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Edited by: Firat Soylu, The University of Alabama, United States

Reviewed by: Maria Grazia Di Bono, Università degli Studi di Padova, Italy; Laura Elizabeth Thomas, North Dakota State University, United States

This article was submitted to Cognition, a section of the journal Frontiers in Psychology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

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

Knowledge about numbers and numerical concepts is acquired through interaction with the world around us (e.g.,

The theoretical account that explains the aforementioned phenomena, also referred to as

Importantly, there is evidence suggesting that associations between numbers and space can be influenced not only by bodily experiences with fingers but also the whole body (

Numerical magnitude has long been thought to be associated with physical space. This association can either be between numerical and physical extensions (e.g.,

The mental number line is assumed to be activated automatically whenever number magnitude information is processed (

One of the most well-known indicators for spatial-numerical associations is the SNARC effect (Spatial Numerical Association of Response Codes,

Another indicator of spatial-numerical associations is the relative numerical congruity effect (NCE) described by

Recently, spatial-numerical associations have received increasing research interest following numerous studies showing that they are associated with bodily movements (

Recent research suggests, however, that bodily movements that interact with spatial-numerical associations generalize from the hands to the whole body (e.g.,

Following the previous studies investigating interactions between numbers and the body, full-body movements have been used to not only measure spatial-numerical associations (

However, the specific working mechanisms of embodied spatial-numerical trainings are not yet fully understood. Previously,

While the success of full-body spatial-numerical trainings has been investigated and supported several times, the respective training studies also raised the question of whether it was indeed the combination of full-body movements and spatial presentation of training content that increased training effects, or whether either the spatially distributed presentation along a number line or the full-body response would have sufficed. The underlying working mechanisms of these trainings were first investigated in an experimental study by

They found that, at least in elementary school children, the strength of spatial-numerical associations was only influenced by response format, but not by the presentation of a number line. In particular, a SNARC effect was observed irrespective of response conditions, whereas the NCE was only observed in conditions requiring physical response movements. Thereby, physical response movements seemingly increased spatial-numerical associations, but only when magnitude processing was necessary as reflected by the NCE in magnitude comparison with a variable standard.

However, while

Measuring both SNARC effect and NCE and building directly on the study by

The extent of bodily movement was varied in three different response formats: Verbal, manual, and full-body responses. Although responses were all spatially oriented (to the left or right), we expected that active bodily movement should increase spatial-numerical associations, whereas they should be smaller in verbal responses as previously observed (

Also in line with previous work, we varied stimulus presentation. In embodied numerical training studies, activation of the mental number line was often additionally enhanced by presenting a number line along with the task (for overviews see

Finally, we were interested in whether response format and mode of stimulus presentation would interact in affecting spatial-numerical associations. Assuming that both presentation and response format impact spatial-numerical associations, there should be an additive effect on SNARC effect and NCE, with the strongest effects being present when a number line was presented and a full-body movement is required as the response.

Previous results also indicated that the SNARC effect might not only reflect spatial-numerical associations but also aspects of verbal coding. In turn, the SNARC effect should occur regardless of response format. However, the NCE might be more exclusively determined by spatial-numerical associations, which is why we expected it to increase steadily with the extent of the required response movement.

Prior to testing, we conducted an ^{∗}Power 3.1.9.2 (

Forty-five university students took part in the study. Out of these, five had to be excluded from the analysis due to missing data. In two cases, the voice key software did not recognize the participants’ voice onset correctly, and in three cases, technical difficulties lead to missing data files. Out of the remaining 40 participants (13 male; age:

To measure SNARC effect and NCE, we used two types of numerical comparison tasks. In both tasks, participants decided whether the magnitude of a target number was smaller or larger than a simultaneously presented comparison standard. To distinguish the target from the standard, the rectangle surrounding the standard was marked by additional cross-shaped lines (see Figure

Variation of response and visual presentation formats demonstrated on the magnitude classification of 4 vs. 2. Here, the number 2 is marked as the standard for comparison, so participants had to decide whether 4 is larger or smaller than 2.

