To assess arithmetic skills, the children were asked to solve calculations presented horizontally on a computer screen. No guidance on calculation strategies was provided; therefore, spontaneous finger use could be observed. The stimuli comprised 36 items of increasing difficulty (18 additions and 18 subtractions mixed; Table 1), half of which involved carrying or borrowing.

Item difficulty was determined using a staircase procedure (inspired by Geurten et al., 2021) established through the pre-testing of 30 children enrolled in kindergarten (n = 11, mean age = 5.2 ± 0.2 years), in Grade 1 (n = 10, mean age = 6.4 ± 0.2 years), or in Grade 2 (n = 9, mean age = 7.3 ± 0.2 years). Prior to the pre-test sessions, the 36 items were classified into three levels of difficulty based on the mathematics curricula for children attending school in the French-speaking part of Belgium. Level 1 items consisted of Unit+Unit (U + U) additions without a carry, which were suitable for kindergarten students. Level 2 items include Tens-Units+Units (TU+U) and TU+TU without carry, U-U and TU-U subtractions without borrow, and U+U and TU+U additions with carry. All these items were considered suitable for first graders. Finally, Level 3 items, intended for second graders, consisted of TU+TU additions with carry and TU-U subtractions with borrows. During the pre-test sessions, kindergartners were asked to solve items from Levels 1 and 2, whereas second-graders were asked to solve items from Levels 2 and 3. First graders were divided into two groups: five were asked to solve Levels 1 and 2 items, and the others were asked to solve Levels 2 and 3 items. Level 1 items, which were not presented to second graders, were deemed successful for these children. In contrast, Level 3 items, which were not administered to kindergartners, were considered to have failed for this younger subgroup. The final order of the items was established according to the average success rate of each item obtained during the pre-test sessions.

When solving the experimental task, the children were asked to provide their answers orally. The answers and the time intervals between the presentation of the item and the child’s responses were recorded. In addition, the strategies used to solve each problem (i.e., mental calculation or finger use) have been reported. The finger-use accuracy score was calculated as the ratio of the number of items correctly solved using the fingers to the number of items processed using the fingers throughout the arithmetic task. Each correct answer was awarded one point. The test was discontinued after three consecutive errors. The internal consistency of the task was high with a Cronbach’s alpha of 0.89.

FMS was assessed using four tasks: three subtests from the Movement Assessment Battery (MABC-2, Henderson et al., 2007) measuring visuomotor precision and one task designed specifically for the present study to assess sequential finger movement coordination. This additional task was included alongside the MABC-2 subtests to evaluate finger coordination in a manner comparable to that for counting fingers.

The first visuomotor task was the Placing Pegs subtest of the MABC-2, which requires unimanual fine motor skills. In this subtest, the child is asked to place 12 pegs as quickly as possible on a pegboard (i.e., a 12-hole board with four lines of three holes). They were instructed to use one hand and manipulate each peg simultaneously. Both hands were tested sequentially, starting with the dominant hand. The subtest started with a training trial in which the child had to place six pegs, followed by test trials requiring the placement of 12 pegs, with the execution time recorded. Two trials were conducted for each hand, and only the best execution time was recorded. The mean execution time for both hands was used as the score for the first subtest.

The second visuomotor task was the Threading Lace subtest of the MABC-2 which requires bimanual motor skills. In this subtest, the child is asked to thread a lace through eight holes drilled into a board as quickly as possible. To do so, he had to insert a lace into the first hole and move it back and forth until the task was completed. The child could choose the hand that guided the lace. The subtest began with a practice trial, in which the child threaded the lace through four holes, followed by test trials that required completing all eight holes, with the execution time recorded. Two trials were required to complete each task. The best time of the two was considered as a measure of the execution time.

The third visuomotor task was the Drawing Trail subtest of the MABC-2, a graphomotor task in which participants had to draw a continuous line within a pathway delimited by two equidistant curved lines. Each child was required to comply with the following rules: (1) The drawn line must not cross boundaries. (2) Pen should not be lifted from the sheet. If so, the drawing had to resume where the pen had been lifted. (3) A line was drawn in a single direction. (4) The exercise sheet could not be tilted by >45°. A training trial was presented to the children before they started the subtest. One error was recorded each time one of the four rules was broken. The child repeated the subtest twice, with the best time reported as the measure of execution time.

