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Numerical distance and size effects (easier number comparison with large distance or small size) are mostly supposed to reflect a single effect, the ratio effect, which is the consequence of the analogue number system (ANS) activation, working according to Weber’s law. In an alternative model, symbolic numbers can be processed by a discrete semantic system (DSS), in which the distance and the size effects could originate in two independent factors: the distance effect depends on the semantic distance of the units, and the size effect depends on the frequency of the symbols. While in the classic view both symbolic and nonsymbolic numbers are processed by the ANS, in the alternative view only nonsymbolic numbers are processed by the ANS, but symbolic numbers are handled by the DSS. The current work contrasts the two views, investigating whether the size of the distance and the size effects correlate in nonsymbolic dot comparison and in symbolic Indo-Arabic comparison tasks. If a comparison is backed by the ANS, the distance and the size effects should correlate, because the two effects are merely two ways to measure the same ratio effect, however, if a comparison is supported by other system, for example the DSS, the two effects might dissociate. In the current measurements the distance and the size effects correlated very strongly in the dot comparison task, but they did not correlate in the Indo-Arabic comparison task. Additionally, the effects did not correlate between the Indo-Arabic and the dot comparison tasks. These results suggest that symbolic number comparison is not handled by the ANS, but by an alternative representation, such as the DSS.
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... It has recently been shown that the distance and size effects do not correlate in symbolic number comparison (Krajcsi, 2017). This result is in contrast with the ANS model: The model proposes that the two effects are simply two ways to measure the single ratio effect, therefore, the two effects should correlate. ...

... In this explanation, the stable and flexible parts of the size effect can come from different sources: the former is rooted in the ANS, and the latter is caused by the frequency of the stimuli. Although this explanation may account for the present results, it cannot explain other features of symbolic comparison: For example, the distance and size effects were found to be independent in symbolic comparison (Krajcsi, 2017); the distance effect can be present without the size effect when the frequency is uniform (Krajcsi et al., 2016); the distance effect follows the statistics of associations between numbers and small-large categories instead of the values of the numbers Krajcsi & Kojouharova, 2017); unlike non-symbolic comparison performance, symbolic Indo-Arabic comparison performance cannot be described properly with psychophysical functions (Krajcsi, Lengyel, & Kojouharova, 2018). Overall, while the ANS model can explain the present results, it cannot explain the independence and flexibility of distance and size effects found in related studies. ...

... Thus, if the frequency explanation is correct for the stable component of the size effect, it is not yet clear why the distance effect is more flexible than the size effect observed here. There could be other explanations for the stable part of the size effect other than the ANS (Dehaene, 1992(Dehaene, , 2007Moyer & Landauer, 1967) or the models that propose the role of the frequency in the size effect, such as the connectionist model of symbolic number processing (Verguts et al., 2005;Verguts & Van Opstal, 2014) or the DSS model (Krajcsi, 2017;Krajcsi et al., 2016). However, we are not aware of any other model that could give a consistent explanation for the size effect. ...

In the symbolic number comparison task, the size effect (better performance for small than for large numbers) is usually interpreted as the result of the more general ratio effect, in line with Weber's law. In alternative models, the size effect might be a result of stimulus frequency: smaller numbers are more frequent, and more frequent stimuli are easier to process. It has been demonstrated earlier, that in artificial new number digits, the size effect reflects the frequencies of those digits. In the present work we investigate whether frequency also directs the size effect in Indo-Arabic numbers, in which notation, unlike in new symbols, the frequencies are already firmly established for the participants. We found that frequency has an effect on the size effect in Indo-Arabic notation, but this influence is limited. However, this limited size effect change is acquired fast at the beginning of the session. We argue that these results are more in line with the frequency-based accounts of the size effect.

... The term "speeded" should be taken in an inclusive sense-it entails cases in which RT-measurement does not appear explicitly in the instructions. Speeded responding is sometimes implicit in the task even when the stimulus presentation is long and even when the observer is given unlimited time to respond (see Krajcsi, 2017). This is attested by the fairly short responses (they rarely exceed 2 s even under these conditions). ...

... One should realize though that, even when the effect is present, it does not necessarily refer to, or become diagnostic of, number processing. Any ordered set of stimuli (height of people, months of the year, or ranks in the military) yields a "distance effect" (Leth-Steensen, & Marley, 2000;Sasanguie, De Smedt, & Reynvoet, 2017;Verguts, Fias, & Stevens, 2005;Verguts, & Van Opstal, 2005, 2014 see also Krajcsi, 2017). Leth-Steensen and Marley (2000) and subsequently Van Opstal (2005, 2014) 1 3 demonstrated in their respective computational models that tasks with ordered sets of items produce a distance effect as a natural derivative. ...

... However, we do not interpret the observed effect as one uniquely associated with number processing. As we recounted, any ordered set of stimuli yields a "distance effect" (Leth-Steensen, & Marley, 2000;Sasanguie et al., 2017;Verguts, & Van Opstal, 2005, 2014; see also Krajcsi, 2017). A distance effect is a necessary derivative of processing ordered items. ...

In the number-to-position methodology, a number is presented on each trial and the observer places it on a straight line in a position that corresponds to its felt subjective magnitude. In the novel modification introduced in this study, the two-numbers-to-two-positions method, a pair of numbers rather than a single number is presented on each trial and the observer places them in appropriate positions on the same line. Responses in this method indicate not only the subjective magnitude of each single number but, simultaneously, provide a direct estimation of their subjective numerical distance. The results of four experiments provide strong evidence for a linear representation of numbers and, commensurately, for the linear representation of numerical distances. We attribute earlier results that indicate a logarithmic representation to the ordered nature of numbers and to the task used and not to a truly non-linear underlying representation.

... Supporting the alternative DSS model, it has been found that the size effect followed the frequency of the digits in an artificial number notation comparison task (Krajcsi et al., 2016). In addition, it has been shown in a correlational study that in symbolic number comparison task, the distance and the size effects were independent (Krajcsi, 2017), reflecting two independent mechanisms generating the two effects. (See a similar prediction for independent distance and size effects in Verguts et al., 2005;Verguts and Van Opstal, 2014). ...

... Together with the present results, several findings converge to the conclusion that the symbolic number comparison task cannot be explained by the ANS. First, unlike the prediction of that model suggesting that distance and size effects are two ways to measure the single ratio effect, symbolic distance and size effects are independent (Krajcsi, 2017), and the distance effect can be present even when no size effect can be observed (shown in the present results and in Krajcsi et al., 2016). Second, the size effect follows the frequency of the numbers as demonstrated in Krajcsi et al. (2016) and also in the present results, where the uniform frequency of the digits induced no size effect (i.e., the slope of the size effect is zero). ...

