^{*}

^{a}

^{b}

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Evidence for number-space associations in implicit and explicit magnitude processing tasks comes from the parity and magnitude SNARC effect respectively. Different spatial accounts were suggested to underlie these spatial-numerical associations (SNAs) with some inconsistencies in the literature. To determine whether the parity and magnitude SNAs arise from a single predominant account or task-dependent coding mechanisms, we adopted an individual differences approach to study their correlation and the extent of their association with arithmetic performance, spatial visualization ability and visualization profile. Additionally, we performed moderation analyses to determine whether the relation between these SNAs depended on individual differences in those cognitive factors. The parity and magnitude SNAs did not correlate and were differentially predicted by arithmetic performance and visualization profile respectively. These variables, however, also moderated the relation between the SNAs. While positive correlations were observed in object-visualizers with lower arithmetic performances, correlations were negative in spatial-visualizers with higher arithmetic performances. This suggests the predominance of a single account for both implicit and explicit SNAs in the two types of visualizers. However, the spatial nature of the account differs between object- and spatial-visualizers. No relation occurred in mixed-visualizers, indicating the activation of task-dependent coding mechanisms. Individual differences in arithmetic performance and visualization profile thus determined whether SNAs in implicit and explicit tasks co-varied and supposedly relied on similar or unrelated spatial coding mechanisms. This explains some inconsistencies in the literature regarding SNAs and highlights the usefulness of moderation analyses for understanding how the relation between different numerical concepts varies between individuals.

An impressive number of studies on numerical cognition hint towards a potential link between numbers and mental space (e.g.,

For instance, the central display of a non-informative digit was shown to facilitate responses to stimuli in either the left or right hemifield depending on its magnitude (

Even though number-space associations have been extensively replicated in tasks with implicit and explicit magnitude processing, the cognitive mechanisms contributing to SNAs are highly debated. Up to date, three spatial coding accounts have been suggested to underlie spatial-numerical interactions, including a visuospatial, verbal-spatial, and working memory (WM) account.

The dominant and most traditional

An alternative view suggests that SNAs arise from categorical

A final explanation for the link between numbers and space was recently provided by

Considering the different spatial coding mechanisms proposed to account for number-space associations, the question arises whether only one of these accounts underlies spatial-numerical interactions regardless of the task or whether different spatial coding mechanisms might play a role depending on whether the task requires implicit or explicit processing of numerical magnitudes.

Several findings in the literature suggest that a single spatial coding mechanism predominates in both implicit and explicit magnitude processing tasks. For instance, number-space associations were shown to mainly arise from verbal-spatial coding mechanisms not only in the parity judgment (

In contrast, the hypothesis that different coding mechanisms might come into play depending on the task assessing number-space associations also receives robust support from the literature. For instance, we recently showed that the spatial mechanisms underlying number-space associations depended on contextual elements such as the task instructions (

To better understand which spatial coding mechanisms potentially contribute to number-space associations, several recent studies have adopted an individual differences approach by investigating how individual differences in SNAs can be explained by differences in other cognitive processes (e.g.,

Variability in the parity SNAs has for instance been related to individual differences in mathematical skills. Participants scoring lower in arithmetic measures displayed more pronounced number-space associations in the parity judgment task (

Interestingly, however, despite these findings associating individual differences in arithmetic and spatial skills with variability in the parity SNAs, corresponding investigations using explicit magnitude judgement tasks are lacking. Furthermore, to the best of our knowledge, there are currently no differential psychology studies examining whether individual differences in numerical and spatial factors can influence the extent to which number-space associations co-vary in implicit and explicit magnitude processing tasks.

