The impact of mathematical proficiency on the number-space association

The role of cognitive control in the SNARC effect: A review

The spatial-numerical association of response codes (SNARC) effect, in which people respond to small numbers faster with the left hand and to large numbers faster with the right hand, is a popular topic in cognitive psychology. Some well-known theoretical accounts explaining this effect include the mental number line model, polarity correspondence principle, dual-route model, and working memory account. However, these fail to explain the finding that the size of the SNARC effect is modulated by cognitive control. Here, we propose a new account-a cognitive control-based view of the SNARC effect. This view argues that the SNARC effect is fundamentally determined by cognitive control in resolving conflicts during stimulus-response mapping. Several subcomponents of cognitive control, such as working memory, mental or task set shifting, inhibition control, and conflict adaptation, can easily modulate the SNARC effect. The cognitive control-based view can account for the flexible SNARC effect observed in diverse task situations while providing new insight into its mechanism.

Children perform better on left than right targets in an ordinal task

Francis Galton first reported that humans mentally organize numbers from left to right on a mental number line (1880). This spatial-numerical association was long considered to result from writing and reading habits. More recently though, newborns and animals showed a left-to-right oriented spatial numerical association challenging the primary role assigned to culture in determining the link between number and space. Despite growing evidence supporting the intrinsic association between number and space in different species, its adaptive value is still largely unknown. Here we tested for an advantage in identification of left versus right target positions in 3- to 6-year-old children. Children watched as a toy was hidden under one of 10 linearly arranged identical cups and were then asked to help a stuffed animal retrieve the toy. On each trial, the toy was hidden in the 2nd, 3rd, or 4th cup, from the left or right. To prevent children from staring at the target cup, they were asked to pick up the stuffed animal from under their chair after witnessing the hiding of the toy and then to help the stuffed animal find the toy. Older children were more accurate than younger children. Children exhibited a serial position effect, with performance higher for more exterior targets. Remarkably, children also showed a left bias: they remembered the left targets better than the right targets. Only the youngest children were dramatically influenced by the location of the experimenter during search. Additional analyses support the hypothesis that children used a left-to-right oriented searching strategy in this spatial/ordinal task.

Heterogeneity of Dyscalculia Risk Dependent on the Type of Number Line Estimation Task and the Number Magnitude

An ability that is impaired in developmental dyscalculia (DD) is related to number line estimation (NLE). However, due to variability in NLE task performance, group differences do not exemplify the real difficulty level observed in the DD population. Thirty-two of the fifty-two participants posing dyscalculia risk (DR) (mean age = 9.88) experienced difficulties in mathematics. All the children performed two number-to-position tasks and two tasks requiring a verbal estimation of a number indicated on a line, utilizing the ranges 0–100 and 0–1000. The results showed that the estimation error in the verbal task was greater in the DR group than in the typically developed (TD) group for the 0–1000 range. In the number-to-position task, group differences were found for both ranges and the variability within both groups was smaller than it was in the verbal tasks. Analyses of each of the 26 numerical magnitudes revealed a more comprehensive pattern. The majority of the group effects were related to the 0–1000 line. Therefore, considerable data variability, especially in the DD group, suggests this issue must be analyzed carefully in the case of other mathematical capacities. It also critically questions some well-established phenomena and norms in experimental and diagnostic practices.

Hand preference and Mathematical Learning Difficulties: New data from Greece, the United Kingdom, and Germany and two meta-analyses of the literature

Increased rates of atypical handedness are observed in neurotypical individuals who are low-performing in mathematical tasks as well as in individuals with special educational needs, such as dyslexia. This is the first investigation of handedness in individuals with Mathematical Learning Difficulties (MLD). We report three new studies (N = 134; N = 1,893; N = 153) and two sets of meta-analyses (22 studies; N = 3,667). No difference in atypical hand preference between MLD and Typically Achieving (TA) individuals was found when handedness was assessed with self-report questionnaires, but weak evidence of a difference was found when writing hand was the handedness criterion in Study 1 (p = .049). Similarly, when combining data meta-analytically, no hand preference differences were detected. We suggest that: (i) potential handedness effects require larger samples, (ii) direction of hand preference is not a sensitive enough measure of handedness in this context, or that (iii) increased rates of atypical hand preference are not associated with MLD. The latter scenario would suggest that handedness is specifically linked to language-related conditions rather than conditions related to cognitive abilities at large. Future studies need to consider hand skill and degree of hand preference in MLD.

