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

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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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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Symbolic number comparison is not processed by the analogue number system: different symbolic and nonsymbolic numerical distance and size effects

10.31234/osf.io/dmau5 ◽

2017 ◽

Cited By ~ 1

Author(s):

Attila Krajcsi ◽

Gabor Lengyel ◽

Petia Kojouharova

Keyword(s):

Size Effects ◽

Goodness Of Fit ◽

Numerical Cognition ◽

Reaction Times ◽

Number System ◽

Error Rates ◽

Numerical Distance ◽

Comparison Task ◽

Number Comparison ◽

Symbolic Number

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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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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Weber’s Law Applies to the Ants’ Visual Perception

Journal of Biology and Life Science ◽

10.5296/jbls.v11i2.16896 ◽

2020 ◽

Vol 11 (2) ◽

pp. 36

Author(s):

Marie-Claire Cammaerts ◽

Roger Cammaerts

Keyword(s):

Visual Perception ◽

Operant Conditioning ◽

Size Effects ◽

Experimental Methodology ◽

Just Noticeable Difference ◽

Weber’S Law ◽

Numerical Distance ◽

Weber's Law

Non-numerical distance and size effects have been previously observed in the ant Myrmica sabuleti. As such effects can be theoretically in line with Weber’s law, we presumed that this law, until now examined in vertebrates, could also apply to ants. Using operant conditioning we trained then tested M. sabuleti workers faced with black circles having fixed diameters of 2, 3 and 4 mm against circles with diameters increasing by 0.5 mm until the ants perceived a difference between the smaller and the larger circles. This just noticeable difference occurred when the larger diameter reached 3.5, 5.5 and 7 mm respectively, what corresponded to a ratio larger/smaller surface of 3.06, 3.36 and 3.06. Owing to the degree of accuracy of the experimental methodology, this ratio is sufficiently constant for being consistent with Weber’s law.

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