Numerical distance and size effects dissociate in Indo-Arabic number comparison

Dominant numerical cognition models suppose that both symbolic and nonsymbolic numbers are processed by the Analogue 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 nonsymbolic 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 nonsymbolic dot comparison and symbolic Indo-Arabic comparison tasks. It was consistently found that while the ANS model’s prediction is relatively good for the nonsymbolic dot comparison, its prediction is poorer and systematically biased for the symbolic Indo-Arabic comparison. We conclude that only nonsymbolic comparison is supported by the ANS, and symbolic number comparisons are processed by other representation.

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Response-Selection-Related Parietal Activation during Number Comparison

Journal of Cognitive Neuroscience ◽

10.1162/0898929042568442 ◽

2004 ◽

Vol 16 (9) ◽

pp. 1536-1551 ◽

Cited By ~ 135

Author(s):

Silke M. Göbel ◽

Heidi Johansen-Berg ◽

Tim Behrens ◽

Matthew F. S. Rushworth

Keyword(s):

Nonhuman Primates ◽

Response Selection ◽

Perceptual Task ◽

Numerical Distance ◽

Single Digit ◽

Selection Processes ◽

Number Comparison ◽

Basic Stimulus ◽

The Right ◽

Double Digit

Neuroimaging studies of number comparison have consistently found activation in the intraparietal sulcus (IPS). Recently, it has been suggested that activations in the IPS vary with the distance between the numbers being compared. In number comparison, the smaller the distance between a number and the reference the longer the reaction time (RT). Activations in the right or left IPS, however, have also been related to attentional and intentional selection. It is possible, therefore, that activity in this region is a reflection of the more basic stimulus and response-selection processes associated with changes in RT. This fMRI experiment investigated the effect of numerical distance independently from RT. In addition, activations during number comparison of single-digit and double-digit stimuli were compared. During number comparison blocks, subjects had to indicate whether digits were greater or smaller than a reference (5 or 65). In control blocks, they were asked to perform a perceptual task (vertical line present/absent) on either numerical or nonnumerical stimuli. Number comparison versus rest yielded a large bilateral parietal-posterior frontal network. However, no areas showed more activation during number comparison than during the control tasks. Furthermore, no areas were more active during comparison of numbers separated by a small distance than comparisons of those separated by a large distance or vice versa. A left-lateralized parietal-posterior frontal network varied significantly with RT. Our findings suggest that magnitude and numerical-distance-related IPS activations might be difficult to separate from fundamental stimulus and response-selection processes associated with RT changes. As is the case with other parameters, such as space, magnitude may be represented in the context of response selection in the parietal cortex. In this respect, the representation of magnitude in the human IPS may be similar to the representation of magnitude in other nonhuman primates.

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On the Perceptual Generality of the Unit-Decade Compatibility Effect

Experimental Psychology (formerly Zeitschrift für Experimentelle Psychologie) ◽

10.1027/1618-3169.51.1.72 ◽

2004 ◽

Vol 51 (1) ◽

pp. 72-79 ◽

Cited By ~ 43

Author(s):

Hans-Christoph Nuerk ◽

Ulrich Weger ◽

Klaus Willmes

Keyword(s):

Compatibility Effect ◽

Number Line ◽

Mental Number Line ◽

Original Experiment ◽

Holistic Model ◽

Magnitude Representation ◽

Number Magnitude ◽

Number Comparison ◽

Arabic Number ◽

Analog Magnitude

Abstract. Number magnitude is assumed to be holistically represented along a single mental number line. Recently, we have observed a unit-decade-compatibility effect which is inconsistent with that assumption (Nuerk, Weger, & Willmes, 2001) . In two-digit Arabic number comparison, we have demonstrated that compatible comparisons in which separate decade and unit comparisons lead to the same decision (32_47, 3 < 4 and 2 < 7) were faster than incompatible trials (37_52, 3 < 5, but 7 > 2). Because overall distance was matched, a holistic model could not account for the compatibility effect. However, one could argue that the compatibility effect was due to the specific vertical perceptual arrangement of the two-digit numbers in Nuerk et al.’s (2001) experiment where the decade digits and unit digits were presented column-wise above each other. To examine this objection, we studied the perceptual generality of the compatibility effect with diagonal presentation. We replicated the compatibility effect with diagonal presentation. It is concluded that the compatibility effect is not due to encoding characteristics imposed by the perceptual setting of the original experiment. In particular, the assumption of an overall analog magnitude representation for two-digit numbers is not consistent with these data.

