Topographic numerosity maps cover subitizing and estimation ranges

Numerosity, the set size of a group of items, helps guide behaviour and decisions. Non-symbolic numerosities are represented by the approximate number system. However, distinct behavioural performance suggests that small numerosities, i.e. subitizing range, are implemented differently in the brain than larger numerosities. Prior work has shown that neural populations selectively responding (i.e. hemodynamic responses) to small numerosities are organized into a network of topographical maps. Here, we investigate how neural populations respond to large numerosities, well into the ANS. Using 7 T fMRI and biologically-inspired analyses, we found a network of neural populations tuned to both small and large numerosities organized within the same topographic maps. These results demonstrate a continuum of numerosity preferences that progressively cover both the subitizing range and beyond within the same numerosity map, suggesting a single neural mechanism. We hypothesize that differences in map properties, such as cortical magnification and tuning width, underlie known differences in behaviour.

Impaired large numerosity estimation and intact subitizing in developmental dyscalculia

It is believed that the approximate estimation of large sets and the exact quantification of small sets (subitizing) are supported by two different systems, the Approximate Number System (ANS) and Object Tracking System (OTS), respectively. It is a current matter of debate whether they are both impaired in developmental dyscalculia (DD), a specific learning disability in symbolic number processing and calculation. Here we tackled this question by asking 32 DD children and 32 controls to perform a series of tasks on visually presented sets, including exact enumeration of small sets as well as comparison of large, uncountable sets. In children with DD, we found poor sensitivity in processing large numerosities, but we failed to find impairments in the exact enumeration of sets within the subitizing range. We also observed deficits in visual short-term memory skills in children with dyscalculia that, however, did not account for their low ANS acuity. Taken together, these results point to a dissociation between quantification skills in dyscalculia, they highlight a link between DD and low ANS acuity and provide support for the notion that DD is a multifaceted disability that covers multiple cognitive skills.

Decoding Location-specific and Location-invariant Stages of Numerosity Processing in Subitizing

Extracting the number of objects in perceived scenes is a fundamental cognitive ability. Number processing is proposed to rely on two consecutive stages: an early object location map that captures individuated objects in a location-specific way and a subsequent location-invariant representation that captures numerosity at an abstract level. However, it is unclear whether this framework applies to small numerosities that can be individuated at once ("subitized"). Here we used EEG-based multivariate pattern decoding to test for location-specific and location-invariant stages of numerosity processing in the subitizing range. In two experiments, 1-3 targets were presented in the left or right hemifield, which allowed for decoding target numerosity within each hemifield separately (location-specific) or across hemifields (location-invariant). Experiment 1 indicated the presence of a location-specific stage (180-200 ms post-stimulus), followed by a location-invariant stage (300 ms post-stimulus). Experiment 2 showed that both location-specific and invariant components are engaged only during tasks that explicitly require numerosity processing, ruling out automatic, passive recording of numerosity. Overall, the results suggest that numerosity coding in subitizing is strongly grounded on an attention-based, location-specific stage. This stage remains active in parallel with the subsequent activation of a location-invariant stage, where a full representation of numerosity is finalized.

Enumerating the forest before the trees: The time courses of estimation-based and individuation-based numerical processing

Ensemble perception refers to the ability to report attributes of a group of objects, rather than focusing on only one or a few individuals. An everyday example of ensemble perception is the ability to estimate the numerosity of a large number of items. The time course of ensemble processing, including that of numerical estimation, remains a matter of debate, with some studies arguing for rapid, “preattentive” processing and other studies suggesting that ensemble perception improves with longer presentation durations. We used a forward-simultaneous masking procedure that effectively controls stimulus durations to directly measure the temporal dynamics of ensemble estimation and compared it with more precise enumeration of individual objects. Our main finding was that object individuation within the subitizing range (one to four items) took about 100–150 ms to reach its typical capacity limits, whereas estimation (six or more items) showed a temporal resolution of 50 ms or less. Estimation accuracy did not improve over time. Instead, there was an increasing tendency, with longer effective durations, to underestimate the number of targets for larger set sizes (11–35 items). Overall, the time course of enumeration for one or a few single items was dramatically different from that of estimating numerosity of six or more items. These results are consistent with the idea that the temporal resolution of ensemble processing may be as rapid as, or even faster than, individuation of individual items, and support a basic distinction between the mechanisms underlying exact enumeration of small sets (one to four items) from estimation.

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.

