strength within the numerosity network reliably predicted individual differences in ANS acuity regarding behavior, as measured using a nonsymbolic number-comparison task. Furthermore, in an independent dataset of the Human Connectome Project (HCP), we found that the summed FC strength within the numerosity network also specifically predicted individual differences in arithmetic skills, but not domain-general cognitive abilities. Therefore, our findings revealed that the identified numerosity network could serve as an applicable neuroimaging-based biomarker of nonverbal number acuity and arithmetic skills.

Predicting children's math skills from task-based and resting-state functional brain connectivity.

A critical goal of cognitive neuroscience is to predict behavior from neural structure and function, thereby providing crucial insights into who might benefit from clinical and/or educational interventions. Across development, the strength of functional connectivity among a distributed set of brain regions is associated with children's math skills. Therefore, in the present study we use connectome-based predictive modeling to investigate whether functional connectivity during numerical processing and at rest "predicts" children's math skills (N = 31, Mage = 9.21years, 14 Female). Overall, we found that functional connectivity during symbolic number comparison and rest, but not during nonsymbolic number comparison, predicts children's math skills. Each task revealed a largely distinct set of predictive connections distributed across canonical brain networks and major brain lobes. Most of these predictive connections were negatively correlated with children's math skills so that weaker connectivity predicted better math skills. Notably, these predictive connections were largely nonoverlapping across task states, suggesting children's math abilities may depend on state-dependent patterns of network segregation and/or regional specialization. Furthermore, the current predictive modeling approach moves beyond brain-behavior correlations and toward building models of brain connectivity that may eventually aid in predicting future math skills.

Sharpening, focusing, and developing: A study of change in nonsymbolic number comparison skills and math achievement in 1st grade.

Children's ability to discriminate nonsymbolic number (e.g., the number of items in a set) is a commonly studied predictor of later math skills. Number discrimination improves throughout development, but what drives this improvement is unclear. Competing theories suggest that it may be due to a sharpening numerical representation or an improved ability to pay attention to number and filter out non-numerical information. We investigate this issue by studying change in children's performance (N=65) on a nonsymbolic number comparison task, where children decide which of two dot arrays has more dots, from the middle to the end of 1st grade (mean age at time 1=6.85 years old). In this task, visual properties of the dot arrays such as surface area are either congruent (the more numerous array has more surface area) or incongruent. Children rely more on executive functions during incongruent trials, so improvements in each congruency condition provide information about the underlying cognitive mechanisms. We found that accuracy rates increased similarly for both conditions, indicating a sharpening sense of numerical magnitude, not simply improved attention to the numerical task dimension. Symbolic number skills predicted change in congruent trials, but executive function did not predict change in either condition. No factor predicted change in math achievement. Together, these findings suggest that nonsymbolic number processing undergoes development related to existing symbolic number skills, development that appears not to be driving math gains during this period. Children's ability to discriminate nonsymbolic number improves throughout development. Competing theories suggest improvement due to sharpening magnitude representations or changes in attention and inhibition. The current study investigates change in nonsymbolic number comparison performance during first grade and whether symbolic number skills, math skills, or executive function predict change. Children's performance increased across visual control conditions (i.e., congruent or incongruent with number) suggesting an overall sharpening of number processing. Symbolic number skills predicted change in nonsymbolic number comparison performance.

Only Approximately the Same: Approximate Number Skills in Typically Developing Children and Children with Down Syndrome

ABSTRACT The approximate number system (ANS) is foundational to numerical cognition. Whereas some studies hint at intact ANS skills in individuals with Down syndrome (DS), others report impaired ANS abilities. To reconcile these discrepant findings, we investigated ANS skills of children with DS and typically developing (TD) children in a non-symbolic number comparison task. Children had to indicate which of two characters possessed more dots. The two sets of dots differed by ratios of 1:2 to 4:5. While overall DS and TD participants’ performance was comparable, accuracy levels differed for the two groups depending on the ratio presented. Both groups performed similarly for ‘easier’ ratios, however, TD children outperformed children with DS when they had to discriminate ‘harder’ ratios. Children with DS may thus face specific problems when discriminating ratios that are more demanding. The deviating ANS skills could potentially be related to problems with other symbolic maths abilities.

First fixation duration as a bottom-up measure during symbolic and non-symbolic numerical comparisons (La duración de la primera fijación como medida bottom-up al comparar cantidades simbólicas y no simbólicas)

ABSTRACT The ability to represent quantities, either in a symbolic or non-symbolic format, predicts calculation skills. Recording the saccades during gazing at these stimuli allows us to infer the cognitive processing taking place at that moment. The purpose of this research study was to analyse the duration of the first fixation during symbolic and non-symbolic numerical stimuli. We evaluated 32 university students through symbolic and non-symbolic number comparison tasks while recording their eye movements. In addition, they answered a calculation task. Participants spent more time looking first at the stimulus of greater magnitude during the execution of the comparison task (regardless of the presentation format). This measure was not a predictor of calculation skills. The duration of the first fixation was interpreted as a bottom-up measure during numerical stimuli perception, and therefore during the initial stage of cognitive processing.

