Experiment 1
A total of 26 undergraduate students participated in the study, of which 16 were female, with a mean age of 21.69 years (SD = 1.76). One subject dropped out of the experiment and was excluded, resulting in a final sample size of 25 participants. All participants included in the study exhibited normal or corrected visual acuity and were right-handed. In adherence to ethical guidelines, informed consent was obtained from all participants involved in this study. Detailed information regarding the study’s purpose, procedures, and potential risks and benefits was provided to each participant. The consent process took place through face-to-face meetings, allowing participants the opportunity to ask questions before expressing their consent. Written consent was obtained from each participant, and a copy of the consent form was provided to them for their records.

The stimulus material was created in accordance with our previous research ( Pan et al., 2021 ). The stimulus arrangement consisted of a grid measuring 6° × 6°, composed of 144 squares, each measuring 0.4° × 0.4°. The stimuli were positioned at the intersections of the grids, resulting in a total of 121 possible positions for the stimulus array (see Figure 1A ). In the grouping condition, the stimuli were divided into four groups, with each group having 12 possible positions. The stimuli within each group were distributed within the same quadrant. The grouping conditions included 2-4 subgroups for each numerosity, with each subgroup containing 2-6 items ( Figure 1A ). The numerosity range was 5-17, and there were 13 different numerosities used in the study, with the following combinations: N5 (2, 2, 1); N6 (3, 3); N7 (3, 3, 1); N8 (2, 2, 2, 2); N9 (3, 3, 3); N10 (3, 3, 3, 1); N11 (3, 3, 3, 2); N12 (3, 3, 3, 3); N13 (5, 5, 3); N14 (4, 4, 3, 3); N15 (4, 4, 4, 3); N16 (4, 4, 4, 4); N17 (5, 4, 4, 4). In the no-grouping condition, each stimulus was randomly distributed within the large grid (see Figure 1B ).

Participants sat 60 cm from a screen monitor (refresh rate: 60 Hz) in a quiet, dimly lit room. Stimulus materials were created and presented using E-Prime 3.0 ( https://pstnet.com/products/e-prime/ ). Each trial commenced with a 6-second adaptation screen. Participants were instructed to tap with their dominant hand (right) on the right side of the screen in mid-air as quickly as possible during the high-frequency adaptation level. During the low-frequency adaptation level, participants were instructed to tap at a much slower frequency of approximately 1 Hz (once per second) within the 6-second duration. The two adaptation levels were tested separately. According to the research by Anobile et al. (2020b) , the motor adaptation effect occurs within a spatially specific range in external (rather than body-centered) coordinates, confined to a 10° distance centered on the tapping hand. Therefore, in this study, the distance for visual stimuli and adaptation regions is also chosen to be 10°. Hand movement was recorded and monitored by infrared motion sensor equipment (Leap motion controller: https://www.leapmotion.com/ ). Subsequently, a red “+” appeared in the center of the screen for 500 ms, and participants were instructed to stop tapping when the red “+” appeared. This was followed by a blank screen lasting 1000 ms to allow sufficient time for the participants to stop moving. The stimulus was then presented for 250 ms. After the stimulus disappeared, a rectangular input box appeared, and participants were asked to estimate the number of stimuli and use the numeric keypad to quickly and accurately input the estimated result into the input box ( Figure 1C ). The grouping and no-grouping conditions were randomly presented within the same block.

All coauthors agreed with the contents of the manuscript. This study was performed in line with the principles of the Declaration of Helsinki. This study was approved by the School of Psychology Ethics Committee at Guizhou Normal University (Approval number: GZNUPSY202111005). The complete raw data, materials, and code used in this experiment are accessible on our Zenodo page: https://doi.org/10.5281/zenodo.10421155 . We express our gratitude to an anonymous reviewer for providing this valuable suggestion. All data generated or analyzed during this study have been included in this published article.

For the data analysis of grouping effects, reaction times (RTs) and coefficients of variation (CVs) were calculated for each participant. CV represents the sensory noise involved in numerosity perception studies and is used to measure variation in the data. An increased CV value indicates greater sensory noise, resulting in less precision in estimations ( Eq. (1) ). Statistical significance was determined using a 2 × 2 × 13 repeated measures ANOVA with test grouping condition (two levels for grouping: grouping and no-grouping), adaptation level (low frequency and high frequency), and numerosity (13 levels ranging from five to 17) as the main factors. Each numerosity was presented three times under each condition, yielding a total of 156 trials per participant. (1) C V = i N i

Where N i is the analyzed numerosities, N i =13 (ranging from five up to 17), i is the standard deviation of the numerosities.

As for the analysis of adaptation effects, we referred to the 2014 study conducted by Arrighi and colleagues ( Arrighi, Togoli & Burr, 2014 ), utilizing the Adaptation Index (AI) to quantify the adaptation magnitude. The adaptation effect is operationally defined as the difference between perceived numerosity following low-frequency adaptation and those following high-frequency adaptation. To quantitatively assess the extent of adaptation, we employed linear regression, fitting the differential curve of adaptation amplitude and calculating its slope. The multiplication of this slope by 100 yields the Adaptation Index (AI), providing an estimate of the magnitude of adaptation ( Arrighi, Togoli & Burr, 2014 ). To assess the statistical significance of the variances in adaptation effects between grouping and no-grouping condition, a two-tailed paired t -test was utilized ( Anobile et al., 2016 ).