... The present and some previous results also characterize the symbolic numerical comparison task; an alternative model should take the following into consideration: (a) symbolic distance and size effects are independent (Krajcsi et al., 2016;Krajcsi, 2017), (b) the effects are notation independent (the present results and Krajcsi et al., 2016), (c) the size effect depends on the frequency of the numbers (the present results and Krajcsi et al., 2016), (d) the distance effect depends on the association between the numbers and the small-large categories (present results), and (e) the distance effect can be described with a logarithm of the difference of the values (present results). ...

In a comparison task, the larger the distance between the two numbers to be compared, the better the performance—a phenomenon termed as the numerical distance effect. According to the dominant explanation, the distance effect is rooted in a noisy representation, and performance is proportional to the size of the overlap between the noisy representations of the two values. According to alternative explanations, the distance effect may be rooted in the association between the numbers and the small-large categories, and performance is better when the numbers show relatively high differences in their strength of association with the small-large properties. In everyday number use, the value of the numbers and the association between the numbers and the small-large categories strongly correlate; thus, the two explanations have the same predictions for the distance effect. To dissociate the two potential sources of the distance effect, in the present study, participants learned new artificial number digits only for the values between 1 and 3, and between 7 and 9, thus, leaving out the numbers between 4 and 6. It was found that the omitted number range (the distance between 3 and 7) was considered in the distance effect as 1, and not as 4, suggesting that the distance effect does not follow the values of the numbers predicted by the dominant explanation, but it follows the small-large property association predicted by the alternative explanations.

... Previous research has indeed provided conflicting evidence about such a relationship and has consequently generated competing theoretical perspectives. That is, whereas some theories have maintained that both the representation of symbolic and non-symbolic numbers can be traced back to the same pre-verbal approximate number system (ANS; Cantlon, Platt & Branno, 2009;Dehaene, 1997;Feigenson, Dehaene & Spelke, 2004;Gallistel & Gelman, 1992;Rinaldi & Marelli, 2020a), other competing theoretical perspectives have rather proposed the existence of two independent systems (e.g., Krajcsi, 2017;Krajcsi, Lengyel & Kojouharova, 2018;Sasanguie, De Smedt & Reynvoet, 2017). ...

... Rather surprisingly, indeed, a ratio-dependency has been as well observed in symbolic number comparison tasks (i.e., either with number words or Arabic digits) (Moyer & Landauer, 1967;Gallistel & Gelman, 1992). Yet, this theoretical account has been challenged in the more recent years by an increasing body of evidence supporting two separate systems for the representation of symbolic and nonsymbolic numbers (Krajcsi, 2017;Krajcsi, Lengyel, & Kojouharova, 2016;Lyons, Ansari & Beilock, 2015;Marinova, Sasanguie & Reynvoet, 2020;Sasanguie et al., 2017). Some studies have shown, indeed, that the performance in number comparison tasks is fully ratiodependent only with non-symbolic numbers (e.g., Marinova et al., 2020). ...

There is an ongoing, vibrant debate about whether numerical information in both nonsymbolic and symbolic notations would be supported by different neurocognitive systems or rather by a common preverbal approximate number system, which is ratio dependent and follows Weber’s law. Here, we propose that the similarities between nonsymbolic and symbolic number processing can be explained based on the principle of efficient coding. To probe this hypothesis we employed a new empirical approach, by predicting the behavioural performance in number comparison tasks with symbolic (i.e., number words) and nonsymbolic (i.e., arrays of dots) information not only from numerical ratio, but for the first time also from natural language data. That is, we used data extracted from vector-space models that are informative about the distributional pattern of numberwords usage in natural language. Results showed that linguistic estimates predicted the behavioural performance in both symbolic and nonsymbolic tasks. However, and critically, our results also showed a task-dependent dissociation: linguistic data better predicted the performance in the symbolic task, whereas real numerical ratio better predicted the performance in the nonsymbolic task. These findings indicate that efficient coding of environmental regularities is an explanatory principle of human behavior in tasks involving numerical information. They also suggest that the ability to discriminate a stimulus from similar ones varies as a function of the specific statistical structure of the considered learning environment.

... In contrast, according to the hybrid ANS-DSS account, this perfect correlation could be expected only in nonsymbolic comparison, while symbolic comparison distance and size effects might be independent (see again Figure 5). In a study, measuring the correlation of the slope of the distance effect and the slope of the size effect in nonsymbolic and symbolic comparison tasks, it was found that in nonsymbolic comparison after correcting for the reliability-caused attenuation of correlation, the correlation was practically 1, while in symbolic comparison the same correlation was not significantly different from 0 (Krajcsi, 2017). Again, this result argues for different types of mechanisms behind symbolic and nonsymbolic comparison, in line with the hybrid ANS-DSS account. ...

... Correlation of the slopes (Krajcsi, 2017) Independent Strongly correlate ...

According to the dominant view in the literature, several numerical cognition phenomena are explained coherently and parsimoniously by the Approximate Number System (ANS) model, which supposes the existence of an evolutionarily old, simple representation behind many numerical tasks. We offer an alternative account that proposes that only nonsymbolic numbers are processed by the ANS, while symbolic numbers, which are more essential to human mathematical capabilities, are processed by the Discrete Semantic System (DSS). In the DSS, symbolic numbers are stored in a network of nodes, similar to conceptual or linguistic networks. The benefit of the DSS model and the benefit of the more general hybrid ANS–DSS framework are demonstrated using the crucial example of the distance and size effects of comparison tasks.

... As found in the numerical domain (Verguts & Van Opstal, 2005), this type of dissociation between distance and size effects might indicate that the size effect did not originate from the MTL but from other mechanisms. At the same time, this finding may indicate that a discrete semantic system (DSS), instead of the analogue time system (ATS in line with the analogue number system or ANS) is involved in temporal processing, as has been proposed for numbers (Krajcsi, 2017). Nevertheless, it cannot be excluded that the creation of two groups according to the assignment between hand and key to be pressed masked any size effect. ...

... This dissociation could advocate the involvement of different mechanisms, as well as that of different models, in the mental representation of time. Thus, future studies should examine the dissociation between distance and size effects, and conduct an investigation, similar to that concerning the numerical domain, into the possible sources of these effects (Krajcsi, 2017). ...

