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Asymptotes and Asymptotic Behaviour in Graphing Functions and Curves: an Analysis of the Croatian Upper Secondary Education Within the Anthropological Theory of the Didactic

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Abstract

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.

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References

  • Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher’s practice: The case of limits of functions in Spanish high schools. Educational Studies in Mathematics, 59(1/3), 235–268.

    Article  Google Scholar 

  • Biza, I., & Zachariades, T. (2010). First year mathematics undergraduates’ settled images of tangent line. The Journal of Mathematical Behavior, 29(4), 218–229.

    Article  Google Scholar 

  • Bosch, M. (2012). Doing research within The anthropological theory of the didactic: The case of school algebra. Paper presented at the 12th International Congress on Mathematical Education, Seul, Korea. Retrieved from http://www.icme12.org/upload/submission/1996_F.pdf. Accessed 16 Nov 2014.

  • Bosch, M., Fonseca, C., & Gascón, J. (2004). Incompletitud de las organizaciones matemáticas locales en las instituciones escolares [Incompleteness of the mathematical organizations in the educational institutions]. Recherches En Didactique Des Mathématiques, 24(2), 205–250.

    Google Scholar 

  • Bosch, M., & Gascón, J. (2014). Introduction to the anthropological theory of the didactic (ATD). In A. Bikner-Ahsbahs & S. Prediger (Eds.), Networking of theories as a research practice in mathematics education (pp. 67–83). Cham, Switzerland: Springer.

    Chapter  Google Scholar 

  • Chevallard, Y. (1980). The didactics of mathematics: Its problematic and related research. Recherches En Didactique Des Mathématiques, 2(1), 146–158.

    Google Scholar 

  • Chevallard, Y. (1992). A theoretical approach to curricula. Journal für Mathematik-Didaktik, 13(2–3), 215–230.

    Article  Google Scholar 

  • Chevallard, Y. (2007). Readjusting didactics to a changing epistemology. European Educational Research Journal, 6(2), 131–134.

    Article  Google Scholar 

  • Chevallard, Y., & Bosch, M. (2014). Didactic transposition in mathematics education. In S. Lerman (Ed.), Encyclopedia of mathematics education (pp. 170–174). Dordrecht, The Netherlands: Springer.

    Google Scholar 

  • Čižmešija, A., Katalenić, A., & Milin Šipuš, Ž. (2017). Asymptote as a body of knowledge to be taught in textbooks for Croatian secondary education. In Z. Kolar- Begović, R. Kolar-Šuper, & L. Jukić-Matić (Eds.), Mathematics education as a science and a profession (pp. 127–147). Osijek: Element.

  • Dahl, B. (2017). First-year non-STEM majors’ use of definitions to solve calculus tasks: Benefits of using concept image over concept definition? International Journal of Science and Mathematics Education, 15(7), 1303–1322.

    Article  Google Scholar 

  • Duval, R. (1993). Registres de représentation sémiotique et fonctionnement cognitif de la pensé [Registers of semiotic representation and cognitive functioning of thinking]. Annales de La Didactique et de Sciences Cognitives, 5(1), 37–65.

    Google Scholar 

  • Glazer, N. (2011). Challenges with graph interpretation: A review of the literature. Studies in Science Education, 47(2), 183–210.

    Article  Google Scholar 

  • Hardy, N. (2009). Students’ perceptions of institutional practices: The case of limits of functions in college level calculus courses. Educational Studies in Mathematics, 72(3), 341–358.

    Article  Google Scholar 

  • James, R. C. (1992). Mathematics dictionary (5th ed.). New York, NY: Chapman & Hall.

    Book  Google Scholar 

  • Kidron, I. (2011). Constructing knowledge about the notion of limit in the definition of the horizontal asymptote. International Journal of Science and Mathematics Education, 9(6), 1261–1279.

    Article  Google Scholar 

  • Koklu, O., & Topcu, A. (2012). Effect of Cabri-assisted instruction on secondary school students’ misconceptions about graphs of quadratic functions. International Journal of Mathematical Education in Science and Technology, 43(8), 999–1011.

