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Neuromodulation with transcranial direct current stimulation (tDCS) is an increasingly popular research tool to experimentally manipulate cortical areas and probe their causal involvements in behavior, but its replicability and regional specificity are not clear. This registered report investigated cathodal tDCS effects on spatial-numerical associations (i.e., the SNARC effect), the numerical distance effect (NDE), and inhibitory control (i.e., stop-signal reaction time; SSRT). Healthy adults (N = 160) were randomly assigned to one of five groups to receive sham tDCS or 1 mA cathodal tDCS to one of four stimulation sites (left/right prefrontal cortex [PFC], left/right posterior parietal cortex) with extracephalic return. We replicated that cathodal tDCS over the left PFC reduced the SNARC effect compared to sham tDCS and to tDCS over the left parietal cortex. However, neither NDE nor SSRT were modulated in the main analyses. Post hoc contrasts and exploratory analyses showed that cathodal tDCS over the right PFC had a time-dependent effect by delayed practice-related improvements in SSRT. Math anxiety moderated changes in the NDE in the groups receiving tDCS to the right parietal cortex. With few exceptions, the replicability and regional specificity of tDCS effects on behavior were weak and partially moderated by individual differences. Future research needs to characterize the parameter settings for effective neuromodulation.

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BACKGROUND: Fractions are known to be difficult for children and adults. Behavioral studies suggest that magnitude processing of fractions can be improved via number line estimation (NLE) trainings, but little is known about the neural correlates of fraction learning. METHOD: To examine the neuro-cognitive foundations of fraction learning, behavioral performance and neural correlates were measured before and after a five-day NLE training. RESULTS: In all evaluation tasks behavioral performance increased after training. We observed a fronto-parietal network associated with number magnitude processing to be recruited in all tasks as indicated by a numerical distance effect. For symbolic fractions, the distance effect on intraparietal activation was only observed after training. CONCLUSION: The absence of a distance effect of symbolic fractions before the training could indicate an initially less automatic access to their overall magnitude. NLE training facilitates processing of overall fraction magnitude as indicated by the distance effect in neural activation.

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Based on a theory of impulsive and reflective human behavior, we test the effects of transcranial direct current stimulation (tDCS) targeting either prefrontal or parietal cortex in either hemisphere. In a confirmatory registered report, cathodal tDCS is administered to conceptually reproduce tDCS modulations of implicit spatial-numerical associations, numerical distance effects, and response inhibition. Those cognitive operations are hypothesized to draw on left prefrontal, parietal, and right prefrontal activations, respectively, thereby susceptible to inhibitory, cathodal tDCS across those regions. Vice versa, the mutual regional and behavioral specificity of tDCS effects on these behavioral indices is examined and expected to produce double dissociations. In a mixed within-subjects (baseline, during tDCS, post-tDCS) and between-subjects (target electrode: left/right prefrontal cortex/posterior parietal cortex, or sham tDCS) design, we collect (a) confirmatory data on the robustness of cathodal tDCS effects on three behavioral effects and (b) differential data on the specificity of regional targets in male and female human participants. Results will provide crucial tests of theories of cortical organization implied by implicit associations and explicit regulation, which can direct future brain stimulation studies.

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Moving seamlessly between spoken number words and Arabic digits is common in everyday life. In this study, we systematically investigated the correspondence between auditory number words and visual Arabic digits in adults. Auditory number words and visual Arabic digits were presented concurrently or sequentially and participants had to indicate whether they described the same quantity. We manipulated the stimulus onset asynchronies (SOAs) between the two stimuli (Experiment 1: -500 ms to +500 ms; Experiment 2: -200 ms to +200 ms). In both experiments, we found a significant cross-modal distance effect. This effect was strongest for simultaneous stimulus presentation and decreased with increasing SOAs. Numerical distance emerged as the most consistent significant predictor overall, in particular for simultaneous presentation. However, physical similarity between the stimuli was often a significant predictor of response times in addition to numerical distance, and at longer SOAs, physical similarity between the stimuli was the only significant predictor. This shows that SOA modulates the extent to which participants access quantity representations. Our results thus support the idea that a semantic quantity representation of auditory and visual numerical symbols is activated when participants perform a concurrent matching task, while at longer SOAs participants are more likely to rely on physical similarity between the stimuli. We also investigated whether individual differences in the efficiency of the cross-modal processing were related to differences in mathematical performance. Our results are inconclusive about whether the efficiency of cross-format numerical correspondence is related to mathematical competence in adults.