Finally, a timed finger movement sequence subtest was conducted to assess the sequential coordination of the individual finger movements. The participants had to reproduce as many finger movement sequences as possible in 30 s. In all trials, sequential finger movements involved tapping a table. Four trials were conducted, all involving sequential finger tapping on the table: (1) Single-hand unidirectional sequence, following the successive order of fingers on the hand (i.e., starting with the thumb and ending with the little finger; five taps). The trial was first conducted with the right hand and then with the left hand. (2) Two-hand unidirectional sequence, following the successive order of fingers on the hand (i.e., starting with the left little finger and ending with the right little finger, 10 taps). (3) Two-hand unidirectional sequence, alternating every other finger (i.e., starting with the little finger, middle finger, and thumb of the left hand, followed by the thumb, middle finger, and little finger of the right hand, six taps). (4) Two-hand bidirectional sequence alternating with every other finger. In this trial, the child was required to perform the sequence from the third trial in both forward and backward directions (12 taps).

Before each trial, the child was asked to repeat the movement three times for practice. One point was awarded for each correctly executed finger movement sequence (i.e., when the fingers were mobilised one after the other in the correct order). The average performance of both hands was used as the score for the first trial. The subtest score was the sum of correctly executed finger sequences across the four trials.

An FMS score was extracted from the four task scores by applying a Principal Component Analysis (PCA). The PCA results confirmed that the four subtests reflected a single construct that accounted for 65% of the total variance across the subtests. The factor loading were .32, .31, .30 and .30 respectively, for the Placing Pegs, the Threading Lace, the Drawing Trail and the Timed Finger Movement Sequence subtests.

The child was asked to place one hand palm-down and flat on the table with the fingers spread out. The patient was shown a reference card on which a hand with coloured fingers was drawn. His hand was covered with cardboard, and out of the participant’s view, the experimenter touched the middle phalange with one finger. The cardboard was then removed and the child was asked to indicate the colour of the finger touched using the reference card. Ten trials were performed for each hand. The first five trials consisted of one touch, whereas the last five trials consisted of two successive touches. One point was awarded when the child correctly reported touching the finger(s) touched. For two-touch trials, one extra point was awarded when the fingers were identified according to the order in which they were touched. The sum of the points was used as the task score. The highest possible score is 30. The internal consistency of the task was acceptable with a Cronbach’s alpha of 0.69.

The Matrix Reasoning subtest of the WISC-V (Wechsler, 2016) was used to control the effect of fluid reasoning in the statistical analysis. In this task, children were asked to complete a visual matrix by selecting the element that followed the underlying rule governing the patterns from several alternatives. Two practice trials were conducted to ensure comprehension of the instructions. Responses were provided either by pointing to the selected element or stating its number aloud. This subtest comprised 32 items and was discontinued after three consecutive errors. Subtest performance was assessed using standardised scores derived from the total number of correct responses based on the WISC-V normative data.

Table 3 reports Pearson’s correlations and partial correlations between finger-gnosia, FMS and arithmetic at the four measurement times (i.e., Autumn and spring and of the Grade 1 (T1 and T2) and of the Grade 2 (T3 and T4)). General cognitive abilities appear to have only a negligible effect on the association between sensorimotor skills and arithmetic abilities. When general cognitive abilities were considered, significant correlations were found between FMS at T1 and finger gnosia at T3 (p = .01) and T4 (p = .004), as well as between FMS at T2 and finger gnosia at T4 (p = .004). There was also a strong correlation between children’s arithmetic skills at T3 and T4 and the immediately preceding measurement time (T2 x T3, p < .001; T3 x T4, p < .001). Finger gnosia was moderately correlated with concurrent arithmetic skills at T2 (p < .001), T3 (p < .001) and T4 (p = .003). A significant correlation was reported between finger gnosia at T1 and later arithmetic skills at T4 (p = .004). By contrast, at each time point, FMS showed a weak correlation with concurrent arithmetic skills (from r = .09 to r = .17), none of which were significant. Moreover, arithmetic skills were found to be unrelated to FMS assessed at the previous time point (from r = .11 to r = .21).

Before conducting the LGM with time-varying covariates, the trajectories of changes in finger gnosia, FMS, and arithmetic were examined using a linear LGM.

