Non-symbolic Numerosities Research Articles

Symbolic and non-symbolic representations of numerical zero in the human brain

Representing the quantity zero as a symbolic concept is considered a unique achievement of abstract human thought.1,2 To conceptualize zero, one must abstract away from the (absence of) sensory evidence to construct a representation of numerical absence: creating "something" out of "nothing."2,3,4 Previous investigations of the neural representation of natural numbers reveal distinct numerosity-selective neural populations that overlap in their tuning curves with adjacent numerosities.5,6 Importantly, a component of this neural code is thought to be invariant across non-symbolic and symbolic numerical formats.7,8,9,10,11 Although behavioral evidence indicates that zero occupies a place at the beginning of this mental number line,12,13,14 in humans zero is also associated with unique behavioral and developmental profiles compared to natural numbers,4,15,16,17 suggestive of a distinct neural basis for zero. We characterized the neural representation of zero in the human brain by employing two qualitatively different numerical tasks18,19 in concert with magnetoencephalography (MEG) recordings. We assay both neural representations of non-symbolic numerosities (dot patterns), including zero (empty sets), and symbolic numerals, including symbolic zero. Our results reveal that neural representations of zero are situated along a graded neural number line shared with other natural numbers. Notably, symbolic representations of zero generalized to predict non-symbolic empty sets. We go on to localize abstract representations of numerical zero to posterior association cortex, extending the purview of parietal cortex in human numerical cognition to encompass representations of zero.10,20.

Young children intuitively organize numbers on straight, horizontal lines from left to right before the onset of formal instruction.

The number line estimation task is frequently used to measure children's numerical magnitude understanding. It is unclear whether the resulting straight, horizontal, left-to-right-oriented estimate patterns indicate task constraints or children's intuitive number-space mapping. Three- to six-year-old children (N = 72, Mage = 4.89, 56% girls, 94% German citizenship) were asked to explain the meaning of numbers to a teddy by laying out a rope and attaching cards showing non-symbolic numerosities (dots) to it. Most children intuitively created straight, horizontal, and left-to-right-oriented representations. Characteristics of the line correlated with age, mathematical competencies, and home numeracy. This demonstrates the usefulness of the number line estimation task for assessing how children intuitively map numbers onto space.

Nonsymbolic Numerosity Maps at the Occipitotemporal Cortex Respond to Symbolic Numbers.

Numerosity, the set size of a group of items, helps guide human and animals' behavior and decisions. Numerosity perception is thought to be a precursor of symbolic numerical cognition. Previously, we uncovered neural populations selectively tuned to numerosities organized in a network of topographic maps in human association cortex. Here we investigate whether these numerosity maps are also involved in the processing of symbolic numbers, using 7T fMRI and a number-detection task. We recruited 7 participants (3 females) and found that the numerosity map at the temporal-occipital cortex (NTO) also responds to symbolic numbers. Furthermore, we found that numerosity-tuned neuronal populations at the NTO map in the left hemisphere are tuned to symbolic numbers. These results reveal different functions of the numerosity maps and support a link between numerosity representation and symbolic number processing in the ventral temporal-occipital cortex.SIGNIFICANCE STATEMENT Humans and other animals share an intuitive "number sense" to approximately represent numerosity. However, humans possess a unique ability to process number symbols (e.g., Arabic numbers). It has been argued that the human understanding of symbolic numbers is rooted in our ability to numerosity perception. Here we investigate whether numerosity-tuned neuronal populations organized at a network of topographic maps also respond to symbolic numbers. We find one of the maps at the temporal-occipital cortex is involved in symbolic numerical cognition and the neuronal populations are tuned to numbers. These results provide evidence for a link between nonsymbolic numerosity and symbolic number processing.