Another concern is that spatial visualization style, a factor related to both arithmetic performance and mental rotation ability, has never been considered as a potential candidate for explaining individual differences in either parity or magnitude SNAs, let alone the extent of their covariance. Among the object and spatial visualization styles defined in the literature (

Considering the debate about the spatial nature of the coding processes underlying number-space associations and also the controversy about whether the activation of these mechanisms might depend on explicit or implicit magnitude processing, we first of all aimed to determine whether a significant correlation can be observed between SNAs in the parity judgment and magnitude classification tasks (

Secondly, we investigated to what extent number-space associations in implicit and explicit magnitude processing tasks can be explained by individual differences in numerical and spatial factors (

Finally, we used moderation analyses to investigate whether individual differences in the aforementioned numerical and spatial factors might not only explain differences in the strengths of the parity and magnitude SNAs, but could also determine the extent to which number-space associations in implicit and explicit magnitude processing tasks co-vary (

Overall, this study should advance our understanding of whether number-space associations in implicit and explicit magnitude processing tasks arise from a single account or multiple unrelated spatial coding mechanisms at the population level. Moreover, studying the extent to which the different number-space associations can be predicted by numerical and spatial factors will inform us about the cognitive mechanisms primarily contributing to each of the SNAs in the entire population. This will be especially informative with regards to magnitude SNAs, since their variability has never been investigated using individual differences in cognitive measures. Moreover, with the inclusion of visualization profile, we will extent previous findings about the relationships between number-space associations and arithmetic and spatial variables. Finally and most importantly, this is the first study using moderation analyses to investigate whether individual differences in cognitive variables can determine the relation between number-space associations in implicit and explicit magnitude processing tasks and thus supposedly the relatedness of their underlying spatial coding mechanisms. This should help clarify some of the inconsistencies in the literature regarding the spatial nature of the cognitive processes accounting for number-space associations.

The study was approved by the local Ethics Review Panel (ERP).

A total of 128 participants were recruited via advertisement through their university e-mail addresses, gave written informed consent and received 30€ for their participation. Half of the students came from study fields with a clear absence of explicit daily number and mathematics use (e.g., social and language studies), while the remaining participants all studied math-related subjects (e.g., mathematics, economics, or engineering).

All students were tested in the context of a larger project evaluating amongst others the effects of attention-deficit/hyperactivity disorder (ADHD) on number processing. However, since the focus of the present study was on healthy individuals, we did not consider the data of participants that were either diagnosed with ADHD (7 participants) or displayed symptoms consistent with ADHD according to the Adult ADHD Self-Report Scale-V1.1 (ASRS-V1.1) (30 participants). In addition to this, one participant had to be excluded due to a diagnosis of dyslexia. This reduced our initial sample to 90 students, of which none reported to have any learning difficulties and/or neuropsychological disorders.

For those 90 participants, outliers were identified for each of the measures included in the present study. A total of 9 participants had to be removed from the population sample, since their performances fell 2.5 standard deviations (

Participants were tested individually during two 90 min testing sessions. Sessions were run on separate days to prevent any possible effects of fatigue. The time difference between the two testing sessions was not fixed, so that students could sign up for the sessions according to their preferences (e.g., during their free-time on campus between two lectures). The upper limit of one week between testing sessions was implemented to avoid too much variability in the range of time differences between sessions across participants.

The present study was conducted in the context of a larger project, assessing amongst others the relation between number-space associations and math anxiety. The latter research findings have recently been published elsewhere (see

The design of the

The design of the

Data from the training sessions was not analyzed. The mean error rate on experimental trials was 2.83% and 2% in the parity judgment and magnitude classification tasks respectively (^{2}_{p} = .15). Errors were not further analyzed. Reaction times (RTs) shorter or longer than 2.5 ^{2}_{p} = .003).

SNARC regression slopes were computed using the individual regression equations method suggested by

In addition to the regression analysis, we also calculated correlations between dRTs and magnitude yielding individual SNARC effect sizes. To have normally distributed scores, Pearson’s r values were Fisher z-transformed. These

An important point worth considering here is that calculating dRTs for individual digits does not prevent a possible bias of parity status on lateralized RTs. This so-called MARC effect (reflecting faster left-/right-sided RTs for odd/even digits respectively, see ^{i}. As such, we collapsed RTs to an even and an odd digit separately for each response side and each participant and computed dRTs for each of the four resulting magnitude categories (i.e., very small [1, 2], small [3, 4], large [6, 7], and very large [8, 9],

To further analyze the pattern of dRTs and to test hypotheses regarding the shape of SNAs in the parity judgment and magnitude classification tasks (i.e., continuous vs. categorical shapes respectively), we performed stepwise multiple linear regression analyses on either the parity or magnitude dRTs including both linear and categorical magnitude predictors.

