Same digits, different magnitudes: manual reaching unveils dual systems in symbolic number representation

Abstract

The role of Approximate Number System (ANS) in adult symbolic number processing remains controversial. Despite evidences for a linear and precise representation of symbolic numbers in adults, this study, based on Dual-Process Theory (DPT), proposes that ANS should still influence symbolic number processing in adults. To examine this possibility, we conducted two hand tracking experiments using a manual reaching numerical comparison task. In Experiment 1, we investigated differences in adults’ performances when comparing small numbers (#1–#9 vs. #5) and large numbers (91–99 vs. 95). The results showed that adults exhibited both higher response accuracy and greater deviation in movement trajectories when comparing small number pairs rather than large number pairs, indicating an implicit size effect in symbolic number representation. In Experiment 2, we ruled out the possibility that the results of Experiment 1 were driven by differences in visual format of stimuli (e.g., 9 vs. #9 vs. 09). Together, these findings provide consistent evidences for the implicit influence of ANS on adults’ symbolic number processing, unveiling the dual representations of symbolic numbers in adulthood and underscoring the potential of visuomotor tasks for probing implicit cognitive processing.

Supplementary Information

The online version contains supplementary material available at 10.1186/s40359-026-04227-9.

Introduction

The rightness of “5+5=10” seems too obvious to be denied for any educated person. However, does this equation always hold true? Mathematically, it’s undoubtedly a stupid question. However, from a psychological perspective, this remains an open question, as humans have two distinct ways of processing numbers. One is the exact representation of numbers, in which two “5”s of a “10” are definitely equal in quantity. Generally, this form of processing is considered to be handled by the exact number system (ENS). Within this system, numerical information is represented in a linear manner and processed via counting and numerical symbols [61, 68, 71, 85]. The other is the analog processing of numerosity, which has been characterized as an imprecise way of estimating that allows quick, analog numerical judgments without relying on serial counting [19], via the innate approximate number system (ANS). In this system, numbers larger than four are represented in a logarithmic manner and obey Weber’s law [7, 8, 16, 20, 34, 41]. In other words, this system treats mathematically equal quantities differently in some cases, such as two “5”s more than one “10” (i.e., 5 + 5 > 10).

Although guided by two distinct numerical processing modes, humans, especially adults, do not appear to be confused when selecting between the approximate and precise representations. One main reason can be that ANS and ENS differ in how and when they involve in non-symbolic and symbolic numerical representations. Specifically, ANS plays a central role in processing non-symbolic quantities and begins influencing individuals’ perception of numerosity and spatial/temporal quantities as early as infancy [5, 49, 50, 84, 88, 89]. In contrast, the ENS develops after individuals acquire symbolic number knowledge and skills, enabling adults to operate digits in a precise and linear way [14, 15]. From a developmental perspective, during preschool and school-age years, individuals show logarithmic (i.e., ANS) or mixed logarithmic-linear characteristics when representing symbolic digits eg., [76]. By adulthood, individuals show linear characteristics consistent with the ENS when processing symbolic numbers [14, 15], and exhibit distinct neural activity when processing symbolic number and non-symbolic quantities [39, 82]. These differentiations suggest that ENS serves as the primary mechanism for symbolic number processing in adults [14, 40, 80, 85], thereby vetoing the disagreement of ANS over “5+5=10”. However, has ANS been exclusively ruled out from adults’ symbolic number processing? To what extent does the idea of “5 + 5 > 10” still hold true in terms of mental numerical representation? Dual-process Theory (DPT) offers a new perspective for exploring this question.

According to DPT, cognitive functions emerge from two interactive systems: System 1, which generates intuitive responses and rapid judgments, and System 2, which oversees monitoring and correction [10, 42]. Analogous to ANS, System 1 is evolutionarily shared with human and animals [26], and is prone to imprecision and biases during information processing. In contrast, System 2 embodies the precision attributed to ENS, although its supervisory capacity may falter under cognitive load or distraction. While it remains unclear whether System 1 possesses the logarithmic characteristics of ANS, the interplay between system 1 and system 2 in DPT provides insights into numerical cognition, suggesting that the numerical information processing of adults may not be solely influenced by a single system. This proposal aligns with Graziano, who first proposed that Dual-Process Theory can apply to numerical cognition, especially the study of the two different number representation systems [36]. Likewise, this view is supported by Norris and colleagues. Their studies found a high correlation between ANS and inhibition [58] and revealed that the relationship between adults’ symbolic mathematical ability and ANS reflects contributions from both inhibitory control and the ANS itself [57]. Finally, akin to the developmental research of Siegler and Booth [76], DPT also offers a new angle for understanding the developmental changes in humans’ symbolic number processing ability. To be exact, from early childhood to adulthood, individuals’ numerical representation may be affected by both ANS and ENS, but the influence of ANS gradually decreases from early life to adulthood, while the influence of the later gradually increases. Namely, ANS may not completely withdraw from adult symbolic number processing, but rather may exert its influence in a more subtle manner. However, this assumption has not yet been verified empirically.

