Non-symbolic Numerosities Research Articles

The role of numerical and non-numerical cues in nonsymbolic number processing: Evidence from the line bisection task.

In line bisection tasks, adults and children bisect towards the numerically larger of two nonsymbolic numerosities [de Hevia, M. D., & Spelke, E. S. ( 2009 ). Spontaneous mapping of number and space in adults and young children. Cognition, 110, 198-207. doi:10.1016/j.cognition.2008.11.003]. However, it is not clear whether this effect is driven by number itself or rather by visual cues such as subtended area [Gebuis, T., & Gevers, W. ( 2011 ). Numbers and space: Indeed a cognitive illusion! A reply to de Hevia and Spelke ( 2009 ). Cognition, 121, 248-252. doi:10.1016/j.cognition.2010.09.008]. Furthermore, this effect has only been demonstrated with flanking displays of two and nine items. Here, we report three studies that examined whether this "spatial bias" effect occurs across a range of absolute and ratio numerosity differences; in particular, we examined whether the bias would occur when both flankers were outside the subitizing range. Additionally, we manipulated the subtended area of the stimulus and the aggregate surface area to assess the influence of visual cues. We found that the spatial bias effect occurred for a range of flanking numerosities and for ratios of 3:5 and 5:6 when subtended area was not controlled (Experiment 1). However, when subtended area and aggregate surface area were held constant, the biasing effect was reversed such that participants bisected towards the flanker with fewer items (Experiment 2). Moreover, when flankers were identical, participants bisected towards the flanker with larger subtended area or larger aggregate surface area (Experiments 2 and 3). On the basis of these studies, we conclude that the spatial bias effect for nonsymbolic numerosities is primarily driven by visual cues.

Numerical Context and Time Perception: Contrast Effects and the Perceived Duration of Numbers.

In the current study, we examined how the contextual repetition of magnitude information presented in either symbolic (Arabic digits) or nonsymbolic (numerosities) formats impacted on the perceived duration of a later occurring target number. The results of the current study demonstrated a time-magnitude bias in which, on average, large magnitude target numbers were judged to last for longer durations relative to small magnitude target numbers, regardless of notation (symbolic number and numerosity). Furthermore, context effects were found, in which a greater discrepancy in the target's magnitude from the initial context led to longer perceived duration ratings. However, this was found to be asymmetrical, occurring only for large magnitude targets. Additionally, the type of context effect was shown to be determined by whether the context was presented in the same notation as the target or a different notation.

Varieties of quantity estimation in children.

In the number-to-position task, with increasing age and numerical expertise, children's pattern of estimates shifts from a biased (nonlinear) to a formal (linear) mapping. This widely replicated finding concerns symbolic numbers, whereas less is known about other types of quantity estimation. In Experiment 1, Preschool, Grade 1, and Grade 3 children were asked to map continuous quantities, discrete nonsymbolic quantities (numerosities), and symbolic (Arabic) numbers onto a visual line. Numerical quantity was matched for the symbolic and discrete nonsymbolic conditions, whereas cumulative surface area was matched for the continuous and discrete quantity conditions. Crucially, in the discrete condition children's estimation could rely either on the cumulative area or numerosity. All children showed a linear mapping for continuous quantities, whereas a developmental shift from a logarithmic to a linear mapping was observed for both nonsymbolic and symbolic numerical quantities. Analyses on individual estimates suggested the presence of two distinct strategies in estimating discrete nonsymbolic quantities: one based on numerosity and the other based on spatial extent. In Experiment 2, a non-spatial continuous quantity (shades of gray) and new discrete nonsymbolic conditions were added to the set used in Experiment 1. Results confirmed the linear patterns for the continuous tasks, as well as the presence of a subset of children relying on numerosity for the discrete nonsymbolic numerosity conditions despite the availability of continuous visual cues. Overall, our findings demonstrate that estimation of numerical and non-numerical quantities is based on different processing strategies and follow different developmental trajectories.

Mathematical Difficulties and White Matter Abnormalities in Subacute Pediatric Mild Traumatic Brain Injury.

