Number Comparison Task Research Articles

Toward a unified account of nonsymbolic and symbolic representations of number: Insights from a combined psychophysical-computational approach.

There is an ongoing, vibrant debate about whether numerical information in both nonsymbolic and symbolic notations would be supported by different neurocognitive systems or rather by a common preverbal approximate number system, which is ratio dependent and follows Weber's law. Here, we propose that the similarities between nonsymbolic and symbolic number processing can be explained based on the principle of efficient coding. To probe this hypothesis we employed a new empirical approach, by predicting the behavioural performance in number comparison tasks with symbolic (i.e., number words) and nonsymbolic (i.e., arrays of dots) information not only from numerical ratio, but for the first time also from natural language data. That is, we used data extracted from vector-space models that are informative about the distributional pattern of number-words usage in natural language. Results showed that linguistic estimates predicted the behavioural performance in both symbolic and nonsymbolic tasks. However, and critically, our results also showed a task-dependent dissociation: linguistic data better predicted the performance in the symbolic task, whereas real numerical ratio better predicted the performance in the nonsymbolic task. These findings indicate that efficient coding of environmental regularities is an explanatory principle of human behavior in tasks involving numerical information. They also suggest that the ability to discriminate a stimulus from similar ones varies as a function of the specific statistical structure of the considered learning environment.

Sharpening, focusing, and developing: A study of change in nonsymbolic number comparison skills and math achievement in 1st grade.

Children's ability to discriminate nonsymbolic number (e.g., the number of items in a set) is a commonly studied predictor of later math skills. Number discrimination improves throughout development, but what drives this improvement is unclear. Competing theories suggest that it may be due to a sharpening numerical representation or an improved ability to pay attention to number and filter out non-numerical information. We investigate this issue by studying change in children's performance (N=65) on a nonsymbolic number comparison task, where children decide which of two dot arrays has more dots, from the middle to the end of 1st grade (mean age at time 1=6.85 years old). In this task, visual properties of the dot arrays such as surface area are either congruent (the more numerous array has more surface area) or incongruent. Children rely more on executive functions during incongruent trials, so improvements in each congruency condition provide information about the underlying cognitive mechanisms. We found that accuracy rates increased similarly for both conditions, indicating a sharpening sense of numerical magnitude, not simply improved attention to the numerical task dimension. Symbolic number skills predicted change in congruent trials, but executive function did not predict change in either condition. No factor predicted change in math achievement. Together, these findings suggest that nonsymbolic number processing undergoes development related to existing symbolic number skills, development that appears not to be driving math gains during this period. Children's ability to discriminate nonsymbolic number improves throughout development. Competing theories suggest improvement due to sharpening magnitude representations or changes in attention and inhibition. The current study investigates change in nonsymbolic number comparison performance during first grade and whether symbolic number skills, math skills, or executive function predict change. Children's performance increased across visual control conditions (i.e., congruent or incongruent with number) suggesting an overall sharpening of number processing. Symbolic number skills predicted change in nonsymbolic number comparison performance.

Pick the smaller number: No influence of linguistic markedness on three-digit number processing

The symbolic number comparison task has been widely used to investigate the cognitive representation and underlying processes of multi-digit number processing. The standard procedure to establish numerical distance and compatibility effects in such number comparison paradigms usually entails asking participants to indicate the larger of two presented multi-digit Arabic numbers rather than to indicate the smaller number. In terms of linguistic markedness, this procedure includes the unmarked/base form in the task instruction (i.e., large). Here we evaluate distance and compatibility effects in a three-digit number comparison task observed in Bahnmueller et al. (2015, https://doi.org/10.3389/fpsyg.2015.01216) using a marked task instruction (i.e., ‘pick the smaller number’). Moreover, we aimed at clarifying whether the markedness of task instruction influences common numerical effects and especially componential processing as indexed by compatibility effects. We instructed German- and English-speaking adults (N = 52) to indicate the smaller number in a three-digit number comparison task as opposed to indicating the larger number in Bahnmueller et al. (2015). We replicated standard effects of distance and compatibility in the new pick the smaller number experiment. Moreover, when comparing our findings to Bahnmueller et al. (2015), numerical effects did not differ significantly between the two studies as indicated by both frequentist and Bayesian analysis. Taken together our data suggest that distance and compatibility effects alongside componential processing of multi-digit numbers are rather robust against variations of linguistic markedness of task instructions.

