Numerical Comparison Task Research Articles

Contextual coherence increases perceived numerosity independent of semantic content.

Number perception emerges from multiple stages of visual processing. Understanding how systematic biases in number perception occur within a hierarchy of increasingly complex feature representations helps uncover the multistage processing underlying our visual number sense. Recent work demonstrated that reducing coherence of low-level visual attributes, such as color and orientation, systematically reduces perceived number. Here, we ask when in the visual processing hierarchy coherence affects numerosity perception and specifically whether the coherence effect is exclusive to low-level visual features or instead whether it can be driven by contextual or semantic relationships. We tested adults in an ordinal numerical comparison task with contextual coherence mathematically manipulated using a statistical model of visual object co-occurrence. Across several experiments, we found that arrays with high contextual coherence were perceived as numerically larger than arrays with low contextual coherence. This contextual coherence effect was not attenuated even when we reduced objects to texforms (unrecognizable images that preserve midlevel visual features) or removed semantic content from the images through box scrambling and diffeomorphic warping. Together, these results suggest that visual coherence derived from natural statistics of object co-occurrence systematically alters perceived numerosity at low-level visual processing, even before later stages at which items can be explicitly categorized and identified. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

The effect of spatial distance on numerical distance processing.

The close relationship between numerical and spatial representation has been widely studied. However, little is known regarding the influence of spatial distance on the processing of numerical distance. The purpose of this study was to examine this relationship by employing a modified numerical Stroop task, in which the spatial distance was either congruent or incongruent with the numerical distance. That is, numerical and spatial distances were either compatible with each other or incompatible. Experiment 1 demonstrated that when participants were directly requested to assess the numerical distance, spatial distance influenced task performance, thereby revealing a novel effect-the spatial-numerical distance congruency effect. Experiment 2 demonstrated that these relations are asymmetrical and revealed that numerical distance did not influence spatial distance when the numerical distance was task-irrelevant. Experiment 3 revealed that the spatial-numerical distance congruency effect can also be obtained automatically by employing a numerical comparison task, which is considered a marker for indirect distance processing. In addition, also tested across the three experiments was whether spatial alignment on the screen (i.e., left, centre, and right) can influence the spatial-numerical distance congruency effect. Results revealed that when numbers were presented more naturally (on the left and centre of the screen), a larger effect was obtained compared with when stimuli were presented on the right side. Together, these findings shed new light regarding the relationship between numerical distance and spatial distance and whether and how these aspects influence each other.

Understanding the social-emotional components of our "number sense": insights from a novel non-symbolic numerical comparison task.

A large body of work has identified a core sense of number supported by the Approximate Number System (ANS) that is present in infancy and across species. Although it is commonly assumed that the ANS directly processes perceptual input and is relatively independent from affective factors, some evidence points at a correlation between ANS performance and math anxiety. However, the evidence is mixed. We tested whether giving participants active control in completing a numerical task would change the relationship between math anxiety on performance. Adult participants (N = 103) completed a novel four-alternative-forced-choice non-symbolic numerical comparison task. In a repeated-measures design, participants either passively viewed different dot arrays or actively chose to view each array (i.e., active information-seeking) before deciding on the largest quantity. Participants also provided confidence judgments during the passive version of the task. We replicated the ratio-dependent signature in participants' accuracy in both the passive and active versions of the task using this novel paradigm, as well as in trial-level confidence judgments and information-seeking behavior. Participants' self-reported math anxiety significantly correlated with their accuracy on the passive version of the task. Critically, the correlation disappeared in the active version of the task. Gender also emerged as a predictor of confidence judgments and a moderator of the effect of task on overall accuracy and the effect of active information seeking on accuracy in the active version of the task. Exploratory analysis of estimated Weber Fraction suggests that these results may be driven by auxiliary factors instead of changes in ANS acuity. These findings have implications for understanding the relationship between math anxiety and performance on numerical tasks.

How does it feel? Passage of time judgments in speeded RT performance.

