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

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.

Practice-induced SNARC: evidence from a null-SNARC sample

The mental representation of numbers inherently involves a spatial organization, often positioning smaller numbers to the left and larger numbers to the right. The SNARC effect, characterized by faster responses to small numbers using the left hand and vice versa for large numbers, is typically attributed to this left-to-right oriented mental number line (MNL). However, the direction of the SNARC effect seems to rely on reading direction, with most research exploring these mechanisms conducted within left-to-right reading cultures where the SNARC effect is prevalent. This study takes advantage of a sample from a left-to-right reading culture that does not exhibit the SNARC effect, allowing us to isolate and elucidate the stand-alone effects of recent experiences on SNARC. Therefore, the current study aims to investigate how MNL-compatible and MNL-incompatible practices induce an effect within a sample lacking the SNARC effect. To accomplish this, we reinvited the individuals from the sample which had previously shown no SNARC, and retested those who agreed to take part in the current study after an MNL-compatible or MNL-incompatible practice manipulation. The findings revealed an absence of the SNARC effect with MNL-compatible practices. Conversely, MNL-incompatible practices yielded a reverse SNARC effect. These results prompt a discussion on SNARC mechanisms within the framework of practice effects.

Numeracy, gist, literal thinking and the value of nothing in decision making

The onus on the average person is greater than ever before to make sense of large amounts of readily accessible quantitative information, but the ability and confidence to do so are frequently lacking. Many people lack practical mathematical skills that are essential for evaluating risks, probabilities and numerical outcomes such as survival rates for medical treatments, income from retirement savings plans or monetary damages in civil trials. In this Review, we integrate research on objective and subjective numeracy, focusing on cognitive and metacognitive factors that distort human perceptions and foment systematic biases in judgement and decision making. Paradoxically, an important implication of this research is that a literal focus on objective numbers and mechanical number crunching is misguided. Numbers can be a matter of life and death but a person who uses rote strategies (verbatim representations) cannot take advantage of the information contained in the numbers because 'rote' strategies are, by definition, processing without meaning. Verbatim representations (verbatim is only surface form, not meaning) treat numbers as data as opposed to information. We highlight a contrasting approach of gist extraction: organizing numbers meaningfully, interpreting them qualitatively and making meaningful inferences about them. Efforts to improve numerical cognition and its practical applications can benefit from emphasizing the qualitative meaning of numbers in context - the gist - building on the strengths of humans as intuitive mathematicians. Thus, we conclude by reviewing evidence that gist training facilitates transfer to new contexts and, because it is more durable, longer-lasting improvements in decision making.

Can a Common Magnitude System Theory Explain the Brain Representation of Space, Time, and Number?

Space, time, and number are important parts of our experiences and they are crucial for maintaining our behaviors in daily life. Comprehending the spatial and numerical features of our environment and perceiving and constructing the temporal framework are critical for healthy cognitive functioning and also survival. Although the problem of how these three perceptual processes work was initially studied separately, the emergence of behavioral interactions between these perceptions led to the idea that they could be run by a "common system". Besides the behavioral interactions for space, time, and number perception, the lesion and neuroimaging studies investigating the neural basis of these perceptions suggest the existence of a common size perception system represented in a fronto-parietal network formed around the intraparietal sulcus. However, on the other side of the coin, there are different views proposed based on findings that contradict this common magnitude system theory. The purpose of this review is to evaluate suggested ideas together and to examine whether the representation of space, time, and number perception in the brain can be explained by a common magnitude system theory.

