Analogue magnitude representation of angles and its relation to geometric expertise.

The distance effect (comparing objects becomes easier with increasing differences in their magnitude) is observed in tasks ranging across domains, and its existence has been interpreted as evidence for analogue magnitude representation. Similarly, associations between response side and magnitude (faster left/right-sided responses to small/large objects, respectively) are observed across domains. We investigated the analogue processing of angles and the association between angle magnitude and response side in relation to geometric expertise. We compared the behavioural pattern of two groups-architects and controls-in a direct angle magnitude classification task (i.e., judge whether a presented angle was greater or less than 90°) and in an indirect task (i.e., judge whether an angle was drawn with a dashed or continuous line). We found a robust distance effect for reaction times and accuracy at the whole sample level and in each group separately. Architects revealed a smaller distance effect for accuracy than controls. This could be interpreted as an argument for a more precise analogue representation of angles in experts compared to non-experts. However, we did not find evidence for an association between angle magnitude and response side in any group.

Don't SNARC me now! Intraindividual variability of cognitive phenomena - Insights from the Ironman paradigm.

Two implicit generalizations are often made from group-level studies in cognitive experimental psychology and their common statistical analysis in the general linear model: (1) Group-level phenomena are assumed to be present in every participant with variations between participants being often treated as random error in data analyses; (2) phenomena are assumed to be stable over time. In this preregistered study, we investigated the validity of these generalizations in the commonly used parity judgment task. In the proposed Ironman paradigm, the intraindividual presence and stability of three popular numerical cognition effects were tested in 10 participants on 30 days: the SNARC (Spatial-Numerical Association of Response Codes, i.e., faster left-/right-sided responses to small/large magnitude numbers, respectively; Dehaene, Bossini, & Giraux, 1993), MARC (Linguistic Markedness of Response Codes; i.e., faster left-/right-sided responses to odd/even numbers, respectively; Nuerk, Iversen, & Willmes, 2004), and Odd (i.e., faster responses to even numbers; Hines, 1990) effects. We replicated the group-level effects; however, they were reliably present in only four to five (SNARC), six (MARC) or five (Odd) of 10 participants. Fluctuations seemed unsystematic, although the SNARC effect decreased over time along with reaction times. No correlation between the SNARC and MARC effects and sleep duration, tiredness, daytime, and consumption of stimulants were detected in most participants. These results challenge the frequent generalizations from group-level phenomena to individual participants and from single sessions to typical behavior. The innovative Ironman paradigm combined with bootstrap analyses permits unique insights into the intraindividual presence and stability of cognitive phenomena.

Regional specificity of cathodal transcranial direct current stimulation effects on spatial-numerical associations: Comparison of four stimulation sites.

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.

Emotions and mathematics: anxiety profiles and their influence on arithmetic performance in university students.

Mathematics anxiety (MA), general and test anxieties affect mathematics performance. However, little is known about how different anxiety profiles (i.e. individual configurations of anxiety forms) influence the relationship between MA and mathematics performance in university students. To the best of our knowledge, studies that have categorized participants based on their anxiety profiles and investigated how such groups differ in mathematics performance and other individual characteristics have all been conducted only with children and adolescents. Using latent profile analysis, we identified five different anxiety profiles in UK university students (N = 328) based on their MA, test anxiety (TA) and trait general anxiety levels (GA). Beyond extreme profiles (high or low levels in all forms of anxiety), we found groups characterized by more specific anxiety forms (MA profile, TA profile and high anxiety with low MA learning profile). These profiles were differentially related to arithmetic performance (but not the performance in a non-mathematics task), and individual factors (e.g. self-concept and self-efficacy). Results can inform the design of interventions tailored to individuals' unique anxiety profiles and highlight the necessity to further study the underpinning mechanisms that drive the MA developmental trajectory from childhood to adulthood.

Cultural similarities and specificities of finger counting and montring: Evidence from Amazon Tsimane' people.