The relationship between duration perception and the feeling of time passing (passage of time) is not yet understood. In the present study, we assessed introspective reaction times (RT) and passage of time judgments in a speeded RT task. Task difficulty was manipulated in a numerical comparison task by numerical distance (distance from the number 45) and notation (digit vs. word). The results showed that both effects were reflected in introspective RTs, replicating previous results. Moreover, passage of time judgments showed a very similar pattern, with slower passage of time for more difficult comparisons. These results suggest that in the millisecond range judgments of duration and passage of time largely mirror each other when participants introspect about their own RT performance.

The plural counts: Inconsistent grammatical number hinders numerical development in preschoolers — A cross-linguistic study

The role of grammar in numerical development, and particularly the role of grammatical number inflection, has already been well-documented in toddlerhood. It is unclear, however, whether the influence of grammatical language structure further extends to more complex later stages of numerical development. Here, we addressed this question by exploiting differences between Polish, which has a complex grammatical number paradigm, leading to a partially inconsistent mapping between numerical quantities and grammatical number, and German, which has a comparatively easy verbal paradigm: 151 Polish-speaking and 123 German-speaking kindergarten children were tested using a symbolic numerical comparison task. Additionally, counting skills (Give-a-Number and count-list), and mapping between non-symbolic (dot sets) and symbolic representations of numbers, as well as working memory (Corsi blocks and Digit span) were assessed. Based on the Give-a-Number and mapping tasks, the children were divided into subset-knowers, CP-knowers-non-mappers, and CP-knowers-mappers.Linguistic background was related to performance in several ways: Polish-speaking children expectedly progressed to the CP-knowers stage later than German children, despite comparable non-numerical capabilities, and even after this stage was achieved, they fared worse in the numerical comparison task. There were also meaningful differences in spatial-numerical mapping between the Polish and German groups.Our findings are in line with the theory that grammatical number paradigms influence.the development of representations and processing of numbers, not only at the stage of acquiring the meaning of the first number-words but at later stages as well, when dealing with symbolic numbers.

Attention to number requires magnitude-specific inhibition

Recent studies have shown that the ability to process number in the face of conflicting dimensions of magnitude is a crucial aspect of numerosity judgments, relying in part on the inhibition of the non-numerical dimensions. Here we report, for the first time, that these inhibitory control processes are specific to the conflicting dimension of magnitude. Using a non-symbolic numerical comparison task adapted to a conflict adaptation paradigm on a group of 82 adults, we show that congruency effects between numerical and non-numerical information were reduced only when the conflicting dimension was the same in the preceding incongruent trial. Attention to number thus involves inhibitory control processes acting at a specific level of information. These results contribute to better characterize the domain general abilities involved in numerical cognition, and provide evidence for a specific interaction between numerosity perception and inhibitory control.

A shared numerical magnitude representation evidenced by the distance effect in frequency-tagging EEG

Humans can effortlessly abstract numerical information from various codes and contexts. However, whether the access to the underlying magnitude information relies on common or distinct brain representations remains highly debated. Here, we recorded electrophysiological responses to periodic variation of numerosity (every five items) occurring in rapid streams of numbers presented at 6 Hz in randomly varying codes—Arabic digits, number words, canonical dot patterns and finger configurations. Results demonstrated that numerical information was abstracted and generalized over the different representation codes by revealing clear discrimination responses (at 1.2 Hz) of the deviant numerosity from the base numerosity, recorded over parieto-occipital electrodes. Crucially, and supporting the claim that discrimination responses reflected magnitude processing, the presentation of a deviant numerosity distant from the base (e.g., base “2” and deviant “8”) elicited larger right-hemispheric responses than the presentation of a close deviant numerosity (e.g., base “2” and deviant “3”). This finding nicely represents the neural signature of the distance effect, an interpretation further reinforced by the clear correlation with individuals’ behavioral performance in an independent numerical comparison task. Our results therefore provide for the first time unambiguously a reliable and specific neural marker of a magnitude representation that is shared among several numerical codes.

