# Constructing conceptual knowledge and promoting “number sense” from computer-managed practice in rounding whole numbers

## Abstract

This study sought to identify how high achievers learn and understand new concepts in arithmetic from computer-based practice which provides full solutions to examples but without verbal explanations. Four high-achieving second graders were observed in their natural school settings throughout all their computer-based practice sessions which involved the concept of rounding whole numbers, a concept which was totally new to them. Immediate post-session interviews inquired into students’ strategies for solutions, errors, and their understanding of the underlying mathematical rules. The article describes the process through which the students construct their knowledge of the rounding concepts and the errors and misconceptions encountered in this process. The article identifies the cognitive abilities that promote student self-learning of the rounding concepts, their number concepts and “number sense.” Differences in the ability to generalise, “mathematical memory,” mindfulness of work and use of cognitive strategies are shown to account for the differences in patterns of, and gains in, learning and in maintaining knowledge among the students involved. Implications for the teaching of estimation concepts and of promoting students’ “number sense,” as well as for classroom use of computer-based practice are discussed.

## Keywords

Conceptual Knowledge Level Versus Number Sense Computer Work Number Concept## Preview

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## References

- Becker, H. J. (1990).
*Computer use in United States schools: 1989. An initial report of U.S. participation in the I.E.A. computers in education survey*. Paper presented at the annual meeting of the American Educational Research Association, Boston.Google Scholar - Becker, H. J. (1991). How computers are used in United States schools: Basic data from the 1989 I.E.A. Computers in Education Survey.
*Journal of Educational Computing Research, 7*(4), 385–406CrossRefGoogle Scholar - Bereiter, C. (1985). Towards a solution of the learning paradox.
*Review of Educational Research, 55*, 201–226.Google Scholar - Carpenter, T. P., & Moser, J. M. (1983). Current research: a process approach. In R. Lesh & M. Landau (Eds.),
*Acquisition of mathematics concepts*(pp. 7–44). N.Y.: Academic Press.Google Scholar - Cobb, P. (1988). The tension between theories of learning and instruction in mathematics education.
*Educational Psychologist, 23*(2), 87–103.CrossRefGoogle Scholar - Cobb, P., Yackel, E. & Wood, T. (1992). A constructivist’s alternative to the representational view of mind in mathematics education.
*Journal for Research in Mathematics Education, 23*(1), 2–33.CrossRefGoogle Scholar - Greeno, J. G. (1991). Number sense as situated knowing in a conceptual domain.
*Journal for Research in Mathematics Education, 22*(3), 170–218.CrossRefGoogle Scholar - Guba, E. G. (1981). Criteria for assessing the trustworthiness of naturalistic inquiries.
*Educational Communications and Technology Journal, 29*(2), 75–91.Google Scholar - Hativa, N. (1988a). Computer-based drill and practice in arithmetic—widening the gap between high and low achieving students.
*American Educational Research Journal, 25*(3), 366–397.Google Scholar - Hativa, N. (1988b). Sigal’s ineffective computer-based practice of arithmetic: A case study.
*Journal for Research in Mathematics Education, 19*(3), 195–214.CrossRefGoogle Scholar - Hativa, N. (1988c). CAI versus paper and pencil—discrepancies in students’ performance.
*Instructional Science, 17*(1), 77–96.CrossRefGoogle Scholar - Hativa, N. (1992).
*Cognitive processes and patterns of learning in computer-based drill- and-practice with below-average students*. Unpublished manuscript.Google Scholar - Hativa, N., & Lesgold, A. (1991). The computer as a tutor—can it adapt to the individual learner?
*Instructional Science, 20*, 49–78.CrossRefGoogle Scholar - Hiebert, J., & Wearne, D. (1988). Instruction and cognitive change in mathematics.
*Educational Psychologist, 23*(2), 105–117.CrossRefGoogle Scholar - Krutetskii, V. A. (1976).
*The psychology of mathematical abilities in schoolchildren*. The University of Chicago Press.Google Scholar - National Research Council (Board on Mathematical Sciences and Mathematical Sciences Education Board). (1989).
*Everybody counts: A report to the nation on the future of mathematics education*. Washington, DC: National Academy of Sciences.Google Scholar - Osin, L. (1984) TOAM: C.A.I. on a national scale. In
*Proceedings of the Fourth Jerusalem Conference on Information Technology*. Jerusalem: IEEE Computer Society Press.Google Scholar - Resnick, L. B., & Ford, W. W. (1981).
*The psychology of mathematics for instruction*. Hillsdale, NJ: Laurence Erlbaum Associates.Google Scholar - Resnick, L. B. (1989). Developing mathematical knowledge.
*American Psychologist, 44*(2), 162–169.CrossRefGoogle Scholar - Reys, B. J. (1986). Teaching computational estimation: Concepts and strategies. In H. L. Schoen, & M. J. Zweng (Eds.),
*Estimation and mental computation: 1986 Yearbook*(pp. 31–44). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Schoen, H. L. (1986). Preface. In H. L. Schoen, & M. J. Zweng (Eds.),
*Estimation and mental computation: 1986 Yearbook*(pp. vii-viii). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Shute, V., Glazer, R. & Raghavan, K. (1988). Inference and discovery in an exploratory laboratory. In P. L. Ackerman, R. J. Sternberg, & R. Glaser (Eds.)
*Learning and individual differences*(pp. 279–326). New York: W.H. Freeman and Company.Google Scholar - Silver, E. A. (1979). Student perceptions of relatedness among mathematical verbal problems.
*Journal for research in Mathematics Education, 10*, 195–210.CrossRefGoogle Scholar - Sowder, J. T. (1989). Developing understanding of computational estimation.
*Arithmetic Teacher, 36*(5), 25–27.Google Scholar - Sowder, J. T. (1990). Relative and absolute error in computational estimation. In G. Booker, P. Cobb, & T. N. deMendicuti (Eds.),
*Proceedings of the Fourteenth Psychology of Mathematics Education Conference*(pp. 321–328). Mexico: International Group for the Psychology of Mathematics Education.Google Scholar - Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra.
*Cognition and Instruction, 2*(1), 59–89.CrossRefGoogle Scholar - Trafton, P. R. (1986). Teaching computational estimation: Establishing an estimation mind-set. In H. L. Schoen, & M. J. Zweng (Eds.),
*Estimation and mental computation: 1986 Yearbook*(pp. 16–30). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Usiskin, Z. (1986). Reasons for estimating. In H. L. Schoen, & M. J. Zweng (Eds.),
*Estimation and mental computation: 1986 Yearbook*(pp. 1–15). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - VanLehn, K. (1986). Arithmetic procedures are induced from examples. In J. Hiebert (Ed.).
*Conceptual and procedural knowledge: The case of mathematics*(pp. 133–179). Hillsdale, NJ: Laurence Erlbaum Associates.Google Scholar - Woodward, A., & Mathinos, D. A. (1987).
*Microcomputer education in an elementary school: The rhetoric versus the reality of an innovation*. Paper presented at the annual meeting of the American Educational Research Association, Washington D.C.Google Scholar - Zhu, X., & Simon, H. A. (1987). Learning mathematics from examples and by doing.
*Cognition and Instruction, 4*(3), 137–166.CrossRefGoogle Scholar