Number Comparison Task Research Articles

Symbolic Number Comparison Is Not Processed by the Analog Number System: Different Symbolic and Non-symbolic Numerical Distance and Size Effects.

HIGHLIGHTS We test whether symbolic number comparison is handled by an analog noisy system.Analog system model has systematic biases in describing symbolic number comparison.This suggests that symbolic and non-symbolic numbers are processed by different systems.Dominant numerical cognition models suppose that both symbolic and non-symbolic numbers are processed by the Analog Number System (ANS) working according to Weber's law. It was proposed that in a number comparison task the numerical distance and size effects reflect a ratio-based performance which is the sign of the ANS activation. However, increasing number of findings and alternative models propose that symbolic and non-symbolic numbers might be processed by different representations. Importantly, alternative explanations may offer similar predictions to the ANS prediction, therefore, former evidence usually utilizing only the goodness of fit of the ANS prediction is not sufficient to support the ANS account. To test the ANS model more rigorously, a more extensive test is offered here. Several properties of the ANS predictions for the error rates, reaction times, and diffusion model drift rates were systematically analyzed in both non-symbolic dot comparison and symbolic Indo-Arabic comparison tasks. It was consistently found that while the ANS model's prediction is relatively good for the non-symbolic dot comparison, its prediction is poorer and systematically biased for the symbolic Indo-Arabic comparison. We conclude that only non-symbolic comparison is supported by the ANS, and symbolic number comparisons are processed by other representation.

A rapid, objective and implicit measure of visual quantity discrimination

There is evidence that accurate and rapid judgments of visual quantities form an essential component of human mathematical ability. However, explicit behavioural discrimination measures of visual quantities are readily contaminated both by variations in low-level physical parameters and higher order cognitive factors, while implicit measures often lack objectivity and sensitivity at the individual participant level. Here, with electrophysiological frequency tagging, we show discrimination differences between briefly presented visual quantities as low as a ratio of 1.4 (i.e., 14 vs. 10 elements). From this threshold, the neural discrimination response increases with parametrically increasing differences in ratio between visual quantities. Inter-individual variability in magnitude of the EEG response at this population threshold ratio predicts behavioural performance at an independent number comparison task. Overall, these findings indicate that visual quantities are perceptually discriminated automatically and rapidly (i.e., at a glance) within the occipital cortex. Given its high sensitivity, this paradigm could provide an implicit diagnostic neural marker of this process suitable for a wide range of fundamental and clinical applications.

Intergenerational associations of the approximate number system in toddlers and their parents.

From birth, humans are able to discriminate quantities using the approximate number system (ANS). However, previous methods have only been suitable to examine ANS functioning in infancy and older children. The goals of this study were twofold: first, to modify an existing method of assessing ANS functioning for toddlerhood; and second, to investigate individual differences in toddlers' ANS performance by examining correlations with their parents' ANS acuity. Using a preferential looking paradigm, we found that 1- to 3-year-olds (N=46) looked significantly longer to numerically changing images compared to numerically constant ones suggesting that the paradigm is a suitable measure of ANS functioning in toddlerhood. Furthermore, we found a positive relation between toddlers' ANS performance and that of their parents (assessed using a non-symbolic number comparison task) independent of children's vocabulary or parents' perceived math ability or preference for math. These findings are consistent with a specific intergenerational transmission of the ANS. Statement of contribution What is already known on this subject? Past methods used to examine ANS functioning were only suitable for infants and older children. Little research has examined sources underlying individual difference in ANS acuity. What does this study add? We developed a preferential looking task to assess ANS functioning in toddlerhood. Individual differences in toddlers' ANS functioning are correlated with their parents' ANS acuity.

