We applaud Clarke and Beck (C&B) for their convincing arguments supporting the presence of an approximate number system (ANS). Most importantly, we agree with their notion that the ANS represents numbers, not numerosities or non-numerical confounds, even if its representations can be derived from computations involving perceptual cues. The ANS has attracted increasingly more attention over the last decade as correlational and training studies suggest a link between the ANS and children's and adults' math abilities. Many (but not all) studies report that children and adults with greater ANS acuity tend to perform better on math assessments both concurrently and longitudinally (Chen & Li, Reference Chen and Li2014; Fazio, Bailey, Thompson, & Siegler, Reference Fazio, Bailey, Thompson and Siegler2014; Schneider et al., Reference Schneider, Beeres, Coban, Merz, Schmidt, Stricker and De Smedt2017) and that training the ANS leads to improvements in children's and adults' math abilities (Bugden, DeWind, & Brannon, Reference Bugden, DeWind and Brannon2016). However, none of these studies have been able to provide a definitive mechanistic explanation for the association between the ANS and math abilities. Previous explanations have (1) largely relied on developmental arguments, (2) invoked the function of the ANS as an error-detection mechanism, or (3) cited possible motivational or affective factors.

Several different possibilities may explain the link between the ANS and math throughout development. On the one hand, it is possible that a more precise ANS may better support children's acquisition of exact number representations (Pinheiro-Chagas et al., Reference Pinheiro-Chagas, Wood, Knops, Krinzinger, Lonnemann, Starling-Alves and Haase2014; Wagner & Johnson, Reference Wagner and Johnson2011). For example, children's ability to map between symbolic and non-symbolic quantities is associated with their math achievement, suggesting that ANS representations are involved in the development of children's math skills via their associations with number symbols (Mundy & Gilmore, Reference Mundy and Gilmore2009). On the other hand, it is possible that a more precise ANS may serve as a foundation to understand ordinal relations between quantities and their relation to arithmetic operations, especially as children acquire these math skills (Libertus, Odic, Feigenson, & Halberda, Reference Libertus, Odic, Feigenson and Halberda2016; Mussolin, Nys, Leybaert, & Content, Reference Mussolin, Nys, Leybaert and Content2016; Park, Bermudez, Roberts, & Brannon, Reference Park, Bermudez, Roberts and Brannon2016). For example, the ability to identify ordered sequences of Arabic numerals mediates the relation between the ANS and adults' mental arithmetic (Lyons & Beilock, Reference Lyons and Beilock2011) and steadily increases in its role between first and sixth grades (Lyons, Price, Vaessen, Blomert, & Ansari, Reference Lyons, Price, Vaessen, Blomert and Ansari2014).

Another explanation is that the ANS may provide a sense of certainty about number-related judgments or serve as an “error detection mechanism” providing rough estimates of arithmetic computations and aiding in the detection of gross miscalculations (Baer & Odic, Reference Baer and Odic2019; Vo, Li, Kornell, Pouget, & Cantlon, Reference Vo, Li, Kornell, Pouget and Cantlon2014). For example, individuals' ability to detect errors in symbolic arithmetic problems is related to their ANS acuity (Wong & Odic, Reference Wong and Odic2021).

Finally, the ANS and math may be linked via motivational or affective factors. For example, greater ANS acuity in childhood may increase children's attention to number or engagement with math-related information (Libertus, Reference Libertus, Geary, Berch and Mann Koepke2019). Alternatively, greater ANS acuity may lead to greater confidence in mathematical reasoning (Wang, Odic, Halberda, & Feigenson, Reference Wang, Odic, Halberda and Feigenson2016) or poorer ANS acuity may lead to increased math anxiety (Lindskog, Winman, & Poom, Reference Lindskog, Winman and Poom2017; Maldonado Moscoso, Anobile, Primi, & Arrighi, Reference Maldonado Moscoso, Anobile, Primi and Arrighi2020; Maloney, Ansari, & Fugelsang, Reference Maloney, Ansari and Fugelsang2011).

Many of these explanations rest on the (albeit, implicit) assumption that the ANS represents natural numbers. As such, extant hypotheses cannot fully explain why the ANS may be correlated, for instance, with adults' performance on college entrance exams in math that require far more than whole number arithmetic (Libertus, Odic, & Halberda, Reference Libertus, Odic and Halberda2012). Even if, for example, the ANS is involved in error monitoring during calculations, how could this system operate to detect errors in calculations that do not depend solely on positive integers? Clarke's and Beck's proposal that the ANS represents rational numbers opens up an exciting additional explanation which may provide a missing link in the theoretical pathway from non-symbolic number representations to math abilities. Specifically, their proposal that the ANS represents rational numbers would provide a compelling explanation of how the ANS may directly support a broader range of math skills that transcend the natural numbers and operations thereon, including fraction understanding and proportional reasoning.

However, as C&B mentioned, there is a dearth of research on non-symbolic ratio processing. Future research should test the sensitivity of the ANS to rational numbers and probe the relation between the ANS and the ratio processing system (RPS), which the authors argue is a component of the ANS. An initial step is to examine the associations between individuals' performance on a wide range of tasks tapping into the ANS and the RPS that have previously only been used in separate studies. Although some research has suggested that the ANS is recruited during tasks that require the RPS or proportional reasoning (Matthews & Chesney, Reference Matthews and Chesney2015), no studies have explicitly established a correlation between the precision of these systems. O'Grady and Xu (Reference O'Grady and Xu2020) posit that children's proportional judgments of non-symbolic dot arrays are reliant on the ANS to represent discrete numbers, which are used to calculate probabilities. However, it is unclear whether the relational processing of whole numbers is supported by the ANS and/or facilitated by the RPS.

Extending beyond ratio processing, recent research on risky decision-making involving non-symbolic quantities demonstrates an association between adults' performance on tasks tapping the ANS and probability understanding (Mueller & Brand, Reference Mueller and Brand2018). For instance, individuals' non-symbolic quantity estimation relates to their abilities to estimate risks presented non-symbolically, and both of these abilities relate to adults' ability to transform and compare symbolic probabilities, an important aspect of math abilities beyond whole number operations (Mueller, Schiebener, Delazer, & Brand, Reference Mueller, Schiebener, Delazer and Brand2018). Thus, Clarke's and Beck's view of the ANS may also provide an explanation for these findings and suggest further interesting research directions, including the development of probability understanding and its link to the ANS.

In sum, the proposal that the ANS represents rational numbers helps in further elucidating the link between the ANS and math abilities. This perspective opens up interesting new directions for future research, including probing the relations between the ANS and RPS as well as understanding the relations between the ANS and decision-making processes.