Non-symbolic and symbolic number lines are dissociated

People use mental number lines for both symbolic numerals and numerosity, but little is known about how these two mental number lines are related. The current study investigated the association in effect size, directionality of the mental number line, and development between symbolic and non-symbolic mental number lines to determine if they were related to or independent from each other. We collected data from numerosity- and digit-matching tasks that used the following numbers: 11, 14, 17, 20, 23, 26, and 29. Tasks were performed by college undergraduates and the fifth-grade primary school students. The results showed that none of the effects for non-symbolic numbers was related to any of the effects for symbolic numbers, and vice versa, in both adults and children. Another notable finding was that the correlation between the SNARC (spatial-numerical association of response code) effect size and mathematical ability was negative in the adult group. These results are consistent with the dissociated processes hypothesis and suggest that mental number lines are notation-dependent. As shown by the SNARC effect, the mental number line might result in interference in the current task by an automatically activated spatial notation-dependent representation of numbers.

The Spatial-Numerical Association of Response Codes (SNARC) effect in highly math-anxious individuals: An ERP study

The Spatial-Numerical Association of Response Codes (SNARC) effect was examined in highly (HMA) and low math-anxious (LMA) individuals performing a number comparison in an ERP study. The SNARC effect consists of faster latencies when the response side is congruent with number location in the mental number line (MNL). Despite the stronger SNARC effect in the HMA group, their responses in incongruent trials were slower than in congruent trials only for the largest numerical magnitudes. Moreover, HMAs showed a less positive centroparietal P3b component in incongruent trials than in congruent ones, but only for the largest magnitudes. Since the SNARC effect arises during response selection and P3b positivity decreases with the difficulty of decision, this result suggests that HMA individuals might find it more difficult than LMAs to control the conflict between the automatically activated location of numbers in the MNL and the response side, especially in more cognitively demanding trials.

The influence of children's mathematical competence on performance in mental number line, time knowledge and time perception

A growing body of research suggests that space, time and number are represented within a common system. Other studies have shown this relationship is related to the mathematical competency. Here we examined the influence of the mathematical capacities of 8-12 years old children, grouped into high (n = 63) and low (n = 58) on performance in mental number line, time knowledge and time perception. The results revealed that mathematical competency influences mental number line and time knowledge, but with regard to time perception the effects were only observed in time production task. In addition, the results of correlation analysis revealed interaction between time knowledge, time production (but not reproduction) and mental number line. Finally, the findings are discussed within the framework of the recent theories regarding representation of space, time and number.

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.

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.

Implicit and explicit spatial-numerical representations diverge in number-form synesthetes

In number-form (NF) synesthesia-a condition in which people report vivid, automatic and consistent mental layouts for numerical sequences-numbers and space are closely linked. These explicit associations are similar to the implicit associations demonstrated by the Spatial-Numerical Association of Response Codes (SNARC) effect. Thus, NF synesthesia offers a unique opportunity to investigate spatial-numerical associations. We tested implicit and explicit representations in NF synesthetes using a multiple case-study design. Over two sessions, synesthetes participated in a semi-structured interview focusing on the nature of their associations, as well as SNARC and number line estimation tasks. Contrary to our hypotheses, only one synesthete demonstrated SNARC effects congruent with her reported form, whereas two others exhibited SNARC effects that were the opposite of their explicit NFs. While this inconsistency between implicit and explicit representations may indicate separate underlying cognitive mechanisms, factors such as task-specific constraints and strategic variability must also be considered.