Some previous studies suggest the covert influence of ANS on adult symbolic number processing. For example, Anobile et al. [1] compared adults’ number line judgment performances in a single task condition (number line estimation alone) and a dual task condition (where participants completed a color-orientation task before performing the number line judgment). They found that participants exhibited a more pronounced logarithmic pattern in the dual-task situation. Similarly, Dotan and Dehaene [23] analyzed hand movements in adults during a symbolic number line estimation task, finding that the mapping between numerals and hand movements exhibited a partially logarithmic pattern in the early stages of hand movement, particularly when participants had limited attention resources. Though these studies did not directly and systematically investigate the role of ANS in symbolic number processing, they provide evidence for the subtle influence of ANS on symbolic number representation from different perspectives: Anobile et al. [1] initially suggested that the effect of ANS on adults’ digit processing may be influenced by the allocation of attention resources, a view consistent with DPT. Dotan and Dehaene, to a certain extent, indicates the possibility that adults activate ANS while processing symbolic digits. Finally, some previous studies linking differences in ANS accuracy or non-symbolic quantity processing in adults with their math performance also support the significant role of ANS in adult symbolic number processing [2, 37, 48, 51, 62, 63, 73, 86]. All of these findings suggest the importance of ANS in adults’ representation of symbolic numbers.

Nonetheless, while DPT and empirical findings might indirectly support the view that ANS retains functionality in adults’ symbolic numerical representation, the findings from a series of studies using number comparison tasks are apparently against this hypothesis because of their failure in figuring out the size effect on symbolic number tasks. The size effect is a robust phenomenon originating from ANS, which refers to the phenomenon that number comparisons are easier for smaller number pairs (e.g., 1 vs. 5) than for larger number pairs (e.g., 91 vs. 95) despite equal distances in numerical magnitude exist in two numbers for comparison [6, 43–45, 55], reflecting the core feature of logarithmic mapping between numbers.

Different from the distance effect, which refers to faster and more accurate judgement of numbers with larger magnitude differences and has been extensively observed for both non-symbolic and symbolic number representation [3, 6, 30, 35, 55, 72] the size effect has been mostly found in non-symbolic quantity processing rather than in symbolic number processing. For instance, Verguts et al. [85] found that the size effect in non-symbolic number processing tasks did not vary across different paradigms, whilst the size effect of symbolic number was only present in digit comparison but not in parity judgment task or number naming task. Additionally, Krajcsi and colleagues proposed in a series of studies that ANS dominates non-symbolic quantity processing but is not involved in symbolic number processing, and defined a Discrete Semantic System (DSS) for symbolic digits processing [43–45]. Specifically, Krajcsi et al. (2016) [44] analyzed the relation between size effect and number word frequency of digit comparison, and found no size effect on reaction time and error rate in adults’ performances after controlling for number word frequency, suggesting that adults’ symbolic number processing should be mediated by the DSS and ANS has a limited influence on symbolic number representation. Moreover, Krajcsi [43]investigated the correlation between the regression slopes of the size effect and the distance effect in adults’ symbolic number/non-symbolic quantity comparison. The results showed that the slopes of the size effect and the distance effect were correlated in dot comparison, but not in digit comparison, and the slopes of size effect and the distance effect were uncorrelated across the two different tasks. In the same way, Krajcsi and colleagues [45] further compared the role of ANS in predicting error rates and reaction times in adults’ dots/digits comparison tasks, finding an ANS effect for non-symbolic numbers only. These results consistently indicate that while ANS significantly influences adults’ processing of non-symbolic numbers, it does not appear to have a significant effect on symbolic number processing.

The lack of evidence on the size effect in symbolic number comparison tasks is so far inconsistent with the aforementioned findings on the involvement of ANS in adults’ symbolic number processing. One possible explanation for these null results is that reaction time analysis based on keyboard responses are not sufficiently sensitive for capturing subtle effects [25]. Moreover, a logarithmic mapping of ANS does not necessarily have significant influence on the speed of symbolic number processing, because this effect may be suppressed or modified, as we have speculated based on DPT. Furthermore, compared to the visual processing of non-symbolic numerical information, the ease of symbolic number processing for educated adults may also make it difficult for researchers to observe a significant size effect on reaction time. Nevertheless, the logarithmic mapping may still feature and can be observed through other approaches which are more sensitive to representation biases.

Recent studies [21–25, 29, 47, 69, 79] suggest that hand tracking technique could serve as a valuable alternative to traditional key-pressing tasks. In our manual reaching task, participants are required to respond by manually moving their fingers to the target positions. The dynamics and movement trajectory of the fingers would potentially provide information to test the size effect during number processing. Song and Nakayama [79] first introduced the manual reaching task for number comparison and observed clear distance effects in movement trajectory. Rugani and Sartori [69] highlighted that numerical magnitude can influence motor execution and emphasized the advantages of hand tracking approach in research on the mental number line. Consistently, Erb and Marcovitch [25] found that, using a manual reaching task, one can not only obtain similar results as using traditional key pressing tasks, but also overcome the shortcoming of reaction time, which is not sensitive to index dynamic processes underlying cognitive control over time. Dotan and colleagues have further revealed adults’ representation of magnitude [22], and cognitive processes in number-to-position mapping tasks [23]. They also developed a number-to-quantity conversion model [24] by analyzing finger trajectories in the manual reaching task. These findings have demonstrated the high sensitivity of hand tracking technique to cognitive processes in number representation and indicated hand movement trajectory as a good window for investigating whether and how the size effect features in symbolic number processing.