The paradigms differed with respect to the comparison standard, which was fixed in the SNARC task (

In magnitude classification, numbers had to be compared to the fixed comparison standard 5 (see also ^{1}, they responded to the right for ‘smaller’ decisions and to the left for ‘larger’ decisions. The SNARC effect was then calculated by comparing the incongruent and congruent response direction condition (for a similar procedure see

In the magnitude classification task, 5 was used as the fixed standard and 1, 4, 6, 9 as targets. All numbers were presented at an equal frequency (each number 12 times per condition) and in random order.

In magnitude comparison, the comparison standard was varied like the comparison probe from trial to trial (see also

The standard was a flexible number in the range between 1 and 9 (excluding 5), and both numbers of a pair were always either smaller than 5 or larger than 5. We used all possible number pairs in the range from 1 to 9, which resulted in a total of 12 number pairs (smaller than 5: 1–2, 1–3, 1–4, 2–3, 2–4, 3–4; larger than 5: 6–7, 6–8, 6–9, 7–8, 7–9, 8–9). Each pair was presented eight times per condition, four of which as a congruent pairing (e.g., 2–4) and four as an incongruent pairing (e.g., 4–2), again in random order.

Participants were tested individually in a university lab. Each participant came in for two sessions that lasted approximately 55 min each. In each session, participants were given the opportunity to take a break in between the different response conditions. To keep experimental conditions as comparable as possible across response conditions, all tasks were presented by projecting them onto a wall in front of participants at a distance of 2.5 m. Tasks were programmed in Java Eclipse and ran on a standard notebook (Fujitsu Siemens Lifebook T 4010).

The three different response formats were implemented using three different types of response media. In the

In the manual response condition, participants were seated at a table and responded on an external numeric keypad with the index fingers of both hands. To avoid the numbers on the keypad interfering with the response and to help participants remember the correct response keys, circular stickers were placed on the keys to cover the numbers. Each trial started with the participant’s fingers on two adjacent keys of the keypad, located centrally in the second row from the bottom (keys ‘2’ and ‘3’) and marked with yellow stickers. To respond, participants had to press the key to the left (key ‘1’) or right (‘enter’ key) from the starting point of the respective index finger, which were marked with blue stickers (see also Figure

In the full-body response condition, we used a digital dance mat (Positive Gaming Impact Dance Pad^{2}) with fields arranged in a 3 × 3 layout. Participants responded by hopping from the central field to the right or left field of the dance mat depending on their decision. Both the external keypad and the dance mat were connected to the notebook via USB.

Visual presentation (number line or no number line) was varied by either presenting the comparison standard correctly placed on a number line (endpoints marked 0 and 10) with the to-be-compared number placed centrally above the number line or by presenting both numbers above each other without a number line (see Figure

The experimental manipulations resulted in a 2 × 3 × 2 design for both tasks. For magnitude classification (fixed standard), the factors were response direction (SNARC compatible/incompatible), response format, and presentation format. For magnitude comparison (variable standard), the factors were congruity, response format, and presentation format. Half of the participants started with magnitude classification (fixed standard), while the other half started with magnitude comparison (variable standard). The order of permutations of the factors was balanced between participants. To this end, we generated 2 × 3 × 2 different task sequences and randomly assigned participants to one of them.

In each of the two tasks, participants completed 576 trials. These trials were presented in 12 blocks of 48 trials in magnitude classification (2 response directions, 3 response formats, and 2 presentation formats), and in 6 blocks of 96 trials in magnitude comparison (3 response formats and 2 presentation formats). Note that response direction was varied in blocks in the magnitude classification task. However, congruity in the comparison task was not blocked and varied on a trial by trial basis, as it was determined by the relationship between the presented magnitudes and thus not dependent on a change of response direction.