Increasing entropy reduces perceived numerosity throughout the lifespan

Numerical illusions may provide a powerful window into the mechanisms that give rise to our visual number sense. Recent research has shown that similarly oriented elements appear more numerous than randomly oriented elements in an array. Here we examine whether the orientation coherence illusion is a more general byproduct of the effect of entropy on numerical information-processing. Participants engaged in an ordinal numerical comparison task where the color entropy of arrays was manipulated. We found that arrays with low color entropy were perceived as more numerous than arrays with high color entropy (Experiments 1 and 2), suggesting that the coherence illusion on numerosity perception is not specific to a particular visual property (e.g., orientation) but instead that the entropy of visual arrays more generally affects numerical processing. In Experiment 3, we explored the developmental trajectory of the color entropy effect in children aged 5 to 17 and found that the strength of the coherence illusion increases into adulthood, raising intriguing questions as to how perceptual experiences influence the progression of this numerosity illusion. We consider a recently proposed resource-rational model as a framework for understanding the entropy effect on numerosity perception under an information-theoretic perspective.

Differential processing of "small" and "large" multidigit numbers.

Little is known about the mental representation of large multidigit numbers that are usually beyond our personal experience. The present study explored the processing mechanisms of these numbers in a series of experiments, using the numerical comparison task. Experiment 1 included within and between-scale comparisons of multidigit numbers varying in their left digits (e.g., 8,000,000), with one group comparing small numbers (tens, hundreds, and thousands) and the other large ones (millions, billions, and trillions). In Experiment 2, comparisons of small (tens, hundreds) and large (millions, billions) multidigit numbers that varied in their left and right digits (e.g., 8,000,003) were presented in separate blocks. Experiment 3 presented small and large multidigit numbers (from tens to trillions) that varied in their left digits in the same block. We found novel compatibility effects between the left digit and scale components, as well as between the left digit, right digit, and scale components, and extended the previously reported unit-decade compatibility effect to larger scales. We also obtained global and scale distance effects for all scales in most conditions. Both compatibility and distance effects showed context dependency in large, but not small, multidigit numbers. Overall, these results demonstrate that small and large multidigit numbers are processed differently. We discuss these differences and propose a processing model that accounts for them. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Sometimes nothing is simply nothing: Automatic processing of empty sets.

Previous work using the numerical comparison task has shown that an empty set, the nonsymbolic manifestation of zero, can be represented as the smallest quantity of the numerical magnitude system. In this study, we examined whether an empty set can be represented as such under conditions of automatic processing in which deliberate processing of stimuli magnitudes is not required by the task. In Experiment 1, participants performed physical and numerical comparisons of empty sets (i.e., empty frames) and of other numerosities presented as framed arrays of 1 to 9 dots. The physical sizes of the frames varied within pairs. Both tasks revealed a size congruity effect (SCE) for comparisons of non-empty sets. In contrast, comparisons to empty sets produced an inverted SCE in the physical comparison task, whereas no SCE was found for comparisons to empty sets in the numerical comparison task. In Experiment 2, participants performed an area comparison task using the same stimuli as Experiment 1 to examine the effect of visual cues on the automatic processing of empty sets. The results replicated the findings of the physical comparison task in Experiment 1. Taken together, our findings indicate that empty sets are not perceived as "zero," but rather as "nothing," when processed automatically. Hence, the perceptual dominance of empty sets seems to play a more important role under conditions of automatic processing, making it harder to abstract the numerical meaning of zero from empty sets.

