Non-symbolic Number Research Articles

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.

Arithmetic knowledge from the spontaneous attention to relations.

Children's knowledge of arithmetic principles is a key aspect of early mathematics knowledge. Knowledge of arithmetic principles predicts how children approach solving arithmetic problems and the likelihood of their success. Prior work has begun to address how children might learn arithmetic principles in a classroom setting. Understanding of arithmetic principles involves understanding how numbers in arithmetic equations relate to another. For example, the Relation to Operands (RO) principle is that for subtracting natural numbers (A-B=C), the difference (C) must be smaller than the minuend (A). In the current study we evaluate if individual differences in arithmetic principle knowledge (APK) can be predicted by the learners' spontaneous attention to relations (SAR) and if feedback can increase their attention to relations. Results suggest that participants' Spontaneous Attention to Number (SAN) does not predict their knowledge of the RO principle for symbolic arithmetic. Feedback regarding the attention to relations did not show a significant effect on SAR or participants' APK. We also did not find significant relations between reports of parent talk and the home environment with individual differences in SAN. The amount of parent's talk about relations was not significantly associated with learner's SAR and APK. We conclude that children's SAR with non-symbolic number does not generalize to attention to relations with symbolic arithmetic.

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.

The relationship between children’s approximate number certainty and symbolic mathematics

Why do some children excel in mathematics while others struggle? A large body of work has shown positive correlations between children’s Approximate Number System (ANS) and school-taught symbolic mathematical skills, but the mechanism explaining this link remains unknown. One potential mediator of this relationship might be children’s numerical metacognition: children’s ability to evaluate how sure or unsure they are in understanding and manipulating numbers. While previous work has shown that children’s math abilities are uniquely predicted by symbolic numerical metacognition, we focus on the extent to which children’s non-symbolic/ANS numerical metacognition, in particular sensitivity to certainty, might be predictive of math ability, and might mediate the relationship between the ANS and symbolic math. A total of 72 children aged 4–6 years completed measures of ANS precision, ANS metacognition sensitivity, and the Test of Early Mathematical Ability (TEMA-3). Our results replicate many established findings in the literature, including the correlation between ANS precision and the TEMA-3, particularly on the Informal subtype questions. However, we did not find that ANS metacognition sensitivity was related to TEMA-3 performance, nor that it mediated the relationship between the ANS and the TEMA-3. These findings suggest either that metacognitive calibration may play a larger role than metacognitive sensitivity, or that metacognitive differences in the non-symbolic number perception do not robustly contribute to symbolic math performance.

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.

Introducing CUSTOM: A customized, ultraprecise, standardization-oriented, multipurpose algorithm for generating nonsymbolic number stimuli.

When evaluating the properties of a set of elements in a natural environment, an increase in numerosity unavoidably corresponds to an increase in the physical properties of the set: Five apples differ from ten apples not only in numerosity, but also in their visual features, such as volume, density, and surface. Since nonsymbolic number processing is typically investigated through the presentation of arrays of elements, it is mandatory to keep track of the visual features characterizing the stimuli. A plethora of solutions have been proposed to address this complex methodological issue; yet, there is no agreed-upon standard for how to measure and control for visual features. Here we present the "customized ultraprecise standardization-oriented multipurpose" (CUSTOM) algorithm for generating nonsymbolic number stimuli. It is characterized by several core features: The absence of fixed parameters or rules-apart from geometrical constraints-lets the user freely manipulate the visual features of the stimuli; control over the visual features of the stimuli is extremely accurate; no modification is required in order to perform different types of manipulation; and users can re-create any set of stimuli described so far in previous experiments on numerical cognition, for a wide variety of tasks, including comparison, estimation, habituation, and match-to-sample. The CUSTOM algorithm could represent an asset in the field of numerical cognition, as a versatile instrument for effectively generating high-precision visual stimuli within an unbiased theoretical framework.

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.

Numerals do not need numerosities: robust evidence for distinct numerical representations for symbolic and non-symbolic numbers.

In numerical cognition research, it has traditionally been argued that the processing of symbolic numerals (e.g., digits) is identical to the processing of the non-symbolic numerosities (e.g., dot arrays), because both number formats are represented in one common magnitude system-the Approximate Number System (ANS). In this study, we abandon this deeply rooted assumption and investigate whether the processing of numerals and numerosities can be dissociated, using an audio-visual paradigm in combination with various experimental manipulations. In Experiment 1, participants performed four comparison tasks with large symbolic and non-symbolic numbers: (1) number word-digit (2) tones-dots, (3) number word-dots, (4) tones-digit. In Experiment 2, we manipulated the number range (small vs. large) and the presentation modality (visual-auditory vs. auditory-visual). Results demonstrated ratio effects (i.e., the signature of ANS being addressed) in all tasks containing numerosities, but not in the task containing numerals only. Additionally, a cognitive cost was observed when participants had to integrate symbolic and non-symbolic numbers. Therefore, these results provide robust (i.e., independent of presentation modality or number range) evidence for distinct processing of numerals and numerosities, and argue for the existence of two independent number processing systems.

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.

The interplay between non-symbolic number and its continuous visual properties revisited: Effects of mixing trials of different types.

In the last few years, the existence of a pure number sense has been challenged. Recent studies suggest that numerosity processing is influenced not only by the number of elements in a display but also by continuous magnitudes, such as the size of the elements. The aim of our study was to replicate and extend the findings by Gebuis and Reynvoet, who systematically manipulated different continuous magnitudes either congruently or incongruently with discrete numerosity. We were particularly interested in finding the same pattern of congruency effects and assess its stability and robustness as this pattern indicates a complex influence of continuous magnitudes on numerosity judgements. We did so by showing stimuli of different conditions either in separate blocks or mixed together while participants solved a dot comparison task. Our results are in line with the notion that discrete number and continuous magnitudes are integrated in numerosity judgements by means of a weighing process. Moreover, our findings suggest that this integration is modified by the mode of presentation (blocked vs. mixed).

