### Abstract

- Top of page
- Abstract
- Introduction
- Method
- Results
- Discussion
- Acknowledgements
- References

Recent findings by Libertus, Feigenson, and Halberda (2011) suggest that there is an association between the acuity of young children's approximate number system (ANS) and their mathematics ability before exposure to instruction in formal schooling. The present study examined the generalizability and validity of these findings in a sample of preschoolers from low-income homes. Children attending Head Start (*N *=* *103) completed measures to assess ANS acuity, mathematics ability, receptive vocabulary, and inhibitory control. Results showed only a weak association between ANS acuity and mathematics ability that was reduced to non-significance when controlling for a direct measure of receptive vocabulary. Results also revealed that inhibitory control plays an important role in the relation between ANS acuity and mathematics ability. Specifically, ANS acuity accounted for significant variance in mathematics ability over and above receptive vocabulary, but only for ANS acuity trials in which surface area conflicted with numerosity. Moreover, this association became non-significant when controlling for inhibitory control. These results suggest that early mathematical experiences prior to formal schooling may influence the strength of the association between ANS acuity and mathematics ability and that inhibitory control may drive that association in young children.

### Introduction

- Top of page
- Abstract
- Introduction
- Method
- Results
- Discussion
- Acknowledgements
- References

Children's early abilities in mathematics predict not only later mathematics achievement (e.g. Jordan, Kaplan, Locuniak, & Ramineni, 2007; Jordan, Kaplan, Ramineni, & Locuniak, 2009), but also later performance in other academic domains such as reading (Duncan *et al*., 2007) and science (Sadler & Tai, 2007). Thus, it is important to identify factors that differentiate children who struggle with early mathematics concepts from those who do not. Research to date has uncovered several predictors of individual differences in early mathematics ability, including socioeconomic status (SES) of the child's family (e.g. Jordan, Kaplan, Oláh, & Locuniak, 2006; Jordan *et al*., 2007), early regulatory skills such as executive functioning or inhibitory control (e.g. Blair & Razza, 2007; Bull, Espy, & Wiebe, 2008; Bull & Scerif, 2001; Clark, Pritchard, & Woodward, 2008; St Clair-Thompson & Gathercole, 2006; Welsh, Nix, Blair, Bierman, & Nelson, 2010), and early language ability (e.g. Fuchs *et al*., 2010; Purpura, Hume, Sims, & Lonigan, 2011). Of increasing interest to the field is the potential role of early developing ‘core’ number systems, particularly the approximate number system (ANS), in facilitating children's mastery of early mathematics concepts (e.g. Condry & Spelke, 2008; Le Corre & Carey, 2007; Libertus, Feigenson, & Halberda, 2011). In this study, we investigated the contribution of ANS acuity to mathematics ability at the start of preschool in a sample of children from low-income homes, for which we know little about ANS acuity performance. We were particularly interested in whether the association between ANS acuity and mathematics ability found in previous studies would hold in this sample, even after controlling for language ability, and if so, how much of that association could be explained by the shared variance with inhibitory control.

#### ANS acuity and early mathematics ability

The ANS is an imprecise ability for large magnitude representation that does not rely on counting (Feigenson, Dehaene, & Spelke, 2004). It operates according to Weber's law of ‘just noticeable difference’, or the idea that at a certain ratio threshold, one can perceive a difference between the numerical amounts of two sets of stimuli (Feigenson *et al*., 2004). The acuity of an individual's ANS is defined as the ratio at which individuals can reliably discriminate between two non-symbolic quantities (e.g. two sets of black dots). This acuity varies both across development and across individuals. Acuity improves with development such that the discriminable ratio changes from 1:2 at six months (Xu & Spelke, 2000), to 2:3 at 10 months (Xu & Arriaga, 2007), to 3:4 in preschool (Halberda & Feigenson, 2008), to 7:8 or higher in adulthood (Barth, Kanwisher, & Spelke, 2003). Even within a given age group, acuity can vary greatly (e.g. Halberda, Mazzocco, & Feigenson, 2008; Inglis, Attridge, Batchelor, & Gilmore, 2011). Researchers have argued that the ANS develops spontaneously (Xu & Spelke, 2000; Xu, Spelke, & Goddard, 2005) and forms the basis of symbolic mathematical knowledge in humans (Condry & Spelke, 2008; Dehaene, 1997; Feigenson *et al*., 2004), with rapid development in acuity occurring during the preschool years (Halberda & Feigenson, 2008). Although age-related and individual variations in ANS acuity have been widely documented, little is known about the factors that contribute to individual differences in children's ANS acuity development beyond those differences accounted for by maturation.

