Comparing Numerical Comparison Tasks: A Meta-Analysis of the Variability of the Weber Fraction Relative to the Generation Algorithm

in Frontiers in Psychology (2018)

Since more than 15 years, researchers have been expressing their interest in evaluating the Approximate Number System (ANS) and its potential influence on cognitive skills involving number processing ... [more ▼]

Since more than 15 years, researchers have been expressing their interest in evaluating the Approximate Number System (ANS) and its potential influence on cognitive skills involving number processing, such as arithmetic. Although many studies reported significant and predictive relations between ANS and arithmetic abilities, there has recently been an increasing amount of published data that failed to replicate such relationship. Inconsistencies lead many researchers to question the validity of the assessment of the ANS itself. In the current meta-analysis of over 68 experimental studies published between 2004 and 2017, we show that the mean value of the Weber fraction (w), the minimal amount of change in magnitude to detect a difference, is very heterogeneous across the literature. Within young adults, w might range from <10 to more than 60, which is critical for its validity for research and diagnostic purposes. We illustrate here the concern that different methods controlling for non-numerical dimensions lead to substantially variable performance. Nevertheless, studies that referred to the exact same method (e.g., Panamath) showed high consistency among them, which is reassuring. We are thus encouraging researchers only to compare what is comparable and to avoid considering the Weber fraction as an abstract parameter independent from the context. Eventually, we observed that all reported correlation coefficients between the value of w and general accuracy were very high. Such result calls into question the relevance of computing and reporting at all the Weber fraction. We are thus in disfavor of the systematic use of the Weber fraction, to discourage any temptation to compare given data to some values of w reported from different tasks and generation algorithms. [less ▲]

A rapid, objective and implicit measure of visual quantity discrimination

in Neuropsychologia (2018), 111

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 ... [more ▼]

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. [less ▲]

Directionality of Attacks in Natural Language Argumentation

in CEUR Workshop Proceedings (2018)

In formal (abstract and structured) argumentation theory, a central notion is that of an attack between a counterargument and the argument that it is challenging. Unlike the notion of an inconsistency ... [more ▼]

In formal (abstract and structured) argumentation theory, a central notion is that of an attack between a counterargument and the argument that it is challenging. Unlike the notion of an inconsistency between two statements in classical logic, this notion of an attack between arguments can be asymmetric, i.e. an argument A can attack an argument B without B attacking A. While this property of the formal systems studied by argumentation theorist has been motivated by considerations about the human practice of argumentation in natural language, there have not been any systematic studies on the connection between the directionality of attacks in argumentation-theoretic formalisms and the way humans actually interpret conflicts between arguments in a non-symmetric way. In this paper, we report on the result of two empirical cognitive studies that aim at filling this gap, one study with ordinary adults (undergraduate students) and one study with adult experts in formal argumentation theory. We interpret the results in light of the notions and distinctions defined in the ASPIC+ framework for structured argumentation, and discuss the relevance of our findings to past and future empirical studies about the link between human argumentation and formal argumentation theory. [less ▲]

Assessing the cerebral correlates of non-symbolic number processing with fast periodic visual stimulation

Poster (2017)

Some authors recently challenged the claim that numerical processes specifically handle non-symbolic magnitudes and they alternately suggested that general visual and/or control executive processes could ... [more ▼]

Some authors recently challenged the claim that numerical processes specifically handle non-symbolic magnitudes and they alternately suggested that general visual and/or control executive processes could explain performance in number comparison tasks. To further investigate this issue, we set up an EEG paradigm in which we recorded neural responses to the passive viewing of different arrays of basic visual forms. The stimuli sequence followed a fast and sinusoidal contrast modulation at the frequency of 10Hz (ten items per second). Visual properties of elements randomly changed from item to item, but their number was manipulated: in a control condition, arrays always contained the same number, and in the experimental conditions, the number periodically changed (each eight iteration, at 1.25Hz). We varied the numerical ratio between the frequent and the rare number throughout the experimental conditions. We recorded significant responses on occipital and parietal electrodes to the oddball frequency and its harmonics during our experimental conditions. Crucially, the strength of the signal was proportionally larger when the numerical ratio was larger. The results suggest that implicit and passive viewing of quick sequence of arrays was sufficient to automatically elicit neural synchronisation to numerical magnitudes without any explicit involvement of higher general cognitive processes. [less ▲]

Cardinal and ordinal processing in spatial neglect

Poster (2016, January)

Patients with spatial neglect do not only have difficulties in orienting attention in physical space but also in representational space, especially with respect to the mental representation of numbers ... [more ▼]

