Despite extensive evidence in adults, the developmental trajectory of number–action coupling remains underexplored. Emerging findings suggest that numerical–motor associations may arise early in life (Decarli et al., 2025). Hand movements appear to scaffold the acquisition of numerical concepts (Novack et al., 2014; Fischer et al., 2015), supporting the hypothesis that the integration of number and action plays a foundational role during childhood (Rusconi et al., 2006; Rugani and Sartori, 2016). Neurocognitive evidence reinforces this claim: parietal regions implicated in numerosity processing also support spatial and motor functions, and infants as young as 3–9 months show sensitivity to congruency between numerosities and hand shapes (Decarli et al., 2022a, 2022b).

These findings support the existence of an early, preverbal number–action mapping, potentially grounded in the embodied role of fingers in numerical development (Sixtus et al., 2023; Krenger and Thevenot, 2024; Orrantia et al., 2024; Roesch et al., 2024; Soylu, 2024; Fu and He, 2025). Notably, the Latin term digit — for both fingers and written numbers — echoes this deep historical and cognitive link.

The present study extends the kinematic paradigm of Rugani et al. (2018) to preschool and school-age children to investigate how non-symbolic numerosity influences motor execution during development. Children wore a miniature soccer shoe on their right index finger and kicked a small ball toward a frontal goal after viewing numerical (2- or 8-Dots arrays) or baseline non-numerical stimuli (a soccer goal). A no-go signal (5-Dots) ensured numerical processing. Movements were recorded with a 3-D motion capture system. Because the motor response was identical in all trials, any systematic kinematic variation can be attributed to numerical processing.

We hypothesized that if the spatially-oriented mental number line influences motor execution, children would exhibit greater leftward deviation after seeing small numerosities (2-Dots) relative to baseline, but not after seeing large ones (8-Dots). Large numbers, linked to right space, would either produce no deviation or a rightward shift. However, as stated above, in this paradigm, rightward effects are weaker or absent.

By combining number presentation with precise action measurement in children, this study aims to shed light on early number–action mappings and on how these mappings are preserved, reshaped, or reweighted across development, distinguishing foundational cognitive mechanisms from those increasingly shaped by formal education and experience.

Twenty-one right-handed children (7 females and 14 males, mean age = 8 years, age range: 4 years and 10 months – 11 years and 10 months) took part in the experiment. Handedness was assessed with a brief questionnaire asking participants which hand they used to grasp a pen, for coloring, and for eating. Concordance was assessed by observing which hand the participant used to insert their index finger into the soccer shoe during the test. Sample size was determined by means of GPOWER 3.1 (Erdfelder et al., 1996) based on previous literature (Rugani et al., 2018). Since we used a repeated-measures ANOVA, we considered an effect size f of 0.3, alpha = 0.05, and power = 0.8. The projected sample size needed with this effect size was N = 20 for within-group comparisons.

To better investigate the relationship between number and movement during the critical age for the gradual acquisition of math skills, we explored the data by dividing the group into two clusters. A cluster of preschool children who had not received formal math education and a cluster of school-age children with corresponding academic math skills. Mathematical abilities were assessed with a Mathematical Proficiency Index, MPI, based on three scales: (i) Enumeration Time (ET) from 1 to 20 in 20″ (expressed as the total time, 20″, minus the time taken to pronounce the entire list without errors), (ii) Additive Calculation Ability (ACA) in 30″ (expressed as the number of additions performed correctly out of a total of 20 in the time interval), (iii) Subtractive Calculation Ability (SCA) in 30″ (expressed as the number of subtractions performed correctly out of a total of 20 in the time interval; see Supplementary Table S1 for the list of demanding math tasks). Synthetically, the formula for calculating the Math Proficiency Index is: MPI = Σ [ ( 2 0 ' − ET ) + ACA + SCA ]

Once it was verified that none of the children in the pre-school group had properly mastered the ability to count, add, or subtract and none reached the criterion – while the school-aged children had an average score (MPI = 33) in the test range (0–60) – we divided the sample into two subgroups accordingly. The resulting clusters were therefore: (i) a Preschool group with no formal math education (4 males and 3 females, mean age = 5 years and 8 months, age range: 4 years and 10 months – 6 years and 10 months); (ii) a group of school-aged children (10 males and 4 females, mean age = 9 years and 1 month, age range: 7 years and 6 months – 10 years and 10 months). All participants had normal or corrected-to-normal vision and were naive about the purpose of the experiment. The data obtained from the school-age participants in this test are shown in Table 1.