Numerical cognition might be embodied, that is, grounded in bodily actions. This claim is supported by the observation that, potentially due to our shared biology, finger counting is prevalent among a variety of cultures. Differences in finger counting are apparent even within Western cultures. Relatively few indigenous cultures have been systematically analyzed in terms of traditional finger counting and montring (i.e., communicating numbers with fingers) routines. Even fewer studies used the same protocols across cultures, allowing for a systematic comparison of indigenous and Western finger counting routines. We analyze the finger counting and montring routines of Tsimane' (N = 121), an indigenous people living in the Bolivian Amazon rainforest, depending on handedness, education level, and exposure to mainstream, industrialized Bolivian culture. Tsimane' routines are compared with those of German and British participants. Tsimane' reveal a greater variation in finger counting and montring routines, which seems to be modified by their education level. We outline a framework on how different factors such as handedness and reading direction might affect cross-cultural and within-cultural variation in finger counting.

The plural counts: Inconsistent grammatical number hinders numerical development in preschoolers - A cross-linguistic study.

The role of grammar in numerical development, and particularly the role of grammatical number inflection, has already been well-documented in toddlerhood. It is unclear, however, whether the influence of grammatical language structure further extends to more complex later stages of numerical development. Here, we addressed this question by exploiting differences between Polish, which has a complex grammatical number paradigm, leading to a partially inconsistent mapping between numerical quantities and grammatical number, and German, which has a comparatively easy verbal paradigm: 151 Polish-speaking and 123 German-speaking kindergarten children were tested using a symbolic numerical comparison task. Additionally, counting skills (Give-a-Number and count-list), and mapping between non-symbolic (dot sets) and symbolic representations of numbers, as well as working memory (Corsi blocks and Digit span) were assessed. Based on the Give-a-Number and mapping tasks, the children were divided into subset-knowers, CP-knowers-non-mappers, and CP-knowers-mappers. Linguistic background was related to performance in several ways: Polish-speaking children expectedly progressed to the CP-knowers stage later than German children, despite comparable non-numerical capabilities, and even after this stage was achieved, they fared worse in the numerical comparison task. There were also meaningful differences in spatial-numerical mapping between the Polish and German groups. Our findings are in line with the theory that grammatical number paradigms influence. the development of representations and processing of numbers, not only at the stage of acquiring the meaning of the first number-words but at later stages as well, when dealing with symbolic numbers.

No fingers, no SNARC? Neither the finger counting starting hand, nor its stability robustly affect the SNARC effect.

The Spatial-Numerical Association of Response Codes (SNARC) effect (i.e., faster left/right sided responses to small/large magnitude numbers, respectively) is considered to be strong evidence for the link between numbers and space. Studies have shown considerable variation in this effect. Among the factors determining individual differences in the SNARC effect is the hand an individual uses to start the finger counting sequence. Left-starters show a stronger and less variable SNARC effect than right-starters. This observation has been used as an argument for the embodied nature of the SNARC effect. For this to be the case, one must assume that the finger counting sequence (especially the starting hand) is stable over time. Subsequent studies challenged the view that the SNARC differs depending on the finger counting starting hand. At the same time, it has been pointed out that the temporal stability of the finger counting starting hand should not be taken for granted. Thus, in this preregistered study, we aimed to replicate the difference in the SNARC between left- and right-starters and explore the relationship between the self-reported temporal stability of the finger counting starting hand and the SNARC effect. In line with the embodied cognition account, left-starters who declare more temporarily stable finger counting habits should reveal a stronger SNARC effect. Results of the preregistered analysis did not show the difference between left- and right-starters. However, further exploratory analysis provided weak evidence that this might be the case. Lastly, we found no evidence for the relationship between finger counting starting hand stability and the SNARC effect. Overall, these results challenge the view on the embodied nature of the SNARC effect.

Mathematics-gender stereotype endorsement influences mathematics anxiety, self-concept, and performance differently in men and women.