Quantitative and Qualitative Differences in the Canonical and the Reverse Distance Effect and Their Selective Association With Arithmetic and Mathematical Competencies

Understanding the relationship between symbolic numerical abilities and individual differences in mathematical competencies has become a central research endeavor in the last years. Evidence on this foundational relationship is often based on two behavioral signatures of numerical magnitude and numerical order processing: thecanonicaland thereverse distance effect.Theformerindicates faster reaction times for the comparison of numerals that are far in distance (e.g., 2 8) compared to numerals that are close in distance (e.g., 2 3). The latter indicates faster reaction times for the ordinal judgment of numerals (i.e., are numerals in ascending/descending order) that are close in distance (e.g., 2 3 4) compared to numerals that are far in distance (e.g., 2 4 6). While a substantial body of literature has reported consistent associations between thecanonical distance effectand arithmetic abilities, rather inconsistent findings have been found for thereverse distance effect. Here, we tested the hypothesis that estimates of thereverse distance effectshow qualitative differences (i.e., not all participants show areverse distance effectin the expected direction) rather than quantitative differences (i.e., all individuals show areverse distance effect, but to a different degree), and that inconsistent findings might be a consequence of this variation. We analyzed data from 397 adults who performed a computerized numerical comparison task, a computerized numerical order verification task (i.e., are three numerals presented in order or not), a paper pencil test of arithmetic fluency, as well as a standardized test to assess more complex forms of mathematical competencies. We found discriminatory evidence for the two distance effects. While estimates of thecanonical distance effectshowed quantitative differences, estimates of thereverse distance effectshowed qualitative differences.Comparisons between individuals who demonstrated an effect and individuals who demonstrated noreverse distance effectconfirmed a significant moderation on the correlation with mathematical abilities. Significantly larger effects were found in the group who showed an effect. These findings confirm that estimates of thereverse distance effectare subject to qualitative differences and that we need to better characterize the underlying mechanisms/strategies that might lead to these qualitative differences.

Numerical Comparisons and Sex-Related Brain Connectivity

Despite the recent literature on sex-related anatomic, maturational and functional brain differences, the study of significant individual developments in math learning and achievement has scarcely approached this perspective. We aimed to compare the influence of sex in functional brain connectivity and behavioral measures in a numerical comparison task. Therefore, a group of school children with ages from 8 to 11 years old was evaluated during a number comparison task. Even though the behavioral performance was similar across the sexes, males distinctly showed a significant correlation between their math WRAT-4 scores and the number of correct responses in the experimental task and working memory scores. Besides, the analysis of the concurrent EEG during task performance showed that males comparatively had a greater brain left intra-hemispheric connectivity, as well as greater interhemispheric connectivity, particularly in Theta and Alpha bands during task performing -as compared to resting-. In contrast, females showed a significantly different decrement of brain connectivity in the Alpha band from resting to task performing. Present results are interpreted as probably reflecting sex-related maturational dissimilarities in neurodevelopment, along with the progressive development of more efficient cognitive strategies, processes running not necessarily parallel in both sexes. 

Mutual influences between numerical and non-numerical quantities in comparison tasks.

Humans possess a numerical intuition that allows them to manipulate large non-symbolic quantities. This ability has been broadly assessed with the help of number comparison tasks involving simultaneously displayed arrays. Many authors pointed out that the manipulation (or the lack thereof) of non-numerical features deeply impacts performance in these tasks, but the specific nature of this influence is not clear. This study investigates the interaction between numerical and non-numerical quantity judgement tasks. Adult participants performed five distinct comparison tasks, each based on a target dimension: numerosity, total area, dot size, convex hull, and mean occupancy. We manipulated the relation between the target and the other dimensions to measure their respective influence on task performance. Results showed that total area and convex hull substantially affected numerosity comparisons. The number of dots conversely acted as an informative dimension when participants had to make a decision based on the total area or the convex hull. Our results illustrate that adults flexibly use non-target dimensions as visual cues to perform comparison judgements. Overall, this suggests that the influence found in numerical comparison tasks is explicit and deliberate rather than due to implicit visual integration processes.

Nonsymbolic and Symbolic Numerical Magnitude Processing in the Brazilian Children with Mathematics Difficulties.