Participants were tested individually in a dimly lit room. The stimuli presentation was implemented using E-prime V2.0. One infrared reflective marker (ultra-light 6 mm diameter semi-spheres) was applied to the right index finger for kinematic analysis. A further marker was placed on the ball to perform a detailed analysis of the deviation movements of the index with respect to this reference point. Six infrared cameras (sampling rate: 70 Hz, adequate to record spatial and low-frequency temporal parameters) were placed in a semicircle at a distance of 1–1.2 m from the center of the room captured the relative position of the markers. The movements were recorded using a 3-D motion analysis system (SMART-D, Bioengineering Technology and Systems [B|T|S] BTS, Milano, Italy). Prior to data collection, the image captured by each camera was optimized by adjusting its zoom, focus, threshold and brightness. The system was then calibrated in order to define an absolute spatial reference system. For the static calibration, a three-axes frame of 5 markers at known distances from each other was placed in the middle of the table. For the dynamic calibration, a three markers wand was moved throughout the workspace of interest for 120 s. The spatial resolution of the recording system was 0.2 mm over the field of view. The standard deviation of the reconstruction error was 0.2 mm for the x, y and z axes.

The experimental stimuli were matrices of non-symbolic numbers: 10 different configurations of two (2-Dots) and eight dots (8-Dots). These numerosities were selected for methodological consistency with prior work and because they are reliably discriminable (Rugani et al., 2017). The dots were black with a diameter of 2 cm and were arranged in a white square of 964×964 pixels (see Supplementary Table S2 for stimulus properties). The arrangement of the dots was such as to avoid both a clumping effect if they were too close together and a scattering effect if they were not (Rugani et al., 2017). The program used to create the stimuli was GeNEsIs, a tool for generating controlled stimuli in non-symbolic numerical experiments (Zanon et al., 2022). A non-numerical stimulus (drawing of a soccer goal) was presented during baseline trials. This picture was a salient stimulus capable of sustaining children’s motivation and attention. It was perfectly symmetrical, thus avoiding any indication of direction. A stimulus with five dots (5-Dots) positioned according to the standard arrangement of a die was adopted as a “no-go” signal to ensure that movements did not occur automatically without cognitively processing the numerosity of the stimulus.

We capitalized on a previous experimental paradigm developed in our laboratory and applied so far only to the adult population (Rugani et al., 2018). Participants sat on a chair in front of a table with their right hand located in the designated start position. Participants’ right index was introduced in a small plastic soccer shoe (3 cm long, 1.5 cm wide) positioned on a footprint. In the start position, participants rested their right wrist on a pillow (16 cm long, 11 cm wide and 6.5 cm high), which was shaped to guarantee a comfortable and repeatable posture of the hand, allowing them to effortlessly kick the ball. A plastic ball (2.3 cm in diameter) was located at 1 cm and a soccer goal (18 cm long, 16 cm high) at 50 cm away from the footprint. Given the novelty of the task, a training phase was implemented so that children could practice or adjust the soccer shoe until they felt comfortable with kicking the ball with their finger. Moreover, in order to increase their motivation towards the task, during familiarization they were shown a sketch of the FIFA soccer video game and told they were about to play a similar kicking game, where dots represented opposing players to be kicked against. A monitor set at eye level (the eye–screen distance was 80 cm) was used to present the experimental stimuli (Figure 1). Participants underwent two sessions (i.e., Baseline and Testing) and were instructed to kick the ball toward the soccer goal following the stimulus presentation, at their own pace. No instruction was given concerning the speed of movement. A black fixation cross appeared for 100 ms and was replaced with a stimulus after 1,000 ms. There were 40 trials in total. The Baseline session was presented for 10 trials. During the Testing session participants kicked the ball upon random presentation of either 2-Dots or 8-Dots (10 trials each). On each trial, after kicking the ball, children were requested to say aloud if they saw a large or a small number of “opponents” (dots). This was meant to ensure children were paying attention to the numerical stimuli and not only focused on the movement itself. In addition, whenever 5-Dots were presented (n = 10 trials), participants were required to refrain from kicking the ball, which prevented participants from falling into a routine response pattern and maintained engagement throughout the session. The configuration of the 5-Dots stimuli was always the same (dice-like).