Several studies have identified an association between tasks assumed to tap ANS proficiency and symbolic mathematical abilities in elementary and middle school children (e.g. De Smedt, Verschaffel & Ghesquiere, 2009; Halberda *et al*., 2008; Holloway & Ansari, 2009; Inglis *et al*., 2011; Mundy & Gilmore, 2009). However, only two of these studies used tasks designed to measure the acuity of the non-symbolic ANS *per se* (Halberda *et al*., 2008; Inglis *et al*., 2011). Halberda and colleagues (2008) found that children's ANS acuity at age 14 retrospectively predicted children's earlier symbolic mathematics ability in kindergarten through sixth grade controlling for many potential covariates, and Inglis *et al*. (2011) found that ANS acuity at age 8 concurrently predicted mathematics achievement controlling for IQ. Taken together, these results support the claim that early ANS acuity is the foundation upon which children build symbolic numerical knowledge; however, it could be just as likely that children's experience with symbolic mathematics affects their ANS acuity (see Butterworth, 2010).

Research examining the relation between ANS acuity and mathematics achievement in children prior to formal schooling has been more limited. Gilmore, McCarthy, and Spelke (2010) found that performance on a non-symbolic approximate addition task correlated with symbolic mathematics ability in a sample of children in their first year of formal schooling. However, these researchers used an approximate addition task that required children to both approximately represent numbers and perform a mathematical operation, so they were not able to assess the association between children's ANS acuity and symbolic mathematics ability independent of children's competency at arithmetic operations. Both Libertus *et al*. (2011) and Mazzocco, Feigenson, and Halberda (2011) addressed this limitation by investigating the relation between ANS acuity and early symbolic mathematics ability in preschoolers using a non-symbolic ANS acuity task that did not require children to perform a mathematical operation. Libertus *et al*. (2011) showed that children's ANS acuity at the start of preschool was positively correlated with performance on a test of early mathematics ability, even when controlling for children's expressive vocabulary, and Mazzocco *et al*. (2011) showed that children's ANS acuity in preschool predicted mathematics achievement – but not IQ – two years later. These findings are consistent with the notion that children's early ANS acuity is foundational for children's early mathematics ability. However, neither of these studies controlled for likely confounding variables, such as receptive language ability or inhibitory control. Libertus and colleagues (2011) did control for a parent report of expressive vocabulary, but the measure was not correlated with children's mathematics ability, so its role as a confounding variable was limited. On the other hand, the Peabody Picture Vocabulary Test (PPVT-4; Dunn & Dunn, 2007) has often been used not only as a measure of receptive vocabulary, but also as a proxy for general mental ability, and it has been shown to correlate significantly with mathematics ability (e.g. Blair & Razza, 2007). Thus, it remains possible that some of the shared variance between ANS acuity and symbolic mathematics ability in the preschool years can be accounted for by receptive language ability, or other confounding variables such as inhibitory control (described later). To address this limitation, we controlled for performance on the PPVT-4 in the present study.

Overall, the findings of Libertus *et al*. (2011) and Mazzocco *et al*. (2011) suggest that there may be an association between ANS acuity and mathematics ability in children even before they begin formal schooling. However, several questions remain concerning the generalizability and validity of these findings, including whether or not this relation holds in children from low-income homes, who likely have less exposure to mathematics prior to preschool entry, even after controlling for potentially confounding variables such as receptive vocabulary and inhibitory control. In the present study, we first examined whether or not ANS acuity is significantly associated with mathematics ability in preschoolers from low-income homes, controlling for receptive language ability. We then examined whether or not a relation between ANS acuity and mathematics ability holds above and beyond the influence of inhibitory control.