In the present study, we adapt this new method to investigate the size effect in adults’ symbolic number comparison. Dotan and Dehaene [23] pointed out that the logarithmic number-to-position mapping during the number matching task was partially due to decision making factors. This complicates interpreting the logarithmic mapping as direct evidence of the involvement of ANS in symbolic number processing. To resolve this issue, we opted for a comparison task instead of a matching task, which helped participants to avoid factors regarding decision making in number-to-position mapping and allowed for a more direct investigation of whether ANS is involved in processing symbolic numbers via examining the potential size effect on symbolic number comparison by means of the participants’ hand movements.

Specifically, this study aims to explore the potential role of ANS in adult symbolic number processing. Based on DPT, we propose that although adults exhibit linear and precise characteristics consistent with the ENS when processing symbolic digits, ANS may still exert a subtle influence on adults’ representation of symbolic number. The experimental task in this study was adapted from [79], which compared the numbers 1–4 and 6–9 with 5 and found a significant distance effect. In the task, participants were required to compare numbers and indicate their choice by manually reaching the target under different conditions. To explore the influence of different number ranges, in Experiment 1, we added larger number pairs (91–99) in addition to Song and Nakayama’s [79] small number pairs. Moreover, to control for differences in the tens place value between two-digit and one-digit numbers, we added a meaningless symbol ‘#’ in front of small number stimuli (#1-#9) in Experiment 1. In Experiment 2, three number comparison tasks with equivalent numeric magnitudes but different formats (i.e., 1–9, #1-#9 and 01–09) were included to further validate the robustness of our findings and rule out the possibility that the results of Experiment 1 were due to visual differences in the stimuli. We hypothesize that, if ANS did affect symbolic number representation in a logarithmic manner, trajectory deviation from center line would be larger when comparing small numbers than when comparing large numbers (i.e., size effect on finger movement trajectory), as illustrated by Fig. 1. Correspondingly, we expected no significant differences in adults’ performance across the three format conditions in Experiment 2.

Fig. 1.

Trajectory deviation from center line in small numbers and large numbers

Note. The solid lines represent the conditions of #1-#9, and the dotted lines represent the conditions of 91–99

Experiment 1

Method

Participants

Forty-eight right-handed college students (24 male, age: 18–28 years, M = 22.06 years) were recruited from a university in Shanghai, China. All the participants had a normal or corrected-to-normal vision and had no musculoskeletal dysfunctions. All the experimental procedures were approved by and comply with East China Normal University Ethics Committee.

Stimuli & apparatus

Figure 2 illustrates the experimental setting. A plexiglass screen was placed vertically, approximately 50 cm away from the participant, with a rear-mounted projector displaying the stimulus in the middle area of the screen (43.5cm × 26.5cm). Three horizontally arranged unfilled squares (2^° × 2^°) were equally displayed on the screen, one of which was located in the central of the screen, and the other two were 12° left or right away from it. A symbolic number was presented on the middle square in each trial. All the squares are white against the black background.

Fig. 2.

The illustration of the experiment setting Note. The setup comprised two 6DoF sensors. One sensor was attached to the back of participants’ right index finger to record finger movements. The other sensor was mounted behind the Plexiglas screen and served as a spatial reference (blue area in the figure). The signal transmitter was placed at the lower left corner of the table. The 3D Guidance trakSTAR signal transmitter was positioned above the table to provide the magnetic field

The experimental program was developed and executed using MATLAB (MathWorks). The 3D positions of the participant’s right index finger were recorded during movements at a rate of 720 Hz using a 3D Guidance trakSTAR system (Northern Digital, Waterloo, ON, Canada). All digit stimuli were presented as 60px × 60px images on a black background (see Fig. 3), with gray numerals in bold 46 pt Times New Roman font. The computer used to run the experiment had a screen resolution of 1920 × 1080 pixels.

Fig. 3.

The illustration of the procedure of a trial in Experiment 1

Procedure

The participant was tested in a quiet, semi-dark room and completed a small number comparison task (#1-#9) in which the standard number is #5, and a large number comparison task (91–99) in which the standard number is 95, in separate blocks. In each task, the participant was required to compare the present number with the standard number and execute manual reaching as quickly as possible. The participant needed to point a) to the left square when the presented number was smaller than the standard number, b) to the right square when the presented number was larger than the standard number, or c) to the center when the presented number was equal to the standard number.

A trial started when the participant placed the right hand index finger on the start position, which was 50 cm away from the screen and marked by a small LEGO piece on the desk, and stayed still for about 300ms. Then a white fixation cross appeared in the central of the screen for 300 ms and, after that, a number was presented in the center of the screen. The participant was required to execute the pointing task as quickly as possible after the presentation of the number, and the movement trajectory of the index finger was recorded during the whole movement (Fig. 3). If the participant answered incorrectly or the total response time exceeds 700ms, a high-pitch sound with the corresponding red text (“Wrong!”/“Too Slow”) would be presented for 1000 ms to warn the participant. To prevent the movement before the presentation of stimulus, the participant was asked to keep their fingers still on the start position until they see the stimulus. Otherwise, the program would warn the participant and restart the trial automatically.