NASCO: A new method and program to generate dot arrays for non-symbolic number comparison tasks

Basic numerical abilities are generally assumed to influence more complex cognitive processes involving numbers, such as mathematics. Yet measuring non-symbolic number abilities remains challenging due to the intrinsic correlation between numerical and non-numerical dimensions of any visual scene. Several methods have been developed to generate non-symbolic stimuli controlling for the latter aspects but they tend to be difficult to replicate or implement. In this study, we describe the NASCO method, which is an extension to the method popularized by Dehaene, Izard, and Piazza (2005). Their procedure originally controlled for two visual dimensions that are mediated by Number: Total Area and Item Size (i.e., N = TA/IS). Here, we additionally propose to control for another twofold dimension related to the array extent, which is also mediated by Number: Convex Hull Area and Mean Occupancy (i.e., N = CH/MO). We illustrate the NASCO method with a MATLAB app—NASCO app—that allows easy generation of dot arrays for a visually controlled assessment of non-symbolic numerical abilities. Results from a numerical comparison task revealed that the introduction of this twofold dimension manipulation substantially affected young adults’ performance. In particular, we did not replicate the relation between non-symbolic number abilities and arithmetic skills. Our findings open the debate about the reliability of previous results that did not take into account visual features related to the array extent. We then discuss the strengths of NASCO method to assess numerical ability, as well as the benefits of its straightforward implementation in NASCO app for researchers.

Children skilled in mental abacus show enhanced non-symbolic number sense

Mental abacus is mental arithmetic with the help of an imagined abacus. Children skilled in mental abacus have been shown to exhibit top-quality arithmetic abilities. The current study investigated whether children with high-level mental abacus ability could outperform untrained control children in non-symbolic number sense, which is considered to be fundamental for arithmetic development. One hundred and fifty children (75 children skilled in mental abacus and 75 controls) took part in this study. Children skilled in mental abacus completed a mental abacus level test. The two groups of children performed serial cognitive tasks, assessing non-symbolic number comparison, arithmetic, language, spatial processing, visual perception, attention, processing speed, working memory, and general intelligence. Results showed that children skilled in mental abacus had significantly better non-symbolic number sense than the other children after controlling for general intelligence. The significant difference in non-symbolic number sense remained after controlling for age, gender, all types of cognitive processing available, and arithmetic performance. A mediation model showed that non-symbolic number sense partially mediated the group difference in arithmetic development. These findings suggest that children skilled in mental abacus have enhanced non-symbolic number sense and raise the possibility that mental abacus training could directly improve children’s non-symbolic numerical skills.

Children's Home Numeracy Environment Predicts Growth of their Early Mathematical Skills in Kindergarten.

This study investigated the longitudinal associations between children's early mathematics and their home numeracy environment (HNE). Chilean children from families who varied widely in socioeconomic status were assessed at the beginning and end of prekindergarten in 2016 (N=419, Mage =4:7 [years:months]), and at the end of kindergarten in 2017 (N=368, Mage =5:10). Children whose parents provided frequent operational numeracy activities (e.g., learning simple sums) at prekindergarten showed better arithmetic performance and growth in nonsymbolic and symbolic number comparison at the end of kindergarten. Parents' knowledge of number-related games predicted children's arithmetic skills and growth in nonsymbolic number comparison. These findings underscore the persistent relations between the HNE and the development of children's mathematical skills.

Network topology of symbolic and nonsymbolic number comparison.

Studies of brain activity during number processing suggest symbolic and nonsymbolic numerical stimuli (e.g., Arabic digits and dot arrays) engage both shared and distinct neural mechanisms. However, the extent to which number format influences large-scale functional network organization is unknown. In this study, using 7 Tesla MRI, we adopted a network neuroscience approach to characterize the whole-brain functional architecture supporting symbolic and nonsymbolic number comparison in 33 adults. Results showed the degree of global modularity was similar for both formats. The symbolic format, however, elicited stronger community membership among auditory regions, whereas for nonsymbolic, stronger membership was observed within and between cingulo-opercular/salience network and basal ganglia communities. The right posterior inferior temporal gyrus, left intraparietal sulcus, and two regions in the right ventromedial occipital cortex demonstrated robust differences between formats in terms of their community membership, supporting prior findings that these areas are differentially engaged based on number format. Furthermore, a unified fronto-parietal/dorsal attention community in the nonsymbolic condition was fractionated into two components in the symbolic condition. Taken together, these results reveal a pattern of overlapping and distinct network architectures for symbolic and nonsymbolic number processing.