#### Association between ANS and mathematics ability in children from low-income homes

Although researchers have argued that children's ANS acuity and mathematics ability are related even prior to formal schooling, an increasingly stronger correlation across development might suggest a role for early experience impacting both ANS acuity and mathematics ability even prior to school. Indeed, the strength of the association between ANS acuity and mathematics ability in preschoolers reported in Libertus *et al*. (2011) was much smaller than that reported for older children in Halberda *et al*. (2008). Libertus and colleagues (2011) suggested that this discrepancy was likely due to the fact that they controlled for response time in their analyses, whereas Halberda *et al*. (2008) did not. However, it is possible that the association between ANS acuity and mathematics performance in younger children is weaker than that reported for older children because younger children do not yet have exposure to mathematics activities that would allow them opportunities to map symbolic mathematics onto their pre-existing ANS acuity. In support of this view, Le Corre and Carey (2007) found that young children who had a verbal count list but had not yet acquired counting principles performed poorly when asked to estimate (without counting) quantities larger than four. In contrast, children who had mastered the counting principles successfully estimated larger quantities. This suggests that the strength of the association between ANS acuity and mathematics ability may increase as children begin to learn mathematical concepts. Of course, as children's ANS acuity and proficiency with symbolic mathematics improves to adult levels, the association may start to decrease again, partly because higher-level mathematics is increasingly abstract and divorced from one's knowledge of concrete quantities. In other words, the relationship between ANS acuity and symbolic mathematics may be represented by a U-shaped function, where once mapping is set and children gain increasingly advanced knowledge of symbolic mathematics, they may not need to rely so heavily on their ANS. Consistent with this view, Inglis *et al*. (2011) found an association between ANS acuity and symbolic mathematics ability at age 8, but not in adults.

If strength of association depends on exposure to mathematics activities, then one might also expect a weaker association between ANS acuity and math ability in children from low-income homes. Children growing up in low-income homes often lag behind their peers on tests of mathematics ability at school entry, and many are at risk for poor mathematics achievement in school. Several studies have shown that children from low-income homes often have less exposure to early number concepts (Bradley & Corwyn, 2002; Ramani & Siegler, 2008; Saxe, Guberman, & Gearhart, 1987; Votruba-Drzal, 2003) and poorer performance on assessments of symbolic mathematics ability (Griffin, Case, & Siegler, 1994; Jordan *et al*., 2006) compared to their peers from middle to high-income backgrounds. Thus, they may not yet have had opportunities to connect their ANS with symbolic mathematics skills.

Unfortunately, it is not currently known if the association between ANS acuity and mathematics ability holds in children who are from low-income homes and likely at greater risk for poor mathematics outcomes. The sample of preschoolers in Libertus *et al*.'s (2011) study scored on average 107.43 on the Test of Early Mathematics Ability (TEMA-3; Ginsburg & Baroody, 2003), and the children in Mazzocco *et al*.'s (2011) study scored on average 114.12, both of which are above the normed average of 100 (*SD *= 15). In fact, children from low-income homes have only been included in two previous studies of the association between ANS proficiency and symbolic mathematics ability, and both of these studies assessed children's performance on an approximate addition task, which does not assess children's ANS acuity independent of their competency at arithmetic operations (Gilmore *et al*., 2010; McNeil, Fuhs, Keultjes, & Gibson, 2011). The present study addressed this gap in the literature by examining the association between ANS acuity and mathematics ability in children from low-income homes.