To assess reliability, we calculated split-half reliabilities for the unstandardized parity and magnitude SNARC regression slopes using the odd–even method to control for systematic influences of practice or tiring within the tasks. Trials were odd–even half-split (based on order of appearance) and two SNARC regression slopes were calculated separately for each participant in each task. The correlation coefficients were Spearman–Brown corrected to get a reliability estimate for the entire set of items. Spearman-Brown corrected correlation coefficients were

To determine whether low reliabilities (especially in the parity judgment task) might be due to the influence of outliers, we performed linear regression analyses between odd and even SNARC regression slopes and subsequently identified influential data points based on the conventional Cook’s distances criterion of > 4/

Split-half reliabilities were also calculated for the parity and magnitude SNARC effect sizes. Spearman-Brown corrected correlation coefficients were

We administered the untimed battery of arithmetic operations (

As in

We administered the 24-item mental rotations test (MRT-A;

Mental rotation skills were given by the number of items where both of the two rotated versions of the target figure were correctly identified (i.e., maximum score = 24). The mental rotations test was internally consistent with a Cronbach’s alpha of .87. This value is comparable to the ones reported in previous studies (e.g.,

We used the object spatial imagery questionnaire (OSIQ;

For each participant, average object and spatial scale scores were calculated. To allow for comparison between the two visualization styles, z-scores were computed for each scale (

All descriptive information can be found in

Variable | All participants |
---|---|

Gender (f/m) | 40/41 |

Age (years) | 23.38 (3.23) |

Handedness (r/l) | 77/4 |

Parity SNARC regression slope | -10.07 (12.82) |

Parity SNARC effect size | -.72 (1.04) |

Magnitude SNARC regression slope | -5.2 (13.1) |

Magnitude SNARC effect size | -.4 (1.14) |

ArithACC (%) | 92.04 (5.42) |

Mental rotation (score) | 13.23 (5.35) |

Visualization profile (z-score difference) | 0 (1.31) |

First of all, we conducted

We then performed two separate

Finally,

The mean ^{2}_{p} = .08), thus indicating stronger SNAs in the parity judgment than the magnitude classification task in terms of the inclination of the regression lines.

A main effect of task was also observed for the ^{2}_{p} = .05), with larger absolute values for mean SNARC effect sizes in the parity judgment (Fisher transformed z-score = -.72,

Significant correlations were observed between SNARC regression slopes and effect sizes for both the parity judgment (

Considering the shape of SNAs, only the continuous predictor accounted for variance in the parity dRTs when considering dRTs computed for individual digits (^{2} = .7, ^{2} = .92, ^{2} = .93, ^{2} = .96,

The correlation between the parity and magnitude SNARC regression slopes trended towards significance (

Cognitive variables | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1. Parity SNARC regression slope | - | ^{†} |
|||

2. Magnitude SNARC regression slope | .31 | - | ^{†} |
||

3. Arithmetic performance | .35 | .16 | - | ||

4. Spatial visualization ability | -.09 | -.04 | .32 | - | |

5. Visualization profile | -.03 | .23 | -.29 | -.30 | - |

^{†}

None of the two regression models reached significance as an overall model (parity SNARC regression slopes as DV: ^{2} = .09, ^{2} = .1,

In accordance with the correlation analyses outcomes, arithmetic performance and visualization profile either significantly predicted or trended towards being significant predictors of SNARC regression slopes in the parity judgment (

Model | ß | ||||
---|---|---|---|---|---|

Constant | -53.15 | 25.12 | -2.12 | .04 | |

Magnitude SNARC regression slope | 0.18 | 0.11 | 0.18 | 1.57 | .12 |

Arithmetic performance | 0.52 | 0.28 | 0.22 | 1.87 | .07 |

Spatial visualization ability | -0.27 | 0.28 | -0.11 | -0.98 | .33 |

Visualization profile | -0.33 | 1.16 | -0.03 | -0.29 | .78 |

^{2} = .09, adj. ^{2} = .04,

Regression analyses thus suggest that SNAs in implicit and explicit tasks rely on different cognitive mechanisms. However, at this point, we also have to consider the non-perfect reliabilities of some of the variables included in the regression models (notably the parity SNARC regression slopes). In multiple regression analysis, low reliabilities can lead to erroneous findings in that the risk of type II errors is increased for the predictors with poor reliability. Underestimation of the predictive validity of the variables with low reliability could then cause the overestimation of the effects of confounders in the regression models, thereby potentially manifesting in type I errors for those variables (

