The Influence of Curriculum on the Concept of Function: An Empirical Study of Pre-Service Teachers

Ljerka Jukić Matić 1, Gabrijela Kehler-Poljak 2 * , Sanja Rukavina 3
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1 Department of Mathematics, University of Osijek, Osijek, CROATIA
2 Faculty of Mathematics, Institute for Didactics of Mathematics, University of Bielefeld, GERMANY
3 Faculty of Mathematics, University of Rijeka, Rijeka, CROATIA
* Corresponding Author
EUR J SCI MATH ED, Volume 10, Issue 3, pp. 380-395. https://doi.org/10.30935/scimath/12042
Published: 22 April 2022
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ABSTRACT

This article reports on the understanding of the function concept by pre-service mathematics teachers from two countries (Germany and Croatia). We focused on investigating students’ concept definition and concept image of the function in relation to their curriculum experiences. Data were collected using a questionnaire in the form of open-ended questions followed by interviews. The results indicate that the curriculum has a great influence on the development of the concept definition and concept image. The curriculum strongly influenced the theoretical background of the function concept and thus the gap between the formal and the personal definition of function. Later and more intensive work with the formal definition of function led to a better development of the function concept in general. The curriculum also had an influence on the range of the concept image developed by the pre-service mathematics teachers, with no proportional dependence in relation to the better developed understanding of the concept of function.
Keywords: concept definition, concept image, function, pre-service mathematics teachers

CITATION

Jukić Matić, L., Kehler-Poljak, G., & Rukavina, S. (2022). The Influence of Curriculum on the Concept of Function: An Empirical Study of Pre-Service Teachers. European Journal of Science and Mathematics Education, 10(3), 380-395. https://doi.org/10.30935/scimath/12042

