Academic literature on the topic 'Numerosity perception'

Background It has been proposed that numerosity perception is the cognitive underpinning of mathematics ability. However, the existence of the association between numerosity perception and mathematics ability is still under debate, especially in adults. The present study examined the relationship between numerosity perception and mathematics ability and the moderating role of dots number (i.e., the numerosity of items in dot set) in adults. Methods Sixty-four adult participants from Anshun University completed behavioral measures that tested numerosity perception of small numbers and large numbers, mathematics ability, inhibition ability, visual-spatial memory, and set-switching ability. Results We found that numerosity perception of small numbers correlated significantly with mathematics ability after controlling the influence of inhibition ability, visual-spatial memory, and set-switching ability, but numerosity perception of large numbers was not related to mathematics ability in adults. Conclusions These findings suggest that the dots number moderates the relationship between numerosity perception and mathematics ability in adults and may contribute to explaining the contradictory findings in the previous literature about the link between numerosity perception and mathematics ability.

What is a number? The number sense hypothesis suggests that numerosity is “a primary visual property” like color, contrast, or orientation. However, exactly what attribute of a stimulus is the primary visual property and determines numbers in the number sense? To verify the invariant nature of numerosity perception, we manipulated the numbers of items connected/enclosed in arbitrary and irregular forms while controlling for low-level features (e.g., orientation, color, and size). Subjects performed discrimination, estimation, and equality judgment tasks in a wide range of presentation durations and across small and large numbers. Results consistently show that connecting/enclosing items led to robust numerosity underestimation, with the extent of underestimation increasing monotonically with the number of connected/enclosed items. In contrast, grouping based on color similarity had no effect on numerosity judgment. We propose that numbers or the primitive units counted in numerosity perception are influenced by topological invariants, such as connectivity and the inside/outside relationship. Beyond the behavioral measures, neural tuning curves to numerosity in the intraparietal sulcus were obtained using functional MRI adaptation, and the tuning curves showed that numbers represented in the intraparietal sulcus were strongly influenced by topology.

There is strong evidence that humans can make rough estimates of the numerosity of a set of items, almost from birth. However, as numerosity covaries with many non-numerical variables, the idea of a direct number sense has been challenged. Here we applied two different psychophysical paradigms to demonstrate the spontaneous perception of numerosity in a cohort of young pre-school children. The results of both tasks showed that even at that early developmental stage, humans spontaneously base the perceptual choice on numerosity, rather than on area or density. Precision in one of these tasks predicted mathematical abilities. The results reinforce strongly the idea of a primary number sense and provide further evidence linking mathematical skills to the sensory precision of the spontaneous number sense, rather than to mechanisms involved in handling explicit numerosity judgements or extensive exposure to mathematical teaching.

Magnitude information is essential to create a representation of the external environment and successfully interact with it. Duration and numerosity, for example, can shape our predictions and bias each other (i.e. the greater the number of people queuing, the longer we expect to wait). While these biases suggest the existence of a generalized magnitude system, asymmetric effects (i.e. numerosity affecting duration but not vice versa) challenged this idea. Here, we propose that such asymmetric integration depends on the stimuli used and the neural processing dynamics they entail. Across multiple behavioural experiments employing different stimulus presentation displays (static versus dynamic) and experimental manipulations known to bias numerosity and duration perceptions (i.e. connectedness and multisensory integration), we show that the integration between numerosity and time can be symmetrical if the stimuli entail a similar neural time-course and numerosity unfolds over time. Overall, these findings support the idea of a generalized magnitude system, but also highlight the role of early sensory processing in magnitude representation and integration.

Enumeration of a dot array is faster and easier if the items form recognizable subgroups. This phenomenon, which has been termed “groupitizing,” appears in children after one year of formal education and correlates with arithmetic abilities. We formulated and tested the hypothesis that groupitizing reflects an ability to sidestep counting by using arithmetic shortcuts, for instance, using the grouping structure to add or multiply rather than just count. Three groups of students with different levels of familiarity with mathematics were asked to name the numerosity of sets of 1–15 dots in various arrangements, for instance, 9 represented as a single group of 9 items, three distinct groups of 2, 3, and 4 items (affording addition 2 + 3 + 4), or three identical groups of 3 items (affording multiplication 3 × 3). Grouping systematically improved enumeration performance, regardless of whether the items were grouped spatially or by color alone, but only when an array was divided into subgroups with the same number of items. Response times and error patterns supported the hypothesis of a multiplication process. Our results demonstrate that even a simple enumeration task involves mental arithmetic.