Spatial-numerical associations without a motor response? Grip force says 'Yes'

In numerical processing, the functional role of Spatial-Numerical Associations (SNAs, such as the association of smaller numbers with left space and larger numbers with right space, the Mental Number Line hypothesis) is debated. Most studies demonstrate SNAs with lateralized responses, and there is little evidence that SNAs appear when no response is required. We recorded passive holding grip forces in no-go trials during number processing. In Experiment 1, participants performed a surface numerical decision task ("Is it a number or a letter?"). In Experiment 2, we used a deeper semantic task ("Is this number larger or smaller than five?"). Despite instruction to keep their grip force constant, participants' spontaneous grip force changed in both experiments: Smaller numbers led to larger force increase in the left than in the right hand in the numerical decision task (500-700 ms after stimulus onset). In the semantic task, smaller numbers again led to larger force increase in the left hand, and larger numbers increased the right-hand holding force. This effect appeared earlier (180 ms) and lasted longer (until 580 ms after stimulus onset). This is the first demonstration of SNAs with passive holding force. Our result suggests that (1) explicit motor response is not a prerequisite for SNAs to appear, and (2) the timing and strength of SNAs are task-dependent. (216 words).

EXPRESS: Attention does not always help: the role of expectancy, divided, and spatial attention on illusory conjunctions

Humans have the subjective impression of a rich perceptual experience, but this perception is riddled with errors that might be produced by top-down expectancies or failures in feature integration. The role of attention in feature integration is still unclear. Some studies support the importance of attention in feature integration (Paul & Schyns, 2003), whereas others suggest that feature integration does not require attention (Humphreys, 2016). Understanding attention as a heterogeneous system, in this study we explored the role of divided (as opposed to focused - Experiment 1) attention, and endogenous-exogenous spatial orienting (Experiments 2 and 3) in feature integration. We also explored the role of feature expectancy, by presenting stimulus features that were completely unexpected to the participants. Results demonstrated that both endogenous and exogenous orienting improved feature integration while divided attention did not. Moreover, a strong and consistent feature expectancy effect was observed, demonstrating perceptual completion when an unexpected perceptual feature was presented in the scene. These results support the feature confirmation account (Humphreys, 2016), which proposes that attention is important for top-down matching of stable representations.

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.

What explains the relationship between spatial and mathematical skills? A review of evidence from brain and behavior

There is an emerging consensus that spatial thinking plays a fundamental role in how people conceive, express, and perform mathematics. However, the underlying nature of this relationship remains elusive. Questions remain as to how, why, and under what conditions spatial skills and mathematics are linked. This review paper addresses this gap. Through a review and synthesis of research in psychology, neuroscience, and education, we examine plausible mechanistic accounts for the oft-reported close, and potentially causal, relations between spatial and mathematical thought. More specifically, this review targets candidate mechanisms that link spatial visualization skills and basic numerical competencies. The four explanatory accounts we describe and critique include the: (1) Spatial representation of numbers account, (2) shared neural processing account, (3) spatial modelling account, and (4) working memory account. We propose that these mechanisms do not operate in isolation from one another, but in concert with one another to give rise to spatial-numerical associations. Moving from the theoretical to the practical, we end our review by considering the extent to which spatial visualization abilities are malleable and transferrable to numerical reasoning. Ultimately, this paper aims to provide a more coherent and mechanistic account of spatial-numerical relations in the hope that this information may (1) afford new insights into the uniquely human ability to learn, perform, and invent abstract mathematics, and (2) on a more practical level, prove useful in the assessment and design of effective mathematics curricula and intervention moving forward.

SNARC effect modulated by central executive control: revealed in a cue-based trisection task

People respond to small numbers faster with the left hand and respond to large numbers faster with the right hand, a phenomenon known as the Spatial-Numerical Association of Response Codes (SNARC) effect. Whether the SNARC effect originates from culturally determined long-term experience or the task-set-influenced temporary associations among spaces, locations, and numerical magnitudes in working memory (WM) is still controversial. In the present study, we used a trisection paradigm in which numbers were divided into three categories (small: 1, 2; middle: 4, 5, 6; and large: 8, 9) to explore whether the central executive control can modulate the SNARC effect. Participants were serially presented with a cue and a target number. The cue denoted a task rule, which informed participants to compare the target number with either 3 or 7. The cue was either switched or repeated across trials. We found that the SNARC effects were observed in the cue-switching condition. In the cue-repeat condition, the SNARC effect disappeared. These findings suggest that the SNARC effect is modulated by set-shifting-related central executive control in WM, supporting the view that the SNARC effect is WM-dependent.