After data acquisition, the 3-D coordinates of the markers were pre-processed using the SMART-D Tracker (BTS) software to reconstruct the trajectories of the index finger. Subsequently, the SMART-D Analyzer (BTS) software was used to filter and analyze the data. The analysis protocol was specifically designed to extract spatial, velocity, and temporal dependent variables. First, signal frequencies above 10 Hz were eliminated using a low-pass filter. Next, key events or time instants were identified in the time traces of distance, velocity (calculated as the derivative of distance), and deceleration (the derivative of velocity). The onset and offset of the movement were determined from the velocity trace of the index finger marker. Specifically, movement onset was defined as the moment when velocity exceeded a threshold of 5 mm/s and remained above this value for at least 500 ms. Movement offset was defined as the moment when the tangential velocity of the finger marker dropped below 5 mm/s after the ball was kicked. To account for inter-individual and inter-trial variability in movement duration, temporal variables were normalized to the total movement time (MT) by dividing the absolute values by MT. The resulting normalized value thus represents the proportion of the movement duration at which each event occurred, with values ranging from 0 to 1. Following the methodology of Rugani et al. (2018), the following spatial, velocity, and temporal parameters were calculated:

Children performed the verbal magnitude judgment task with 100% accuracy, and no-go trials showed fewer than 2% errors. Data analyses were performed in the R environment (R Core Team, 2025). In the data cleaning phase, the 10 kinematic parameters of interest were standardized in z-scores and data were filtered by excluding outliers’ values greater than ±3 SD from the mean, to avoid their potential influence. Overall, less than 5% of the total data were excluded. The final dataset was composed of 576 observations from 21 individuals. Descriptive and inferential statistical investigations were performed on all the 10 kinematic parameters of interest to investigate possible differences in performance between groups (Pre-schoolers as reference level, Schoolers) when observing the numerical stimulus “2-Dots” (as reference level) compared to stimulus “8-Dots” and the baseline (no dots). Figure 2 describes the distribution of each parameter controlling for Group and Stimulus (2, 8-Dots). Pairwise Pearson correlations among the variables of interest were computed (Figure 3), and Bonferroni corrections were applied (corrplot R package; Wei and Simko, 2017). Finally, data were analyzed using generalized linear mixed models (GLMMs) implemented in the glmmTMB R package (Brooks et al., 2017). This approach was chosen for its flexibility in modelling several non-Gaussian outcomes, including beta- and gamma-distributed dependent variables, and for its ability to accommodate random effects structures. Models – one per each kinematic parameter as dependent variable - included fixed effects for Group, Stimulus, and their interaction, with a random intercept for Participant to account for repeated observations. All the models considered the 2-Dots experimental condition as reference level. Although our sample size was limited due to practical constraints, the use of glmmTMB R package and GLMMs is suitable for small to moderate samples and allows flexible specification of distributions while accounting for participant-level random effects. To mitigate small-sample limitations, we reported marginal and conditional R2 (Nakagawa and Schielzeth, 2013) and inspected model fit with DHARMa diagnostics (Hartig, 2022). Post hoc contrasts were conducted on estimated marginal means using emmeans (Lenth, 2024), with p-values adjusted for multiple comparisons. Distribution test on each variable was then performed to determine the reference family distribution to apply in each model (fitdistrplus), and potential alternative distributions were compared using goodness-of-fit statistics provided by the same R package (‘gofstat’ function). Considering the evidence collected and the residual check from each model, in the Result section are presented results from five models which resulted of more interest (m1: MH; m2: MDevLeft; m3: MV; m4: TMDevRight; m5: TMV). Additional descriptives are provided in the Supplementary Figure S1 and Supplementary Table S3. The dataset and the R script are available in the OSF project folder: https://osf.io/unf9y/?view_only=780672a28ec947ba9fa469308c425ace.