Patients with spatial neglect do not only have difficulties in orienting attention in physical space but also in representational space, especially with respect to the mental representation of numbers. Indeed, in a study by Zorzi et al. (2012) neglect patients were particularly slow when asked to compare the number 4 to the standard number 5, suggesting difficulties to process numbers on the left side of an internal standard. This difficulty was observed in a magnitude judgement, but not in a parity task, implying a dissociation between explicit and implicit processing of numerical magnitude. The present study aimed at replicating these findings and extending them to non-numerical sequences in order to complement the data obtained on bisection tasks (Zamarian, et al., 2007). Sixteen right-sided brain damaged patients with neglect (N+ =6; 4 females; all right handers; mean age: 55 +/- 8,7) and without neglect (N- =10; 2 females; all right hander; mean age: 48 +/- 6.2) participated in the study. They were administered the following tasks: a magnitude and a parity judgement task; an ordinal judgement task on numbers and on letters and a consonant/vowel classification task. For each task and each patient, a linear regression was computed in which the difference between the response times for the left effector (index finger) and the right effector (middle finger) was predicted by number magnitude. A negative slope will indicate the presence of a SNARC-like effect. We compared the negative slopes of the two patient groups using a Chi-square. Considering the proportion of SNARC-like effects, it appeared that, on one hand, N+ patients showed fewer SNARC-like effects than N- patients during magnitude judgements on numbers. Thus confirming the findings by Zorzi et al. (2012). On the other hand, N+ patients behaved similarly to N- patients for the parity judgements on numbers and for the order judgements both on numbers and letters. This last result suggest a dissociation between the spatial representation of magnitude and of order in N+ patients. These results point towards a specific impairment in explicit access to number magnitude in spatial hemineglect. [less ▲]

Functional connectivity and structural analyses in the bilingual brain: implications for arithmetic.

Poster (2015, June)

Do bilinguals use the same brain networks than monolinguals when they solve arithmetic problems? We investigated this question by using resting-state functional connectivity and cortical thickness ... [more ▼]

Do bilinguals use the same brain networks than monolinguals when they solve arithmetic problems? We investigated this question by using resting-state functional connectivity and cortical thickness measurements. Recent studies highlighted differences of functional connectivity (e.g. Grady et al., 2015) and of brain structure (e.g. Klein et al., 2014) between bilinguals and monolinguals. However, no study so far has linked these differences to arithmetic problem solving, a cognitive skill that may at least partially rely on language processing. Our study population was composed of carefully selected German-French bilinguals (N = 20) who acquired each language at the same age, leading to high proficiency levels in both languages. These bilinguals all attended university in their second language at the time of the experiment, namely French. Therefore we selected a control group of French-speaking monolinguals (N = 12). Structural and functional images of brain activity were collected using a 3T MRI scanner. Functional scans of resting-state were acquired during a 6-minute session, with eyes closed. A 3D T1-weighted data set encompassing the whole brain was acquired to provide detailed anatomy (1 mm3), which was used both for the co-registration of functional data and for morphometric analyses. Prior to the scanning session, all participants took a behavioral test measuring their arithmetic skill. For the resting-state part of the study, we generated spheres based on ROIs reported in the literature as magnitude manipulation- and language-related areas during arithmetic problem solving (Klein et al. 2013), and addition-related areas reported in a recent meta-analysis (Arsalidou & Tayor, 2011). We used these spheres as seed regions for the analyses. We correlated resting activations between these regions and compared these correlations in bilinguals versus monolinguals. Results showed significantly higher correlations between the three seed regions in monolinguals than in bilinguals (all ts > 2.306; ps < .05), suggesting that regions used to solve arithmetic problems form a different network in bilinguals than in monolinguals. To control for general differences between both populations, we also created two spheres in areas not specifically related to neither arithmetic nor language regions. There were no significant differences between groups in terms of correlations of these regions with resting-state activations. These results suggest that the differences observed in arithmetic problem solving regions could not account for by general differences between groups. In the second part of the study, we aimed at verifying whether the differences in functional connectivity we observed between bilinguals and monolinguals coincide with structural brain differences. We measured and compared cortical thickness in both groups. Then we compared the correlations between cortical thickness and arithmetic skill in both groups (considering differences with corrected p < .001). Cortical thickness of areas commonly associated to language or number processing correlated differently with arithmetic skill as a function of the group: Higher cortical thickness of left pars triangularis, bilateral superior parietal gyri and precuneus positively correlated with arithmetic skill in monolinguals but negatively correlated with arithmetic skill in bilinguals. These results highlight that there are different relations between brain structure and arithmetic skills in bilinguals and monolinguals. In conclusion the current study provides new evidence for differences between bilinguals’ and monolinguals’ brain networks engaged in arithmetic problem solving, even without any arithmetic task during the data acquisition. These findings based on functional connectivity and brain structure analyses also reveal the general involvement of language in arithmetic problem solving in bilingual as well as non-bilingual individuals. [less ▲]