Abstract Premature birth is associated with a high risk of damage in the parietal cortex, a key area for numerical and non-numerical magnitude perception and mathematical reasoning. Children born preterm have higher rates of learning difficulties for school mathematics. In this study, we investigated how preterm newborns (born at 28–34 weeks of gestation age) and full-term newborns respond to visual numerosity after habituation to auditory stimuli of different numerosities. The results show that the two groups have a similar preferential looking response to visual numerosity, both preferring the incongruent set after crossmodal habituation. These results suggest that the numerosity system is resistant to prematurity.

Numerosity is the set size of a group of items. Numerosity perception is a trait shared across numerous species. Numerosity-selective neural populations are thought to underlie numerosity perception. These neurons have been identified primarily using electrical recordings in animal models and blood oxygen level dependent (BOLD) functional magnetic resonance imaging (fMRI) in humans. Here we use electrical intracranial recordings to investigate numerosity tuning in humans, focusing on high-frequency transient activations. These recordings combine a high spatial and temporal resolution and can bridge the gap between animal models and human recordings. In line with previous studies, we find numerosity-tuned responses at parietal sites in two out of three participants. Neuronal populations at these locations did not respond to other visual stimuli, i.e. faces, houses, and letters, in contrast to several occipital sites. Our findings further corroborate the specificity of numerosity tuning of in parietal cortex, and further link fMRI results and electrophysiological recordings.

Illusion of numerosity can be observed in many of the classical illusions of linear extent by replacing the uninterrupted lines with rows of dots. Using the method of constant stimuli both length and numerosity illusions move in the same direction, whereas using a magnitude-estimation method the two illusions move in opposite directions. Two experiments show that this inversion occurs also in the Müller-Lyer illusion.

Although humans are the only species with a linguistically mediated code for numbers, we share a nonverbal representation of numerical quantities with many animals species. We posses a “visual sense of numbers” that allow us to roughly estimate the numerosity of objects present in the visual scene. The efficency of this capacity, has been demonstrated to be a strong predictor of school acquired formal math skills. Numerosity perception can been interpreted has a “start-up” tool for the development of formal math achievements. In this thesis, we studied the capacity to visually estimate numerosity by using psychophysical paradigms and we were able to uncover new and unexpected features of this ability. In the first part of the thesis we have shown that numerosity estimation is not a monolithic entity but is composed by several different and separable, although closely overlapped, sub-systems. In particular, our results clearly show that three different systems are engaged, governed by different psychophysical laws. In the second part of the thesis we dealt with the spatial nature of numbers. The ability to map numbers onto space evolves during development: preschool children as well as illetterate adults show a performance characterized by a compressed, simil-logarithmic shape, which is then linearized by increasing chronological age and formal math skills. The current literature interprets the logarithmic shape as a reflection of the 'innate shape’ (not yet culturally modified) of the numerical representation. Our studies show that instead, the logarithmic shape results from a different process based on a dynamic mechanisms, that interprets and combines the to be mapped stumuli. This phenomenon, by itself, produces a pattern of results well describable by a logarithmic function but logarithmic in its nature. In the final part of the thesis, in a sample of school-age children, we have dealt with the relationship between numerosity perception, visual attention and the development of formal mathematical skills. Through a correlational approach, we have shown that these functions are closely related. Importantly, we demonstrated that the proper functioning of the attentional system, plays a key role in the development of mathematical skills. These results have an important impact on the recent researches regarding children suffering from difficulty in learning arithmetic (developmental dyscalculia).

