The relationship between numerosity perception and mathematics ability in adults: the moderating role of dots number

Background It has been proposed that numerosity perception is the cognitive underpinning of mathematics ability. However, the existence of the association between numerosity perception and mathematics ability is still under debate, especially in adults. The present study examined the relationship between numerosity perception and mathematics ability and the moderating role of dots number (i.e., the numerosity of items in dot set) in adults. Methods Sixty-four adult participants from Anshun University completed behavioral measures that tested numerosity perception of small numbers and large numbers, mathematics ability, inhibition ability, visual-spatial memory, and set-switching ability. Results We found that numerosity perception of small numbers correlated significantly with mathematics ability after controlling the influence of inhibition ability, visual-spatial memory, and set-switching ability, but numerosity perception of large numbers was not related to mathematics ability in adults. Conclusions These findings suggest that the dots number moderates the relationship between numerosity perception and mathematics ability in adults and may contribute to explaining the contradictory findings in the previous literature about the link between numerosity perception and mathematics ability.


INTRODUCTION
Mathematics ability is a unique cognitive ability for humans, which is essential to our everyday life. Yet, there is a subgroup of people who have trouble acquiring mathematics skills. What factors influence mathematics ability? Previous studies have found that many domain-general factors could predict individual differences in mathematics ability, including memory (Raghubar, Barnes & Hecht, 2010), executive function (Kroesbergen et al., 2009) and early language ability (Hecht et al., 2001). In addition, symbolic number system (SNS) ability was an essential domain-specific predictor of mathematical ability (Lau et al., 2021;Lyons et al., 2018). For example, Geary & Vanmarle (2016) found that young children's core symbolic abilities significantly correlated with later mathematics adults. In particular, we conducted three measurements to assess participants' mathematics ability, including an arithmetical test, a subjective mathematics ability evaluation, and their mathematics course scores. In the meantime, we conducted the numerosity discrimination tasks of small and large number conditions separately, which allowed us to evaluate the numerosity perception of different dot numbers. In addition, to control the potential effects of other general cognitive abilities on the relationship between numerosity perception and mathematics ability (Fuhs & McNeil, 2013;Gilmore et al., 2013), all participants performed three measurements on their inhibition ability, visual-spatial memory, and set-switching ability. We predicted that the number of dots would moderate the relationship between numerosity perception and mathematics ability in adults, and only the numerosity perception of small numbers would be related to mathematics ability.

Participants
Sixty-four undergraduate students from Anshun University (28 males and 36 females, M age = 20.27, SDage = 1.11) participated in this study and received financial compensation for their time. This study was approved by the ethics committee of the School of Education Science, Anshun University (ID number: ASU-JYXY-201903) and all participants provided written informed consent at the beginning of the experiment. Three participants were excluded from the analyses due to either misunderstanding experimental task demand or missing data caused by computer errors.

Apparatus and software
All measurements were completed in a dimly lit room. Computer-based experiments were completed using MATLAB (R2016b; MathWorks, Cambridge, MA, USA) and PsychToolbox (Brainard, 1997) on a 23-inch LED monitor (Dell: U2312 HM) with 1,600 × 1,080 resolution at 60 Hz and a standard desk (width: 70 cm; length: 150 cm), viewed binocularly from a distance of 60 cm.

Measures
Numerosity perception, mathematics ability, inhibition ability, visual-spatial memory, and set-switching ability were measured in the following sequential order. The whole experimental procedure took approximately 85 min.

Numerosity perception
A modified version of a numerosity discrimination task adapted from Anobile, Cicchini & Burr (2014) was used to measure numerosity perception. Two sets of dots were of equal size and presented simultaneously for 250 ms on either left or right side of a central fixation point. Each set was constrained to a 14 • -diameter virtual circle and comprised a number of 0.3 • (in visual angle) diameter dots, half white and half black. For each trial, the set on the right side of the fixation point was the reference, and the left set was the probe. In separate blocks, the reference always contained 11 dots (small number condition) or 152 dots (large number condition), while the numerosity of the probe was changed from trial to trial, following the Method of Constant stimuli that varied the number of dots from 50% to 200% of the number of the reference that was split into seven equal log unit steps. For small number condition, the probe patches contained 6, 7, 9, 11, 14, 17, or 22 dots. For large number condition, the probe patches contained 76, 96, 121, 152, 192, 241, or 304 dots. Each of the seven probe stimulus levels was tested for 16 trials, and each block consisted of 112 trials. For each trial, participants were asked to judge as accurately and quickly as possible which set had more dots. Two blocks were presented in counterbalanced order. Participants gave their responses using a keyboard. Based on the pilot study, 14 practice trials were given to participants with feedback before the formal data collection.
''The proportion of probe greater trials'' was plotted against the reference number and fitted with the Methods of Probits (Sun et al., 2013). The 50% point estimated the point of subjective equality (PSE), and the difference in numerosity between the 25% and 75% points provided the just notable difference (JND), which was then divided by PSE to estimate the Weber fraction (w).