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Neural Correlates of Symbolic Number Comparison in Developmental Dyscalculia

Journal of Cognitive Neuroscience ◽

10.1162/jocn.2009.21237 ◽

2010 ◽

Vol 22 (5) ◽

pp. 860-874 ◽

Cited By ~ 111

Author(s):

Christophe Mussolin ◽

Anne De Volder ◽

Cécile Grandin ◽

Xavier Schlögel ◽

Marie-Cécile Nassogne ◽

...

Keyword(s):

Number Processing ◽

Numerical Distance ◽

Comparison Task ◽

Developmental Dyscalculia ◽

Disability History ◽

Magnitude Processing ◽

Number Magnitude ◽

Number Comparison ◽

The Right ◽

Symbolic Number

Developmental dyscalculia (DD) is a deficit in number processing and arithmetic that affects 3–6% of schoolchildren. The goal of the present study was to analyze cerebral bases of DD related to symbolic number processing. Children with DD aged 9–11 years and matched children with no learning disability history were investigated using fMRI. The two groups of children were controlled for general cognitive factors, such as working memory, reading abilities, or IQ. Brain activations were measured during a number comparison task on pairs of Arabic numerals and a color comparison task on pairs of nonnumerical symbols. In each task, pairs of stimuli that were close or far on the relevant dimension were constituted. Brain activation in bilateral intraparietal sulcus (IPS) was modulated by numerical distance in controls but not in children with DD. Moreover, although the right IPS responded to numerical distance only, the left IPS was influenced by both numerical and color distances in control children. Our findings suggest that dyscalculia is associated with impairment in areas involved in number magnitude processing and, to a lesser extent, in areas dedicated to domain-general magnitude processing.

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The Source of the Symbolic Numerical Distance and Size Effects

10.20944/preprints201609.0025.v1 ◽

2016 ◽

Cited By ~ 1

Author(s):

Attila Krajcsi ◽

Gábor Lengyel ◽

Petia Kojouharova

Keyword(s):

Size Effect ◽

Size Effects ◽

Mathematical Knowledge ◽

Semantic Network ◽

Number System ◽

Numerical Distance ◽

Alternative Account ◽

Semantic System ◽

Alternative Approach ◽

Conceptual Network

Human number understanding is thought to rely on the analogue number system (ANS), working according to Weber’s law. We propose an alternative account, suggesting that symbolic mathematical knowledge is based on a discrete semantic system (DSS), a representation that stores values in a semantic network, similar to the mental lexicon or to a conceptual network. Here, focusing on the phenomena of numerical distance and size effects in comparison tasks, first we discuss how a DSS model could explain these numerical effects. Second, we demonstrate that DSS model can give quantitatively as appropriate a description of the effects as the ANS model. Finally, we show that symbolic numerical size effect is mainly influenced by the frequency of the symbols, and not by the ratios of their values. This last result suggests that numerical distance and size effects cannot be caused by the ANS, while the DSS model might be the alternative approach that can explain the frequency-based size effect.

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Can working memory training improve preschoolers’ numerical abilities?

