Search

Solving arithmetic word problems requires individuals to create a correct mental representation, and this involves both text processing and number processing. The latter comprises understanding the semantic meaning of numbers (i.e., their magnitudes) and potentially executing the appropriate mathematical operation. However, it is not yet clear whether number processing occurs after text processing or both take place simultaneously. We hypothesize that number processing occurs early and simultaneously with other problem-solving processes such as text processing. To test this hypothesis, we created non-solvable word problems that do not require any number processing and we manipulated the calculation difficulty using carry/borrow vs. non-carry/non-borrow within addition and subtraction problems. According to a strictly sequential model, this manipulation should not matter, because when problems are non-solvable, no calculation is required. In contrast, according to an interactive model, attention to numbers would be higher when word problems require a carry/borrow compared to a non-carry/non-borrow operation. Eye-tracking was used to measure attention to numbers and text in 63 adults, operationalized by static (duration and count of fixations and regressions) and dynamic measures (count of transitions). An interaction between difficulty and operation was found for all static and dynamic eye-tracking variables as well as for response times and error rates. The observed number processing in non-solvable word problems, which indicates that it occurs simultaneously with text processing, is inconsistent with strictly sequential models., Competing Interests: Declarations. Confict of interest: The authors have no conflict of interest to be declared., (© 2025. The Author(s).)

Spatial-Numerical Associations (SNAs) refer to the demonstrations of spatial processing of numbers. The Mental Number Line (MNL) is a representation model describing numbers as aligning left-to-right (LR) and was suggested to account for directional biases in participants' responses during numerical tasks. One common behavioral demonstration of this is the Spatial-Numerical Associations of Response Codes (SNARC) effect, which describes faster left-/right-hand responses to smaller/larger numbers, respectively. The MNL, and, consequently, directional SNAs, show variabilities across different cultures. Reading direction is considered to be the main factor in explaining these differences. In line with this, individuals with right-to-left (RL) reading habits show a weaker or even reverse SNARC effect. In the present study, we investigated whether SNAs are influenced not only by reading direction, but also by cultural directional preferences such as drawing lines, arranging objects, imagining objects (i.e., rightward or leftward facing), or representing events in time (i.e., mentally representing the past/future on the left/right, respectively). To test this hypothesis, we measured the cultural directional preferences and the SNARC effect across three cultures in an online setup; German, Turkish, and Iranian. LR preferences in the Cultural Directional Preferences Questionnaire were most prominent in German participants, intermediate in Turkish participants, and least prominent in Iranian participants. In line with this, the LR SNARC effect was strongest in German, intermediate in Turkish, and weakest (but not RL) in Iranian culture. These findings suggest that cultural directional preferences are involved in the emergence of adult SNAs in addition to the reading direction., Competing Interests: Declarations. Conflict of Interest: The authors declare no competing interests., (© 2024. The Author(s).)

While arithmetic training reduces fronto-temporo-parietal activation related to domain-general processes in typically developing (TD) children, we know very little about the training-related neurocognitive changes in children with mathematical disabilities (MD), who seek evidenced-based educational interventions. In a within-participant design, a group of 20 children (age range = 10-15 years old) with MD underwent 2 weeks of arithmetic training. Brain activation was measured using functional near-infrared spectroscopy (fNIRS) before and after training to assess training-related changes. Two weeks of training led to both behavioral and brain changes. Training-specific change for trained versus untrained (control) simple multiplication solving was observed as activation increase in the bilateral temporo-parietal region including angular gyrus and middle temporal gyrus. Training-specific change for trained versus untrained (control) complex multiplication solving was observed as activation increase in the bilateral parietal region including intraparietal sulcus, superior parietal lobule, and supramarginal gyrus. Unlike the findings of a similar study in TD children, 2 weeks of multiplication training led to brain activation increase in the fronto-parietal network in children with MD. Interestingly, these brain activation differences between the current findings and a recent similar study in TD children underlie a rather similar behavioral improvement as regards response time and accuracy after 2 weeks of training. This finding provides valuable insights into underlying mechanisms of mathematics learning in special samples and suggests that the findings in TD children may not be readily generalized to children with MD., (© 2022. The Author(s).)

