Methods

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

5.3.1 Participants

A total of 68 children participated in this study. Pupils were recruited from the “3^rd cycle”

(corresponding to grade 3-4 of the German school system) of two different public elementary schools in Esch-sur-Alzette, Luxembourg. Parents’ informed consent was obtained prior to the start of the study, and all children participated voluntarily. Participants came from various backgrounds including different mother languages, of which Luxembourgish and Portuguese were the most common. Roughly two third of the participants had a single mother tongue, while the remaining children were exposed to more than one language at home.

Nonetheless, all participants had good comprehension of Luxembourgish and/or German.

None of the children suffered from any learning difficulties like dyscalculia, dyslexia, and/or dyspraxia.

Children for whom descriptive information was missing (N = 1) or who did not yet fully understand the concept of parity since they asked for assistance while attempting to complete the parity judgment task and/or committed more than 25% of errors on this task (N

= 10) were removed prior to data analyses. This reduced the study sample to 57 participants. Among these children, two were additionally excluded since their parity SNARC effects fell 2.5 standard deviations (SD) below or above the mean parity SNARC regression slope (for a similar exclusion procedure with adults, see Georges, Hoffmann, & Schiltz, 2016). All analyses were thus conducted on 55 healthy elementary school children for whom descriptive information is displayed in Table 1.

5.3.2 Procedure and tasks

The tasks were administered during two different testing sessions, which were run on separate days to prevent any possible effects of fatigue. The first testing session comprised paper-and-pencil tests and questionnaires administered collectively in class during approximately 120 min. The second testing session included computerized tasks (programmed in E-prime version 2.0 and administered using a Lenovo ThinkPad). These were completed collectively in groups of 5-6 children over approximately 60 min. The time between sessions depended on the teachers’ and children’s availabilities and was on average 5.96 weeks (SD = 3.66, range 1-11).

Since the present study was conducted in the context of a larger project, a battery of different tests and questionnaires was implemented during the two testing sessions. Only those experiments required to answer the current research questions will be mentioned here and described in more detail below. The Heidelberg Mathematics Test (Heidelberger Rechentest, HRT 1-4, Haffner, Baro, Parzer, & Resch, 2009) was administered during the first testing session, while the parity judgment task was performed during the second testing session. Explanations for both tasks were given in Luxembourgish and German.

5.3.2.1 The Heidelberg Mathematics Test

The Heidelberg Mathematics Test (Heidelberger Rechentest, HRT 1-4, Haffner et al., 2009) was used to assess mathematical competencies. It is a standardized speeded math test battery for primary school children in Germany, consisting of two subscales that evaluate different mathematical components. All subtests within each subscale started with a couple of practice items. After completion of the practice trials, children had 2 min to complete each of the subtests.

The arithmetical ability subscale comprises six subtests: mental addition (e.g., 17 + 15 = _), mental subtraction (e.g., 50 – 14 = _), mental multiplication (e.g., 6 x 7 = _), mental division (e.g., 28 ÷ 4 = _), number equations filling (e.g., 4 + _ = 3 + 7) and number comparison (e.g., 2 + 9 _ 20). Trials in each subtest were presented serially with an order of increasing difficulty.

The visuospatial ability subscale consists of five subtests. In the length estimation subtest, children were required to estimate the length (i.e., number of steps) of a series of two-dimensional black lines by comparing each line with three one-two-dimensional bolder black lines presented on the top of the test sheet corresponding to 1, 5 or 10 steps respectively. In the object counting subtest, children were instructed to count the number of small objects included in each of 21 presented frames. In the cubes counting subtest, children had to

sequences subtest, children had to complete number sequences (e.g., 1 2 1 2 1 2 _ _ _) by applying a rule established through deductive reasoning. In the connecting numbers subtest, numbers from 1 to 20 were randomly presented in each of 10 frames and participants were required to connect the numbers in their increasing order.

Data analysis and descriptive information

Children received one point for every correctly solved item. Sum scores of arithmetical and visuospatial abilities were then computed across all six arithmetical and five visuospatial subscales subtests respectively and expressed as percentage accuracies. Performances did not differ between the two ability subscales (F(1, 54) = 0.01, p = .95, ηp2 = .00, see Table 1).

5.3.2.2 The parity judgment task

The parity judgment task was used to assess number-space associations (i.e., the SNARC effect). Children were required to indicate whether a centrally presented single Arabic digit (1-9, excluding 5) was odd or even by pressing the “A” or “L” key on a QWERTZ keyboard respectively. This stimulus-response mapping was reversed for all children in a second block. Trial sequence was identical for all participants, but pseudo-randomized in a way that no digit could appear twice in a row, and the correct response could not be on the same side more than three times consecutively. For a more detailed description of this task see Georges et al. (2016).

Data analysis and descriptive information

Data from the training sessions was not analysed. The mean error rate across all 55 children on experimental trials was 3.55%. Errors were not further analysed. Reaction times (RTs) shorter or longer than 2.5 SD from the individual mean were considered as outliers and removed prior to data analysis (2.96%).

Number-space associations were determined using the individual regression equations method (Fias, Brysbaert, Geypens, & D’ Ydewalle, 1996), which provides a single SNARC

effect value for each participant. First, RTs were averaged separately for each digit and each response side (left/right) for every participant. Individual RT differences (dRTs) were then calculated by subtracting for each digit the mean left-sided RT from the mean right-sided RT.

Subsequently, dRTs were submitted to a regression analysis, using the magnitude of individual digits as predictor variable. Unstandardized regression slopes were taken as a measure of the SNARC effect. Negative regression slopes reflect number-space associations in the expected direction, with a more negative slope corresponding to a stronger SNARC effect. The SNARC effect was significant at the group level, since unstandardized regression slopes significantly differed from zero (t(54) = -5.07, p < .001, see Table 1).

Individual parity judgment RTs were determined by averaging response times across all trials included in the analysis for each participant (see Table 1).

5.3.3 Statistical analyses

First of all, we conducted correlation analyses to determine the relationship between all included variables.

Next, two separate multiple linear regression analyses were performed on either HRT arithmetical or visuospatial subscale scores including the parity SNARC effect, parity judgment RTs and age as independent variables. This will inform us about the strongest predictor of each subscale.

Finally, two simple moderation analyses were performed using Hayes’ PROCESS macro for SPSS to investigate whether the relation between the parity SNARC effect and each of the HRT subscale scores was conditional upon age. Moderation will be depicted by the significant effect of the product term between the parity SNARC effect and the moderator variable age on the HRT subscale score, while controlling for the effects of the two factors included in the product term. Parity judgment RTs were also included as covariate in each

the analysis. Significance was determined at 95% bias-corrected confidence intervals. To avoid multicollinearity issues, all variables were mean centered prior to analyses. Only unstandardized regression coefficients were reported. The Johnson-Neyman computational technique was used to identify the values of the moderator for which the parity SNARC effect and the different HRT subscale scores showed a significant association. This technique identifies the value(s) within the measurement range of the moderator, where the conditional effect of the parity SNARC effect transitions between not statistically significant to statistically significant.