The space-time interaction suggests a left-to-right directionality in the mind’s representation of elapsing time. However, studies showing a possible vertical time representation are scarce and contradictory. In Experiment 1, 32 participants had to judge the duration (200, 300, 500 or 600 milliseconds) of the target stimulus that appeared at the top, centre, or bottom of the screen, compared to a reference stimulus (400 milliseconds) always appeared in the centre of the screen. In Experiment 2, 32 participants were administered with the same procedure, but the reference stimulus appeared at the top, centre, or bottom of the screen and the target stimulus was fixed in the centre location. In both experiments, a space-time interaction was found with an association between short durations and bottom response key as well as between long durations and top key. The evidence of a vertical mental timeline was further confirmed by the distance effect with a lower level of performance for durations close to that of the reference stimulus. The results suggest a bottom-to-top mapping of time representation, more in line with the metaphor “more is up”.

... On the other hand, our findings can be accounted for by another framework for magnitude processing, namely Krajcsi's DSS framework 34,91,92 . It postulates that the symbolic NDE and NSE emerge from two independent mechanisms. ...

... This is substantiated in the fact that the symbolic NDE is uncorrelated with the non-symbolic NDE 96 . Moreover, Krajcsi's framework discussed above assumes that symbolic and non-symbolic numerical magnitudes are handled by independent cognitive systems (namely non-symbolic by ANS and symbolic by DSS) 34,91 . Therefore, a generalization of the behavioural pattern we found in the symbolic processing domain to the non-symbolic one would be unsound. ...

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.

... Alternative mechanisms have been suggested to explain distance effects due to symbolic numerical stimuli. For example, Krajcsi (2017) suggested instead of the ANS, a discrete semantic system (DSS) underlies symbolic number representation. Here, symbolic numbers exist as nodes that are connected through semantic associations. ...

... Instead, the DSS, in which numbers are represented discretely with semantically associated nodes, seems to better fit symbolic numerical comparison behavioral data, and thus may reflect a more suitable explanation for the NDE in symbolic numerical tasks than the ANS. Fitting with this hypothesis that different mechanisms underlie symbolic and nonsymbolic numerical representation, both Krajcsi (2017) and Lyons, Nuerk, and Ansari (2015) did not find a significant association between measures from symbolic and nonsymbolic comparison tasks within-participants. If these tasks are tapping into representations that have a shared underlying mechanism (i.e., the ANS), one would expect an association between the nonsymbolic and symbolic measures. ...

How are number symbols (e.g., Arabic digits) represented in the brain? Functional resonance imaging adaptation (fMRI‐A) research has indicated that the intraparietal sulcus (IPS) exhibits a decrease in activation with the repeated presentation of the same number, that is followed by a rebound effect with the presentation of a new number. This rebound effect is modulated by the numerical ratio or difference between presented numbers. It has been suggested that this ratio‐dependent rebound effect is reflective of a link between the symbolic numerical representation system and an approximate magnitude system. Experiment 1 used fMRI‐A to investigate an alternative hypothesis: that the rebound effect observed in the IPS is related to the ordinal relationships between symbols (e.g., 3 comes before 4; C after B). In Experiment 1, adult participants exhibited the predicted distance‐dependent parametric rebound effect bilaterally in the IPS for number symbols during a number adaptation task, however, the same effect was not found anywhere in the brain in response to letters. When numbers were contrasted with letters (numbers > letters), the left intraparietal lobule remained significant. Experiment 2 demonstrated that letter stimuli used in Experiment 1 generated a behavioral distance effect during an active ordinality task, despite the lack of a neural distance effect using fMRI‐A. The current study does not support the hypothesis that general ordinal mechanisms underpin the neural parametric recovery effect in the IPS in response to number symbols. Additional research is needed to further our understanding of mechanisms underlying symbolic numerical representation in the brain.

... In the present experiment we investigated whether size effects can dissociate from distance effect if the frequency of the symbols is manipulated. (See another type of test for the dissociation of the two effects in Krajcsi, 2016) To manipulate the frequency of the symbols, it might be more appropriate to use new symbols, instead of the well-known Indo-Arabic symbols, because the frequency of the already known symbols might be well established and learned. ...

... For example, symbolic and non-symbolic performance seems to be independent on many behavioral (Holloway and Ansari, 2009;Sasanguie et al., 2014;Schneider et al., 2016) and neural level (Damarla and Just, 2013;Bulthé et al., 2014Bulthé et al., , 2015Lyons et al., 2015). In a correlational study it has been shown that distance and size effects dissociate in Indo-Arabic comparison task (Krajcsi, 2016). Some results show that the numerical representation is not analog: Functional activation in the brain while processing symbolic numbers seems to be discrete (Lyons et al., 2015), and symbolic numbers can also interfere with the discrete yes-no responses (Landy et al., 2008). ...

Human number understanding is thought to rely on the analog 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 the 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.

... In the present experiment we investigated whether size effects can dissociate from distance effect if the frequency of the symbols is manipulated. (See another type of test for the dissociation of the two effects in Krajcsi, 2016) To manipulate the frequency of the symbols, it might be more appropriate to use new symbols, instead of the well-known Indo-Arabic symbols, because the frequency of the already known symbols might be well established and learned. ...

... For example, symbolic and non-symbolic performance seems to be independent on many behavioral (Holloway and Ansari, 2009;Sasanguie et al., 2014;Schneider et al., 2016) and neural level (Damarla and Just, 2013;Bulthé et al., 2014Bulthé et al., , 2015Lyons et al., 2015). In a correlational study it has been shown that distance and size effects dissociate in Indo-Arabic comparison task (Krajcsi, 2016). Some results show that the numerical representation is not analog: Functional activation in the brain while processing symbolic numbers seems to be discrete (Lyons et al., 2015), and symbolic numbers can also interfere with the discrete yes-no responses (Landy et al., 2008). ...

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. (Find preprint at http://www.preprints.org/manuscript/201609.0025/v1)

... In a DSS, symbolic numerical magnitudes are stored within a large semantic network, with each symbolic numerical magnitude acting as a node within that network. A DSS would produce a 'distance effect' because the strength of the associations between symbolic numerical magnitudes (i.e., nodes) would correlate with the strength of the semantic relations between the numbers (Krajcsi, 2017;Krajcsi et al., 2016). Evidence that symbolic numerical magnitudes may be supported by a DSS rather than an approximate magnitude system has accumulated both behaviourally (Krajcsi et al., 2016(Krajcsi et al., , 2018 and at the neural level of analysis (Lyons & Beilock, 2018). ...