The effects of risk magnitude training on mapping risks on space

Risk perception has recently been shown to reveal a mental spatial representation, with people responding faster to low-risk items on the left side, and high-risk items on the right side. Subjective risk perception has a stronger spatial representation than objective risk perception; however, both reveal small effect sizes. With risk magnitude being a new domain within spatial mapping literature, we sought to explore its nuances. Following discussion surrounding the relationship between spatial mapping and level of expertise, this study investigated the effect of training an objective risk magnitude sequence on mental spatial representations. Participants (n = 34) used their left and right hands to indicate whether eight risk stimuli were lower or higher risk than a referent activity, both before and after training. Training involved repetitively learning the objectively correct order of the same eight risk stimuli for approximately 15 min. Pre-training results demonstrated the expected spatial representations. Contrary to our predictions, the spatial representation did not get stronger post-training, but instead disappeared. Previous research has demonstrated a loss of spatial-numerical mappings with increased task load. An increase in post-training reaction times could reflect an increase in task load due to a lack of adequate knowledge of risk stimulus order; thus revealing no mental spatial representation. However, failure to find training effects highlights the flexibility of weaker spatial representations, and supports research demonstrating spatial representation flexibility.

Individual Differences in Implicit and Explicit Spatial Processing of Fractions

Recent studies have explored the foundations of mathematical skills by linking basic numerical processes to formal tests of mathematics achievement. Of particular interest is the relationship between spatial-numerical associations-specifically, the Spatial Numerical Association of Response Codes (SNARC) effect-and various measures of math ability. Thus far, studies investigating this relationship have yielded inconsistent results. Here, we investigate how individual implicit and explicit spatial representations of fractions relate to fraction knowledge and other formal measures of math achievement. Adult participants (n = 105) compared the magnitude of single digit, irreducible fractions to ½, a task that has previously produced a reliable SNARC effect. We observed a significant group-level SNARC effect based on overall fraction magnitude, with notable individual variability. While individual SNARC effects were correlated with performance on a fraction number-line estimation (NLE) task, only NLE significantly predicted scores on a fractions test and basic standardized math test, even after controlling for IQ, mean accuracy, and mean reaction time. This suggests that-for fractions-working with an explicit number line is a stronger predictor of math ability than implicit number line processing. Neither individual SNARC effects nor NLE performance were significant predictors of algebra scores; thus, the mental number line may not be as readily recruited during higher-order mathematical concepts, but rather may be a foundation for thinking about simpler problems involving rational magnitudes. These results not only characterize the variability in adults' mental representations of fractions, but also detail the relative contributions of implicit (SNARC) and explicit (NLE) spatial representations of fractions to formal math skills.

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.

Judging risk magnitude: walking to the left and base jumping to the right

When thinking about quantifiable domains such as numbers, pitch, and size, they are implicitly mapped on to representational space with small/low/less and large/high/more of the respective domain represented on the left and right sides of representational space, respectively. Recent research has also demonstrated that more abstract domains (colours, language, political party names) are also mapped in the same way. This study investigated a new abstract domain, risk, to examine if this same pattern of effects is apparent (left = low risk/right = high risk) to get a better understanding of how risk magnitudes are processed. Experiment 1 (n = 26) presented objective, statistically calculated risk stimuli (micromorts) to participants, who indicated if the stimuli had lower or higher risks than a referent, with their left and right hands. Experiment 2 (n = 25) utilised the same task, but the risk stimuli were generated by the participants themselves. Both experiments found the expected association of risk with space-indicated by faster left-hand responses to low-risk stimuli and faster right-hand responses to high-risk stimuli. Risks appear to fit onto a standard left-right spatial association; however, the effect sizes for all analyses were small. The results of this study are not only in line with the idea of a generalised magnitude processing system, but might also inform best practices in effective communications of risks.

Non-symbolic magnitudes are represented spatially: Evidence from a non-symbolic SNARC task

A core proposition in numerical cognition is numbers are represented spatially. Evidence for this proposition comes from the "spatial numerical association of response codes" effect (SNARC) in which faster responses are made by the left/right hand judging whether one of a pair of Arabic digits is smaller/larger than the other. Less is known if a similar SNARC effect exists for non-symbolic magnitudes; and research that has been conducted used stimuli which could be translated into symbolic terms. To overcome this limitation, we employed a referent-to-target judgment paradigm in which a referent dot array (n = 30 dots) was follow by a second array of dots (e.g., n = 45 or 15 dots)-participants judged if the second array contained fewer or more dots than the referent array. Dot arrays with fewer dots were judged more quickly with the left hand compared to the right hand (i.e., a SNARC effect). Not all participants demonstrated a SNARC effect, however. Neither visuospatial working memory nor math ability was associated with the presence/absence of a non-symbolic SNARC effect. Implications of the non-symbolic SNARC effect for accounts of numerical cognition are discussed.