The SNARC (Spatial-Numerical Association of Response Codes) effect (i.e., a tendency to associate small/large magnitude numbers with the left/right hand side) is prevalent across the whole lifespan. Because the ability to relate numbers to space has been viewed as a cornerstone in the development of mathematical skills, the relationship between the SNARC effect and math skills has been frequently examined. The results remain largely inconsistent. Studies testing groups of people with very low or very high skill levels in math sometimes found relationships between SNARC and math skills. So far, however, studies testing such extreme math skills level groups were mostly investigating the SNARC effect in individuals revealing math difficulties. Groups with above average math skills remain understudied, especially in regard to children. Here, we investigate the SNARC effect in gifted children, as compared to normally developing children (overall n = 165). Frequentist and Bayesian analysis suggested that the groups did not differ from each other in the SNARC effect. These results are the first to provide evidence for the SNARC effect in a relatively large sample of gifted (and mathematically highly skilled) children. In sum, our study provides another piece of evidence for no direct link between the SNARC effect and mathematical ability in childhood.

Research on multi-digit number processing suggests that, in Arabic numerals, their place-value magnitude is automatically activated, whenever a magnitude-relevant task was employed. However, so far, it is unknown, whether place-value is also activated when the target task is magnitude-irrelevant. The current study examines this question using the parity congruency effect in two-digit numbers: It describes that responding to decade-digit parity congruent numbers (e.g., 35, 46; same parity of decades and units) is faster than to decade-digit parity incongruent numbers (e.g., 25; 36; different parities of decades and units). Here we investigate the (a-) symmetry of the parity congruency effect; i.e. whether it makes a difference whether participants are assessing the parity of the unit digit or the decade digit. We elaborate, how and why such an asymmetry is related to place-value processing, because the parity of the unit digit only interferes with the parity of the decade digit, while the parity of the decade digit interferes with both the parity of the unit digit and the integrated parity of the whole two-digit number. We observed a significantly larger parity congruency effect in the decade parity decision than in the unit parity decision. This suggests that automatic place-value processing also takes place in a typical parity judgment task, in which magnitude is irrelevant. Finally, because of the cross-lingual design of the study, we can show that these results and their implications were language-independent.

Symbolic magnitude comparison is one of the most well-studied cognitive processes in research on numerical cognition. However, while the cognitive mechanisms of symbolic magnitude processing have been intensively studied, previous studies have paid less attention to individual differences influencing symbolic magnitude comparison. Employing a two-digit number comparison task in an online setting, we replicated previous effects, including the distance effect, the unit-decade compatibility effect, and the effect of cognitive control on the adaptation to filler items, in a large-scale study in 452 adults. Additionally, we observed that the most influential individual differences were participants' first language, time spent playing computer games and gender, followed by reported alcohol consumption, age and mathematical ability. Participants who used a first language with a left-to-right reading/writing direction were faster than those who read and wrote in the right-to-left direction. Reported playing time for computer games was correlated with faster reaction times. Female participants showed slower reaction times and a larger unit-decade compatibility effect than male participants. Participants who reported never consuming alcohol showed overall slower response times than others. Older participants were slower, but more accurate. Finally, higher grades in mathematics were associated with faster reaction times. We conclude that typical experiments on numerical cognition that employ a keyboard as an input device can also be run in an online setting. Moreover, while individual differences have no influence on domain-specific magnitude processing-apart from age, which increases the decade distance effect-they generally influence performance on a two-digit number comparison task.

In recent research, a systematic association of musical pitch with space has been described in the so-called Spatial-Pitch-Association-of-Response Codes-effect (SPARC). Typically, high pitch is associated with upper/right and low pitch with lower/left space. However, a theoretical classification of these associations regarding their experiential sources is difficult. Therefore, we applied a theoretical framework of numerical cognition classifying similar Space-Associated Response Codes (SARC) effects according to their groundedness, embodiedness and situatedness. We tested these attributes with a group of non-musicians and with a group of highly skilled cello players playing high tones with lower hand positions (i.e., reverse SPARC alignment) in a standard SPARC context of a piano and a reversed SPARC context of a cello. The results showed that SPARC is grounded, in general. However, for cello player SPARC is also situated and embodied. We conclude that groundedness, embodiedness and situatedness provide general characteristics of mapping cognitive representations to space.