Mathematics anxiety (MA) is negatively associated with mathematics performance. Although some aspects, such as mathematics self-concept (M self-concept), seem to modulate this association, the underlying mechanism is still unclear. In addition, the false gender stereotype that women are worse than men in mathematics can have a detrimental effect on women. The role that the endorsement of this stereotype (mathematics-gender stereotype (MGS) endorsement) can play may differ between men and women. In this study, we investigated how MA and mathematics self-concept relate to arithmetic performance when considering one's MGS endorsement and gender in a large sample (n = 923) of university students. Using a structural equation modeling approach, we found that MA and mathematics self-concept mediated the effect of MGS endorsement in both men and women. For women, MGS endorsement increased their MA level, while in men, it had the opposite effect (albeit weak). Specifically, in men, MGS endorsement influenced the level of the numerical components of MA, but, unlike women, it also positively influenced their mathematics self-concept. Moreover, men and women perceived the questions included in the considered instruments differently, implying that the scores obtained in these questionnaires may not be directly comparable between genders, which has even broader theoretical and methodological implications for MA research.

Spatialization in working memory and its relation to math anxiety.

Working memory (WM) is one of the most important cognitive functions that may play a role in the relation between math anxiety (MA) and math performance. The processing efficiency theory proposes that rumination and worrisome thoughts (induced by MA) result in less available WM resources (which are needed to solve math problems). At the same time, high MA individuals have lower verbal and spatial WM capacity in general. Extending these findings, we found that MA is also linked to the spatial coding of serial order in verbal WM: subjects who organize sequences from left-to-right in verbal WM show lower levels of MA compared with those who do not spatialize. Furthermore, these spatial coders have higher verbal WM capacity, better numerical order judgment abilities, and higher math scores. These findings suggest that spatially structuring the verbal mind is a promising cognitive correlate of MA and opens new avenues for exploring causal links between elementary cognitive processes and MA.

Mathematics anxiety-where are we and where shall we go?

In this paper, we discuss several largely undisputed claims about mathematics anxiety (MA) and propose where MA research should focus, including theoretical clarifications on what MA is and what constitutes its opposite pole; discussion of construct validity, specifically relations between self-descriptive, neurophysiological, and cognitive measures; exploration of the discrepancy between state and trait MA and theoretical and practical consequences; discussion of the prevalence of MA and the need for establishing external criteria for estimating prevalence and a proposal for such criteria; exploration of the effects of MA in different groups, such as highly anxious and high math-performing individuals; classroom and policy applications of MA knowledge; the effects of MA outside educational settings; and the consequences of MA on mental health and well-being.

A gifted SNARC? Directional spatial-numerical associations in gifted children with high-level math skills do not differ from controls.

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.

Automatic place-value activation in magnitude-irrelevant parity judgement.

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.

Spatialization in working memory: can individuals reverse the cultural direction of their thoughts?

A recent study based on the SPoARC effect (spatial position association response codes) showed that culture heavily shapes cognition and more specifically the way thought is organized; when Western adults are asked to keep in mind a sequence of colors, they mentally organize them from left to right, whereas right-to-left reading/writing adults spatialize them in the opposite direction. Here, we investigate if the spontaneous direction of spatialization in Westerners can be reversed. Lists of five consonants were presented auditorily at a rate of 3 s per item, participants were asked to mentally organize the memoranda from right to left. Each list was followed by a probe. Participants had to indicate whether the probe was part of the sequence by pressing a "yes" key or a "no" key with the left or right index finger. Left/right-hand key assignment was switched after half of the trials were completed. The results showed a reverse SPoARC effect that was comparable in magnitude to the spontaneous left-to-right SPoARC effect found in a previous study. Overall, our results suggest that individuals can reverse the cultural direction of their thoughts.

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

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.

The spatial-numerical association of response codes effect and math skills: why related?