Mathematical difficulties have been documented following pediatric mild traumatic brain injury (mTBI), yet a precise characterization of these impairments and their neural correlates is currently unavailable. We aimed to characterize these impairments by comparing behavioral and neuroimaging (i.e., diffusion tensor imaging [DTI]) outcomes from children with subacute mTBI to typically-developing controls. Twenty subacute pediatric mTBI patients and 20 well-matched controls underwent cognitive assessment and DTI examination. DTI tractography was used to detect white matter abnormalities in the corpus callosum (CC) and superior and inferior longitudinal fasciculi; these tracts are involved in mathematical performance and they are often damaged after mTBI. Behavioral results revealed that children with mTBI performed significantly more poorly on rapid apprehension of small numbers of objects (or "subitizing"), processing of non-symbolic numerosities, and procedural problem solving. These group differences were explained by differences in visuospatial working memory, which suggests that the observed mathematical difficulties may be a consequence of impairments in visuospatial abilities. DTI analysis revealed subtle group differences in the CC genu and splenium (i.e., higher fractional anisotropy and lower mean and radial diffusivity in children with mTBI) but the observed white matter abnormalities of the CC were not significantly associated with the observed mathematical difficulties in the mTBI patients.

Small numerosities are associated with the left, large numerosities are associated with the right: Evidence from a SNARC task

Event Abstract Back to Event Small numerosities are associated with the left, large numerosities are associated with the right: Evidence from a SNARC task Fiona Nemeh1, Mark Yates2*, Tobias Loetscher3, Anna Ma-Wyatt4 and Michael E. Nicholls3 1 University of Melbourne, Melbourne School of Psychological Sciences, Australia 2 University of Melbourne, School of Psychological Sciences, Australia 3 Flinders University, School of Psychology, Australia 4 University of Adelaide, School of Psychology, Australia A central finding within numerical cognition is that symbolic numbers (e.g. Arabic numerals) are represented spatially with smaller numbers on the left and larger numbers on the right (Spatial-Numerical Association of Response Codes, or 'SNARC' effect). This study explored whether non-symbolic numbers, i.e. dot cloud stimuli of varying numerosities, are also represented spatially. Participants judged whether a briefly presented dot cloud stimulus contained more or less dots than a reference dot cloud. It was predicted that dot clouds with less (more) dots would be categorised more quickly with the left (right) hand. This effect was observed, but may have been due to numerosity per se, or total dot surface area, which co-varied with numerosity in the initial experiment. To disambiguate between these two possibilities, a follow-up experiment was conducted in which total dot surface area was held constant as numerosity increased. The effect remained, suggesting that non-symbolic numbers are also represented spatially. Keywords: SNARC effect, magnitude representation, Keywords: Numerical Cognition, Non-symbolic numerosity, Number- space associations Conference: XII International Conference on Cognitive Neuroscience (ICON-XII), Brisbane, Queensland, Australia, 27 Jul - 31 Jul, 2014. Presentation Type: Poster Topic: Cognition and Executive Processes Citation: Nemeh F, Yates M, Loetscher T, Ma-Wyatt A and Nicholls ME (2015). Small numerosities are associated with the left, large numerosities are associated with the right: Evidence from a SNARC task. Conference Abstract: XII International Conference on Cognitive Neuroscience (ICON-XII). doi: 10.3389/conf.fnhum.2015.217.00225 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 19 Feb 2015; Published Online: 24 Apr 2015. * Correspondence: Dr. Mark Yates, University of Melbourne, School of Psychological Sciences, Melbourne, Australia, mjyates@unmelb.edu.au Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Fiona Nemeh Mark Yates Tobias Loetscher Anna Ma-Wyatt Michael E Nicholls Google Fiona Nemeh Mark Yates Tobias Loetscher Anna Ma-Wyatt Michael E Nicholls Google Scholar Fiona Nemeh Mark Yates Tobias Loetscher Anna Ma-Wyatt Michael E Nicholls PubMed Fiona Nemeh Mark Yates Tobias Loetscher Anna Ma-Wyatt Michael E Nicholls Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Development of brain systems for nonsymbolic numerosity and the relationship to formal math academic achievement.