Model | ß | ||||
---|---|---|---|---|---|

Constant | -33.53 | 26.02 | -1.29 | .20 | |

Parity SNARC regression slope | 0.18 | 0.12 | 0.18 | 1.57 | .12 |

Arithmetic performance | 0.32 | 0.29 | 0.13 | 1.13 | .26 |

Spatial visualization ability | 0.04 | 0.29 | 0.02 | 0.15 | .89 |

Visualization profile | 2.38 | 1.15 | 0.24 | 2.06 | .04 |

^{2} = .1, adj. ^{2} = .05,

Considering that the correlation and multiple linear regression analyses failed to provide unequivocal evidence for a relation between number-space associations in the parity judgment and magnitude classification tasks, we tested whether the relationship between the two SNARC regression slopes could potentially be moderated by the cognitive factors also related to their respective strengths (i.e., arithmetic performance and visualization profile).

We therefore calculated interaction terms between the magnitude SNARC regression slopes and arithmetic performance as well as visualization profile. We then evaluated in separate models whether any of these interaction terms significantly predicted the parity SNARC regression slopes, while controlling for the variables included in the respective product terms.

The ^{2} = .07,

Simple moderation analysis results for arithmetic performance with unstandardized regression coefficients.

The ^{2} = .14,

Simple moderation analysis results for visualization profile with unstandardized regression coefficients.

Considering the moderating effects of arithmetic performance and visualization profile when tested in separate models, a ^{2} (^{2} = .14, ^{2} = .07,

Multiple additive moderation analysis results with unstandardized regression coefficients.

Visualization profile thus moderated the relationship between number-space associations in the parity judgment and magnitude classification tasks even after controlling for the moderating effect of arithmetic performance. Conversely, the addition of the magnitude SNARC regression slopes x arithmetic performance product term to the regression equation, which already included the interaction term between the magnitude SNARC regression slopes and visualization profile, only caused a non-significant 2% increase in the model’s capacity to predict variability in the parity SNAs (^{2} = .02,

The relationship between parity and magnitude SNARC regression slopes at the means and at one standard deviation below and above the means of arithmetic performance and visualization profile.

The present study aimed to determine whether number-space associations in tasks with implicit and explicit magnitude processing result from a single predominant spatial account or from multiple task-dependent spatial coding mechanisms. We adopted an individual differences approach to study the relation between parity and magnitude SNAs and the extent of their associations with arithmetic performance, spatial visualization ability and visualization profile at the population level. Additionally, we performed moderation analyses to determine whether the relation between number-space associations in implicit and explicit tasks depended on individual differences in the aforementioned cognitive factors.

A tendency for a positive correlation between the parity and magnitude SNARC regression slopes was observed at the population level, which is in line with the positive association between SNAs in implicit and explicit tasks recently reported by

On the other hand, the present results also showed that SNAs in the parity judgment and magnitude classification tasks were correlated with and predicted by different cognitive variables, namely arithmetic performance and visualization profile respectively. In addition, number-space associations in the magnitude classification task did not significantly predict spatial-numerical interactions in the parity judgment task (and vice-versa) when controlling for the effects of arithmetic performance, spatial visualization ability, and visualization profile. Those findings thus rather suggest that number-space associations in tasks with implicit and explicit magnitude processing arise from at least partially unrelated spatial coding mechanisms. This outcome would be in line with the principle component analysis of