Evidence from multiple studies conducted in the past few decades converges on the conclusion that numerical properties can be associated with specific directions in space. Such spatial-numerical associations (SNAs), as a signature of elementary number processing, seem to be a likely correlate of math skills. Nevertheless, almost three decades of research on the spatial-numerical association of response codes (SNARC) effect, the hallmark of SNAs, has not provided conclusive results on whether there is a relation with math skills. Here, going beyond reviewing the existing literature on the topic, we try to answer a more fundamental question about why the SNARC effect should (and should not) be related to math skills. We propose a multiroute model framework for a SNARC-math skills relationship. We conclude that the relationship is not straightforward and that several other factors should be considered, which under certain circumstances or in certain groups can cause effects of opposite directions. The model can account for conflicting results, and thus may be helpful for deriving predictions in future studies.

Regional specificity of cathodal transcranial direct current stimulation (tDCS) effects on spatial-numerical associations: Comparison of four stimulation sites.

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.

Corrigendum: A Mental Odd-Even Continuum Account: Some Numbers May Be "More Odd" Than Others and Some Numbers May Be "More Even" Than Others.

[This corrects the article DOI: 10.3389/fpsyg.2018.01081.].

The SNARC and MARC effects measured online: Large-scale assessment methods in flexible cognitive effects.

The Spatial-Numerical Association of Response Codes (SNARC) effect (i.e., faster reactions to small/large numbers on the left-/right-hand side) is usually observed along with the linguistic Markedness of Response Codes (MARC) effect-that is, faster left-/right-hand responses to odd/even numbers. The SNARC effect is one of the most thoroughly investigated phenomena in numerical cognition. However, almost all SNARC and MARC studies to date were conducted with sample sizes smaller than 100. Here we report on a study with 1,156 participants from various linguistic and cultural backgrounds performing a typical parity judgment task. We investigated whether (1) the SNARC and MARC effects can be observed in an online setup, (2) the properties of these effects observed online are similar to those observed in laboratory setups, (3) the effects are reliable, and (4) they are valid. We found robust SNARC and MARC effects. Their magnitude and reliabilities were comparable to values previously reported in in-lab studies. Furthermore, we reproduced commonly observed validity correlations of the SNARC and MARC effects. Namely, SNARC and MARC correlated with mean reaction times and intraindividual variability in reaction times. Additionally, we found interindividual differences in the SNARC and MARC effects (e.g., finger-counting routines for the SNARC and handedness for the MARC). Large-scale testing via web-based data acquisition not only produces SNARC and MARC effects and validity correlations similar to those from small, in-lab studies, but also reveals substantial insights with regard to interindividual differences that usually cannot be revealed in the offline laboratory, due to power considerations.

A large-scale survey on finger counting routines, their temporal stability and flexibility in educated adults.

A strong link between bodily activity and number processing has been established in recent years. Although numerous observations indicate that adults use finger counting (FC) in various contexts of everyday life for different purposes, existing knowledge of FC routines and their use is still limited. In particular, it remains unknown how stable the (default) FC habits are over time and how flexible they can be. To investigate these questions, 380 Polish participants completed a questionnaire on their FC routines, the stability of these routines, and the context of FC usage, preceded by the request to count on their fingers from 1 to 10. Next, the test-retest stability of FC habits was examined in 84 participants 2 months following the first session. To the best of our knowledge, such a study design has been adopted for the first time. The results indicate that default FC routines of the majority of participants (75%) are relatively stable over time. At the same time, FC routines can flexibly adapt according to the situation (e.g., when holding an object). As regards prevalence, almost all participants, in line with previous findings on Western individuals, declared starting from the closed palm and extending consecutive fingers. Furthermore, we observed relations between FC preferences and handedness (more left-handers start from the left hand) and that actual finger use is still widespread in healthy adults for a variety of activities (the most prevalent uses of FC are listing elements, presenting arguments and plans, and calendar calculations). In sum, the results show the practical relevance of FC in adulthood, the relative stability of preferences over time along with flexible adaptation to a current situation, as well as an association of FC routines with handedness. Taken together our results suggest that FC is the phenomenon, which is moderated or mediated by multiple embodied factors.