A central question in cognitive and educational neuroscience is whether brain operations supporting nonlinguistic intuitive number sense (numerosity) predict individual acquisition and academic achievement for symbolic or "formal" math knowledge. Here, we conducted a developmental functional magnetic resonance imaging (MRI) study of nonsymbolic numerosity task performance in 44 participants including 14 school age children (6-12 years old), 14 adolescents (13-17 years old), and 16 adults and compared a brain activity measure of numerosity precision to scores from the Woodcock-Johnson III Broad Math index of math academic achievement. Accuracy and reaction time from the numerosity task did not reliably predict formal math achievement. We found a significant positive developmental trend for improved numerosity precision in the parietal cortex and intraparietal sulcus specifically. Controlling for age and overall cognitive ability, we found a reliable positive relationship between individual math achievement scores and parietal lobe activity only in children. In addition, children showed robust positive relationships between math achievement and numerosity precision within ventral stream processing areas bilaterally. The pattern of results suggests a dynamic developmental trajectory for visual discrimination strategies that predict the acquisition of formal math knowledge. In adults, the efficiency of visual discrimination marked by numerosity acuity in ventral occipital-temporal cortex and hippocampus differentiated individuals with better or worse formal math achievement, respectively. Overall, these results suggest that two different brain systems for nonsymbolic numerosity acuity may contribute to individual differences in math achievement and that the contribution of these systems differs across development.

Concurrent validity of approximate number sense tasks in adults and children

Reasoning with non-symbolic numerosities is suggested to be rooted in the Approximate Number System (ANS) and evidence pointing to a relationship between the acuity of this system and mathematics is available. In order to use the acuity of this ANS as a screening instrument to detect future math problems, it is important to model ANS acuity over development. However, whether ANS acuity and its development have been described accurately can be questioned. Namely, different tasks were used to examine the developmental trajectory of ANS acuity and studies comparing performances on these different tasks are scarce. In the present study, we examined whether different tasks designed to measure the acuity of the ANS are comparable and lead to related ANS acuity measures (i.e., the concurrent validity of these tasks). We contrasted the change detection task, which is used in infants, with tasks that are more commonly used in older children and adults (i.e., comparison and same-different tasks). Together, our results suggest that ANS acuity measures obtained with different tasks are not related. This poses serious problems for the comparison of ANS acuity measures derived from different tasks and thus for the establishment of the developmental trajectory of ANS acuity.

The association between higher education and approximate number system acuity.

Humans are equipped with an approximate number system (ANS) supporting non-symbolic numerosity representation. Studies indicate a relationship between ANS-precision (acuity) and math achievement. Whether the ANS is a prerequisite for learning mathematics or if mathematics education enhances the ANS remains an open question. We investigated the association between higher education and ANS acuity with university students majoring in subjects with varying amounts of mathematics (mathematics, business, and humanities), measured either early (First year) or late (Third year) in their studies. The results suggested a non-significant trend where students taking more mathematics had better ANS acuity and a significant improvement in ANS acuity as a function of study length that was mainly confined to the business students. The results provide partial support for the hypothesis that education in mathematics can enhance the ANS acuity.

Enumeration of small and large numerosities in adolescents with mathematical learning disorders

The accuracy and speed in an enumeration task were investigated in adolescents with typical and atypically poor development of arithmetic skills. The number naming performances on small and large non-symbolic numerosities of 18 adolescents with mathematical learning disorders (MLD) and 28 typically achieving age-matched (TA) adolescents were compared. A mixed logistic regression model showed that adolescents with MLD were not significantly less accurate on numbers within the subitizing range than control peers. Moreover, no significant differences in reaction times were found between both groups. Nevertheless, we found that within the control group adolescents with higher ability tended to respond faster when taking into account the whole range (1-9) of numerosities. This correlation was much weaker in the MLD group. When looking more closely at the data, however, it became clear that the correlation between accuracy and speed within the control group differed in direction dependent on the range (subitizing or counting) of the numerosities. As such, our findings did not support a limited capacity of subitizing in MLD. However, the data stressed a different correlation between speed and accuracy for both groups of adolescents and a different behavioral pattern depending on the numerosity range as well. Implications for the understanding and approach of MLD are considered.

