A delta-rule model of numerical and non-numerical order processing.

Lengyel²,

2016

Preprint

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.Keywords: Numerical cognition; Numerical distance effect; Numerical size effect; Analogue number system; Discrete semantic system 1 An alternative to the analogue number system According to the current models understanding numbers is supported by an evolutionary ancient representation shared by many species (Dehaene, Dehaene-Lambertz, & Cohen, 1998;Gallistel & Gelman, 2000;Hauser & Spelke, 2004), the analogue number system (ANS). One defining feature of the ANS is that it works similarly to some perceptual representations in which the ratio of the stimuli's intensity determines the performance (Weber's law) (Cantlon, Platt, & Brannon, 2009;Moyer & Landauer, 1967;Walsh, 2003). Two critical phenomena supporting the ratio based performance is the distance and the size effects: when two numbers are compared, the smaller the distance between the two values is or the larger the two numbers are, the slower and more error prone the comparison is (Moyer & Landauer, 1967) (Figure 1 and 2). Thus, in the literature, the numerical distance and size effects are considered to be the sign of an analogue noisy numerical processing system working according to Weber's law. The distance and the size effects are observable both in non-symbolic and symbolic number processing, reflecting that the same type of system processes numerical information, independent of the number notations (Dehaene, 1992;Eger, Sterzer, Russ, Giraud, & Kleinschmidt, 2003).However, the distance and size effects in symbolic comparison can also be explained by a different representation. Quite intuitively, one might think that symbolic and abstract mathematical concepts, Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 7 September 2016 doi:10.20944/preprints201609.0025.v1 PREPRINTThe source of the symbolic numerical distance and size effects 2 like numbers could be handled by a discrete semantic system (DSS), similar to conceptual networks or to the mental lexicon, i.e., representations that process symbolic and abstract concepts. In this DSS model, numbers are stor...

Section: Related Models For Symbolic Number Processingmentioning

confidence: 99%

Section: Related Models For Symbolic Number Processingmentioning

confidence: 99%

Section: Dss Explanation For Other Numerical Effectsmentioning

confidence: 99%

mentioning

confidence: 99%

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confidence: 99%

See 3 more Smart Citations

The Source of the Symbolic Numerical Distance and Size Effects

Lengyel²,

2016

Preprint

Processing Symbolic Numbers: The Example of Distance and Size Effects

Language, Cognition, and Mind

Lengyel³

2021

In 1967, Moyer and Landauer published a relatively simple experiment in Nature that, despite its simplicity, profoundly changed how cognitive scientists thought about mathematical abilities. Before Moyer and Landauer's work, it was supposed that mathematics is a complicated subject for most people and that, from a cognitive perspective, numerical cognition is a high-level, human-specific, culture-dependent capability, which relies on complex mental processes. The work of Moyer and Landauer (1967) changed this view radically; they proposed that even symbolic number tasks may rely on a mechanism as simple as the perceptual representations described in psychophysics. In their experiment, it was found that, when participants compare single-digit numbers (e.g., "which one is larger, 5 or 8?"), the their performance depended on the ratio of the two numbers, with a behavioral pattern resembling the performance that the well-known Weber's law would generate. This meant that understanding even symbolic mathematics, which had been believed to be a high-level and culture-dependent cognitive process, may be supported by a very simple, evolutionarily old representation, which was later named the Approximate Number System (ANS).In line with this discovery, many later studies demonstrated that the functioning of the ANS can be observed in young children or even infants (e.g., Izard, Sann, Spelke, & Streri, 2009), as well as in non-human animals (e.g., Hauser, 2000). Many further works proposed that the ANS played pivotal roles in many phenomena, such as the interference of numerical and spatial information (Dehaene, Bossini,

The Indo-Arabic distance effect originates in the response statistics of the task

2018

Psychological Research

In the number comparison task distance effect (better performance with larger distance between the two numbers) and size effect (better performance with smaller numbers) are used extensively to find the representation underlying numerical cognition. According to the dominant analog number system (ANS) explanation, both effects depend on the extent of the overlap between the noisy representations of the two values. An alternative discrete semantic system (DSS) account supposes that the distance effect is rooted in the association between the numbers and the "small-large" properties with better performance for numbers with relatively high differences in their strength of association, and that the size effect depends on the everyday frequency of the numbers with smaller numbers being more frequent and thus easier to process. A recent study demonstrated that in a new, artificial digit notation-where both association and frequency can be arbitrarily manipulated-the distance and size effects change according to the DSS account. Here, we investigate whether the same manipulations modify the distance and size effects in Indo-Arabic notation, for which associations and frequency are already well established. We found that the distance effect depends on the association between the numbers and the "small-large" responses. It was also found that while the distance effect is flexible, the size effect seems to be unaltered, revealing a dissociation between the two effects. This result challenges the ANS view, which supposes a single mechanism behind the distance and size effects, and supports the DSS account, supposing two independent, statistics-based mechanisms behind the two effects.