Numbers Uniquely Bias Spatial Attention: A Novel Paradigm for UnderstandingSpatial-Numerical Associations

Over the past two-and-a-half decades, numerous empiricalstudies have demonstrated a relationship between numbersand space. A classic interpretation is that these spatial-numerical associations (SNAs) are a product of a stablemental number line (MNL) in the mind, yet others haveargued that SNAs are a product of transient mappings thatoccur in working memory. Importantly, although the latterinterpretation has no implications for the representation ofnumber, the former suggests that the representation ofnumber is inherently spatial. Here, we tease apart questionsof spatial representation (à la an MNL perspective) andspatial strategy (à la alternative accounts). In a novel place-the-number task, we demonstrate that numbersautomatically bias spatial attention whereas other ordinalsequences (i.e., letters) do not. We argue that this isevidence of an inherently spatial representation of numberand explore how this work may help answer futurequestions about the relationship between space andnumber

Visual adaptation reveals multichannel coding for numerosity

Visual numerosity is represented automatically and rapidly, but much remains unknown about the computations underlying this perceptual experience. For example, it is unclear whether numerosity is represented with an opponent channel or multichannel coding system. Within an opponent channel system, all numerical values are represented via the relative activity of two pools of neurons (i.e., one pool with a preference for small numerical values and one pool with a preference for large numerical values). However, within a multichannel coding system, all numerical values are represented directly, with separate pools of neurons for each (discriminable) numerical value. Using an adaptation paradigm, we assessed whether the visual perception of number is better characterized by an opponent channel or multichannel system. Critically, these systems make distinct predictions regarding the pattern of aftereffects exhibited when an observer is adapted to an intermediate numerical value. Opponent channel coding predicts no aftereffects because both pools of neurons adapt equally. By contrast, multichannel coding predicts repulsive aftereffects, wherein numerical values smaller than the adapter are underestimated and those larger than the adapter are overestimated. Consistent with multichannel coding, visual adaptation to an intermediate value (50 dots) yielded repulsive aftereffects, such that participants underestimated stimuli ranging from 10–50 dots, but overestimated stimuli ranging from 50–250 dots. These findings provide novel evidence that the visual perception of number is supported by a multichannel, not opponent channel, coding system, and raise important questions regarding the contributions of different cortical regions, such as the ventral and lateral intraparietal areas, to the representation of number

The Developing Mental Number Line: Does Its Directionality Relate to 5- to 7-Year-Old Children’s Mathematical Abilities?

Spatial representations of number, such as a left-to-right oriented mental number line, are well documented in Western culture. Yet, the functional significance of such a representation remains unclear. To test the prominent hypothesis that a mental number line may support mathematical development, we examined the relation between spatial-numerical associations (SNAs) and math proficiency in 5- to 7-year-old children. We found evidence of SNAs with two tasks: a non-symbolic magnitude comparison task, and a symbolic “Where was the number?” (WTN) task. Further, we found a significant correlation between these two tasks, demonstrating convergent validity of the directional mental number line across numerical format. Although there were no significant correlations between children’s SNAs on the WTN task and math ability, children’s SNAs on the magnitude comparison task were negatively correlated with their performance on a measure of cross-modal arithmetic, suggesting that children with a stronger left-to-right oriented mental number line were less competent at cross-modal arithmetic, an effect that held when controlling for age and a set of general cognitive abilities. Despite some evidence for a negative relation between SNAs and math ability in adulthood, we argue that the effect here may reflect task demands specific to the magnitude comparison task, not necessarily an impediment of the mental number line to math performance. We conclude with a discussion of the different properties that characterize a mental number line and how these different properties may relate to mathematical ability

Numerosity and cumulative surface area are perceived holistically as integral dimensions

Human and non-human animals have the 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 non-numerical 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 non-numerical magnitude

The relative salience of numerical and non-numerical dimensions shifts over development: A re-analysis of Tomlinson, DeWind, and Brannon (2020)

Visual displays of objects include information about number and other magnitudes such as cumulative surface area. Despite the confluence of cues, a prevalent view is that number is uniquely salient within multidimensional stimuli. Consistent with this view, Tomlinson, DeWind, and Brannon (2020) report that, in addition to greater acuity for number than area among both children and adults, number biases area judgments more than the reverse, at least in childhood. However, a failure to consider perceived area, undermines these results. To address this concern, we used an index of perceived area when assessing acuity and bias of number and area. In this context, number and area were comparable in acuity among children and adults. Bias, however, differed across development. Although adults showed greater bias of number on area judgments than the reverse, children experienced greater area bias on number judgments. Thus, contra Tomlinson et al., when differences in mathematical and perceived area are accounted for, area is more salient than number early in development. However, number does become the more salient dimension by adulthood, suggesting a role for experience with symbolic number and education in drawing attention to number within multidimensional visual stimuli

Files for: Numerosity and cumulative surface area are perceived holistically as integral dimensions

Supplemental information, stimuli, & data for: Aulet, L. S., & Lourenco, S. F. (2020). Numerosity and cumulative surface area are perceived holistically as integral dimensions. Journal of Experimental Psychology: General

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 non-numerical 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 read-out of a multidimensional (i.e., integral) representation vis-à-vis selective attention, not the independent encoding of number

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

Spatial-numerical associations from a novel paradigm support the mental number line account

Multiple tasks have been used to demonstrate the relation between numbers and space. The classic interpretation of these directional spatial-numerical associations (d-SNAs) is that they are the product of a mental number line (MNL), in which numerical magnitude is intrinsically associated with spatial position. The alternative account is that d-SNAs reflect task demands, such as explicit numerical judgments and/or categorical responses. In the novel ‘Where was The Number?’ task, no explicit numerical judgments were made. Participants were simply required to reproduce the location of a numeral within a rectangular space. Using a between-subject design, we found that numbers, but not letters, biased participants’ responses along the horizontal dimension, such that larger numbers were placed more rightward than smaller numbers, even when participants completed a concurrent verbal working memory task. These findings are consistent with the MNL account, such that numbers specifically are inherently left-to-right oriented in Western participants