Mathematics ability
Three measures were used to assess participants' mathematics ability, including a modifiedversion arithmetical test (AT), a subjective mathematics ability evaluation, and the mathematics course score. The arithmetical test, which was designed based on Shalev et al. (2001), assessed basic arithmetic skills. The types of items that were tested included number facts, complex arithmetic, decimals calculation, and fraction calculation. In the self-developed measure of subjective mathematics ability evaluation, participants were instructed to evaluate their mathematics ability compared with other people using a 10-point scale (1 = the worst level, 10 = the best level). Furthermore, participants' college entrance examination score was used as the mathematics course score. Following the method of convention of Holloway & Ansari (2009), the scores for the three tasks were converted into z-scores and averaged to create an overall mathematics ability score for each participant, which was then used in the following analyses as an index of participants' mathematics ability.

Inhibition ability
Inhibition ability is the ability to suppress dominant, automatic, or prepotent response for irrelevant or no-longer-relevant information (Toll et al., 2011). The classical Go/No-go task was used to assess inhibition ability (Falkenstein, Hoormann & Hohnsbein, 1999). In this task, participants were told to respond to a yellow square (Go stimuli) and stop responding to a yellow circle (No-go stimuli). Participants were given five practice trials with feedback. The official task had 200 test trials without feedback, including 150 Go trials and 50 No-go trials. For each participant, the accuracy for No-go trials was recorded as the inhibition score.

Visual-spatial memory
Visual-spatial memory is a system for retaining location and object information (Wood, 2011). To assess participants' visual-spatial memory ability, a widely used memory span task for visual patterns was administered (Mejias, Gregoire & Noel, 2012). The task was developed by Wilson, JH & Power (1987) and consisted of the presentation of boxes with some boxes being filled in. The participants were instructed to remember the pattern of the boxes, which was displayed for 2 s. After a blank interval of 2 s, the boxes would be presented again with one box missing. The participants had to recall which box was missing. The initial pattern involved filling in two boxes, and upon receiving two correct responses out of three attempts, the complexity of the pattern would increase. The final number of successfully recalled boxes was recorded as a memory span score.

Set-switching ability
Set-switching ability is the ability to direct actions and thoughts to selected goals; it has been thought of as a kind of executive control. In this study, a modified Trail Making Test was presented in the paper-and-pencil version to measure set-switching ability (Arbuthnott & Frank, 2000). The test consisted of three trials: Part A-1, Part A-2, and Part B. Part A-1 included drawing a line connecting consecutive numbers from 1 to 25. Part A-2 included drawing a line connecting consecutive letters from A to Y. Part B included drawing a line connecting alternating numbers and letters in sequence (i.e., 1-A-2-B and so on). Participants were shown three trials in order. In each trial, a timer was set when the participants began drawing a line on paper and stopped when they put the pencil on the desk. The time to complete each trial was recorded. The ratio of completion time between Part B and Part A (average completion time of Part A-1 and Part A-2) was calculated as a set-switching score. A higher set-switching score indicated a poorer set-switching ability.

Data analysis
The statistical analyses of data were performed using SPSS software (Version 23.0). The correlations between all variables were measured using Zero-order correlation analyses. Then, we used hierarchical regression analysis to explore the predictors of mathematics ability.

