Cognitive arithmetic: A review of data and theory

Bending the power law: A CMPL theory of strategy shifts and the automatization of cognitive skills.

“…There was again a reliable effect of log block, F(1, 20) = 69.5, p < .0001, but there was no reliable effect either of addend, F(2, 40) = .08, p = .92, or of the interaction log block and addend, F(2, 40) = .06, p = .94. In sum, then, in line with thepredictions of the Two experiments provide new evidence in support of the general claim of Logan (1988) and Siegler (1988) that practice on skills that originally require execution of sequential algorithms can produce a strategy shift to direct memory retrieval (see also, Ashcraft, 1992). However, the results from both experiments also suggest that the CMPL model may provide a better account of the mechanisms underlying this strategy shift than does the version of the instance theory proposed by Logan (1988).…”

“…These results constitute the 1st clear demonstration of a skill for which the power law does not apply overall. The results do not support the instance theory of automatization (G. D. Logan, 1988) but are consistent with an alternative component 12ower laws (CMPL) theory that assumes that because of intrinsic attentional limitations, only 1 strategy can be executed at a time.One of the fundamental processes of human skill acquisition is the strategy shift with practice from use of generic, multistep procedures to direct retrieval of answers from memory (Ashcraft, 1992;Lemaire & Siegler, 1995;Logan, 1988;Reder and Ritter, 1992;Rickard & Bourne, 1996;Siegler, 1988). Examples are numerous in both the natural environment and the laboratory.…”

Effects of problem size, operation, and working-memory span on simple-arithmetic strategies: differences between children and adults?

“…The problem-size eVect, which refers to slower and more error-prone performance on large problems (e.g., 8 £ 9) than on small problems (e.g., 2 £ 3), is one of the most robust eVects observed in mental-arithmetic research (Ashcraft 1992;ZbrodoV 1995). According to Campbell and Xue (2001), there are three strategy-related sources of the problem-size eVect in adults: less frequent retrieval use for large than for small problems, lower retrieval eYciency for large than for small problems, and lower procedural eYciency for large than for small problems.…”

A neural model of how the brain represents and compares multi-digit numbers: spatial and categorical processes

“…The most influential and widely cited models developed in the last 20 years include models by McCloskey and colleagues (McCloskey, 1992;McCloskey & Macaruso, 1995), Ashcraft (1987Ashcraft ( , 1992, the encoding complex hypothesis by Campbell and Clark (1992;Clark & Campbell, 1991), and the triple-code model of Dehaene (1992). These are functional models of numerical abilities that generally consider high-level verbal, phonological, graphemic representations and complicated cognitive tasks, including numerical calculations.…”