Numerosity Estimation Task Research Articles

Response boosts serial dependence in the numerosity estimation task

Perceptions of current stimuli are sometimes biased toward or away from past perceptions. This phenomenon is called serial dependence. However, the strength of the effect of past responses on serial dependence has not been fully elucidated. We conducted experiments with a task in which participants estimated the number of dot arrays (numerosity estimation task) and directly compared whether the strength of serial dependence changed in the numerosity estimation task when participants responded or did not respond in the immediately preceding trial. We also examined whether the strength of serial dependence affected the accuracy of the numerosity estimation. We found that attractive serial dependence was stronger when participants responded in the immediately preceding trial than when they only saw the stimulus. The results suggest that the information from the previous stimulus must reach the higher-level processes associated with perceptual decisions to influence the estimation of the current stimulus. However, it is possible that the results of this study are specific to tasks in which participants respond with numeric symbols. The magnitude of the serial dependence effect was not observed to affect numerosity estimation performance, and no evidence was found that serial dependence enhances accuracy in the numerosity estimation task.

EEG signature of grouping strategies in numerosity perception.

The moment we see a group of objects, we can appreciate its numerosity. Our numerical estimates can be imprecise for large sets (>4 items), but they become much faster and more accurate if items are clustered into groups compared to when they are randomly displaced. This phenomenon, termed groupitizing, is thought to leverage on the capacity to quickly identify groups from 1 to 4 items (subitizing) within larger sets, however evidence in support for this hypothesis is scarce. The present study searched for an electrophysiological signature of subitizing while participants estimated grouped numerosities exceeding this range by measuring event-related potential (ERP) responses to visual arrays of different numerosities and spatial configurations. The EEG signal was recorded while 22 participants performed a numerosity estimation task on arrays with numerosities in the subitizing (3 or 4) or estimation (6 or 8) ranges. In the latter case, items could be spatially arranged into subgroups (3 or 4) or randomly scattered. In both ranges, we observed a decrease in N1 peak latency as the number of items increased. Importantly, when items were arranged to form subgroups, we showed that the N1 peak latency reflected both changes in total numerosity and changes in the number of subgroups. However, this result was mainly driven by the number of subgroups to suggest that clustered elements might trigger the recruitment of the subitizing system at a relatively early stage. At a later stage, we found that P2p was mostly modulated by the total numerosity in the set, with much less sensitivity for the number of subgroups these might be segregated in. Overall, this experiment suggests that the N1 component is sensitive to both local and global parcelling of elements in a scene suggesting that it could be crucially involved in the emergence of the groupitizing advantage. On the other hand, the later P2p component seems to be much more bounded to the global aspects of the scene coding the total number of elements while being mostly blind to the number of subgroups in which elements are parsed.

The effect of abstract representation and response feedback on serial dependence in numerosity perception.

Serial dependence entails an attractive bias based on the recent history of stimulation, making the current stimulus appear more similar to the preceding one. Although serial dependence is ubiquitous in perception, its nature and mechanisms remain unclear. Here, in two independent experiments, we test the hypothesis that this bias originates from high-level processing stages at the level of abstract information processing (Exp. 1) or at the level of judgment (Exp. 2). In Experiment 1, serial dependence was induced by a task-irrelevant "inducer" stimulus in a numerosity discrimination task, similarly to previous studies. Importantly, in this experiment, the inducers were either arrays of dots similar to the task-relevant stimuli (e.g., 12 dots), or symbolic numbers (e.g., the numeral "12"). Both dots and symbol inducers successfully yielded attractive serial dependence biases, suggesting that abstract information about an image is sufficient to bias the perception of the current stimulus. In Experiment 2, participants received feedback about their responses in each trial of a numerosity estimation task, which was designed to assess whether providing external information about the accuracy of judgments would modulate serial dependence. Providing feedback significantly increased the attractive serial dependence effect, suggesting that external information at the level of judgment may modulate the weight of past perceptual information during the processing of the current image. Overall, our results support the idea that, although serial dependence may operate at a perceptual level, it originates from high-level processing stages at the level of abstract information processing and at the level of judgment.