Reactive and proactive cognitive control as underlying processes of number processing in children

Cognitive control is crucial to resolve conflict in tasks such as the flanker task. Reactive control is used when conflict is rare, whereas proactive control is more efficient in situations where conflict is frequent. Macizo and Herrera (Psychological Research, 2013, Vol. 77, pp. 651–658) found that these two control processes can also underlie two-digit number comparison in adults. Specifically, they observed that the unit–decade compatibility effect decreased in a block containing many conflict trials as compared with a block containing few conflict trials (i.e., a list-wide proportion congruency effect). In the current study, we assessed whether this finding also applies to children (7-, 9-, and 11-year-olds). Participants performed a flanker task and a two-digit number comparison task. In both tasks, the proportion of conflict was manipulated (80% vs. 20%). Results from the flanker task showed a typical list-wide proportion congruency effect in reaction times in all participating age groups. In the number comparison task, we observed list-wide proportion congruency effects in both reaction times and error rates, which did not interact with age. Our findings support the assumption that children as young as 7 years can effectively use proactive and reactive control strategies. We showed that this effect is not limited to standardized artificial laboratory tasks, such as the flanker task, but also underlies more daily life tasks, such as the processing of Arabic numbers.

The interaction between numerical and continuous non-numerical magnitudes in a double change detection paradigm.

In the field of numerical cognition, researchers conventionally assess nonsymbolic numerical abilities with the help of number comparison tasks, in which participants need to compare two arrays. Many studies emphasized that visual (non-numerical) dimensions can serve as strategic cues and influence the decision on numerosity in these tasks. In this study, we suggest the use of a novel paradigm based on the change detection paradigm. Here, participants had to simultaneously pay attention to numerical changes and visual changes on a target non-numerical dimension (individual area, total area, field area, or density). Participants had to detect changes relative to the two dimensions and press response keys indicating either number change or visual change or press both keys. In such a double change detection paradigm, and unlike number comparison tasks, target and covarying dimensions cannot serve as cues to influence the numerical decision. We found that numerical change detection was excellent and stable across the conditions. Further, participants were more likely to falsely consider visual changes as numerical changes than the other way around. Lastly, when both dimensions varied, participants more frequently missed visual changes than numerical changes. Overall, our findings show that numerosity was the most salient visual dimension. From a methodological perspective, such a double change detection paradigm could be of critical interest to assess numerical abilities for future studies. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Detours increase local knowledge-Exploring the hidden benefits of self-control failure.

Self-control enables people to override momentary thoughts, emotions, or impulses in order to pursue long-term goals. Good self-control is a predictor for health, success, and subjective well-being, as bad self-control is for the opposite. Therefore, the question arises why evolution has not endowed us with perfect self-control. In this article, we draw some attention to the hidden benefits of self-control failure and present a new experimental paradigm that captures both costs and benefits of self-control failure. In an experiment, participants worked on three consecutive tasks: 1) In a transcription task, we manipulated how much effortful self-control two groups of participants had to exert. 2) In a number-comparison task, participants of both groups were asked to compare numbers and ignore distracting neutral versus reward-related pictures. 3) After a pause for recreation, participants were confronted with an unannounced recognition task measuring whether they had incidentally encoded the distracting pictures during the previous number-comparison task. The results showed that participants who exerted a high amount of effortful self-control during the first task shifted their priorities and attention toward the distractors during the second self-control demanding task: The cost of self-control failure was reflected in worse performance in the number-comparison task. Moreover, the group which had exerted a high amount of self-control during the first task and showed self-control failure during the second task was better in the unannounced third task. The benefit of self-control failure during number comparison was reflected in better performance during the recognition task. However, costs and benefits were not specific for reward-related distractors but also occurred with neutral pictures. We propose that the hidden benefit of self-control failure lies in the exploration of distractors present during goal pursuit, i.e. the collection of information about the environment and the potential discovery of new sources of reward. Detours increase local knowledge.