Are number symbols (e.g., 3) and numerically equivalent quantities (e.g., •••) processed similarly or distinctly? If symbols and quantities are processed similarly then processing one format should activate the processing of the other. To experimentally probe this prediction, we assessed the processing of symbols and quantities using a Stroop-like paradigm. Participants (NStudy1 = 80, NStudy2 = 63) compared adjacent arrays of symbols (e.g., 4444 vs 333) and were instructed to indicate the side containing either the greater quantity of symbols (nonsymbolic task) or the numerically larger symbol (symbolic task). The tasks included congruent trials, where the greater symbol and quantity appeared on the same side (e.g. 333 vs. 4444), incongruent trials, where the greater symbol and quantity appeared on opposite sides (e.g. 3333 vs. 444), and neutral trials, where the irrelevant dimension was the same across both sides (e.g. 3333 vs. 333 for nonsymbolic; 333 vs. 444 for symbolic). The numerical distance between stimuli was systematically varied, and quantities in the subitizing and counting range were analyzed together and independently. Participants were more efficient comparing symbols and ignoring quantities, than comparing quantities and ignoring symbols. Similarly, while both symbols and quantities influenced each other as the irrelevant dimension, symbols influenced the processing of quantities more than quantities influenced the processing of symbols, especially for quantities in the counting rage. Additionally, symbols were less influenced by numerical distance than quantities, when acting as the relevant and irrelevant dimension. These findings suggest that symbols are processed differently and more automatically than quantities.

... A first group of relevant empirical studies investigated the correlation of the CDE and the PDE. According to the ANS account, because both the CDE and the PDE rely on the same representation and the performance in both effects depends on the Weber fraction (i.e., sensitivity) of the ANS, the correlation coefficient of the two effects ideally should be 1 (see a similar approach in Krajcsi, 2017, where the expected perfect correlation between nonsymbolic numerical comparison effects has been observed). This prediction was not confirmed by empirical studies: Significant nonzero correlation was found neither in children with symbolic numbers (Reynvoet et al., 2009) nor in adults with nonsymbolic values (Sasanguie et al., 2011). ...

In elementary symbolic number processing, the comparison distance effect (in a comparison task, the task is more difficult with smaller numerical distance between the values) and the priming distance effect (in a number processing task, actual number is easier to process with a numerically close previous number) are two essential phenomena. While a dominant model, the approximate number system model, assumes that the two effects rely on the same mechanism, some other models, such as the discrete semantic system model, assume that the two effects are rooted in different generators. In a correlational study, here we investigate the relation of the two effects. Critically, the reliability of the effects is considered; therefore, a possible null result cannot be attributed to the attenuation of low reliability. The results showed no strong correlation between the two effects, even though appropriate reliabilities were provided. These results confirm the models of elementary number processing that assume distinct mechanisms behind number comparison and number priming.

... In a DSS, symbolic numerical magnitudes are stored within a large semantic network, with each symbolic numerical magnitude acting as a node within that network. A DSS would produce a 'distance effect' because the strength of the associations between symbolic numerical magnitudes (i.e., nodes) would correlate with the strength of the semantic relations between the numbers (Krajcsi, 2017;Krajcsi et al., 2016). Evidence that symbolic numerical magnitudes may be supported by a DSS rather than an approximate magnitude system has accumulated both behaviourally (Krajcsi et al., 2016(Krajcsi et al., , 2018 and at the neural level of analysis (Lyons & Beilock, 2018). ...

Are number symbols (e.g., 3) and numerically equivalent quantities (e.g., •••) processed similarly or distinctly? If symbols and quantities are processed similarly then processing one format should activate the processing of the other. To experimentally probe this prediction, we assessed the processing of symbols and quantities using a Stroop-like paradigm. Participants (NStudy1 = 80, NStudy2 = 63) compared adjacent arrays of symbols (e.g., 4444 vs 333) and were instructed to indicate the side containing either the greater quantity of symbols (nonsymbolic task) or the numerically larger symbol (symbolic task). The tasks included congruent trials, where the greater symbol and quantity appeared on the same side (e.g. 333 vs. 4444), incongruent trials, where the greater symbol and quantity appeared on opposite sides (e.g. 3333 vs. 444), and neutral trials, where the irrelevant dimension was the same across both sides (e.g. 3333 vs. 333 for nonsymbolic; 333 vs. 444 for symbolic). The numerical distance between stimuli was systematically varied, and quantities in the subitizing and counting range were analyzed together and independently. Participants were more efficient comparing symbols and ignoring quantities, than comparing quantities and ignoring symbols. Similarly, while both symbols and quantities influenced each other as the irrelevant dimension, symbols influenced the processing of quantities more than quantities influenced the processing of symbols, especially for quantities in the counting rage. Additionally, symbols were less influenced by numerical distance than quantities, when acting as the relevant and irrelevant dimension. These findings suggest that symbols are processed differently and more automatically than quantities.

... In addition to this, behavioral studies have shown that symbolic and non-symbolic representations of numbers may be qualitatively different from one another (Krajcsi, 2017;Krajcsi, Lengyel, & Kojouharova, 2016;Lyons, Nuerk, & Ansari, 2015). Similarly, a complete overlap in regions subserving symbolic and non-symbolic numerical quantity processing has been questioned by recent neuroimaging evidence (Bulthé, De Smedt, & Op de Beeck, 2015;Damarla & Just, 2013; To appear in Journal of Experimental Psychology: General 6 Lyons, Ansari, & Beilock, 2015). ...

It has been suggested that the origins of number words can be traced back to an evolutionarily ancient approximate number system, which represents quantities on a compressed scale and complies with Weber’s law. Here, we use a data-driven computational model, which learns to predict one event (a word in a text corpus) from associated events, to characterize verbal behavior relative to number words in natural language, without appeal to perception. We show that the way humans use number words in spontaneous language reliably depends on numerical ratio - a clear signature of Weber’s law - thus perfectly mirroring the human and non-human psychophysical performance in comparative judgments of numbers. Most notably, the adherence to Weber’s law is robustly replicated in a wide range of different languages. Together, these findings suggest that the everyday use of number words in language rests upon a pre-verbal approximate number system, which would thus affect the handling of numerical information not only at the input level but also at the level of verbal production.

... Second, the distance and size effects are independent of each other, thus they cannot be the two consequences of the same ratio effect (Weber's law) as suggested by the ANS model. The independence of the two effects was also demonstrated in a correlational study, where the slopes of the distance and size effects of a symbolic comparison did not correlate (Krajcsi, 2016), and in a new, artificial digit comparison task where the distance effect was observable even when the size effect was absent . Similar to that correlational study, the present data also confirm the low correlation of the association-based distance and the size effect slopes in the reaction time: r s (18) = 0.347, p = 0.133, 95% CI [− 0.112, 0.685]. ...

