In sections 3 and 4 of their proposal, Clarke and Beck (C&B) discuss a body of literature challenging the claim that the approximate number system (ANS) represents genuine numerical content coming from studies demonstrating the presence of congruency effects and interference from non-numerical confounds in non-symbolic number comparison (e.g., Gebuis, Cohen Kadosh, & Gevers, Reference Gebuis, Cohen Kadosh and Gevers2016; Gebuis & Reynvoet, Reference Gebuis and Reynvoet2012; Smets, Sasanguie, Szucs, & Reynvoet, Reference Smets, Sasanguie, Szucs and Reynvoet2015). We argue that in their current paper, C&B somewhat misinterpreted these articles' key message. In fact, the propositions put forward by these studies are very similar to the indirect model of number perception C&B suggest.

To begin with, the abovementioned studies do not claim that the ANS “[…] merely represents a mishmash of non-numerical magnitudes” (C&B, sect. 4, para. 11; emphasis added) – at least not in the way C&B conceive the ANS. These studies contested the what is called here, direct model of number perception – the idea that “number” is a primary feature of a set, which can be directly perceived from the environment. Instead, these studies argued that “[…] number judgements are based on the integration of information […]” (Gebuis & Reynvoet, Reference Gebuis and Reynvoet2012, abstract; italic added). In our view, this idea is very similar to the indirect model of the ANS C&B appeal to themselves.

Researchers in numerical cognition are well aware that the number of a set is confounded with non-numerical magnitudes such as size in the visual modality and pitch length in the auditory modality. Consequently, various algorithms have been developed to control these non-numerical magnitudes by making them uninformative for the decision (e.g., Gebuis & Reynvoet, Reference Gebuis and Reynvoet2011; Halberda, Mazzocco, & Feigenson, Reference Halberda, Mazzocco and Feigenson2008; Marinova, Sasanguie, & Reynvoet, Reference Marinova, Sasanguie and Reynvoet2021). Nevertheless, even when these confounds are accounted for, they still affect participants' performance in non-symbolic number comparison and lead to congruency effects (i.e., lower accuracies when numerical and non-numerical magnitudes conflict; see Reynvoet et al., Reference Reynvoet, Ribner, Elliot, Van Steenkiste, Sasanguie and Libertusin press; Smets et al., Reference Smets, Sasanguie, Szucs and Reynvoet2015). Some studies even observed that the size of the congruency effects depends on the interrelation of different non-numerical magnitudes. For instance, the congruency between one non-numerical magnitude (e.g., the convex hull) and number and another non-numerical magnitude (e.g., size of the individual dots) and number can result in an additive effect. Alternatively, they can also cancel out each other (Gebuis & Reynvoet, Reference Gebuis and Reynvoet2012). Based on these findings, researchers proposed that participants integrate the information from multiple visual cues into one weighted sum (Gebuis et al., Reference Gebuis, Cohen Kadosh and Gevers2016; see also Picon, Dramkin, & Odic, Reference Picon, Dramkin and Odic2019).

The idea of integrated information is also very similar to the excellent example of the representation of depth C&B describe in their article. Here, the authors argue that the representation of depth is constructed based on various visual inputs, which can be weighted differently depending on the context (e.g., some inputs may be less informative in particular situations and given less weight). Therefore, the main difference between our previous study and the indirect model proposed by C&B does not lie in the pre-assumed underlying perceptual and cognitive processes. Rather, it lies in what one considers a “number.” In our view, the weighted sum of the visual inputs can be regarded as a representation of number, similar to the representation of depth (see also Halberda, Reference Halberda2019 who describes a similar position from a reductionist/empiricist and rationalist point of view). Our previous study shows that in many situations, the representation of number on which numerical decisions are based is influenced by the (incongruent) non-numerical magnitudes. As C&B and some proponents of the direct model of number perception rightfully argue, this interference may arise at a response stage (similar to a classic Stroop effect). In contrast to this claim, in their study, Picon et al. (Reference Picon, Dramkin and Odic2019) demonstrated that interference between non-numerical magnitudes and number occurs early in the processing stream – an observation rather in line with an indirect account of number perception. It is worth noting that the idea of “weighted” number representation does not entirely rule out the possibility of a direct model of number perception. Concretely, a substantial body of literature suggests that automatic and direct extraction of number is possible (e.g., Burr & Ross, Reference Burr and Ross2008; Van Rinsveld et al., Reference Van Rinsveld, Wens, Guillaume, Beuel, Gevers, De Tiège and Content2021). The latter case is especially plausible in adult participants because of the extensive focus on “number” in the educational curricula.

Finally, instead of remaining agnostic to one of the most crucial questions in numerical cognition about whether the number is extracted directly or indirectly, we believe it is possible to reconcile the direct and the indirect models of number perception. Concretely, whether the numerical decision would be based on direct and automatic extraction of number or indirect weighted number representation possibly depends on various factors such as developmental differences (e.g., Piazza, De Feo, Panzeri, & Dehaene, Reference Piazza, De Feo, Panzeri and Dehaene2018), stimulus set (e.g., Reynvoet et al., Reference Reynvoet, Ribner, Elliot, Van Steenkiste, Sasanguie and Libertusin press), and so on.

In sum, in this commentary, we argued that previous studies describing congruency effects resulting from contrasting numerical and non-numerical cues could be easily reconciled with the indirect model of the ANS as proposed by C&B. That is, multiple sources of information may be integrated into a higher-order representation of number. We also acknowledged that in some circumstances, the number might be extracted directly. Whether the number will be processed directly or indirectly, as a weighted sum of visual inputs, depends mainly on the individual and the context in which the numerical decision arises. Finally, it is also worth noting that, thanks to its exceptionally well-presented framework, the article by C&B provides an excellent starting point to disentangle further the conditions under which direct and weighted number representations occur.