Overlapping but separate number representations in the intraparietal sulcus—Probing format- and modality-independence in sighted Braille readers

The Triple-Code Model stipulates that numerical information from different formats and modalities converges on a common magnitude representation in the Intraparietal Sulcus (IPS). To what extent the representations of all numerosity forms overlap remains unsolved. It has been postulated that the representation of symbolic numerosities (for example, Arabic digits) is sparser and grounded in an existing representation that codes for non-symbolic numerosity information (i.e., sets of objects). Other theories argue that numerical symbols represent a separate number category that emerges only during education. Here, we tested a unique group of sighted tactile Braille readers with numerosities 2, 4, 6 and 8 in three number notations: Arabic digits, sets of dots, tactile Braille numbers. Using univariate methods, we showed a consistent overlap in activations evoked by these three number notations. This result shows that all three used notations are represented in the IPS, which may suggest at least a partial overlap between the representations of the three notations used in this experiment. Using MVPA, we found that only non-automatized number information (Braille and sets of dots) allowed successful number classification. However, the numerosity of one notation could not be predicted above chance from the brain activation patterns evoked by another notation (no cross-classification). These results show that the IPS may host independent number codes in overlapping cortical circuits. In addition, they suggest that the level of training in encoding a given type of number information is an important factor that determines the amount of exploitable information and needs to be controlled for in order to identify the neural code underlying numerical information per se.

Acquisition of new arithmetic skills based on prior arithmetic skills: A cross-sectional study in primary school from grade 2 to grade 5.

In several countries, children's math skills have been declining at an alarming rate in recent years and decades, and one of the explanations for this alarming situation is that children have difficulties in establishing the relations between arithmetical operations. In order to address this question, our goal was to determine the predictive power of previously taught operations on newly taught ones above general cognitive skills and basic numerical skills. More than one hundred children in each school level from Grades 2 to 5 from various socio-cultural environments (N= 435, 229 girls) were tested. Children were assessed on their abilities to solve the four basic arithmetic operations. They were also tested on their general cognitive abilities, including working memory, executive functions (i.e., inhibition and flexibility), visual attention and language. Finally, their basic numerical skills were measured through a matching task between symbolic and nonsymbolic numerosity representations. Additions and subtractions were presented to children from Grade 2, multiplications from Grade 3 and divisions from Grade 4. We show that addition predicts subtraction and multiplication performance in all grades. Moreover, multiplication predicts division performance in both Grades 4 and 5. Finally, addition predicts division in Grade 4 but not in Grade 5 and subtraction and division are not related whatever the school grade. These results are examined considering the existing literature, and their implications in terms of instruction are discussed.

The database for extracting numerical and visual properties of numerosity processing in the Chinese population

The ability to handle non-symbolic numerosity has been recurrently linked to mathematical abilities. The accumulated data provide a rich resource that can reflect the underlying properties (i.e., dot ratio, area, convex hull, perimeters, distance, and hash) of numerosity processing. This article reports a database of numerosity processing in the Chinese population. The database contains five independent datasets with 7459, 4902, 415, 671, 414 participants respectively. For each dataset, all data were collected in the same online computerized test, examination room, professorial tester, and using the same protocols. Computational modeling method could be used to extract the dot ratio and visual properties of numerosity from five types of dot stimuli. This database enables researchers to test the theoretical hypotheses regarding numerosity processing using a large sample population. The database can also indicate the individual difference of non-symbolic numerosity in mathematical abilities.

Recognition ability of untrained neural networks to symbolic numbers.

Although animals can learn to use abstract numbers to represent the number of items, whether untrained animals could distinguish between different abstract numbers is not clear. A two-layer spiking neural network with lateral inhibition was built from the perspective of biological interpretability. The network connection weight was set randomly without adjustment. On the basis of this model, experiments were carried out on the symbolic number dataset MNIST and non-symbolic numerosity dataset. Results showed that the model has abilities to distinguish symbolic numbers. However, compared with number sense, tuning curves of symbolic numbers could not reproduce size and distance effects. The preference distribution also could not show high distribution characteristics at both ends and low distribution characteristics in the middle. More than half of the network units prefer the symbolic numbers 0 and 5. The average goodness-of-fit of the Gaussian fitting of tuning curves increases with the increase in abscissa non-linearity. These results revealed that the concept of human symbolic number is trained on the basis of number sense.

