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

Adult’s simple-arithmetic strategy use depends on problem-related characteristics, such as problem size and operation, and on individual-difference variables, such as working-memory span. The current study investigates (a) whether the effects of problem size, operation, and working-memory span on children’s simple-arithmetic strategy use are equal to those observed in adults, and (b) how these effects emerge and change across age. To this end, simple-arithmetic performance measures and a working-memory span measure were obtained from 8-year-old, 10-year-old, and 12-year old children. Results showed that the problem-size effect in children results from the same strategic performance differences as in adults (i.e., size-related differences in strategy selection, retrieval efficiency, and procedural efficiency). Operation-related effects in children were equal to those observed in adults as well, with more frequent retrieval use on multiplication, more efficient strategy execution in addition, and more pro-nounced changes in multiplication. Finally, the advantage of having a large working-memory span was also present in children. The differences and similarities across children’s and adult’s strategic performance and the relevance of arithmetic models are discussed

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

In the present study, the authors tested the effects of working-memory load on math problem solving in 3 different cultures: Flemish-speaking Belgians, English-speaking Canadians, and Chinese-speaking Chinese currently living in Canada. Participants solved complex addition problems (e.g., 58 + 76) in no-load and working-memory load conditions, in which either the central executive or the phonological loop was loaded. The authors used the choice/no-choice method to obtain unbiased measures of strategy selection and strategy efficiency. The Chinese participants were faster than the Belgians, who were faster and more accurate than the Canadians. The Chinese also required fewer working-memory resources than did the Belgians and Canadians. However, the Chinese chose less adaptively from the available strategies than did the Belgians and Canadians. These cultural differences in math problem solving are likely the result of different instructional approaches during elementary school (practice and training in Asian countries vs. exploration and flexibility in non-Asian countries), differences in the number language, and informal cultural norms and standards. The relevance of being adaptive is discussed as well as the implications of the results in regards to the strategy choice and discovery simulation model of strategy selection (J. Shrager & R. S. Siegler, 1998)

Efficient PML for the wave equation

In the last decade, the perfectly matched layer (PML) approach has proved a flexible and accurate method for the simulation of waves in unbounded media. Most PML formulations, however, usually require wave equations stated in their standard second-order form to be reformulated as first-order systems, thereby introducing many additional unknowns. To circumvent this cumbersome and somewhat expensive step, we instead propose a simple PML formulation directly for the wave equation in its second-order form. Inside the absorbing layer, our formulation requires only two auxiliary variables in two space dimensions and four auxiliary variables in three space dimensions; hence it is cheap to implement. Since our formulation requires no higher derivatives, it is also easily coupled with standard finite difference or finite element methods. Strong stability is proved while numerical examples in two and three space dimensions illustrate the accuracy and long time stability of our PML formulation.Comment: 16 pages, 6 figure

The influence of problem features and individual differences on strategic performance in simple arithmetic

The present study examined the influence of features differing across problems (problem size and operation) and differing across individuals (daily arithmetic practice, the amount of calculator use, arithmetic skill, and gender) on simple-arithmetic performance. Regression analyses were used to investigate the role of these variables in both strategy selection and strategy efficiency. Results showed that more-skilled and highly practiced students used memory retrieval more often and executed their strategies more efficiently than less-skilled and less practiced students. Furthermore, calculator use was correlated with retrieval efficiency and procedural efficiency but not with strategy selection. Only very small associations with gender were observed, with boys retrieving slightly faster than girls. Implications of the present findings for views on models of mental arithmetic are discussed

Do multiplication and division strategies rely on executive and phonological working memory resources?

The role of executive and phonological working-memory resources in simple arithmetic was investigated in two experiments. Participants had to solve simple multiplication problems (e.g., 4 x 8; Experiment 1) or simple division problems (e.g., 42 : 7; Experiment 2) under no-load, phonological-load, and executive-load conditions. The choice/no-choice method was used to investigate strategy execution and strategy selection independently. Results on strategy execution showed that executive working memory resources were involved in direct memory retrieval of both multiplication and division facts. Executive working-memory resources were also needed to execute nonretrieval strategies. Phonological working-memory resources, on the other hand, tended to be involved in non-retrieval strategies only. Results on strategy selection showed no effects of working-memory load. Finally, correlation analyses showed that both strategy execution and strategy selection correlated with individual-difference variables such as gender, math anxiety, associative strength, calculator use, arithmetic skill, and math experience