The development of the SNARC effect: Evidence for early verbal coding

Children grow upwards, and so does the number line: Evidence from a directional number line paradigm

Technological advancements give researchers the opportunity to explore the internal metric that allows to mentally place numbers in a spatial and ordered way to establish relationships between quantities. In this study, we implement the cMNL, an embodied number line paradigm to investigate the configuration of children's number space mappings under multiple conditions. A sample of 185 primary school children aged 8-10years old completed digitally an embodied number line task encompassing directionality and modality as variables. Contrary to the premise of a fixed internal number line moving from left to right in many Western scripts, our results suggest that children's number-space mapping is more robust along a vertical axis. In addition, children's embodied number line estimation differed depending on input modality. The findings provide insight into the variability in children's number line estimation, and the usability of digital assessment in understanding the mechanisms of the developing number-space system.

Children's spatial-numerical associations on horizontal, vertical, and sagittal axes

There is substantial evidence linking numerical magnitude to the physical properties of space. The most influential support for this connection comes from the SNARC effect (spatial-numerical association of response codes), in which responses to small/large numbers are faster on the left/right side of space, respectively. The SNARC effect has been extensively replicated, and is understood as horizontal mapping of numerical magnitude. However, much less is known about how numbers are represented on the vertical and sagittal axes, and whether spatial-numerical associations on different axes emerge during childhood. To that end, we tested two groups of children, aged 5-7 years and 8 and 9 years, on a single-digit magnitude comparison task with response buttons positioned either upper/lower (vertical), left/right (horizontal) or near/far (sagittal). Our results provide evidence of spatial-numerical mapping on all three axes for both age groups that are similar in strength. This indicates that, even at an early stage of formal education, children can flexibly assign numerical magnitude to any spatial dimension. To examine the contribution of extracorporeal space and spatio-anatomical mapping to the SNARC effect across axes, these sources were pitted against each other by swapping the position of the response hands in Experiment 1b. Switching hand position did not reveal convincing evidence for SNARC effects on any axis. Results are discussed with respect to the utility of three-dimensional mental number lines, and potential avenues for future research are outlined.

A purely power-space association without spatial and strategic biases

Previous studies have shown that powerful and powerless concepts are metaphorically associated with top and bottom spaces respectively. However, this association might be contaminated by spatial and strategic biases due to the involvement of spatialized stimuli or responses. It is unknown whether power by itself can automatically activate spatial representations. To eliminate spatial and strategic biases, Experiment 1 separately presented power and spatial stimuli at the center of the screen, and participants had to classify power words and HIGH/LOW labels (Experiment 1a) or indicate up/down arrows (Experiment 1b) using a single key and follow different response rules that were combined with a Go/no-go task. Experiment 2 replicated the non-spatialized design but used an implicit power judgment task. Based upon the non-spatialized method, the results provide the first evidence that a power-space association effect still exists when eliminating spatial and strategic biases, revealing the nonconscious spatial nature of power.

The importance of visuospatial abilities for verbal number skills in preschool: Adding spatial language to the equation

Children's verbal number skills set the foundation for mathematical development. Therefore, it is central to understand their cognitive origins. Evidence suggests that preschool children rely on visuospatial abilities when solving counting and number naming tasks despite their predominantly verbal nature. We aimed to replicate these findings when controlling for verbal abilities and sociodemographic factors. Moreover, we further characterized the relation between visuospatial abilities and verbal number skills by examining the role of spatial language. Because spatial language encompasses the verbalization of spatial thinking, it is a key candidate supporting the interplay between visuospatial and verbal processes. Regression analysis indicated that both visuospatial and verbal abilities, as assessed by spatial perception and phonological awareness, respectively, uniquely predicted verbal number skills when controlling for their respective influences, age, gender, and socioeconomic status. This confirms the spatial grounding of verbal number skills. Interestingly, adding spatial language to the model abolished the predictive effects of visuospatial and verbal abilities, whose influences were completely mediated by spatial language. Verbal number skills thus concurrently depend on specifically those visuospatial and verbal processes jointly indexed through spatial language. The knowledge of spatial terms might promote verbal number skills by advancing the understanding of the spatial relations between numerical magnitudes on the mental number line. Promoting spatial language in preschool thus might be a successful avenue for stimulating mathematical development prior to formal schooling. Moreover, measures of spatial language could become an additional promising tool to screen preschool children for potential upcoming difficulties with mathematical learning.