The Association between Approximate Number Sense (ANS) and Math Achievement Depends on the Format of the ANS Test

ABSTRACT There are two lines of discussion regarding the function of the Approximate Number System (ANS). The first line focuses on the extent to which visual cues affect the estimation of numerosity. The second line investigates the extent to which ANS precision is associated with symbolic math performance. The current study combined these two lines of discussion. Data from the nonsymbolic comparison test and the math achievement of 313 first graders (mean age: 7.6 years; 51% girls) from Russia were analyzed. First, we estimated the extent to which the effect of nonnumerical visual cues on nonsymbolic numerosity estimation varied across the four formats of stimulus presentation: separated/homogenous, separated/heterogeneous, mixed/homogeneous and mixed/heterogeneous. The results revealed that in the mixed/heterogeneous format of stimulus presentation, the congruency effect was not significant with respect to accuracy and was negative with respect to reaction time (RT). The reduction in the congruency effect may indicate that in this format, the participants tended to directly estimate numerosity, ignoring visual cues. Second, we tested the association between math achievement and precision on the ANS test in each format. The results demonstrated that accuracy in the separated/homogenous format had no significant association with math performance, while accuracy in other formats had significant associations with math performance. Moreover, accuracy in the mixed/heterogeneous format had a larger association than that found in other formats. This result might indicate that ANS accuracy has a significant association with math achievement in formats that demonstrate a smaller congruency effect and greater involvement of the direct estimation of numerosity.

Circling around number: People can accurately extract numeric values from circle area ratios.

It has long been known that people have the ability to estimate numerical quantities without counting. A standard account is that people develop a sense of the size of symbolic numbers by learning to map symbolic numbers (e.g., 6) to their corresponding numerosities (e.g. :::) and concomitant approximate magnitude system (ANS) representations. However, we here demonstrate that adults are capable of extracting fractional numerical quantities from non-symbolic visual ratios (i.e., labeling a ratio of two circle areas with the appropriate symbolic fraction). Not only were adult participants able to perform this task, but they were remarkably accurate: linear regressions on median estimates yielded slopes near 1, and accounted for 97% of the variability. Participants also performed at least as well on line-estimation and ratio-estimation tasks using non-numeric circular stimuli as they did in earlier experiments using non-symbolic numerosities, which are frequently considered to be numeric stimuli. We discuss results as consistent with accounts suggesting that non-symbolic ratios have the potential to act as a reliable and stable ground for symbolic number, even when composed of non-numeric stimuli.

The Sense of Number in Fish, with Particular Reference to Its Neurobiological Bases.

Simple SummaryThe ability to deal with quantity, both discrete (numerosities) and continuous (spatial or temporal extent) developed from an evolutionarily conserved system for approximating numerical magnitude. Non-symbolic number cognition based on an approximate sense of magnitude has been documented in a variety of vertebrate species, including fish. Fish, in particular zebrafish, are widely used as models for the investigation of the genetics and molecular mechanisms of behavior, and thus may be instrumental to development of a neurobiology of number cognition. We review here the behavioural studies that have permitted to identify numerical abilities in fish, and the current status of the research related to the neurobiological bases of these abilities with special reference to zebrafish. Combining behavioural tasks with molecular genetics, molecular biology and confocal microscopy, a role of the retina and optic tectum in the encoding of continuous magnitude in larval zebrafish has been reported, while the thalamus and the dorso-central subdivision of pallium in the encoding of discrete magnitude (number) has been documented in adult zebrafish. Research in fish, in particular zebrafish, may reveal instrumental for identifying and characterizing the molecular signature of neurons involved in quantity discrimination processes of all vertebrates, including humans.It is widely acknowledged that vertebrates can discriminate non-symbolic numerosity using an evolutionarily conserved system dubbed Approximate Number System (ANS). Two main approaches have been used to assess behaviourally numerosity in fish: spontaneous choice tests and operant training procedures. In the first, animals spontaneously choose between sets of biologically-relevant stimuli (e.g., conspecifics, food) differing in quantities (smaller or larger). In the second, animals are trained to associate a numerosity with a reward. Although the ability of fish to discriminate numerosity has been widely documented with these methods, the molecular bases of quantities estimation and ANS are largely unknown. Recently, we combined behavioral tasks with molecular biology assays (e.g c-fos and egr1 and other early genes expression) showing that the thalamus and the caudal region of dorso-central part of the telencephalon seem to be activated upon change in numerousness in visual stimuli. In contrast, the retina and the optic tectum mainly responded to changes in continuous magnitude such as stimulus size. We here provide a review and synthesis of these findings.

