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

2021 ◽  
pp. 174702182110664
Author(s):  
Yam Zagury ◽  
Rut Zaks-Ohayon ◽  
Joseph Tzelgov ◽  
Michal Pinhas
Keyword(s):  
Visual Cues ◽  
Automatic Processing ◽  
Congruity Effect ◽  
Numerical Magnitude ◽  
Numerical Comparison ◽  
Comparison Task ◽  
Numerical Comparisons ◽  
Perceptual Dominance ◽  
Size Congruity Effect

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 the present 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, while 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.

scholarly journals Modeling the latent structure of individual differences in the numerical size-congruity effect

2020 ◽  
Author(s):  
Thomas J. Faulkenberry ◽  
Kristen Bowman
Keyword(s):  
Individual Differences ◽  
Numerical Cognition ◽  
Congruity Effect ◽  
Numerical Magnitude ◽  
Data Sets ◽  
Comparison Task ◽  
Physical Size ◽  
The Individual ◽  
Size Congruity Effect ◽  
Individual Size

When people are asked to choose the physically larger of a pair of numerals, they are often slower when relative physical size is incongruent with numerical magnitude. This size-congruity effect is usually assumed as evidence for automatic activation of numerical magnitude. In this paper, we apply the methods of Haaf and Rouder (2017) to look at the size-congruity effect through the lens of individual differences. Here, we simply ask whether everyone exhibits the effect. We develop a class of hierarchical Bayesian mixed models with varying levels of constraint on the individual size- congruity effects. The models are then compared via Bayes factors, telling us which model best predicts the observed data. We then apply this modeling technique to three data sets. In all three data sets, the winning model was one in which the size-congruity effect was constrained to be positive. This indicates that, at least in a physical comparison task with numerals, everyone exhibits a positive size-congruity effect. We discuss these results in the context of measurement fidelity and theory-building in numerical cognition.

scholarly journals Modeling response times in the size-congruity effect: Early versus late interaction

2020 ◽  
Author(s):  
Kristen Bowman ◽  
Thomas J. Faulkenberry
Keyword(s):  
Drift Rate ◽  
Response Times ◽  
Decision Time ◽  
Congruity Effect ◽  
Response Threshold ◽  
Numerical Magnitude ◽  
Comparison Task ◽  
Single Digit ◽  
Paired Samples ◽  
Size Congruity Effect

The size-congruity effect occurs when numerical magnitude interferes with judgments of physical size. Various accounts propose that this interference is either encoding-related or decision-related, though at present a clear consensus is lacking. In our study, we administered a single-digit physical comparison task (i.e., which digit is physically larger?) and applied four different mathematical models (ex-Gaussian, ex-Wald, shifted Wald and EZ-diffusion) to the observed response times. The aim of this modeling was to index the underlying cognitive processes via estimates of drift rate, response threshold, and non-decision time. The collection of estimates for each individual was then subjected to Bayesian paired samples t-tests. We found that the drift rate for incongruent trials was smaller than for congruent trials, indicating that congruent trials had a faster rate of information uptake. The response threshold for incongruent trials was generally larger than for congruent trials, indicating that for incongruent trials more information needed to be accumulated before a response could be initiated. Critically, we found evidence of an invariance in non-decision times between incongruent and congruent trials. This combination of results provides support for a late interaction account of the size-congruity effect, shedding further light onto models of decision making in number processing.

scholarly journals Quantitative and qualitative differences in the canonical and the reverse distance effect and their selective association with arithmetic and mathematical competencies

2021 ◽  
Author(s):  
Stephan Vogel ◽  
Thomas J. Faulkenberry ◽  
Roland H. Grabner
Keyword(s):  
Standardized Test ◽  
Reaction Times ◽  
Distance Effect ◽  
Numerical Magnitude ◽  
Numerical Comparison ◽  
Central Research ◽  
Comparison Task ◽  
Mathematical Competencies ◽  
Numerical Order ◽  
Underlying Mechanisms

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: the canonical and the reverse distance effect. The former indicates 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 the canonical distance effect and arithmetic abilities, rather inconsistent findings have been found for the reverse distance effect. Here, we tested the hypothesis that estimates of the reverse distance effect show qualitative differences (i.e., not all participants show a reverse distance effect in the expected direction) rather than quantitative differences (i.e., all individuals show a reverse 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 the canonical distance effect showed quantitative differences, estimates of the reverse distance effect showed qualitative differences. Comparisons between individuals who demonstrated an effect and individuals who demonstrated no reverse distance effect confirmed 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 the reverse distance effect are subject to qualitative differences and that we need to better characterize the underlying mechanisms/strategies that might lead to these qualitative differences.

