Abstract
Our research project aimed at understanding the complexity of the construction of knowledge in a CAS environment. Basing our work on the French instrumental approach, in particular the Task–Technique–Theory (T–T–T) theoretical frame as adapted from Chevallard’s Anthropological Theory of Didactics, we were mindful that a careful task design process was needed in order to promote in students rich and meaningful learning. In this paper, we explore further Lagrange’s (2000) conjecture that the learning of techniques can foster conceptual understanding by investigating at close range the task-based activity of a pair of 10th grade students—activity that illustrates the ways in which the use of symbolic calculators along with appropriate tasks can stimulate the emergence of epistemic actions within technique-oriented algebraic activity.
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Notes
As a perspective for future research, it could be of interest to consider elaborating a task design for older students that promotes discussion regarding the number of factors in a complete factorization of x n−1, the number of factors depending on the number of divisors of n. If q divides n, then n = pq, p being an integer, and x n − 1 = x pq − 1 = (x q)p − 1. Thus, the complete factorization of x n − 1 has all the factors of x q − 1.
Balacheff (1987, p. 148) has said: “We call a ‘preuve’ an explanation that is accepted by a given community at a given moment. This decision can be the object of a debate whose significance lies in the need to determine a validation system that is common to the interlocutors” [our translation]. Note that the French word ‘preuve’ does not translate into the English word ‘proof’, the French word for ‘proof’ being ‘démonstration’. The French word ‘preuve’ has at times been translated into English as ‘evidence’, ‘warrant’, ‘supporting argument’, and ‘justification’.
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Acknowledgments
Special thanks are extended to André Boileau and Denis Tanguay (Université du Québec à Montréal), José Guzmán (CINVESTAV-IPN), and Luis Saldanha (now at Arizona State University) for their contributions as part of the group in the task-design process. We would also like to acknowledge the support of Paul Drijvers (Freudenthal Institute), especially for his suggestions regarding an initial version of this paper. The research presented herein was made possible by a grant from the Social Sciences and Humanities Research Council of Canada (INE Grant # 501-2002-0132). We express our appreciation as well to the students who participated in this study.
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Hitt, F., Kieran, C. Constructing Knowledge Via a Peer Interaction in a CAS Environment with Tasks Designed from a Task–Technique–Theory Perspective. Int J Comput Math Learning 14, 121–152 (2009). https://doi.org/10.1007/s10758-009-9151-0
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DOI: https://doi.org/10.1007/s10758-009-9151-0