Previous research has proposed that the approximate number system (ANS) constitutes a building block for later mathematical abilities. Therefore, numerous studies investigated the relationship between ANS acuity and mathematical performance, but results are inconsistent. Properties of the experimental design have been discussed as a potential explanation of these inconsistencies. In the present study, we investigated the influence of set size and presentation duration on the association between non-symbolic magnitude comparison and math performance. Moreover, we focused on strategies reported as an explanation for these inconsistencies. In particular, we employed a non-symbolic magnitude comparison task and asked participants how they solved the task. We observed that set size was a significant moderator of the relationship between non-symbolic magnitude comparison and math performance, whereas presentation duration of the stimuli did not moderate this relationship. This supports the notion that specific design characteristics contribute to the inconsistent results. Moreover, participants reported different strategies including numerosity-based, visual, counting, calculation-based, and subitizing strategies. Frequencies of these strategies differed between different set sizes and presentation durations. However, we found no specific strategy, which alone predicted arithmetic performance, but when considering the frequency of all reported strategies, arithmetic performance could be predicted. Visual strategies made the largest contribution to this prediction. To conclude, the present findings suggest that different design characteristics contribute to the inconsistent findings regarding the relationship between non-symbolic magnitude comparison and mathematical performance by inducing different strategies and additional processes.

Mathematical abilities are essential for an individual, as they predict career prospects among many other abilities. However, little is known about whether neural correlates of arithmetic problem difficulty differ between individuals with high and low math ability. For instance, the difficulty of two-digit addition and subtraction increases whenever a carry or borrow operation is required. Therefore, we systematically investigated the spatial and temporal neural correlates of the carry and borrow effects for high and low performers in a written production paradigm using combined functional near-infrared spectroscopy (fNIRS) and event-related potential (ERP) measurements. Effects of arithmetic difficulty interacted with an individual's math ability. High performers showed increased frontal activation especially in the left inferior frontal gyrus associated with the carry and borrow effects, whereas low performers did not. Furthermore, high and low performers even differed in their early processing of the borrow effect, as reflected by differences in slow waves at 1000-1500 ms at frontal sites. We conclude that the processing of arithmetic difficulty relies on an individual's mathematical ability, and suggest that individual differences should be taken into account when investigating mental arithmetic in an ecologically valid assessment.

Being able to adequately process numbers is a key competency in everyday life. Yet, self-reported negative affective responses towards numbers are known to deteriorate numerical performance. Here, we investigated how physiological threat responses predict numerical performance. Physiological responses reflect whether individuals evaluate a task as exceeding or matching their resources and in turn experience either threat or challenge, which influences subsequent performance. We hypothesized that, the more individuals respond to a numerical task with physiological threat, the worse they would perform. Results of an experiment with cardiovascular indicators of threat/challenge corroborated this expectation. The findings thereby contribute to our understanding of the physiological mechanism underlying the influence of negative affective responses towards numbers on numerical performance.

While mathematically impaired individuals have been shown to have deficits in all kinds of basic numerical representations, among them spatial-numerical associations, little is known about individuals with exceptionally high math expertise. They might have a more abstract magnitude representation or more flexible spatial associations, so that no automatic left/small and right/large spatial-numerical association is elicited. To pursue this question, we examined the Spatial Numerical Association of Response Codes (SNARC) effect in professional mathematicians which was compared to two control groups: Professionals who use advanced math in their work but are not mathematicians (mostly engineers), and matched controls. Contrarily to both control groups, Mathematicians did not reveal a SNARC effect. The group differences could not be accounted for by differences in mean response speed, response variance or intelligence or a general tendency not to show spatial-numerical associations. We propose that professional mathematicians possess more abstract and/or spatially very flexible numerical representations and therefore do not exhibit or do have a largely reduced default left-to-right spatial-numerical orientation as indexed by the SNARC effect, but we also discuss other possible accounts. We argue that this comparison with professional mathematicians also tells us about the nature of spatial-numerical associations in persons with much less mathematical expertise or knowledge.