#### Contributions of inhibitory control

An important confounding variable that might explain the association between ANS acuity and symbolic mathematics ability in young children is inhibitory control. Inhibitory control, or the ability to suppress a prepotent response, develops gradually during childhood, and much like the trajectory of ANS acuity, children show significant improvements in inhibitory control during the preschool years (Carlson, 2005; Garon, Bryson, & Smith, 2008). Similar to the achievement gap we see between young children from varying SES backgrounds on mathematics ability, children from low-income homes also show difficulties with neurocognitive abilities such as inhibitory control (Noble, Norman, & Farah, 2005). Neither Libertus *et al*. (2011) nor Mazzocco *et al*. (2011) controlled for inhibitory control in their studies, but Halberda *et al*. (2008) found a unique relation between older children's ANS acuity and earlier mathematics ability controlling for many potential covariates and confounding variables, including a Stroop-like inhibitory control measure. However, because Halberda and colleagues (2008) assessed ANS acuity and inhibitory control at age 14, it is not clear how children's earlier, and likely weaker, ANS acuity and inhibitory control related to their mathematics ability. Moreover, in research conducted with children of similar ages (11–12), St Clair-Thompson and Gathercole (2006) found that a Stroop-like measure of inhibitory control was not significantly related to mathematics achievement, whereas a stop-signal measure of inhibitory control was. These discrepancies highlight a broader issue when measuring inhibitory control known as the ‘task impurity problem’ (e.g. Miyake, Friedman, Emerson, Witzki & Howerter, 2000). That is, because inhibitory control measures also tap other cognitive constructs such as language or motor skills, any single measure of inhibitory control is likely to have considerable measurement error. Thus, in the present study, we examined the associations among ANS acuity, mathematics ability, and inhibitory control using a composite inhibitory control measure created from tasks with varying response modalities to get a more reliable inhibitory control measure.

In addition to St Clair-Thompson and Gathercole's (2006) study, a substantial body of research has linked children's inhibitory control to their symbolic mathematics performance (Blair & Razza, 2007; Bull *et al*., 2008; Bull & Scerif, 2001; Clark *et al*., 2010; Welsh *et al*., 2010). Because mathematics involves problem solving, even in more basic activities, it likely requires children to identify relevant information in the face of conflicting or distracting irrelevant information. For example, in a word problem, a child must identify the numbers to be added, for example, and hold those in mind and ignore other presented information that is not necessary for solving the problem. Another example involves the common practice of using manipulative objects for counting and addition. Children must attend to counting principles and identify the objects numerically while ignoring their other properties. Mathematics ability evolves across time to accommodate increasingly complex problem solving, and thus, children likely continue to rely heavily on their inhibitory control abilities for mathematical processing throughout childhood.

We postulate that children's performance on tasks designed to measure ANS acuity represent not only children's ability to discriminate numerosities, but also their ability to attend to numerical information and filter out extraneous information. In other words, children's ANS acuity could be indicative of both their ability to approximate numerosity, and their inhibitory control. One of the variables that must be filtered out by children when participating in an ANS acuity task is surface area. Researchers have shown that young children show sensitivity to this dimension and are able to make discriminations based on surface area (Feigenson, Carey & Spelke, 2002), although they may be somewhat noisier estimates compared to numerosity discriminations (Cordes & Brannon, 2008). Halberda and Feigenson (2008) studied ANS acuity development from early childhood into adulthood, and they included controls for area such that during half of the ANS acuity trials, area was correlated with numerosity, and during the other ANS acuity trials, area and numerosity were inversely related such that selecting a response based solely on area would always yield an incorrect answer. They found that young children performed significantly better on trials in which area was correlated with numerosity, but this pattern did not hold for older children and adults. They concluded that if surface area had a major effect on children's performance, children would have scored significantly below chance on area inverse trials. However, even if children did not show evidence of responses based solely on the distractor surface area feature of the stimuli, it is still possible that variance across children's performance on inverse trials may have represented not only differences in children's discrimination of numerosity, but also their ability to focus on numerosity in the face of conflicting surface area information (i.e. inhibitory control). Thus, one explanation for an association between ANS acuity and mathematics ability in young children is that in the absence of significant symbolic mathematics instruction, inhibitory control may drive performance in young children's ANS acuity, such that children who are better able to focus attention on numerosity in the face of extraneous information might show evidence of greater ANS acuity as well as greater mathematics ability.

#### Current study

Two primary hypotheses guided the current study. The first hypothesis was that the relation between ANS acuity and mathematics ability would be weaker in a sample of children from low-income homes compared to previously reported results for children who are not at-risk for poor mathematics performance (e.g. Libertus *et al*., 2011; Mazzocco *et al*., 2011). Preschoolers from low-income homes exhibit lower performance on tests of early mathematics performance, and this may suggest that they have not yet integrated their ANS acuity and mathematics ability. Perhaps a unique relation between ANS acuity and symbolic mathematics unfolds across time as children acquire counting principles (e.g. Le Corre & Carey, 2007) as well as understanding of mathematical concepts. The association between children's ANS acuity and mathematics may be smaller for children who have less exposure to mathematical concepts in their environment. This association may also not be robust when controlling for potential confounding variables such as receptive vocabulary.

