Teaching practices with the derivative concept. A problematic meeting between Algebra and Calculus in secondary school.

Panero, Monica (2015) Teaching practices with the derivative concept. A problematic meeting between Algebra and Calculus in secondary school. UNSPECIFIED. PhD thesis, Università degli Studi di Torino.

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The derivative is one of the crucial notions related to functions in secondary teaching. Some rooted algebraic practices intervene in the development of practices that are proper to Calculus, such as the use of limit. Hence, the introduction of the derivative is a delicate moment for both students and teachers. In this thesis, the problematics of the transition from Algebra to Calculus is approached through an historical and epistemological analysis. In entering the Calculus domain, the work on functions becomes increasingly grounded on local properties, which are valid in the neighbourhood of a point. This is the case of the derivative introduction that requires the activation of a local perspective on functions and poses an additional difficulty, since from a global perspective the derivative of a function is a function itself. This thesis investigates the intervention of the local perspective in the secondary teaching of the derivative, within the Italian context. Framing our research in the Anthropological Theory of the Didactic (ATD), we study the didactic transposition of the derivative notion, when it is considered both as a tool for studying a function and as a function itself. In our analysis, we network three different theoretical elements: we focus on two types of task and the related mathematical and didactic praxeologies; we identify the perspectives activated on the involved functions (i.e., pointwise, global and local); we analyse the employed semiotic resources (e.g., speech, gestures, symbols, drawings) to convey these perspectives and to construct such praxeologies. The didactic transposition of the derivative concept is presented through the analysis of the intended curriculum and the implemented curriculum, with insights into the attained curriculum. At each stage, we specifically concentrate on the presence and the role given to the local perspective. In particular, the core of the thesis is the analysis of three case studies involving three teachers who introduce the derivative to their grade 13 students of scientific high school. After an interview, we have observed each teacher in her classroom and finally we have proposed two activities to the students. One of the main results is the identification of two different derivative-related praxeologies. They are based on different concept images of the tangent line: on the one hand, the limit of secant lines, and on the other hand, the best linear approximation. We discuss them by distinguishing the different levels of intervention of the local perspective in the work on functions.

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