The SNARC effect is associated with worse mathematical intelligence and poorer time estimation

Interactions between the ways we process space, numbers and time may arise from shared and innate generic magnitude representations. Alternatively or concurrently, such interactions could be due to the use of physical magnitudes, like spatial extent, as metaphors for more abstract ones, like number and duration. That numbers might be spatially represented along a mental number line is suggested by the SNARC effect: faster left-side responses to small single digits, like 1 or 2, and faster right-side responses to large ones, like 8 or 9. Previously, we found that time estimation predicts mathematical intelligence and speculated that it may predict spatial ability too. Here, addressing this issue, we test-on a relatively large sample of adults and entirely within subjects-the relationships between (a) time: proficiency at producing and evaluating durations shorter than one second, (b) space: the ability to mentally rotate objects, (c) numbers: mathematical reasoning skills, and (d) space-number associations: the SNARC effect. Better time estimation was linked to greater mathematical intelligence and better spatial skills. Strikingly, however, stronger associations between space and numbers predicted worse mathematical intelligence and poorer time estimation.

The Developing Mental Number Line: Does Its Directionality Relate to 5- to 7-Year-Old Children's Mathematical Abilities?

Spatial representations of number, such as a left-to-right oriented mental number line, are well documented in Western culture. Yet, the functional significance of such a representation remains unclear. To test the prominent hypothesis that a mental number line may support mathematical development, we examined the relation between spatial-numerical associations (SNAs) and math proficiency in 5- to 7-year-old children. We found evidence of SNAs with two tasks: a non-symbolic magnitude comparison task, and a symbolic "Where was the number?" (WTN) task. Further, we found a significant correlation between these two tasks, demonstrating convergent validity of the directional mental number line across numerical format. Although there were no significant correlations between children's SNAs on the WTN task and math ability, children's SNAs on the magnitude comparison task were negatively correlated with their performance on a measure of cross-modal arithmetic, suggesting that children with a stronger left-to-right oriented mental number line were less competent at cross-modal arithmetic, an effect that held when controlling for age and a set of general cognitive abilities. Despite some evidence for a negative relation between SNAs and math ability in adulthood, we argue that the effect here may reflect task demands specific to the magnitude comparison task, not necessarily an impediment of the mental number line to math performance. We conclude with a discussion of the different properties that characterize a mental number line and how these different properties may relate to mathematical ability.

Implicit and Explicit Number-Space Associations Differentially Relate to Interference Control in Young Adults With ADHD

Behavioral evidence for the link between numerical and spatial representations comes from the spatial-numerical association of response codes (SNARC) effect, consisting in faster reaction times to small/large numbers with the left/right hand respectively. The SNARC effect is, however, characterized by considerable intra- and inter-individual variability. It depends not only on the explicit or implicit nature of the numerical task, but also relates to interference control. To determine whether the prevalence of the latter relation in the elderly could be ascribed to younger individuals' ceiling performances on executive control tasks, we determined whether the SNARC effect related to Stroop and/or Flanker effects in 26 young adults with ADHD. We observed a divergent pattern of correlation depending on the type of numerical task used to assess the SNARC effect and the type of interference control measure involved in number-space associations. Namely, stronger number-space associations during parity judgments involving implicit magnitude processing related to weaker interference control in the Stroop but not Flanker task. Conversely, stronger number-space associations during explicit magnitude classifications tended to be associated with better interference control in the Flanker but not Stroop paradigm. The association of stronger parity and magnitude SNARC effects with weaker and better interference control respectively indicates that different mechanisms underlie these relations. Activation of the magnitude-associated spatial code is irrelevant and potentially interferes with parity judgments, but in contrast assists explicit magnitude classifications. Altogether, the present study confirms the contribution of interference control to number-space associations also in young adults. It suggests that magnitude-associated spatial codes in implicit and explicit tasks are monitored by different interference control mechanisms, thereby explaining task-related intra-individual differences in number-space associations.