Emotional SNARC: emotional faces affect the impact of number magnitude on gaze patterns

The Spatial-Numerical Association of Response Codes (SNARC) effect is characterised by a spatial cognitive representation of low numbers to the left side of space and high numbers to the right side of space. This effect has been found using a diversity of stimuli and experimental paradigms. However, the influence of emotional stimuli on this effect remains unclear. In this study, the SNARC effect is analysed in relation to pairs of emotional facial stimuli (happy-neutral, sad-neutral and happy-sad pairs). Gaze patterns of 151 participants were analysed when exposed to a free-viewing eye-tracking paradigm consisting of pairs of emotional faces preceded by small and large numbers. Replicating previous results, a standard SNARC effect was found independently of the emotional expressions of the faces (i.e., there was a significant linear trend of number magnitude in the frequency of first fixations of the gaze to the left side space). However, specific slope analyses revealed that the SNARC effect was influenced by the spatial position of each emotion presented in the emotional pairs. Specifically, the effect disappeared in happy-neutral trials, when the happy faces were allocated in the right position and also in happy-sad trials when two emotional stimuli were simultaneously displayed. The study revealed that the SNARC effect is sensitive to the spatial position of emotional stimuli which further adds to other known limits of the phenomenon. The limitations of the study and its implications in the area of cognition and emotion are also discussed.

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.

The cognitive mechanisms of the power-space associations: an individual differences approach

A power-space interaction, which denotes the phenomenon that people responded faster to powerful words with up cursor keys and faster to powerless words with down cursor keys, has been repeatedly found. In the present study, we took an individual differences approach to investigate how the power-space interaction is modulated by the spatial cognition. First, we found that the amplitude of power-space interaction was relatively stable within individuals across different stimuli. And, this individual difference in power-space interaction was correlated with the individual's spatial cognition, in such a way that participants with faster speed of mental rotation showed stronger power-space interactions. Our results shed new light on the cognitive mechanisms of the power-space associations by suggesting that spatial codes play an important role in the expression of such effect.

Factoring in the spatial effects of symbolic number representation

Spatial constituents of adult symbolic number representation produce effects of size-value congruity, Spatial Numerical Association of Response Codes (SNARC), and numerical distance. According to behavioral experiments, these effects belong to distinct processing stages. Yet, these effects evoke overlapping responses in both early and late Event Related Potentials (ERPs). To probe whether these overlaps indicate sharing of resources, all relevant stimulus and response conditions were factorially combined in a numerical value comparison task. To secure ERP validity, same numbers were compared against variable reference values. This design resulted in previously unobserved interactions in behavior but inhibited late ERP effects. All effects arose early in the P1 component (around 100 ms) and most showed hemispheric specificity. Independency of congruity and SNARC effects was observed, whereas SNARC and numerical distance were closely intertwined. Differences in hemispheric specificity, rather than stage-wise separation, were key to independence.

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.

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.

Relations between numerical, spatial, and executive function skills and mathematics achievement: A latent-variable approach

