We reviewed the literature on the role of working memory in the solution of arithmetic problems such as 3 + 4 or 345 + 29. The literature was neither comprehensive nor systematic, but a few conclusions are tenable. First, all three components of the working memory system proposed by Baddeley (i.e., central executive, phonological loop, and visual‐spatial sketchpad) play a role in mental arithmetic, albeit under different conditions. Second, mental arithmetic requires central executive resources, even for single‐digit problems. Third, further progress in understanding the role of working memory in arithmetic requires that researchers systematically manipulate factors such as presentation conditions (e.g., operand duration, format), problem complexity, task requirements (e.g., verification vs production), and response requirements (e.g., spoken vs written); and that they consider individual differences in solution procedures. Fourth, the encoding‐complex model (Campbell, 1994) seems more likely to account f...

The role of working memory in mental arithmetic

Canadian university students either of Chinese origin (CC) or non-Asian origin (NAC) and Chinese university students educated in Asia (AC) solved simple-arithmetic problems in the 4 basic operations (e.g., 3 + 4, 7 - 3, 3 X 4, 12 •*• 3) and reported their solution strategies. They also completed a standardized test of more complex multistep arithmetic. For complex arithmetic, ACs outperformed both CCs and NACs. For simple arithmetic, however, ACs and CCs were equal and both performed better than NACs. The superior simple-arithmetic skills of CCs relative to NACs implies that extracurricular culture-specific factors rather than differences in formal education explain the simple-arithmetic advantage for Chinese relative to non-Asian North American adults. NAC's relatively poor simple-arithmetic performance resulted both from less efficient retrieval skills and greater use of procedural strategies. Nonetheless, all 3 groups reported using procedures for the larger simple subtraction and division problems, confirming the importance of procedural knowledge in skilled adults' performance of elementary mathematics. Knowledge of elementary arithmetic (i.e., simple addition, multiplication, subtraction, and division) is a pervasive requirement of everyday modern life, providing the essential means for dealing with a widely diverse variety of numerical-problem-solving situations. Basic arithmetic also provides the foundation for the more advanced mathematical skills that are central to all modern scientific disciplines. Consequently, understanding this fundamental intellectual skill is an important goal for cognitive science (Ashcraft, 1995; Geary, 1994). In this study, Chinese adults educated in the People's Republic of China and Canadian adults educated in Canada, either of Chinese or non-Asian origin, solved simple arithmetic problems involving the four basic operations (e.g., 3 + 4, 7 - 3, 3 X 4, 12 -=- 3). Participants also reported their strategy (i.e., direct memory retrieval vs. procedural strategies) after each problem. The purpose was to address three important questions of current research in cognitive arithmetic. First, when adults solve simple-arithmetic problems, what is the relative balance of direct memory retrieval versus use of procedural strategies such as counting or transformation (e.g., 6 + 7 = 6 + 6+1 = 13)? Recent evidence suggests that even skilled adults make substantial use of procedures (e.g., LeFevre, Sadesky, & Bisanz, 1996), but no study has attempted to assess this for the entire domain of elementary arithmetic. Second, what determines the problem-size effect (PSE) in cognitive arithmetic? The PSE is the virtually ubiquitous phenomenon that the difficulty of simplearithmetic problems increases as problem size increases. The PSE has been recognized and studied systematically for over 75 years (e.g., Clapp, 1924), but our study was the first to estimate the

/pdf/cognitive-arithmetic-across-cultures-18fjoynvev.pdf

Cognitive arithmetic across cultures.

http://www.unicog.org/publications/Lemer_approxexact_neuropsychologia_2003.pdf

Approximate quantities and exact number words: dissociable systems.

Learning by strategies and learning by drill--evidence from an fMRI study.

Neuroscience of learning arithmetic--evidence from brain imaging studies.

Adults (N=32) solved simple multiplication (e.g., 8×7) and corresponding division problems (e.g., 56/8). Self-reports of solution processes were given by half of the participants. Latency patterns and error rates were closely related across operations and were similar in self-report and no-report conditions. Solution of division problems, however, facilitated solution of multiplication problems more than the reverse. On large division problems, participants reported that they “recast” problems as multiplication (e.g., 56/8 as 8×?=56). These results support the hypothesis that multiplication and division are stored in separate mental representations but that solution of difficult division problems sometimes involves access to multiplication.

/pdf/more-on-the-relation-between-division-and-multiplication-in-2u2r9lqm5n.pdf

More on the relation between division and multiplication in simple arithmetic: evidence for mediation of division solutions via multiplication.

In 2 experiments participants solved division problems presented in multiplication-based formats (e.g., 8 x _ = 72) more quickly than division problems presented in division-based formats (e.g., 72 / 8 = _). In contrast, participants solved multiplication problems presented in a division-based format (e.g., _ / 8 = 9) slowly and made many errors. In both experiments, the advantage for multiplication-based formats on division problems was found only for large problems (i.e., those with products or dividends greater than 25). These findings provide support for the view that large single-digit division facts are mediated via multiplication-based representations and that multiplication is the primary mode of representation for both division and multiplication facts.

Jason Morris

Papers

More on the relation between division and multiplication in simple arithmetic: evidence for mediation of division solutions via multiplication.

Effects of problem format on division and multiplication performance: division facts are mediated via multiplication-based representations.