Nonsymbolic numerosity in sets with illusory-contours exploits a context-sensitive, but contrast-insensitive, visual boundary formation process

The visual mechanisms underlying approximate numerical representation are still intensely debated because numerosity information is often confounded with continuous sensory cues (e.g., texture density, area, convex hull). However, numerosity is underestimated when a few items are connected by illusory contours (ICs) lines without changing other physical cues, suggesting in turn that numerosity processing may rely on discrete visual input. Yet, in these previous works, ICs were generated by black-on-gray inducers producing an illusory brightness enhancement, which could represent a further continuous sensory confound. To rule out this possibility, we tested participants in a numerical discrimination task in which we manipulated the alignment of 0, 2, or 4 pairs of open/closed inducers and their contrast polarity. In Experiment 1, aligned open inducers had only one polarity (all black or all white) generating ICs lines brighter or darker than the gray background. In Experiment 2, open inducers had always opposite contrast polarity (one black and one white inducer) generating ICs without strong brightness enhancement. In Experiment 3, reverse-contrast inducers were aligned but closed with a line preventing ICs completion. Results showed that underestimation triggered by ICs lines was independent of inducer contrast polarity in both Experiment 1 and Experiment 2, whereas no underestimation was found in Experiment 3. Taken together, these results suggest that mere brightness enhancement is not the primary cause of the numerosity underestimation induced by ICs lines. Rather, a boundary formation mechanism insensitive to contrast polarity may drive the effect, providing further support to the idea that numerosity processing exploits discrete inputs.

Number is not just an illusion: Discrete numerosity is encoded independently from perceived size

While seminal theories suggest that nonsymbolic visual numerosity is mainly extracted from segmented items, more recent views advocate that numerosity cannot be processed independently of nonnumeric continuous features confounded with the numerical set (i.e., such as the density, the convex hull, etc.). To disentangle these accounts, here we employed two different visual illusions presented in isolation or in a merged condition (e.g., combining the effects of the two illusions). In particular, in a number comparison task, we concurrently manipulated both the perceived object segmentation by connecting items with Kanizsa-like illusory lines, and the perceived convex-hull/density of the set by embedding the stimuli in a Ponzo illusion context, keeping constant other low-level features. In Experiment 1, the two illusions were manipulated in a compatible direction (i.e., both triggering numerical underestimation), whereas in Experiment 2 they were manipulated in an incompatible direction (i.e., with the Ponzo illusion triggering numerical overestimation and the Kanizsa illusion numerical underestimation). Results from psychometric functions showed that, in the merged condition, the biases of each illusion summated (i.e., largest underestimation as compared with the conditions in which illusions were presented in isolation) in Experiment 1, while they averaged and competed against each other in Experiment 2. These findings suggest that discrete nonsymbolic numerosity can be extracted independently from continuous magnitudes. They also point to the need of more comprehensive theoretical views accounting for the operations by which both discrete elements and continuous variables are computed and integrated by the visual system.

Perceiving numerosity does not cause automatic shifts of spatial attention

It is debated whether the representation of numbers is endowed with a directional-spatial component so that perceiving small-magnitude numbers triggers leftward shifts of attention and perceiving large-magnitude numbers rightward shifts. Contrary to initial findings, recent investigations have demonstrated that centrally presented small-magnitude and large-magnitude Arabic numbers do not cause leftward and rightward shifts of attention, respectively. Here we verified whether perceiving small or large non-symbolic numerosities (i.e., clouds of dots) drives attention to the left or the right side of space, respectively. In experiment 1, participants were presented with central small (1, 2) vs large-numerosity (8, 9) clouds of dots followed by an imperative target in the left or right side of space. In experiment 2, a central cloud of dots (i.e., five dots) was followed by the simultaneous presentation of two identical dot-clouds, one on the left and one on the right side of space. Lateral clouds were both lower (1, 2) or higher in numerosity (8, 9) than the central cloud. After a variable delay, one of the two lateral clouds turned red and participants had to signal the colour change through a unimanual response. We found that (a) in Experiment 1, the small vs large numerosity of the central cloud of dots did not speed up the detection of left vs right targets, respectively, (b) in Experiment 2, the detection of colour change was not faster in the left side of space when lateral clouds were smaller in numerosity than the central reference and in the right side when clouds were larger in numerosity. These findings show that perceiving non-symbolic numerosity does not cause automatic shifts of spatial attention and suggests no inherent association between the representation of numerosity and that of directional space.