Current evidence suggests that numerical, spatial, and executive function (EF) skills each play critical and independent roles in the learning and performance of mathematics. However, these conclusions are largely based on isolated bodies of research and without measurement at the latent variable level. Thus, questions remain regarding the latent structure and potentially shared and unique relations between numerical, spatial, EF, and mathematics abilities. The purpose of the current study was to (i) confirm the latent structure of the hypothesized constructs of numerical, spatial, and EF skills and mathematics achievement, (ii) measure their unique and shared relations with one another, and (iii) test a set of novel hypotheses aimed to more closely reveal the underlying nature of the oft reported space-math association. Our analytical approach involved latent-variable analyses (structural equation modeling) with a sample of 4- to 11-year-old children (N = 316, M_age = 6.68 years). Results of a confirmatory factor analysis demonstrated that numerical, spatial, EF, and mathematics skills are highly related, yet separable, constructs. Follow-up structural analyses revealed that numerical, spatial, and EF latent variables explained 84% of children's mathematics achievement scores, controlling for age. However, only numerical and spatial performance were unique predictors of mathematics achievement. The observed patterns of relations and developmental trajectories remained stable across age and grade (preschool - 4th grade). Follow-up mediation analyses revealed that numerical skills, but not EF skills, partially mediated the relation between spatial skills and mathematics achievement. Overall, our results point to spatial visualization as a unique and robust predictor of children's mathematics achievement.

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.

There's a SNARC in the Size Congruity Task

The size congruity effect involves interference between numerical magnitude and physical size of visually presented numbers: congruent numbers (either both small or both large in numerical magnitude and physical size) are responded to faster than incongruent ones (small numerical magnitude/large physical size or vice versa). Besides, numerical magnitude is associated with lateralized response codes, leading to the Spatial Numerical Association of Response Codes (SNARC) effect: small numerical magnitudes are preferably responded to on the left side and large ones on the right side. Whereas size congruity effects are ascribed to interference between stimulus dimensions in the decision stage, SNARC effects are understood as (in)compatibilities in stimulus-response combinations. Accordingly, size congruity and SNARC effects were previously found to be independent in parity and in physical size judgment tasks. We investigated their dependency in numerical magnitude judgment tasks. We obtained independent size congruity and SNARC effects in these tasks and replicated this observation for the parity judgment task. The results confirm and extend the notion that size congruity and SNARC effects operate in different representational spaces. We discuss possible implications for number representation.

Clock Walking and Gender: How Circular Movements Influence Arithmetic Calculations

Starting from a rich body of evidence on the strict bidirectional relationship between numerical cognition and action processes, the present study aims at deepening the existing knowledge of the influence of body movement on arithmetic calculation. Numerous studies have shown that moving the body along the vertical or the horizontal axis could facilitate calculations such as additions and subtractions. More specifically, results showed an effect of congruence between the type of operation (additions vs. subtractions) and the direction of the movement performed (up/right or down/left). While this congruence effect is present for both additions and subtractions when the axis of action is vertical, when the axis of action is horizontal, the effect appears only for additions. The purpose of this study is to investigate the influence of circular motion, which has so far not been explored, on counting. Participants were asked to count by adding or subtracting “three,” while performing a circular motion (i.e., a clockwise or counterclockwise movement), in an active (i.e., walking) or passive mode (i.e., being pushed on a wheelchair). Results showed a congruence effect for additions calculated in the active modality and only for male participants. Implications of the results for theories of embodied cognition and for the debate on gender differences in mathematical skills are discussed in this paper.

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.

On the genesis of spatial-numerical associations: Evolutionary and cultural factors co-construct the mental number line

Mapping numbers onto space is a common cognitive representation that has been explored in both behavioral and neuroimaging contexts. Empirical work probing the diverse nature of these spatial-numerical associations (SNAs) has led researchers to question 1) how the human brain links numbers with space, and 2) whether this link is biologically vs. culturally determined. We review the existing literature on the development of SNAs and situate that empirical work within cognitive and neuroscientific theoretical frameworks. We propose that an evolutionarily-ancient frontal-parietal circuit broadly tuned to multiple magnitude dimensions provides the phylogenetic substrate for SNAs, while enculturation and sensorimotor experience shape their specific profiles. We then use this perspective to discuss educational implications and highlight promising avenues for future research.