REFERENCES

  • Alajmi, A. H., & Al-Kandari, M. M. (2020). Calculus 1 college students’ concept of function. International Journal of Mathematical Education in Science and Technology, 53(2), 251-268. https://doi.org/10.1080/0020739X.2020.1798526
  • Bannister, V. R. P. (2014). Flexible conceptions of perspectives and representations: An examination of pre-service mathematics teachers’ knowledge. International Journal of Education in Mathematics, Science and Technology, 2(3), 223-233. https://doi.org/10.18404/ijemst.23592
  • Bossé, M., Adu-Gyamfi, K., & Cheetham, M. (2011). Assessing the difficulty of mathematical translations: Synthesizing the literature and novel findings. International Electronic Journal of Mathematics Education, 6(3), 113-133. https://doi.org/10.29333/iejme/264
  • Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. In J. J. Kaput, A. H. Schoenfeld, & E. Dubinsky (Eds.), Research in collegiate mathematics education, 3, CBMS issues in mathematics education (pp. 114-162). Mathematical Association of America. https://doi.org/10.1090/cbmath/007/04
  • Carlson, M., & Oehrtman, M. (2005). Key aspects of knowing and learning the concept of function. Research sampler series. Mathematical Association of America.
  • Chesler, J. (2012). Pre-service secondary mathematics teachers making sense of definitions of functions. Mathematics Teacher Education and Development, 14(1), 27-40.
  • Cohen, L., Manion, L., & Morrison, K. (2018). Research methods in education. Routledge. https://doi.org/10.4324/9781315456539
  • Cooney, T. J., Beckman, S., & Lloyd, G. M. (2010). Developing essential understanding of functions for teaching mathematics in grades 9-12. NCTM.
  • Cooney, T., & Wiegel, H. (2003). Examining mathematics in mathematics teacher education. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.), Second international handbook of mathematics education (pp. 795-828). Kluwer Academic Publishers. https://doi.org/10.1007/978-94-010-0273-8_26
  • Dede, Y., & Soybas, D. (2011). Pre-service mathematics teachers’ experiences about function and equation concepts. EURASIA Journal of Mathematics, Science and Technology, 7(2), 89-102. https://doi.org/10.12973/ejmste/75183
  • Denzin, N. K. (2015). Triangulation. In G. Ritzer (Ed.), The Blackwell encyclopedia of sociology. https://doi.org/10.1002/9781405165518.wbeost050.pub2
  • Dubinsky, E., & Wilson, R. T. (2013). High school students’ understanding of the function concept. The Journal of Mathematical Behavior, 32(1), 83-101. https://doi.org10.1016/j.jmathb.2012.12.001
  • Edwards, B. S., & Ward, M. B. (2004). Surprises from mathematics education research: Student (mis)use of mathematical definitions. The American Mathematical Monthly, 111(5), 411-424. https://doi.org/10.1080/00029890.2004.11920092
  • Elia, I., & Spyrou, P. (2006). How students conceive function: A triarchic conceptual-semiotic model of the understanding of a complex concept. The Montana Mathematics Enthusiast, 3(2), 256-272. https://doi.org/10.54870/1551-3440.1053
  • Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24(2), 94-111. https://doi.org/10.2307/749215
  • FitzSimons, G. E. (2002). What counts as mathematics? Technologies of power in adult and vocational education. Springer. https://doi.org/10.1007/0-306-47683-5
  • Gusić, M., & Milin Šipuš, Ž. (2019). Teorijski okvir za razvoj pojma funkcije. Primjer kvadratne funkcije [Theoretical framework for the development of the concept of function. Example of a quadratic function]. https://www.huni.hr/wp-content/uploads/2019/05/Zadar_2504_Teorijski-okvir-za-razvoj-pojma-funkcije.pdf
  • Hadjidemetriou, C., & Williams, J. (2002). Children’s graphical conceptions. Research in Mathematics Education, 4(1), S. 69-87. https://doi.org/10.1080/14794800008520103
  • Hansson, Ö. (2006). Studying the views of pre-service teachers on the concept of function [PhD thesis, Luleå University].
  • Hatisaru, V., & Erbas, A. K. (2017). Mathematical knowledge for teaching the function concept and student learning outcomes. International Journal of Science and Mathematics Education, 15(4), 703-722. https://doi.org/10.1007/s10763-015-9707-5
  • KMK. (2004). Bildungsstandards im Fach Mathematik für den Mittleren Schulabschluss [Educational standards in mathematics for the middle school leaving certificate]. Kultusministerkonferenz [Conference of Ministers of Education].
  • Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1-64. https://doi.org/10.3102/00346543060001001
  • Mayring, P. (2002). Einführung in die qualitative Sozialforschung [Introduction to qualitative research]. Beltz.
  • MSJK. (2004). Kernlehrplan für die Realschule in Nordrhein- Westfalen–Mathematik [Core curriculum for the Realschule in North Rhine-Westphalia–Mathematics]. Ministerium für Schule, Jugend und Kinder des Landes NRW [Ministry for Schools, Youth and Children of the State of North Rhine-Westphalia].
  • MZOS. (2003). Kurikularni pristup promjenama u gimnaziji [Curriculum approach to changes in high school]. Ministry of Science, Education and Sport.
  • MZOS. (2006). Nastavni plan i program [Educational plan and program]. Ministry of Science, Education and Sport.
  • NCTM. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
  • Niss, M. (1996). Goals of mathematics teaching. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.), International handbook of mathematics education (pp. 11-41). Springer. https://doi.org/10.1007/978-94-009-1465-0_2
  • Nitsch, R. (2015). Diagnose von Lernschwierigkeiten im Bereich funktionaler Zusammenhänge. Research [Diagnosis of learning difficulties in the area of functional relationships. Research]. Springer Spektrum. https://doi.org/10.1007/978-3-658-10157-2
  • Pepin, B., Gueudet, G., & Trouche, L. (2013). Re-sourcing teachers’ work and interactions: A collective perspective on resources, their use and transformation. ZDM–The International Journal of Mathematics Education, 45(7), 929-943. https://doi.org/10.1007/s11858-013-0534-2
  • Rezat, S., & Sträßer, R. (2012). From the didactical triangle to the socio-didactical tetrahedron: Artifacts as fundamental constituents of the didactical situation. ZDM–The International Journal on Mathematics Education, 44(5), 641-651. https://doi.org/10.1007/s11858-012-0448-4
  • Ron, G., Dreyfus, T., & Hershkowitz, R. (2010). Partially correct constructs illuminate students’ inconsistent answers. Educational Studies in Mathematics, 75(1), 65-87. https://doi.org/10.1007/s10649-010-9241-x
  • Schwarz, B. B., & Hershkowitz, R. (1999). Prototypes: Brakes or levers in learning the function concept? The role of computer tools. Journal for Research in Mathematics Education, 30, 362-389. https://doi.org/10.2307/749706
  • Sierpinska, A. (1992). On understanding the notion of function. In E. Dubinsky, & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 22-58). The Mathematical Association of America,
  • Sintema, E. J., & Marban, J. M. (2020). Pre-service secondary teachers’ mathematical pedagogical content knowledge self-concept related to their content knowledge of functions and students. International Electronic Journal of Mathematics Education, 15(3), em0598. https://doi.org/10.29333/iejme/8327
  • Stewart, S., & Reeder, S. (2017). Algebra underperformances at college level: What are the consequences? In S. Stewart (Ed.), And the rest is just algebra (pp. 3-18). Springer. https://doi.org/10.1007/978-3-319-45053-7_1
  • Stölting, P. (2008). Die Entwicklung funktionalen Denkens in der Sekundarstufe I–Vergleichende Analysen und empirische Studien zum Mathematikunterricht in Deutschland und Frankreich [The development of functional thinking in secondary school I–Comparative analyzes and empirical studies on mathematics teaching in Germany and France] [PhD dissertation, Universität Regensburg]. https://doi.org/10.5283/epub.10725
  • Strauss, A., & Corbin, J. M. (1990). Basics of qualitative research: Grounded theory procedures and techniques. SAGE.
  • Tabach, M., & Nachlieli, T. (2015). Classroom engagement towards using definitions for developing mathematical objects: The case of function. Educational Studies in Mathematics, 90(2), 163-187. https://doi.org/10.1007/s10649-015-9624-0
  • Tall, D., & Bakar, M. (1992). Students’ mental prototypes for functions and graphs. International Journal of Mathematical Education in Science and Technology, 23(1), 39-50. https://doi.org/10.1080/0020739920230105
  • Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169. https://doi.org/10.1007/BF00305619
  • Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). National Council of Teachers of Mathematics.
  • Viholainen, A. (2008). Incoherence of a concept image and erroneous conclusions in the case of differentiability. The Montana Mathematics Enthusiast, 5(2), 231-248. https://doi.org/10.54870/1551-3440.1104
  • Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall (Ed.), Advanced mathematical thinking (pp. 65-81). Kluwer Academic Publishers. https://doi.org/10.1007/0-306-47203-1_5
  • Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(4), 356-366. https://doi.org/10.5951/jresematheduc.20.4.0356
  • Wilson, M. (1994). One pre-service secondary teacher’s understanding of function: The impact of a course integrating mathematical content and pedagogy. Journal for Research in Mathematics Education, 25(4), 346-370. https://doi.org/10.2307/749238
  • Zaslavsky, O., & Shir, K. (2005). Students’ conceptions of a mathematical definition. Journal for Research in Mathematics Education, 36(4), 317-346.