Pre-schoolers exhibited greater leftward trajectory deviation after observing small numerosities compared to the baseline condition, indicating that their motor actions were modulated by a spatial–numerical mapping consistent with the Mental Number Line (MNL). Comparable effects have been documented in adults, who showed prolonged leftward deviations after viewing small symbolic numbers (Rugani et al., 2018). A more pronounced leftward tilt in the 2-Dots condition indicates that participants oriented further to the left following the presentation of small numbers compared to large numbers. Comparable findings were obtained using an index-finger pointing paradigm (Fischer, 2003). Subsequent investigations in adults (Ishihara et al., 2006) and in 7-year-old children (Möhring et al., 2019) suggest that this bias may reflect the automatic activation of spatially congruent representations during motor preparation.

Despite differences across studies in age and stimulus type, the influence of numerical magnitude on spatial aspects of action appears robust and can be reliably detected via kinematic parameters such as trajectory deviation—an index known to reflect motor intentions and action planning (Georgiou et al., 2007; Becchio et al., 2008). This suggests that the motor system integrates abstract task-relevant information, including numerical magnitude, into action plans.

Evidence that numerical magnitude affects action selection even in very early infancy further suggests that these effects may arise independently of extensive experience (McCrink and Opfer, 2014; Rugani and De Hevia, 2017). Our findings support this view: the association between numerical magnitude and spatial direction emerges before reading and writing are acquired. Crucially, the present task avoided intrinsic spatial–numerical biases common in manual response paradigms (e.g., left/right key presses). While adult associations between small/large numbers and left/right responses may partly stem from overlearned mappings, the present action-based task—novel for children and identical across conditions—reduces this concern and leverages fine-grained kinematic measures to isolate the contribution of numerical processing.

These results align with ATOM theory (Walsh, 2003), supporting the idea of a shared magnitude-processing system across domains such as number, space, and action. Importantly, our data are consistent with developmental accounts proposing that such common coding frameworks emerge early and may later be refined, reshaped, or reweighted through interaction with symbolic systems and formal education (Decarli et al., 2022a, 2022b, 2025).

Preschoolers also displayed more efficient movement trajectories after observing smaller numerosities (2-Dots arrays), reflected in lower trajectory height – an index of enhanced sensorimotor integration. Increased trajectory height typically reflects prolonged deceleration phases, which occur when the motor system must satisfy higher precision constraints (Marteniuk et al., 1987). By contrast, adults previously tested with the same task showed improved performance after viewing larger symbolic numbers (8), consistent with overlearned associations linking numerical magnitude to physical weight (Rugani et al., 2018).

Although the same kinematic parameter was sensitive to numerical magnitude in both age groups, the direction of the effect differed: preschoolers optimized movements after smaller non-symbolic numerosities, whereas adults did so after larger symbolic numbers. This developmental dissociation may reflect several, non-mutually exclusive factors.

First, differences in stimulus type (non-symbolic vs. symbolic) may play a role. In adults, non-symbolic stimuli often fail to reliably modulate performance (Rugani et al., 2017), a pattern replicated here in school-aged children, suggesting that non-symbolic effects may diminish or become masked with schooling. Second, general motor maturation is unlikely to fully account for the observed pattern. Kinematic scaling during reaching is broadly comparable between children and adults (Kuhtz-Buschbeck et al., 1998), and our paradigm intentionally excluded grasping components to minimize developmental motor confounds.