Mathematics ability score
We looked for the correlations of the participants' performance on the three mathematics ability tests. Subjective mathematics ability evaluation was significantly correlated with AT accuracy (r = 0.32, p = 0.013) and the mathematics course score (r = 0.32, p = 0.012). However, AT accuracy was not significantly correlated with mathematics course scores, r = 0.15, p = 0.257. Given that mathematics ability is a complex construct and includes different skills (Mannamaa et al., 2012), these results showed that performances in three tests measured not only common aspects but also different aspects of mathematics ability. This might, in part, explain the previous contradictory relationships between numerosity perception and mathematics ability. To provide an evaluation of overall mathematics ability, we followed Holloway & Ansari (2009) and used the mathematics ability score for each participant in further analyses.

The correlations between numerosity perception and mathematics ability
A zero-order correlation matrix was calculated to investigate the relationship between all measurements (See Table 1). Results showed that mathematics ability significantly correlated with w small (r = −0.42, p = 0.001; the alternative Pearson r = −0.36, p = 0.004, see Fig. 1), but not with w large (r = 0.10, p = 0.444; the alternative Pearson r = −0.11, p = 0.414), indicating that lower mathematical ability was associated with higher w (reflecting poorer numerosity discrimination precision) in the small number condition.
Further correlation analysis showed that w small was significantly correlated with AT accuracy and the mathematics course score (r = −0.32, p = 0.017; r = −0.37, p = 0.005), but not with subjective mathematics ability evaluation (r = −0.21, p = 0.125). Conversely, we found no significant correlations between w large and three mathematics ability tests, rs < 0.24, ps > 0.075. It should be noted that there was also a significant correlation between w small and w large (r = 0.47, p < 0.001; the alternative Pearson r = 0.53, p < 0.001).
Here, we further examined the unique effect of dots number (i.e., small number and large number conditions) on the relationship between numerosity perception and mathematics ability. We used a specific test of the difference between two dependent correlations with one variable in common developed by Lee & Preacher (2013) to compare the correlations between w small , w large and mathematics ability. Results showed that there was a significant difference between the correlation of w small and mathematics ability and the correlation of w large and mathematics ability, p < 0.001. These results indicated the dots number moderated the relationship between numerosity perception and mathematics ability.

The correlations among other variables
The correlation between visual-spatial memory and mathematics ability reached marginal significance (r = 0.26, p = 0.054). Inhibition ability significantly correlated with w small (r = −0.27, p = 0.044) but not with w large (r = −0.15, p = 0.273). The correlations between all other variables also did not reach significance (|rs| < 0.19, ps > 0.149).

Hierarchical regression analysis
To measure the extent to which mathematics ability could be predicted by numerosity perception of small numbers, we ran a hierarchical regression analysis with w small as a Figure 1 The Pearson correlations between mathematics ability score and ws (w small and w large ). A filled symbol represents a participant's original data. See Table 1   predictor and mathematics ability as the dependent variable. The w large , visual-spatial memory, inhibition ability, and set-switching ability were entered as control variables. As shown in Table 2, all controlling variables only explained 8% of mathematics ability variance, F change(56) = 1.24, p = 0.304. The numerosity perception of small numbers (w small ) still explained a significant portion of mathematics ability variance (17%), F change(55) = 12.06, p = 0.001.