Our second hypothesis was that shared variance between ANS acuity and mathematics ability would be explained, in part, by the role of inhibitory control in both ANS acuity performance and mathematics ability in early childhood. Thus, we expected that children's inhibitory control would be significantly correlated with both their ANS acuity and their mathematics ability. Importantly, if inhibitory control drives the relation between ANS acuity and early mathematics, then trials for which the demands for maintaining attention on numerosity in the face of conflicting information are highest (i.e. area inverse trials) should be most predictive of mathematics ability. In addition, if inhibitory control is the mechanism explaining the relation between ANS acuity and mathematics ability, then the association between ANS acuity and mathematics ability should be significantly reduced when controlling for inhibitory control.

### Discussion

- Top of page
- Abstract
- Introduction
- Method
- Results
- Discussion
- Acknowledgements
- References

In the present study, we found a small, marginally significant association between ANS acuity and mathematics ability in a sample of children attending Head Start. However, that association disappeared when we controlled for a direct measure of receptive vocabulary. These results suggest that, as predicted, the association between ANS acuity and mathematics ability is weaker and less robust in this low-income sample than it has been in previous studies with middle and high-income samples. Consistent with our predictions, we also found that inverse trials were the only trial type on the ANS acuity task that significantly predicted mathematics ability over and above receptive vocabulary, but even this more robust association disappeared when controlling for inhibitory control. These results suggest that inhibitory control likely plays a key role in the relation between ANS acuity and early mathematics ability. In the following paragraphs, we first discuss a potential explanation of why the association between ANS acuity and mathematics ability is weaker in a sample of children from low-income homes. We then discuss how inhibitory control may drive the relation between ANS acuity and mathematics ability.

Prior studies have found a significant association between ANS proficiency and mathematics ability; however, many of these studies have been limited to older children who had received formal mathematics instruction (e.g. Gilmore *et al*., 2010; Halberda *et al*., 2008; Holloway & Ansari, 2009; Inglis *et al*., 2011; Mundy & Gilmore, 2009), or to younger children who scored well on tests of early mathematics ability (Libertus *et al*., 2011; Mazzocco *et al*., 2011). Although the association between ANS acuity and mathematics ability was robust in these prior studies (e.g. *r*_{p} = .36 in Libertus *et al*. [2011] after controlling for an indirect measure of expressive vocabulary), the association was only marginal in the present study, and it disappeared after controlling for a direct measure of receptive vocabulary. One interpretation of these results is that children's early mathematical experiences prior to the start of formal schooling influence the relation between ANS acuity and mathematics performance such that children who have fewer opportunities to engage in early mathematics activities do not use their ANS as readily when participating in mathematical tasks. Indeed, experience with mathematical activities has previously been proposed as an explanation of why ANS and mathematics ability are related in childhood (Butterworth, 2010).

Several studies have found significant differences in the mathematical experiences of children from low-income and middle-income backgrounds prior to the start of schooling, such that children from low-income homes have less access to mathematical materials in the home and less exposure to mathematical activities (e.g. Bradley & Corwyn, 2002; Ramani & Siegler, 2008; Saxe *et al*., 1987; Votruba-Drzal, 2003). Thus, even assessing children at the start of preschool cannot rule out the possibility that early mathematical experiences are driving the associations observed in prior studies of the relation between ANS acuity and symbolic mathematics performance.

