Abstract
This study was designed to gain information about the understandings children in Israel and the United States have about multiplication and division of whole numbers that may be useful in building accurate understandings of these operations with decimals and the extent to which they hold conceptions about these operations that may interfere with their work with decimals. Data from interviews of the fourth and fifth graders indicate that students of this age already hold misconceptions such as “multiplication always makes bigger.” However they also hold conceptions that are prerequisite to understanding the area model of multiplication and the measurement model of division. These early conceptions might be used to build understanding of multiplication and division by decimals. Implications for the content and sequencing of instructional activities are presented.
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References
BellA. W.: 1982, ‘Diagnosing students misconceptions’, Australian Mathematics Teacher 38, 6–10.
BellA. W., E.Fischbein, and B.Greer: 1984, ‘Choice of operation in verbal arithmetic problems: The effects of number size, problem structure and content’, Educational Studies in Mathematics 15, 129–148.
BellA., M.Swan, and G.Taylor: 1981, ‘Choice of operation in verbal problems with decimal numbers’, Educational Studies in Mathematics 12, 399–420.
BrownM.: 1981, ‘Place value and operations’, in K. M.Hart (ed.), Children's Understanding of Mathematics: 11–16, pp. 48–65, John Murray, London.
CarpenterT. C., M. K.Corbitt, H. S.Kepner, M. M.Lindquist, and R. E.Reys: 1981, ‘Decimals: Results and implications from national assessment’, Arithmetic Teacher 28(8), 34–37.
CaseR.: 1985, Intellectual Development: Birth to Adulthood, Plenum, New York.
DavisR.: 1984, Learning Mathematics: The Cognitive Science Approach to Mathematics Education, Ablex Publishing, Norwood, NJ.
EkenstamA.: 1977, ‘On children's quantitative understanding of numbers’, Educational Studies in Mathematics 8, 317–332.
FischbeinE., M.Deri, M. S.Nello, and M. S.Marino: 1985, ‘The role of implicit models in solving problems in multiplication and division’, Journal for Research in Mathematics Education 16, 3–17.
FischbeinE.: 1987, Intuition in science and mathematics: An educational approach, Kluwer Academic, Boston.
Graeber, A. O. and K. Baker: (in press), ‘Curriculum materials and misconceptions concerning multiplication and division’, Focus on Learning Problems in Mathematics.
GraeberA. O., D.Tirosh, and R.Glover: 1989, ‘Preservice teachers' misconceptions in solving verbal problems in multiplication and division’, Journal for Research in Mathematics Education 20(1), 95–102.
GreerB. and C.Mangan: 1986, ‘Choice of operations: From 10-year-olds to student teachers’, in Proceedings of the Tenth International Conference Psychology of Mathematics Education, pp. 25–30, University of London, Institute of Education, London, England.
HartK. (ed.): 1981, Children's Understanding of Mathematics: 11–16, John Murray, London.
Hiebert, J. and D. Wearne: 1983, April, ‘Students' conceptions of decimal numbers’, Paper presented at the annual meeting of the American Educational Research Association, Montreal.
KieranT. E.: 1980, ‘Knowing rational numbers: Ideas and symbols’, in M. M.Lindquist (ed.), Selected Issues in Mathematics Education, pp. 69–81, McCutchan, Berkeley, CA.
McKnightC., J.Crosswhite, J.Dossey, E.Kifer, J.Swafford, K.Travers, and T.Cooney: 1987, The Underachieving Curriculum: Assessing U.S. School Mathematics from an International Perspective, Stipes Publishing, Champaign, IL.
SemadeniZ.: 1984, ‘A principle of concretization permanence for the formation of arithmetical concepts’, Educational Studies in Mathematics 15, 379–395.
SowderL.: 1986, ‘Strategies children use in solving problems’, in Proceedings of the Tenth International Conference Psychology of Mathematics Education, pp. 469–474, University of London, Institute of Education, London, England.
Steffe, L.: (in press), ‘Children's construction of meaning for arithmetical words: A curriculum problem’, in D. Tirosh (ed.), Implicit and Explicit Knowledge: An Educational Approach.
SwanM.: 1983, Teaching decimal place value: A comparative study of ‘conflict’ and ‘positive only’ approaches, Shell Centre for Mathematical Education, University of Nottingham, Nottingham, England.
ThorndikeE.: 1921, The New Methods in Arithmetic, Rand McNally, New York.
TiroshD.: 1989, ‘Division and multiplication: What is in textbooks?’ (unpublished manuscript), Tel-Aviv University, (in Hebrew, Tel-Aviv.
TiroshD. and A. O.Graeber: 1989, ‘Preservice elementary teachers' explicit beliefs about multiplication and division’, Educational Studies in Mathematics, 20(1), 79–96.
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Graeber, A.O., Tirosh, D. Insights fourth and fifth graders bring to multiplication and division with decimals. Educ Stud Math 21, 565–588 (1990). https://doi.org/10.1007/BF00315945
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DOI: https://doi.org/10.1007/BF00315945