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Teachers’ beliefs towards teaching calculus

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Abstract

This paper focuses on the belief systems towards teaching calculus of 29 upper secondary mathematics teachers. Firstly, we discuss different educational trends in teaching calculus, and the theoretical approach based on the construct of belief systems. Afterwards we describe the method of our study used to analyse the belief systems of the calculus teachers. Referring to findings of our research, we firstly focus on central and peripheral beliefs and thus discuss the structure of the teachers’ belief systems. Further, we compare the teachers’ belief systems to established educational trends of teaching calculus. Finally, we conclude the paper by reflecting on our main findings and by discussing possible directions of further research.

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References

  • Artigue, M. (1991). Analysis. In D. Tall (Ed.), Advanced mathematical thinking (pp. 167–198). Dordrecht: Kluwer.

    Google Scholar 

  • Artigue, M. (1996). Learning and teaching elementary analysis. In C. Alsina, J. M. Alvarez, M. Niss, A. Pérez, L. Rico, & A. Sfard (Eds.), 8th International Congress on Mathematics Education—Selected Lectures (pp. 15–30). Sevilla: S.A.E.M. Thalès.

    Google Scholar 

  • Artigue, M., Batanero, C., & Kent, P. (2007). Mathematics thinking and learning at post-secondary level. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1011–1050). Reston: National Council of Teachers of Mathematics (NCTM).

    Google Scholar 

  • Blum, W., & Törner, G. (1983). Didaktik der analysis. Göttingen: Vandenhoeck & Ruprecht.

    Google Scholar 

  • Calderhead, J. (1996). Teachers: beliefs and knowledge. In D. C. Berliner (Ed.), Handbook of education (pp. 709–725). New York: Macmillan.

    Google Scholar 

  • Danckwerts, R., & Vogel, D. (1992). Quo vadis Analysisunterricht? MNU, 45(6), 370–374.

    Google Scholar 

  • Danckwerts, R., & Vogel, D. (2010). Analysis verständlich unterrichten. Heidelberg: Spektrum.

    Google Scholar 

  • Doorman, L. M., & Gravemeijer, K. P. E. (2009). Emergent modeling: discrete graphs to support the understanding of change and velocity. ZDM - The International Journal on Mathematics Education, 41(1–2), 199–211.

    Article  Google Scholar 

  • Dreyfus, T. (1991). Advanced mathematical thinking processes. In D. Tall (Ed.), Advanced mathematical thinking (pp. 25–41). Dordrecht: Kluwer.

    Google Scholar 

  • Eichler, A. (2011). Teaching statistics in school mathematics: challenges for teaching and teacher education. In C. Batanero, G. Burril, & C. Reading (Eds.), Statistics teachers and classroom practices (15th ed., pp. 175–186)., New ICMI Study Series New York: Springer.

    Google Scholar 

  • Eichler, A., & Erens, R. (2014). From beliefs and affect to dynamic systems: a mosaic of relationships and interactions. In B. Roesken & B. Pepin (Eds.), Domain specific belief systems of secondary mathematics teachers. New York: Springer.

    Google Scholar 

  • Felbrich, A., Kaiser, G., & Schmotz, C. (2012). The cultural dimension of beliefs: an investigation of future primary teachers’ epistemological beliefs concerning the nature of mathematics in 15 countries. ZDM - The International Journal on Mathematics Education, 44(3), 355–366.

    Article  Google Scholar 

  • Förster, F. (2011). Trends in teaching and learning of mathematical modelling. In G. Kaiser, W. Blum, R. Borromeo Ferri, & G. Stillman (Eds.), Secondary teachers’ beliefs about teaching applications design and selected results of a qualitative case study (pp. 65–74). Dordrecht: Springer.

    Google Scholar 

  • Fothergill, L. (2011). Aspects of calculus for preservice teachers. The Mathematics Educator, 21(1), 23–31.

    Google Scholar 

  • Fox, T. B. (1998). Teacher change in reform calculus curriculum: concepts related to derivative. In S. B. Berensen (Ed.), Proceedings of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 221–225). Columbus: Clearinghouse.

    Google Scholar 

  • Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Kluwer.

