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Mental Computation Strategies

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Context and Mental Computation   dot   Mental Computation Strategies   dot   References


Context and Mental Computation

Context and Mental Computation. Carraher, Carraher, and Schliemann (1985) conducted a qualitative analysis of how Brazilian street-market children invoke effective computational procedures (algorithms) in real-life contexts in contrast to traditional school mathematics and abstract computational problems. The Brazilian researchers predicted that participants would often perform mathematics computation differently in informal settings than in school, and this would often be more effective. For example, if children could efficiently compute the costs of variety of market produce involving addition, subtraction, and multiplication, how would they fare in performing abstract, school-based problems? Participants were poor children ranging in age from nine to fifteen, with little formal schooling, ranging from one to eight years. These are the kind of children who do not have things that we take for granted like school uniform, Mobile Broadband and computer consoles. The children first were asked to perform computations mentally by interviewers posing as customers in their natural (informal) setting, the street market. A follow-up test asked them to perform word problems, using market items and the same numbers, but under the unnatural (formal) condition of being given a pencil-and-paper test in their homes. Results showed that participants correctly solved 98 percent of the context-embedded problems in the informal setting, but only 74 percent of imaginary, context-embedded items and 37 percent of abstract items in the formal setting. It was also learned that children apply different computational algorithms when presented with problems orally than with the pencil-and-paper test. Interpretation of these results argue that mental computational algorithms may be more effective when applied to a real-life context, but also that the strategies (oral or written) invoked are context dependent. This analysis supports the notion of the present study that children are helped when their employment of mental computational strategies occur in a natural context. Recall, however, that mental computation strategies are appropriate for certain kinds of problems under certain conditions, however, more formal algorithms are appropriate for more complex, abstract operations.

The proposition that the use of written versus oral (mental) computation strategies is context dependent was further analyzed by Carraher, Carraher, and Schlieman (1987) in a study involving third-grade Brazilian school children. The researchers predicted that the type of mental or written computational strategy children would employ would depend on whether the context was concrete or abstract. Specifically, they wanted to test whether formal mathematical operations would predominate in school-type settings, while informal operations would predominate in natural contexts. As in a previous study (Carraher, Carraher, & Schliemann, 1985), children were tested in different settings. A major difference in this design, however, was that the street market was simulated, in school, as a store situation. Three settings were analyzed: 1) the simulated store, 2) problems embedded in story problems, and 3) computational exercises. Problems were presented orally, however, children were allowed to perform operations mentally or with pencil-and-paper. Children more often chose to perform the operations mentally, however, this was not significantly different than the number of operations performed with pencil and paper. Significant results revealed that children in the simulation setting accurately calculated more often than the other conditions. Qualitative analysis of posttest interviews with the participants, also revealed that children used different algorithms to perform operations mentally than on paper. Specifically, students often performed mental computation algorithms by decomposing quantities (e.g. solving portions of the calculation at a time) and repeated grouping (e.g. using repeated addition instead of multiplication). That study thus contributes to the idea in the present study that mental computation strategies are not simply written algorithms performed mentally, and that these strategies may be more effective in certain contexts: either real or simulated, as in text-based stories.

Story-based Context. The notion that story-based instruction aids learning is supported. Anderson, Spiro, and Anderson (1978) conducted an experiment to test whether text is better interpreted, that is, learned and recalled, in story structure form. Two story passages were created in two contexts: One involved dining at a fancy restaurant and the other shopping in a supermarket. They predicted that the restaurant context would provide more structure, due to the natural temporal order of appetizers, main course, and dessert, and would therefore be more effectively interpreted. Participants were 75 college students randomly assigned to read either passage followed by a posttest recalling food items, food order, and character names situated in the passages. All actors and most food items in the passages were identical. Posttest results supported the hypothesis that the restaurant context significantly predicted better recall of food items. The restaurant context also significantly enabled better recall of characters attributed to certain food items. Results of the study support the long-held notion that context schemata significantly aid the interpretation of textual information; that is, situations in which the presentation of information occurs in a natural way is a worthwhile aid to learning.
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Mental Computation Strategies

During my own childhood, I was taught that the one (and only) way to compute mentally was to imagine a 'chalkboard in my mind' and then to 'see' the numbers and 'carry' as I would with paper and pencil. Unfortunately, the numbers always disappeared before I could finish the calculation. (Richards, 1991, p. 109)

The author of the passage above is now a college mathematics instructor. To finish the passage, she goes on to write that she was convinced she was not "good at mathematics," but after learning a mental arithmetic strategy (left-to-right operation) she gained a "renewed sense of confidence" in her mathematics capabilities.