Prior to analyses, any response times (RT) below or above 3 standard deviations of each participant’s individual mean and all RT faster than 200 ms were removed to control for outliers. Only RT for correct responses were analyzed. RT for magnitude classification (fixed standard) and magnitude comparison (variable standard) were then entered into a within-subject repeated measures design. We conducted separate 2 × 3 × 2 (2

Datasets are available on request.

Overall, participants were faster in the magnitude classification task with a fixed standard (^{3}

Figure

Mean effects of spatial-numerical associations in each condition in the magnitude classification task (fixed standard) measuring the SNARC effect

Analyses revealed a significant overall SNARC effect as indicated by a main effect of response direction _{fullbody} = 1165 ms vs. RT_{manual} = 708 ms vs. RT_{verbal} = 773 ms. The full-body movement condition led to slower responses than both the manual and the verbal condition, and responses in the manual condition were faster than in the verbal condition.^{4}

Number line presentation also yielded a main effect of RT, as responses were slower when a number line was presented than when it was not, _{nl} _{presented} = 889 ms vs. RT_{nonl} = 875 ms.

Only the interaction between response direction and presentation format was significant,

No other interactions reached significance (all

Because we had hypothesized that the SNARC effect should differ between the response formats, but found no significant interaction between response direction and response format to support this hypothesis, we followed up the ANOVA with a Bayesian analysis. This analysis tested the alternative hypothesis that there should be an interaction against the null hypothesis of no interaction between response direction and response format. Using the SPSS_BAYES_ANOVA expansion pack for SPSS Statistics 25.0 (

In magnitude comparison, we observed no overall significant NCE as indicated by a non-significant main effect of congruity, _{congruent} = 1087 ms vs. RT_{incongruent} = 1095 ms. Because we had expected a main effect of congruity, we followed this up with a Bayesian analysis. This analysis (alternative/null) revealed that the data were 0.38:1 in favor of the null hypothesis, or 2.62 times more likely under the null than under the alternative hypothesis. This presents only anecdotal evidence in favor of the null hypothesis that there is no overall NCE in the data (e.g.,

As in the magnitude classification task with a fixed standard, there was a main effect of response format, _{fullbody} = 1369 ms vs. RT_{manual} = 935 ms vs. RT_{verbal} = 969 ms. Responses on the dance mat were slower than manual and verbal responses, whereas verbal and manual response condition did not differ in response speed. Presentation format did not yield a main effect,

However, congruity interacted significantly with presentation format,

Furthermore, the interaction between response format and presentation format was significant,

No other interactions reached significance (all

For the first time, the current study investigated the interplay of response and presentation formats for spatial-numerical associations in adult participants. Following up on previous developmental studies (

Most notably, there were no differences in the strengths of SNARC effect and NCE in the three response conditions. However, unlike in children, adult participants were influenced by the presentation of a number line along with the task, which was not always beneficial. We discuss the theoretical impact of these findings in the following.

In line with previous work (

Regarding the NCE, the picture was more inconsistent, as it was not observed overall, but only when no number line was presented. However, overall RT did not differ depending on number line presentation. A closer inspection of the marginal means revealed that participants performed at roughly the same speed whenever a trial was incongruent (with NL: 1093 ms; without NL: 1098 ms); and even on congruent trials when a number line was presented (1101 ms). However, when a congruent trial was presented without a number line, response speed increased (1073 ms). Accordingly, the absence of a number line seemed to help participants to solve congruent trials faster. This finding might either indicate that participants did not refer to any spatial-numerical directional representation for solving the congruent trials, and therefore benefited from not having to process redundant visual information. Alternatively, participants might in general rely more on their internal mental number line for the magnitude comparison task, potentially ‘zooming in’ on the relevant section of the number line (i.e., 0–5 when comparing 2 and 4), and can do so more efficiently when they do not have to inhibit an externally presented number line of a non-fitting larger range (i.e., 0–10). However, in the latter case, this should also result in processing advantages for incongruent trials with no number line presentation, which was not supported by the data. Here, future studies would be desirable to further differentiate spatial and numerical aspects of the presentation format.