Spatial bias in symbolic and non-symbolic numerical comparison in neglect

When asked to bisect mentally numerical intervals, neglect patients show a displacement of the numerical midpoint similar to the one observed in physical line bisection. This spatial-numerical bias has been taken as evidence of the spatial nature of numerical magnitude representations. However, to date, neuropsychological studies in neglect patients have only used symbolic numerical material. Here, we compare the results of patients with right-hemisphere damage with and without unilateral left neglect and age-matched healthy control participants in two numerical comparison tasks using symbolic and non-symbolic materials, in order to determine whether the representation of non-symbolic numerosities was altered or not by the presence of neglect. When asked to judge if an Arabic digit or a sequence of flashed dots was smaller or larger than a reference value (i.e., 5), the responses of neglect patients to smaller magnitudes (i.e., 4) were impaired. Moreover, only neglect patients presented an asymmetrical distance effect (i.e., an enhanced effect only for stimuli of smaller numerical magnitude than the reference). These results provide the first direct evidence of a spatial bias in non-symbolic numerosity in neglect patients, and support the existence of common processing mechanisms and/or a representational system for symbolic and non-symbolic inputs.

Left parietal TMS disturbs priming between symbolic and non-symbolic number representations

An amodal number representation activated by all types of numerical input, irrespective of the input notation, has often been proposed to be located in the left or right intraparietal sulcus (IPS). Two cross-notational priming experiments were carried out to test the existence of a notation-independent magnitude representation in the left or right parietal lobes. In Experiment 1, stimuli were Arabic digits and number words. Results revealed no significant effect of repetitive transcranial magnetic stimulation (rTMS) over left or right IPS during prime presentation. In contrast, in Experiment 2, digits and dot patterns were intermixed and here the priming distance effect (PDE) was reduced in the right TMS condition and absent for stimulation over left IPS. These findings suggest: (1) that TMS over left but not right IPS disrupts processes that are crucial for priming when symbolic and non-symbolic stimuli are intermixed, and (2) that disruption of the left IPS on its own is not sufficient to disrupt cross-notational priming when purely symbolic number notations are used. Our results point towards a crucial role of the left hemisphere for the mapping between small symbolic and non-symbolic numerosities.

Developmental dyscalculia and low numeracy in Chinese children

Children struggle with mathematics for different reasons. Developmental dyscalculia and low numeracy - two kinds of mathematical difficulties - may have their roots, respectively, in poor understanding of exact non-symbolic numerosities and of symbolic numerals. This study was the first to explore whether Chinese children, despite cultural and linguistic factors supporting their mathematical learning, also showed such mathematical difficulties and whether such difficulties have measurable impact on children's early school mathematical performance. First-graders, classified as dyscalculia, low numeracy, or normal achievement, were compared for their performance in various school mathematical tasks requiring a grasp of non-symbolic numerosities (i.e., non-symbolic tasks) or an understanding of symbolic numerals (i.e., symbolic tasks). Children with dyscalculia showed poorer performance than their peers in non-symbolic tasks but not symbolic ones, whereas those with low numeracy showed poorer performance in symbolic tasks but not non-symbolic ones. As hypothesized, these findings suggested that dyscalculia and low numeracy were distinct deficits and caused by deficits in non-symbolic and symbolic processing, respectively. These findings went beyond prior research that only documented generally low mathematical achievements for these two groups of children. Moreover, these deficits appeared to be persistent and could not be remedied simply through day-to-day school mathematical learning. The present findings highlighted the importance of tailoring early learning support for children with these distinct deficits, and pointed to future directions for the screening of such mathematical difficulties among Chinese children.

