Perceived number is not abstract

To support the claim that the ANS represents rational numbers, C&B argue that number perception is abstract and characterized by a second-order character. However, converging evidence from visual illusions and psychophysics suggests that perceived number is not abstract, but rather, is perceptually interdependent with other magnitudes. Moreover, number, as a concept, is second-order, but number, as a percept, is not.

The Influence of the Place Value System on Symbolic Number Perception

Past research on numerical cognition has suggested that both symbolic and non-symbolic numbers are mapped onto the same compressed mental analogue representation. However, experiments using magnitude estimation tasks show logarithmic compression of symbolic numbers while the compression of non-symbolic numbers has a power-function shape. This warrants closer inspection of what differentiates the two processes. In this study, we hypothesized that estimates of symbolic numbers are systematically shaped by the format in which they are represented, namely, the place value system. To investigate this, we tested adults (N = 188) on a magnitude estimation task with unfamiliar base-26 and base-5 scales. Results reveal that adults showed systematic, logarithmic-looking underestimation on both scales, indicating that the place value system itself can cause the pattern. Additionally, the observed shape of participants’ estimates on both scales could be well explained with a simple model that assumed insufficient understanding of exponential growth (i.e., a characteristic of place value systems). Taken together, our results suggest that the discrepancy between symbolic and non-symbolic number compression can be explained by taking the effect of the place value system into account.

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.

Do children estimate area using an ‘Additive-Area Heuristic’?

A large and growing body of work has documented large, robust illusions of area perception in adults. To date, however, there has been surprisingly little in-depth investigation into children’s area perception, despite the importance of this topic to the study of quantity perception more broadly (and to the many studies that have been devoted to studying children’s number perception). Here, in order to understand the interactions of number and area on quantity perception, we study both dimensions in tandem. First, inspired by recent work showing that human adults appear to rely on an 'Additive Area Heuristic', we test whether children may rely on this same kind of heuristic. Indeed, ‘additive area’ explains children’s area judgments better than true, mathematical area. Second, we show that children’s use of ‘additive area’ biases number judgments. Finally, to isolate ‘additive area’ from number, we test children’s area perception in a task where number is held constant across all trials. We find something surprising: even when there is no overall effect of ‘additive area’ or ‘mathematical area’, individual children adopt, and stick to, specific strategies throughout the task. In other words, some children appear to rely on ‘additive area’, while others appear to rely on true, mathematical area — a pattern of results that may be best explained by a misunderstanding about the concept of cumulative area. We discuss how these findings raise both theoretical and practical challenges of studying quantity perception in young children.

The psychophysics of number arise from resource-limited spatial memory

People can identify the number of objects in small sets rapidly and without error but become increasingly noisy for larger sets. However, the cognitive mechanisms underlying these ubiquitous psychophysics are poorly understood. We present a model of a limited-capacity visual system optimized to individuate and remember the location of objects in a scene which gives rise to all key aspects of number psychophysics, including error-free small number perception and scalar variability for larger numbers. We therefore propose that number psychophysics can be understood as an emergent property of primitive perceptual mechanisms --- namely, the process of identifying and representing individual objects in a scene. To test our theory, we ran two experiments: a change-localization task to measure participants' memory for the locations of objects (Experiment 1) and a numerical estimation task (Experiment 2). Our model accounts well for participants' performance in both experiments, despite only being optimized to efficiently encode where objects are present in a scene. Our results demonstrate that the key psychophysical features of numerical cognition do not arise from separate modules or capacities specific to number, but rather from lower-level constraints on perception which are manifested even in non-numerical tasks.

Perceived number is not abstract

To support the claim that the approximate number system (ANS) represents rational numbers, Clarke and Beck (C&B) argue that number perception is abstract and characterized by a second-order character. However, converging evidence from visual illusions and psychophysics suggests that perceived number is not abstract, but rather, is perceptually interdependent with other magnitudes. Moreover, number, as a concept, is second-order, but number, as a percept, is not.

Weighted numbers

Clarke and Beck (C&B) discuss in their sections on congruency and confounds (sects. 3 and 4) literature that has challenged the claim that the approximate number system (ANS) represents numerical content. We argue that the propositions put forward by these studies aren't that far from the indirect model of number perception suggested by C&B.

Number Adaptation Can Be Dissociated From Density Adaptation

Rapidly judging the number of objects in a scene is an important perceptual ability. Recent debates have centered on whether number perception is accomplished by dedicated mechanisms and, in particular, on whether number-adaptation aftereffects reflect adaptation of number per se or adaptation of related stimulus properties, such as density. Here, we report an adaptation experiment ( N = 8) for which the predictions of number and density theories are diametrically opposed. We found that when a reference stimulus has higher density than an adaptation stimulus but contains fewer elements, adaptation reduces the perceived number of elements in the reference stimulus. This is consistent with number adaptation and inconsistent with density adaptation. Thus, number-adaptation aftereffects are more than a by-product of density adaptation: When density and number are dissociated, adaptation effects are in the direction predicted by adaptation to number, not density.

Infants’ intermodal perception of numerosity in an experimental study with objects and socially salient stimuli

In the present cross-sectional experimental study we investigated infants’ early ability to intermodally detect numerosity of visual-auditory object-like and social stimuli. We assumed that presentation of face – voice stimuli would distract infants’ attention from detection of numerical invariant. Seventy-eight infants (aged 5, 7 and 9 months) participated in four experimental Conditions (simultaneously projected pairs of identical objects, non-identical objects, objects projected together with familiar face and objects projected together with unfamiliar face). Visual stimuli in each trial varied in numerosity (1 -2 / 1-3 / 2 -3) and they were accompanied by piano sounds or voice sounds also varying in numerosity (one, two or three sounds in La tonality). By means of preferential looking technique, we measured infants’ fixation of attention to the visual stimulus that numerically matched with the sound. When object-like stimuli were projected, infants –except 5-month-old boys –tended to intermodally detect numerical invariant. Shape similarity of the objects facilitated infants’ intermodal detection of numerosity. When socially salient stimuli were co-presented with object-like stimuli, infants preferred to look at the face, ignoring numerosity of the auditory stimulus. Nor sound quality (piano vs. voice) neither familiarity of the face (mother’s face vs. stranger woman’s face) affected infants’ perception. Although intermodal detection of perceptual cues is a primary function of both face and number perception, each one of these perceptual systems seems to follow a different developmental path.

The relationship between children’s approximate number certainty and symbolic mathematics

Why do some children excel in mathematics while others struggle? A large body of work has shown positive correlations between children’s Approximate Number System (ANS) and school-taught symbolic mathematical skills, but the mechanism explaining this link remains unknown. One potential mediator of this relationship might be children’s numerical metacognition: children’s ability to evaluate how sure or unsure they are in understanding and manipulating numbers. While previous work has shown that children’s math abilities are uniquely predicted by symbolic numerical metacognition, we focus on the extent to which children’s non-symbolic/ANS numerical metacognition, in particular sensitivity to certainty, might be predictive of math ability, and might mediate the relationship between the ANS and symbolic math. A total of 72 children aged 4–6 years completed measures of ANS precision, ANS metacognition sensitivity, and the Test of Early Mathematical Ability (TEMA-3). Our results replicate many established findings in the literature, including the correlation between ANS precision and the TEMA-3, particularly on the Informal subtype questions. However, we did not find that ANS metacognition sensitivity was related to TEMA-3 performance, nor that it mediated the relationship between the ANS and the TEMA-3. These findings suggest either that metacognitive calibration may play a larger role than metacognitive sensitivity, or that metacognitive differences in the non-symbolic number perception do not robustly contribute to symbolic math performance.