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.

... Even if this viewpoint might seem unusual, it still could be valid. In this case, another types of tests should be found (see for alternative approaches for these tests in Krajcsi et al., 2016;Krajcsi, 2017). But if one thinks that the works that have proposed that ratio-based performance were valid, the present test should be considered to be valid, too. ...

HIGHLIGHTSWe test whether symbolic number comparison is handled by an analog noisy system.
Analog system model has systematic biases in describing symbolic number comparison.
This suggests that symbolic and non-symbolic numbers are processed by different systems.
Dominant numerical cognition models suppose that both symbolic and non-symbolic numbers are processed by the Analog 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 non-symbolic 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 non-symbolic dot comparison and symbolic Indo-Arabic comparison tasks. It was consistently found that while the ANS model's prediction is relatively good for the non-symbolic dot comparison, its prediction is poorer and systematically biased for the symbolic Indo-Arabic comparison. We conclude that only non-symbolic comparison is supported by the ANS, and symbolic number comparisons are processed by other representation.

17. Az áttekintő írás a magyar kognitív pszichológia és kognitív tudomány utóbbi 30 évét mutatja be. Intézmé-nyesen sokat jelentett a '90-es években a Soros Alapítvány támogatása az egyetemi kognitív programokban, melynek egyik következménye, hogy ma Budapesten három kognitív tanszék működik. Az intézményes fej-lődés második oldala a sok szakmát érintő konferenciák (MAKOG) sorozata és a bekapcsolódás a nemzet-közi kognitív oktatási programokba. Tudományos tartalmában a magyar kognitív kutatás is elmozdult a lehorgonyzatlan tiszta kognitív modellektől az idegrendszeri, fejlődési, szociális és evolúciós értelmezés irá-nyába, részben hazai hagyományokat is folytatva. Fontosabb sikeres területei az észlelés, elsősorban a látás és hallás fejlődésének vizsgálata (Kovács, Winkler), az emlékezeti gátlás és az implicit emlékezeti rendszerek neuropszichológiai értelmezése (Racsmány, Németh), a pszicholingvisztikában a magyar mondatszerkezet és az alaktan megértési modellekbe illesztése (Pléh, Lukács, Gergely), a magyar téri nyelv fejlődési és patológiás jellemzése (Pléh, Lukács), a képes beszéd elemzése pszichopatológiai folyamatokban (Schnell), és a metafo-rikusság és gyakoriság neuropszichológiai szétválasztása (Forgács). A fejlődési pszicholingvisztika legfon-tosabb eredményei a korai tudatelmélet nyelvelsajátítási szerepével kapcsolatosak (Kovács, Téglás, Király, Forgács). Tisztázták azt is, hogy a nyelvi fejlődés zavarai Williams-szindrómában és az ún. specifikus nyelvi zavarban (Lukács, Racsmány, Ladányi) a munkaemlékezeti rendszer moduláló szerepével, illetve általá-nos tanulási zavarokkal kapcsolatosak, különös tekintettel a procedurális rendszerek zavaraira (Lukács, Racsmány, Ladányi). Az utóbbi érintettségét számos neurológiai nyelvi zavarban is kimutatták (Janacsek, Németh, Lukács).

What is the nature of mathematics? What are its foundations and origins? Many readers would perhaps agree that a reasonable place to begin such quest is with the concept of “number.” But where do numbers come from? The field of “numerical cognition,” which gathers efforts from multiple disciplines, ranging from experimental psychology to cognitive linguistics, and from animal cognition to cognitive neuroscience, has studied these questions for several decades. The field has made progress in many directions (see, for example, Campbell, 2005; Cohen Kadosh & Dowker, 2015; Henik, 2016), but it has also proposed various accounts that are deeply incompatible and inconsistent. For instance, while it is very well established that there are hundreds of human cultures that do not exhibit arithmetic and lack words for quantities beyond three or four (Bowern & Zentz, 2012; Epps, Bowern, Hansen, Hill, & Zentz, 2012; Everett, 2017), studies in nonhuman animal cognition, in stark contrast, unproblematically report that there are “mathematical abilities” in monkeys (Beran, Perdue, & Evans, 2015), “numerical and arithmetic abilities in non-primate species” (Agrillo, 2015), “spontaneous number representation in mosquitofish” (Dadda, Piffer, Agrillo, & Bisazza, 2009), and that “numerical cognition in honeybees enables addition and subtraction” (Howard, Avarguès-Weber, Garcia, Greentree, & Dyer, 2019). And while historians of mathematics document with painstaking detail the struggles and difficulties that were involved in the (relatively recent) development of the concept of “zero” (Ifrah, 1985; Menninger, 1969), a recent article in Science magazine states that bees, despite their relatively simple nervous system and small cortex-less brains, are not prevented “from knowing how to understand numbers, including zero” (Nieder, 2018, p. 1970). How can insects with tiny cortex-less brains and nonsymbolic language be involved in “counting” and understanding “numbers” and “zero” while healthy humans from ancestral, environmentally adapted cultures, with brains that have about 90 billion more neurons, sophisticated symbolic languages and cultural practices do not? Clearly, there is something fundamental that does not square.

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 the analogue number magnitude representation. 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, nothing is known about the prevalence of these effects at the individual level. Here we compared professional mathematicians, engineers, social scientists, and a reference sample using the classical magnitude classification task. The groups did not differ with respect to numerical distance and size effects in frequentist or Bayesian analysis. 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 prevalent in mathematicians, suggesting that extremely high mathematical skills neither rely on nor have special consequences on analogue number magnitude processing.

HIGHLIGHTSWe test whether symbolic number comparison is handled by an analog noisy system.
Analog system model has systematic biases in describing symbolic number comparison.
This suggests that symbolic and non-symbolic numbers are processed by different systems.
Dominant numerical cognition models suppose that both symbolic and non-symbolic numbers are processed by the Analog 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 non-symbolic 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 non-symbolic dot comparison and symbolic Indo-Arabic comparison tasks. It was consistently found that while the ANS model's prediction is relatively good for the non-symbolic dot comparison, its prediction is poorer and systematically biased for the symbolic Indo-Arabic comparison. We conclude that only non-symbolic comparison is supported by the ANS, and symbolic number comparisons are processed by other representation.

Human number understanding is thought to rely on the analog 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 the 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.