DISCUSSION
The present study examined the relationship between numerosity perception and mathematics ability in adults and found the moderating role of dots number in this relationship. We showed that a significant association existed only between numerosity perception of small numbers and mathematics ability in adults. Even after controlling other cognitive variables, the association remained stable and robust.
Prior studies have not found a significant correlation between numerosity perception and mathematics ability in adults (Gilmore et al., 2013;Inglis et al., 2011). However, the inconsistency might be attributed to that many of these studies did not take into account the specific effect of the methodological differences in numerosity perception tasks and other cognitive abilities (Dietrich et al., 2019;Fuhs & McNeil, 2013;Norris & Castronovo, 2016). The present study found that the relationship between numerosity perception and mathematics ability might depend on the dots number. In addition, even after controlling other cognitive variables (i.e., inhibition ability, visual-spatial memory, and set-switching ability), the numerosity perception of small numbers could still explain 17% of mathematics ability variance. These results were consistent with a similar study in children (Anobile et al., 2016), in which only small numerosity perception thresholds were significantly correlated with mathematics ability. Indeed, it is suggested that small numerosity perception was based on numerosity mechanism, and large numerosity perception was based on density mechanism (Anobile, Cicchini & Burr, 2014;Zimmermann, 2018). Thus, our findings supported the notion that dots number could influence the association between numerosity perception and mathematics ability in adults. Future studies with more levels of number of dots are required in order to further explore how dots number influences the relationship between numerosity perception and mathematics ability. In addition, our current findings may have particular practical implications for math learning. They can guide psychological interventions to help children and adults with math learning difficulties improve mathematics ability.
In this study, the role of the cognitive abilities in the relationship between numerosity perception and mathematics ability, particularly the inhibition ability, has not been validated. Fuhs & McNeil (2013) found correlations between numerosity perception, inhibition ability, and mathematics ability. They argued that the numerosity perception explained the mathematical achievement only when considering the contribution of inhibition ability. Our study only found correlations between numerosity perception of small numbers and mathematics ability but not between inhibition ability and mathematics skills. One possibility was that the participants' age influenced the relationship between the two constructs. Several studies supported the relationship between inhibition ability and mathematics ability in preschool and school-age children (Bull & Scerif, 2001). Yet, the nature of inhibition ability and mathematics ability profoundly changed with age (Espy et al., 2004). These variations inherently affected how inhibition ability related to mathematics ability (Ahmed et al., 2019). As discussed above, we speculated that the association of numerosity perception and mathematics ability might be stable and robust and exist in both children and adults; whereas the association of inhibition ability and mathematics ability might be diverse and labile and change with age. Future research could examine this issue.
Another critical finding pertained the selective relationship between inhibition ability and numerosity perception of small numbers. The results implied that adults with high levels of inhibition ability could better cope with small numerosity perception tasks. Adults needed to exact abstract numerical properties and inhibit extraneous continuous properties in small number discrimination tasks. But they only needed to exact continuous properties to complete large number discrimination tasks (Fuhs & McNeil, 2013). It was not surprising that inhibition ability played a key role in numerosity perception of small numbers. In addition, although the correlation between visual-spatial memory and mathematics ability was significant in children (Caviola, Mammarella & Dénes Szcs, 2020), the present study found a marginal association in adults. We speculated that the relationship between visual-spatial memory and mathematics ability would be much less pronounced in adults since they had many years of schooling.
It was also important to note that the numerosity perception performances across two conditions were similar in the present study. These findings contradicted with the results of Anobile, Cicchini & Burr (2014), who found that the numerosity perception threshold of small numbers was lower compared to large numbers. These inconsistencies could be partially attributed to individual differences in numerosity perception performance (Halberda, Mazzocco & Feigenson, 2008;Ross, 2003). For example, Ross (2003) used numbers from 8 to 64 as base numbers and asked participants to complete a numerosity discrimination task. Four of five observers' thresholds initially increased almost linearly with base numbers and decreased with base numbers of 30 or above. However, the thresholds of one observer were consistently higher than any of the others and increased with base numbers. The author suggested that there might not be a universally available numerosity discrimination pattern. Given the large individual differences in patterns of numerosity perception, the differences in numerosity perception performances between the small and large number conditions might be cancelled out in the present study. Despite the lack of significant differences between small and large numerosity perception, only small numerosity perception correlated with mathematics ability and inhibition ability. These results implied that there might be separate mechanisms for small and large numbers, but more sensitive indexes other than the numerosity perception threshold were needed to detect the difference between small and large number perception. In addition, it appeared likely that the difference between psychophysical methods used by the previous study and the current study to measure w could explain the inconsistent findings. Specifically, Anobile, Cicchini & Burr (2014) used the adaptive Quest method, and we used the Method of Constant Stimuli. Future studies should compare the two different methods.

CONCLUSIONS
This article demonstrated the relationship between numerosity perception and mathematics ability in adults and the influence of dots number on this relationship. The results suggested that mathematics ability was significantly associated with the numerosity perception of small numbers rather than large numbers. It seemed likely that the numerosity perception of small and large numbers were encoded differently and had a distinct effect on mathematics ability. Future studies should consider other potential domain-specific factors, such as SNS, to further explore the relationship between numerosity perception and mathematics ability. In addition, the neural mechanisms underlying small and large numerosity perception might also provide strong evidence for the effect of dots number and should be explored further.