On the problem-size effect in small additions: Can we really discard any counting-based account?

Abstract

a b s t r a c tThe problem-size effect in simple additions, that is the increase in response times (RTs) and error rates with the size of the operands, is one of the most robust effects in cognitive arithmetic. Current accounts focus on factors that could affect speed of retrieval of the answers from long-term memory such as the occurrence of interference in a memory network or the strength of memory traces that would differ from problem to problem. The present study analyses chronometric data from a sample of 91 adults solving very small additions (operands from 1 to 4) that are generally considered as being solved by retrieval. The results reveal a monotonic linear increase in RTs with the magnitude of both operands. This pattern is at odds with the retrieval-based accounts of the problem-size effect and challenges the well-established view that small additions are solved through retrieval of the answer from long-term memory. Our results are more compatible with the hypothesis that even very small additions are solved using compacted fast procedures that scroll an ordered representation such as a number line or a verbal number sequence. This interpretation is corroborated by the analysis of individual differences.

Fast automated counting procedures in addition problem solving: When are they used and why are they mistaken for retrieval?

“…The hypothesis of automated procedures was reinforced by a recent study in which we had adult participants solving very small additions with operands varying from 1 to 4 (Barrouillet & Thevenot, 2013). This study revealed a quasi perfect monotonic size effect in non-tie problems, with RTs increasing with the size of both the first and the second operand (mean slopes of 16 ms and 23 ms respectively).…”

“…It is also difficult to account for by a hypothesis of exposure frequency. As we noted in Barrouillet and Thevenot (2013), the potential effect of differences in frequency has probably been overestimated in accounting for retrieval times in the domain of mental arithmetic. In the linguistic domain, dramatic differences in word frequency (from about 3000 to 60 per million) result in rather small differences in RTs (i.e., 15 ms in a lexical decision task, Ferrand et al, 2011), whereas the observed difference in RTs between probably very frequent additions such as 2 + 1 and 2 + 4 was higher than 90 ms.…”

“…However, the widespread hypothesis of a solving of frequent and simple additions by direct retrieval of problem-answer associations cannot be jettisoned before a close investigation. Although Barrouillet and Thevenot's (2013) results seemed to point toward the hypothesis of compiled procedures, their study had limitations that make premature any firm conclusion. First, it was assumed, but not verified, that the additions studied would have been considered as solved through direct retrieval by defenders of retrieval models.…”

“…Thus, we concluded that the problem-size effect in these very small additions was better explained by the use of automated fast procedures as hypothesized by Baroody (1994). We suggested that these procedures could consist of scrolling an ordered representation such as a number line or a verbal number sequence (Barrouillet & Thevenot, 2013). For example, solving 3 + 2 would involve moving forward of three and then two steps in the verbal number sequence, the arrival point of the process giving the answer.…”

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a b s t r a c tContrary to a widespread assumption, a recent study suggested that adults do not solve very small additions by directly retrieving their answer from memory, but rely instead on highly automated and fast counting procedures (Barrouillet & Thevenot, 2013). The aim of the present study was to test the hypothesis that these automated compiled procedures are restricted to small quantities that do not exceed the size of the focus of attention (i.e., 4 elements).

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