The present dissertation investigated visual perception of numerosity. In the first part I reviewed the prominent literature about the topic. In the second chapter I described the first experiment, in which I measured confidence and reaction times to study the origins of the well-established visual and motor adaptation effects on numerosity perception. The results reinforce the evidence for a shared mechanism that encodes the quantity of both internally and externally generated events, and shows that the adaptation effects result from changes in sensory encoding, rather than perceptual decisions. More generally, in the study was introduced a novel and useful technique for investigating the mechanisms of numerosity adaptation and sensory adaptation in general. The third chapter investigated the effects of grouping cues on sensory precision of numerosity estimation. The results provide strong evidence that “grouping”, which can improve performance by up to 20%, can be induced by color and/or spatial proximity and occurs in temporal sequences as well as spatial arrays. In the fourth chapter I further examined the groupitizing phenomenon, by testing the hypothesis that the advantage provided by clustering stimuli relies on subitizing. This was achieved by manipulating attention, which is known to strongly affect the subitizing system. In the same chapter I discussed an additional explorative analysis on the relationship between calculation skills and estimation precision of grouped and ungrouped arrays. Taken together, the results showed that groupitizing is truly an attention-based process that leverages on the subitizing system. Furthermore, the outcome of the study suggested that measuring numerosity estimation thresholds with grouped stimuli may be a sensitive correlate of math abilities. In the fifth chapter I went on investigating the neural correlates of the groupitizing phenomenon with both a behavioral and a fMRI study. Similarly to the previous study I measured acuity in estimation of grouped and ungrouped stimuli and additionally I also examined whether the two tasks shared or not the same neural substrate. The results showed that the estimation of grouped and ungrouped stimuli activates similar regions in the right lateralized fronto-parietal network, however, only the presentation of grouped stimuli in the numerosity task elicited the additional activation of regions linked with calculations strategies, for instance the angular gyrus. Moreover, a multivariate pattern analysis showed that parietal activation patterns for individual numerosities could be accurately decoded in the parietal regions independently of the spatial arrangement of the stimuli. Finally, I correlated fMRI decoding accuracy of primary visual areas and angular gyrus with Wfs calculated in the grouped estimation task. Results suggested that the numerical representation in angular gyrus, but not in primary visual areas, is strongly linked with numerical performance and behavior. Overall, the results confirmed psychophysical studies highlighting that groupitizing shares the same regions and neural pattern mechanism of the estimation of ungrouped stimuli, but, furthermore, it also activates brain regions typically activated during calculation tasks. The last part of the dissertation is dedicated to investigating the link between numerosity precision, math abilities and a non-cognitive factor affecting mathematical learning: mathematical anxiety. To this aim, university students with low (< 25th percentile) and high (> 75th percentile) score in the Abbreviate Math Anxiety Scale were tested in multiple domains: a) math proficiency assessed using a standardized test (Mathematics Prerequisite for Psychometrics), b) visuo-spatial attention capacity, measured via a Multiple Object Tracking task, and c) the sensory precision for non-numerical quantities. The results confirmed previous studies showing that math abilities and numerosity precision correlate in subjects with high math anxiety. Furthermore, neither precision in size-discrimination nor visuo-spatial attentional capacity were found to correlate with math capacities. However, within the group with high MA the data also revealed a relationship between numerosity precision and math anxiety, with math anxiety playing a key role in mediating the correlation between participants’ numerosity precision and their math achievement. Taken together, this last study suggests an interplay between extreme levels of MA and sensory precision in the processing of non-symbolic numerosity, giving further insight into the processes (and the variables affecting these processes) behind the acquisition of formal mathematical abilities. In conclusion, the present work assessed the ability to perceive non-symbolic quantities in adults while providing new experimental evidence suggesting its perceptual nature and its link with cognitive and affective factors.

