Near optimal encoding of numerosity in typical and dyscalculic development

The effects of feedback and task accuracy in serial dependence to orientation

Assimilative serial dependence in perception occurs where responses about a stimulus (e.g., orientation) are biased towards previously seen perceptual information (e.g., the orientation of the stimulus shown on the previous trial). This bias may occur to perceptual information from the previous trial, or to the response or decision made on the previous trial. We asked whether providing response feedback could change the serial dependence effect on the following trial. Twenty-one participants completed a task in which they adjusted an on-screen pointer to reproduce the orientation of a briefly-presented Gabor stimulus. They received feedback about the accuracy of their response that either reflected their actual accuracy or was random. We found significant positive biases to the stimulus and response only when the participant had received positive ("correct!") feedback on that trial. When the inducer response had been incorrect, the effect was significant only to the response itself and not to the stimulus. Overall, we suggest that our participants demonstrated a bias towards the percept from the previous trial, which is better represented by the response than the stimulus for incorrect trials, and that this effect can be modulated post-perceptually by feedback.

The radial-tangential anisotropy of numerosity perception

Humans can estimate the number of visually presented items without counting. In most studies on numerosity perception, items are uniformly distributed across displays, with identical distributions in central and eccentric parts. However, the neural and perceptual representation of the human visual field differs between the fovea and the periphery. For example, in peripheral vision, there are strong asymmetries with regard to perceptual interferences between visual items. In particular, items arranged radially usually interfere more strongly with each other than items arranged tangentially (the radial-tangential anisotropy). This has been shown for crowding (the deleterious effect of clutter on target identification) and redundancy masking (the reduction of the number of perceived items in repeating patterns). In the present study, we tested how the radial-tangential anisotropy of peripheral vision impacts numerosity perception. In four experiments, we presented displays with varying numbers of discs that were predominantly arranged radially or tangentially, forming strong and weak interference conditions, respectively. Participants were asked to report the number of discs. We found that radial displays were reported as less numerous than tangential displays for all radial and tangential manipulations: weak (Experiment 1), strong (Experiment 2), and when using displays with mixed contrast polarity discs (Experiments 3 and 4). We propose that numerosity perception exhibits a significant radial-tangential anisotropy, resulting from local spatial interactions between items.

Serial dependence bias can predict the overall estimation error in visual perception

Although visual feature estimations are accurate and precise, overall estimation errors (i.e., the difference between estimates and actual values) tend to show systematic patterns. For example, estimates of orientations are systematically biased away from horizontal and vertical orientations, showing an oblique illusion. Additionally, many recent studies have demonstrated that estimations of current visual features are systematically biased toward previously seen features, showing a serial dependence. However, no study examined whether the overall estimation errors were correlated with the serial dependence bias. To address this question, we enrolled three groups of participants to estimate orientation, motion speed, and point-light-walker direction. The results showed that the serial dependence bias explained over 20% of overall estimation errors in the three tasks, indicating that we could use the serial dependence bias to predict the overall estimation errors. The current study first demonstrated that the serial dependence bias was not independent from the overall estimation errors. This finding could inspire researchers to investigate the neural bases underlying the visual feature estimation and serial dependence.

Partially Separable Aspects of Spatial and Temporal Estimations in Virtual Navigation as Revealed by Adaptation

Recent studies claim that estimating the magnitude of the spatial and temporal aspects of one's self-motion shows similar characteristics, suggesting shared processing mechanisms between these two dimensions. While the estimation of other magnitude dimensions, such as size, number, and duration, exhibits negative aftereffects after prolonged exposure to the stimulus, it remains to be elucidated whether this could occur similarly in the estimation of the distance travelled and time elapsed during one's self-motion. We sought to fill this gap by examining the effects of adaptation on distance and time estimation using a virtual navigation task. We found that a negative aftereffect occurred in the distance reproduction task after repeated exposure to self-motion with a fixed travel distance. No such aftereffect occurred in the time reproduction task after repeated exposure to self-motion with a fixed elapsed time. Further, the aftereffect in distance reproduction occurred only when the distance of the adapting stimulus was fixed, suggesting that it did not reflect adaptation to time, which varied with distance. The estimation of spatial and temporal aspects of self-motion is thus processed by partially separable mechanisms, with the distance estimation being similar to the estimation of other magnitude dimensions.