Interference Between Number Magnitude and Parity

Interference between number magnitude and other properties can be explained by either an analogue magnitude system interfering with a continuous representation of the other properties or by discrete, categorical representations in which the corresponding number and property categories interfere. In this study, we investigated whether parity, a discrete property which supposedly cannot be stored on an analogue representation, could interfere with number magnitude. We found that in a parity decision task the magnitude interfered with the parity, highlighting the role of discrete representations in numerical interference. Additionally, some participants associated evenness with large values, while others associated evenness with small values, therefore, a new interference index, the dual index was introduced to detect this heterogeneous interference. The dual index can be used to reveal any heterogeneous interference that were missed in previous studies. Finally, the magnitude-parity interference did not correlate with the magnitude-response side interference (Spatial-Numerical Association of Response Codes [SNARC] effect) or with the parity-response side interference (Markedness Association of Response Codes [MARC] effect), suggesting that at least some of the interference effects are not the result of the stimulus property markedness.

Developmental Changes in the Effect of Active Left and Right Head Rotation on Random Number Generation

Numbers are thought to be spatially organized along a left-to-right horizontal axis with small/large numbers on its left/right respectively. Behavioral evidence for this mental number line (MNL) comes from studies showing that the reallocation of spatial attention by active left/right head rotation facilitated the generation of small/large numbers respectively. While spatial biases in random number generation (RNG) during active movement are well established in adults, comparable evidence in children is lacking and it remains unclear whether and how children’s access to the MNL is affected by active head rotation. To get a better understanding of the development of embodied number processing, we investigated the effect of active head rotation on the mean of generated numbers as well as the mean difference between each number and its immediately preceding response (the first order difference; FOD) not only in adults (n = 24), but also in 7- to 11-year-old elementary school children (n = 70). Since the sign and absolute value of FODs carry distinct information regarding spatial attention shifts along the MNL, namely their direction (left/right) and size (narrow/wide) respectively, we additionally assessed the influence of rotation on the total of negative and positive FODs regardless of their numerical values as well as on their absolute values. In line with previous studies, adults produced on average smaller numbers and generated smaller mean FODs during left than right rotation. More concretely, they produced more negative/positive FODs during left/right rotation respectively and the size of negative FODs was larger (in terms of absolute value) during left than right rotation. Importantly, as opposed to adults, no significant differences in RNG between left and right head rotations were observed in children. Potential explanations for such age-related changes in the effect of active head rotation on RNG are discussed. Altogether, the present study confirms that numerical processing is spatially grounded in adults and suggests that its embodied aspect undergoes significant developmental changes.

Mapping of non-numerical domains on space: a systematic review and meta-analysis

The spatial numerical association of response code (SNARC) effect is characterized by low numbers mapped to the left side of space and high numbers mapped to the right side of space. In addition to numbers, SNARC-like effects have been found in non-numerical magnitude domains such as time, size, letters, luminance, and more, whereby the smaller/earlier and larger/later magnitudes are typically mapped to the left and right of space, respectively. The purpose of this systematic and meta-analytic review was to identify and summarise all empirical papers that have investigated horizontal (left-right) SNARC-like mappings using non-numerical stimuli. A systematic search was conducted using EMBASE, Medline, and PsycINFO, where 2216 publications were identified, with 57 papers meeting the inclusion criteria (representing 112 experiments). Ninety-five of these experiments were included in a meta-analysis, resulting in an overall effect size of d = .488 for a SNARC-like effect. Additional analyses revealed a significant effect size advantage for explicit instruction tasks compared with implicit instructions, yet yielded no difference for the role of expertise on SNARC-like effects. There was clear evidence for a publication bias in the field, but the impact of this bias is likely to be modest, and it is unlikely that the SNARC-like effect is a pure artefact of this bias. The similarities in the response properties for the spatial mappings of numerical and non-numerical domains support the concept of a general higher order magnitude system. Yet, further research will need to be conducted to identify all the factors modulating the strength of the spatial associations.