Attractive serial dependence between memorized stimuli

Attractive serial dependence – a bias whereby the current stimulus appears more similar to the previous ones – is thought to reflect a stability mechanism integrating past and current visual signals. Prior work suggests that serial dependence originates from both perceptual and cognitive mechanisms, but the conditions under which this attractive bias occurs remain to be studied. In particular, whether serial dependence can occur solely from memory interference remains unclear. Here, we address this question by testing the hypothesis that if memory interference is sufficient to generate serial dependence, it should occur within memorized stimuli irrespective of the order of stimulus presentation. In Exp. 1, we used a numerosity estimation task in which participants estimated the number of dots of a briefly flashing dot-array comprising 8 to 32 dots. The pattern of serial dependence was found in that numerical estimates of a dot array were biased towards the numerosity of the preceding dot array. In Exp. 2, we presented a series of three such dot arrays, and cued the one to be estimated only after the whole series was presented, making the participants first form a memorized representation of the three dot arrays. The results show a pattern of attractive biases both in the forward (the stimulus presented before biases the one presented after) and the backward (the stimulus presented after biases the one presented before) directions. Overall, our results demonstrate that serial dependence can be induced solely from memory interference and that this interference can operate irrespective of the chronological order of the stimulus presentation.

The knowledge of the preceding number reveals a mature understanding of the number sequence

There is an ongoing debate concerning how numbers acquire numerical meaning. On the one hand, it has been argued that symbols acquire meaning via a mapping to external numerosities as represented by the approximate number system (ANS). On the other hand, it has been proposed that the initial mapping of small numerosities to the corresponding number words and the knowledge of the properties of counting list, especially the order relation between symbols, lead to the understanding of the exact numerical magnitude associated with numerical symbols. In the present study, we directly compared these two hypotheses in a group of preschool children who could proficiently count (most of the children were cardinal principle knowers). We used a numerosity estimation task to assess whether children have created a mapping between the ANS and the counting list (i.e., ANS-to-word mapping). Children also completed a direction task to assess their knowledge of the directional property of the counting list. That is, adding one item to a set leads to he next number word in the sequence (i.e., successor knowledge) whereas removing one item leads to the preceding number word (i.e., predecessor knowledge). Similarly, we used a visual order task to assess the knowledge that successive and preceding numbers occupy specific spatial positions on the visual number line (i.e., preceding: [?], [13], [14]; successive: [12], [13], [?]). Finally, children's performance in comparing the magnitude of number words and Arabic numbers indexed the knowledge of exact symbolic numerical magnitude. Approximately half of the children in our sample have created a mapping between the ANS and the counting list. Most of the children mastered the successor knowledge whereas few of them could master the predecessor knowledge. Children revealed a strong tendency to respond with the successive number in the counting list even when an item was removed from a set or the name of the preceding number on the number line was asked. Crucially, we found evidence that both the mastering of the predecessor knowledge and the ability to name the preceding number in the number line relate to the performance in number comparison tasks. Conversely, there was moderate/anecdotal evidence for a relation between the ANS-to-word mapping and number comparison skills. Non-rote access to the number sequence relates to knowledge of the exact magnitude associated with numerical symbols, beyond the mastering of the cardinality principle and domain-general factors.

Malleability of mappings between Arabic numerals and approximate quantities: Factors underlying individual differences and the relation to math

Humans tend to be inaccurate and inconsistent when estimating a large number of objects. Furthermore, we modify our estimates when feedback or a reference array is provided, indicating that the mappings between perceived numerosity and their corresponding numerals are largely malleable in response to calibration. However, there is great variability in response to calibration across individuals. Using uncalibrated and calibrated numerosity estimation conditions, the current study explored the factors underlying individual differences in the extent and nature of the malleability of numerosity estimation performance as a result of calibration in a sample of 71 undergraduate students. We found that individual differences in performance were reliable across conditions, and participants' responses to calibration varied greatly. Participants who were less consistent or had more proportionally spaced (i.e., linear) estimates before calibration tended to shift the distributions of their estimates to a greater extent. Higher calculation competence also predicted an increase in how linear participants' estimates were after calibration. Moreover, the effect of calibration was not continuous across numerosities within participants. This suggests that the mechanisms underlying numeral-numerosity mappings may be less systematic than previously thought and likely depend on cognitive mechanisms beyond representation of numerosities. Taken together, the mappings between numerosities and numerical symbols may not be stable and direct, but transient and mediated by task-related (e.g., strategic) mechanisms. Rather than estimation skills being foundational for math competence, math competence may also influence estimation skills. Therefore, numerosity estimation tasks are not a pure measure of number representations.