The experiment included two task conditions, each involving 9 different number stimuli presented 14 times, resulting in 126 trials per task. Consequently, each participant completed a total of 252 trials. Trials of different conditions were fully randomized within each task, and the order of the two tasks was counterbalanced. 24 participants completed the large number comparison task first, followed by the small-number comparison task (#1-#9), while the remaining participants completed the tasks in the reverse order. We did not use interleaved trials in different task conditions because of the concern that different magnitudes of standard numbers in a same block might introduce unexpected confusion for participants’ judgments. Participants performed 30 practice trials before the first task, and no more practice was done before the second task. Each participant spent about 30 minutes to finish the whole experiment.

Data analysis

To analyze the finger movement, we adapted the analysis methods from previous studies [4, 25, 28, 70, 79].

Specifically, we first calculated the area under the curve (AUC) to index the movement deviation. As illustrated in Fig. 4, the AUC was defined by the area between the finger movement trajectory and the y axis. To remove individual biases in hand movement, each AUC was subtracted by the mean AUC of the standard number condition for each individual participant. We also analyzed the kinematics of individual participants by investigating the initiation time (IT) of finger movement in each trial, defined as the time interval from the presentation of time to the onset of the movement (i.e. the first time when the finger movement speed reached 10% of the maximum finger movement speed), and the movement time (MT), defined as the time interval from the onset of the movement to when the finger reached the target position, in different conditions

Fig. 4.

The area under the curve in tasks. Note. Light gray areas indicate negative AUC values when participants judge numbers smaller than #5 or 95, relative to the center line. Dark gray areas indicate positive AUC values when participants judge numbers larger than #5 or 95

Trials were excluded from analysis if 1) the responses were incorrect, 2) the finger moved backward during the movement, or 3) the initiation times exceeded 3 standard deviations from the mean movement initiation time. 751 of 12096 trials (6.21%) in total were excluded. Statistical analyses in this study were performed using IBM SPSS Statistics 25.0, with additional Bayesian robustness analyses (Bayesian ANOVA) conducted in JASP 0.95.4.0, and effects were evaluated using Bayes factors for inclusion (BF_incl). Results of Bayesian model comparisons are provided in the supplementary materials (Table S1 to Table S4).

Results

Trials of different number conditions were collapsed according to the numerical distances between the target number and the standard number (i.e., #5 or 95), making five distance conditions, which were referred as conditions of D0 (#5 or 95), D1 (#4/#6 or 94/96), D2 (#3/#7 or 93/97), D3 (#2/#8 or 92/98) and D4 (#1/#9 or 91/99), respectively. The means of response accuracy, AUC, IT and MT in different conditions of number magnitude and distance are listed in Table 1.

Table 1.

Means of hand movement measures in Experiment 1

RA = Response accuracy, AUC= Area under curve, IT =Initiation time, MT =Movement time

Response accuracy

Figure 5 shows the mean response accuracy in different conditions of task and distance. The grand mean accuracy was 97.6% (± 2.4% s.d.), indicating that the participants performed well in the tasks. A 2 (number range, #_ vs. 9_) × 5 (distance) × 2 (task order) mixed-design ANOVA was performed and we found a significant main effect of number range (F(1,46) = 13.02, p =.001, η_p² =.221) supported by Bayesian analysis (BF_incl = 2.82), indicating that response accuracy was higher in the small number conditions (i.e. #1-#9) than in the large number conditions (i.e. 91–99). The distance effect was significant as well (F(4,184) = 6.77, p <.001, η_p² =.128), with extremely strong Bayesian support (BF_incl = 133.20), indicating that accuracy increased as the distance between the standard and target numbers grew. The main effect of task order was not significant (F(1,46) =.190, p =.665, η_p² =.004). No significant interactions were found on response accuracy (F <.99, p >.417, η_p² <.022).

Fig. 5.

Mean response accuracy in different tasks and distances

AUC

The mean trajectories of finger movements and the mean AUC in different conditions of number range and distance are plotted in Fig. 6. A 2 (number range) × 4 (distance) × 2 (task order) mixed-design ANOVA was performed. As mentioned above, the D0 conditions (with standard number as target number) were reference conditions and not included for analysis. The main effect of number range was significant (F(1,46) = 10.54, p =.002, η_p² =.186) with very strong Bayesian support (BF_incl = 42.02), indicating that AUC was larger with small numbers than with large numbers. Likewise, the main effect of distance was significant (F(3,138) = 64.25, p <.001, η_p² =.579) and decisively supported by Bayesian analysis (BF_incl = 2.30 × 10¹⁴). As shown in Fig. 6, AUC increased with larger distance between the standard number and the target number. The main effect of task order was not significant (F(1,46) =.576, p =.452, η_p² =.012).

Fig. 6.

Mean AUC in different number ranges and distances. Note. AUC=area under curve. A negative AUC indicates trials where participants judged the numbers less than #5 or 95, whereas a positive AUC corresponds to trials where the stimuli were from #6 to #9 and 96 to 99

There was a significant interaction between number range and task order (F(1,46) = 11.62, p =.001, η_p² =.202, BF_incl = 25.21). The post-hoc simple-effects analysis showed that the effect of number range was significant for the participants who first did the task with small numbers (F(1,23) = 18.77, p <.001, η_p² =.449), but insignificant for the group who first did the task with large numbers (F(1,23) =.016, p =.900, η_p² =.001). No other significant interactions were observed (F < 1.39, p >.25, η_p² <.030).