Humans and other species are endowed with perceptual mechanisms dedicated to estimating approximate quantity, an ability that has been defined as a sense of number. Converging evidence gathered from neurophysiological, behavioural and imaging studies, support the idea that this number sense has a truly abstract nature, being capable of encoding the numerosity of any set of discrete elements, displayed simultaneously or sequentially, and across different sensory modalities (Nieder et al., 2006; Piazza et al., 2006; Burr & Ross, 2008). It has been shown that numerosity, like most other primary visual attributes, is highly susceptible to adaptation: visually inspecting for a few seconds a large number of items, simultaneously presented, results in the perceived numerosity of a subsequent ensemble to be strongly underestimated, and vice-versa after adaptation to low numbers (Burr & Ross, 2008). Given that processing numerical information is also fundamental for the motor system to program sequences of self- movement, a further level of generalization of the number sense would be the possibility that a shared numerical representation exists between action and perception – that is, according to this view, the number sense would be generalized across presentation formats, sensory modalities, and perceptual and motor domains. In this work, we investigate numerosity perception within this theoretical framework. The first study was designed to investigate the perception of numerosity for stimuli presented sequentially by using an adaptation paradigm. This study tested whether, and to what extent, adaptation to a high or low number of events distorts the perceived numerosity of a subsequent sequence of visual events presented in the adapted location. In line with the typical dynamics of adaptation aftereffects, adapting to few events caused an overestimation of the perceived numerosity of the test stimuli, whilst adaptation to high-numerosity yielded a robustly underestimation. We further showed that adaptation effects transcend the sensory modality and presentation format: adapting to sequences of tones affected the perceived numerosity of a subsequently presented series of flashes (and vice versa), and adapting to sequences of flashes affected the perceived numerosity of spatial arrays of items. Similar results were obtained with tactile stimuli. Moreover, adaptation occurred only when test and adaptor positions were presented at the same location in spatiotopic (external world) coordinates, as demonstrated by introducing a saccadic eye movement between the offset of the adapting stimuli and the onset of the test stimuli (Arrighi et al., 2014). In the second part of this work, we present a subsequent work examining the possibility that the perceptual and the motor system might share a common numerical representation by using again the psychophysical technique of adaptation. In different sessions, we asked the subjects to produce either a fast (high number) or slow (low number) tapping routine. At the end of this adaptation phase subjects had to estimate the number of pulses presented sequentially, or of a cloud of dots simultaneously presented either on the same side where the motor actions were performed or on the opposite side. We found that motor adaptation strongly affected numerosity estimation of the test stimuli only when they were presented on the congruent side, with no effect when the visual stimuli were displayed on the neutral, not adapted, location. Moreover, to verify the robustness of the spatial selectivity, we repeated the experiment with a new subject pool, changing the tapping hand and location. Again, the spatial selectivity of the adaptation resulted to be in external – not hand-based – coordinates (Anobile, Arrighi et al., 2016). In the third part of this work we present another work where we evaluated the possibility that vision could drive the development of an external coordinate system for perceived numbers. In this study, congenitally blind (CB) and sighted controls (SC) were asked to evaluate the numerosity of sounds after performing either slow or fast motor adaptation (tapping), with the dominant hand, either in an uncrossed or in a crossed posture. Robust adaptation effects were observed in both groups of participants: an underestimation of the numerosity presented was observed after the execution of fast movements and an overestimation of the numerosity was observed after the execution of slow movements, in the crossed as well as in the uncrossed posture. Taken together, these results expand previous findings showing that adaptation to self-produced actions distorts perceived numerosity of sounds. Moreover, we demonstrate that visual experience is not necessary for the development of an external coordinate system for the shared numerical representation across action and perception. Finally, in the last part of this work, we examine the possibility of a common neural mechanism for different magnitude dimensions. Indeed, it has been recently proposed that space, time, and number might share a common representation in the human brain. For example, adaptation to visual motion affects both perceived position and duration of subsequent stimuli presented in the adapted location, suggesting that adaptation to visual motion distorts spatial maps as well as time processing (Johston et al. 2006, Burr et al., 2007; Fornaciai et al., 2016). In this study, we tested whether motion adaptation also affects perceived numerosity. Adaptation to fast translational motion yielded a significant reduction in the apparent numerosity of the adapted stimulus (of about 25%), while adaptation to slow translational or circular motion (both 20Hz and 5Hz) yielded a weaker but still significant compression of perceived numerosity. Our results generally support the idea of a common system for processing of space, time and number. However, as changes in perceived numerosity co-varied with both adapting motion profiles and speed, our evidence suggest a complex and asymmetric interactions between the representations of space, time and number in the brain. Taken together, the results obtained across these studies point to the existence of a generalized mechanism for numerical representation in the brain that is amodal, independent of the presentation format, shared between the perceptual and the motor systems, and based on external coordinate system.