The adaptation effect is measured by adaptation index. We calculated the adaptation index separately for the grouping and no-grouping conditions. The results are presented in Figure 2A , which illustrates the adaptation index under both grouping and no-grouping conditions. Notably, the grouping condition exhibits a more pronounced adaptation effect compared to the no-grouping condition ( t = 3.233, p = 0.23, Cohen’s d = 1.572). Figure 2B illustrates the mean perceived numerosity (averaged across all subjects) plotted against the physical number of pulses. The mean estimates without adaptation, represented by the gray dotted line, closely aligned with the veridical values. A zero-anchored linear regression model ( R 2 = 0.99) effectively described the data, exhibiting a best-fitting slope of 0.99. The introduction of high-frequency adaptation consistently led to a 9% reduction in apparent numerosity across all tested numerosities (linear regression slope of 0.90), as can be seen from Figure 2B , the line shifted towards the upper side compared to the baseline (no adaptation), indicating an overestimation of perceived numerosity. Conversely, low-frequency adaptation resulted in a 4% increase in numerosity (linear regression slope of 1.03, compared to the baseline slope of 0.99), the line shifted towards the upper side compared to the baseline (no adaptation), indicating an overestimation of perceived numerosity ( Figure 2B ). Given that the zero-anchored linear regressions explained over 98% of the variance across all conditions, it suggests that adaptation influenced numerosities uniformly across the entire range.

We analyzed the coefficient of variation (CVs) under the conditions of high and low frequency motor adaptation. The statistical analysis employed in this study involved conducting a repeated measures analysis of variance (ANOVA), which revealed a significant main effect of adaptation level, F (1,24) =13.128, p = 0.001**, the high frequency hand tapping was more accurate and had less sensory noise than low frequency hand tapping conditions ( Figure 3A ). The main effect of numerosity was also significant, and the sensory precision of small numerosities was more accurate. Different from previous studies, no grouping advantage was found in this study, that is, the main effect of group conditions was not significant, F (1,24) = 0.599, p = 0.447.

For the ANOVA on RTs, We found a significant main effect of adaptation level, F (1,24) =83.004, p < 0.001***, the reaction time of the high frequency hand tapping was shorter than that of low frequency hand tapping conditions. A significant main effect of group condition was found, F (1,24) = 184.833, p < 0.001***, The grouping condition resulted in faster reaction times compared to the no-grouping condition. There was also a significant main effect of numerosity, F (12,23) = 2.657, p = 0.047*, small numerosities react faster than large numerosities. And the interaction between adaptation level and grouping was significant, F (1,24) = 13.641, p = 0.001**. Due to the significant interaction, we conducted additional simple effects analyses to examine differences in grouping conditions under different adaptation frequencies. The results revealed that high-frequency adaptation exhibited a stronger grouping effect, with grouped conditions showing faster reaction times compared to no-grouping conditions under high-frequency adaptation ( Figure 3B ).

In Experiment 1, our findings were consistent with previous studies ( Anobile et al., 2016 ; Anobile et al., 2020a ; Anobile et al., 2020b ), demonstrating that the adaptation to high frequency hand tapping was underestimated in the subsequent numerosity estimation task, while the adaptation to low frequency hand tapping was overestimated ( Figure 3A ). These results indicate that the adaptation effect persists even when subjects adopt the “groupitizing” strategy. Previous research has established that visually grouped arrays tend to result in faster and more accurate numerosity perception compared to ungrouped arrays, a phenomenon referred to as “groupitizing” ( Anobile et al., 2020a ; Anobile et al., 2020b ; Ciccione & Dehaene, 2020 ). However, contrary to these findings, our results did not reveal a significant grouping effect on perception precision. The sensory precision under the grouping condition was not significantly higher than that under the non-grouping condition ( Figure 3A ), the reason for this result is that the grouping effect is affected by the motor adaptation, so that no grouping advantage is found. Interestingly, our data did show a grouping advantage in reaction time. As shown in Figure 3B , the reaction time in the grouping condition was significantly faster than that in the no-grouping condition.

Experiment 1 further reveals a stronger adaptation effect under grouping conditions. As illustrated in Figure 2A , the adaptation index in the grouping condition is significantly higher than that in the no-grouping condition, indicating a more pronounced deviation in numerosity estimation induced by motor adaptation under grouping conditions. This finding contradicts our initial expectations, as we anticipated that grouping conditions might enhance the precision of numerosity perception. Instead, it resulted in a greater estimation bias under the influence of motor adaptation.This unexpected observation prompts a deeper consideration of the interactive mechanisms between motor adaptation and grouping effects. One possible explanation is that motor adaptation may alter the weighting allocation for processing grouping information, making the adaptation effect more prominent. On the other hand, motor adaptation and groupitizing may involve distinct neural mechanisms, which could lead to mutual competition or enhancement effects in numerosity perception. Furthermore, our study results serve as a reminder that numerosity perception is a complex process influenced by multiple factors, to comprehensively understand the neural underpinnings of numerosity perception, future research can delve into exploring the interactions among these factors in greater detail.