The Association of Number and Space Under Different Tasks: Insight From a Process Perspective

We investigated the Spatial Numerical Association of Response Codes (SNARC) effect in 240 adults using a parity judgment and a magnitude classification task. Eight numbers from 1 to 9 except 5 were randomly presented one at a time, half of the participants were asked to compare these number with the target number 5 in the magnitude classification task; the other half of the participants were asked to judge whether these numbers were even or odd. It was called a phase when all eight numbers were tested; there were in total 16 phases. Detailed analyses of the changes in response times across the range of numbers and the chronological trend of the SNARC effect size over 16 phases estimated by a curvilinear regression model were reported. This phase-to-phase design and analyses provide an insight into the process of the SNARC effect in different tasks. We found that the SNARC effect emerged earlier and stayed more stable in magnitude classification task than in the parity task during the time course. Furthermore, the size of SNARC effect increased with time in the magnitude classification task, whereas it fluctuated up and down over time in the parity task. These findings indicate that the association of the number and space is dynamic and the process of the SNARC effect varies across tasks.

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.

Time dependency of the SNARC effect for different number formats: evidence from saccadic responses

In line with the suggestion that the strength of the spatial numerical association of response codes (SNARC) effect was time dependent, the aim of the present study was to assess whether the association strength depends on the processing time of numerical quantity and/or of the time to initiate responses. More specifically, we examined whether and how the SNARC effect could be modulated by number format and effector type. Experiment 1 compared the effect induced by Arabic numbers and number words on the basis of saccadic responses in a parity judgment task. Indeed, previous studies have shown that Arabic numbers lead to faster processing than number words. The results replicated the SNARC effect with Arabic numbers, but not with number words. Experiment 2 was similar to Experiment 1, but this time manual responses (i.e., responses far slower than saccadic ones) were recorded. A strong SNARC effect was observed for both number formats. Further analyses revealed a correlation between mean individual response times and the strength of the SNARC effect. We proposed that the initiation times for saccadic responses may be too short for the SNARC effect to appear, in particular with the written number format for which activation of magnitude takes time. We conclude in terms of time variations resulting from processing specificities related with number format, effector type and also individual reaction and processing speed.

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.

Switching between Multiple Codes of SNARC-Like Associations: Two Conceptual Replication Attempts with Anodal tDCS in Sham-Controlled Cross-Over Design

In societies with left-to-right reading direction, left-side vs. right-side behavioral decisions are faster for relatively small vs. large number magnitudes, and vice versa, a phenomenon termed Spatial-Numerical Associations of Response Codes (SNARC) effect. But also for non-numerical sequential items, SNARC-like effects were observed, suggesting a common neurocognitive mechanism based on the ordinal structures of both numbers and sequences. Modulation of prefrontal networks that are involved in providing spatial associations during cognitive behavior can contribute to elaborate their neuropsychological theoretical foundations. With transcranial direct current stimulation (tDCS) directed to the left prefrontal cortex, we recently showed that (i) cathodal tDCS can block the emergence of spatial-numerical associations and that (ii) anodal tDCS can reverse spatial associations of sequential order, most likely based on markedness correspondence. Two conceptual replication attempts of the latter reversal of space-order associations are presented in the current sham-controlled experiment, using either weekdays (Monday-Friday) or month names (January-December) as stimuli in the temporal order classification task. In addition, to control for possible influences of notation, number stimuli were presented as written German names (One-Five). We report on a successful modulation of spatial-numerical associations of response codes (SNARC) effects with month stimuli induced by anodal tDCS, but failed to observe the same reversal of SNARC effects for weekday stimuli. The former stimulation effect was orthogonal to the small anodal tDCS effect on written number words, which replicates the dissociation of SNARC effects for numbers vs. non-numerical sequences. Moreover, this result reinforces the hypothesis that the ordinal item and task structure was the source of dissociation (as opposed to verbal presentation). We suggest that the diverging results can be explained by the markedness correspondence account of spatial associations in a multiple coding framework. Left-hemispheric prefrontal excitation from anodal tDCS renders verbal markedness relatively more dominant, but this effect is not absolute. We discuss task contagion, study design, and individual differences in performance measures or tDCS response as possible contributors to systematic variation of the weights of multiple coding parameters for spatial-numerical associations.