Number-related Brain Potentials Are Differentially Affected by Mapping Novel Symbols on Small versus Large Quantities in a Number Learning Task

The nature of the mapping process that imbues number symbols with their numerical meaning—known as the “symbol-grounding process”—remains poorly understood and the topic of much debate. The aim of this study was to enhance insight into how the nonsymbolic–symbolic number mapping process and its neurocognitive correlates might differ between small (1–4; subitizing range) and larger (6–9) numerical ranges. Hereto, 22 young adults performed a learning task in which novel symbols acquired numerical meaning by mapping them onto nonsymbolic magnitudes presented as dot arrays (range 1–9). Learning-dependent changes in accuracy and RT provided evidence for successful novel symbol quantity mapping in the subitizing (1–4) range only. Corroborating these behavioral results, the number processing related P2p component was only modulated by the learning/mapping of symbols representing small numbers 1–4. The symbolic N1 amplitude increased with learning independent of symbolic numerical range but dependent on the set size of the preceding dot array; it only occurred when mapping on one to four item dot arrays that allow for quick retrieval of a numeric value, on the basis of which, with learning, one could predict the upcoming symbol causing perceptual expectancy violation when observing a different symbol. These combined results suggest that exact nonsymbolic–symbolic mapping is only successful for small quantities 1–4 from which one can readily extract cardinality. Furthermore, we suggest that the P2p reflects the processing stage of first access to or retrieval of numeric codes and might in future studies be used as a neural correlate of nonsymbolic–symbolic mapping/symbol learning.

Does subitizing require attention? A systematic review and meta-analysis

Whether enumeration of small number of objects requires attention remains controversial. Although most recent studies argue for a role of attention in subitizing, these studies include varied stimuli and different methods of manipulating attention. It is unclear if the observed attention effects in different studies are real effects. It is also unclear whether there is publication bias in these studies. To answer this question, a systematic review and meta-analysis was performed to evaluate the attention effects on enumeration of small numbers. A total of 14 studies (22 experiments, 35 comparisons) were included in a meta-analysis to compare the attention effects on subitizing. Results from the meta-analysis suggest that a manipulation of attention can evidently modulate the behavioural performance in the subitizing range (response time, accuracy and Weber fraction). These findings were consistently observed in various experimental designs and different stimuli (p < .010; p < .001; p < .001; respectively), suggesting attention does play a role in subitizing. A new model was proposed to explain the mechanism of subitizing and enumeration. Findings in this study may contribute to the understanding of “number module” in brain and contribute to models of numerical cognition in education. However, a publication bias was observed in this study, suggesting the observed effects might not be very accurate. To better estimate the effect of attention manipulations in the subitizing range, studies with larger samples, or future meta-analyses including unpublished outcomes and unpublished studies may be required.

A new method for calculating individual subitizing ranges

A large body of research has shown that human adults are fast and accurate at enumerating arrays of ~1-4 items. This phenomenon has been called subitizing. Above this range, enumeration is slower and less accurate. The subitizing range has been related to individual differences in variables such as mathematical abilities, working memory, etc. The two most common methods for calculating subitizing range today – bilinear fit and sigmoid fit – have their strengths and weaknesses. By combining these two methods, we overcome their biggest limitations and come up with a novel way for calculating Individual Subitizing Range (ISR). This paper introduces this new method as well as empirical studies designed to test the new method. We replicated classic effects from the literature and obtain a high correlation with the sigmoid fit method. This paper includes a Matlab code for easy calculation of ISR as well as a ready-to-use experimental file for testing ISR. We hope that these tools would be of use to researchers studying individual differences in the subitizing range.

Non-symbolic number does not automatically activate spatial-numerical associations: Evidence from the SNARC effect.

The SNARC (spatial numerical association of response codes) effect is the finding that people are generally faster to respond to smaller numbers with left-sided responses and larger numbers with right-sided responses. The SNARC effect has been widely reported for responses to symbolic representations of number such as digits. However, there is mixed evidence as to whether it occurs for non-symbolic representations of number, particularly when magnitude is irrelevant to the task. Mitchell et al. (2012) reported a SNARC effect when participants were asked to make orientation decisions to arrays of one-to-nine triangles (pointing upwards versus pointing downwards) and concluded that SNARC effects occur for non-symbolic, non-canonical representations of number. They additionally reported that this effect was stronger in the subitizing range. However, here we report four experiments that do not replicate either of these findings. Participants made upwards / inverted decisions to one-to-nine triangles where total surface area was either controlled across numerosities (Experiments 1, 2 and 4) or increased congruently with numerosity (Experiment 3). There was no evidence of a SNARC effect either across the full range, or within the subset of the subitizing range. The results of Experiment 4 (in which we presented the original stimuli of Mitchell et al.) suggested that visual properties of non-symbolic displays can prompt SNARC-like effects driven by visual cues rather than numerosity. Taken in the context of other recent findings, we argue that non-symbolic representations of number do not offer a direct and automatic route to numerical-spatial associations.

Individual Subitizing Range (ISR)

contains a matlab code for calculating individual subitizing range (ISR), instructions for using the code, and a ready to run experiment - number naming task that is suitable for both children and adults (requires OpenSesame 3.07).