Kinematics

A 2 (number range) × 5 (distance) × 2 (task order) mixed-design ANOVA was conducted for IT. The result showed that there was no significant main effect or interaction (F < 2.83, p >.099, η_p² <.058). For MT, we did not include D0 conditions because of the different movement distances between D0 conditions and other distance conditions. We then performed a 2 (number range) × 4 (distance) × 2 (task order) ANOVA. The main effect of distance was significant (F(3,138) = 22.22, p <.001, η_p² =.326, BF_incl = 1.36 × 10⁷). As Fig. 7 shows, MT decreased when the distance between standard number and target number increased. There was no other significant main effect or interaction (F <1.45, p >.23, η_p² <.031).

Fig. 7.

Mean MT in different number ranges and distances. Note. MT=movement time

Discussion

The most important finding of this experiment was that a significant size effect (i.e. #1-#9 vs. 91–99) was found in both response accuracy and movement trajectory deviation in the number comparison task. Participants made more accurate responses and exhibited larger deviations from the center line when comparing small numbers (#1-#9) than larger numbers (91–99). These results demonstrated that, given the same numerical magnitude distances, the perceived differences between mental representations of two symbolic numbers were significantly larger when the magnitudes of the numbers were smaller, which has been documented as the size effect [6, 43–45, 55]. This effect seemed to indicate a logarithmic characteristic of ANS influence on symbolic number representation. If symbolic number comparison was exclusively due to processing of ENS, such size effect would not be observed. In other words, these findings provide important empirical evidence that ANS is still involved in adults’ magnitude comparison of symbolic numbers to a certain extent.

In addition, the size effect on trajectory deviation was significantly modulated by task order. Specifically, the effect was only observed when the small-number conditions were presented first, whereas trajectory deviations were similar across conditions when the large-number conditions were presented first. One possible reason for this difference is that the participants who first carried out tasks with larger numbers may have gradually developed a strategy to ignore the digit (“9_”) or symbol (“#_”) information in the position of tens e.g., [60], thereby reducing the differences between numerical representations in the conditions of larger and smaller numbers.

A distance effect was also observed in both response accuracy and trajectory deviations in the present experiment. These results are partially consistent with the findings of [79], although in their study, the effect was not significant for response accuracy. Such difference may be due to fact that Song and Nakayama used a much smaller sample (9 people), which did not provide sufficient statistical power to test the effect. These results are also consistent with the previous findings from the studies using key-pressing tasks e.g., [17, 35, 38, 83], further supporting the validity of the hand tracking technique in unveiling the mechanisms of numerical representation and processing [69].

In term of kinematics, we found a number distance effect on MT which was consistent with the previous findings from Song and Nakayama [79] as well as Erb and Marcovitch [25]. The number distance effect on movement trajectory and MT were consistent with each other. As the magnitude distance between standard number and target number increased, participants showed both more deviated movement trajectory and shorter MT. No other effects were found on either IT or MT. These results excluded the possibility of speed-accuracy trade-off in this experiment.

Experiment 2

The results of Experiment 1 showed the number range effects on response accuracy and trajectory deviations, indicating the possible involvement of ANS in symbolic number representation. However, alternative explanations remain for the results of Experiment 1, in which a small number comparison task (#1–#9) and a large number comparison task (91–99) differed not only in magnitude but also in structure. It could be difference between the one-digit vs. two-digit formats rather than the magnitudes of numbers that led to the significant differences of participants’ response accuracy and movement trajectory. Alternatively, participants may have treated symbol ‘#’ in the tens position as an extra cue, which promoted their hand movements.

Therefore, we conducted Experiment 2 as a follow-up control experiment to test these possibilities by comparing performances across different number format conditions, while keeping numerical magnitude constant across conditions. To be specific, we tested three number format conditions, namely one-symbol condition (1–9), with-# condition (#1-#9), and with-0 condition (01–09). If the size effects observed in Experiment 1 were exclusively driven by numerical magnitude rather than number format, no significant differences in performance would be expected among these conditions.

Method

In Experiment 2, the stimuli and procedure were the same as in Experiment 1 with two exceptions. First, three number format conditions, namely one-symbol, with-# and with-0 conditions, were used in this experiment, as mentioned above. Second, considering the target numbers used in Experiment 2 referred to same magnitudes, trials of different number format conditions were not separately tested but were interleaved in a randomized order in the same blocks. In this experiment, each stimulus was repeated 14 times, making a total of 378 trials. Each participant practiced 30 trials before the formal test. The entire experiment lasted approximately 40 minutes.

Thirty right-handed college students (M=20.8 years old, SD=2.27 years,16 females) were recruited from a university in Shanghai, China. All the participants had normal or corrected-to-normal vision and reported no musculoskeletal dysfunctions. All the experimental procedures were approved by and comply with the East China Normal University Ethics Committee.

The trajectories and the kinematics of the hand movements were analyzed by the same means as in Experiment 1.