Subitizing is a fast and accurate enumeration process of small sets of usually less than four objects. Several models were proposed in the literature. Critically, only pattern recognition theory suggests that subitizing performance is sensitive to the arrangement of the array. In our study, arrays of dots in random or canonical arrangements were enumerated. The subitizing range was larger and the reaction time slope was less steep in the canonical arrangements. When noise was added to the canonical pattern, the reaction time slope was proportional to the amount of noise. Moreover, arrangement has a stronger effect on sets with more than four objects. These results support the pattern recognition model of subitizing.

It has been difficult to determine how cognitive systems change over the grand time scale of an entire life, as few cognitive systems are well enough understood; observable in infants, adolescents, and adults; and simple enough to measure to empower comparisons across vastly different ages. Here we address this challenge with data from more than 10,000 participants ranging from 11 to 85 years of age and investigate the precision of basic numerical intuitions and their relation to students' performance in school mathematics across the lifespan. We all share a foundational number sense that has been observed in adults, infants, and nonhuman animals, and that, in humans, is generated by neurons in the intraparietal sulcus. Individual differences in the precision of this evolutionarily ancient number sense may impact school mathematics performance in children; however, we know little of its role beyond childhood. Here we find that population trends suggest that the precision of one's number sense improves throughout the school-age years, peaking quite late at ∼30 y. Despite this gradual developmental improvement, we find very large individual differences in number sense precision among people of the same age, and these differences relate to school mathematical performance throughout adolescence and the adult years. The large individual differences and prolonged development of number sense, paired with its consistent and specific link to mathematics ability across the age span, hold promise for the impact of educational interventions that target the number sense.

There is considerable interest in how humans estimate the number of objects in a scene in the context of an extensive literature on how we estimate the density (i.e., spacing) of objects. Here, we show that our sense of number and our sense of density are intertwined. Presented with two patches, observers found it more difficult to spot differences in either density or numerosity when those patches were mismatched in overall size, and their errors were consistent with larger patches appearing both denser and more numerous. We propose that density is estimated using the relative response of mechanisms tuned to low and high spatial frequencies (SFs), because energy at high SFs is largely determined by the number of objects, whereas low SF energy depends more on the area occupied by elements. This measure is biased by overall stimulus size in the same way as human observers, and by estimating number using the same measure scaled by relative stimulus size, we can explain all of our results. This model is a simple, biologically plausible common metric for perceptual number and density.

Human mathematical competence emerges from two representational systems. Competence in some domains of mathematics, such as calculus, relies on symbolic representations that are unique to humans who have undergone explicit teaching. More basic numerical intuitions are supported by an evolutionarily ancient approximate number system that is shared by adults, infants and non-human animals-these groups can all represent the approximate number of items in visual or auditory arrays without verbally counting, and use this capacity to guide everyday behaviour such as foraging. Despite the widespread nature of the approximate number system both across species and across development, it is not known whether some individuals have a more precise non-verbal 'number sense' than others. Furthermore, the extent to which this system interfaces with the formal, symbolic maths abilities that humans acquire by explicit instruction remains unknown. Here we show that there are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual differences in the present correlate with children's past scores on standardized maths achievement tests, extending all the way back to kindergarten. Moreover, this correlation remains significant when controlling for individual differences in other cognitive and performance factors. Our results show that individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense. Further research will determine whether early differences in number sense acuity affect later maths learning, whether maths education enhances number sense acuity, and the extent to which tertiary factors can affect both.

Numbers are an integral part of our everyday life - we use them to quantify, rank and identify objects. The verbal number concept allows humans to develop superior mathematical and logic skills that define technologically advanced cultures. However, basic numerical competence is rooted in biological primitives that can be explored in animals, infants and human adults alike. We are now beginning to unravel its anatomical basis and neuronal mechanisms on many levels, down to its single neuron correlate. Neural representations of numerical information can engage extensive cerebral networks, but the posterior parietal cortex and the prefrontal cortex are the key structures in primates.

To account for the size effect in numerical comparison, three assumptions about the internal structure of the mental number line (e.g., Dehaene, 1992) have been proposed. These are magnitude coding (e.g., Zorzi & Butterworth, 1999), compressed scaling (e.g., Dehaene, 1992), and increasing variability (e.g., Gallistel & Gelman, 1992). However, there are other tasks besides numerical comparison for which there is clear evidence that the mental number line is accessed, and no size effect has been observed in these tasks. This is contrary to the predictions of these three assumptions. Moreover, all three assumptions have difficulties explaining certain symmetries in priming studies of number naming and parity judgment. We propose a neural network model that avoids these three assumptions but, instead, uses place coding, linear scaling, and constant variability on the mental number line. We train the model on naming, parity judgment, and comparison and show that the size effect appears in comparison, but not in naming or parity judgment. Moreover, no asymmetries appear in primed naming or primed parity judgment with this model, in line with empirical data. Implications of our findings are discussed.

The vast majority of studies into visual processing are conducted using computer display technology. The current paper describes a new free suite of software tools designed to make this task easier, using the latest advances in hardware and software. PsychoPy is a platform-independent experimental control system written in the Python interpreted language using entirely free libraries. PsychoPy scripts are designed to be extremely easy to read and write, while retaining complete power for the user to customize the stimuli and environment.
Tools are provided within the package to allow everything from stimulus presentation and response collection (from a wide range of devices) to simple data analysis such as psychometric function fitting. Most importantly, PsychoPy is highly extensible and the whole system can evolve via user contributions. If a user wants to add support for a particular stimulus, analysis or hardware device they can look at the code for existing examples, modify them and submit the modifications back into the package so that the whole community benefits.

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.

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. (Find preprint at http://www.preprints.org/manuscript/201609.0025/v1)

Mechanisms for thinking about quantities range from crude estimation to precise counting, arithmetic, and high-level mathematics. Current evidence implicates cognitive and neural continuity among these mechanisms due to overlapping developmental and evolutionary origins. This chapter reviews current research on the nature continuities in numerical processes across studies on child development, the organization of the human brain, and animal cognition. The results are important for understanding the history and organization of one of our most basic conceptual faculties: the number system. The implications of this research are far reaching, including how we approach the development of formal mathematics in human children.

Note:
Republished in: Am J Psychol. 100(3-4) 441-71 (1987).
Republished in: Int J Epidemiol. 39(5):1137-50 (2010).

The relation between the approximate number system (ANS) and symbolic number processing skills remains unclear. Some theories assume that children acquire the numerical meaning of symbols by mapping them onto the preexisting ANS. Others suggest that in addition to the ANS, children also develop a separate, exact representational system for symbolic number processing. In the current study, we contribute to this debate by investigating whether the nonsymbolic number processing of kindergarteners is predictive for symbolic number processing. Results revealed no association between the accuracy of the kindergarteners on a nonsymbolic number comparison task and their performance on the symbolic comparison task six months later, suggesting that there are two distinct representational systems for the ANS and numerical symbols.