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The architecture of the numerical cognition system is currently not well understood, but at a general level, assumptions are made about two core components: a quantity processor and an identity processor. The quantity processor is concerned with accessing and using the stored magnitude denoted by a given digit, and the identity processor is concerned with recovery of the corresponding digit's identity. Blanc-Goldhammer and Cohen (Journal of Experimental Psychology: Learning, Memory, and Cognition, 40, 1389-1403, 2014) established that the recovery and use of quantity information operates in an unlimited-capacity fashion. Here we assessed whether the identity processor operates in a similar fashion. We present two experiments that were digit identity variations of Blanc-Goldhammer and Cohen's magnitude estimation paradigm. The data across both experiments reveal a limited-capacity identity processor whose operation reflects cross-talk with the quantity processor. Such findings provide useful evidence that can be used to adjudicate between competing models of the human number-processing system.

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Following the classical work of Moyer and Landauer (1967), experimental studies investigating the way in which humans process and compare symbolic numerical information regularly used one of two experimental designs. In selection tasks, two numbers are presented, and the task of the participant is to select (for example) the larger one. In classification tasks, a single number is presented, and the participant decides if it is smaller or larger than a predefined standard. Many findings obtained with these paradigms fit in well with the notion of a mental analog representation, or an Approximate Number System (ANS; e.g., Piazza 2010). The ANS is often conceptualized metaphorically as a mental number line, and data from both paradigms are well accounted for by diffusion models based on the stochastic accumulation of noisy partial numerical information over time. The present study investigated a categorization paradigm in which participants decided if a number presented falls into a numerically defined central category. We show that number categorization yields a highly regular, yet considerably more complex pattern of decision times and error rates as compared to the simple monotone relations obtained in traditional selection and classification tasks. We also show that (and how) standard diffusion models of number comparison can be adapted so as to account for mean and standard deviations of all RTs and for error rates in considerable quantitative detail. We conclude that just as traditional number comparison, the more complex process of categorizing numbers conforms well with basic notions of the ANS.

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When objects are manually lifted to compare their weight, then smaller objects are judged to be heavier than larger objects of the same physical weights: the classical size-weight illusion (Gregory, 2004). It is also well established that increasing numerical magnitude is strongly associated with increasing physical size: the number-size congruency effect e.g., (Besner & Coltheart Neuropsychologia, 17, 467-472 1979); Henik & Tzelgov Memory & Cognition, 10, 389-395 1982). The present study investigates the question suggested by combining these two classical effects: if smaller numbers are associated with smaller size, and objects of smaller size appear heavier, then are numbered objects (balls) of equal weight and size also judged as heavier when they carry smaller numbers? We present two experiments testing this hypothesis for weight comparisons of numbered (1 to 9) balls of equal size and weight, and report results which largely conform to an interpretation in terms of a new "number-weight illusion".

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Individual differences in the ability to compare and evaluate nonsymbolic numerical magnitudes-approximate number system (ANS) acuity-are emerging as an important predictor in many research areas. Unfortunately, recent empirical studies have called into question whether a historically common ANS-acuity metric-the size of the numerical distance effect (NDE size)-is an effective measure of ANS acuity. NDE size has been shown to frequently yield divergent results from other ANS-acuity metrics. Given these concerns and the measure's past popularity, it behooves us to question whether the use of NDE size as an ANS-acuity metric is theoretically supported. This study seeks to address this gap in the literature by using modeling to test the basic assumption underpinning use of NDE size as an ANS-acuity metric: that larger NDE size indicates poorer ANS acuity. This assumption did not hold up under test. Results demonstrate that the theoretically ideal relationship between NDE size and ANS acuity is not linear, but rather resembles an inverted J-shaped distribution, with the inflection points varying based on precise NDE task methodology. Thus, depending on specific methodology and the distribution of ANS acuity in the tested population, positive, negative, or null correlations between NDE size and ANS acuity could be predicted. Moreover, peak NDE sizes would be found for near-average ANS acuities on common NDE tasks. This indicates that NDE size has limited and inconsistent utility as an ANS-acuity metric. Past results should be interpreted on a case-by-case basis, considering both specifics of the NDE task and expected ANS acuity of the sampled population.

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The approximate number system (ANS) is an innate cognitive template that allows for the mental representation of approximate magnitude, and has been controversially linked to symbolic number knowledge and math ability. A series of recent studies found that an approximate arithmetic training (AAT) task that draws upon the ANS can improve math skills, which not only supports the existence of this link, but suggests it may be causal. However, no direct transfer effects to any measure of the ANS have yet been reported, calling into question the mechanisms by which math improvements may emerge. The present study investigated the effects of a 7-day AAT and successfully replicated previously reported transfer effects to math. Furthermore, our exploratory analyses provide preliminary evidence that certain ANS-related skills may also be susceptible to training. We conclude that AAT has reproducible effects on math performance, and provide avenues for future studies to further explore underlying mechanisms - specifically, the link between improvements in math and improvements in ANS skills.