Results

In this experiment, 597 of 11340 trials (5.26%) were excluded from analysis based on the same criteria as in Experiment 1. Trials of different number conditions were collapsed according to the numerical distances from the standard number as in Experiment 1. The means of response accuracy, AUC, IT and MT in different conditions of number format and distance are listed in Table 2.

Table 2.

Means of hand movement measures in Experiment 2

RA=Response accuracy, AUC= Area under curve, IT= Initiation time, MT= Movement time

Response accuracy

Figure 8 shows the mean response accuracy in different conditions of number format and distance. A 3 (number format) × 5 (distance) repeated-measures ANOVA showed that the main effect of distance was significant (F(4,116) = 6.76, p <.001, η_p² =.189). This effect was extremely strongly supported by Bayesian analysis (BF_incl = 219.38), indicating that the accuracy became higher when the distance between standard number and target number increased. The main effect of number format was not significant (F(2,58) = 1.02, p =.368, η_p² =.034), with strong evidence in favor of the null hypothesis according to Bayesian analysis (BF_incl =.047). The interaction between number format and distance was not significant either (F(2,58) = 1.43, p =.185, η_p² =.047, BF_incl =.036).

Fig. 8.

Mean response accuracy in different number formats and distances

AUC

Figure 9 shows the mean AUC in different conditions of number format and distance, and D0 conditions were not included for analysis for the same reason as in Experiment 1. A 3 (number format) × 4 (distance) repeated-measures ANOVA was conducted. The ANOVA revealed a significant main effect of number format (F(2,58) = 10.43, p =.006, η_p² =.161), but Bayesian analysis provided only anecdotal evidence for this effect (BF_incl = 1.12). The post-hoc tests found that AUC were smaller in with-0 conditions compared to one-symbol and with-# conditions (with-0 vs. one-symbol: F(1,29) = 10.43, p =.003, η_p² =.265; with-0 vs. with-#: F(1,29) = 4.41, p =.045, η_p² =.132), but did not significantly differ between one-symbol and with-# conditions (F(1,29) = 1.06, p =.312, η_p² =.035). The main effect of distance was significant as well (F(1,29) = 72.14, p <.001, η_p² =.713, BF_incl = 2.50 × 10¹⁴), indicating that AUC increased when the distance between target number and standard number increased. The interaction between number format and distance was not significant (F(6,174) =.772, p =.593, η_p² =.026, BF_incl =.098).

Fig. 9.

Mean AUC in different number formats and distances. Note. AUC=area under curve

Kinematics

A 3 (number format) × 5 (distance) repeated-measures ANOVA was conducted for IT. The main effect of number format was not significant (F(2,58) =.098, p =.907, η_p² =.003), with strong Bayesian evidence supporting the null hypothesis (BF_incl =.028). The main effect of distance was marginally significant (F(4,116) = 2.42, p =.052, η_p² =.077), suggesting the trend that IT became higher when the distance between standard number and target number was smaller. The interaction between number format and distance was not significant (F(8,232) =.737, p =.659, η_p² =.025, BF_incl = 6.30 × 10⁻⁴).

For MT, D0 conditions were not included for analysis as in Experiment 1 and we conducted a 3 (number format) × 4 (distance) repeated-measures ANOVA. The main effect of number format was not significant (F(2,58) =.204, p =.816, η_p² =.013) and the null hypothesis was strongly supported by Bayesian analysis (BF_incl =.036). The main effect of distance was significant (F(3,87) =.204, p <.001, η_p² =.462, BF_incl = 6.00 × 10⁷), indicating that MT decreased when the distance between standard number and target number increased (Fig. 10). The interaction between number format and distance was not significant (F(6,174) =.526, p =.788, η_p² =.018, BF_incl =.005).

Fig. 10.

Mean MT in different number formats and distances. Note. MT=movement time

Discussion

One important finding from Experiment 2 was that the format effect on trajectory deviation was not detected between the one-symbol condition and the with-# condition. This supports the conclusion drawn from Experiment 1, thereby ruling out the possibility that participants treated ‘#’ as an extra cue to facilitate performance in the comparison task.

However, the finding that the trajectory deviations were smaller in the with-0 conditions than in the two other conditions suggests potential different mechanisms underlying the processing of one-digit and two-digit symbolic numbers. Multiple studies [24, 32, 54, 60, 65] have shown that, different from one-digit number processing, two-digit number processing is more complicated and might involve multiple processes (e.g., a mixture of parallel and serial processes), which might account for the trajectory differences observed between the with-0 condition and the other two conditions in this experiment. Furthermore, According to Prpic et al. [66], multi-digit numbers can be viewed as hierarchical structures composed of individual elements. Therefore, unlike the processing of one-digit numbers, people may be influenced by both the overall magnitude (global information) and the value of each constituent digit (local information) when representing multi-digit numbers. Thus, the distinct trajectory deviations observed in the with-0 condition may reflect differences in participants’ processing of local and holistic information between two-digit and one-digit numbers.

The distance effects observed in Experiment 1 were well replicated in Experiment 2. Again, these findings indicate the validity of the manual reaching task in unveiling and comparing representations of symbolic numbers. In addition, we observed a marginally significant effect of numerical distance on participants’ IT in this experiment (p =.052). This distance effect, as revealed by IT, was not found in Experiment 1 or in [79]. One possible explanation for such differences is that Experiment 2 included more number formats in same blocks, which may have made the decision-making process more difficult and, consequently, more distinguishable across different conditions.