In this chapter, I put together the first elements of a mathematical theory relating neuro- biological observations to psychological laws in the domain of numerical cognition. The starting point is the postulate of a neuronal code whereby numerosity—the cardinal of a set of objects—is represented approximately by the firing of a population of numerosity detectors. Each of these neurons fires to a certain preferred numerosity, with a tuning curve which is a Gaussian function of the logarithm of numerosity. From this log- Gaussian code, decisions are taken using Bayesian mechanisms of log-likelihood compu- tation and accumulation. The resulting equations for response times and errors in classical tasks of number comparison and same-different judgments are shown to tightly fit behavioral and neural data. Two more speculative issues are discussed. First, new chronometric evidence is presented supporting the hypothesis that the acquisition of number symbols changes the mental number line, both by increasing its precision and by changing its coding scheme from logarithmic to linear. Second, I examine how symbolic and nonsymbolic representations of numbers affect performance in arithmetic compu- tations such as addition and subtraction.

Humans and nonhuman animals share the capacity to estimate, without counting, the number of objects in a set by relying on an approximate number system (ANS). Only humans, however, learn the concepts and operations of symbolic mathematics. Despite vast differences between these two systems of quantification, neural and behavioral findings suggest functional connections. Another line of research suggests that the ANS is part of a larger, more general system of magnitude representation. Reports of cognitive interactions and common neural coding for number and other magnitudes such as spatial extent led us to ask whether, and how, nonnumerical magnitude interfaces with mathematical competence. On two magnitude comparison tasks, college students estimated (without counting or explicit calculation) which of two arrays was greater in number or cumulative area. They also completed a battery of standardized math tests. Individual differences in both number and cumulative area precision (measured by accuracy on the magnitude comparison tasks) correlated with interindividual variability in math competence, particularly advanced arithmetic and geometry, even after accounting for general aspects of intelligence. Moreover, analyses revealed that whereas number precision contributed unique variance to advanced arithmetic, cumulative area precision contributed unique variance to geometry. Taken together, these results provide evidence for shared and unique contributions of nonsymbolic number and cumulative area representations to formally taught mathematics. More broadly, they suggest that uniquely human branches of mathematics interface with an evolutionarily primitive general magnitude system, which includes partially overlapping representations of numerical and nonnumerical magnitude.

The greater of 2 multidigit integers can be chosen by generating and comparing internal representations of the integers (holistic models), sequentially comparing corresponding digits in the integers (sequential place-value models), or simultaneously comparing corresponding digits (parallel place-value models). In Exps I and II, 44 undergraduates chose the greater of 2 4-digit or 2 6-digit integers. As predicted by sequential place-value models, latencies did not depend on the number of unequal digits in the 2 numbers and latencies increased linearly with the position of the leftmost unequal digits, except when only the rightmost digits were unequal. In Exp III (17 Ss), latencies increased linearly for all positions when 2 letters were presented to the right of the integers. Results imply that multidigit integers are compared by sequentially comparing digits. (15 ref) (PsycINFO Database Record (c) 2012 APA, all rights reserved)

We examine the frequency of numerals and ordinals in seven different languages and/or cultures. Many cross-cultural and cross-linguistic patterns are identified. The most striking is a decrease of frequency with numerical magnitude, with local increases for reference numerals such as 10, 12, 15, 20, 50 or 100. Four explanations are considered for this effect: sampling artifacts, notational regularities, environmental biases and psychological limitations on number representations. The psychological explanation, which appeals to a Fechnerian encoding of numerical magnitudes and to the existence of numerical points of reference, accounts for most of the data. Our finding also has practical importance since it reveals the frequent confound of two experimental variables: numerical magnitude and numeral frequency.

To date, researchers investigating nonsymbolic number processes devoted little attention to the visual properties of their stimuli. This is unexpected, as nonsymbolic number is defined by its visual characteristics. When number changes, its visual properties change accordingly. In this study, we investigated the influence of different visual properties on nonsymbolic number processes and show that the current assumptions about the relation between number and its visual characteristics are incorrect. Similar to previous studies, we controlled the visual cues: Each visual cue was not predictive of number. Nevertheless, participants showed congruency effects induced by the visual properties of the stimuli. These congruency effects scaled with the number of visual cues manipulated, implicating that people do not extract number from a visual scene independent of its visual cues. Instead, number judgments are based on the integration of information from multiple visual cues. Consequently, current ways to control the visual cues of the number stimuli are insufficient, as they control only a single variable at the time. And, more important, the existence of an approximate number system that can extract number independent of the visual cues appears unlikely. We therefore propose that number judgment is the result of the weighing of several distinct visual cues. (PsycINFO Database Record (c) 2011 APA, all rights reserved).

Attaching meaning to arbitrary symbols (i.e. words) is a complex and lengthy process. In the case of numbers, it was previously suggested that this process is grounded on two early pre-verbal systems for numerical quantification: the approximate number system (ANS or 'analogue magnitude'), and the object tracking system (OTS or 'parallel individuation'), which children are equipped with before symbolic learning. Each system is based on dedicated neural circuits, characterized by specific computational limits, and each undergoes a separate developmental trajectory. Here, I review the available cognitive and neuroscientific data and argue that the available evidence is more consistent with a crucial role for the ANS, rather than for the OTS, in the acquisition of abstract numerical concepts that are uniquely human.

The comparison distance effect (CDE), whereby discriminating between two numbers that are far apart is easier than discriminating between two numbers that are close, has been considered as an important indicator of how people represent magnitudes internally. However, the underlying mechanism of this CDE is still unclear. We tried to shed further light on how people represent magnitudes by using priming. Adults have been shown to exhibit a priming distance effect (PDE), whereby numbers are processed faster when they are preceded by a close number than when they are preceded by a more distant number. Surprisingly, there are no studies available that have investigated this effect in children. The current study examined this effect in typically developing first, third, and fifth graders and in adults. Our findings revealed that the PDE already occurs in first graders and remains stable across development. This study also documents the usefulness of number priming in children, making it an interesting tool for future research.

