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

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 achievement. Recently, several studies have found a potential role of numerosity perception in facilitating individuals’ formal mathematical skills (e.g., Elliott et al., 2019; Halberda, Mazzocco & Feigenson, 2008; Mazzocco, Feigenson & Halberda, 2011; Park & Brannon, 2013). Furthermore, numerosity perception has been found to be a more important predictor of mathematics ability than other general cognitive functions (Chen & Li, 2014; Halberda, Mazzocco & Feigenson, 2008).

Numerosity perception is the elementary numerical ability in both humans and animals, representing nonverbal information of quantity without counting (Dehaene, 1997; Dehaene & Changeux, 1993; Feigenson, Dehaene & Spelke, 2004). A numerosity discrimination task is commonly applied to measure the precision of numerosity perception in which two dot sets are displayed briefly, and the participants are asked to judge which set is more numerous (Halberda & Feigenson, 2008; Inglis et al., 2011). The numerosity discrimination performance, such as weber fraction (w), was used as an index of numerosity perception. The w is an abstract measure of discrimination threshold that specifies the precision of the numerosity representation and represents the width of numerosity tuning curve (Kersey & Cantlon, 2017; Piazza et al., 2004), where higher w reflects less accurate numerosity representation.

The supporting evidence for the relationship between numerosity perception and mathematics ability came from the fact that numerosity discrimination performance (which reflects numerosity perception) correlated with mathematics ability (Halberda, Mazzocco & Feigenson, 2008; Libertus, Feigenson & Halberda, 2011; Mazzocco, Feigenson & Halberda, 2011), which was measured by different tasks (Holloway & Ansari, 2009; Fagerlin et al., 2007). Furthermore, it has been shown that training on numerosity perception tasks improved formal mathematics abilities in both children and adults (Hyde, Khanum & Spelke, 2014; Park & Brannon, 2013). However, it should be noted that many studies did not find such a relationship (Gilmore et al., 2013; Lyons & Beilock, 2011; Lyons et al., 2014). The inconsistent results might ascribe to two reasons. First, the inconsistency might be caused by age differences of participants recruited (Gilmore et al., 2013; Inglis et al., 2011; Schneider et al., 2017). For instance, Inglis et al. (2011) found that the association between numerosity perception and mathematics skills only existed in children but not in adults. Second, a methodological difference which refers to the different numerosity of items in the dot set (i.e., dots number) used in previous studies, might also contribute to the inconsistency. There is evidence that the number of stimuli that has to be estimated (i.e., dots number) could influence the perception mechanism (Anobile et al., 2016; Anobile, Cicchini & Burr, 2014; Anobile, Cicchini & Burr, 2016; Pome et al., 2019; Fornaciai & Park, 2017; Zimmermann, 2018). It seems that small numbers of stimuli (relatively sparse dot-pattern) were sensed by the numerosity mechanism, but large numbers (relatively dense dot-pattern) were sensed by the density-texture mechanism (Anobile et al., 2016; Anobile, Cicchini & Burr, 2014; Zimmermann, 2018). Actually, previous studies found that only the numerosity perception of small numbers correlated with mathematics ability in children (Anobile et al., 2016).

Given the above, this study aimed to investigate the relationship between numerosity perception and mathematics ability and the effect of dots number on this relationship in 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.

Materials & Methods

Results

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.

Table 1:

Descriptive statistics and Zero-order correlations.

	Range	M	SD	1	2	3	4	5	6
1. wsmall	0.12–0.98	0.34	0.15	1.00
2. wlarge	0.11–0.89	0.34	0.17	0.47***	1.00
3. Mathematics ability	−2.47–1.29	0.00	0.71	−0.42***	0.10	1.00
4. Visual-spatial memory	2–16	8.43	3.34	0.14	−0.05	0.26	1.00
5. Inhibition ability	0.64–0.96	0.84	0.08	−0.27*	−0.15	−0.19	−0.03	1.00
6. Set-switching ability	0.07–5.35	1.75	0.94	0.08	−0.15	0.11	0.06	0.00	1.00

DOI: 10.7717/peerj.12660/table-1

Notes:

***p < 0.001.

**p < 0.01.

*p < 0.05.

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.

Supplemental Information