Unbounded number line estimation as a measure of numerical estimation.

Number magnitude estimation has been investigated over the last decades using different tasks including non-symbolic numerosity but also number line estimation tasks. Recently, a bi-directional mapping process was suggested for numerosity estimation accounting for underestimation in a perception version of the task (i.e., indicating the number of non-symbolic dots in a set) and overestimation in the corresponding production task (i.e., produce the number of dots indicated by a symbolic number). In the present study, we evaluated the generalizability of these estimation biases in perception and production tasks to bounded and unbounded number line estimation. Importantly, target numbers were underestimated/overestimated by participants in the perception/production version of numerosity estimation as well as unbounded number line estimation. However, this pattern was reversed for bounded number line estimation. Thereby, the present data indicate a conceptual similarity of unbounded number line estimation and the established non-symbolic numerosity estimation task as a measure of numerical estimation. Accordingly, this corroborates the notion that unbounded number line estimation may reflect a purer measure of number magnitude representation than the bounded task version. Furthermore, our findings strengthen the bi-directional mapping hypothesis for numerical estimation by providing evidence for its generalizability to unbounded number line estimation for the first time.

Task-irrelevant spatial dividers facilitate counting and numerosity estimation

Counting is characterized as a slow and error-prone action relying heavily on serial allocation of focused attention. However, quick and accurate counting is required for many real-world tasks (e.g., counting heads to ensure everyone is evacuated to a safe place in an emergency). Previous research suggests that task-irrelevant spatial dividers, which segment visual displays into small areas, facilitate focused attention and improve serial search. The present study investigated whether counting, which is also closely related to focused attention, can be facilitated by spatial dividers. Furthermore, the effect of spatial dividers on numerosity estimation, putatively dependent upon distributed attention, was also examined to provide insights into different types of number systems and different modes of visual attention. The results showed profound performance improvement by task-irrelevant spatial dividers in both counting and numerosity estimation tasks, indicating that spatial dividers may activate interaction between number and visual attention systems. Our findings provide the first evidence that task-irrelevant spatial dividers can be used to facilitate various types of numerical cognition.

Repulsive Serial Effects in Visual Numerosity Judgments.

We investigated how the approximate perceived numerosity of ensembles of visual elements is modulated by the numerosity of previously viewed ensembles depending on whether the first ensemble is held in visual working memory or not. We show that the numerosity of the previously seen ensemble has a repulsive effect, that is, a stimulus with high numerosity induces an underestimation of the following one and vice versa. This repulsive effect is present regardless of whether the first stimulus is memorized or not. While subtle changes of the experimental paradigm can have major consequences for the nature of interstimulus dependencies in perception, generally speaking the fact that we found such effects in a visual numerosity estimation task confirms that the process by which human observers produce estimates of the number of elements bears analogies to the processes that lead to the perception of visual dimensions such as orientation.

Counteracting estimation bias and social influence to improve the wisdom of crowds.

Aggregating multiple non-expert opinions into a collective estimate can improve accuracy across many contexts. However, two sources of error can diminish collective wisdom: individual estimation biases and information sharing between individuals. Here, we measure individual biases and social influence rules in multiple experiments involving hundreds of individuals performing a classic numerosity estimation task. We first investigate how existing aggregation methods, such as calculating the arithmetic mean or the median, are influenced by these sources of error. We show that the mean tends to overestimate, and the median underestimate, the true value for a wide range of numerosities. Quantifying estimation bias, and mapping individual bias to collective bias, allows us to develop and validate three new aggregation measures that effectively counter sources of collective estimation error. In addition, we present results from a further experiment that quantifies the social influence rules that individuals employ when incorporating personal estimates with social information. We show that the corrected mean is remarkably robust to social influence, retaining high accuracy in the presence or absence of social influence, across numerosities and across different methods for averaging social information. Using knowledge of estimation biases and social influence rules may therefore be an inexpensive and general strategy to improve the wisdom of crowds.