Preschoolers' Associations Between Space and Numbers Can Be Biased by Surface Area

ABSTRACTPeople associate numbers and horizontal space. This association has been demonstrated by the so‐called SNARC (Spatial‐Numerical Association of Response Codes)‐effect, with Western participants responding faster to larger numbers with their right hand and vice versa for smaller numbers. SNARC‐like effects have also been reported for preschoolers. However, it remains unclear whether children's SNARC‐effects are exclusively based on number or whether nonnumerical factors such as surface area can also play a role. To address this question, we asked children to compare quantities in a nonsymbolic number comparison task. On half of the trials, the larger set took up more total surface area (congruent trials), and on the other half, the smaller number had more surface area (incongruent trials). We obtained a significant interaction of SNARC‐like effects and congruency. SNARC‐effects were only observed for congruent trials, demonstrating that preschoolers' spatial‐numerical associations can be affected by nonnumerical magnitude.

Mathematics Competence Level: The Contribution of Non-symbolic and Spatial Magnitude Comparison Skills.

Magnitude comparison skills have been related to mathematics competence, although results in this area vary. The current study aimed to describe the performance of 75 children (aged 4–5 years) in two comparison tasks; and examine the strength of the relationship between each of the two tasks and mathematics competence level (MCL). Participants were assessed with the Early Numeracy Test which provides a global MCL score. Magnitude comparison skills were assessed with two tasks: a non-symbolic number comparison task and a spatial comparison task. Results of the Pearson correlation analysis showed a relationship between the two tasks with better performance in the spatial comparison task. Regression analysis with the stepwise method showed that only the non-symbolic number comparison task had a significant value in the prediction of the MCL pointing to the need to take these kinds of tasks into account in the first years of school.

Dyscalculia and Typical Math Achievement Are Associated With Individual Differences in Number-Specific Executive Function.

Deficits in numerical magnitude perception characterize the mathematics learning disability developmental dyscalculia (DD), but recent studies suggest the relation stems from inhibitory control demands from incongruent visual cues in the nonsymbolic number comparison task. This study investigated the relation among magnitude perception during differing congruency conditions, executive function, and mathematics achievement measured longitudinally in children (n=448) from ages 4 to 13. This relation was investigated across achievement groups and as it related to mathematics across the full range of achievement. Only performance on incongruent trials related to achievement. Findings indicate that executive function in a numerical context, beyond magnitude perception or executive function in a non-numerical context, relates to DD and mathematics across a wide range of achievement.

Using Hierarchical Linear Models to Examine Approximate Number System Acuity: The Role of Trial-Level and Participant-Level Characteristics.

The ability to intuitively and quickly compare the number of items in collections without counting is thought to rely on the Approximate Number System (ANS). To assess individual differences in the precision of peoples’ ANS representations, researchers often use non-symbolic number comparison tasks in which participants quickly choose the numerically larger of two arrays of dots. However, some researchers debate whether this task actually measures the ability to discriminate approximate numbers or instead measures the ability to discriminate other continuous magnitude dimensions that are often confounded with number (e.g., the total surface area of the dots or the convex hull of the dot arrays). In this study, we used hierarchical linear models (HLMs) to predict 132 adults’ accuracy on each trial of a non-symbolic number comparison task from a comprehensive set of trial-level characteristics (including numerosity ratio, surface area, convex hull, and temporal and spatial variations in presentation format) and participant-level controls (including cognitive abilities such as visual-short term memory, working memory, and math ability) in order to gain a more nuanced understanding of how individuals complete this task. Our results indicate that certain trial-level characteristics of the dot arrays contribute to our ability to compare numerosities, yet numerosity ratio, the critical marker of the ANS, remains a highly significant predictor of accuracy above and beyond trial-level characteristics and across individuals with varying levels of math ability and domain-general cognitive abilities.

Attention to number: The convergence of numerical magnitude processing, attention, and mathematics in the inferior frontal gyrus.