Arithmetic in the bilingual brain: an fMRI study

Scientific Conference (2015, March)

Using fMRI we observed that solving addition and multiplication problems induced activation in several fronto-parietal regions in both German-French bilingual and French monolingual adults. However ... [more ▼]

Using fMRI we observed that solving addition and multiplication problems induced activation in several fronto-parietal regions in both German-French bilingual and French monolingual adults. However, during complex addition frontal regions showed systematically higher activation levels in bilinguals than monolinguals, both when bilinguals computed in German (math-acquisition language) and in French. [less ▲]

Assessing the Approximate Number System: no relation between numerical comparison and estimation tasks

in Psychological Research (2015)

Whether our general numerical skills and the mathematical knowledge that we acquire at school are entwined is a debated issue, which many researchers are still striving to investigate. The findings ... [more ▼]

Whether our general numerical skills and the mathematical knowledge that we acquire at school are entwined is a debated issue, which many researchers are still striving to investigate. The findings reported in the literature are actually inconsistent; some studies emphasised the existence of a relationship between the acuity of the Approximate Number System (ANS) and arithmetic competence, while some others did not observe any significant correlation. One potential explanation of the discrepancy might stem from the evaluation of the ANS itself. In the present study, we correlated two measures used to index ANS acuity with arithmetic performance. These measures were the Weber fraction (w), computed from a numerical comparison task and the Coefficient of Variation (CV), computed from a numerical estimation task. Arithmetic performance correlated with estimation CV but not with comparison w. We further investigated the meaning of this result by taking the relationship between w and CV into account. We expected a tight relation as both these measures are believed to assess ANS acuity. Crucially however, w and CV did not correlate with each other. Moreover, the value of w was modulated by the congruity of the relation between numerical magnitude and non-numerical visual cues, potentially accounting for the lack of correlation between the measures. Our findings thus challenge the overuse of w to assess ANS acuity and more generally put into question the relevance of correlating this measure with arithmetic without any deeper understanding of what they are really indexing. [less ▲]

Arithmetic in the Bilingual Brain: an fMRI study

Scientific Conference (2014, May)

How do bilinguals solve arithmetic problems in their different languages? We investigated this question with functional magnetic resonance imaging (fMRI) by exploring the neural substrates of arithmetic ... [more ▼]

How do bilinguals solve arithmetic problems in their different languages? We investigated this question with functional magnetic resonance imaging (fMRI) by exploring the neural substrates of arithmetic processing in bilinguals in comparison to monolinguals. Bilingual participants were highly proficient both in German and French as they attended primary school in German and secondary school and higher education in French. This bilingual combination is particularly interesting because the order of two-digit number words is inversed in these languages: decade-unit in French but unit-decade in German. 21 German-French bilinguals and 12 French-speaking monolinguals were scanned while performing different types of arithmetic problems: additions of different complexity levels (from simple to complex additions) and multiplication facts. We presented different types of operations in order to disentangle arithmetic computation from pure memory retrieval that occurs in very simple additions or multiplications. Arithmetic problems were presented via headsets in a verification paradigm and bilinguals performed the tasks in both languages. Results showed that all arithmetic tasks elicited a broad fronto-parietal network in both groups and for both of bilinguals’ language sessions. However, we observed that complex additions involved more left frontal activity (i.e. inferior frontal gyrus, anterior cingulate gyrus) in bilinguals than in monolinguals. It is important to notice that these frontal activation differences occurred both for the arithmetic acquisition language (i.e. German) and the second language (i.e. French). These BOLD differences between bilingual and monolingual participants were observed despite the fact that both groups solved the arithmetic problems with equivalent accuracy rates. Moreover, localization of the regions activated by complex additions in bilinguals differed from the typical activation pattern reported for mental arithmetic in recent meta-analyses (Arsalidou & Taylor, 2011). Taken together, our results indicate that highly proficient bilinguals rely on differential activation patterns than monolinguals to solve complex additions. The differences in left frontal activations might reflect different degrees of language-related automaticity when computing complex arithmetic problems. Executive functions that are necessary to control language context and access for bilinguals’ respective languages might also play a role. Further insights about the role of language in arithmetic solving process in bilingual and non-bilingual individuals will be discussed. [less ▲]

Procedural and magnitude-based strategies in solving complex additions

Scientific Conference (2013)