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 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.

scholarly journals Effects of attentional shifts along the vertical axis on number processing: an eye-tracking study with optokinetic stimulation

2021 ◽  
Author(s):  
Arianna Felisatti ◽  
Mariagrazia Ranzini ◽  
Elvio Blini ◽  
Matteo Lisi ◽  
Marco Zorzi
Keyword(s):  
Eye Tracking ◽  
Numerical Cognition ◽  
Response Times ◽  
Eye Gaze ◽  
Distance Effect ◽  
Vertical Axis ◽  
Number Processing ◽  
Numerical Magnitude ◽  
Optokinetic Stimulation ◽  
Comparison Task

Previous studies suggest that associations between numbers and space are mediated by shifts of visuospatial attention along the horizontal axis. In this study, we investigated the effect of vertical shifts of overt attention, induced by optokinetic stimulation (OKS) and monitored through eye-tracking, in two tasks requiring explicit (number comparison) or implicit (parity judgment) processing of number magnitude. Participants were exposed to black-and-white stripes (OKS) that moved vertically (upward or downward) or remained static (control condition). During the OKS, participants were asked to verbally classify auditory one-digit numbers as larger/smaller than 5 (comparison task; Exp. 1) or as odd/even (parity task; Exp. 2). OKS modulated response times in both experiments. In Exp.1, downward attentional displacement increased the Magnitude effect (slower responses for large numbers) and reduced the Distance effect (slower responses for numbers close to the reference). In Exp.2, we observed a parity by magnitude interaction that was amplified by downward OKS. Moreover, eye tracking analyses revealed an influence of number processing on eye movements both in Exp. 1, with eye gaze shifting downwards during the processing of numbers 1-2 as compared to 8-9; and in Exp. 2, with leftward shifts after large even numbers (6,8) and rightward shifts after large odd numbers (7,9). These results provide evidence of bidirectional links between number and space and extend them to the vertical dimension. Moreover, they document the influence of visuo-spatial attention on processing of numerical magnitude, numerical distance and parity. Together, our findings are in line with grounded and embodied accounts of numerical cognition.

scholarly journals Neural coding partially accounts for the relation between children’s number line estimation and numerical comparison performance

2018 ◽  
Author(s):  
Richard Prather
Keyword(s):  
Neural Coding ◽  
Standardized Test ◽  
Number System ◽  
Number Line ◽  
Standardized Test Scores ◽  
Numerical Comparison ◽  
Comparison Task ◽  
Mathematics Interventions ◽  
Number Line Estimation ◽  
Number Perception

Numerical comparison is a primary measure of the acuity of children’s approximate number system (ANS). ANS acuity is associated with key developmental outcomes such as symbolic number skill, standardized test scores and even employment outcomes(Halberda, Mazzocco, & Feigenson, 2008; Parsons & Bynner, 1997). We examine the relation between children’s performance on the numerical comparison task and the number line estimation task. It is important to characterize the relation between tasks in order to develop mathematics interventions that lead to transfer across tasks. We find that number line performance is significantly predicted by non-symbolic comparison performance for participants ranging in age from 5 to 8 years. We also evaluate, using a computational model, if the relation between the two tasks can be adequately explained based on known neural correlates of number perception. Data from humans and non-human primates characterizes neural activity corresponding to the perception of numerosities. Results of behavioral experimentation and computational modeling suggest that though neural coding of number predicts a correlation in participants’ performance on the two tasks, it cannot account for all of the variability in the human data. This is interpreted as consistent with accounts of number line estimation in which number line estimation does not rely solely on participants’ numerical perception.

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.

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 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.

Order and Magnitude Share a Common Representation in Parietal Cortex

2009 ◽  
Vol 21 (11) ◽  
pp. 2114-2120 ◽  
Author(s):  
Michael S. Franklin ◽  
John Jonides
Keyword(s):  
Distance Effect ◽  
Number Line ◽  
Judgment Task ◽  
Mental Number Line ◽  
Numerical Magnitude ◽  
Intraparietal Sulcus ◽  
Ordinal Information ◽  
Common Representation ◽  
Distance Effects

The role of the intraparietal sulcus (IPS) in the representation of numerical magnitude is well established. Recently, there has also been speculation that the IPS is involved in the representation of ordinal information as well. These claims, however, overlook the fact that all neuroimaging paradigms in which participants make judgments about either magnitude and/or order result in a behavioral distance effect (i.e., the comparison is easier when the stimuli span a greater distance). This leaves open two possibilities: It may be that activation of the IPS is due to the mechanism that yields distance effects, or it may be that the IPS is involved in the representation of information about both magnitude and order. The current study used fMRI to compare a magnitude task in which participants show distance effects to an order-judgment task that yields reverse-distance effects. The results reveal activation of the IPS for both the magnitude and order tasks that is based on participants' strategies as opposed to the actual distance between the numbers. This leads to the conclusion that the IPS represents a mental number line, and that accessing this line can lead to distance effects when participants compare magnitudes and to reverse-distance effects when participants check for order.

scholarly journals Imaging Informational Conflict: A Functional Magnetic Resonance Imaging Study of Numerical Stroop

2006 ◽  
Vol 18 (12) ◽  
pp. 2049-2062 ◽  
Author(s):  
J. Tang ◽  
H. D. Critchley ◽  
D. E. Glaser ◽  
R. J. Dolan ◽  
B. Butterworth
Keyword(s):  
Imaging Study ◽  
Distance Effect ◽  
Irrelevant Information ◽  
Numerical Magnitude ◽  
Neural Basis ◽  
Neural Fate ◽  
Numerical Magnitudes ◽  
Informational Conflict ◽  
Task Irrelevant ◽  
Theoretical Issues

We employed a parametric version of the comparison Stroop paradigm to investigate the processing of numerical magnitude and physical size under task-relevant and -irrelevant conditions to investigate two theoretical issues: (1) What is the neural fate of task-irrelevant information? (2) What is the neural basis of the resolution of the conflict between task-relevant and -irrelevant information? We show in 18 healthy adults that numerical magnitudes of numbers call for higher processing requirements than physical sizes. The enhanced activation elicited by numerical magnitudes is not modulated by task relevance, indicating autonomous processing. Moreover, the normal behavioral distance effect when the numerical dimension is task relevant and reversed distance effect when it is not show that autonomous processing fully encodes numerical magnitudes. Conflict trials elicited greater activation in bilateral inferior frontal gyri, right middle frontal gyri, and right superior frontal gyri. We postulate two sources to the conflict, namely, at cognitive and response levels.

scholarly journals Numerical Comparisons and Sex-Related Brain Connectivity

2021 ◽  
Vol 13 (2) ◽  
pp. 62
Author(s):  
Fabiola R. Gómez-Velázquez ◽  
Andrés A. González-Garrido ◽  
Ricardo A. Salido-Ruiz ◽  
Sulema Torres-Ramos ◽  
Aurora Espinoza-Valdez ◽  
...  
Keyword(s):  
Brain Connectivity ◽  
Cognitive Strategies ◽  
Numerical Comparison ◽  
Math Learning ◽  
Comparison Task ◽  
Behavioral Measures ◽  
Functional Brain ◽  
Progressive Development ◽  
Interhemispheric Connectivity ◽  
Number Comparison

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. 

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