Research indicates that the neurocognitive system representing nonsymbolic numerical magnitudes is foundational for the development of mathematical competence. However, recent studies found that the most common task used to measure numerical acuity, the nonsymbolic number comparison task, is heavily influenced by non-numerical visual parameters of stimuli that increase executive function demands. Further, this influence may be a confound invalidating theoretical accounts of the relation between number comparison performance and mathematical competence. Instead of acuity, the relation may depend on one's ability to attend to numerical information in the face of competing, non-numerical cues. The current study investigated this issue by measuring neural activity associated with numerical magnitude processing acuity, domain-general attention, and selective attention to number via functional magnetic resonance imaging while children 8-11 years old completed a nonsymbolic number comparison task and a flanker task. Results showed that activation in the right inferior frontal gyrus during incongruent versus congruent trials of the comparison task, our construct for attention to number, predicted mathematics achievement after controlling for verbal IQ, flanker accuracy rate, and the neural congruency effect from the flanker task. In contrast, activity in frontal and parietal regions responding to differences in difficulty of numerical comparisons, our construct for numerical magnitude processing acuity, did not correlate with achievement. Together, these findings suggest a need to reframe existing models of the relation between number processing and math competence to include the interaction between attention and use of numerical information, or in other words "attention to number."

What predicts mathematics achievement? Developmental change in 5- and 7-year-old children

Many changes occur in general and specific cognitive abilities in children between 5 and 7 years of age, the period coinciding with entrance into formal schooling. The current study focused on the relative contributions of approximate number system (ANS) acuity, mapping precision between numeral symbols and their corresponding magnitude (mapping precision) and working memory (WM) capacity to mathematics achievement in 5- and 7-year-olds. Children’s performance was examined in different tasks: nonsymbolic number comparison, number line estimation, working memory, mathematics achievement, and vocabulary. This latter task was used to determine whether predictors were general or specific to mathematics achievement. The results showed that ANS acuity was a significant specific predictor of mathematics achievement only in 5-year-olds, mapping precision was a significant specific predictor at the two ages, and WM was a significant general predictor only in 7-year-olds. These findings suggest that a general cognitive ability, especially WM, becomes a stronger predictor of mathematics achievement after entrance into formal schooling, whereas ANS acuity, a specific cognitive ability, loses predictive power. Moreover, mediation analyses showed that mapping precision was a partial mediator of the relation between ANS acuity and mathematics achievement in 5-year-olds but not in 7-year-olds. Conversely, in 7-year-olds but not in 5-year-olds, WM fully mediated the relation between ANS acuity and mathematics achievement. These results showed that between 5 and 7 years of age, the period of transition into formal mathematical learning, important changes occurred in the relative weights of different predictors of mathematics achievement.

Expanding the Home Numeracy Model to Chilean children: Relations among parental expectations, attitudes, activities, and children’s mathematical outcomes

We used structural equation modeling to evaluate an enhanced version of the Home Numeracy Model proposed by Skwarchuk and colleagues (2014, Journal of Experimental Child Psychology, 121, 63–84). Participants were 390 Chilean preschool children and their parents. Children completed numeracy and literacy tasks at the beginning of preschool (mean age: 4 years and 7 months) and approximately 8 months later. Parents reported on the home numeracy activities they engaged in with their children, including formal (i.e., mapping and operational), informal (i.e., parents’ number-game knowledge), and home literacy activities (i.e., code-related and meaning-related), as well as on numeracy and literacy attitudes and expectations for children’s performance prior to Grade 1. We found that parents with more positive numeracy attitudes and higher academic expectations reported a higher frequency of formal numeracy (mapping and operational) activities. In turn, formal operational activities predicted number line estimation and applied problem-solving skills. In contrast, informal activities (i.e., parents’ number-game knowledge) predicted children’s non-symbolic arithmetic and non-symbolic number comparison tasks, as well as their applied problem-solving skills. The links between home activities and numeracy outcomes were domain specific: Parents’ reports of literacy activities did not predict early numeracy skills. We discuss how our results support the enhanced Home Numeracy Model and thus provide a more complete framework connecting parents’ engagement in numeracy activities and children’s mathematical outcomes.

Intergenerational associations of the approximate number system in toddlers and their parents.

From birth, humans are able to discriminate quantities using the approximate number system (ANS). However, previous methods have only been suitable to examine ANS functioning in infancy and older children. The goals of this study were twofold: first, to modify an existing method of assessing ANS functioning for toddlerhood; and second, to investigate individual differences in toddlers' ANS performance by examining correlations with their parents' ANS acuity. Using a preferential looking paradigm, we found that 1- to 3-year-olds (N=46) looked significantly longer to numerically changing images compared to numerically constant ones suggesting that the paradigm is a suitable measure of ANS functioning in toddlerhood. Furthermore, we found a positive relation between toddlers' ANS performance and that of their parents (assessed using a non-symbolic number comparison task) independent of children's vocabulary or parents' perceived math ability or preference for math. These findings are consistent with a specific intergenerational transmission of the ANS. Statement of contribution What is already known on this subject? Past methods used to examine ANS functioning were only suitable for infants and older children. Little research has examined sources underlying individual difference in ANS acuity. What does this study add? We developed a preferential looking task to assess ANS functioning in toddlerhood. Individual differences in toddlers' ANS functioning are correlated with their parents' ANS acuity.