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Models of the spatial-numerical association of response codes (SNARC) effect-faster responses to small numbers using left effectors, and the converse for large numbers-diverge substantially in localizing the root cause of this effect along the numbers' processing chain. One class of models ascribes the cause of the SNARC effect to the inherently spatial nature of the semantic representation of numerical magnitude. A different class of models ascribes the effect's cause to the processing dynamics taking place during response selection. To disentangle these opposing views, we devised a paradigm combining magnitude comparison and stimulus-response switching in order to monitor modulations of the SNARC effect while concurrently tapping both semantic and response-related processing stages. We observed that the SNARC effect varied nonlinearly as a function of both manipulated factors, a result that can hardly be reconciled with a unitary cause of the SNARC effect.

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Premature birth is a significant risk factor for learning disabilities in general and mathematics learning difficulties in particular. However, the exact reasons for this relation are still unknown. While typical numerical development is associated with a frontal-to-parietal shift of brain activation with increasing age, influences of gestational age have hardly been considered so far. Therefore, we investigated the influence of gestational age on the neural correlates of number processing in 6- and 7-year-old children born prematurely (n=16). Only the numerical distance effect - as a measure of intentional number magnitude processing - elicited the fronto-parietal activation pattern typically observed for numerical cognition. On the other hand, the size congruity effect - as a measure of automatic number magnitude processing - was associated with activation of brain areas typically attributed to cognitive control. Most importantly, however, we observed that gestational age reliably predicted the frontal-to-parietal shift of activation observed for the numerical distance effect. Our findings seem to indicate that human numerical development may start even before birth and prematurity might hamper neural facilitation of the brain circuitry subserving numerical cognition. In turn, this might contribute to the high risk of premature children to develop mathematical learning difficulties.

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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.

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Number processing is a cognitive domain particularly sensitive to prenatal alcohol exposure, which relies on intact parietal functioning. Alcohol-related alterations in brain activation have been found in the parietal lobe during symbolic number processing. However, the effects of prenatal alcohol exposure on the neural correlates of non-symbolic number comparison and the numerical distance effect have not been investigated. Using functional magnetic resonance imaging (fMRI), we examined differences in brain activation associated with prenatal alcohol exposure in five parietal regions involved in number processing during a non-symbolic number comparison task with varying degrees of difficulty. fMRI results are presented for 27 Cape Colored children (6 fetal alcohol syndome (FAS)/partial FAS, 5 heavily exposed (HE) non-sydromal, 16 controls; mean age ± SD = 11.7 ± 1.1 years). Fetal alcohol exposure was assessed by interviewing mothers using a timeline follow-back approach. Separate subject analyses were performed in each of five regions of interest, bilateral horizontal intraparietal sulci (IPS), bilateral posterior superior parietal lobules (PSPL), and left angular gyrus (left AG), using the general linear model with predictors for number comparison and difficulty level. Mean percent signal change for each predictor was extracted for each subject for each region to examine group differences and associations with continuous measures of alcohol exposure. Although groups did not differ in performance, controls activated the right PSPL more during non-symbolic number comparison than exposed children, but this was not significant after controlling for maternal smoking, and the right IPS more than children with fetal alcohol syndrome (FAS) or partial FAS. More heavily exposed children recruited the left AG to a greater extent as task difficulty increased, possibly to compensate, in part, for impairments in function in the PSPL and IPS. Notably, in non-syndromal heavily exposed children activation was impaired in the right PSPL, but spared in the right IPS. These results extend previous findings of poor right IPS recruitment during symbolic number processing in FAS/PFAS, indicating that mental representation of relative quantity is affected by prenatal alcohol exposure for both symbolic and non-symbolic representations of quantity.

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Numerical distance and size effects (easier number comparisons with large distance or small size) are mostly supposed to reflect a single effect, the ratio effect, which is a consequence of activation of the analog number system (ANS), 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 size effects could originate in two independent factors: the distance effect depending on the semantic distance of the units, and the size effect depending on the frequency of the symbols. Whereas 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 present work contrasts the two views, investigating whether the sizes of the distance and 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 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 another system-for example, the DSS-the two effects might dissociate. In the present measurements, the distance and 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 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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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.

