Non-numerical Magnitudes Research Articles

Disparate processing of numerosity and associated continuous magnitudes in rats.

The studies of number sense in different species are severely hampered by the inevitable entanglement of non-numerical attributes inherent in nonsymbolic stimuli representing numerosity, resulting in contrasting theories of numerosity processing. Here, we developed an algorithm and associated analytical methods to generate stimuli that not only minimized the impact of non-numerical magnitudes in numerosity perception but also allowed their quantification. We trained number-naïve rats with these stimuli as sound pulses representing two or three numbers and demonstrated that their numerical discrimination ability mainly relied on numerosity. Also, studying the learning process revealed that rats used numerosity before using magnitudes for choices. This numerical processing could be impaired specifically by silencing the posterior parietal cortex. Furthermore, modeling this capacity by neural networks shed light on the separation of numerosity and magnitudes extraction. Our study helps dissect the relationship between magnitude and numerosity processing, and the above different findings together affirm the independent existence of innate number and magnitudes sense in rats.

The right side of price: evidence of a SNARC-like effect for economic value

It is well known that both numerical and non-numerical magnitudes can be represented horizontally from left to right. Building on this knowledge, here we explored whether a similar spatial representation exists for the economic value of goods. Participants were presented with images of a reference and a target product and classified the economic value of the target as higher or lower than that of the reference (Experiments 1 and 2), or classified the target product as belonging to the same or different semantic category as the reference (Experiment 3). Responses were collected using lateralized keys. Evidence of a SNARC-like effect for economic value emerged, whereby low economic value was associated with the left side of space, and high economic value was associated with the right side of space. Importantly, this spatial representation appeared to be based on external spatial coordinates and only emerged when the economic value was treated as an explicit dimension. Regression analyses also ruled out the potential contributions of other dimensions, such as the presumed physical weight of the target products or their valence. These findings support the hypothesis of a general magnitude representation system.

The role of numerical and nonnumerical magnitudes in pigeons' conditional discrimination behavior.

Research on approximate numerical estimation suggests that numerical representations can be influenced by nonnumerical magnitudes. Current theories of numerical cognition differ on the nature of this interaction. The present project evaluated the effect of task requirements on the stimulus control exerted by numerical and nonnumerical magnitudes on pigeons' numerical discrimination behavior. In a series of experiments, we explored the effects of cumulative area and item size on pigeons' numerical discrimination. The effect of cumulative area was assessed by presenting visual displays in which cumulative area and item number were either positively correlated, uncorrelated, or negatively correlated. The effect of item size was evaluated by presenting displays in which the size of individual items was varied across trials. Results confirmed that pigeons' numerical discrimination behavior accorded with Weber's law, a prime indicator of nonsymbolic numerical representation. The results further indicated that pigeons did not use numerical information when nonnumerical magnitudes also provided reliable information to solve the discrimination task. However, task manipulations that rendered the information provided by nonnumerical magnitudes unreliable successfully shifted stimulus control toward numerical magnitudes. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Supplemental Material for The Role of Numerical and Nonnumerical Magnitudes in Pigeons’ Conditional Discrimination Behavior

Supplemental Material for The Role of Numerical and Nonnumerical Magnitudes in Pigeons’ Conditional Discrimination Behavior

A theory of perceptual number encoding.

There has long been interest in how the mind represents numerical magnitude, particularly in the absence of symbols. For humans and nonhuman animals, number represents a core dimension of perceptual experience by which objects in the physical world are delineated. The physical world is also well characterized by other dimensions, many of which covary with number. Yet, the general consensus is that number is perceived independently of other magnitudes that co-occur with it. Here, we present evidence against the independence of number perception. In particular, we use evidence from neuroimaging, computational modeling, visual illusions, and psychophysics to introduce a novel theory of visual number encoding, wherein nonnumerical magnitude information such as cumulative surface area is encoded along with number and sustained throughout visual perception. Moreover, we propose that the experience of number per se reflects the readout of a multidimensional (i.e., integral) representation vis-à-vis selective attention, not the independent encoding of number. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Does auditory numerosity and non-numerical magnitude affect visual non-symbolic numerical representation?

In the present study, we investigated the effects of auditory numerosity and magnitude (loudness) on visual numerosity processing. Participants compared numerosities of two sequential dot arrays. The second dot array was paired with a tone array that was independent of visual comparison. The numerosity (One-tone vs. Multiple-tone) and the non-numerical magnitude of tones (loudness) were manipulated in Experiments 1 and 2, respectively. In Experiment 1, participants' inverse efficiency score (IES), that is, the quotient between response time and accuracy, was significantly smaller in the One-tone and Multiple-tone conditions than that in the No-tone condition, and linear trend analyses showed that the IES decreased with the number of tones. In Experiment 2, the IES in the Loud-tone condition was significantly smaller than that in the No-tone condition, and the IES decreased as the loudness of the tones increased. In Experiment 3, both auditory numerosity and magnitude were manipulated. For soft tones, the IES was smaller in the Multiple-tone condition than in the One-tone condition, whereas no significant difference was found between two conditions in loud tones. In sum, these findings suggest that the visual numerical representation can be spontaneously affected by the numerosity and non-numerical magnitude of stimuli from another modality.

