In this paper, we will examine the mathematical knowledge that prospective mathematics teachers draw upon when graphing function graphs and curves, with a special focus on the occurrence of asymptotes. Three tasks which involved a graph of a rational and exponential function and a hyperbola as a conic section were designed and administered to students. We performed this study within the framework of Anthropological Theory of the Didactic to examine the relationship of prospective mathematics teachers' available knowledge with the knowledge to be taught in upper secondary schools and scholarly knowledge relevant for teaching. By studying prospective mathematics teachers' knowledge, we aim to understand the feasibility of our proposed reference epistemological model for graphing functions and curves in the upper secondary school. Our findings reveal students' shortcomings with respect to the choice of the appropriate graphing praxeology for given tasks. Students' graphing strategies relied mostly on plotting points obtained by evaluating a formula, which is a dominant approach in the textbooks we analysed. Plotting points did not lead students to examine asymptotic behaviour, along with the observed monotonicity of a function. Their graphing strategies were found to be predominantly dependent on the particular setting in which the task was presented. Additionally, in our study, the idea of an asymptote as a tangent line at infinity in the geometric setting was questioned.
Graphical representation is one of the fundamental and widely spread representations. We performed a comprehensive research of the didactic transposition of asymptote and asymptotic behaviour in the upper secondary education in Croatia, within the framework of the anthropological theory of the didactics. Our study included textbook analysis, questionnaires with university students and interviews with two mathematicians. In this poster, we present a part of our research with an emphasis on university students’ graphing praxeologies. Results showed that students’ graphing praxeologies differ from scholarly praxeologies. Further, students grounded their praxeologies mostly on their high-school graphing knowledge, even after being exposed to advanced mathematics that could foster their autonomous thinking.Keywords: Prospective mathematics teachers, Function graph, Asymptotes and asymptotic behaviour.RésuméLa représentation graphique est l'une des représentations fondamentales et largement répandues. Nous avons effectué une recherche approfondie de la transposition didactique de l'asymptote et du comportement asymptotique dans l'enseignement secondaire supérieur en Croatie, dans le cadre de la théorie anthropologique du didactique. Notre étude comprenait une analyse de livres scolaires, des questionnaires avec des étudiants de l’universités et un entretien avec deux mathématiciens. Dans cette affiche, nous présenterons une partie de notre recherche en mettant l'accent sur les praxéologies graphiques des étudiants de l’université. Les résultats montrent que les praxéologies graphiques des élèves diffèrent des praxéologies académiques. En outre, les élèves ont fondé leurs praxéologies principalement sur leur connaissance graphique de lycée, même après avoir été exposés à des mathématiques avancées qui pourraient favoriser leur réflexion autonome.Mots-clés: Futurs professeurs de mathématiques, Graphe fonctionnel, Asymptotes et comportement asymptotique.
In this paper we present a review of the basic ideas and results concerning asymptotic lines of plane curves. We discuss their different definitions, namely that of a limiting position of tangent lines, of the tangent line at infinity, and finally the one that requires that the distance between points of a curve and asymptotic line tends to 0 as the point moves along an infinite branch of the curve. We also recall the method of determining asymptotes of algebraic curves from the leading coefficients in their equation and provide examples.
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