Regarding finger gnosia, the children had an average score of 20.84 at the intercept. The average slope was .67. The estimated parameters of the linear LGM for finger gnosia showed that the variance of the intercept (Estimate = 14.83, p < .001) and the slope (Estimate = 2.26, p = .01) both differed significantly from 0, confirming the presence of inter-individual differences in finger gnosia at T1, as well as age-related changes observed up to the end of Grade 2. Fluid reasoning did not affect the variance of the intercept (Estimate = .09, p = .51) or the slope (Estimate = .06, p = .71) of finger gnosia. The correlation between the variance of the intercept and the slope of finger gnosia was significant (Estimate = –.45, p = .01), even when controlling for general cognitive abilities, indicating that higher initial values of finger gnosia were associated with a slower rate of increase.

At the first measurement point, children had an average FMS score of -.42 points²2

FMS score was derived using Principal Component Analysis (PCA), which can yield negative values.

Finally, the children had an arithmetic mean score of 2.89 points at the intercept, while the mean slope was 4.87. The estimated parameters of the linear LGM for arithmetic showed that the variance of the intercept did not statistically differ from 0 (Estimate = 5.95, p = .24), suggesting that at T1, there were no notable individual differences in arithmetic skills among the children. Conversely, the slope variance (Estimate = 3.90, p = .006) was significantly different from 0, suggesting a heterogeneous developmental trend among children. General cognitive abilities did not significantly affect either the intercept (Estimate = -.02, p = .62) or the slope (Estimate = .05, p = .15) of arithmetic. The correlation between the intercept and the slope was not significant (Estimate = -.24, p = .89), indicating that the children's improvement in arithmetic between Grade 1 and 2 was independent of their skills at T1.

As all three variables of interest showed significant changes over time, two linear LGMs with time-varying covariates were estimated to examine the predictive value of the intercept and slope of finger gnosia (Model 1) and the FMS (Model 2) for age-related changes in arithmetic skills while controlling for general cognitive abilities.

According to the first model, the intercept of finger gnosia was not a significant predictor of the intercept of arithmetic skills (Estimate = .15, p = .19). In contrast, the intercept of finger gnosia was a significant predictor of the arithmetic slope (Estimate = .43, p < .001). The slope of the finger gnosia latent variable was not significantly associated with arithmetic development (Estimate = .03, p = .95). The R-square analysis further indicated that the intercept and slope of finger gnosia accounted for 50% of the development of arithmetic skills between the beginning of Grade 1 and the end of Grade 2 once fluid reasoning was considered, which represents a significantly large explanatory share (p = .02).

A second linear LGM with the FMS as the time-variant covariate was conducted. The intercept of the FMS significantly predicted neither the intercept (Estimate = .38, p = .55) nor the slope of arithmetic skills (Estimate = .67, p = .35). Likewise, improvements in FMS scores over time did not predict age-related changes in arithmetic skills (Estimate = .34, p = .89).

Table 3 reports partial correlations between finger use, finger-gnosia, FMS and arithmetic, at the four measurement time points. General cognitive abilities appears to have only a negligible effect on the association between sensorimotor skills and arithmetic abilities. Children's rates of correct finger use were not related to their abilities at the previous measurement time (from r = -.17 to r = .03). Finger gnosia and the rate of correct finger use negatively correlated at T4 (p = .01). Finger gnosia at T3 and T4 was related to the rate of correct finger use at T1 (T3xT1, p = .04; T4xT1, p = .04). On the other hand, FMS did not correlate with concurrent rates of correct finger use at any specific time point (from r = -.18 to r = .24). Finger use moderately correlated with concurrent arithmetic skills at T2 (p < .001), T3 (p = .006) and T4 (p = .004). Finally, no significant relationships were found between the rate of correct finger use and the subsequent arithmetic performance (from r = .05 to r = .28).

The linear LGM, examining the trajectory of change in the rate of correct finger use, showed that the variance in the slope (Estimate = -.009, p = .10) was not significant, indicating that there were no inter-individual differences in the progression of the rate of correct finger use between the beginning of Grade 1 and the end of Grade 2. As the variability in the rate of change in finger use was not significant, it was not relevant to conduct multivariate analyses to examine its predictive value for the development of children's arithmetic skills or its mediating role in the relationship between sensorimotor finger skills and arithmetic development.