Mirror-normal difference in the late phase of mental rotation: An ERP study

Mirror-normal letter discriminations are thought to require mental rotation in order to transform the rotated alphanumeric character into its canonical orientation. Moreover, out-of-plane rotation is likely to occur after in-plane rotation to fully normalize the mirror version before the final mirror-normal judgment. The so-called rotation-related negativity, which varies with orientation, is found in both ERPonset (averaged with respect to stimulus onset) and ERPRT (averaged with respect to response time), representing the involvement of mental rotation in both time windows. Additionally, the mean amplitude of ERPRT correlates with individual performance. We performed a comprehensive analysis of the mirror-normal differences in the early and late phases of mental rotation and deduced that out-of-plane rotation is more likely to occur in the late phase and interacts with both in-plane rotation and the decision-making process, as indicated by both behavioral and electrophysiological findings.

The effect of working memory load on the SNARC effect: Maybe tasks have a word to say

We investigated the effect of working memory load on the SNARC (spatial-numerical association of response codes) effect under different number judgment tasks (parity judgment and magnitude comparison), using a novel dual task. Instead of exerting load over the whole block of number judgment trials, in this dual task, number judgment trials were inserted into each interstimulus interval of an n-back task, which served as the working memory load. We varied both load type (verbal and spatial) and amount (1-load, 2-load, and 3-load). The results indicated that the SNARC effect disappeared even under the 1-load condition for a parity judgment, regardless of the type of load. However, during the magnitude comparison task, the SNARC effect increased with increasing load amounts under spatial load conditions; under verbal load conditions, the SNARC effect decreased with increasing amounts of load, and disappeared during the 3-load task. The difference between the parity and magnitude tasks was not attributable to the interval stimuli or task switching. These findings confirm that different spatial-numerical associations for comparing numerical magnitudes and judgments of parity have different needs with respect to working memory resources.

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.

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.

Which numbers do you have in mind? Number generation is influenced by reading direction

In Western participants, small numbers are associated with left and larger numbers with right space. A biological account proposes that brain asymmetries lead to these attentional asymmetries in number space. In contrast, a cultural account proposes that the direction of this association is shaped by reading direction. We explored whether number generation is influenced by reading direction in participants from a left-to-right (UK) and a right-to-left (Arab) reading culture. Participants generated numbers randomly while lying on their left and right side. The mean number generated by participants from a left-to-right reading culture was smaller when they lay on their left than on their right side, and the opposite was found for participants from a right-to-left reading culture. Asymmetries in number space observed in number generation are more compatible with a cultural than biological account.

SNARC Effect in Different Effectors

The SNARC (spatial numerical association of response codes) effect, indicating that subjects react faster to the left for small numbers and to the right for large numbers, is used as evidence for the idea that humans use space to organize number representations. Previous studies compared the SNARC effect across sensory modalities within participants and concluded modality independence. So far, it is unknown what sensory-to-motor mappings are involved in generating the SNARC effect and whether these mappings are identical for different effectors within subjects. Hence, we tested whether the SNARC effect is effector specific. Participants performed an auditory parity judgment task and responded with three different effectors: finger (button release), eyes (saccades), and arm (pointing). The SNARC effect occurred in each effector but varied in strength across the effectors. Across subjects, we found a significant correlation of SNARC strength for finger and arm responses suggesting the use of a shared sensory-to-motor mapping. SNARC strength did not correlate, however, between finger and eyes or arm and eyes. An additional statistical analysis based on conditional probabilities provided further evidence for SNARC-effector specificity. Taken together, our results suggest that the sensory-to-motor mapping is not as tight as it would be expected if the SNARC effect was effector independent.

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.