Measuring acuity of the approximate number system reliably and validly: the evaluation of an adaptive test procedure

Two studies investigated the reliability and predictive validity of commonly used measures and models of Approximate Number System acuity (ANS). Study 1 investigated reliability by both an empirical approach and a simulation of maximum obtainable reliability under ideal conditions. Results showed that common measures of the Weber fraction (w) are reliable only when using a substantial number of trials, even under ideal conditions. Study 2 compared different purported measures of ANS acuity as for convergent and predictive validity in a within-subjects design and evaluated an adaptive test using the ZEST algorithm. Results showed that the adaptive measure can reduce the number of trials needed to reach acceptable reliability. Only direct tests with non-symbolic numerosity discriminations of stimuli presented simultaneously were related to arithmetic fluency. This correlation remained when controlling for general cognitive ability and perceptual speed. Further, the purported indirect measure of ANS acuity in terms of the Numeric Distance Effect (NDE) was not reliable and showed no sign of predictive validity. The non-symbolic NDE for reaction time was significantly related to direct w estimates in a direction contrary to the expected. Easier stimuli were found to be more reliable, but only harder (7:8 ratio) stimuli contributed to predictive validity.

The spatial–numerical congruity effect in preschoolers

Number-to-space mapping and its directionality are compelling topics in the study of numerical cognition. Usually, literacy and math education are thought to shape a left-to-right number line. We challenged this claim by analyzing performance of preliterate precounting preschoolers in a spatial–numerical task. In our experiment, children exhibited a spatial–numerical congruity (SNC) effect during a nonsymbolic numerosity comparison (quicker reaction times to smaller sets presented on the left side of the screen and to larger ones presented on the right side). These findings suggest that left-to-right number ordering may also have some sources that are independent of reading and math education. We argue that the current explanations of the spatial–numerical link need to be reconsidered.

Numerical and nonnumerical estimation in children with and without mathematical learning disabilities

There are currently multiple explanations for mathematical learning disabilities (MLD). The present study focused on those assuming that MLD are due to a basic numerical deficit affecting the ability to represent and to manipulate number magnitude (Butterworth, 1999, 2005; A. J. Wilson & Dehaene, 2007) and/or to access that number magnitude representation from numerical symbols (Rousselle & Noël, 2007). The present study provides an original contribution to this issue by testing MLD children (carefully selected on the basis of preserved abilities in other domains) on numerical estimation tasks with contrasting symbolic (Arabic numerals) and nonsymbolic (collection of dots) numbers used as input or output. MLD children performed consistently less accurately than control children on all the estimation tasks. However, MLD children were even weaker when the task involved the mapping between symbolic and nonsymbolic numbers than when the task required a mapping between two nonsymbolic numerical formats. Moreover, in the estimation of nonsymbolic numerosities, MLD children relied more than control children on perceptual cues such as the cumulative area of the dots. Finally, the task requiring a mapping from a nonsymbolic format to a symbolic format was the best predictor of MLD. In order to explain these present results, as well as those reported in the literature, we propose that the impoverished number magnitude representation of MLD children may arise from an initial mapping deficit between number symbols and that magnitude representation.

Mode-dependent and mode-independent representations of numerosity in the right intraparietal sulcus

In humans, areas around the intraparietal sulcus (IPS) have been found to play a crucial role in coding nonsymbolic numerosities (i.e., number of elements in a collection). In the parietal cortex of monkeys, some populations of neurons were found to respond selectively to sequentially- or simultaneously-presented numerosities, whereas other populations showed similar activation in both modes of presentation. However, whether such mode-dependent and -independent representations of numerosity also exist in humans is still unknown. Here, we used fMRI to identify the areas involved in numerosity processing while participants classified linear arrays of dots (simultaneous stimuli) or flashed dot sequences (sequential stimuli). The processing of simultaneous numerosities induced activations bilaterally in several areas of the IPS, whereas activations during the processing of sequential numerosities were restricted to the right hemisphere. A conjunction analysis showed that only the right IPS and precentral gyrus showed overlapping activations during the judgement of sequential and simultaneous stimuli. Voxelwise correlations confirmed the highly similar pattern of activation found in these regions during both tasks. This pattern was weaker or absent in mode-dependent regions, like the right inferior frontal cortex and the lateral occipital complex. Finally, a close look at the right IPS revealed an anterior-to-posterior gradient of activation with selective activation for sequential and simultaneous stimuli in the anterior and posterior areas, respectively, and overlapping activations in-between. This study provides the first direct evidence that, in humans, the right IPS contains both mode-dependent and mode-independent representations of numerosity.