The present study aimed at investigating the underlying cognitive processes and language specificities of three-digit number processing. More specifically, it was intended to clarify whether the single digits of three-digit numbers are processed in parallel and/or sequentially and whether processing strategies are influenced by the inversion of number words with respect to the Arabic digits [e.g., 43: dreiundvierzig ("three and forty")] and/or by differences in reading behavior of the respective first language. Therefore, English- and German-speaking adults had to complete a three-digit number comparison task while their eye-fixation behavior was recorded. Replicating previous results, reliable hundred-decade-compatibility effects (e.g., 742&#95;896: hundred-decade compatible because 7 < 8 and 4 < 9; 362&#95;517: hundred-decade incompatible because 3 < 5 but 6 > 1) for English- as well as hundred-unit-compatibility effects for English- and German-speaking participants were observed, indicating parallel processing strategies. While no indices of partial sequential processing were found for the English-speaking group, about half of the German-speaking participants showed an inverse hundred-decade-compatibility effect accompanied by longer inspection time on the hundred digit indicating additional sequential processes. Thereby, the present data revealed that in transition from two- to higher multi-digit numbers, the homogeneity of underlying processing strategies varies between language groups. The regular German orthography (allowing for letter-by-letter reading) and its associated more sequential reading behavior may have promoted sequential processing strategies in multi-digit number processing. Furthermore, these results indicated that the inversion of number words alone is not sufficient to explain all observed language differences in three-digit number processing.

The current study provides a generalizable account of the anatomo-functional associations as well as the connectivity of representational codes underlying numerical processing as suggested by the triple code model (TCM) of numerical cognition. By evaluating the neural networks subserving numerical cognition in two specific and substantially different numerical tasks with regard to both grey matter localizations as well as white matter tracts we (1) considered the possibility of additional memory-related cortex areas crucial for arithmetic fact retrieval (e.g., the hippocampus); (2) specified the functional involvement of prefrontal areas in number magnitude processing, and, finally; (3) identified the connections between these anatomo-functional instantiations of the representations involved in number magnitude processing and arithmetic fact retrieval employing probabilistic fiber tracking. The resulting amendments to the TCM are summarized in a schematic update, and ideas concerning the possible functional interplay between number magnitude processing and arithmetic fact retrieval are discussed.

Parity is important semantic information encoded by numbers. Interestingly, there are hand-based effects in parity judgment tasks: right-hand responses are faster for even and left-hand responses for odd numbers. As this effect was initially explained by the markedness of the words even vs. odd and right vs. left, it was denoted as the linguistic markedness of response codes (MARC) effect. In the present study, we investigated whether the MARC effect differs for right and left handers. We conducted a parity judgment task, in which right- and left-handed participants had to decide whether a presented single or two-digit number was odd or even by pressing a corresponding response key. We found that handedness modulated the MARC effect for unit digits. While we replicated a regular MARC effect for right handers, there was no evidence for a MARC effect for left handers. However, closer inspection revealed that the MARC effect in left handers depended on the degree of left-handedness with a reversed MARC effect for most left-handed participants. Furthermore, although parity of tens digits interfered with the processing of unit digits, the MARC effect for tens digits was not modulated by handedness. Our findings are discussed in the light of three different accounts for the MARC effect: the linguistic markedness account, the polarity correspondence principle, and the body-specificity hypothesis.