One interpretation of why early experience may promote an association between ANS acuity and symbolic mathematics is that participation in mathematical tasks allows children an opportunity to engage their ANS in a context in which the ANS is likely to be of particular benefit. In other words, although studies have shown that very young children are capable of approximately discriminating quantities even in infancy (e.g. Xu & Spelke, 2000), the potential for them to actively engage their ANS more regularly increases markedly during mathematical activities. For example, a parent may ask a child to determine which of two bowls has more Cheerios, which of two jars has more coins, or which of two hands has more raisins. Participating in mathematical activities like these and others that engage the ANS may facilitate further development of the ANS and increase the likelihood that children rely on their ANS to learn new mathematical concepts. This explanation is consistent with Le Corre and Carey (2007), who found that only after children had mastered counting principles did they show evidence of consistently engaging their ANS to estimate numerical quantities without counting. It is also consistent with McNeil and colleagues (2011), who found that children from low-income homes performed significantly worse on an approximate addition task compared to children from middle-to-high-income homes. This suggests that this relation could even be reciprocal such that engagement in early mathematics activities allows children an opportunity to engage their ANS, which likely enhances their mathematics learning. In turn, children's mathematics learning may also facilitate greater precision of their ANS acuity. This relationship may not be linear, however, and recent evidence suggests that in fact there may be a U-shaped relationship between ANS acuity and mathematics ability such that children increasingly use their ANS during their schooling years, but rely on it less once their mathematics knowledge becomes more abstract. For example, Inglis *et al*. (2011) found that the relation between ANS acuity and mathematics achievement in children was significant, but it was not significant in adults. Although speculative, this could suggest that we can see the strongest relations between children's ANS acuity and mathematics ability during formal schooling where the relation begins to diminish once children move on to more advanced mathematical concepts such as algebra. Future longitudinal work is therefore needed to better understand the role of children's early environment in ANS acuity as it may not only affect how children use their ANS when engaging in symbolic mathematics, but also shape the developmental trajectory of their ANS acuity at a more basic level.

Although we have interpreted the present results as support for the idea that children's early exposure to mathematical activities affects the strength of the association between ANS acuity and mathematics ability, other interpretations are also possible. For example, it is possible that the strength of the association between ANS acuity and mathematics ability is dependent on children's exposure to early learning tasks more generally, rather than to mathematics activities *per se*. There are many environmental factors unique to growing up in a low-income household that may put children at risk for poor academic outcomes, including lower levels of maternal education, less access to quality child care (e.g. Burchinal, Peisner-Feinberg, Pianta, & Howes, 2002; Peisner-Feinberg & Burchinal, 1997), and exposure to a high-stress environment (e.g. Blair, 2010; Noble *et al*., 2005). Future work is necessary to examine possible environmental mechanisms that may explain the weaker association found between ANS acuity and mathematics ability in an at-risk sample.

Equally important, the current study used a somewhat different methodology from that used in prior studies with children from middle-income homes, so it is possible that the weaker association observed in the current study is due to these methodological differences, such as using a paper version of the ANS acuity task instead of a computerized task, or using black stars instead of colored dots or child-friendly objects as the items to be discriminated in the ANS acuity task. However, we still found the signature ratio effect that is typically seen in other ANS acuity tasks for younger children, suggesting that this task is tapping similar if not identical abilities. Future research directly comparing groups from varying SES backgrounds on the same ANS acuity measure will advance our understanding of the impact of SES on early correlates of mathematics ability in preschoolers.

In addition to finding a weaker association between ANS acuity and mathematics ability in a sample of children from low-income homes, we also found four pieces of evidence to suggest that inhibitory control plays a key role in this association. First, children's inhibitory control was significantly correlated with their ANS acuity both in terms of accuracy across all trials as well as accuracy by control trial type. Second, the association between inhibitory control and ANS acuity was stronger for inverse trials than it was for mean area and total area trials. The inverse trials placed the most demands on inhibitory control, as children had to compare numerosity in the face of conflicting surface area information. Third, children's performance on the inverse trials – but not on the mean area and total area trials – predicted mathematics ability, even when controlling for a direct assessment of receptive vocabulary. Finally, the relation between ANS inverse trials and mathematics ability became non-significant once inhibitory control was added to the model.

