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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). For this purpose, we analyzed the response times of ninety adult participants when solving the 81 additions with operands from 1 to 9. Even when focusing on small problems (i.e. with sums 610) reported by participants as being solved by direct retrieval, chronometric analyses revealed a strong size effect. Response times increased linearly with the magnitude of the operands testifying for the involvement of a sequential multistep procedure. However, this size effect was restricted to the problems involving operands from 1 to 4, whereas the pattern of response times for other small problems was compatible with a retrieval hypothesis. These findings suggest that very fast responses routinely interpreted as reflecting direct retrieval of the answer from memory actually subsume compiled automated procedures that are faster than retrieval and deliver their answer while the subject remains unaware of their process, mistaking them for direct retrieval from long-term memory.

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.

The aim of this study was to investigate the strategies used by third graders in solving the 81 elementary subtractions that are the inverses of the one-digit additions with addends from 1 to 9 recently studied by Barrouillet and Lépine. Although the pattern of relationship between individual differences in working memory, on the one hand, and strategy choices and response times, on the other, was the same in both operations, subtraction and addition differed in two important ways. First, the strategy of direct retrieval was less frequent in subtraction than in addition and was even less frequent in subtraction solving than the recourse to the corresponding additive fact. Second, contrary to addition, the retrieval of subtractive answers is confined to some peculiar problems involving 1 as the subtrahend or the remainder. The implications of these findings for developmental theories of mental arithmetic are discussed.

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The authors showed that for primary school children, response times (RTs) for simple addition problems (e.g., 4 3) increase linearly with the size of the smaller operand. This result was the first to provide evidence for the use of the min strategy by children. This strategy consists in counting on from the larger of the two operands by the number indicated by the smaller of the operands (Carr & Jessup, 1995;Siegler, 1987;Siegler & Crowley, 1994). Adults also show a significant increase in response latencies as a function of the size of the operands, but this increase is much smaller than that in children. Moreover, unlike in children, adults' RTs form a curvilinear function that is best explained by the square of the sum or the product of the operands than by the size of the operands. These differences between adults and children have been interpreted as evidence that, unlike children, adults use fast and efficient retrieval from memory to solve simple addition problems.However, LeFevre, Sadesky, and Bisanz (1996) stressed that, as has already been mentioned for children (Siegler, 1987(Siegler, , 1989, averaging solution latencies across trials that involve different procedures can result in misleading conclusions about how adults solve problems. These authors note that the researchers who have used the more direct approach of asking participants to report their procedure have challenged the strong assumption that adults always use retrieval for simple addition problems. For example, Svenson (1985) showed that adults were certain that they had used a retrieval strategy on simple addition problems on only 78% of the trials (for similar results, see also Geary, Frensch, & Wiley, 1993;Geary & Wiley, 1991). Then, to address the issue of how the selection of procedures varies across problems and participants, LeFevre et al. collected trial-by-trial reports of procedure, in addition to the classical chronometric data. The authors concluded that the importance of retrieval had been overemphasized in models of adult performance. Indeed, in the case of addition problems, 81% of their participants had used two or more of the counting, retrieval, and decomposition procedures in order to solve simple problems (i.e., both the addend and the augend inferior to 10). In fact, retrieval was the most frequently used procedure for problems with sums lower According to LeFevre, Sadesky, and Bisanz (1996), averaging solution latencies in order to study individuals' arithmetic strategies can result in misleading conclusions. Therefore, in addition to classical chronometric data, they collected verbal reports and challenged the assumption that adults rely systematically on retrieval of arithmetic facts from memory to solve simple addition problems. However, Kirk and Ashcraft (2001) questioned the validity of such a methodology and concluded that a more appropriate method has to be found. Thus, we developed an operand recognition paradigm that does not rely on verbal reports or on solution latencies. In accordance with LeFevre et al.,...

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The authors used the operand-recognition paradigm (C. Thevenot, M. Fanget, & M. Fayol, 2007) in order to study the strategies used by adults to solve subtraction problems. This paradigm capitalizes on the fact that algorithmic procedures degrade the memory traces of the operands. Therefore, greater difficulty in recognizing them is expected when calculations have been solved by reconstructive strategies rather than by retrieval of number facts from long-term memory. The present results suggest that low- and high-skilled individuals differ in their strategy when they solve problems involving minuends from 11 to 18. Whereas high-skilled individuals retrieve the results of such subtractions from long-term memory, lower skilled individuals have to resort to reconstructive strategies. Moreover, the authors directly confront the results obtained with the operand-recognition paradigm and those obtained with the more classical method of verbal report collection and show clearly that this second method of investigation fails to reveal this differential pattern. The rationale behind the operand-recognition paradigm is then discussed.