Sound sample detection and numerosity estimation using auditory display

This article investigates the effect of various design parameters of auditory information display on user performance in two basic information retrieval tasks. We conducted a user test with 22 participants in which sets of sound samples were presented. In the first task, the test participants were asked to detect a given sample among a set of samples. In the second task, the test participants were asked to estimate the relative number of instances of a given sample in two sets of samples. We found that the stimulus onset asynchrony (SOA) of the sound samples had a significant effect on user performance in both tasks. For the sample detection task, the average error rate was about 10% with an SOA of 100 ms. For the numerosity estimation task, an SOA of at least 200 ms was necessary to yield average error rates lower than 30%. Other parameters, including the samples' sound type (synthesized speech or earcons) and spatial quality (multichannel loudspeaker or diotic headphone playback), had no substantial effect on user performance. These results suggest that diotic, or indeed monophonic, playback with appropriately chosen SOA may be sufficient in practical applications for users to perform the given information retrieval tasks, if information about the sample location is not relevant. If location information was provided through spatial playback of the samples, test subjects were able to simultaneously detect and localize a sample with reasonable accuracy.

Over-estimation in numerosity estimation tasks: More than an attentional bias?

Over- and under-estimation have been observed in numerosity estimation and approximate arithmetic tasks. Two different models have been proposed to account for these reverse patterns of performance: 1) the bi-directional mapping account (Crollen, Castronovo, & Seron, 2011); 2) the operational momentum hypothesis (McCrink, Dehaene, & Dehaene-Lambertz, 2007). Our study was designed to examine whether the operational momentum could account for the over-estimation found in numerosity estimation tasks. To this aim, a series of 3 experiments involving a symbolic to non-symbolic numerical mapping and a rightward or leftward displacement along the mental number line were designed. Over-estimation was observed in these three tasks irrespective of the direction and size of the displacement to be done on the mental number line. These results thus clearly demonstrated that overestimation was not merely due to an attentional bias, but rather relied on the cognitive operation of mapping two differently scaled numerical representations.

Aging and Numerosity Estimation

In two experiments, young and older participants were asked to find the approximate number of dots in collections including between 40 and 460 dots. Experiment 1 showed that both age groups had comparable performance and no age-related differences in the power-function exponents for numerosity. Experiment 2 found that these age-related similarities were not due to speed-accuracy trade-offs or to compensation by older adults for potential age-related decline in numerosity estimation processes. Furthermore, young and older participants' estimation performance was influenced by physical features of stimuli only for very large numerosities, presumably because these are poorly represented in long-term memory. Implications of these findings for the further understanding of how participants accomplish numerosity estimation tasks and effects of aging in this domain are discussed.

Interactions between area and numerosity

The concept of filled area, i.e. the impressionistic ensemble of parts of a stimulus field occupied by dots, is used to account for various kinds of numerosity illusions due to perceptual interaction between a number of dots and their spatial arrangement. A measure of filled area is derived from a model for the perceptual clustering of dots on the basis of relative proximity. It is shown that the quantified concept of filled area successfully predicts illusion data from earlier studies. Subsequently, a two-alternatives forced-choice numerosity experiment is reported, the data of which gave further evidence of the predictive power of the filled-area hypothesis. Our approach is discussed with respect to spatial features of dot stimuli that might be in rivalry with the filled-area factor in numerosity-estimation tasks.

Differential Adoption of Norms by High-Status and Low-Status Members of Informal Groups

Individuals who were high-status and low-status members of real-groups of teenaged males were placed in task situations with E's 4 collaborators who gave prescribed arbitrary ranges of estimates in autokinesis and numerosity-judgment tasks. Individuals who held high status responded to the social clues in the situations more efficiently. A new judgment situation, “the shotgun,” provides a numerosity-estimation task which is statistically comparable to the classic autokinetic situation and is ego-involving for American teenaged males.