Investigating the respective contribution of sensory modalities and spatial disposition in numerical training

Recent studies have suggested that multisensory redundancy may improve cognitive learning. According to this view, information simultaneously available across two or more modalities is highly salient and, therefore, may be learned and remembered better than the same information presented to only one modality. In the current study, we wanted to evaluate whether training arithmetic with a multisensory intervention could induce larger learning improvements than a visual intervention alone. Moreover, because a left-to-right-oriented mental number line was for a long time considered as a core feature of numerical representation, we also wanted to compare left-to-right-organized and randomly organized arithmetic training. Therefore, five training programs were created and called (a) multisensory linear, (b) multisensory random, (c) visual linear, (d) visual random, and (e) control. A total of 85 preschoolers were randomly assigned to one of these five training conditions. Whereas children were trained to solve simple addition and subtraction operations in the first four training conditions, story understanding was the focus of the control training. Several numerical tasks (arithmetic, number-to-position, number comparison, counting, and subitizing) were used as pre- and post-test measures. Although the effect of spatial disposition was not significant, results demonstrated that the multisensory training condition led to a significantly larger performance improvement than the visual training and control conditions. This result was specific to the trained ability (arithmetic) and is discussed in light of the multisensory redundancy hypothesis.

Compatibility between response position and either object typical size or semantic category: SNARC- and MARC-like effects in primary school children

In two experiments, we administered two binary classification tasks to third- to fifth-grade students and found that either the conceptual size (Experiment 1) or semantic category (Experiment 2) of the target picture interacted with response position: Participants emitted faster and/or more accurate right- and left-side responses to stimuli referring to typically large and small objects (i.e., a SNARC [spatial-numerical association of response codes]-like effect) or to living and nonliving objects (i.e., a MARC [markedness association of response codes]-like effect), respectively. The effect of either stimulus dimension was present only when this dimension was the task-relevant one. These findings may be accounted for by the polarity compatibility principle. In binary classification tasks, one of the two relevant stimulus values (i.e., the dominant one) and one of the two response values would be coded as having a positive polarity, whereas the other would be coded as having a negative polarity. Response selection would be faster and/or more accurate when stimulus and response polarities are compatible. According to a less parsimonious hypothesis, the polarity principle would only explain the effect of the stimulus semantic category in Experiment 2, whereas the effect of stimulus conceptual size in Experiment 1 might be traced back to the fact that children spatially code the typical size of stimulus referents (even if only when the task requires accessing this stimulus dimension). Small and large objects would be represented relatively on the left and right, respectively, which would result in faster and/or more accurate responses when response position is compatible with the stimulus referent's position in this spatial representation.

Development of number-space associations: SNARC effects and spatial attention in 7- to 11-year-olds

The spatial numerical association of response codes effect, referred to as the SNARC effect, reveals that small numbers elicit faster left than right responses, and conversely, large numbers elicit faster right responses. Here, we explored the development of this number-space association by assessing how 7-, 9-, 11-year-olds, and adults differed in spatial orienting of attention on Posner' paradigm. Compared with the previous research, we examined how the cues would affect the level and strength of the SNARC effect in children under the different attentional conditions. Subjects made parity decisions for endogenous attention (Experiment 1) and exogenous attention (Experiment 2). The results showed that adults displayed the SNARC effect in both experiments, relatively speaking, 7- to 11-year-old Chinese children's ability of spatial numerical association progressed gradually. With endogenous attention, the SNARC effect appeared in all age groups except for 7-year-olds for invalid cues. Compared with the endogenous attention condition, the SNARC effect was more significantly affected by cues in the exogenous attention condition. This result might be owing to the fact that the SNARC effect was not demonstrated in 7-year-olds with either valid or invalid cues. Our results suggest that the differences in the spatial orienting of attention are based on the cognitive load associated with processing number information and that this process can be affected by cues. Further, there may be cross-cultural influences on the SNARC effect, as early family training may explain the results seen in this sample of Chinese 7-year-olds. Thus, reaction times decreased with increasing age in the parity judgment task, and reaction times for valid cues were faster than for invalid cues regardless of the age group in both experiments. The SNARC effect was only present for 7-year-olds for valid cues, for endogenous attention.

