## Introduction

Mathematical competence, from managing a budget to calculating a restaurant tip, is essential to everyday activity in most modern cultures. Indeed, previous research suggests that math ability is an important factor in determining career success, income, and psychological well-being (Paglin & Rufolo, 1990; Parsons & Bynner, 2005; Rivera-Batiz, 1992; Rose & Betts, 2004). Yet, wide variety exists in the level of mathematical competence that people achieve, even starting early in development. Investigations of school math ability, hereafter referred to as math ability, in kindergarteners and elementary school children find stable individual differences in performance on tasks relevant to school success (e.g. verbal counting, simple arithmetic, ordinal comparison of numerals, and story problems) (Jordan, Kaplan, Locuniak & Ramineni, 2007; Jordan, Kaplan, Olah & Locuniak, 2006; Jordan, Kaplan, Ramineni & Locuniak, 2009; Mazzocco & Thompson, 2005). These studies reveal individual differences in mathematical ability already present from the earliest years of formal education, and highlight the importance of investigating their sources. Moreover, identifying predictors of weak math abilities might allow for early detection of future math difficulties and a hastening of intervention.

What factors lead to early mathematical competence? In addition to social dimensions such as income level (Griffin, Case & Siegler, 1994; Jordan *et al*., 2009), amount of number-relevant teacher input (Klibanoff, Levine, Huttenlocher, Vasilyeva & Hedges, 2006), and home learning environment (Melhuish, Sylva, Sammons, Siraj-Blatchford, Taggart, Phan & Malin, 2008), cognitive capacities contribute significantly. General cognitive abilities such as short-term and working memory have been suggested to play an important role in mathematical abilities (Geary, 2004; Mabott & Bisanz, 2008; McLean & Hitch, 1999; Passolunghi & Siegel, 2001; Wilson & Swanson, 2001). In addition, recent attention has been given to the possibility of individual differences in an unlearned, number-specific competence used by children and adults.

Educators and researchers often refer to a ‘number sense’ that includes a variety of competences, including the ability to subitize and count, to discriminate quantities, to discern number patterns, to rule out unreasonable results of arithmetic operations, and to move flexibly between different numerical formats (Berch, 2005; Gersten, Jordan & Flojo, 2005; Jordan *et al*., 2007; Kalchman, Moss & Case, 2001). This number sense supports math achievement and is a focus of many United States math curricula (NCTM, 2000; NMAP, 2008). One central component of the number sense is the Approximate Number System (ANS). A focus of research in cognitive psychology and neuroscience, the ANS has been shown to support a primitive sense of number in infants, children, and adults (for reviews see Dehaene, 1997; Feigenson, Dehaene & Spelke, 2004; Libertus & Brannon, 2009). It is present at birth (Izard, Sann, Spelke & Streri, 2009) and has been documented in many non-human animal species (for review see Brannon, Jordan & Jones, 2010), supporting the notion that the ANS is independent from language and other acquired number symbols. In humans, the ANS is active across the entire lifespan, from infancy to old age (Dehaene, 1997; Halberda, Germine, Ly, Naiman, Nakyama & Willmer, 2011). Finally, a wealth of brain imaging studies has identified the intraparietal sulcus as the neural locus of the ANS (for review see Nieder & Dehaene, 2009).

The ANS has been shown to produce imperfect ‘noisy’ estimates of numbers of items from input across all sensory modalities (e.g. beeps, visually or tactilely presented objects, taps of a finger). These numerical estimates support quantitative computations such as ‘greater-than, less-than’, addition, subtraction, multiplication, and division (Barth, Kanwisher & Spelke, 2003; Barth, La Mont, Lipton, Dehaene, Kanwisher & Spelke, 2006; Barth, La Mont, Lipton & Spelke, 2005; Eger, Sterzer, Russ, Giraud & Kleinschmidt, 2003; McCrink & Spelke, 2010; McCrink & Wynn, 2004). The inherent noisiness of the ANS means that the accuracy of observers’ numerical estimates, and hence their performance at comparing or computing over ANS representations, accords with Weber’s Law, with larger numerical estimates being increasingly imprecise. As a result, the discriminability of any two ANS representations is a function of the ratio between them (e.g. 5 is as discriminable from 10 as 10 is discriminable from 20). Importantly, the amount of variability associated with representing a particular number (e.g. with approximating how many items are present when flashed an array containing exactly 10 dots) is not fixed over development. Infants and young children have much noisier ANS representations than adults, with the acuity of ANS representations sharpening throughout childhood, eventually supporting adult discriminations of about 9:10 (Halberda & Feigenson, 2008; Libertus & Brannon, 2010; Lipton & Spelke, 2003; Piazza, Facoetti, Trussardi, Berteletti, Conte, Lucangeli, Dehaene & Zorzi, 2010; Xu & Spelke, 2000).