Sex differences and functional hemispheric asymmetries during number comparison

BackgroundGlobal-local stimuli are hierarchical structures consisting of a larger global structure which is composed of smaller local stimuli. Numbers are also constructed hierarchically, with multi-digit numbers being made up from single digits. During two-digit number comparison, compatible items (larger number contains larger unit digit, e.g., 53 vs. 68) are processed faster and more accurately than incompatible items (smaller number contains larger unit digit, e.g., 58 vs. 63). This so-called unit-decade-compatibility effect has challenged the holistic model of number processing and suggests that the processing of number magnitudes occurs at least in part, decomposed, i.e., separately for each digit. Thus, the compatibility effect is indicative of how decomposed numbers are processed, thereby sharing similarities with traditional global-local processing of hierarchical stimuli.The goal of this study was to investigate whether factors that have been shown to reliably influence global-local processing also affect the compatibility effect during number comparison. Those include visual hemifield, sex, and menstrual cycle phase in women.MethodOne hundred sixty participants, 77 naturally cycling women and 83 men, completed a two-digit number comparison task twice, with test-sessions time-locked to the early follicular or mid-luteal cycle phase in women. Number comparison stimuli were presented to the right or left hemifield, respectively.ResultsWe observed a stronger compatibility effect in the right visual hemifield compared to left visual hemifield and in women compared to men, but no evidence for an influence of menstrual cycle phase in women could be found.ConclusionHemispheric asymmetries in holistic versus decomposed number processing could be demonstrated for the first time, suggesting a similar hemispheric modulation for number magnitude processing as for global-local processing.

Altered Parietal Activation during Non-symbolic Number Comparison in Children with Prenatal Alcohol Exposure.

Number processing is a cognitive domain particularly sensitive to prenatal alcohol exposure, which relies on intact parietal functioning. Alcohol-related alterations in brain activation have been found in the parietal lobe during symbolic number processing. However, the effects of prenatal alcohol exposure on the neural correlates of non-symbolic number comparison and the numerical distance effect have not been investigated. Using functional magnetic resonance imaging (fMRI), we examined differences in brain activation associated with prenatal alcohol exposure in five parietal regions involved in number processing during a non-symbolic number comparison task with varying degrees of difficulty. fMRI results are presented for 27 Cape Colored children (6 fetal alcohol syndome (FAS)/partial FAS, 5 heavily exposed (HE) non-sydromal, 16 controls; mean age ± SD = 11.7 ± 1.1 years). Fetal alcohol exposure was assessed by interviewing mothers using a timeline follow-back approach. Separate subject analyses were performed in each of five regions of interest, bilateral horizontal intraparietal sulci (IPS), bilateral posterior superior parietal lobules (PSPL), and left angular gyrus (left AG), using the general linear model with predictors for number comparison and difficulty level. Mean percent signal change for each predictor was extracted for each subject for each region to examine group differences and associations with continuous measures of alcohol exposure. Although groups did not differ in performance, controls activated the right PSPL more during non-symbolic number comparison than exposed children, but this was not significant after controlling for maternal smoking, and the right IPS more than children with fetal alcohol syndrome (FAS) or partial FAS. More heavily exposed children recruited the left AG to a greater extent as task difficulty increased, possibly to compensate, in part, for impairments in function in the PSPL and IPS. Notably, in non-syndromal heavily exposed children activation was impaired in the right PSPL, but spared in the right IPS. These results extend previous findings of poor right IPS recruitment during symbolic number processing in FAS/PFAS, indicating that mental representation of relative quantity is affected by prenatal alcohol exposure for both symbolic and non-symbolic representations of quantity.

The fractions SNARC revisited: Processing fractions on a consistent mental number line.

The ability to understand fractions is key to establishing a solid foundation in mathematics, yet children and adults struggle to comprehend them. Previous studies have suggested that these struggles emerge because people fail to process fraction magnitude holistically on the mental number line (MNL), focusing instead on fraction components. Subsequent studies have produced evidence for default holistic processing but examined only magnitude processing, not spatial representations. We explored the spatial representations of fractions on the MNL in a series of three experiments. Experiment 1 replicated Bonato et al.; 30 naïve undergraduates compared unit fractions (1/1-1/9) to 1/5, resulting in a reverse SNARC (Spatial-Numerical Association of Response Codes) effect. Experiment 2 countered potential strategic biases induced by the limited set of fractions used by Bonato et al. by expanding the stimulus set to include all irreducible, single-digit proper fractions and asked participants to compare them against 1/2. We observed a classic SNARC effect, completely reversing the pattern from Experiment 1. Together, Experiments 1 and 2 demonstrate that stimulus properties dramatically impact spatial representations of fractions. In Experiment 3, we demonstrated within-subjects reliability of the SNARC effect across both a fractions and whole number comparison task. Our results suggest that adults can indeed process fraction magnitudes holistically, and that their spatial representations occur on a consistent MNL for both whole numbers and fractions.

