Abstract
Six fifth-grade children participated in a four-day teaching experiment on complex additively-structured problems, which was followed by in-depth interviews of individual children. The teaching experiment was meant to investigate children's difficulties in holding in mind at once situations in which one or more items played multiple roles. Two important difficulties were identified: (1) distinguishing between “difference” as the result of subtracting and “difference” as the amount by which one quantity exceeded another; and (2) indirect evaluation of an additive comparison. Sources of these difficulties, along with pedagogical and curricular recommendations for addressing them, are discussed.
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References
Balacheff, N.: 1990, ‘Towards a problematique for research on teaching’,Journal for Research in Mathematics Education 21, 258–272.
Carpenter, T.: 1985, ‘Learning to Add and Subtract: An Exercise in Problem Solving’, in E. A. Silver (ed.),Teaching and Learning Mathematical Problem Solving: Multiple Research Pespectives, Hillsdale, NJ: Erlbaum, pp. 17–40.
Carpenter, T. and Moser, J.: 1982, ‘The Development of Addition and Subtraction Problem-Solving Skills’, in T. P. Carpenter, J. M. Moser, and T. A. Romberg (eds.),Addition and Subtraction: A Cognitive View, Hillsdale, NJ: Erlbaum, pp. 9–24.
Carpenter, T. and Moser, J.: 1983, ‘The Acquisition of Addition and Subtraction Concepts’, in R. Lesh and M. Landau (eds.),Acquisition of Mathematics Concepts and Processes, New York: Academic Press.
Cobb, P., Yackel, E., and Wood, T.: 1989, ‘Young Children's Emotional Acts While Doing Mathematical Problem Solving’, in D. McLeod and V. Adams (eds.),Affect and Mathematical Problem Solving: A New Perspective, New York: Springer-Verlag, pp. 119–148.
Dreyfus, T. and Thompson, P.: 1985, ‘Microworlds and van Hiele levels’, in L. Streefland (ed.),Proceedings of the Ninth International Conference for the Psychology of Mathematics Education, Utrecht, The Netherlands: University of Utrecht, Research Group on Mathematics Education and Educational Computer Center, Vol. 1, pp. 5–11.
Greeno, J., Riley, M., and Gelman, R.: 1984, ‘Conceptual Competence and Children's Counting’,Cognitive Psychology 16, 93–143.
Janvier, C.: 1983, ‘The Understandng of Directed Number’, in J. C. Bergeron and N. Herscovics (eds.),Proceedings of the Fifth Annual Conference of the North American Chapter of the Internatinal Group for the Psychology of Mathematics Education, Montréal: Université de Montréal, Faculté de Sciences de l'Education.
Steffe, L. and Cobb, P.: 1988,Construction of Arithmetical Meanings and Strategies, New York: Springer-Verlag.
Steffe, L., von Glasersfeld, E., Richards, J., and Cobb, P.: 1984,Children's Counting Types: Philosophy, Theory, and Application, New York: Praeger Scientific.
Steffe, L., Thompson, P., and Richards, J.: 1982, ‘Children's Counting in Arithmetical Problem Solving’, in T. P. Carpenter, J. M. Moser, and T. A. Romberg (eds.),Addition and Subtraction: A Cognitive View, Hillsdale, NJ: Erlbaum, pp. 83–98.
Thompson, P.: 1989,A Cognitive Model of Quantity-Based Reasoning in Algebra, Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA, 27–37 March, 1989.
Thompson, P.: 1990,A Theoretical Model of Quantity-Based Reasoning in Arithmetic and Algebra, Progress report to the National Science Foundation, San Diego State University, Center for Research in Mathematics and Science Education.
Thompson, P and Thompson, A.: 1992,Images of Rate, paper presented at the Annual Meeting of the American Educational Research Association, San Francisco.
Thompson, P.: (in press), 'The Development of the Concept of Speed and its Relationship to Concepts of Rate', in G. Harel and J. Confey (eds.),The Development of Multiplicative Reasoning in the Learning of Mathematics, New York: SUNY Press.
Thompson, P. and Dreyfus, T.: 1988, ‘Integers as Transformations’,Journal for Research in Mathematics Education 19, 115–133.
Vergnaud, G.: 1982, ‘A Classification of Cognitive Tasks and Operations of Thought Involved in Addition and Subtraction Problems’, in T. P. Carpenter, J. M. Moser, and T. A. Romberg (eds.),Addition and Subtraction: A Cognitive View, Hillsdale, NJ: Erlbaum, pp. 39–59.
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Research reported in this paper was supported by National Science Foundation Grants NO. MDR 89-50311 and 90-96275, and by a grant of equipment from Apple Computer, Inc., Office of External Research. Any conclusions or recommendations stated here are those of the author and do not necessarily reflect official positions of NSF or Apple Computer.
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Thompson, P.W. Quantitative reasoning, complexity, and additive structures. Educ Stud Math 25, 165–208 (1993). https://doi.org/10.1007/BF01273861
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DOI: https://doi.org/10.1007/BF01273861