No intrinsic number bias: Evaluating the role of perceptual discriminability in magnitude categorization.

Accumulating evidence suggests that there is a spontaneous preference for numerical, compared to non-numerical (e.g., cumulative surface area), information. However, given a paucity of research on the perception of non-numerical magnitudes, it is unclear whether this preference reflects a specific bias towards number, or a general bias towards the more perceptually discriminable dimension (i.e., number). Here, we found that when the number and area of visual dot displays were matched in mathematical ratio, number was more perceptually discriminable than area in both adults and children. Moreover, both adults and children preferentially categorized these ratio-matched stimuli based on number, consistent with previous work. However, when number and area were matched in perceptual discriminability, a different pattern of results emerged. In particular, children preferentially categorized stimuli based on area, suggesting that children's previously observed number bias may be due to a mismatch in the perceptual discriminability of number and area, not an intrinsic salience of number. Interestingly, adults continued to categorize the displays on the basis of number. Altogether, these findings suggest a dominant role for area during childhood, refuting the claim that number is inherently and uniquely salient. Yet they also reveal an increased salience of number that emerges over development. Potential explanations for this developmental shift are discussed. RESEARCH HIGHLIGHTS: Previous work found that children and adults spontaneously categorized dot array stimuli by number, over other magnitudes (e.g., area), suggesting number is uniquely salient. However, here we found that when number and area were matched by ratio, as in prior work, number was significantly more perceptually discriminable than area. When number and area were made equally discriminable ('perceptually-matched'), children, contra adults, spontaneously categorized stimuli by area over number (and other non-numerical magnitudes). These findings suggest that area may be uniquely salient early in childhood, with the previously-observed number bias not emerging until later in development.

Context-Dependent Modulation of Early Visual Cortical Responses to Numerical and Nonnumerical Magnitudes.

Whether and how the brain encodes discrete numerical magnitude differently from continuous nonnumerical magnitude is hotly debated. In a previous set of studies, we orthogonally varied numerical (numerosity) and nonnumerical (size and spacing) dimensions of dot arrays and demonstrated a strong modulation of early visual evoked potentials (VEPs) by numerosity and not by nonnumerical dimensions. Although very little is known about the brain's response to systematic changes in continuous dimensions of a dot array, some authors intuit that the visual processing stream must be more sensitive to continuous magnitude information than to numerosity. To address this possibility, we measured VEPs of participants viewing dot arrays that changed exclusively in one nonnumerical magnitude dimension at a time (size or spacing) while holding numerosity constant and compared this to a condition where numerosity was changed while holding size and spacing constant. We found reliable but small neural sensitivity to exclusive changes in size and spacing; however, exclusively changing numerosity elicited a much more robust modulation of the VEPs. Together with previous work, these findings suggest that sensitivity to magnitude dimensions in early visual cortex is context dependent: The brain is moderately sensitive to changes in size and spacing when numerosity is held constant, but sensitivity to these continuous variables diminishes to a negligible level when numerosity is allowed to vary at the same time. Neurophysiological explanations for the encoding and context dependency of numerical and nonnumerical magnitudes are proposed within the framework of neuronal normalization.

The interaction between numerical and continuous non-numerical magnitudes in a double change detection paradigm.

In the field of numerical cognition, researchers conventionally assess nonsymbolic numerical abilities with the help of number comparison tasks, in which participants need to compare two arrays. Many studies emphasized that visual (non-numerical) dimensions can serve as strategic cues and influence the decision on numerosity in these tasks. In this study, we suggest the use of a novel paradigm based on the change detection paradigm. Here, participants had to simultaneously pay attention to numerical changes and visual changes on a target non-numerical dimension (individual area, total area, field area, or density). Participants had to detect changes relative to the two dimensions and press response keys indicating either number change or visual change or press both keys. In such a double change detection paradigm, and unlike number comparison tasks, target and covarying dimensions cannot serve as cues to influence the numerical decision. We found that numerical change detection was excellent and stable across the conditions. Further, participants were more likely to falsely consider visual changes as numerical changes than the other way around. Lastly, when both dimensions varied, participants more frequently missed visual changes than numerical changes. Overall, our findings show that numerosity was the most salient visual dimension. From a methodological perspective, such a double change detection paradigm could be of critical interest to assess numerical abilities for future studies. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Magnitude integration in the Archerfish