Individual differences influence two-digit number processing, but not their analog magnitude processing: a large-scale online study.

Symbolic magnitude comparison is one of the most well-studied cognitive processes in research on numerical cognition. However, while the cognitive mechanisms of symbolic magnitude processing have been intensively studied, previous studies have paid less attention to individual differences influencing symbolic magnitude comparison. Employing a two-digit number comparison task in an online setting, we replicated previous effects, including the distance effect, the unit-decade compatibility effect, and the effect of cognitive control on the adaptation to filler items, in a large-scale study in 452 adults. Additionally, we observed that the most influential individual differences were participants' first language, time spent playing computer games and gender, followed by reported alcohol consumption, age and mathematical ability. Participants who used a first language with a left-to-right reading/writing direction were faster than those who read and wrote in the right-to-left direction. Reported playing time for computer games was correlated with faster reaction times. Female participants showed slower reaction times and a larger unit-decade compatibility effect than male participants. Participants who reported never consuming alcohol showed overall slower response times than others. Older participants were slower, but more accurate. Finally, higher grades in mathematics were associated with faster reaction times. We conclude that typical experiments on numerical cognition that employ a keyboard as an input device can also be run in an online setting. Moreover, while individual differences have no influence on domain-specific magnitude processing-apart from age, which increases the decade distance effect-they generally influence performance on a two-digit number comparison task.

What is in a reach? Domain-general spatial modulation of motor responses by number representations

Gaze, pointing, and reaching movements are thought to provide a window to internal cognitive states. In the case of numerical cognition, it has been found that the left-right deviation of a reaching movement is modulated by the relative magnitude of values in a number comparison task. Some have argued that these patterns directly reflect the representation of a logarithmically compressed mental number line (direct mapping view). However, other studies suggest that the modulation of motor outputs by numerical value could be a more general decision-making phenomenon (response competition view). Here we test the generality of interactions between the motor system and numerical processing by comparing subjects' reach trajectories during two different nonverbal tasks: numerosity comparison and facial expression comparison. We found that reaching patterns were practically identical in both tasks – reach trajectories were equally sensitive to stimulus similarity in the numerical and face comparisons. The data provide strong support for the response competition view that motor outputs are modulated by domain-general decision processes, and reflect generic decision confidence or accumulation of evidence related to mental comparison.

Evidence for hippocampal dependence of value-based decisions

Consistent decisions are intuitively desirable and theoretically important for utility maximization. Neuroeconomics has established the neurobiological substrate of value representation, but brain regions that provide input to this network is less explored. The constructed-preference tradition within behavioral decision research gives a critical role to associative cognitive processes, suggesting a hippocampal role in making consistent decisions. We compared the performance of 31 patients with mediotemporal lobe (MTL) epilepsy and hippocampal lesions, 30 patients with extratemporal lobe epilepsy, and 30 healthy controls on two tasks: binary choices between candy bars based on their preferences and a number-comparison control task where the larger number is chosen. MTL patients made more inconsistent choices than the other two groups for the value-based choice but not the number-comparison task. These inconsistencies correlated with the volume of compromised hippocampal tissue. These results add to increasing evidence on a critical involvement of the MTL in preference construction and value-based choices.

The many ways quantifiers count: Children’s quantifier comprehension and cardinal number knowledge are not exclusively related

Children’s comprehension of quantifiers (like all, many, some) correlates with their cardinal number knowledge. However, it is unclear whether children’s quantifier comprehension is exclusively related to cardinal number knowledge or whether the relationship extends to approximate number skills. To find out, we tested German-speaking children in a quantifier comprehension task, a ‘Give-n task’, and a non-symbolic number comparison task (‘ANS task’). While we observed a significant correlation between quantifier and number knowledge in German-speaking children (age- and IQ-controlled), quantifiers did not mediate the relationship between cardinal number knowledge and age. This result suggests that quantifiers do not support number acquisition, contrasting with previous findings in English. Our results also revealed links between quantifier knowledge and approximate number skills, suggesting that quantifier and cardinal number skills are not exclusively related. Rather, children’s quantifier knowledge seems to be linked to number skills in a more general way, including approximate number skills.

Deficits in Approximate Number System Acuity and Mathematical Abilities in 6.5-Year-Old Children Born Extremely Preterm.