There is considerable controversy as to how the brain extracts numerosity information from a visual scene and as to how much attention is needed for this process. Traditionally, it has been assumed that visual enumeration is subserved by two functionally distinct mechanisms: the fast and accurate apprehension of 1 to about 4 items, a process termed “subitizing”, and the slow and error-prone appraisal of larger numerosities referred to as “estimation”. Further to a functional dichotomy between these two mechanisms, an attentional dichotomy has been proposed. Subitizing has been thought of as a pre-attentive and parallel process, whereas estimation is supposed to require serial attention. In this thesis, the hypothesis of a parallel and pre-attentive subitizing mechanism was tested. In the first part of the thesis to this aim, the amount of attention that could be allocated to an Estimation task was experimentally manipulated. We shown that numerosity estimation is composed by different and separable, sub- systems. Results indicated that subitizing strongly depends on attentional resources, while estimation of larger quantities does not. Exactly the same results were found when the attentional resources dedicated to the visual numerical estimation task were limited on other sensory modalities: indeed visual, auditory and also haptic attentional load strongly and similarly impair visual subitizing but much less high numbers. We also demonstrated that visual adaptation to numerosity, absent in the subitizing range under normal condition, emerges under attentional load with a magnitude of the effect highly comparable to that measured for high numbers. Moreover we first demonstrate that the ability to accurately map numbers onto space also depends on attentional resources, showing that the assumption that performance on the ‘numberline task’ is the direct reflection of the internal numeric representation form could be misleading. In last part of the thesis we study how number adaptation affects number perception in two different population; high-functioning autistic and typically developing children. We demostrated that ASD children discriminated numerosity with the same precision as the typical children, but showed much less (about half) the levels of adaptation to number than the control group. These new results show that adaptation, processes, fundamental for efficient processing of variable sensory inputs, is diminished in autism.

“What is a number that a man may know it?” (Warren McCulloch). A wave model can determine “numerosity” (number of identical items) for small numbers of items. Identity and numerosity can be explained through similar mechanisms. Can there be a biology of number? Imaging studies find a topographic map of number magnitude in the human brain. Higher mathematics is based in part on refined perception. Classic mathematical philosophy—Platonism and formalism—may be usefully extended with perceptual components both learned and unlearned. Perceptual involvement suggests why mathematics is surprisingly good at dealing with the physical world. We find perceptual involvement even in simple integer multiplication. We can use “active” perceptual-based nets to program elementary abstract mathematical operations. A “brain-like” program is described for the “greater-than” program done by a digital computer in Chapter 4

In this study, we present a class of directed graphs with bounded degree sequences, which embodies the physical phenomenon of numerosity found in the collective behavior of large animal groups. Behavioral experiments show that an animal’s perception of number is capped by a critical limit, above which an individual perceives a nonspecific “many”. This species-dependent limit plays a pivotal role in the decision making process of large groups, such as fish schools and bird flocks. Here, we consider directed graphs whose edges model information-sharing between individual vertices. We incorporate the numerosity phenomenon as a critical limit on the intake of information by bounding the degree sequence and include the variability of cognitive processes by using a random variable in the network construction. We analytically compute measures of the expected structure of this class of graphs based on cycles, clustering, and sorting among vertices. Theoretical results are verified with numerical simulation.

In this work, we study a discrete-time consensus protocol for a group of agents which communicate over a class of stochastically switching networks inspired by fish schooling. The network model incorporates the phenomenon of numerosity that has a prominent role on the collective behavior of animal groups by defining the individuals’ perception of numbers. The agents comprise leaders, which share a common state, and followers, which update their states based on information exchange among neighboring agents. We write a closed form expression for the asymptotic convergence factor of the protocol, which measures the decay rate of disagreement among the followers’ and the leaders’ states. Numerical simulations are conducted to validate analytical results and illustrate the consensus dynamics as a function of the group size, number of leaders in the group, and the numerosity.