### Abstract

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

Previous research shows a correlation between individual differences in people’s school math abilities and the accuracy with which they rapidly and nonverbally approximate how many items are in a scene. This finding is surprising because the Approximate Number System (ANS) underlying numerical estimation is shared with infants and with non-human animals who never acquire formal mathematics. However, it remains unclear whether the link between individual differences in math ability and the ANS depends on formal mathematics instruction. Earlier studies demonstrating this link tested participants only after they had received many years of mathematics education, or assessed participants’ ANS acuity using tasks that required additional symbolic or arithmetic processing similar to that required in standardized math tests. To ask whether the ANS and math ability are linked early in life, we measured the ANS acuity of 200 3- to 5-year-old children using a task that did not also require symbol use or arithmetic calculation. We also measured children’s math ability and vocabulary size prior to the onset of formal math instruction. We found that children’s ANS acuity correlated with their math ability, even when age and verbal skills were controlled for. These findings provide evidence for a relationship between the primitive sense of number and math ability starting early in life.

### Introduction

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

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).

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.

### Discussion

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

Our findings reveal that the acuity of preschoolers’ Approximate Number System (ANS) correlates with their school math ability. The robustness of this relationship was established in two ways. First, we found that three different estimates of ANS acuity (accuracy, *w,* and RT) correlate with math ability as measured by the Test of Early Mathematics Ability (TEMA-3; Ginsburg & Baroody, 2003) in preschoolers. Second, these correlations between ANS acuity and math ability remain significant when controlling for age, vocabulary size, and speed–accuracy trade-offs in the ANS acuity task. These findings thus provide strong evidence for a link between ANS acuity and math ability early in life.

Previous studies testing older children left open the possibility that differences in instructional experience mediated both increases in symbolic math abilities and in ANS acuity (Halberda *et al*., 2008; Piazza *et al*., 2010). However, unlike previous studies, the present results show that the link between ANS acuity and math ability is already present before the beginning of formal math instruction.

Our findings also demonstrate that the relationship between ANS acuity and math ability holds even when these are measured using highly non-overlapping tasks. Previous reports of a relationship between primitive numerical abilities and math often used tasks that overlapped in the need to process Arabic digits, or to perform explicit addition and subtraction (Booth & Siegler, 2006, 2008; De Smedt *et al*., 2009; Durand *et al*., 2005; Gilmore *et al*., 2010; Holloway & Ansari, 2008). Therefore, our results more directly bolster the conclusion that individual differences in the noisiness of people’s ANS representations *per se* are linked to individual differences in their math ability.

However, our study leaves open the root cause of the link between ANS acuity and math ability. One possibility is that the ANS is foundational for acquiring symbolic numerical abilities (e.g. Dehaene, 1997). It is conceivable that while learning the meaning of number words, children need to map these to corresponding representations in the ANS (Dehaene, 1997; Gallistel & Gelman, 1992; Mundy & Gilmore, 2009) (but see Le Corre & Carey, 2007). On such a view, children’s ANS acuity may have important consequences for the ease with which children acquire the counting sequence and other subsequent symbolic numerical skills, and thereby could affect the robustness of symbolic number representations. Another possibility is that noisier ANS representations may cause difficulties in performing and evaluating arithmetic operations (Gilmore *et al*., 2007, 2010). Yet a third possibility is that less accurate representations in the ANS may lead to decreased engagement in number-related activities, which might cause an increase in math anxiety and a subsequent decrease in math performance (Maloney, Ansari & Fugelsang, 2011; Maloney, Risko, Ansari & Fugelsang, 2010). Although here we show a connection between ANS acuity and symbolic numerical abilities in preschoolers, our study cannot yet elucidate the origin of this link.

It is also possible that the size of the relationship between math ability and ANS acuity observed in this study is smaller than what has been reported previously. For example, Halberda and colleagues (2008) found that *w* explained between 14 and 20% of the variance in math ability on a standardized math test when controlling for other cognitive abilities such as intelligence, task demands, memory, and language. In the present study, while accuracy, *w,* and RT explained between 7 and 18% of the variance before controlling for age and vocabulary size, entering these factors into a single model resulted in smaller effect sizes for *w* (6% explained) and accuracy (13% explained) while RT continued to predict between 5 and 8% of the variance in math ability. An important consideration is that both RT and estimates of accuracy (percent correct and *w*) appear to carry information about ANS acuity in the current dataset, and RT was not previously analyzed in adolescents (Halberda *et al*., 2008). Considering the predictive power of the conjunction of these factors for the current dataset suggests that the effect size for the relationship between ANS acuity and math ability in the present sample is no less powerful than what was previously demonstrated for adolescents (Halberda *et al*., 2008). Future studies that rely on parallel methods across ages will be needed to determine the relative changes in the effect size of this relationship throughout the lifespan; the current results highlight that considering both *w* and RT will be important for such analyses.

A few further limitations of our study raise the need for future research. First, as noted above, it is unclear whether acuity of the ANS *predicts* math abilities and what role the ANS plays in acquiring symbolic numerical abilities. Our cross-sectional design does not shed light on this important question, and future longitudinal studies will be needed to fully address this issue. Second, we did not directly control for overall intelligence, information-processing speed, working memory, or other cognitive abilities that have been found to correlate with math achievement (e.g. Bull & Johnston, 1997; Geary, 1993; Koontz & Berch, 1996), although many of these correlate with vocabulary size which we did control for. It remains possible that the correlations between ANS acuity and math ability shown here were mediated by a third unknown factor. Third, our work leaves open the possibility that other cognitive factors beyond ANS acuity make significant contributions to children’s overall number sense. Indeed the ANS is only one core cognitive ability relevant to success in school mathematics. Future work will be needed to determine how each of these factors relates to success in school.

In summary, we found that the acuity of preschoolers’ Approximate Number System correlated significantly with their math ability even when controlling for age and vocabulary size. Our study thereby supports the notion of a tight link between a primitive sense of number and more formal math abilities.