Spatial Presentations, but Not Response Formats Influence Spatial-Numerical Associations in Adults

According to theories of embodied numerosity, processing of numerical magnitude is anchored in bodily experiences. In particular, spatial representations of number interact with movement in physical space, but it is still unclear whether the extent of the movement is relevant for this interaction. In this study, we compared spatial-numerical associations over response movements of differing spatial expansion. We expected spatial-numerical effects to increase with the extent of physical response movements. In addition, we hypothesized that these effects should be influenced by whether or not a spatial representation of numbers was presented. Adult participants performed two tasks: a magnitude classification (comparing numbers to the fixed standard 5), from which we calculated the Spatial Numerical Association of Response Codes (SNARC) effect; and a magnitude comparison task (comparing two numbers against each other), from which we calculated a relative numerical congruity effect (NCE), which describes that when two relatively small numbers are compared, responses to the smaller number are faster than responses to the larger number; and vice versa for large numbers. A SNARC effect was observed across all conditions and was not influenced by response movement extent but increased when a number line was presented. In contrast, an NCE was only observed when no number line was presented. This suggests that the SNARC effect and the NCE reflect two different processes. The SNARC effect seems to represent a highly automated classification of numbers as large or small, which is further emphasized by the presentation of a number line. In contrast, the NCE likely results from participants not only classifying numbers as small or large, but also processing their relative size within the relevant section of their mental number line representation. An additional external presentation of a number line might interfere with this process, resulting in overall slower responses. This study follows up on previous spatial-numerical training studies and has implications for future spatial-numerical trainings. Specifically, similar studies with children showed contrasting results, in that response format but not number line presentation influenced spatial-numerical associations. Accordingly, during development, the relative relevance of physical experiences and presentation format for spatial-numerical associations might change.

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.

Flexible Verbal–Spatial Mapping in the Horizontal and Vertical SNARC Effects of Mainland Chinese Readers

Traditionally, the spatial–numerical association of response codes (SNARC) effect was explained by assuming that the processing of numbers automatically activates internal visuospatial codes. In contrast, Gevers et al. (2010) suggested that the SNARC effect can result from both visuospatial and verbal–spatial processing, with verbal–spatial being the dominant processing mechanism. The current study tested this assumption in two experiments with the SNARC effects obtained from the horizontal and the vertical directions. Consistent with the assumption of Gevers et al., the horizontal SNARC effect is dominantly driven by verbal–spatial coding in Arabic numerals and simplified and complex Chinese numbers. More specifically, in the horizontal direction, mainland Chinese participants more dominantly associate small numbers with the verbal–spatial “left” and large numbers with the verbal–spatial “right.” But inconsistent with key predictions of Gevers et al., a remarkable association was found between small numbers with the visuospatial “top” and large numbers with the visuospatial “bottom” in the vertical SNARC effect for numbers in Arabic numerals and simplified Chinese numbers and a dominant association between small numbers with the verbal–spatial coding “bottom” and large numbers with the verbal–spatial coding “top” for complex Chinese numbers, also in the vertical SNARC effect. These findings demonstrate that verbal–spatial coding of numbers is flexible, which is plausible because of the direction-dissociative and notation-specific nature of number–space mapping.

Verbal-Spatial and Visuospatial Coding of the SNARC Effect: A Comparative Study in Cultures With Different Reading Directions

Using the magnitude comparison task, the current study tested verbal-spatial and visuospatial coding accounts by comparing the spatial-numerical association of response codes effect in 30 Chinese-speaking and 60 bilingual Uighur Chinese participants. The experimental tasks were presented using Chinese words for 30 Chinese-speaking and 30 bilingual Uighur Chinese participants, while only Uighur words were presented to the remaining 30 bilingual Uighur Chinese participants. Overall, the results of the current study showed that verbal-spatial coding could be applied to the experimental task when presented using Chinese words, but visuospatial coding could be applied to the experimental task when presented using Uighur words. The findings demonstrated that spatial coding was flexible and depended upon the language context in which the experimental task was presented.

Finger counting habit and spatial-numerical association in children and adults

Sensory-motor experiences are known to build up concrete and abstract concepts during the lifespan. The present study aimed to test how finger counting habits (right-hand vs. left-hand starters) could influence the spatial-numerical representation in number-to-position (explicit) and digit-string bisection (implicit) tasks. The subjects were Italian primary school children (N=184, from the first to the fifth year) and adults (N=42). No general preference for right- or left-starting in the finger counting was found. In the explicit task, right- or left-starting did not affect performance. In the implicit task, the right-hand starters shifted from the left to the right space when bisecting small and large numbers respectively, while the left-hand starters shifted from the right to the left space with higher leftward bias for large numbers. The finger configuration in Italian children and adults influences the spatial-numerical representation, but only when implicit number processing is required by the task.