Another unexpected finding was the interaction between presentation and response format in the magnitude comparison task measuring the NCE. Here, we observed that when participants responded with their entire body, number line presentation led to faster responses compared to a presentation without a number line. However, the opposite was observed for verbal and manual responses, which were descriptively slower with than without number line presentation. While unexpected, this result fits in with previous explanations for why embodied numerical trainings for children have been efficient in the past. For example,

Previous studies implementing embodied spatial-numerical trainings suggested that combining spatial-numerical presentation (e.g., a number line) with full-body spatial responses could increase training success (for overviews see

Within the context of spatial-numerical trainings, the differences in the findings for children and adults might mean that the relevance of each training component (response and presentation) may vary depending on the age of the participants. This possible effect of age should be taken into account when designing future trainings, as older participants might not benefit from an embodied spatial-numerical training in the same way that the young children in previous studies did. To this point, studies on embodied numerical trainings and their underlying mechanisms have only been conducted with children from kindergarten up to fourth grade. It is possible that for children above this age, a full-body response format might not improve training gains, and a presentation of a number line could even hinder training progress. Considering our results, embodied numerical trainings might not even be effective at all for adult participants. However, seeing as the idea behind embodied spatial-numerical trainings is mostly to convey basic numerical competencies, these trainings are not targeted at adult participants. Future studies will be necessary to determine the age at which a full-body response might no longer be adequate. In this vein, longitudinal studies testing the effects of different types of spatial-numerical trainings throughout childhood development would also be informative.

The current study builds on a previous experimental study conducted by

Another aspect to be considered is that task difficulty was possibly not comparable for children and adults. Although the study by

A promising avenue for future research could be to test participants across different age groups, while also combining an experimental approach such as the one implemented in the current study with different types of spatial-numerical trainings. Firstly, comparing different age groups within the same paradigm would be informative with regard to what type of training would be most beneficial at what age. Secondly, by measuring spatial-numerical associations before and after trainings, the relevance of the SNARC effect and NCE as measures of spatial-numerical associations may be further clarified. Furthermore, in case these trainings were to vary in whether they include only a spatial response, a spatial presentation, or both, the effect of each training component on spatial-numerical associations could be distinguished more clearly.

The present findings indicated that adult participants, unlike children, show stable spatial-numerical associations that are independent of the effector with which a task was performed. This suggests that in adults, the strength of spatial-numerical associations is no longer as strongly associated with bodily experiences. Accordingly, while full-body numerical trainings are beneficial for young children, it is possible that trainings for older participants need to take a different approach.

Contrary to previous results of studies with children, visual presentation seemed to play more of a role in adults. However, it was mostly interfering, suggesting that adults’ magnitude representations are either (1) more abstract (see e.g.,

This study was carried out in accordance with the recommendations of the ‘Ethische Richtlinien der Deutschen Gesellschaft für Psychologie e.V.’ (Ethical guidelines of the German Psychological Society) with written informed consent from all subjects. All subjects gave written informed consent in accordance with the Declaration of Helsinki. The protocol was approved by the ‘Lokale Ethikkommission am IWM’ (Local ethics committee at KMRC) in Tuebingen, Germany.

UF, SH, H-CN, UC, and KM conceptualized the study and designed the experiment. SH programmed the experiment. UF conducted the study, analyzed the data, and wrote the first draft of the manuscript. UF, H-CN, UC, and KM wrote the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at:

^{∗}Power 3.1: tests for correlation and regression analyses.

Note that we tested German participants, who grow up with a left-to-right reading/writing direction and therefore in their majority associate larger numbers with the right side and small numbers with left. Congruity might be defined differently in participants from right-to-left reading/writing cultures.

Comparisons between left- and right-handed participants revealed no differences depending on handedness in SNARC effect [

Because previous studies (see