This paper provides a tutorial introduction to numerical cognition, with a review of essential findings and current points of debate. A tacit hypothesis in cognitive arithmetic is that numerical abilities derive from human linguistic competence. One aim of this special issue is to confront this hypothesis with current knowledge of number representations in animals, infants, normal and gifted adults, and brain-lesioned patients. First, the historical evolution of number notations is presented, together with the mental processes for calculating and transcoding from one notation to another. While these domains are well described by formal symbol-processing models, this paper argues that such is not the case for two other domains of numerical competence: quantification and approximation. The evidence for counting, subitizing and numerosity estimation in infants, children, adults and animals is critically examined. Data are also presented which suggest a specialization for processing approximate numerical quantities in animals and humans. A synthesis of these findings is proposed in the form of a triple-code model, which assumes that numbers are mentally manipulated in an arabic, verbal or analogical magnitude code depending on the requested mental operation. Only the analogical magnitude representation seems available to animals and preverbal infants.

Apart from replicating findings of other experimenters, the present authors show that Ss can make a fast "countability" judgment indicating whether or not they could, if requested, give an accurate numerosity response. These judgments were fast and produced a "yes" response within the subitizing range and a "no" response thereafter. Developmental data have indicated that children count arrays as small as 2 and 3; adults seem to give a more automatic response, shown also in the fast RTs to those array sizes. The suggestion that this response is an acquired one to certain frequently appearing canonical patterns of 2 and 3 events (pairs/lines and triples/triangles) was explored in an experiment in which Ss were given canonical patterns of arrays of up to 10. Results show that within few trials, the response to these canonical patterns was usually as fast and accurate as the response to the smaller (1–3) array sizes. Data also demonstrate that the slope for array sizes from 4 to 6 with short exposure time was indistinguishable from the slope for array sizes from 4 to 15 under an unlimited exposure condition. It is concluded, on the basis of 5 studies with 48 adults, that the RT function found in subitizing consisted of 3 processes: a response to arrays of 1–3 that was fast and accurate and was based on acquired canonical patterns; a response to arrays 4 to 6 or 7 that was based on mental counting; and an estimating response for arrays larger than 6 that could be held in consciousness for mental counting. (36 ref)

How do we use place information in a number comparison task involving multidigit numbers? In four experiments, subjects identified
stimulus numbers as larger or smaller than the number 5,000 in a choice reaction time task. As the difference between the
number of places in the stimulus number and the number of places in the standard decreased, response time and errors significantly
increased. When the number of places was held constant, the type of numeric information extracted depended on the value of
the standard (5,000 or 5,555). In some cases, irrelevant place information affected choice time.

Graded interference effects were tested in a naming task, in parallel for objects and actions. Participants named either object or action pictures presented in the context of other pictures (blocks) that were either semantically very similar, or somewhat semantically similar or semantically dissimilar. We found that naming latencies for both object and action words were modulated by the semantic similarity between the exemplars in each block, providing evidence in both domains of graded semantic effects.

The triple-code theory of numerical processing postulates an abstract-semantic "number sense." Neuropsychology points to intraparietal cortex as a potential substrate, but previous functional neuroimaging studies did not dissociate the representation of numerical magnitude from task-driven effects on intraparietal activation. In an event-related fMRI study, we presented numbers, letters, and colors in the visual and auditory modality, asking subjects to respond to target items within each category. In the absence of explicit magnitude processing, numbers compared with letters and colors across modalities activated a bilateral region in the horizontal intraparietal sulcus. This stimulus-driven number-specific intraparietal response supports the idea of a supramodal number representation that is automatically accessed by presentation of numbers and may code magnitude information.

Although it is often assumed that abilities that reflect basic numerical understanding, such as numerical comparison, are related to children's mathematical abilities, this relationship has not been tested rigorously. In addition, the extent to which symbolic and nonsymbolic number processing play differential roles in this relationship is not yet understood. To address these questions, we collected mathematics achievement measures from 6- to 8-year-olds as well as reaction times from a numerical comparison task. Using the reaction times, we calculated the size of the numerical distance effect exhibited by each child. In a correlational analysis, we found that the individual differences in the distance effect were related to mathematics achievement but not to reading achievement. This relationship was found to be specific to symbolic numerical comparison. Implications for the role of basic numerical competency and the role of accessing numerical magnitude information from Arabic numerals for the development of mathematical skills and their impairment are discussed.

Subitizing is the rapid and accurate enumeration of small sets (up to 3-4 items). Although subitizing has been studied extensively since its first description about 100 years ago, its underlying mechanisms remain debated. One hypothesis proposes that subitizing results from numerical estimation mechanisms that, according to Weber's law, operate with high precision for small numbers. Alternatively, subitizing might rely on a distinct process dedicated to small numerosities. In this study, we tested the hypothesis that there is a shared estimation system for small and large quantities in human adults, using a masked forced-choice paradigm in which participants named the numerosity of displays taken from sets matched for discrimination difficulty; one set ranged from 1 through 8 items, and the other ranged from 10 through 80 items. Results showed a clear violation of Weber's law (much higher precision over numerosities 1-4 than over numerosities 10-40), thus refuting the single-estimation-system hypothesis and supporting the notion of a dedicated mechanism for apprehending small numerosities.

Analog origins of numerical concepts Evolutionary origins and early development of number processing

- J F Cantlon

Cantlon, J. F. (2015). Analog origins of numerical concepts. In D. C.
Geary, D. B. Berch, & K. Mann Koepke (Eds.), Evolutionary origins and early development of number processing (Vol. 1, pp. 225251). London: Academic Press.

Symbolic number processing: Analogue number system or discrete semantic system?

- A Krajcsi
- G Lengyel
- P Kojouharova

Krajcsi, A., Lengyel, G., & Kojouharova, P. (2016c). Symbolic number
processing: Analogue number system or discrete semantic system?
Manuscript submitted for publication.

A model of exact smallnumber representation Semantic distance effects on object and action naming

- T Verguts
- W Fias
- M G Stevens
- D P Vinson
- M F Damian
- W Levelt

Verguts, T., Fias, W., & Stevens, M. (2005). A model of exact smallnumber representation. Psychonomic Bulletin & Review, 12, 6680. doi:10.3758/BF03196349
Vigliocco, G., Vinson, D. P., Damian, M. F., & Levelt, W. (2002).
Semantic distance effects on object and action naming. Cognition,
85, B61-B69. doi:10.1016/S0010-0277(02)00107-5

A computational model of number comparison

- M Zorzi
- B Butterworth

Zorzi, M., & Butterworth, B. (1999). A computational model of
number comparison. In M. Hahn & S. C. Stoness (Eds.),
Proceedings of the Twenty-First Annual Conference of the
Cognitive Science Society (pp. 778-783). Mahwah: Erlbaum.
Retrieved from http://eprints.ucl.ac.uk/91024/