Conflict processing of symbolic and non-symbolic numerosity

It is commonly assumed that the processing of magnitudes occurs independent of modality or notation. Several studies have reported similar behavioural as well as neurophysiological responses to magnitudes presented in distinct modalities as well as notations, but a direct assessment of possible interactions between different modalities and notations, using measures of electro-cortical processing, is lacking. The present study investigates whether the neural activity underlying symbolic and non-symbolic numerosity processing interacts with the neural activity underlying physical size processing before, or proceeds independently until, selective activation of the motor system. We used a symbolic (Arabic numbers) and non-symbolic (arrays of dots) size congruency task and instructed subjects to judge either the numerical or the physical size of the stimuli, while event related potentials were recorded. Longer reaction times as well as a decrease in accuracy were obtained for incongruent compared to congruent trials. For the event related potential data, this congruency effect was also found with respect to the latency of the P3 component reflecting an interaction at the level of stimulus evaluation. Moreover, incongruence delayed the stimulus-locked but not the response-locked lateralized readiness potential. Together these results suggest that, irrespective of notation, the interaction between different magnitudes occurs before selective response activation.

Brain Correlates of Non-Symbolic Numerosity Estimation in Low and High Mathematical Ability Children

Previous studies have implicated several brain areas as subserving numerical approximation. Most studies have examined brain correlates of adult numerical approximation and have not considered individual differences in mathematical ability. The present study examined non-symbolic numerical approximation in two groups of 10-year-olds: Children with low and high mathematical ability. The aims of this study were to investigate the brain mechanisms associated with approximate numerosity in children and to assess whether individual differences in mathematical ability are associated with differential brain correlates during the approximation task. The results suggest that, similarly to adults, multiple and distributed brain areas are involved in approximation in children. Despite equal behavioral performance, there were differences in the brain activation patterns between low and high mathematical ability groups during the approximation task. This suggests that individual differences in mathematical ability are reflected in differential brain response during approximation.

Cross-notational semantic priming between symbolic and nonsymbolic numerosity

Symbolic and nonsymbolic numerosities produce similar behavioural effects and activate the same brain areas. These results have usually been interpreted in terms of a common, notation-independent magnitude representation. However, semantic priming between symbolic and nonsymbolic inputs has been somehow elusive (e.g., Koechlin, Naccache, Block, & Dehaene, 1999). In Experiment 1, we looked at whether cross-notational semantic priming depends on exact numerical meaning. Dice faces and digits were mixed as prime and target. Semantic priming occurred when prime and target were in the same notation as much as when they were in different notation. In Experiment 2, we found cross-notation semantic priming even when the nonsymbolic numerosity was presented as a set of random dots. Priming, however, occurred only from sets of dots to digit, not vice versa. These data support the computational model recently proposed by Verguts and Fias (2004; Verguts, Fias, & Stevens, 2005).

Automatic quantity processing in 5-year olds and adults

In this study adults performed numerical and physical size judgments on a symbolic (Arabic numerals) and non-symbolic (groups of dots) size congruity task. The outcomes would reveal whether a size congruity effect (SCE) can be obtained irrespective of notation. Subsequently, 5-year-old children performed a physical size judgment on both tasks. The outcomes will give a better insight in the ability of 5-year-olds to automatically process symbolic and non-symbolic numerosities. Adult performance on the symbolic and non-symbolic size congruity tasks revealed a SCE for numerical and physical size judgments, indicating that the non-symbolic size congruity task is a valid indicator for automatic processing of non-symbolic numerosities. Physical size judgments on both tasks by children revealed a SCE only for non-symbolic notation, indicating that the lack of a symbolic SCE is not related to the mathematical or cognitive abilities required for the task but instead to an immature association between the number symbol and its meaning.