Preterm children are at increased risk for poor academic achievement, especially in math. In the present study, we examined whether preterm children differ from term-born children in their intuitive sense of number that relies on an unlearned, approximate number system (ANS) and whether there is a link between preterm children’s ANS acuity and their math abilities. To this end, 6.5-year-old extremely preterm (i.e., <27 weeks gestation, n = 82) and term-born children (n = 89) completed a non-symbolic number comparison (ANS acuity) task and a standardized math test. We found that extremely preterm children had significantly lower ANS acuity than term-born children and that these differences could not be fully explained by differences in verbal IQ, perceptual reasoning skills, working memory, or attention. Differences in ANS acuity persisted even when demands on visuo-spatial skills and attention were reduced in the ANS task. Finally, we found that ANS acuity and math ability are linked in extremely preterm children, similar to previous results from term-born children. These results suggest that deficits in the ANS may be at least partly responsible for the deficits in math abilities often observed in extremely preterm children.

Accumulation of non-numerical evidence during nonsymbolic number processing in the brain: An fMRI study.

Behavioral evidence has shown that when performing a nonsymbolic number comparison task (e.g., deciding which of two dot arrays contains more dots), participants' responses are sensitive to affected by both numerical (e.g., number of items) and non-numerical magnitudes (i.e., area, density, etc.). Thus far it is unclear what brain circuits support this process of accumulating non-numerical variables during nonsymbolic number processing. To investigate this, 21 adult participants were asked to engage in a dot comparison task. To measure the neural correlates of accumulating numerical and non-numerical variables, we manipulated the number of the non-numerical magnitudes that were congruent (correlated with number) or incongruent (anticorrelated with number). In a control task, participants were asked to choose the darker of two gray rectangles (brightness task). The tasks were matched in terms of their difficulty. The results of a whole brain analysis for regions sensitive to the congruity of numerical and non-numerical magnitudes revealed a region in the right inferior frontal gyrus (rIFG). Activation in this region was found to be correlated with the relative congruency of numerical and non-numerical magnitudes. In contrast, this region was not modulated by difficulty of the brightness control task. Accordingly in view of these findings, we suggest that the rIFG supports the accumulation of non-numerical magnitudes that are positively correlated with number. Therefore taken together, this study reveals a brain region whose pattern of activity is influenced by the congruency between numerical and non-numerical variables during nonsymbolic number judgments. Hum Brain Mapp 38:4908-4921, 2017. © 2017 Wiley Periodicals, Inc.

Effects of Healthy Aging and Mild Cognitive Impairment on a Real-Life Decision-Making Task.

In this study, we investigated the effects of age and of mild cognitive impairment (MCI) on decision making under risk by adopting a task representing real-life health-related situations and involving complex numerical information. Moreover, we assessed the relationship of real-life decision making to other cognitive functions such as number processing, executive functions, language, memory, and attention. For this reason, we compared the performance of 19 healthy, relatively younger adults with that of 18 healthy older adults and the performance of the 18 healthy older adults with that of 17 patients with MCI. Results indicated difficulties in real-life decision making for the healthy older adults compared with the healthy, relatively younger adults. Difficulties of patients with MCI relative to the healthy older adults arose in particular in difficult items requiring processing of frequencies and fractions. Significant effects of age and of MCI in processing frequencies were also evident in a ratio number comparison task. Decision-making performance of healthy participants and of the patient group correlated significantly with number processing. There was a further significant correlation with executive functions for the healthy participants and with reading comprehension for the patients. Our results suggest that healthy older individuals and patients with MCI make less advantageous decisions when the information is complex and high demands are put on executive functions and numerical abilities. Moreover, we show that executive functions and numerical abilities are not only essential in laboratory gambling tasks but also in more realistic and ecological decision situations within the health context.