Journal of Numerical Cognition ◽

10.5964/jnc.v3i2.54 ◽

2017 ◽

Vol 3 (2) ◽

pp. 516-539 ◽

Cited By ~ 6

Author(s):

Nastasya Honoré ◽

Marie-Pascale Noël

Keyword(s):

Working Memory ◽

Spatial Working Memory ◽

Verbal Working Memory ◽

Working Memory Training ◽

Cognitive Factors ◽

Memory Training ◽

Numerical Development ◽

Number Comparison ◽

Arabic Number

A large number of studies have pointed out the role of working memory throughout numerical development. Working memory capacities seem to be improved after training and some studies have observed an impact of working memory training on academic performance. In our study, we examined whether training visuo-spatial working memory (with Cogmed) enhances working memory abilities and numerical development in the short and middle term in 5-6 year-old children. Fourty six children were randomly assigned to the experimental condition (adaptive working memory training) or the control condition (non-adaptive, demo version). The program was implemented daily for a period of five weeks in both groups. We observed an immediate impact of the adaptive version on visuo-spatial sketchpad and visuo-spatial central executive abilities and a small impact on Arabic number comparison. No training effect was observed in verbal working memory, in counting, collection comparison and addition. Furthermore, the observed effects were not sustained ten weeks later. These results are discussed in the context of specific and general cognitive factors that support numerical development and we argue against the idea of developing general cognitive factors to efficiently boost numerical development.

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Professional mathematicians do not differ from others in the symbolic numerical distance and size effects

10.31234/osf.io/evzpa ◽

2020 ◽

Author(s):

Mateusz Hohol ◽

Klaus Willmes ◽

Edward Nęcka ◽

Bartosz Brożek ◽

Hans-Christoph Nuerk ◽

...

Keyword(s):

Size Effect ◽

Size Effects ◽

Reference Group ◽

Distance Effect ◽

Numerical Distance ◽

Single Digit ◽

Mathematical Skills ◽

Numerical Distance Effect ◽

Individual Level ◽

The Individual

The numerical distance effect (it is easier to compare numbers that are further apart) and size effect (for a constant distance, it is easier to compare smaller numbers) characterize symbolic number processing. However, evidence for a relationship between these two basic phenomena and more complex mathematical skills is mixed. Previously this relationship has only been studied in participants with normal or poor mathematical skills, not in mathematicians. Furthermore, the prevalence of these effects at the individual level is not known. Here we compared professional mathematicians, engineers, social scientists, and a reference group using the symbolic magnitude classification task with single-digit Arabic numbers. The groups did not differ with respect to symbolic numerical distance and size effects in either frequentist or Bayesian analyses. Moreover, we looked at their prevalence at the individual level using the bootstrapping method: while a reliable numerical distance effect was present in almost all participants, the prevalence of a reliable numerical size effect was much lower. Again, prevalence did not differ between groups. In summary, the phenomena were neither more pronounced nor more prevalent in mathematicians, suggesting that extremely high mathematical skills neither rely on nor have special consequences for analogue processing of symbolic numerical magnitudes.

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Professional mathematicians do not differ from others in the symbolic numerical distance and size effects

Scientific Reports ◽

10.1038/s41598-020-68202-z ◽

2020 ◽

Vol 10 (1) ◽

Cited By ~ 1

Author(s):

Mateusz Hohol ◽

Klaus Willmes ◽

Edward Nęcka ◽

Bartosz Brożek ◽

Hans-Christoph Nuerk ◽

...

Keyword(s):

Size Effects ◽

Numerical Distance

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Non-numerical Distance and Size Effects in an Ant

Journal of Biology and Life Science ◽

10.5296/jbls.v11i2.16895 ◽

2020 ◽

Vol 11 (2) ◽

pp. 13

Author(s):

Marie-Claire Cammaerts ◽

Roger Cammaerts

Keyword(s):

Size Effect ◽

Operant Conditioning ◽

Size Effects ◽

Visual Cues ◽

Distance Effect ◽

Just Noticeable Difference ◽

Numerical Distance ◽

Weber's Law ◽

Different Dimensions ◽

Similar Elements

The distance effect (the fact that the individuals’ discrimination between two similar elements increases with the magnitude of the distance between them) as well as the size effect (the fact that the individuals’ discrimination between two similar elements decreases with the size of these elements) have been largely reported in vertebrates but not in invertebrates. Here, we demonstrate their existence in an ant, using operant conditioning to visual cues (black circles) of different dimensions. The two effects were obvious and differed from one another. Both effects could be accounted for Weber’s law, but it was here not tempted to verify if they are in line with this law by defining the just noticeable difference the ants can perceive between the cues.

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