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.

Comparing Implicit and Synaesthetic Number–Space Associations: Visuospatial and Verbal Spatial–Numerical Associations of Response Codes

In the theory of the mental number line, number, and space are implicitly associated, giving rise to the spatial–numerical association of response codes (SNARC) effect, in which smaller numbers are more readily associated with the left side of space and larger numbers with the right, during a parity judgement task. Others, however, have argued that the SNARC effect is flexible and better explained by verbal rather than spatial associations. A few single-case studies on the SNARC effect have tested number–space synaesthetes, who make explicit associations between number and space. Here, we present data from experiments conducted on groups of synaesthetes and nonsynaesthetes on the classic SNARC parity judgement task with lateralized response keys and a modified version in which they responded to labels appearing on screen. Synaesthetes’ behaviour was expected to differ from nonsynaesthetes’ behaviour due to the explicit, fixed nature of their number–space associations, but both experiments show the two groups behaving in the same way, indicating that parity judgement tasks may not be tapping the same representation of number that gives rise to synaesthetic number–space experiences.

The impact of mathematical proficiency on the number-space association

A specific instance of the association between numerical and spatial representations is the SNARC (Spatial Numerical Association of Response Codes) effect. The SNARC effect describes the finding that during binary classification of numbers participants are faster to respond to small/large numbers with the left/right hand respectively. Even though it has been frequently replicated, important inter-individual variability has also been reported. Mathematical proficiency is an obvious candidate source for inter-individual variability in numerical judgments, but studies investigating its influence on the SNARC effect remain scarce. The present experiment included a total of 95 University students, divided into three groups differing significantly in their mathematical proficiency levels. Using group analyses, it appeared that the three groups differed significantly in the strength of their number-space associations in a parity judgment task. This result was further confirmed on an individual level, with higher levels in arithmetic leading to relatively weaker SNARC effects. To explain this negative relationship we propose accounts based on differences in access to qualitatively different numerical representations and also consider more domain general factors, with a focus on inhibition capacities.

Spatial representations of numbers and letters in children

Different lines of evidence suggest that children's mental representations of numbers are spatially organized in form of a mental number line. It is, however, still unclear whether a spatial organization is specific for the numerical domain or also applies to other ordinal sequences in children. In the present study, children (n = 129) aged 8–9 years were asked to indicate the midpoint of lines flanked by task-irrelevant digits or letters. We found that the localization of the midpoint was systematically biased toward the larger digit. A similar, but less pronounced, effect was detected for letters with spatial biases toward the letter succeeding in the alphabet. Instead of assuming domain-specific forms of spatial representations, we suggest that ordinal information expressing relations between different items of a sequence might be spatially coded in children, whereby numbers seem to convey this kind of information in the most salient way.

Examining the presence and determinants of operational momentum in childhood

The operational momentum (OM) effect describes a systematic bias in estimating the outcomes of simple addition and subtraction problems. Outcomes of addition problems are overestimated while outcomes of subtraction problems are underestimated. The origin of OM remains debated. First, a flawed uncompression of numerical information during the course of mental arithmetic is supposed to cause OM due to linear arithmetic operations on a compressed magnitude code. Second, attentional shifts along the mental number line are thought to cause OM. A third hypothesis explains OM in 9-month olds by a cognitive heuristic of accepting more (less) than the original operand in addition (subtraction) problems. The current study attempts to disentangle these alternatives and systematically examines potential determinants of OM, such as reading fluency which has been found to modulate numerical-spatial associations. A group of 32 6- and 7-year-old children was tested in non-symbolic addition and subtraction problems, in which they had to choose the correct outcome from an array of several possible outcomes. Reading capacity was assessed for half of the children while attentional measures were assessed in the other half. Thirty-two adults were tested with the identical paradigm to validate its potential of revealing OM. Children (and adults) were readily able to solve the problems. We replicated previous findings of OM in the adults group. Using a Bayesian framework we observed an inverse OM effect in children, i.e., larger overestimations for subtraction compared to addition. A significant correlation between children's level of attentional control and their propensity to exhibit OM was observed. The observed pattern of results, in particular the inverse OM in children is hard to reconcile with the previously proposed theoretical frameworks. The observed link between OM and the attentional system might be interpreted as evidence partially supporting the attentional shift hypothesis.