Visual and proprioceptive feedback differently modulate the spatial representation of number and time in children

There has been compelling evidence favoring the idea that human adults similarly represent number and time along a horizontal mental number line (MNL) and mental time line (MTL), respectively. Yet, analogies drawn between the MNL and MTL have been challenged by recent studies suggesting that adults’ representations of number and time arise from different spatial frames of reference; whereas the MNL relies on both hand-centered and object-centered coordinates, the MTL appears to be exclusively anchored on object-centered coordinates. To directly test this possibility, here we explored the extent to which visual feedback and proprioceptive feedback affect children’s performance in a Number Comparison task (Experiment 1) and a Time Comparison task (Experiment 2), in which participants needed to associate a lateralized key with numerical and temporal words, respectively. Children (5- and 6-year-olds) performed the task with their hands either uncrossed or crossed over the body midline (i.e., manipulation of proprioceptive feedback) and with either visual control over their hands allowed or precluded under blindfolds (i.e., manipulation of visual feedback). Results showed that children were facilitated in associating smaller/larger numbers with the left/right side of the external space, but only when hands were uncrossed and visual feedback was available. On the contrary, blindfolding and crossing their hands over the midline did not affect spatial time mapping, with 6-year-olds showing facilitation in associating words referring to the past/future with the left/right side of the external space irrespective of visual and proprioceptive feedback. This same effect was also present in 5-year-olds despite their difficulty in performing the Time Comparison task. Together, these findings show, for the first time, that—just like adults—young children (a) map temporal events onto space in a rightward direction as they do for numbers and (b) anchor their spatial representation of time and numbers to different spatial frames of reference.

Number sense in the transition from natural to rational numbers.

Rational numbers are of critical importance both in mathematics and in other fields of science. However, they form a stumbling block for learners. One widely known source of the difficulty learners have with rational numbers is the natural number bias, that is the tendency to (inappropriately) apply natural number properties in rational number tasks. Still, it has been shown that a good understanding of natural numbers is highly predictive for mathematics achievement in general, and for performance on rational number tasks in particular. In this study, we further investigated the relation between learners' natural and rational number knowledge, specifically in cases where a natural number bias may lead to errors. Participants were 140 sixth graders from six different primary schools. Participants completed a symbolic and a non-symbolic natural number comparison task, a number line estimation task, and a rational number sense test. Learners' natural number knowledge was found to be a good predictor of their rational number knowledge. However, after first controlling for learners' general mathematics achievement, their natural number knowledge only predicted the subaspect of operations with rational numbers. The results of this study suggest that the relation between learners' natural and rational number knowledge can largely be explained by their relation with learners' general mathematics achievement.

Different Neural Mechanisms for the Comparison and Priming Distance Effects: An fMRI Study

Using functional magnetic resonance imaging, the present study examined whether the comparison distance effect (CDE) and the priming distance effect (PDE) in number processing had the same underlying neural mechanisms. 24 healthy participants completed a number comparison task and a number priming task in the scanner. Imaging data were examined for brain regions selected based on a meta-analysis of previous studies of number processing. Results revealed robust CDE and PDE at both behavioral and neural levels. The CDE had a significant hemodynamic signature in the right parietal cortex but not in the left parietal cortex, although a direct test of this hemispheric laterality did not reach statistical significance. In contrast, the PDE showed significant left-hemisphere laterality with a significant hemodynamic signature in the left parietal cortex but not in the right parietal cortex. These results suggested that the CDE and PDE had different underlying neural mechanisms.

Conscious and unconscious performance monitoring: Evidence from patients with schizophrenia

The ability to detect our own errors is an essential component of action monitoring. Using a masking paradigm in normal adults, we recently discovered that some error-detection processes can proceed without awareness, while other markers of performance monitoring such as the Error-Related Negativity (ERN) are tightly linked to conscious perception. Interestingly, research on cognitive deficit in schizophrenia has shown that the ERN is altered in these patients. In the present study, we therefore tested if the error detection impairment in schizophrenia is specific to conscious perception or is also found under non-conscious conditions, probing whether these performance monitoring processes are truly distinct. Thirteen patients with schizophrenia and thirteen age-matched healthy control subjects performed a speeded number comparison task on masked stimuli while EEG and MEG signals were recorded. Conscious perception and error-detection were assessed on a trial-by-trial basis using subjective reports of visibility and confidence. We found that patients with schizophrenia presented altered cingulate error-detection responses in conscious trials, as reflected by a decreased ERN. By contrast, on unconscious trials, both controls and schizophrenia patients performed above chance in evaluating the likelihood of having made an error. This dissociation confirms the existence of two distinct performance monitoring systems, and suggests that conscious metacognition in schizophrenia is specifically altered while non-conscious performance monitoring remains preserved.