A spatial representation of number magnitude, aka the mental number line, is considered one of the basic numerical representations. One way to assess it is number line estimation (e.g., positioning 43 on a number line ranging from 0 to 100). Recently, a new unbounded version of the number line estimation task was suggested: without labeled endpoints but a predefined unit, which was argued to provide a purer measure of spatial numerical representations. To further investigate the processes determining estimation performance in the unbounded number line task, we used an adapted version with variable units other than 1 to evaluate influences of (i) the size of a given unit and (ii) multiples of the units as target numbers on participants' estimation pattern. We observed that estimations got faster and more accurate with increasing unit sizes. On the other hand, multiples of a predefined unit were estimated faster, but not more accurately than non-multiples. These results indicate an influence of multiplication fact knowledge on spatial numerical processing.

The unit-decade compatibility effect describes longer response times and higher error rates for incompatible (e.g., 37&#95;52) than compatible (e.g., 42&#95;57) number comparisons. Recent research indicated that the effect depends on the percentage of same-decade filler items. In the present study, we further examined this relationship by recording participants' eye-fixation behaviour. In four conditions, participants had to compare item sets with different filler item types (i.e., same-decade and same-unit filler items) and different numbers of same-decade filler items (i.e., 25, 50, and 75%). We found a weaker unit-decade compatibility effect with most fixations on tens in the condition with same-unit filler items. Moreover, the compatibility effect increased with the percentage of same-decade filler items which was accompanied by less fixations on tens and more fixations on units. Thus, our study provides first eye-tracking evidence for the influence of cognitive control in number processing.

Currently, there are three competing theoretical accounts concerning the nature of two-digit number magnitude representation: a holistic, a strictly decomposed, and a hybrid model. Observation of the unit-decade compatibility effect (Nuerk et al. in Cognition 82:B25-B33, 2001) challenged the view of two-digit number magnitude to be represented as one integrated entity. However, at the moment there is no study distinguishing between the decomposed and the hybrid model. The present study addressed this issue using a computational modelling approach. Three network models complying with the constraints of all three theoretical models were programmed and trained on two-digit number comparison. Models were compared as to how well they accounted for empirical effects in the most parsimonious way. Generally, this evaluation indicated that the empirical data were simulated best by the strictly decomposed model. Implications of these results for our understanding of the nature of human number magnitude representation are discussed.

Effects of semantic processing of crowded characters were investigated using numbers as stimuli. In an identification task, typical spacing effects in crowding were replicated. Using the same stimuli in a magnitude comparison task, a smaller effect of spacing was observed as well as an effect of response congruency. These effects were replicated in a second experiment with varying stimulus-onset asynchronies. In addition, decreasing performance with increasing onset-asynchrony (so-called type-B masking) for incongruent flankers indicates semantic processing of target and flankers. The data show that semantic processing takes place even in crowded stimuli. This argues strongly against common accounts of crowding in terms of early stimulus-driven impairments of processing.

A perceptual frequency variant of the orthographic cue (OC) hypothesis (Peressotti, Cubelli, & Job, 2003) was tested in two perceptual identification experiments using the variable viewing position technique: German nouns and non-nouns that are most frequently perceived with or without initial letter capitalization, respectively, were tachistoscopically presented in upper-case, lower-case, or with initial capitalization. The results indicated that words were best recognized in the form they are most frequently perceived in, which suggests that during reading acquisition abstract as well as case- and item-specific OCs may be learned and used for recognition.

Our aim was to determine whether age, sex, the degree of weakness, anticonvulsants, the histology of the underlying lesion(s), the presence of oedema or the distance of the lesion from the motor region have an impact on the blood oxygenation level-dependent (BOLD) signal strength and therefore on the validity of functional MRI (fMRI). We studied 98 patients with masses near the central region imaged for surgical planning at 1.5 tesla, employing a BOLD sequence during a motor task. We calculated percentage signal change in the primary motor cortex between rest and activation and carried out multiple linear regression to examine the impact of the above factors on signal strength. Using a stepwise analysis strategy, the distance of the lesion from the motor region had the strongest influence (r=0.653, P<0.001). The factor with largest uncorrelated additional impact on signal change was the presence of oedema. Both predictors together formed a highly significant multiple r=0.739 ( P<0.001). No other predictive factor was identified (all P>0.20). Disturbances of cerebral blood flow and metabolism induced by the tumour were presumed to be the causes of a decrease in signal in the adjacent cortex.