It remains unknown exactly when these noisy ANS representations integrate with more formal math abilities, and what role they may play. One provocative hypothesis is that the ANS is instrumental for acquiring symbolic numerical skills such as counting and arithmetic (Condry & Spelke, 2008; Dehaene, 1997; Dehaene, Dehaene-Lambertz & Cohen, 1998; Gallistel & Gelman, 2000; Gilmore, McCarthy & Spelke, 2007) (but see Butterworth, 2010; Carey, 2000). Another possibility is that the ANS is not required for early math understanding and only later is integrated with symbolic number representations (Le Corre & Carey, 2007).

By adolescence, the ANS appears to play a role in school mathematics performance. Halberda, Mazzocco and Feigenson (2008) tested 14-year-old adolescents on a non-symbolic number comparison task in which participants saw rapidly flashed arrays of spatially intermixed blue and yellow dots, and pressed a key to indicate whether there were more blue or more yellow dots. As predicted by Weber’s Law, adolescents’ accuracy was modulated by the ratio between the numerical values – the closer the two numerical values relative to each other, the lower the group accuracy. The authors then used psychophysical modeling to estimate each individual participant’s Weber fraction (*w,* i.e. the amount of noise in each participant’s ANS representations). ANS acuity measured by this simple task at age 14 years was found to significantly correlate with individual math ability all the way back to kindergarten, as measured by standardized math tasks administered throughout participants’ schooling. Furthermore, this relationship remained robust even when controlling for non-numerical factors such as general IQ, spatial abilities, and working memory. Thus, it appears that individual differences in the ANS are linked to individual differences in performance on school math tasks, at least in adolescents.

This link also appears to modulate performance in participants who struggle with math. Piazza and colleagues (2010) and Mazzocco, Feigenson and Halberda (2011) demonstrated that children with dyscalculia have significantly worse ANS acuity than age-matched peers without dyscalculia. This suggests that less accurate ANS representations may be related to difficulty in school mathematics for children from the lowest end of math achievement.

Several investigations of children’s ability to perform number line estimations suggest that the nature of ANS representations may be linked to math achievement even earlier in life. Booth and Siegler (2006, 2008) found that the spatial representation of the mental number line affects children’s mathematical performance from as early as 5 years of age. In these studies, children were given Arabic numerals to place in their approximate spatial position on a schematic number line with only its anchor points (smallest and largest numbers) marked. The more linear (as opposed to logarithmic) children’s representations were, the better their math ability. Hence the spatial organization of number representations may affect math abilities (Siegler & Ramani, 2009).

Other recent studies found evidence consistent with the idea that the ANS might affect ordinal decisions over numerical symbols, and thereby impact math ability. These studies tested children between 6 and 10 years of age on their rapid judgments of symbolic quantity (e.g. indicating whether the Arabic numeral 7 or 9 is numerically greater), and found that this ability was related to math abilities on calculation tests (De Smedt, Verschaffel & Ghesquiere, 2009; Durand, Hulme, Larkin & Snowling, 2005; Holloway & Ansari, 2008; Rousselle & Noel, 2007).

However, these number line and ordinal judgment tasks required children to process numerical symbols (e.g. Arabic numerals). As such, these studies do not yet answer the question of whether individual differences in ANS acuity *per se*, as opposed to individual differences in the processing of number symbols or integrating ANS representations with number symbols, correlate with formal math ability in early childhood, prior to the large amounts of formal math instruction that children receive in primary school.

Another recent piece of evidence for a link between the ANS and math ability comes from a study by Gilmore, McCarthy and Spelke (2010). The authors tested 5-year-old children on standardized math tests as well as a non-symbolic addition task, in which children saw one quantity of blue dots added to another quantity of blue dots, and then had to report whether their approximate sum was more or less numerous than a comparison quantity of red dots. Children’s accuracy on the non-symbolic addition task was positively correlated with their math ability scores, suggesting that individual differences in performance on non-symbolic arithmetic are related to math ability.

These previous studies are exciting and suggestive of a relationship between ANS and math ability starting from early in life. However, to date no study has focused on ANS acuity in young children in isolation of competence in other math-relevant abilities (e.g. representing a physical number line, processing Arabic digits, performing addition and subtraction). In the present investigation we aimed to test the relationship between ANS acuity and early math ability in young children who have received only minimal formal mathematics instruction. To this end, we tested 200 3- to 5-year-old children in a simple non-symbolic number comparison task that was used previously to demonstrate a relationship between ANS acuity and math achievement in 14-year-old children (Halberda *et al*., 2008). We also tested children on a standardized test of math ability (Test of Early Mathematics Ability, TEMA-3; Ginsburg & Baroody, 2003) and an assessment of verbal ability (Developmental Vocabulary Assessment for Parents, DVAP; Libertus, Stevenson, Odic, Feigenson & Halberda, in preparation). Of primary interest was whether, even in young children with little or no formal mathematical instruction, individual differences in ANS acuity would correlate with math ability, with non-mathematical abilities controlled for.