Finding the SNARC Instead of Hunting It: A 20∗20 Monte Carlo Investigation

The Spatial Numerical Association of Response Codes (SNARC) effect describes a stimulus-response association of left with small magnitude and right with large magnitude. Usually, it is estimated by means of regression slopes, where the independent variable only has a limited number of levels. Inspection of the literature reveals that it is not difficult to detect a SNARC effect within a group, but it has been quite unusual to find group differences. Is the SNARC effect as it is usually estimated using regression slopes largely insensitive to group differences, and are there design parameters necessary to increase sensitivity in group comparison analyses? Using numerical simulations, we provide evidence that both sample size and the number of stimulus repetitions, as well as intra-individual variability, contribute in a substantial way to the probability of detecting an existing SNARC effect. Our results show that the adequate choice of either sample size or number of repetitions per experimental cell does not fully compensate for a poor choice of the other parameter. Moreover, repeated failures to find significant group differences in the SNARC effect can be explained by insufficient power. Fortunately, increasing the number of repetitions to about 20 and testing at least 20 participants provides in most cases sufficient sensitivity to reliably detect the SNARC effect as well as group differences. Power plots are provided, which may help to improve both the economy and sensitivity of experimental design in future SNARC experiments, or, more generally when regression slopes are estimated intra-individually.

The fractions SNARC revisited: Processing fractions on a consistent mental number line

The ability to understand fractions is key to establishing a solid foundation in mathematics, yet children and adults struggle to comprehend them. Previous studies have suggested that these struggles emerge because people fail to process fraction magnitude holistically on the mental number line (MNL), focusing instead on fraction components. Subsequent studies have produced evidence for default holistic processing but examined only magnitude processing, not spatial representations. We explored the spatial representations of fractions on the MNL in a series of three experiments. Experiment 1 replicated Bonato et al.; 30 naïve undergraduates compared unit fractions (1/1-1/9) to 1/5, resulting in a reverse SNARC (Spatial-Numerical Association of Response Codes) effect. Experiment 2 countered potential strategic biases induced by the limited set of fractions used by Bonato et al. by expanding the stimulus set to include all irreducible, single-digit proper fractions and asked participants to compare them against 1/2. We observed a classic SNARC effect, completely reversing the pattern from Experiment 1. Together, Experiments 1 and 2 demonstrate that stimulus properties dramatically impact spatial representations of fractions. In Experiment 3, we demonstrated within-subjects reliability of the SNARC effect across both a fractions and whole number comparison task. Our results suggest that adults can indeed process fraction magnitudes holistically, and that their spatial representations occur on a consistent MNL for both whole numbers and fractions.

Mental Numerosity Line in the Human's Approximate Number System

Previous studies have demonstrated existence of a mental line for symbolic numbers (e.g., Arabic digits). For nonsymbolic number systems, however, it remains unresolved whether a spontaneous spatial layout of numerosity exists. The current experiment investigated whether SNARC-like (Spatial-Numerical Association of Response Codes) effects exist in approximate processing of numerosity, as well as of size and density. Participants were asked to judge whether two serially presented stimuli (i.e., dot arrays, pentagons) were the same regarding numbers of dots, sizes of the pentagon, or densities of dots. Importantly, two confounds that were overlooked by most previous studies were controlled in this study: no ordered numerosity was presented, and only numerosity in the approximate number system (beyond the subitizing range) was used. The results demonstrated that there was a SNARC-like effect only in the numerosity-matching task. The results suggest that numerosity could be spontaneously aligned to a left-to-right oriented mental line according to magnitude information in human's approximate number system.

Counteracting Implicit Conflicts by Electrical Inhibition of the Prefrontal Cortex