Number is not just an illusion: Discrete numerosity is encoded independently from perceived size

While seminal theories suggest that nonsymbolic visual numerosity is mainly extracted from segmented items, more recent views advocate that numerosity cannot be processed independently of nonnumeric continuous features confounded with the numerical set (i.e., such as the density, the convex hull, etc.). To disentangle these accounts, here we employed two different visual illusions presented in isolation or in a merged condition (e.g., combining the effects of the two illusions). In particular, in a number comparison task, we concurrently manipulated both the perceived object segmentation by connecting items with Kanizsa-like illusory lines, and the perceived convex-hull/density of the set by embedding the stimuli in a Ponzo illusion context, keeping constant other low-level features. In Experiment 1, the two illusions were manipulated in a compatible direction (i.e., both triggering numerical underestimation), whereas in Experiment 2 they were manipulated in an incompatible direction (i.e., with the Ponzo illusion triggering numerical overestimation and the Kanizsa illusion numerical underestimation). Results from psychometric functions showed that, in the merged condition, the biases of each illusion summated (i.e., largest underestimation as compared with the conditions in which illusions were presented in isolation) in Experiment 1, while they averaged and competed against each other in Experiment 2. These findings suggest that discrete nonsymbolic numerosity can be extracted independently from continuous magnitudes. They also point to the need of more comprehensive theoretical views accounting for the operations by which both discrete elements and continuous variables are computed and integrated by the visual system.

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. 

The missing link: Global-local processing relates to number-magnitude processing in women.

Number processing and visual hierarchical processing (global-local processing) have much in common. However, the shared aspects of number processing and global-local processing have not been investigated so far. Most visual stimuli are hierarchical with global structures made up of local parts. Processing of global and local aspects occurs in parallel and the global advantage effect indicates faster reactions to global than local aspects. Likewise, multidigit numbers can be represented holistically (whole number magnitudes) or in a decomposed fashion (single digit magnitudes). During comparison of 2-digit numbers, the unit-decade compatibility effect indicates slower responses when the larger number contains the smaller unit digit and has been suggested as a measure of how strongly participants rely on decomposed number representations. However, this interpretation of the compatibility effect is still controversial and a link between global-local processing and the individual tendency to rely on decomposed representations of multidigit numbers remains to be established. To that end we assessed whether the compatibility effect during number comparison was related to various measures of global advantage. To answer this question we drew upon existing data from participants who had completed both, the number comparison task and 2 global-local tasks. Results show that the compatibility effect is indeed negatively related to several measures of global advantage in women, with no evidence for such a relationship in men. These results demonstrate that global-local processing transcends into the numerical domain but also suggest that the compatibility effect reflects different mechanisms in men and women. (PsycInfo Database Record (c) 2021 APA, all rights reserved).

Does Location Uncertainty Modulate Unconscious Processing Under Continuous Flash Suppression?

Previous research suggests that selective spatial attention is a determining factor for unconscious processing under continuous flash suppression (CFS), and specifically, that inattention toward stimulus location facilitates its unconscious processing by reducing the depth of CFS (Eo et al., 2016). The aim of our study was to further examine this modulation-by-attention model of CFS using a number priming paradigm. Participants (N = 26) performed a number comparison task on a visible target number (“compare target to five”). Prime-target pairs were either congruent (both smaller or larger than five) or incongruent. Spatial attention toward the primes was varied by manipulating the uncertainty of the primes’ location. Based on the modulation-by-attention model, we hypothesized the following: In trials with uncertain prime location, RTs for congruent prime-target pairs should be faster than for incongruent ones. In trials with certain prime location, RTs for congruent versus incongruent prime-target pairs should not differ. We analyzed our data with sequential Bayes factors (BFs). Our data showed no effect of location uncertainty on unconscious priming under CFS (BF0+ = 5.16). However, even visible primes only weakly influenced RTs. Possible reasons for the absence of robust number priming effects in our study are discussed. Based on exploratory analyses, we conclude that the numerical order of prime and target resulted in a response conflict and interfered with the predicted priming effect.

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.