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.

The relation between subitizable symbolic and non-symbolic number processing over the kindergarten school year.

A long-standing debate in the field of numerical cognition concerns the degree to which symbolic and non-symbolic processing are related over the course of development. Of particular interest is the possibility that this link depends on the range of quantities in question. Behavioral and neuroimaging research with adults suggests that symbolic and non-symbolic quantities may be processed more similarly within, relative to outside of, the subitizing range. However, it remains unclear whether this unique link exists in young children at the outset of formal education. Further, no study has yet taken numerical size into account when investigating the longitudinal influence of these skills. To address these questions, we investigated the relation between symbolic and non-symbolic processing inside versus outside the subitizing range, both cross-sectionally and longitudinally, in 540 kindergarteners. Cross-sectionally, we found a consistently stronger relation between symbolic and non-symbolic number processing within versus outside the subitizing range at both the beginning and end of kindergarten. We also show evidence for a bidirectional relation over the course of kindergarten between formats within the subitizing range, and a unidirectional relation (symbolic→non-symbolic) for quantities outside of the subitizing range. These findings extend current theories on symbolic and non-symbolic magnitude development by suggesting that non-symbolic processing may in fact play a role in the development of symbolic number abilities, but that this influence may be limited to quantities within the subitizing range.

Differences and Associations in Symbolic and Non-Symbolic Early Numeracy Competencies of Chilean Kinder Grade Children, considering Socioeconomic Status of Schools

ABSTRACTThe central purpose of the present study is to assess the early numeracy skills of Chilean kindergarten students, taking into consideration the socioeconomic status (SES) of the schools they attend. A sample of 19 urban schools and 498 students were used for the study. Research Findings: The data analysis showed statistical differences in the results of students from low SES schools compared with medium and high SES schools in four symbolic number tasks: counting fluency, oral number identification, oral number comparison, and non-oral number comparison. Hierarchical regression analyses revealed a direct contribution of school vulnerability index, non-symbolic number tasks, and symbolic number tasks in number comparison performance. These analyses also revealed that school vulnerability index made a loose predictive contribution when symbolic number tasks were introduced into the model. Practice or Policy: The achievement gap detected in the early stages of formal education will most likely increase with time. Early detection of mathematics learning difficulties, effective intervention, and training of teaching staff may be means of equalizing the performance of students from different socioeconomic backgrounds.

Impairment of non-symbolic number processing in children with mathematical learning disability

The functional origin of Mathematical Learning Disability (MLD), an impairment affecting mathematics learning, remains controversial. We aimed to study the number sense deficit hypothesis in children with MLD. We explored the processing of non-symbolic numbers in Quebec French-speaking 8‒9-year-old children with three non-symbolic tasks (number comparison, number matching and dot set selection task). Results showed that children with MLD were as successful as typically developing children in a comparison task, but less successful in a matching task. Their performance in numerical re-production was also similar to that of typically developing children for large (10-99) numerosities, while they were less successful in transcoding small (1-4) and medium (5-9) numerosities. Children with MLD also presented with general cognitive and reading difficulties. Results suggest a deficit in processing small and medium numerosities in children with MLD that could be largely attributed to their poorer cognitive skills.

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.

Disentangling Mathematics from Executive Functions by Investigating Unique Functional Connectivity Patterns Predictive of Mathematics Ability.

What are the underlying neurocognitive mechanisms that give rise to mathematical competence? This study investigated the relationship between tests of mathematical ability completed outside the scanner and resting-state functional connectivity (FC) of cytoarchitectonically defined subdivisions of the parietal cortex in adults. These parietal areas are also involved in executive functions (EFs). Therefore, it remains unclear whether there are unique networks for mathematical processing. We investigate the neural networks for mathematical cognition and three measures of EF using resting-state fMRI data collected from 51 healthy adults. Using 10 ROIs in seed to whole-brain voxel-wise analyses, the results showed that arithmetical ability was correlated with FC between the right anterior intraparietal sulcus (hIP1) and the left supramarginal gyrus and between the right posterior intraparietal sulcus (hIP3) and the left middle frontal gyrus and the right premotor cortex. The connection between the posterior portion of the left angular gyrus and the left inferior frontal gyrus was also correlated with mathematical ability. Covariates of EF eliminated connectivity patterns with nodes in inferior frontal gyrus, angular gyrus, and middle frontal gyrus, suggesting neural overlap. Controlling for EF, we found unique connections correlated with mathematical ability between the right hIP1 and the left supramarginal gyrus and between hIP3 bilaterally to premotor cortex bilaterally. This is partly in line with the "mapping hypothesis" of numerical cognition in which the right intraparietal sulcus subserves nonsymbolic number processing and connects to the left parietal cortex, responsible for calculation procedures. We show that FC within this circuitry is a significant predictor of math ability in adulthood.

Development of a Possible General Magnitude System for Number and Space.

There is strong evidence for a link between numerical and spatial processing. However, whether this association is based on a common general magnitude system is far from conclusive and the impact of development is not yet known. Hence, the present study aimed to investigate the association between discrete non-symbolic number processing (comparison of dot arrays) and continuous spatial processing (comparison of angle sizes) in children between the third and sixth grade (N = 367). Present findings suggest that the processing of comparisons of number of dots or angle are related to each other, but with angle processing developing earlier and being more easily comparable than discrete number representations for children of this age range. Accordingly, results favor the existence of a more complex underlying magnitude system consisting of dissociated but closely interacting representations for continuous and discrete magnitudes.

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.