General discussion

This study examined the role of ANS in adults’ symbolic number processing. Based on DPT and prior empirical findings, we proposed that ANS does not completely withdraw from adults’ symbolic number representation, but rather influences digit processing in a more subtle manner. Through the implementation of a manual reaching paradigm, Experiment 1 revealed a significant size effect on both response accuracy and trajectory deviation in adults. In Experiment 2, we then ruled out the possibility that the size effect was originated from stimulus format differences, confirming the proposition that ANS still influences adults’ symbolic number processing.

In addition to the size effect, this study also found the distance effects in the digit comparison tasks, which was consistent with findings from previous studies using keyboard responses with reaction time analyses e.g., [6, 25, 30, 35, 38, 55, 72, 74, 83]. These results do not only demonstrate the feasibility of using hand trajectories to analyze individual representation of numerals, but also provide insights for future research of symbolic number representation in adults and individual development in symbolic number processing.

The results of this study provide new evidence for the potential application of DPT [10, 26, 42] in the field of numerical cognition. According to prior research, although DPT has been widely applied to various cognitive domains, including learning [81], reasoning [26, 27, 33], social cognition [46, 78], and decision-making [42], its potential application in individual mathematical learning and numerical representation has only recently been proposed [36] and remains empirically underexplored. By employing hand tracking techniques, this study revealed a dual processing mechanism in adult symbolic number representation, demonstrating that adults’ digit processing is not determined by a single system, but rather by the joint contribution of ANS and ENS. This not only aligns with DPT’s postulation that a given cognitive function does not arise from a single system but from a pair of systems (i.e., System 1 and System 2), but also provides a solid foundation for future research. Specifically, according to Kahneman and Frederick [42], System 1 and System 2 in Dual-Process Theory may interact and play distinct roles in individual cognitive processes: System 1 generates intuitive judgments, while System 2 monitors the quality of these responses. By revealing the dual processing mechanism underlying adult symbolic number representation, this study paves the way for future research to investigate the dynamic interaction between the ANS and ENS in symbolic number processing and individual mathematical learning.

In addition, this study offers valuable insights into the lifelong development of symbolic number representation, underscoring the necessity for further investigation into the distinct influences of the ANS and ENS on individual numeral processing abilities. As mentioned in the introduction, substantial prior research [1, 52, 53, 68, 71, 85] has indicated that ENS plays a central role in symbolic number processing of adults. Combined with the findings of this study, an intriguing possibility, paralleling DPT, is that ENS may function akin to System 2, monitoring and correcting ANS performance to refine individual numerical representations. This hypothesis warrants future investigation, particularly regarding the factors that govern the interaction dynamics between ANS and ENS. For example, Anobile et al. [1] observed that adult participants displayed more pronounced nonlinear patterns in symbolic mapping under dual-task conditions, thereby underscoring the critical role of attentional resources in facilitating linear number mapping. Consequently, future researchers could systematically examine how varying levels of attentional demand modulate the respective contributions of the ENS and ANS to symbolic number processing in adults.

From a developmental perspective, the findings of this study will inspire future research on the relative influences and mechanisms of ANS and ENS at different developmental stages. According to Siegler and Booth [76], individuals exhibit three distinct representational stages in symbolic number representation tasks with age: logarithmic, mixed linear-logarithmic and linear. In light of the current study, this might suggest that ENS and ANS do not replace each other. Instead, as individuals age and the acquired ENS matures, ENS gradually inhibits ANS, and the influence of ANS on symbolic number representation becomes increasingly implicit. This interpretation aligns not only with DPT but also resonates with Siegler’s Overlapping Waves Model [75]. The latter postulated that individuals across developmental stages may co-deploy multiple cognitive processing and representational strategies within tasks, with final strategy selection contingent upon context and accumulated experience. Supporting this framework, Siegler and Opfer [77] demonstrated that number lines with a wider range amplify the logarithmic representation while diminishing the linear aspect of symbolic number processing. Similarly, Dotan and colleagues [22, 23] found that adults’ finger trajectories exhibit a logarithmic pattern in the early stages and are modified in the later stages. Furthermore, this study highlights the need for future research to examine the role of domain-general abilities in symbolic number processing in both children and adults, as well as how these abilities modulate the relationship between ANS and ENS. For instance, previous studies have shown that executive function significantly affects children’s symbolic mathematics abilities [13], and this influence might persist into adulthood e.g., [11], potentially providing insights into the developmental trajectories and interactions of ANS and ENS. Nevertheless, all these interpretations require further empirical validation in future research.

