The target article by Cohen Kadosh & Walsh (CK&W) can be considered as very timely in light of increasing evidence suggesting a strong impact of input format on numerical information processing (e.g., Campbell Reference Campbell1994). One explanation for such findings is CK&W's proposal that different neuronal networks represent numerical quantity depending on the notation. The authors do not only distinguish between non-symbolic (e.g., dot patterns) and symbolic formats, but also suggest separate representations for different symbol systems (e.g., Arabic digits vs. number words). In their dual-stage model, they argue that the (initial) automatic processing of numerical information relies on these non-abstract, notation-specific representations, whereas (later) intentional processing could lead to the creation of connections between them and give rise to result patterns congruent with the abstract-representation view. Therefore, a large part of the findings reported in favor of their hypothesis comes from studies requiring automatic number processing, which should be “unaffected by task demands and intentional strategies” (sect. 5, para. 3).
A critical issue that was not considered by CK&W is the role of expertise in processing different numerical notations. This primarily holds true for number symbols, which, in contrast to non-symbolic representations, have an arbitrary relationship to the numerical magnitude to which they refer (Peirce Reference Peirce and Buchler1955). Over development, we learn to map these cultural symbols onto semantic magnitude information (cf. Ansari Reference Ansari2008) and attain different levels of expertise for different notations. For instance, we are generally more proficient in the numerical processing of Arabic digits compared to number words, since the former are the standard visual notation for quantity processing and calculation. The more expertise we have acquired for a specific type of numerical symbol, the more likely its presentation leads to an automatic activation of the underlying semantic magnitude representation. A similar account was put forward by Campbell and colleagues in their encoding-complex hypothesis: “task-specific retrieval processes will be more efficient when numerical stimuli appear in a familiar, well-practiced format, relative to the retrieval processes activated by an unfamiliar surface form” (Campbell & Epp Reference Campbell and Epp2004, p. 231). If differences in symbol-referent mapping expertise are not considered in tasks drawing on automatic number processing, notation-related effects are not conclusive with regard to the question of non-abstract representations. This point should be illustrated by briefly reviewing some behavioral key findings reported in the target article.
CK&W cited two studies in which interactions between notation and automatic number processing were observed in the size-congruity paradigm (Cohen Kadosh et al. Reference Cohen Kadosh, Henik and Rubinsten2008e; Ito & Hatta Reference Ito and Hatta2004). These studies revealed that numerical information from number words did not interfere with physical size comparison, whereas such an interference emerged for Arabic digits, indicating that “numerical information was not processed automatically” (sect. 6, para. 1). However, if the semantic representation was not automatically accessed by the number words, presumably due to less symbol-referent mapping expertise for verbal numbers, this result cannot be interpreted as evidence for notation-specific representations. Fias and colleagues investigated the well-established SNARC effect using Arabic digits (Fias et al. Reference Fias, Brysbaert, Geypens and d'Ydewalle1996) and verbal numbers (Fias Reference Fias2001). Automatic number processing was probed by requiring participants to monitor the occurrence of certain phonemes in the numbers; in the intentional number-processing task parity judgments were required. The latter task elicited SNARC effects for both, Arabic digits and number words. In the phoneme-monitoring task, however, a SNARC effect was only observed for Arabic digits, not for number words. A lack of expertise in mapping number words to semantic magnitude representations can again explain the absence of a SNARC effect in the phoneme-monitoring task. Also, the reported results from the priming study by Koechlin et al. (Reference Koechlin, Naccache, Block and Dehaene1999) can be interpreted in a similar vein. Participants were presented a prime number before they had to compare a numerical stimulus to a standard. They observed notation-related effects only when the prime number was subliminally presented (for 66 msec). Specifically, a quantity priming effect (i.e., shorter response times in the number comparison task with decreasing numerical distance between prime and target) emerged for Arabic digits but not for verbal numbers. Koechlin and colleagues discussed these findings in terms of notation-specific representations under demanding temporal conditions, which may converge at a later stage of processing. However, it also appears plausible that the number words were not presented long enough to sufficiently activate the underlying semantic quantity representation. Taken together, notation-related differences in symbol-referent mapping expertise, which result in differential activation of the underlying semantic magnitude representation, can explain the reviewed findings without the need for assuming notation-specific representations.
Symbol-referent mapping expertise may also loom large in understanding individual differences in mathematical competence. Recent studies suggest that little expertise or even deficits in processing numerical symbols are related to poor mathematical performance. Rousselle and Noël (Reference Rousselle and Noël2007), for example, reported that children with developmental dyscalculia displayed similar performance as matched controls in a non-symbolic magnitude comparison task, whereas they performed more poorly in the symbolic task version. Holloway and Ansari (Reference Holloway and Ansari2009) investigated typically developing children and found that only the symbolic, but not the non-symbolic, distance effect predicted later mathematical achievement (but see also Halberda et al. Reference Halberda, Mazzocco and Feigenson2008).
The point raised in this commentary is compatible with both the abstract and the non-abstract view of numerical representation in the brain. We may learn to map numerical symbols onto a unitary abstract representation (Dehaene et al. Reference Dehaene, Piazza, Pinel and Cohen2003), or we may develop separate representations for these symbols as is proposed by CK&W. At present, our knowledge about the neurocognitive processes involved in the acquisition of symbol processing expertise is very limited. Recent neuroimaging studies revealing developmental (e.g., Ansari et al. Reference Ansari, Garcia, Lucas, Hamon and Dhital2005) and training-related (e.g., Diester & Nieder Reference Diester and Nieder2007) activation changes during basic number processing, however, suggest a high degree of plasticity in neuronal networks coding numerical quantity. Whether evidence for abstract or non-abstract representations is found could thus partly depend on the acquired level of expertise in numerical symbol-referent mapping.