Although interesting per se, the aforementioned correlation and regression analyses outcomes seem fairly inconclusive and provide somehow conflicting results with regard to whether number-space associations in implicit and explicit tasks result from a single predominant or multiple task-dependent spatial coding mechanism(s). On the one hand, a tendency for a correlation was revealed between the parity and magnitude SNAs, suggesting the activation of a single spatial coding process. Conversely, these two variables were each associated with different cognitive factors, thereby rather indicating task-dependent spatial coding mechanisms. One explanation for this discrepancy might be the non-perfect reliabilities of some of the variables included in this study, especially of the parity SNARC regression slopes. Low reliability usually results in the underestimation of bivariate correlations, consequently increasing the risk of type II errors. Moreover, it potentially biases unstandardized coefficients and/or leads to erroneous statistical significance levels in regression analysis (

Considering that the aforementioned analyses did not allow us to fully resolve inconsistencies in the literature regarding the cognitive origins of number-space associations, we finally performed moderation analyses to determine whether the relation between SNAs in implicit and explicit magnitude processing tasks and as such the relatedness of their underlying spatial coding mechanisms might be conditional upon individual differences in the cognitive factors also explaining individual variations in the strengths of number-space associations (namely arithmetic performance and visualization profile). Interestingly, the relation between the two SNAs was indeed moderated by visualization profile and arithmetic performance. This outcome thus sheds a completely new light onto the reasons why some studies provide evidence for the predominance of a single account (e.g.,

By investigating to what extent number-space associations in implicit and explicit magnitude processing tasks can be explained by individual differences in arithmetic performance, spatial visualization ability, and visualization profile, we not only attempted to shed further light onto the relatedness of the spatial coding mechanisms underlying the different SNAs, but also aimed to advance our understanding of the cognitive processes primarily contributing to each of the SNAs at the population level.

Considering number-space associations in the

In contrast to the parity SNAs, number-space associations in the

On the other hand, no relation was observed between magnitude SNAs and arithmetic performance. One possible explanation for this is that number-space associations in the magnitude classification task do not depend (or to a lesser extent) on executive control, which might mediate the relationship between the parity SNAs and arithmetic performance (see

Another point worth addressing here is that number-space associations were never affected by spatial visualization ability (i.e., mental rotation skills) regardless of the task. This observation is in accordance with the study of

An important point worth mentioning here is that moderation analysis assumes a causal relationship in that its application requires a causal theory and design behind the data (e.g., ^{rd} grade (

Moreover, we also need to bear in mind the non-perfect reliabilities of some of the variables included in this study, especially of the parity SNARC regression slopes. The relatively low reliability of this variable could for instance be explained by the small number of repetitions per digit for each response side (i.e., 9 repetitions), since a considerably higher split-half reliability of

On another note, the present study only determined how individual differences in numerical and spatial factors predicted variability in the parity and magnitude SNAs in the entire study population (i.e., comprising all types of individuals). An interesting idea for future research might thus be to investigate how arithmetic performance, spatial visualization ability and visualization profile relate to number-space associations in implicit and explicit magnitude processing tasks in either object-, spatial-, or mixed-visualizers. This should shed further light onto the spatial nature of the cognitive mechanisms contributing to spatial-numerical interactions in each of the different kinds of visualizers. One might for instance assume that number-space associations in both implicit and explicit magnitude processing tasks are predicted by the same cognitive variable in individuals where SNAs co-varied. However, considering that number-space associations co-varied positively and negatively in object- and spatial-visualizers respectively, the main cognitive predictor of the two SNAs should differ between the former and latter individuals.

Moreover, despite the fact that the present study provided evidence for the activation of visuospatial coding mechanisms in tasks with explicit magnitude processing, no assumptions can be made about the additional contribution of the WM account, since our analyses focused on the effects of numerical and spatial factors rather than executive control. To evaluate the WM account, one could for instance investigate how individual differences in (verbal and/or visuospatial) WM predict number-space associations in implicit and explicit magnitude processing tasks. This question could be addressed at the population level as well as in the different types of visualizers. Moreover, individual differences in (verbal and/or visuospatial) WM might be another factor moderating the relation between number-space associations in the parity judgment and magnitude classification tasks.