The format effect observed in Experiment 2 is another interesting finding of the current study, indicating the different mechanisms may underlie the processing of one-digit numbers (i.e., single-digit condition and with-# condition) and two-digit numbers (i.e., with-0 condition). The existing literature has indicated that people process two-digit number in a hybrid manner. For example, Nuerk et al. [59] proposed that the decades and units may be presented in different number bin in mental number line. Similarly, Dotan and Dehaene [24] found that the delayed presentation of the unit digit of a two-digit number modulated both the effects of unit and decade on finger movement trajectories and suggesting more complicated mechanisms in two-digit number representations than in one-digit number representations. Currently, we have no clear idea about the exact relationship between two-digit numbers processing and the two number representation systems. However, the number format effect observed in Experiment 2 suggests two possible interpretations: First, the number format effect has raised the possibility that the processes of two-digit numbers rely more on ENS than the processes of one-digit numbers do. In other words, two-digit numbers might induce more exact numerical representations than one-digit number, even when they represent same quantities (e.g., 09 vs. 9/#9). This difference may stem from the developmental timeline and learning processes associated with one-digit and two-digit numbers. One-digit numbers are encountered more frequently in early childhood, making them more familiar to individuals. In contrast, two-digit numbers are typically acquired later and require an understanding of both counting routines [9] and the place value system e.g., [12]. Second, the number format effect may reflect participants’ preferences for processing local numerical information rather than global information when dealing with two-digit numbers. According to Prpic and colleagues [66], multi-digit numbers can be considered a hierarchical structure composed of multiple digits, in which individual digits convey local numerical values and their spatial arrangement determines the global magnitude. Unlike the holistic encoding of single-digit numbers (i.e., the with-# condition and the one-symbol condition), two-digit numbers in the with-0 condition may elicit a tendency to engage in local processing strategies, and such differences in processing preference e.g., [64] may contribute to differential responses across different number formats, even when the numerical magnitudes are equivalent. Future research should further examine how digit format influences multi-digit number processing and how these differences relate to the involvement of ANS and ENS.

The present study has several limitations. First, while this study explored how the ANS influences adults’ symbolic number processing under the DPT framework, its emphasis on detecting the size effect rendered the experimental design inadequate for disentangling the relative roles of linear and logarithmic representations. This limits the theoretical implications of this study for extending DPT. Future research could further address this issue by incorporating manual trajectory tracking or other complementary paradigms. For instance, prior studies have shown that the position pointing task plays a distinctive role in numerical processing [67] and provides a valuable means for distinguishing between logarithmic and linear accounts of number representation e.g., [31]. Accordingly, future research could integrate this task with number representation paradigms and apply dynamic analyses of participants’ performance, thereby further validating and extending the DPT framework for number processing. Second, this study did not control for other potential variables, such as word frequency and response hand. Previous studies e.g., [43, 44] have shown that word frequency can influence adults’ reaction times in numerical symbol comparison, thereby affecting size effect and distance effect. In addition, prior research e.g., [18, 56, 87] has reported differences between left-hand and right-hand responses to digits of different magnitudes. However, we only collected right-hand responses in this study, which limited control over the potential influence of hand dominance. Future research may benefit from implementing more comprehensive controls for these variables and conducting further experiments to clarify the internal mechanisms of adult number cognition. Third, the present work did not provide a comprehensive comparison for existing models of adult symbolic number representation. For example, differences between DSS model [44] and ANS in adult symbolic number processing were not examined. As a result, the findings do not fully rule out the possibility that adult symbolic number processing may be explained by alternative models such as DSS. Nevertheless, these results provide a basis for future model comparisons. Finally, this study used MT and IT as indicators without examining other potential kinematic features, such as moment-to-moment velocity, or finer-grained segmentation of movement time. This may limit the analysis of size effects across the full process of adult symbolic number processing. Future research is therefore encouraged to further investigate this issue.

Conclusions

This study investigated the involvement of ANS in adults’ symbolic number processing. Through a manual reaching task, Experiment 1 showed that adults had higher response accuracy and more deviated movement trajectories in small number conditions than in large number conditions, demonstrating the size effect on symbolic number comparison and indicating a critical role of ANS in adults’ representation of symbolic number. Moreover, in Experiment 2, we ruled out the possibility that the observed ANS effects were due to the differences in number formats rather than in number range. Collectively, these findings provide compelling evidence for the implicit influence of ANS on symbolic number processing, underscore the potential of visuomotor tasks for probing implicit cognitive processes, and lay a foundation for understanding the interaction between the ENS and ANS across the lifespan within the framework of Dual-Process Theory.

Abbreviations

ANS

Approximate Number System

ENS

Exact Number System

DPT

Dual-Process Theory

DSS

Discrete Semantic System

AUC

Area under the curve

IT

Initiation time

MT

Movement time

Authors’ contributions

DL: Visualization, Formal analysis, Writing- Original draft preparation. YW: Conceptualization, Investigation, Formal analysis. ZC: Methodology, Writing-Original draft preparation, Funding acquisition. CD: Writing-Reviewing and Editing, Supervision. All authors read and approved the final manuscript for submission to this journal.

Funding

This work was supported by the Shanghai Municipal Natural Science Foundation.

Data availability

The data supporting the findings of this study are available on the Open Science Framework (OSF) at https://osf.io/t4rju (Dataset title: Same Digits, Different Magnitudes: Manual Reaching Unveils Dual Systems in Symbolic Number Representation).

Declarations

Ethics approval and consent to participate

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Declaration of Helsinki and its later amendments or comparable ethical standards. This study was approved by the University Committee on Human Research Protection of East China Normal University (ID: HR 263-2019). All participants in this study signed a paper informed consent form before the start of the experiment and were informed that all data would be stored anonymously and they had the right to withdraw from the experiment at any time.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.