Uncertainty and Prior Assumptions, Rather Than Innate Logarithmic Encoding, Explain Nonlinear Number-to-Space Mapping

Mapping number to space is natural and spontaneous but often nonveridical, showing a clear compressive nonlinearity that is thought to reflect intrinsic logarithmic encoding of numerical values. We asked 78 adult participants to map dot arrays onto a number line across nine trials. Combining participant data, we confirmed that on the first trial, mapping was heavily compressed along the number line, but it became more linear across trials. Responses were well described by logarithmic compression but also by a parameter-free Bayesian model of central tendency, which quantitatively predicted the relationship between nonlinearity and number acuity. To experimentally test the Bayesian hypothesis, we asked 90 new participants to complete a color-line task in which they mapped noise-perturbed color patches to a "color line." When there was more noise at the high end of the color line, the mapping was logarithmic, but it became exponential with noise at the low end. We conclude that the nonlinearity of both number and color mapping reflects contextual Bayesian inference processes rather than intrinsic logarithmic encoding.

Nonsymbolic-Magnitude Deficit in Adults With Developmental Dyscalculia: Evidence of Impaired Size Discrimination but Intact Size Constancy

Although researchers have debated whether a core deficit of nonsymbolic representation of magnitude underlies developmental dyscalculia (DD), research has mostly focused on numerosity processing. We probed the possibility of a general magnitude deficit in individuals with DD and asked whether sensitivity to size varied in contexts of depth ordering and size constancy. We measured full psychometric functions in size-discrimination tasks in 12 participants with DD and 13 control participants. Results showed that although people with DD exhibited veridical perceived magnitude, their sensitivity to size was clearly impaired. In contrast, when objects were embedded in depth cues allowing size-constancy computations, participants with DD demonstrated typical sensitivity to size. These results demonstrate a deficit in the perceptual resolutions of magnitude in DD. At the same time, the finding of an intact size constancy suggests that when magnitude perception is facilitated by implicit mandatory computations of size constancy, this deficit is no longer evident.

Grouping strategies in number estimation extend the subitizing range

When asked to estimate the number of items in a visual array, educated adults and children are more precise and rapid if the items are clustered into small subgroups rather than randomly distributed. This phenomenon, termed "groupitizing", is thought to rely on the recruitment of the subitizing system (dedicated to the perception of very small numbers), with the aid of simple arithmetical calculations. The aim of current study is to verify whether the advantage for clustered stimuli does rely on subitizing, by manipulating attention, known to strongly affect attention. Participants estimated the numerosity of grouped or ungrouped arrays in condition of full attention or while attention was diverted with a dual-task. Depriving visual attention strongly decreased estimation precision of grouped but not of ungrouped arrays, as well as increasing the tendency for numerosity estimation to regress towards the mean. Additional explorative analyses suggested that calculation skills correlated with the estimation precision of grouped, but not of ungrouped, arrays. The results suggest that groupitizing is an attention-based process that leverages on the subitizing system. They also suggest that measuring numerosity estimation thresholds with grouped stimuli may be a sensitive correlate of math abilities.

Math Anxiety Mediates the Link Between Number Sense and Math Achievements in High Math Anxiety Young Adults

In the past few years, many studies have suggested that subjects with high sensory precision in the processing of non-symbolic numerical quantities (approximate number system; ANS) also have higher math abilities. At the same time, there has been interest in another non-cognitive factor affecting mathematical learning: mathematical anxiety (MA). MA is defined as a debilitating emotional reaction to mathematics that interferes with the manipulation of numbers and the solving of mathematical problems. Few studies have been dedicated to uncovering the interplay between ANS and MA and those have provided conflicting evidence. Here we measured ANS precision (numerosity discrimination thresholds) in a cohort of university students with either a high (>75th percentile; n = 49) or low (<25th percentile; n = 39) score on the Abbreviate Math Anxiety Scale (AMAS). We also assessed math proficiency using a standardized test (MPP: Mathematics Prerequisites for Psychometrics), visuo-spatial attention capacity by means of a Multiple Objects Tracking task (MOT) and sensory precision for non-numerical quantities (disk size). Our results confirmed previous studies showing that math abilities and ANS precision correlate in subjects with high math anxiety. Neither precision in size-discrimination nor visuo-spatial attentional capacity were found to correlate with math capacities. Interestingly, within the group with high MA, our data also revealed a relationship between ANS precision and MA, with MA playing a key role in mediating the correlation between ANS and math achievement. Taken together, our results suggest an interplay between extreme levels of MA and the sensory precision in the processing of non-symbolic numerosity.