Children With Dyscalculia Show Hippocampal Hyperactivity During Symbolic Number Perception.

Dyscalculia is a learning disability affecting the acquisition of arithmetical skills in children with normal intelligence and age-appropriate education. Two hypotheses attempt to explain the main cause of dyscalculia. The first hypothesis suggests that a problem with the core mechanisms of perceiving (non-symbolic) quantities is the cause of dyscalculia (core deficit hypothesis), while the alternative hypothesis suggests that dyscalculics have problems only with the processing of numerical symbols (access deficit hypothesis). In the present study, the symbolic and non-symbolic numerosity processing of typically developing children and children with dyscalculia were examined with functional magnetic resonance imaging (fMRI). Control (n = 15, mean age: 11.26) and dyscalculia (n = 12, mean age: 11.25) groups were determined using a wide-scale screening process. Participants performed a quantity comparison paradigm in the fMRI with two number conditions (dot and symbol comparison) and two difficulty levels (0.5 and 0.7 ratio). The results showed that the bilateral intraparietal sulcus (IPS), left dorsolateral prefrontal cortex (DLPFC) and left fusiform gyrus (so-called “number form area”) were activated for number perception as well as bilateral occipital and supplementary motor areas. The task difficulty engaged bilateral insular cortex, anterior cingulate cortex, IPS, and DLPFC activation. The dyscalculia group showed more activation in the left orbitofrontal cortex, left medial prefrontal cortex, and right anterior cingulate cortex than the control group. The dyscalculia group showed left hippocampus activation specifically for the symbolic condition. Increased left hippocampal and left-lateralized frontal network activation suggest increased executive and memory-based compensation mechanisms during symbolic processing for dyscalculics. Overall, our findings support the access deficit hypothesis as a neural basis for dyscalculia.

Non-Symbolic Numerosity and Symbolic Numbers are not Processed Intuitively in Children: Evidence From an Event-Related Potential Study

The approximate number system (ANS) theory and the ANS mapping account have been the most prominent theories on non-symbolic numerosity processing and symbolic number processing respectively, over the last 20 years. Recently, there is a growing debate about these theories, mainly based on research in adults. However, whether the ANS theory and ANS mapping account explain the processing of non-symbolic numerosity and symbolic number in childhood has received little attention. In the current ERP study, we first examined whether non-symbolic numerosity processing in 9-to-12-year-old children (N = 34) is intuitive, as proposed by the ANS theory. Second, we examined whether symbolic number processing is rooted in non-symbolic numerosity processing, as proposed the ANS mapping account. ERPs were measured during four same-different match-to-sample tasks with non-symbolic numerosities, symbolic numbers, and combinations of both. We found no evidence for intuitive processing of non-symbolic numerosity. Instead, children processed the visual features of non-symbolic stimuli more automatically than the numerosity itself. Moreover, children do not seem to automatically activate non-symbolic numerosity when processing symbolic numbers. These results challenge the ANS theory and ANS mapping account in 9-to-12-year-old children.

Topographic numerosity maps cover subitizing and estimation ranges

Numerosity, the set size of a group of items, helps guide behaviour and decisions. Non-symbolic numerosities are represented by the approximate number system. However, distinct behavioural performance suggests that small numerosities, i.e. subitizing range, are implemented differently in the brain than larger numerosities. Prior work has shown that neural populations selectively responding (i.e. hemodynamic responses) to small numerosities are organized into a network of topographical maps. Here, we investigate how neural populations respond to large numerosities, well into the ANS. Using 7 T fMRI and biologically-inspired analyses, we found a network of neural populations tuned to both small and large numerosities organized within the same topographic maps. These results demonstrate a continuum of numerosity preferences that progressively cover both the subitizing range and beyond within the same numerosity map, suggesting a single neural mechanism. We hypothesize that differences in map properties, such as cortical magnification and tuning width, underlie known differences in behaviour.