A robust electrophysiological marker of spontaneous numerical discrimination

Humans have a Number Sense that enables them to represent and manipulate numerical quantities. Behavioral data suggest that the acuity of numerical discrimination is predictively associated with math ability—especially in children—but some authors argued that its assessment is problematic. In the present study, we used frequency-tagged electroencephalography to objectively measure spontaneous numerical discrimination during passive viewing of dot or picture arrays in healthy adults. During 1-min sequences, we introduced periodic numerosity changes and we progressively increased the magnitude of such changes every ten seconds. We found significant brain synchronization to the periodic numerosity changes from the 1.2 ratio over medial occipital regions, and amplitude strength increased with the numerical ratio. Brain responses were reliable across both stimulus formats. Interestingly, electrophysiological responses also mirrored performances on a number comparison task and seemed to be linked to math fluency. In sum, we present a neural marker of numerical acuity that is passively evaluated in short sequences, independent of stimulus format and that reflects behavioural performances on explicit number comparison tasks.

Only Approximately the Same: Approximate Number Skills in Typically Developing Children and Children with Down Syndrome

ABSTRACT The approximate number system (ANS) is foundational to numerical cognition. Whereas some studies hint at intact ANS skills in individuals with Down syndrome (DS), others report impaired ANS abilities. To reconcile these discrepant findings, we investigated ANS skills of children with DS and typically developing (TD) children in a non-symbolic number comparison task. Children had to indicate which of two characters possessed more dots. The two sets of dots differed by ratios of 1:2 to 4:5. While overall DS and TD participants’ performance was comparable, accuracy levels differed for the two groups depending on the ratio presented. Both groups performed similarly for ‘easier’ ratios, however, TD children outperformed children with DS when they had to discriminate ‘harder’ ratios. Children with DS may thus face specific problems when discriminating ratios that are more demanding. The deviating ANS skills could potentially be related to problems with other symbolic maths abilities.

First fixation duration as a bottom-up measure during symbolic and non-symbolic numerical comparisons (La duración de la primera fijación como medida bottom-up al comparar cantidades simbólicas y no simbólicas)

ABSTRACT The ability to represent quantities, either in a symbolic or non-symbolic format, predicts calculation skills. Recording the saccades during gazing at these stimuli allows us to infer the cognitive processing taking place at that moment. The purpose of this research study was to analyse the duration of the first fixation during symbolic and non-symbolic numerical stimuli. We evaluated 32 university students through symbolic and non-symbolic number comparison tasks while recording their eye movements. In addition, they answered a calculation task. Participants spent more time looking first at the stimulus of greater magnitude during the execution of the comparison task (regardless of the presentation format). This measure was not a predictor of calculation skills. The duration of the first fixation was interpreted as a bottom-up measure during numerical stimuli perception, and therefore during the initial stage of cognitive processing.

Eye-movements during number comparison: Associations to sex and sex hormones

Multi-digit numbers are of a hierarchical nature with whole number magnitudes depending on digit magnitudes. Processing of multi-digit numbers can occur in a holistic or decomposed fashion. The unit-decade compatibility effect during number comparison is often used as a measure of decomposed processing. It refers to the fact that performance is reduced when the larger number contains the smaller unit digit (e.g. 73 vs. 26). It has been demonstrated that women show a larger compatibility effect than men, which is in accordance with their general tendency towards focusing on stimulus details during processing of visual hierarchical stimuli (local processing style). Such a local processing style has been related to higher progesterone and lower testosterone levels. One method to study individual processing styles is eye-tracking. The aim of the present study was to examine whether sex and sex hormones (estradiol, progesterone, testosterone) relate to eye movement behavior in the number comparison task. Unlike previous studies we found no evidence for sex differences in the behavioral compatibility effect. Nevertheless, women look more often and longer at individual digits and show a stronger compatibility effect in fixation durations compared to men, while men show more saccades between numbers than women. Estradiol and progesterone were related to fewer fixations and shorter fixation durations and more saccades between numbers in men, but not in women. Furthermore, the compatibility effect in the number of fixations and fixation durations was negatively related to testosterone in women. In summary, this is the first study to demonstrate sex differences and sex hormone influences on eye gaze behavior during number comparison.

Shared Numerosity Representations Across Formats and Tasks Revealed with 7 Tesla fMRI: Decoding, Generalization, and Individual Differences in Behavior.