    Article  Google Scholar 

  • Kop, P. M. G. M., Janssen, F. J. J. M., Drijvers, P. H. M., & van Driel, J. H. (2017). Graphing formulas: Unraveling experts’ recognition processes. The Journal of Mathematical Behavior, 45, 167–182.

    Article  Google Scholar 

  • Lundberg, A. L. V., & Kilhamn, C. (2018). Transposition of knowledge: Encountering proportionality in an algebra task. International Journal of Science and Mathematics Education, 16(3), 559–579.

    Article  Google Scholar 

  • Ministarstvo znanosti, obrazovanja i športa. (2011). Nacionalni okvirni kurikulum za predškolski odgoj i obrazovanje te opće obvezno i srednjoškolsko obrazovanje [The national curriculum framework for pre-school education and general compulsory and secondary education]. Retrieved from http://digarhiv.gov.hr/arhiva/36/33329/Nacionalni_okvirni_kurikulum.pdf. Accessed 4 Mar 2015.

  • Mudaly, V., & Rampersad, R. (2010). The role of visualisation in learners’ conceptual understanding of graphical functional relationships. African Journal of Research in Mathematics, Science and Technology Education, 14(1), 36–48.

    Article  Google Scholar 

  • Nachlieli, T., & Tabach, M. (2012). Growing mathematical objects in the classroom – The case of function. International Journal of Educational Research, 51–52, 10–27.

    Article  Google Scholar 

  • Öçal, M. F. (2017). Asymptote misconception on graphing functions: Does graphing software resolve it? Malaysian Online Journal of Educational Technology, 5(1), 21–33.

    Google Scholar 

  • Rutter, J. W. (2000). Geometry of curves. Boca Raton, FL: Chapman & Hall.

    Google Scholar 

  • Serrano, L., Bosch, M., & Gascón, J. (2018). An overview of “bridging courses” from the ATD perspective. Presented at the 6th International Conference on the Anthropological Theory of the Didactic, Grenoble, France.

  • Swinyard, C., & Larsen, S. (2012). Coming to understand the formal definition of limit: Insights gained from engaging students in reinvention. Journal for Research in Mathematics Education, 43(4), 465–493.

    Article  Google Scholar 

  • Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151–169.

    Article  Google Scholar 

  • Vandebrouck, F. (2011). Perspectives et domaines de travail pour l’étude des fonctions [perspectives and working domains for functions’ studies]. Annales de Didactiques et de Sciences Cognitives, 16, 149–185.

    Google Scholar 

  • Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65–81). New York, NY: Kluwer Academic Publishers.

    Google Scholar 

  • Williams, S. R. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22(3), 219–236.

    Article  Google Scholar 

  • Winsløw, C., & Grønbæk, N. (2014). Klein’s double discontinuity revisited: Contemporary challenges for universities preparing teachers to teach calculus. Recherches En Didactique Des Mathématiques, 34(1), 59–86.

    Google Scholar 

  • Yerushalmy, M. (1997). Reaching the unreachable: Technology and the semantics of asymptotes. International Journal of Computers for Mathematical Learning, 2(1),1–25.

  • Zarhouti, M. K., Mouradi, M., & Maroufi, A. E. (2014). The teaching of the function at high school: The graphic representation of a function at first year, section experimental sciences. IOSR Journal of Research & Method in Education, 4(3), 56–65.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Education, J. J. Strossmayer University of Osijek Faculty of Education, Cara Hadrijana 10, HR-31000, Osijek, Croatia

    Ana Katalenić

  2. Department of Mathematics, University of Zagreb Faculty of Science, Bijenička cesta 30, HR-10000, Zagreb, Croatia

    Željka Milin Šipuš & Aleksandra Čižmešija

Authors
  1. Ana Katalenić
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  2. Željka Milin Šipuš
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  3. Aleksandra Čižmešija
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Correspondence to Ana Katalenić.

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Katalenić, A., Milin Šipuš, Ž. & Čižmešija, A. Asymptotes and Asymptotic Behaviour in Graphing Functions and Curves: an Analysis of the Croatian Upper Secondary Education Within the Anthropological Theory of the Didactic. Int J of Sci and Math Educ 18, 1185–1205 (2020). https://doi.org/10.1007/s10763-019-10020-5

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