Exploring basic numerical capacities in children with difficulties in simple arithmetical achievement

Introduction: Current cognitive theories suggest that mathematical learning disabilities may be caused by a dysfunction in the ability to represent non-symbolic numerosity (non-symbolic skills), by impairments in the ability to associate symbolic numerical representations with the underlying analogic non-numerical magnitude representation (symbolic and numerical mapping skills), or by a combination of both deficits. The aim of this study was to contrast the number sense hypothesis and the access deficit hypothesis, to identify the possible origin of the varying degrees of arithmetical difficulties. Method: We compared the performance of children with very low arithmetic achievement (VLA), children with low arithmetical achievement (LA), and typically achieving peers (TA), in non-symbolic, symbolic and numerical mapping tasks. Intellectual capacity and working memory were also evaluated as control variables. The sample comprised 85 Chilean children (3rd to 6th grades) from the Public General Education System. Data were included in several covariance analyses to identify potentially different behavioural profiles between groups. Results: The results showed deficits in both non-symbolic numerosity processing and number-magnitude mapping skills in children with VLA, whereas children with LA exhibited deficits in numerical mapping tasks only. Conclusions: These findings support the hypothesis of impaired non-symbolic numerical representations as the cognitive foundation of severe arithmetical difficulties. Low arithmetical achievement, in contrast, seems to be better explained by defective numerical mapping skills, which fits the access deficit hypothesis. The results presented here provide new evidence regarding the cognitive mechanisms underlying the different behavioural profiles identified in children with varying degrees of arithmetical difficulties.

Nonsymbolic and symbolic representations of null numerosity.

Previous research has shown that null numerosity can be processed as a numerical entity that is represented together with non-null numerosities on the same magnitude system. The present study examined which conditions enable perceiving nonsymbolic (i.e., an empty set) and symbolic (i.e., 0) representations of null numerosity as a numerical entity, using distance and end effects. In Experiment 1, participants performed magnitude comparisons of notation homogeneous pairs (both numerosities appeared in nonsymbolic or symbolic format), as well as heterogeneous pairs (a nonsymbolic numerosity versus a symbolic one). Comparisons to 0 resulted in faster responses and an attenuated distance effect in all conditions, whereas comparisons to an empty set produced such effects only in the nonsymbolic and symbolic homogeneous conditions. In Experiments 2 and 3, participants performed same/different numerosity judgments with heterogeneous pairs. A distance effect emerged for "different" judgments of 0 and sets of 1 to 9 dots, but not for those with an empty set versus digits 1-9. These findings indicate that perceiving an empty set, but not 0, as a numerical entity is determined by notation homogeneity and task requirements.

The ratio effect in visual numerosity comparisons is preserved despite spatial frequency equalisation

How non-symbolic numerosity is visually extracted remains a matter of intense debate. Most evidence suggests that numerosity is directly extracted on individual objects following Weber’s law, at least for a moderate numerical range. Alternative accounts propose that, whatever the range, numerosity is indirectly derived from summary texture-statistics of the raw image such as spatial frequency (SF). Here, to disentangle these accounts, we tested whether the well-known behavioural signature of numerosity encoding (ratio effect) is preserved despite the equalisation of the SF content. In Experiment 1, participants had to select the numerically larger of two briefly presented moderate-range numerical sets (i.e., 8–18 dots) carefully matched for SF; the ratio between numerosities was manipulated by levels of increasing difficulty (e.g., 0.66, 0.75, 0.8). In Experiment 2, participants performed the same task, but they were presented with both the original and SF equalised stimuli. In both experiments, the results clearly showed a ratio-dependence of the performance: numerosity discrimination became harder and slower as the ratio between numerosities increased. Moreover, this effect was found to be independent of the stimulus type, although the overall performance was better with the original rather than the SF equalised stimuli (Experiment 2). Taken together, these findings indicate that the power spectrum per se cannot explain the main behavioural signature of Weber-like encoding of numerosities (the ratio effect), at least over the tested numerical range, partially challenging alternative indirect accounts of numerosity processing.

A Sensorimotor Numerosity System.

Incoming sensory input provides information for the planning and execution of actions, which yield motor outcomes that are themselves sensory inputs. One dimension where action and perception strongly interact is numerosity perception. Many non-human animals can estimate approximately the number of external elements as well as their own actions, and neurons have been identified that respond to both. Recent psychophysical adaptation studies on humans also provide evidence for neural mechanisms responding to both the number of externally generated events and self-produced actions. Here we advance the idea that these strong connections may arise from dedicated sensorimotor mechanisms in the brain, part of a more generalized system interfacing action with the processing of other quantitative magnitudes such as space and time.