In addition to this, considering the effect of visualization profile on lower-level numerical processes, such as number-space associations during explicit magnitude judgments, one might wonder whether this cognitive factor also influences non-symbolic number comparisons or plays a role in the emergence of mathematical difficulties (e.g., dyscalculia). Furthermore, the effect of the verbal cognitive style might be examined especially with regards to the parity judgment task, given that parity SNAs are commonly assumed to arise from verbal-spatial coding mechanisms (e.g.,

Finally, since the spatial nature of the coding mechanisms underlying number-space associations was shown to depend on task instruction (

The present findings show that individual differences in visualization profile and arithmetic performance determined whether number-space associations in implicit and explicit magnitude processing tasks co-varied and supposedly relied on similar or unrelated spatial coding mechanisms. Significantly positive and negative associations between the parity and magnitude SNAs were observed in object-visualizers with lower arithmetic performance and spatial-visualizers with higher arithmetic performance respectively. These findings thus suggest the predominance of a single spatial coding account in both types of visualizers. The spatial nature of the account, however, differs between object- and spatial-visualizers. No association between the parity and magnitude SNAs was revealed in mixed-visualizers, suggesting the activation of task-dependent spatial coding processes. Moreover, arithmetic performance and visualization profile differentially related to the parity and magnitude SNAs respectively, suggesting the contribution of visuospatial coding mechanisms only to number-space associations in explicit but not implicit tasks (at least at the population level). Ideally, these results should, however, be confirmed by future studies with longer testing sessions allowing to obtain more reliable measurements.

Overall, this study helps explain some of the inconsistencies in the literature regarding the cognitive processes contributing to spatial-numerical interactions. It also highlights the usefulness of moderation analyses for unravelling how the relation between different numerical concepts varies between individuals, thereby potentially clarifying further inconsistencies in the numerical cognition literature.

The correlation between the parity and magnitude SNARC effect sizes was not significant (

Cognitive variables | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

1. Parity SNARC effect size | - | ||||

2. Magnitude SNARC effect size | .22 | - | |||

3. Arithmetic performance | .22 | .03 | - | ||

4. Spatial visualization ability | -.25 | -.05 | - | ||

5. Visualization profile | .36 | .36 | - |

**

None of the two regression models computed with the SNARC effect sizes reached significance as an overall model (parity SNARC effect sizes as DV: ^{2} = .09, ^{2} = .11,

Moreover, the parity SNARC effect sizes were not significantly predicted by any of the numerical or spatial factors included in the regression model (see

Model | ß | ||||
---|---|---|---|---|---|

Constant | -3.74 | 2.03 | -1.85 | .07 | |

Magnitude SNARC effect size | 0.04 | 0.11 | 0.04 | 0.35 | .73 |

Arithmetic performance | 0.04 | 0.02 | 0.19 | 1.64 | .11 |

Spatial visualization ability | -0.03 | 0.02 | -0.13 | -1.10 | .28 |

Visualization profile | 0.16 | 0.10 | 0.20 | 1.66 | .10 |

^{2} = .09; adj. ^{2} = .04;

Model | ß | ||||
---|---|---|---|---|---|

Constant | -2.26 | 2.21 | -1.02 | .31 | |

Parity SNARC effect size | 0.04 | 0.12 | 0.04 | 0.35 | .73 |

Arithmetic performance | 0.02 | 0.02 | 0.09 | 0.80 | .43 |

Spatial visualization ability | 0.01 | 0.03 | 0.04 | 0.31 | .76 |

Visualization profile | 0.29 | 0.10 | 0.33 | 2.83 | <.01 |

^{2} = .11; adj. ^{2} = .06;

The ^{2} = .01,

On the other hand, the ^{2} = .1,

This work was supported by the National Research Fund Luxembourg (FNR;

It is certified that there is no conflict of interest with any financial organization regarding the material discussed in the manuscript.

^{2}and G

^{2}reconsidered.

In the present study, a tendency for a main effect of parity status on dRTs was revealed for the parity judgment (^{2}_{p} = .04, odd dRT = 10.46 ms, even dRT = -19.9 ms), but not the magnitude classification task (^{2}_{p} = .0, odd dRT = -5.85 ms, even dRT = -6.28 ms), indicating the presence of a MARC effect in the former but not the latter task. The presence of a MARC effect only in the parity judgment task is clearly in line with predictions regarding this effect (

The authors have no support to report.