Third, and most plausibly, accumulated experience with symbolic numbers and culturally shaped spatial conventions may progressively modulate early non-symbolic spatial–numerical mappings. Children appear to possess an early neurocognitive structure linking numerosities and actions, which becomes increasingly differentiated through maturation and formal education. Toddlers’ spontaneous use of finger counting and pointing to scaffold numerical understanding (Noël, 2005; Lafay et al., 2013; Gunderson et al., 2015; Soylu et al., 2018; Bagnoud et al., 2025; Poletti et al., 2025; Thevenot et al., 2025) likely strengthens number–action associations and supports the acquisition of cardinality and one-to-one correspondence (Gallistel and Gelman, 1992; Butterworth, 1999; Andres et al., 2008). As symbolic number knowledge is acquired, neural activation patterns shift from right-lateralized to more bilateral involvement, including left-hemisphere regions (Salillas et al., 2023; Visibelli et al., 2024). Such reorganization may not eliminate number–action mappings, but may reshape or reweight their expression and accessibility—potentially accounting for the reduced or absent non-symbolic effects observed in older children and adults.

These results should be interpreted cautiously due to the relatively small sample with gender imbalance and high inter-individual variability in numerical abilities, as reflected by the MPI index (Table 1). Furthermore, the baseline stimulus differed in its perceptual and semantic properties from the dot arrays, which might have influenced the responses. However, if this were the case, differences should have emerged for both numerical stimuli. Instead, the fact that only the small stimulus, and not the large one, induced movements that differed from those observed in responding to the baseline stimulus leads us to hypothesize the presence of a genuine numerical effect, rather than an effect driven solely by a semantically or visually different stimulus. Additionally, the visual dimensions of the stimuli were not controlled, thus the observed effects may reflect responses to numerosity-correlated visual information rather than to numerical magnitude. The decision not to equate the total surface area of the stimuli and to adopt a soccer goal as the baseline stimulus was motivated by ecological considerations: during the familiarization phase, the children were told that the dots represented the opposing players they were kicking against, and maintaining a constant dot size—in addition to adopting a soccer goal—supported this narrative. Although these limitations were partly mitigated by precise kinematic measurement and advanced statistical modelling, larger samples and more controlled stimuli will be crucial to deepen understanding of the links between numerical representation, numerical skill, and motor performance, and to inform ongoing theoretical debates.

A further open question concerns whether the cardinality of small sets is represented through the Approximate Number System (ANS; Feigenson et al., 2004) or through a specialized system for small quantities (Fu et al., 2022). Subitizing—the rapid, errorless apprehension of up to four items (Kaufman et al., 1949)—differs from estimation of larger sets, where variability increases proportionally with numerosity underlying small- and large-number processing (Jevons, 1871).

However, this dichotomy is not consistently observed. Several studies report continuous behavioral signatures across small and large numerosities, with no clear boundary between subitizing and estimation (Cantlon and Brannon, 2006; Cordes and Brannon, 2009; Rugani et al., 2013, 2014; Reoyo-Serrano et al., 2025), suggesting that apparent dissociations may depend on task demands, attentional load, or response requirements.

Although the present study focused on non-symbolic numerosities, it was not designed to directly dissociate ANS-based estimation from subitizing mechanisms, and we therefore refrain from making strong claims regarding the specific representational system involved. Nevertheless, the adoption of a kinematic approach provides a complementary perspective by capturing fine-grained spatial–numerical biases as they unfold during action execution, rather than relying exclusively on accuracy or response times. Because the task imposes concurrent cognitive and motor demands, this approach may be particularly sensitive to how attentional and sensorimotor constraints modulate numerical representations across development.

Finally, this framework may also help elucidate the contribution of shared neural substrates. Meta-analytic evidence reveals substantial overlap within parietal regions supporting both symbolic number processing and goal-directed hand actions (Ranzini et al., 2022), consistent with accounts proposing partially shared magnitude and action representations. Such convergence offers a plausible neural basis for the influence of numerical magnitude on motor performance observed in the present and previous studies.