Across-Notation Automatic Processing of Two-Digit Numbers

2011 ◽  
Vol 58 (2) ◽  
pp. 147-153 ◽  
Author(s):  
Dana Ganor-Stern ◽  
Joseph Tzelgov
Keyword(s):  
Congruency Effect ◽  
Automatic Processing ◽  
Numerical Comparison ◽  
Abstract Representation ◽  
Comparison Task ◽  
Arabic Speakers ◽  
Common Representation ◽  
Whole Number ◽  
Arabic Digits ◽  
Size Congruency

The existence of across-notation automatic numerical processing of two-digit (2D) numbers was explored using size comparisons tasks. Participants were Arabic speakers, who use two sets of numerical symbols – Arabic and Indian. They were presented with pairs of 2D numbers in the same or in mixed notations. Responses for a numerical comparison task were affected by decade difference and unit-decade compatibility and global distance in both conditions, extending previous findings with Arabic digits ( Nuerk, Weger, & Willmes, 2001 ). Responses for a physical comparison task were affected by congruency with the numerical size, as indicated by the size congruency effect (SiCE). The SiCE was affected by unit-decade compatibility but not by global distance, thus suggesting that the units and decades digits of the 2D numbers, but not the whole number value were automatically translated into a common representation of magnitude. The presence of similar results for same- and mixed-notation pairs supports the idea of an abstract representation of magnitude.

Holistic Representation of Unit Fractions

2011 ◽  
Vol 58 (3) ◽  
pp. 201-206 ◽  
Author(s):  
Dana Ganor-Stern ◽  
Irina Karasik-Rivkin ◽  
Joseph Tzelgov
Keyword(s):  
Distance Effect ◽  
Congruity Effect ◽  
Complex Numbers ◽  
Numerical Comparison ◽  
Comparison Task ◽  
Semantic Congruity ◽  
Stimulus Set ◽  
Unit Fractions ◽  
Semantic Congruity Effect ◽  
Holistic Representation

The present study examined the processing of unit fractions and the extent to which it is affected by context. Using a numerical comparison task we found evidence for a holistic representation of unit fractions when the immediate context of the fractions was emphasized, that is when the stimuli set included in addition to the unit fractions also the numbers 0 and 1. The holistic representation was indicated by the semantic congruity effect for comparisons of pairs of fractions and by the distance effect in comparisons of a fraction and 0 and 1. Consistent with previous results (Bonato, Fabbri, Umilta, & Zorzi, 2007) there was no evidence for a holistic representation of unit fractions when the stimulus set included only fractions. These findings suggest that fraction processing is context-dependent. Finally, the present results are discussed in the context of processing other complex numbers beyond the first decade.

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

2021 ◽  
Vol 6 ◽  
Author(s):  
Stephan E. Vogel ◽  
Thomas J. Faulkenberry ◽  
Roland H. Grabner
Keyword(s):  
Standardized Test ◽  
Reaction Times ◽  
Distance Effect ◽  
Numerical Magnitude ◽  
Numerical Comparison ◽  
Central Research ◽  
Comparison Task ◽  
Mathematical Competencies ◽  
Numerical Order ◽  
Underlying Mechanisms

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: the canonical and the reverse distance effect. The former indicates 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 the canonical distance effect and arithmetic abilities, rather inconsistent findings have been found for the reverse distance effect. Here, we tested the hypothesis that estimates of the reverse distance effect show qualitative differences (i.e., not all participants show a reverse distance effect in the expected direction) rather than quantitative differences (i.e., all individuals show a reverse 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 the canonical distance effect showed quantitative differences, estimates of the reverse distance effect showed qualitative differences. Comparisons between individuals who demonstrated an effect and individuals who demonstrated no reverse distance effect confirmed 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 the reverse distance effect are subject to qualitative differences and that we need to better characterize the underlying mechanisms/strategies that might lead to these qualitative differences.

scholarly journals The plural still counts: Cross-linguistic study of the symbolic numerical magnitude comparison task in Polish- and German-speaking preschoolers

2020 ◽  
Author(s):  
Maciej Haman ◽  
Katarzyna Lipowska ◽  
Mojtaba Soltanlou ◽  
Krzysztof Cipora ◽  
Frank Domahs ◽  
...  
Keyword(s):  
Digit Span ◽  
Numerical Magnitude ◽  
Number Word ◽  
Numerical Comparison ◽  
Main Task ◽  
Comparison Task ◽  
Number Words ◽  
Symbolic Representations ◽  
German Speaking ◽  
Grammatical Number

