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Asymptote and asymptotic behavior as bodies of knowledge in the praxeologies of graphing functions and curves

Asymptote and asymptotic behavior are bodies of knowledge present in various areas of mathematics. Moreover, the notion of an asymptote appears also in the secondary mathematics education - as a theoretical concept, a part of a procedure or as a self-sufficient task in different contexts and different educational cycles. Hence, connections with a wide range of teaching contents and mathematical bodies of knowledge makes it an interesting object of research. In this presentation we elaborate some results from our survey on the didactic transposition of this body of knowledge in the general secondary education in Croatia. The survey is conducted within the theoretical framework of the anthropological theory of the didactics, developed by a French mathematician Y. Chevallard especially for research in mathematics education. The main idea of this theory is to determine the relation R_I(p, O) between a body of knowledge O and a person that occupies position p in a institution I. For this purpose mathematical knowledge and activities are described in terms of a praxeology [T, τ, θ, Θ], where its practical component is represented with task T and technique τ and discursive or theoretical component with technology θ and theory Θ. In our setting, we questioned the relations R_B(p, O) and R_S(p, O), where O consists of the task of graphing elementary functions or curves and corresponding techniques, while considered insitutions are mathematics textbooks B and prospective mathematics teachers S. Our results show that: (1) dominant techniques are drawing a curve through corresponding points and drawing a graph on the account of function properties determined using calculus, (2) chosen techniques are not the most efficient for the task in question, (3) asymptotic behavior is available but not fully utilized in praxeologies relevant to graphing functions or curves.

asymptote ; asymptotic behavior ; anthropological theory of the didactics (ATD)

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