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- W1535476989 abstract "We present in this paper five case studies of instrumental genesis of number-theoretic concepts and techniques involving multiples, divisors, and numerical decomposition. The pupils, who were 12 to 15 years old, used the multi-line screen display of the graphing calculator to explore numerical tasks related to the “Five steps to zero” problem. The paper begins with a mathematical task analysis of various techniques that could be used to determine divisibility by 9. Findings show that the use of the technological tool permits students of this age to spontaneously develop variants of these techniques within tasks that simultaneously afford the co-emergence of related conceptual notions. A body of current research in computer-supported mathematics learning environments is oriented toward the theoretical development and empirical study of the interactions among the following concepts: artefact, instrument, technique, task, and theory (e.g., Artigue, 2001; Guin & Trouche, 1999; Guzman & Kieran 2002; Lagrange, 2000; Mariotti, 2002; Trouche, 2000; Varillon & Rabardel, 1995). In particular, these studies discuss the role played by the instrument in the process of instrumental genesis within the context of specific tasks and with pupils of different ages and technological experience. In a paper presented last year at PME (Guzman & Kieran, 2002), we reported, for example, on the perceptions expressed by case-study students of the role played by the calculating tool in the evolution of their numerical conceptual schemas. The present paper, which is quite different, analyses the nature of the mathematical techniques that emerged spontaneously in representative participants of the study. By means of two case studies from each of Secondary 1 and 2, and one from Secondary 3, we illustrate the number-based interactions between students and their calculators and show how, within a given set of tasks, the students developed numerical techniques that were afforded by the technology-supported environment in which their numerical explorations occurred. The emergence of these techniques provides further evidence of the intertwining nature of the development of procedural and conceptual learning in mathematics. THE RESEARCH STUDY" @default.
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- W1535476989 date "2005-07-01" @default.
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- W1535476989 title "The Spontaneous Emergence of Elementary Number-Theoretic Concepts and Techniques in Interaction with Computing Technology" @default.
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