The Approximate Number System Acuity Redefined: A Diffusion Model Approach.

While all humans are capable of non-verbally representing numerical quantity using so-called the approximate number system (ANS), there exist considerable individual differences in its acuity. For example, in a non-symbolic number comparison task, some people find it easy to discriminate brief presentations of 14 dots from 16 dots while others do not. Quantifying individual ANS acuity from such a task has become an essential practice in the field, as individual differences in such a primitive number sense is thought to provide insights into individual differences in learned symbolic math abilities. However, the dominant method of characterizing ANS acuity—computing the Weber fraction (w)—only utilizes the accuracy data while ignoring response times (RT). Here, we offer a novel approach of quantifying ANS acuity by using the diffusion model, which accounts both accuracy and RT distributions. Specifically, the drift rate in the diffusion model, which indexes the quality of the stimulus information, is used to capture the precision of the internal quantity representation. Analysis of behavioral data shows that w is contaminated by speed-accuracy tradeoff, making it problematic as a measure of ANS acuity, while drift rate provides a measure more independent from speed-accuracy criterion settings. Furthermore, drift rate is a better predictor of symbolic math ability than w, suggesting a practical utility of the measure. These findings demonstrate critical limitations of the use of w and suggest clear advantages of using drift rate as a measure of primitive numerical competence.

Processing multi-digit numbers: a translingual eye-tracking study.

The present study aimed at investigating the underlying cognitive processes and language specificities of three-digit number processing. More specifically, it was intended to clarify whether the single digits of three-digit numbers are processed in parallel and/or sequentially and whether processing strategies are influenced by the inversion of number words with respect to the Arabic digits [e.g., 43: dreiundvierzig ("three and forty")] and/or by differences in reading behavior of the respective first language. Therefore, English- and German-speaking adults had to complete a three-digit number comparison task while their eye-fixation behavior was recorded. Replicating previous results, reliable hundred-decade-compatibility effects (e.g., 742_896: hundred-decade compatible because 7 < 8 and 4 < 9; 362_517: hundred-decade incompatible because 3 < 5 but 6 > 1) for English- as well as hundred-unit-compatibility effects for English- and German-speaking participants were observed, indicating parallel processing strategies. While no indices of partial sequential processing were found for the English-speaking group, about half of the German-speaking participants showed an inverse hundred-decade-compatibility effect accompanied by longer inspection time on the hundred digit indicating additional sequential processes. Thereby, the present data revealed that in transition from two- to higher multi-digit numbers, the homogeneity of underlying processing strategies varies between language groups. The regular German orthography (allowing for letter-by-letter reading) and its associated more sequential reading behavior may have promoted sequential processing strategies in multi-digit number processing. Furthermore, these results indicated that the inversion of number words alone is not sufficient to explain all observed language differences in three-digit number processing.

The effect of oppositional parietal transcranial direct current stimulation on lateralized brain functions.

Cognitive functions such as numerical processing and spatial attention show varying degrees of lateralization. Transcranial direct current stimulation (tDCS) can be used to investigate how modulating cortical excitability affects performance of these tasks. This study investigated the effect of bi‐parietal tDCS on numerical processing, spatial and sustained attention. It was hypothesized that tDCS would have distinct effects on these tasks because of varying lateralization (numerical processing left, spatial attention right) and that these effects are partly mediated by modulation of sustained attention. A single‐blinded, crossover, sham‐controlled study was performed. Eighteen healthy right‐handed participants performed cognitive tasks during three sessions of oppositional parietal tDCS stimulation: sham; right anodal with left cathodal (RA/LC); and right cathodal with left anodal (RC/LA). Participants performed a number comparison task, a modified Posner task, a choice reaction task (CRT) and the rapid visual processing task (RVP). RA/LC tDCS impaired number comparison performance compared with sham, with slower responses to numerically close numbers pairs. RA/LC and RC/LA tDCS had distinct effects on CRT performance, specifically affecting vigilance level during the final block of the task. No effect of stimulation on the Posner task or RVP was found. It was demonstrated that oppositional parietal tDCS affected both numerical performance and vigilance level in a polarity‐dependent manner. The effect of tDCS on numerical processing may partly be due to attentional effects. The behavioural effects of tDCS were specifically observed under high task demands, demonstrating the consequences of an interaction between stimulation type and cognitive load.