ABSTRACTIt is still debated if the main deficit in mathematical difficulties (MD) is nonsymbolic or symbolic numerical magnitude processing.Objectives:In the present study, our main goal was to investigate nonsymbolic and symbolic numerical magnitude processing in MD and the relationship between these abilities and arithmetic.Methods:The Brazilian school-age children with MD completed a nonsymbolic and a symbolic numerical magnitude comparison task and an arithmetic task. We compared their performance with a group of children with typical achievement (TA) and investigated the association between numerical magnitude processing and arithmetic with a series of regression analyses.Results:Results indicated that children with MD had low performance in the nonsymbolic numerical magnitude comparison task. Performance in both nonsymbolic and symbolic numerical magnitude comparison tasks predicted arithmetic abilities in children with TA, but not in children with MD.Conclusions:These results indicate that children with MD have difficulties in nonsymbolic numerical magnitude processing, and do not engage basic numerical magnitude representations to solve arithmetic.

NASCO: A new method and program to generate dot arrays for non-symbolic number comparison tasks

Basic numerical abilities are generally assumed to influence more complex cognitive processes involving numbers, such as mathematics. Yet measuring non-symbolic number abilities remains challenging due to the intrinsic correlation between numerical and non-numerical dimensions of any visual scene. Several methods have been developed to generate non-symbolic stimuli controlling for the latter aspects but they tend to be difficult to replicate or implement. In this study, we describe the NASCO method, which is an extension to the method popularized by Dehaene, Izard, and Piazza (2005). Their procedure originally controlled for two visual dimensions that are mediated by Number: Total Area and Item Size (i.e., N = TA/IS). Here, we additionally propose to control for another twofold dimension related to the array extent, which is also mediated by Number: Convex Hull Area and Mean Occupancy (i.e., N = CH/MO). We illustrate the NASCO method with a MATLAB app—NASCO app—that allows easy generation of dot arrays for a visually controlled assessment of non-symbolic numerical abilities. Results from a numerical comparison task revealed that the introduction of this twofold dimension manipulation substantially affected young adults’ performance. In particular, we did not replicate the relation between non-symbolic number abilities and arithmetic skills. Our findings open the debate about the reliability of previous results that did not take into account visual features related to the array extent. We then discuss the strengths of NASCO method to assess numerical ability, as well as the benefits of its straightforward implementation in NASCO app for researchers.

Non-symbolic representation is modulated by math anxiety and cognitive inhibition while symbolic representation not.

The present study aimed to investigate whether there are basic numerical processing deficits in high math anxiety (HMA) individuals and examine the effects of cognitive inhibition on the performance of high and low math anxiety (LMA) individuals. 35 undergraduate students were recruited to perform a numerical comparison task, a numerical Stroop task, a dot comparison task, and a dot Stroop task. Results showed the following: (1) Compared with LMA group, HMA group reacted more slowly and exhibited more fixation counts in non-symbolic representation task. (2) Inhibition condition significantly increased HMA individuals' fixation durations, whereas the fixation durations of LMA individuals were similar between inhibition and non-inhibition condition. The results demonstrated that MA might temporarily occupy individuals' working memory resources. (3) In non-symbolic representation task, the effect of numerical ratio on fixation counts was larger for HMA group than that for LMA group under inhibition condition, indicating MA might arise from a basic level deficit in numerosity processing.

Magnitude processing in populations with spina-bifida: The role of visuospatial and working memory processes

People with Spina Bifida usually experience difficulties with mathematics. In a series of other developmental disorders, a magnitude processing deficit was considered to be the main source of subsequent difficulties in mathematics. The processing of magnitude could be numerical (which is the larger number) or non-numerical such as spatial (e.g., which is the longer?) or temporal (which one last longer?) for instance. However, no study yet has examined directly magnitude processes in a population with Spina Bifida. On the other hand, recent studies in people with genetic syndromes have suggested that visuospatial and working memory processes play an important role in magnitude processing, including number magnitude. Therefore, in this study we explored for the first time magnitude representation using several tasks with different visuospatial and working memory processing requirements, cognitive skills frequently impaired in Spina Bifida. Results showed children with SB presented a global magnitude processing deficit for non-numerical and numerical comparison tasks, but not in symbolic number magnitude tasks compared to controls. Importantly, visuospatial skills and working memory abilities could partially explain the differences between groups in comparison and estimation tasks. This study proposes that magnitude processing difficulties in children with SB could be due to higher cognitive factors such as visuospatial and working memory processes.