If ANS acuity in early childhood represents children's ability to approximate and compare numerosities, as well as their ability to inhibit, or filter out conflicting or extraneous information, then we should expect that inhibitory control might drive performance in both ANS tasks as well as assessments of early mathematics ability in young children. This idea is consistent with empirical evidence suggesting a strong association between children's inhibitory control and their mathematical performance, as well as their change in mathematical achievement across time (e.g. Blair & Razza, 2007; Bull *et al*., 2008; Bull & Scerif, 2001; Clark *et al*., 2010; St Clair-Thompson & Gathercole, 2006; Welsh *et al*., 2010). These results also support recent neuroimaging findings suggesting that children rely more heavily on pre-frontal brain regions, most known for their association with executive functioning or inhibitory control, in early mathematics (Houdé, Rossi, Lubin, & Joliot, 2010). Only later do they exhibit more specialized mathematical processing regions, notably the intraparietal sulcus (Houdé *et al*., 2010), which has been connected specifically with ANS acuity (Ansari, 2008). According to this account, younger children may be more strongly affected by inhibitory control abilities when performing both approximate number tasks as well as exact calculations compared to older children and adults who have more developed mathematics ability. In older children, Halberda and colleagues (2008) found that a unique relation between ANS acuity and mathematics ability held after controlling for children's executive functioning on a Stroop-like task, which could suggest that children's ability to engage their ANS in symbolic mathematics tasks requires less global domain-general processing as children's mathematics ability increases. However, a Stroop-like measure of inhibitory control was not significantly correlated with mathematics achievement in 11- and 12-year-olds in a previous study, whereas a stop-signal measure of inhibitory control was (St Clair-Thompson & Gathercole, 2006). In fact, the correlation between the stop-signal task and mathematics performance was twice the size of the correlation between the Stroop-like task and mathematics performance. Importantly, in the current study, we used a battery of inhibitory control measures to account for the measurement issues (e.g. task impurity problem) surrounding the measurement of inhibitory control. This battery may be more likely than the measure used by Halberda *et al*. (2008) to capture the types of inhibitory skills that are involved in children's mathematics performance.

Although we found that children's inhibitory control was strongly related to their mathematics ability, even after controlling for both receptive vocabulary and performance on ANS inverse trials, it is possible that additional confounds may account for some of the shared variance among these constructs. For example, more basic perceptual processes may influence the relation between inhibitory control, ANS, and mathematics ability. Perceptual processes have been found to influence middle and high school students’ mathematics performance (Kellman, Massey, & Son, 2009) and earlier perceptual processes such as differentiation, which is the ability to distinguish stimuli differences, have been proposed as correlates of ANS proficiency (DeWind & Brannon, 2012). Presumably, if a child has well-developed differentiation abilities (Goldstone, 1998), he or she may be better able to distinguish numerosity from surface area on ANS trials and may also be better able to distinguish two stimuli (e.g. sun and moon) on inhibitory control tasks. Motivational processes could also be at work, as assessments conducted in a one-on-one testing situation with sticker rewards may reflect children's response to reward conditions and may differ from their behavior in everyday contexts. It will be important to understand how these constructs operate in children's real-world learning environments to inform early prevention and intervention efforts in preschool.

Although results supported our hypotheses, there were at least two additional limitations of this study that suggest directions for future work. First, we used a paper version of the computerized ANS acuity task in Halberda and Feigenson (2008). Use of a paper version with only 30 trials precluded us from obtaining reliable estimates of children's Weber fractions. However, Libertus and colleagues (2011) found similar results in their regression analysis using a Weber fraction score and a total accuracy score, and Mazzocco *et al*. (2011) argued that percent correct is a less volatile proxy for the Weber fraction when dealing with preschoolers who have been tested on only a few trials at each ratio. Second, we used cross-sectional data, so we cannot be certain about the temporal relations between ANS acuity, inhibitory control, and math ability. Longitudinal research may aid researchers who are interested in the potential role of ANS acuity in mathematics abilities and disabilities determine how risk factors might affect if and when children from low-income homes engage their ANS when performing symbolic mathematics.

In sum, our results suggest that children who are from low-income homes and who are likely at-risk for poor mathematics achievement exhibit a weak association between ANS acuity and mathematics ability. One interpretation of these results is that children who have less experience with mathematical activities prior to beginning preschool may not yet have opportunities to engage their ANS in mathematics activities and therefore individual variation in ANS acuity may not be significantly related to their mathematics ability. Thus, researchers cannot rule out the possibility that early experiences, even prior to the start of formal schooling, support an association between ANS acuity and mathematics ability. The present results also suggest that other confounding factors, such as inhibitory control, may drive associations seen between ANS acuity and mathematics in young children. Future experimental research may better elucidate causal factors implicated in ANS acuity development as well as the directionality of the relation between ANS acuity and mathematics skills across time.