We make magnitude-related decisions every day, for example, to choose the shortest queue at the grocery store. When making such decisions, which magnitudes do we consider? The dominant theory suggests that our focus is on numerical quantity, i.e., the number of items in a set. This theory leads to quantity-focused research suggesting that discriminating quantities is automatic, innate, and is the basis for mathematical abilities in humans. Another theory suggests, instead, that non-numerical magnitudes, such as the total area of the compared items, are usually what humans rely on, and numerical quantity is used only when required. Since wild animals must make quick magnitude-related decisions to eat, seek shelter, survive, and procreate, studying which magnitudes animals spontaneously use in magnitude-related decisions is a good way to study the relative primacy of numerical quantity versus non-numerical magnitudes. We asked whether, in an animal model, the influence of non-numerical magnitudes on performance in a spontaneous magnitude comparison task is modulated by the number of non-numerical magnitudes that positively correlate with numerical quantity. Our animal model was the Archerfish, a fish that, in the wild, hunts insects by shooting a jet of water at them. These fish were trained to shoot water at artificial targets presented on a computer screen above the water tank. We tested the Archerfish's performance in spontaneous, untrained two-choice magnitude decisions. We found that the fish tended to select the group containing larger non-numerical magnitudes and smaller quantities of dots. The fish selected the group containing more dots mostly when the quantity of the dots was positively correlated with all five different non-numerical magnitudes. The current study adds to the body of studies providing direct evidence that in some cases animals’ magnitude-related decisions are more affected by non-numerical magnitudes than by numerical quantity, putting doubt on the claims that numerical quantity perception is the most basic building block of mathematical abilities.

Symbols Are Special: An fMRI Adaptation Study of Symbolic, Nonsymbolic, and Non-Numerical Magnitude Processing in the Human Brain.

How are different formats of magnitudes represented in the human brain? We used functional magnetic resonance imaging adaptation to isolate representations of symbols, quantities, and physical size in 45 adults. Results indicate that the neural correlates supporting the passive processing of number symbols are largely dissociable from those supporting quantities and physical size, anatomically and representationally. Anatomically, passive processing of quantities and size correlate with activation in the right intraparietal sulcus, whereas symbolic number processing, compared with quantity processing, correlates with activation in the left inferior parietal lobule. Representationally, neural patterns of activation supporting symbols are dissimilar from neural activation patterns supporting quantity and size in the bilateral parietal lobes. These findings challenge the longstanding notion that the culturally acquired ability to conceptualize symbolic numbers is represented using entirely the same brain systems that support the evolutionarily ancient system used to process quantities. Moreover, these data reveal that regions that support numerical magnitude processing are also important for the processing of non-numerical magnitudes. This discovery compels future investigations of the neural consequences of acquiring knowledge of symbolic numbers.

Typical Crossmodal Numerosity Perception in Preterm Newborns.

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.

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.

Do Students Respond Faster to Inequalities with a Greater than Sign or to Inequalities with a Less than Sign: Spatial-Numerical Association in Inequalities

ABSTRACT Previous studies have suggested that there is a left-to-right mental number line, which is based on individuals responding faster when a smaller magnitude is presented on the left visual field and a larger magnitude is presented on the right visual field. This study examined whether the Spatial-Numerical Associations could influence individuals’ responses in an inequality judgment task. Specifically, this study tested the differences in reaction time between judging inequalities with a greater than sign and inequalities with a less than sign and investigated individuals’ subjective liking and perceived fluency in judging these inequalities. Three types of stimuli were used in this study: Arabic number inequalities (numerical and symbolic magnitudes), dot array inequalities (numerical and non-symbolic magnitudes), and square inequalities (non-numerical magnitudes). Results showed that participants reacted quicker to inequalities with a greater than sign than a less than sign when responding to true inequalities in all of the three tasks and reported more subjective liking and perceived fluency in judging inequalities with a greater than sign in symbolic task. This study has implications for studies that are interested in Spatial-Numerical Associations and mathematical education.

Second-order characteristics don't favor a number-representing ANS.

Clarke and Beck argue that the ANS doesn't represent non-numerical magnitudes because of its second-order character. A sensory integration mechanism can explain this character as well, provided the dumbbell studies involve interference from systems that segment by objects such as the Object Tracking System. Although currently equal hypotheses, I point to several ways the two can be distinguished.

Numerosity and cumulative surface area are perceived holistically as integral dimensions.