The semantic control network mediates the relationship between symbolic numerical order processing and arithmetic performance in children

Behavioural and neuroimaging studies have recently demonstrated that symbolic numerical order processing (i.e., deciding whether numbers are in an increasing/decreasing sequence or not) and symbolic numerical magnitude processing (e.g., deciding which of two numerals is larger) engage different cognitive mechanisms and brain regions. Because of this dissociation, growing interest has emerged to better understand the neurocognitive mechanisms of symbolic numerical order processing and their relationship to individual differences in arithmetic performance. In the present functional imaging work, we further investigated this link in a group of thirty children (7.2–10.25 years) from elementary school, who completed a symbolic numerical order verification (are the numbers going up? e.g., 1-2-3) and a symbolic numerical magnitude comparison task (which is the larger number? e.g., 5–7) inside the scanner, as well as an arithmetic fluency test outside the scanner. Behavioural results demonstrated the unique role of numerical order to predict children's arithmetic skills and confirmed its mediating power to explain the association between numerical magnitude and arithmetic performance. Imaging results showed a significant association between numerical order and arithmetic in the intersection of the right inferior frontal gyrus and insula, as well as the posterior middle temporal gyrus. An age-dependent change in brain activity was found in the left intraparietal sulcus. These findings solidify the developmental importance of symbolic numerical order processing in children and provide new evidence that the semantic control network mediates the relationship with arithmetic performance.

Deconstructing South African Grade 1 learners’ awareness of number in terms of cardinality, ordinality and relational understandings

The cardinal and ordinal aspects of number have been widely written about as key constructs that need to be brought together in children’s understanding in order for them to appreciate the idea of numerosity. In this paper, we discuss similarities and differences in the ways in which understandings not only of ordinality, cardinality but also additive and multiplicative relations have been theorized. We examine how the connections between these can be considered through a focus on number line representations and children positioning and comparing numbers. The responses of a cohort of South African Grade 1 learners’ (6- and 7-year-olds) to a numerical magnitude estimation task and to a numerical comparison task are analysed and the findings compared to those in the international literature, some of which argue that children’s early, informal, understandings of cardinality and ordinality are underpinned by an intuitive logarithmic model relating number order and size. A main finding presented here is that the responses from learners in this study exhibited a better fit with an exponential model of the relationship between cardinality and ordinality. These findings raise questions about whether some of the findings in previous research are as universal as sometimes claimed.

A brain connectivity characterization of children with different levels of mathematical achievement based on graph metrics.

Recent studies aiming to facilitate mathematical skill development in primary school children have explored the electrophysiological characteristics associated with different levels of arithmetic achievement. The present work introduces an alternative EEG signal characterization using graph metrics and, based on such features, a classification analysis using a decision tree model. This proposal aims to identify group differences in brain connectivity networks with respect to mathematical skills in elementary school children. The methods of analysis utilized were signal-processing (EEG artifact removal, Laplacian filtering, and magnitude square coherence measurement) and the characterization (Graph metrics) and classification (Decision Tree) of EEG signals recorded during performance of a numerical comparison task. Our results suggest that the analysis of quantitative EEG frequency-band parameters can be used successfully to discriminate several levels of arithmetic achievement. Specifically, the most significant results showed an accuracy of 80.00% (α band), 78.33% (δ band), and 76.67% (θ band) in differentiating high-skilled participants from low-skilled ones, averaged-skilled subjects from all others, and averaged-skilled participants from low-skilled ones, respectively. The use of a decision tree tool during the classification stage allows the identification of several brain areas that seem to be more specialized in numerical processing.