Cognitive conflicts and distractions by task-irrelevant information often counteract effective and goal-directed behaviors. In some cases, conflicting information can even emerge implicitly, without an overt distractor, by the automatic activation of mental representations. For instance, during number processing, magnitude information automatically elicits spatial associations resembling a mental number line. This spatial–numerical association of response codes (SNARC) effect can modulate cognitive-behavioral performance but is also highly flexible and context-dependent, which points toward a critical involvement of working memory functions. Transcranial direct current stimulation to the PFC, in turn, has been effective in modulating working memory-related cognitive performance. In a series of experiments, we here demonstrate that decreasing activity of the left PFC by cathodal transcranial direct current stimulation consistently and specifically eliminates implicit cognitive conflicts based on the SNARC effect, but explicit conflicts based on visuospatial distraction remain unaffected. This dissociation is polarity-specific and appears unrelated to functional magnitude processing as classified by regular numerical distance effects. These data demonstrate a causal involvement of the left PFC in implicit cognitive conflicts based on the automatic activation of spatial–numerical processing. Corroborating the critical interaction of brain stimulation and neurocognitive functions, our findings suggest that distraction from goal-directed behavior by automatic activation of implicit, task-irrelevant information can be blocked by the inhibition of prefrontal activity.

How Math Anxiety Relates to Number-Space Associations

Given the considerable prevalence of math anxiety, it is important to identify the factors contributing to it in order to improve mathematical learning. Research on math anxiety typically focusses on the effects of more complex arithmetic skills. Recent evidence, however, suggests that deficits in basic numerical processing and spatial skills also constitute potential risk factors of math anxiety. Given these observations, we determined whether math anxiety also depends on the quality of spatial-numerical associations. Behavioral evidence for a tight link between numerical and spatial representations is given by the SNARC (spatial-numerical association of response codes) effect, characterized by faster left-/right-sided responses for small/large digits respectively in binary classification tasks. We compared the strength of the SNARC effect between high and low math anxious individuals using the classical parity judgment task in addition to evaluating their spatial skills, arithmetic performance, working memory and inhibitory control. Greater math anxiety was significantly associated with stronger spatio-numerical interactions. This finding adds to the recent evidence supporting a link between math anxiety and basic numerical abilities and strengthens the idea that certain characteristics of low-level number processing such as stronger number–space associations constitute a potential risk factor of math anxiety.

The Spatial Representation of Angles

We investigated whether angle magnitude, similarly to numerical quantities (i.e., the spatial-numerical association of response codes effect), is associated to the side of response execution. In addition, we investigated whether this association has the properties of a spatially oriented mental line, since angles are taught in a right-to-left progression. We tested two groups of participants: civil engineering students (high familiarity with angles) and psychology students (low familiarity with angles). In Experiment 1, participants were asked to judge the continuity of the angles' arms (continuous vs. dashed). Magnitude of the angles was task-irrelevant. In Experiment 2, they were asked to judge whether the presented angles were smaller or larger than a right angle (90°). Therefore, the angle magnitude was relevant for performing the task. Overall, engineering students responded faster with their left hand to large angles and with their right hand to small angles. Conversely, psychology students did not show any reliable differences between left- and right-hand responses. In the case of engineering students, the spatial association has a right-to-left (counter clockwise) direction, suggesting the influence of education and practice on the mental representation of angle magnitude.

Numbers in the eye of the beholder: What do eye movements reveal about numerical cognition?

The eyes, often called the window to our minds, reveal the focus of spatial attention and are therefore a powerful research tool for the study of spatial processing and spatially related higher cognitive functions. The aim of this paper is to highlight the potential of eye movement analysis in the domain of numerical cognition, to review several relevant findings, and to provide an outlook for future research.

Math Anxiety Assessment with the Abbreviated Math Anxiety Scale: Applicability and Usefulness: Insights from the Polish Adaptation

Math anxiety has an important impact on mathematical development and performance. However, although math anxiety is supposed to be a transcultural trait, assessment instruments are scarce and are validated mainly for Western cultures so far. Therefore, we aimed at examining the transcultural generality of math anxiety by a thorough investigation of the validity of math anxiety assessment in Eastern Europe. We investigated the validity and reliability of a Polish adaptation of the Abbreviated Math Anxiety Scale (AMAS), known to have very good psychometric characteristics in its original, American-English version as well as in its Italian and Iranian adaptations. We also observed high reliability, both for internal consistency and test-retest stability of the AMAS in the Polish sample. The results also show very good construct, convergent and discriminant validity: The factorial structure in Polish adult participants (n = 857) was very similar to the one previously found in other samples; AMAS scores correlated moderately in expected directions with state and trait anxiety, self-assessed math achievement and skill as well temperamental traits of emotional reactivity, briskness, endurance, and perseverance. Average scores obtained by participants as well as gender differences and correlations with external measures were also similar across cultures. Beyond the cultural comparison, we used path model analyses to show that math anxiety relates to math grades and self-competence when controlling for trait anxiety. The current study shows transcultural validity of math anxiety assessment with the AMAS.