Mental computation of mathematics is the deriving of exact answers to mathematical calculations without the use of recording devices such as computers, calculators, or writing instruments (Reys, B., 1985; Reys & Barger, 1994). Up to 80 percent of mathematical computations performed in non-technical settings, such as the exchange of money or the determination of times and distances, are done mentally (Reys & Nohda, 1994). Most often, we do not take the time or have the opportunity to use recording devices in making computations. Many times it is simply too embarrassing to use recording devices at the check stand or restaurant and, therefore, people avoid carrying calculators everywhere they go (Lucas, 1991).

Mental computation is still important in this age of high technology. The National Council for Teachers of Mathematics (NCTM) is calling for renewed emphasis on computational alternatives, including mental computation and estimation, as necessary strategies to complement advances in technology (Reys & Nohda, 1994; Shumway, 1994). It is all too easy to make mathematical errors using technological devices, and the ability to compute or estimate numbers mentally assists children in checking calculations.

Mathematics competency is often displayed in the classroom by those efficacious learners who have acquired mental computation strategies. Peers performing mathematical computations swiftly and accurately in their heads is associated with high mathematical capability. Students, however, who never learned these strategies compare their inadequate performance with those of skilled performers and often judge their mathematics capabilities in general as inferior. Attributing their lack of skill to lack of capability is what Bandura (1986) calls "faulty self-knowledge." Relevant to this, one of the goals of the instruction examined in the present research is for the models to gain in perceptions of self-efficacy as they improve performance on the criterial task. Mental computation is the criterial task because it requires an exact answer and therefore reduces guessing and alleviates the difficulty of judging estimation. Importantly, it also assists the learner in making accurate self-appraisals of performance. Mental computation is something the student must perform alone without the aid of external devices and so a correct answer is clearly the result of one's own internal processing. In judging the relationship between levels of self-efficacy and performance, mental computation provides a fairly exact measure.

There are, therefore, a number of major influences affecting mathematics self-efficacy of children. These influences include domain-stereotyping and gender-stereotyping, which often result from social comparison with classmates. Mathematics capabilities are usually displayed socially by mental computation performance in the classroom, making this an important problem area for enhancing mathematics self-efficacy.
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References

Bandura, A. (1986). Social foundations of thought and action: a social cognitive theory. Englewood Cliffs, NJ: Prentice-Hall.

Carraher, T. N. Carraher, D. W. & Schliemann, A. D. (1985). Mathematics in the streets and in schools, British Journal of Developmental Psychology, 3, 21-29.

Carraher, T. N. Carraher, D. W. & Schliemann, A. D. (1987). Written and oral mathematics, Journal for Research in Mathematics Education, 18(2), 83-97.

Lucas, J. (1991). Becoming a mental math wizard. White Hall, VA: Shoe Tree Press.

Reys, B. J. & Barger, R. H. (1994). Mental Computation: Issues from the United States. In R. E. Reys & N. Nohda (Eds.), Computational Alternatives for the Twenty-first Century: Cross Cultural Perspectives from Japan and the United States, (pp. 31-47). Reston, VA: The National Council of Teachers of Mathematics.

Reys, B. J. (1985). Mental Computation, Arithmetic Teacher, 32(6), 43-46.

Reys, B. J., & Reys, R. E. (1993). Mental Computation Performance and Strategy Use of Japanese Students in Grades 2, 4, 6, and 8 (EDRS: ED 365 532): University of Missouri.

Reys, B., Reys, R. E., & Hope, J. A. (1993). Mental computation: A snapshot of second, fifth and seventh grade student performance, Journal for Research in Mathematics Education, 93(6), 306-315.

Reys, R. E. & Nohda, N. (1994). Computation and the need for change. In R. E. Reys & N. Nohda (Eds.), Computational Alternatives for the Twenty-first Century: Cross Cultural Perspectives from Japan and the United States, (pp. 1-11). Reston, VA: The National Council of Teachers of Mathematics.

Richards, J. J. (1991). Commentary. In B. Means, C. Chelemer, & M. S. Knapp (Eds.), Teaching advanced skills to at-risk students: Views from research and practice, (pp. 102-111). San Francisco: Jossey-Bass Publishers.

Shumway, R. J. (1994). Some common directions for future research related to computational alternatives. In R. E. Reys & N. Nohda (Eds.), Computational Alternatives for the Twenty-first Century: Cross Cultural Perspectives from Japan and the United States, (pp. 187-195). Reston, VA: The National Council of Teachers of Mathematics.

Spiro, R. J. & Jehng, J. C. (1990). Cognitive flexibility and hypertext: Theory and technology for nonlinear and multidimensional traversal of complex subject matte. In D. Nix & R. J. Spiro (Eds.), Cognition, education, and multimedia: Exploring ideas in high technology, (pp. 163-205). Hillsdale NJ: Erlbaum.
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Update: 2006-04-25T11:42:38-07:00