Debate continues on whether encoding of symbolic number is grounded in nonsymbolic numerical magnitudes. Nevertheless, fluency of perceiving both number formats, and translating between them, predicts math skills across the life span. Therefore, this study asked if numbers share cortical activation patterns across formats and tasks, and whether neural response to number predicts math-related behaviors. We analyzed patterns of neural activation using 7 Tesla functional magnetic resonance imaging in a sample of 39 healthy adults. Discrimination was successful between numerosities 2, 4, 6, and 8 dots and generalized to activation patterns of the same numerosities represented as Arabic digits in the bilateral parietal lobes and left inferior frontal gyrus (IFG) (and vice versa). This indicates that numerosity-specific neural resources are shared between formats. Generalization was also successful across tasks where participants either identified or compared numerosities in bilateral parietal lobes and IFG. Individual differences in decoding did not relate to performance on a number comparison task completed outside of the scanner, but generalization between formats and across tasks negatively related to math achievement in the parietal lobes. Together, these findings suggest that individual differences in representational specificity within format and task contexts relate to mathematical expertise.

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.

Number Magnitude Processing and Verbal Working Memory in Children with Mild Intellectual Disabilities

ABSTRACT This study examined if children (Mage = 14.60) with Mild Intellectual Disabilities (MID) display weaknesses in number processing and verbal working memory. An age-matched and mental age-matched (MA, Mage = 6.17) design extended by a group of 9–10-year-olds, and a group of 11–12-year-olds were used. The MID children’s working memory was equal to the MA group but poorer than the other groups. On number tasks, the MID group was faster than the MA group but slower than the other groups. All groups obtained equal Weber fraction scores and distance effects on the number comparison tasks. The MID group performed subitizing and counting faster than the MA group, but slower than the 11–12-year-olds. The results demonstrate that number processing and working memory in children with MID is characterized by a developmental delay, not a deficit. Their main problem is to access the quantitative meaning of Arabic numerals. The development of different types of cognitive abilities is differently affected by educational experience and intellectual ability. The innate number system appears to be unaffected by intellectual capacity or educational experience, while the innate working memory ability is affected by intellectual capacity but not by educational experience. Culturally acquired symbolic number abilities are strongly affected by educational experience.

Evaluating car license auction mechanisms: Theory and experimental evidence

In Singapore and many fast-growing cities of China, auctions are used to implement a quota system of car ownership. Three such cities where influential auction formats have been developed for allocating car licenses are the Asian metropolises of Singapore, Shanghai and Guangzhou, with other cities following suit. The current paper examines car auction formats both theoretically and experimentally, with the purpose of maximizing social welfare by improving efficiency and reducing misunderstandings. Reaction time is introduced as an essential non-economic factor of model bidding behaviors in these car license auctions. At the theoretical level, this paper finds that reaction time causes inefficient allocations in the Shanghai auction, but not in the Singapore or Guangzhou auctions. The experimental results are consistent with the theoretical prediction that late bids prevail in all of these auction formats, but only lead to inefficient allocations in the Shanghai auction. Additionally, the subjects' reaction time in the Shanghai treatment was scored by conducting a number comparison task, finding a positive correlation between the probability of winning and the reaction time score.

Various ways to determine rational number size: an exploration across primary and secondary education

Understanding rational numbers is a complex task for primary and secondary school students. Previous research has shown that a possible reason is students’ tendency to apply the properties of natural numbers (inappropriately) when they are working with rational numbers (a phenomenon called natural number bias). Focusing on rational number comparison tasks, recent research has shown that other incorrect strategies such as gap thinking or reverse bias can also explain these difficulties. The present study aims to investigate students’ different ways of thinking when working on fraction and decimal comparison tasks. The participants were 1,262 primary and secondary school students. A TwoStep Cluster Analysis revealed six different student profiles according to their way of thinking. Results showed that while students’ reasoning based on the properties of natural numbers decreased along primary and secondary school, almost disappearing at the end of secondary school, students’ reasoning based on gap thinking increased along these grades. This result seems to indicate that when students overcome their reliance on natural numbers, they enter a stage of qualitatively different errors before finally reaching the stage of correct understanding.