Professional mathematicians differ from controls in their spatial-numerical associations

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.

Task instructions determine the visuospatial and verbal-spatial nature of number-space associations

Evidence for number-space associations comes from the spatial-numerical association of response codes (SNARC) effect, consisting in faster reaction times to small/large digits with the left/right hand, respectively. Two different proposals are commonly discussed concerning the cognitive origin of the SNARC effect: the visuospatial account and the verbal-spatial account. Recent studies have provided evidence for the relative dominance of verbal-spatial over visuospatial coding mechanisms, when both mechanisms were directly contrasted in a magnitude comparison task. However, in these studies, participants were potentially biased towards verbal-spatial number processing by task instructions based on verbal-spatial labels. To overcome this confound and to investigate whether verbal-spatial coding mechanisms are predominantly activated irrespective of task instructions, we completed the previously used paradigm by adding a spatial instruction condition. In line with earlier findings, we could confirm the predominance of verbal-spatial number coding under verbal task instructions. However, in the spatial instruction condition, both verbal-spatial and visuospatial mechanisms were activated to an equal extent. Hence, these findings clearly indicate that the cognitive origin of number-space associations does not always predominantly rely on verbal-spatial processing mechanisms, but that the spatial code associated with numbers is context dependent.

Number-induced shifts in spatial attention: a replication study

In a spatial attention paradigm, Fischer et al. (2003) showed that merely perceiving a number shifted attention according to the magnitude of the number. Low numbers shifted attention to the left and high numbers shifted attention to the right. This suggests that numbers are represented by the mental number line – a spatial image schema that is ordered from left to right with increasing magnitude. In six experiments, we used the spatial attention paradigm of Fischer et al. (2003) to investigate if and when such mental representations are activated. Participants detected visual targets that were preceded by low and high numbers. Between experiments we manipulated how participants processed the number. Participants either merely perceived the number, as in the experiments by Fischer et al. (2003) processed the number’s parity, or processed the number’s magnitude. Our results provide little support for the idea that numbers shift spatial attention. Only in one of the two experiments in which participants processed number magnitude did participants respond faster to targets in congruent locations (left for low magnitudes and right for high magnitudes) than in incongruent locations. In the other five experiments number magnitude did not affect spatial attention. This shows, in contrast to Fischer et al.’s (2003) results, that the mental number line is not activated automatically but at best only when it is contextually relevant. Furthermore, these results suggest that image schemas in general may be context-dependent rather than fundamental to mental concepts.

The cognitive mechanisms of the SNARC effect: an individual differences approach

Access to mental representations of smaller vs. larger number symbols is associated with leftward vs. rightward spatial locations, as represented on a number line. The well-replicated SNARC effect (Spatial-Numerical Association of Response Codes) reveals that simple decisions about small numbers are facilitated when stimuli are presented on the left, and large numbers facilitated when on the right. We present novel evidence that the size of the SNARC effect is relatively stable within individuals over time. This enables us to take an individual differences approach to investigate how the SNARC effect is modulated by spatial and numerical cognition. Are number-space associations linked to spatial operations, such that those who have greater facility in spatial computations show the stronger SNARC effects, or are they linked to number semantics, such that those showing stronger influence of magnitude associations on number symbol decisions show stronger SNARC effects? Our results indicate a significant correlation between the SNARC effect and a 2D mental rotation task, suggesting that spatial operations are at play in the expression of this effect. We also uncover a significant correlation between the SNARC effect and the distance effect, suggesting that the SNARC is also related to access to number semantics. A multiple regression analysis reveals that the relative contributions of spatial cognition and distance effects represent significant, yet distinct, contributions in explaining variation in the size of the SNARC effect from one individual to the next. Overall, these results shed new light on how the spatial-numerical associations of response codes are influenced by both number semantics and spatial operations.