Abstract
In this chapter, I investigate how children understand concepts and the meaning of symbolic representations when working with conceptual models. The particular focus of this study is the nature of children’s conversation when collaborating to solve tasks embodied in what we call an intermediate abstraction. The analysis is based on theoretical considerations developed by Greeno and his colleagues (Greeno, 1989; Greeno, Engle, Kerr, & Moore, 1993) and on a line of research carried out by Resnick and Schwarz (Schwarz, Kohn, & Resnick, 1993; Schwarz & Nathan, 1993; Schwarz, Nathan, & Resnick, 1996; Schwarz, in press) with several computerized systems. My approach focuses theoretical attention on the nature of conversational turn-taking between children working with an intermediate abstraction of mathematical operators. This study shows that conversational processes of acceptance stemmed from collaborative learning, and that those processes enabled the construction of shared meaning resulting in the development of the concept of a mathematical operator. Three such processes were found. The first one is a process of creation of a shared meaning through reference to objects of the system. This process is initiated by one participant describing a property of the intermediate abstraction. The second participant then accepts the meaning uttered by the first participant and refines it. The second one is a process of convergence of a different nature that is initiated by an impasse within one particular representation and is characterized by a change of representation.
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References
Clark, H. H., & Schaefer, E. F. (1989). Contributing to discourse. Cognitive Science, 1., 259–294.
Clark, H. H., & Wilkes-Gibbs, D. (1986). Referring as a collaborative process. Cognition, 2., 1–39.
Clement, J. (1982). Algebra word problem solutions: Thought processes underlying a common misconception. Journal for Research in Mathematical Education, 1., 16–30.
Greeno, J. G. (1989). Situation models, mental models, and generative knowledge. In D. Klahr & K. Kotovsky (Eds.), Complex information processing: The impact of Herbert Simo. (pp. 285–318). Hillsdale, NJ: Erlbaum.
Greeno, J. G., Engle, R. A., Kerr, L. K., & Moore, J. L. (1993, July). Understanding symbols: Situativity-theory analysis of constructing mathematical meaning. In Proceedings of the Fifteenth Annual Meeting of the Cognitive Science Societ. (pp. 504–509). Boulder, CO.
Kirshner, D. (1989). Critical issues in current representation system theory. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebr. (pp. 195–198). Hillsdale, NJ: Lawrence Erlbaum Associates.
Larkin, J. L. (1989). Robust performance in algebra: The role of the problem representation. In S. Wagner & C. Kieran (Eds.), Research issues in the learning and teaching of algebr. (pp. 120–134). Hillsdale, NJ: Lawrence Erlbaum Associates.
Matz, M. (1980). Towards a computational theory of algebraic competence. Journal of Mathematical Behavior., 93–166.
Roschelle, J. (1992). Learning by collaboration: Convergent conceptual change. The Journal of the Learning Sciences., 235–276.
Schoenfeld, A. H. (1987). When good teaching leads to bad results: The disaster of “well taught” mathematics courses. Educational Psychologist, 2., 145–166.
Schwarz, B. B. (in press). Why can intermediate abstractions help acquiring robust representations? Interactive Learning Environments.
Schwarz, B. B., Kohn, A. S., & Resnick, L. B. (1993). Positives about negatives. The Journal of the learning Sciences.(1), 37–92.
Schwarz, B. B., & Nathan, M. J. (1993, July). Assessing conceptual understanding of arithmetic structure and language. In Proceedings of the Fifteenth Annual Meeting of the Cognitive Science Societ. (pp. 912–917). Boulder, CO.
Schwarz, B. B., Nathan, M. J., & Resnick, L. B. (1996). Acquisition of meaning for arithmetical structures with the Planner. In S. Vosniadou, E. De Corte, R. Glaser, & H. Mandl (Eds.), International perspectives on the construction of technology-based learning environment. (pp. 61–80). Mahwah, NJ: Erlbaum.
Smith, C., Snir, J., & Grosslight, L. (1992). Using conceptual models to facilitate conceptual change: The case of weight-density differentiation. Cognition and Instruction., 221–283.
White, B. Y. (1993). Thinkertools: Causal models, conceptual change, and science education. Cognition and Instruction, 1., 1–100.
Wiser, M. (1988, April). The differentiation of heat and temperature: An evaluation of the effect of microcomputer models on students’ misconceptions. Paper presented at the meeting of the American Educational Research Association, New Orleans.
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Schwarz, B.B. (1997). Understanding Symbols with Intermediate Abstractions: An Analysis of the Collaborative Construction of Mathematical Meaning. In: Resnick, L.B., Säljö, R., Pontecorvo, C., Burge, B. (eds) Discourse, Tools and Reasoning. NATO ASI Series, vol 160. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03362-3_14
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DOI: https://doi.org/10.1007/978-3-662-03362-3_14
Publisher Name: Springer, Berlin, Heidelberg
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