The Problem-size Effect in Mental Addition: Developmental and Cross-national Trends

Abstract

Across two experiments, the magnitude of the problem-size effect in mental addition was examined for kindergarten and elementary school children, as well as adults, from mainland China and the United States. In North American samples, the problem-size effect represents the finding that arithmetic problems consisting of larger-valued numbers (e.g. 8+7) take longer to solve and are more error prone than are problems consisting of smaller-valued numbers (e.g. 2+3). This standard finding was found for the kindergarten, elementary school, and adult samples from the United States. For the Chinese children, the problem size effect was evident in kindergarten and at the beginning of first grade. However, the effect had disappeared at the end of first grade and had reversed (i.e. largervalued addition problems were solved more quickly than smaller-valued problems) by the end of third grade. However, the standard problem-size effect "reappeared" for the Chinese adults. The results are interpreted in terms of theoretical models of the nature of the memory representation for arithmetic facts and in terms of the mechanisms that govern the development of these representations.In the nearly 25 years since Groen and Parkman's (1972) seminal study of the mental processes underlying the solution of simple addition problems, cognitive arithmetic has emerged as a vibrant area of research. Scientists in this area have mapped the cognitive processes and neurological correlates that govern the mental solution of simple and complex arithmetic problems and have extended these basic findings to more applied issues, such as mathematical anxiety and mathematical disabilities

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

“…As soon as children Wnish elementary school, such eVects disappear, resulting in less eYcient procedure execution, especially for large problems. Comparable eVects have been reported by Geary (1996), who observed that the problem-size eVect disappeared and reversed between 1st and third grade Chinese children, but re-appeared in Chinese adults. The investigation of the appearance, disappearance, and re-appearance of the problem-size eVect across lifetime provides interesting ideas for future research.…”

The role of working memory in the carry operation of mental arithmetic: Number and value of the carry

“…The larger this outcome, the more difficult this component would be. This derives from earlier findings and theorizing about the problem size effect in mental calculation (e.g., Ashcraft, 1992Ashcraft, , 1995Ashcraft & Battaglia, 1978;Butterworth et al, 2001;Campbell, 1995;Geary, 1996;Groen & Parkman, 1972). Second, difficulty also depends on the number of operations to be performed.…”

“…Therefore, not the number of operations (which is the same in both examples), but the total amount of the outcome varies with the value of the carry. From the literature on simple mental arithmetic it is known that solution time and accuracy depend on the size of the outcome (the so-called problem-size effect; Ashcraft, 1992Ashcraft, , 1995Ashcraft & Battaglia, 1978;Butterworth, Zorzi, Girelli, & Jonckheere, 2001;Campbell, 1995;Geary, 1996;Groen & Parkman, 1972). Consequently, it may be predicted that the larger the value of the carry or carries, the slower the solution and the less accurate on average.…”

Cultural differences in complex addition: Efficient Chinese versus adaptive Belgians and Canadians.

“…Interestingly, cultural differences in strategy efficiency have been shown in the domain of simple arithmetic. Asians solve simple arithmetic problems (e.g., 7 + 5) more quickly and accurately than North Americans (Campbell & Xue, 2001;Geary, 1996b;Geary, Bow-Thomas, Fan, & Siegler, 1993;Geary, BowThomas, Liu, & Siegler, 1996;Geary, Salthouse, Chen, & Fan, 1996;Geary et al, 1997;LeFevre & Liu, 1997;Penner-Wilger, Leth-Steensen, & LeFevre, 2002). This effect was, in part, a result of cultural differences in strategy selection: Asians retrieved the answers from long-term memory more frequently than North Americans, who were more inclined to use non-retrieval strategies Complex Arithmetic -6 such as transformation (e.g., 7 + 5 = 7 + 3 + 2 = 10 + 2 = 12) or counting (e.g., 7 + 2 = 7, 8, 9).…”