Human and nonhuman animals have a remarkable capacity to rapidly estimate the quantity of objects in the environment. The dominant view of this ability posits an abstract numerosity code, uncontaminated by nonnumerical visual information. The present study provides novel evidence in contradiction to this view by demonstrating that number and cumulative surface area are perceived holistically, classically known as integral dimensions. Whether assessed explicitly (Experiment 1) or implicitly (Experiment 2), perceived similarity for dot arrays that varied parametrically in number and cumulative area was best modeled by Euclidean, as opposed to city-block, distance within the stimulus space, comparable to other integral dimensions (brightness/saturation and radial frequency components) but different from separable dimensions (shape/color and brightness/size). Moreover, Euclidean distance remained the best-performing model, even when compared to models that controlled for other magnitude properties (e.g., density) or image similarity. These findings suggest that numerosity perception entails the obligatory processing of nonnumerical magnitude. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Supplemental Material for The Interaction Between Numerical and Continuous Non-Numerical Magnitudes in a Double Change Detection Paradigm

Supplemental Material for The Interaction Between Numerical and Continuous Non-Numerical Magnitudes in a Double Change Detection Paradigm

Developmental Dyscalculia in Adults

Abstract. Developmental dyscalculia (DD) is a chronic condition that poses not only a barrier to employment and socio-emotional wellbeing but that also persists into adulthood. Thus, understanding the neuro-cognitive foundations of DD is relevant for both children and adults with DD. However, so far the vast majority of scientific research endeavours has been dedicated to the study of DD in children only. Consequently, our current understanding of DD in adulthood is rather patchy. The main aim of the present review is to summarize the scientific findings on DD in adults by focusing on its cognitive manifestations and neural substrates in adults. For instance, research on DD in adulthood suggests that – beyond an outstanding deficiency in number processing – the processing of non-numerical magnitudes and domain-general skills seem to be also impaired in adults suffering from persistent DD. A secondary aim of this review is to delineate future lines of research that will provide us with a more elaborate understanding of the neurocognitive underpinnings of DD in adults (thus fostering the development of sensitive diagnostic marker tasks), and to formulate potential intervention areas targeting deficiencies frequently characterizing DD in adults.

The neural signature of numerosity by separating numerical and continuous magnitude extraction in visual cortex with frequency-tagged EEG

The ability to handle approximate quantities, or number sense, has been recurrently linked to mathematical skills, although the nature of the mechanism allowing to extract numerical information (i.e., numerosity) from environmental stimuli is still debated. A set of objects is indeed not only characterized by its numerosity but also by other features, such as the summed area occupied by the elements, which often covary with numerosity. These intrinsic relations between numerosity and nonnumerical magnitudes led some authors to argue that numerosity is not independently processed but extracted through a weighting of continuous magnitudes. This view cannot be properly tested through classic behavioral and neuroimaging approaches due to these intrinsic correlations. The current study used a frequency-tagging EEG approach to separately measure responses to numerosity as well as to continuous magnitudes. We recorded occipital responses to numerosity, total area, and convex hull changes but not to density and dot size. We additionally applied a model predicting primary visual cortex responses to the set of stimuli. The model output was closely aligned with our electrophysiological data, since it predicted discrimination only for numerosity, total area, and convex hull. Our findings thus demonstrate that numerosity can be independently processed at an early stage in the visual cortex, even when completely isolated from other magnitude changes. The similar implicit discrimination for numerosity as for some continuous magnitudes, which correspond to basic visual percepts, shows that both can be extracted independently, hence substantiating the nature of numerosity as a primary feature of the visual scene.

Slow and fast beat sequences are represented differently through space.

The Spatial-Numerical Association of Response Codes (SNARC) suggests the existence of an association between number magnitude and response position, with faster left-hand responses to small numbers and faster right-hand responses to large numbers. Recent studies have revealed similar spatial association effects for non-numerical magnitudes, such as temporal durations and musical stimuli. In the present study we investigated whether a spatial association effect exists between music tempo, expressed in beats per minutes (bpm), and response position. In particular, we were interested in whether this effect is consistent through different bpm ranges. We asked participants to judge whether a target beat sequence was faster or slower than a reference sequence. Three groups of participants judged beat sequences from three different bpm ranges, a wide range (40, 80, 160, 200 bpm) and two narrow ranges ("slow" tempo, 40, 56, 88, 104 bpm; "fast" tempo 133, 150, 184, 201 bpm). Results showed a clear SNARC-like effect for music tempo only in the narrow "fast" tempo range, with faster left-hand responses to 133 and 150 bpm and faster right-hand responses to 184 and 201 bpm. Conversely, a similar association did not emerge in the wide nor in the narrow "slow" tempo ranges. This evidence suggests that music tempo is spatially represented as other continuous quantities, but its representation might be narrowed to a particular range of tempos. Moreover, music tempo and temporal duration might be represented across space with an opposite direction.