Already in toddlerhood, children begin to master the system of number word meanings. The role of grammar, and in particular grammatical number inflection, in early stage of this process has been well documented. It is not clear, however, whether the influence of the grammatical language structure also extends to more complex later stages. In the current study, we have addressed this problem by using differences in the grammatical number paradigms between Polish and German, in particular, the inconsistency of the grammatical number of the verb and the noun for numbers above four. One-hundred-fifty-three Polish-speaking children and 124 German-speaking three-to-six-year-old children took part in the study. Their main task was to compare symbolic numbers (Arabic numerals and spoken number-words) in the range of small numbers (2-4) large numbers (5-9) and between ranges. In addition, counting skills (Give-a-number and count-list) and mapping between non-symbolic (dot sets) and symbolic representations of numbers were checked. The children also performed working memory tests (Corsi-blocks and digit span). Based on Give-a-number and mapping tasks, participants were divided into subset-knowers, CP-knowers-non-mappers and CP-knowers-mappers (cf. LeCorre, 2014). As expected, grammatical number structure influenced performance: Polish-speaking children, later than the German ones, achieved the CP-knowers stage and, after it was achieved, they fared worse in the numerical comparison task, which was further mediated by response side. Importantly, however, there were no significant differences in the mapping task between non-symbolic and symbolic representations of numbers between Polish and German groups. We conclude that cross-linguistic differences in the grammatical number paradigms can significantly affect the development of representations and processing of numbers not only at the stage of acquiring the meaning of the first number-words, but also at later stages, when dealing with symbolic numbers.

Exploring the Boundaries of the Number–Temporal Order Association

2015 ◽  
Vol 62 (3) ◽  
pp. 198-205 ◽  
Author(s):  
Dana Ganor-Stern
Keyword(s):  
Embodied Cognition ◽  
Mental Representation ◽  
Temporal Order ◽  
Past Research ◽  
Numerical Comparison ◽  
Negative Number ◽  
Comparison Task ◽  
Negative Numbers ◽  
The Past

Past research has shown that numbers are associated with order in time such that performance in a numerical comparison task is enhanced when number pairs appear in ascending order, when the larger number follows the smaller one. This was found in the past for the integers 1–9 ( Ben-Meir, Ganor-Stern, & Tzelgov, 2013 ; Müller & Schwarz, 2008 ). In the present study we explored whether the advantage for processing numbers in ascending order exists also for fractions and negative numbers. The results demonstrate this advantage for fraction pairs and for integer-fraction pairs. However, the opposite advantage for descending order was found for negative numbers and for positive-negative number pairs. These findings are interpreted in the context of embodied cognition approaches and current theories on the mental representation of fractions and negative numbers.

scholarly journals Bottom-up and top-down attentional contributions to the size congruity effect

2016 ◽  
Vol 78 (5) ◽  
pp. 1324-1336 ◽  
Author(s):  
Kenith V. Sobel ◽  
Amrita M. Puri ◽  
Thomas J. Faulkenberry
Keyword(s):  
Congruity Effect ◽  
Top Down ◽  
Bottom Up ◽  
Size Congruity Effect

Shoaling decisions in female swordtails: how do fish gauge group size?

Behaviour ◽  
2007 ◽  
Vol 144 (11) ◽  
pp. 1333-1346 ◽  
Author(s):  
Bob Wong ◽  
Gil Rosenthal ◽  
Jessica Buckingham
Keyword(s):  
Gauge Group ◽  
Group Size ◽  
Group Membership ◽  
Size Ratio ◽  
Numerical Comparison ◽  
Weber’S Law ◽  
Significant Preference ◽  
Xiphophorus Helleri ◽  
Weber's Law ◽  
Numerical Comparisons

AbstractLittle is known about the mechanisms individuals might use to compare group sizes when making decisions about group membership. One possibility is that animals use ratio to determine differences in group sizes. Weber's Law states that the ease of any numerical comparison is based on the ratio between the stimuli compared; as the ratio becomes smaller the comparison becomes more difficult. We set out to test this prediction by offering female green swordtails, Xiphophorus helleri, dichotomous choices between different shoal sizes, varying both in ratios and absolute numbers of fish. Swordtails attended to the ratio of group size between stimulus shoals, rather than the numerical difference between shoals, when making shoaling decisions. Where group size ratio was 2:1, subjects showed a significant preference for the larger shoal, independent of the numerical difference between the shoals. When the ratio was 1.5:1, subjects showed no preference. The ratio between group sizes may, thus, be an important factor in shoaling decisions. More broadly, ratio could prove to be a widespread mechanism for animals to make numerical comparisons in group assessments.

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