Cardinal number is an abstract property of a set of elements because it is invariant for all possible kinds of elements or referents. The basic arithmetic operations (i.e., addition, multiplication, subtraction, and division), at least when viewed as formal arithmetical functions, are similarly abstract because the kinds or referents of problem operands are irrelevant. For this reason, both number and arithmetic seem to be natural candidates for abstract instantiation in the brain. Not surprisingly, then, the issue of abstraction has engaged cognitive arithmetic researchers in much the same debate that has occupied research on quantity representation. In the target article, Cohen Kadosh & Walsh (CK&W) have focused on effects of numerical format (e.g., 4 vs. four) in tasks that directly or indirectly tap quantity processing, but they have not discussed in any detail the substantial body of research that has examined effects of numeral format on elementary arithmetic (e.g., 4+5 vs. four+five).

In fact, the prominent models of number processing advocated by Dehaene et al. (Reference Dehaene, Dehaene-Lambertz and Cohen1998a) and McCloskey and Macaruso (Reference McCloskey and Macaruso1995), which assume that different numeral formats activate a common quantity representation, similarly assume that elementary arithmetic is abstracted away from surface form (see also Venkatraman et al. Reference Venkatraman, Ansari and Chee2005). Alternatively, cognitive processes for arithmetic could vary with format (see Campbell & Epp [Reference Campbell, Epp and Campbell2005] for a review of relevant literature). In this alternative view, there are two senses in which elementary arithmetic could be non-abstract. First, calculation performance could be based on discrete, format- and operation-specific processes (McNeil & Warrington Reference McNeil and Warrington1994). This implies non-abstract representation in the same sense defined in the target article (i.e., different formats recruit distinct neuronal populations). Second, calculation might be based on overlapping representations across formats, but calculation efficiency is format specific. This would occur if problem-encoding processes and calculation processes were interactive rather than strictly additive (Szűcs & Csépe Reference Szűcs and Csépe2004). As the following paragraphs illustrate, there is evidence that performance of elementary arithmetic by educated adults is non-abstract in both senses.

In experimental research examining format effects on elementary calculation, the most common contrast has involved arithmetic problems in Arabic digit format (5+6) versus written word format (five+six). It is consistently found that performance is much slower and more error prone with written-word operands compared to Arabic digits. Campbell and Epp (Reference Campbell, Epp and Campbell2005) proposed that relatively poor performance with arithmetic problems in written-word format occurs because number-fact retrieval processes are less efficient with word compared to digit operands. Arithmetic problems are encountered more frequently in digit format (e.g., 2+6, 4×5) than in written number-word format (two+six, four×five). Consequently, digit problems are more likely to activate a visual retrieval path than would a written-word problem (McNeil & Warrington Reference McNeil and Warrington1994). Retrieval, given number words, presumably requires phonological recoding of problems so that the proximal retrieval cue is based on auditory-phonological codes. Retrieval with digits, therefore, would be more efficient because it is mediated both by well-established visual and phonological routes, whereas retrieval with number-word format would not provide a direct visual basis for retrieval.

The proposal that arithmetic-fact retrieval efficiency is lower with problems in word format than digit format is also supported by the well-replicated finding that non-retrieval strategies (i.e., procedural strategies such as counting or decomposition) are reported more often given word format (six+seven) than digit format (6+7) (Campbell & Epp Reference Campbell, Epp and Campbell2005). Educated adults report procedural strategies for simple arithmetic up to 50% more with problems in written-word format than digit format (Campbell & Alberts, in press). Format-induced strategy shifts imply that different formats often recruit different neural processes for elementary arithmetic. Indeed, imaging research suggests that retrieval of arithmetic facts is associated with linguistic representations in the left angular gyrus, whereas procedural strategies requiring semantic quantity processing recruit bilateral components of the intraparietal sulcus (Dehaene et al. Reference Dehaene, Molko, Cohen and Wilson2004). As direct retrieval and procedural strategies activate distinct brain regions (see also Dehaene et al. Reference Dehaene, Piazza, Pinel and Cohen2003), the effects of format on strategy choice for elementary arithmetic imply that calculation is not generally abstracted away from surface form.

Format-related strategy shifts demonstrate that calculation performance sometimes involves discrete, format-specific processes, but calculation also appears to be non-abstract in the second sense mentioned earlier; namely, that format-specific encoding processes or context can interact with calculation processes. One source of evidence for this comes, again, from research examining format effects on simple arithmetic: When procedural strategy trials are removed from analysis and only retrieval trials are analyzed, there remain substantial word-format costs relative to digit format, and word-format retrieval costs tend to increase with problem difficulty (Campbell et al. Reference Campbell, Parker and Doetzel2004; Campbell & Penner-Wilger Reference Campbell and Penner-Wilger2006). This reinforces the conclusion that arithmetic retrieval processes are not abstracted away from surface format.

The non-abstractness of elementary arithmetic is demonstrated further by context-dependent activation of arithmetic facts. Bassok et al. (Reference Bassok, Pedigo and Oskarsson2008) found evidence for obligatory activation of addition facts (4+2=6) when problems were primed by word pairs semantically aligned with addition (e.g., tulips-daisies, which afford addition as a collection of flowers), but not when they were primed by pairs misaligned with addition (hens-radios, records-songs). The automaticity of arithmetic fact retrieval thereby depended on the analogical consistency of the semantic context activated by the prime and the specific arithmetic operation to be performed. This implies that the kinds and referents of problem operands are relevant to cognitive arithmetic, despite being irrelevant to arithmetic as a formal operation. Like the effects of surface form, semantic alignment phenomena demonstrate that cognitive arithmetic is not abstracted away from the conditions of problem encoding. Research on elementary arithmetic thereby aligns with the theoretical perspective represented in the target article, and points toward integrated, multimodal mechanisms in favor of abstract or amodal representations and processes (e.g., Barsalou Reference Barsalou2008; Clark & Campbell Reference Clark and Campbell1991).