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Although several studies have compared the representation of fractions and decimals, no study has investigated whether fractions and decimals, as two types of rational numbers, share a common representation of magnitude. The current study aimed to answer the question of whether fractions and decimals share a common representation of magnitude and whether the answer is influenced by task paradigms. We included two different number pairs, which were presented sequentially: fraction-decimal mixed pairs and decimal-fraction mixed pairs in all four experiments. Results showed that when the mixed pairs were very close numerically with the distance 0.1 or 0.3, there was a significant distance effect in the comparison task but not in the matching task. However, when the mixed pairs were further apart numerically with the distance 0.3 or 1.3, the distance effect appeared in the matching task regardless of the specific stimuli. We conclude that magnitudes of fractions and decimals can be represented in a common manner, but how they are represented is dependent on the given task. Fractions and decimals could be translated into a common representation of magnitude in the numerical comparison task. In the numerical matching task, fractions and decimals also shared a common representation. However, both of them were represented coarsely, leading to a weak distance effect. Specifically, fractions and decimals produced a significant distance effect only when the numerical distance was larger.

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In a comparison task involving numbers, the size congruity effect refers to the general finding that responses are usually faster when there is a match between numerical size and physical size (e.g., 2-8) than when there is a mismatch (e.g., 2-8). In the present study, we used computer mouse tracking to test two competing models of the size congruity effect: an early interaction model, where interference occurs at an early representational stage, and a late interaction model, where interference occurs as dynamic competition between response options. In three experiments, we found that the curvature of responses for incongruent trials was greater than for congruent trials. In Experiment 2 we showed that this curvature effect was reliably modulated by the numerical distance between the two stimulus numbers, with large distance pairs exhibiting a larger curvature effect than small distance pairs. In Experiment 3 we demonstrated that the congruity effects persist into response execution. These findings indicate that incongruities between numerical and physical sizes are carried throughout the response process and result from competition between parallel and partially active response options, lending further support to a late interaction model of the size congruity effect.

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Objective To investigate numerical distance effect of numerical processing in amblyopic children aged from 6 to 13. Methods 46 amblyopic children and 43 control children with normal sight were divided into 3 groups respectively:7Y group (6~8 years old),9Y group (9~10 years old) and 12Y group (11~13 years old). The numerical comparison task (judging the magnitude of number) was used in this study and error rate( ER) and reaction times ( RTs) were recorded. Results ( 1) RTs of judging the num-ber 4 (7Y group (995±100)ms,9Y group (964±141)ms,12Y group (701±125)ms) were significant lon-ger(P5).The develop-ment of numerical processing in amblyopic children is slower than that in children with normal sight.

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Previous studies have suggested that numerical processing relates to mathematical performance, but it seems that such relationship is more evident for intentional than for automatic numerical processing. In the present study we assessed the relationship between the two types of numerical processing and specific mathematical abilities in a sample of 109 children in grades 1-6. Participants were tested in an ample range of mathematical tests and also performed both a numerical and a size comparison task. The results showed that numerical processing related to mathematical performance only when inhibitory control was involved in the comparison tasks. Concretely, we found that intentional numerical processing, as indexed by the numerical distance effect in the numerical comparison task, was related to mathematical reasoning skills only when the task-irrelevant dimension (the physical size) was incongruent; whereas automatic numerical processing, indexed by the congruency effect in the size comparison task, was related to mathematical calculation skills only when digits were separated by small distance. The observed double dissociation highlights the relevance of both intentional and automatic numerical processing in mathematical skills, but when inhibitory control is also involved.

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A growing body of evidence has indicated a link between individual differences in children's symbolic numerical magnitude discrimination (e.g., judging which of two numbers is numerically larger) and their arithmetic achievement. In contrast, relatively little is known about the processing of numerical order (e.g., deciding whether two numbers are in ascending or descending numerical order) and whether individual differences in judging numerical order are related to the processing of numerical magnitude and arithmetic achievement. In view of this, we investigated the relationships among symbolic numerical magnitude comparison, symbolic order judgments, and mathematical achievement. Data were collected from a group of 61 first-grade children who completed a magnitude comparison task, an order judgment task, and two standardized tests of arithmetic achievement. Results indicated a numerical distance effect (NDE) in both the symbolic numerical magnitude discrimination and the numerical order judgment condition. However, correlation analyses demonstrated that although individual differences in magnitude comparison correlated significantly with arithmetic achievement, performance on the order judgment task did not. Moreover, the NDE of the magnitude and order comparison performance was also found to be uncorrelated. These findings suggest that order and numerical magnitude processing may be underpinned by different processes and relate differentially to arithmetic achievement in young children.

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