    Google Scholar 

  • Furinghetti, F., & Morselli, F. (2011). Beliefs and beyond: hows and whys in the teaching of proof. ZDM – The International Journal on Mathematics Education, 43(4), 587–599.

    Article  Google Scholar 

  • Green, T. (1971). The activities of teaching. New York: McGraw-Hill.

    Google Scholar 

  • Grigutsch, S., Raatz, U., & Törner, G. (1998). Einstellungen gegenüber Mathematik bei Mathematiklehrern. Journal für Mathematikdidaktik, 19(1), 3–45.

    Article  Google Scholar 

  • Hannula, M. S. (2012). Exploring new dimensions of mathematics-related affect: embodied and social theories. Research in Mathematics Education, 14(2), 137–161.

    Article  Google Scholar 

  • Heckhausen, H., & Gollwitzer, P. M. (1987). Thought contents and cognitive functioning in motivational versus volitional states of mind. Motivation and Emotion, 11(2), 101–120.

    Article  Google Scholar 

  • Hillel, J. (2001). Trends in curriculum: a working group report. In D. Holten (Ed.), The teaching and learning of mathematics at university level (pp. 59–70). Berlin: Springer.

    Google Scholar 

  • Hofer, B. K., & Pintrich, P. R. (1997). The development of epistemological theories: beliefs about knowledge and knowing in their relation to learning. Review of Educational Research, 67, 88–140.

    Article  Google Scholar 

  • Huberman, M. (1989). The professional life cycle of teachers. Teachers College Record, 91(1), 31–57.

    Google Scholar 

  • Kaiser, G., Blum, W., Borromeo Ferri, R., & Stillman, G. (2011). Trends in teaching and learning of mathematical modelling. Dordrecht: Springer.

    Book  Google Scholar 

  • Kendal, M., & Stacey, K. (2002). Teachers in transition: moving towards CAS-supported classrooms. ZDM – The International Journal on Mathematics Education, 34(5), 196–203.

    Google Scholar 

  • KMK (2012). Bildungsstandards im Fach Mathematik für die Allgemeine Hochschulreife. Online: http://www.kmk.org. Accessed 16 June 2014.

  • Kuckartz, U. (2012). Qualitative Inhaltsanalyse, Methoden Praxis, Computerunterstützung. Weinheim/Basel: Beltz Juventa.

    Google Scholar 

  • Niss, M., Blum, W., & Galbraith, P. (2007). Modelling and applications in mathematics education. The 14th ICMI study. In W. Blum, P. L. Galbraith, H. W. Henn, & M. Niss (Eds.), Introduction (pp. 3–32). New York: Springer.

    Google Scholar 

  • Oates, G. (2012). Applications and implications of recent research for teaching calculus: a report from the 2011 Delta Conference on Undergraduate Mathematics Teaching and Learning. Proceedings of the 12th International Congress on Mathematical Education (8 July–15 July, 2012), COEX, Seoul, Korea. Seoul: International Congress on Mathematical Education ICME-12.

  • Orton, A. (1983). Students’ understanding of differentiation. Educational Studies in Mathematics, 14, 235–250.

    Article  Google Scholar 

  • Pajares, F. M. (1992). Teachers’ beliefs and educational research: cleaning up a messy construct. Review of Educational Research, 62(3), 307–332.

    Article  Google Scholar 

  • Putnam, R. T., & Borko, H. (2000). What do new views of knowledge and thinking have to say about research on teacher learning? Educational Researcher, 29(1), 4–15.

    Article  Google Scholar 

  • Robert, A., & Speer, A. (2001). Research on the teaching and learning of calculus/elementary analysis. In D. Holten (Ed.), The teaching and learning of mathematics at university level (pp. 283–299). Berlin: Springer.

    Google Scholar 

  • Schoenfeld, A. H. (2011). Toward professional development for teachers grounded in a theory of decision making. ZDM – The International Journal on Mathematics Education, 43(4), 457–469.

    Article  Google Scholar 

  • Selden, A., & Selden, J. (2001). Tertiary mathematics education research and its future. In D. Holton (Ed.), The teaching and learning of mathematics at university level (pp. 237–254). Dordrecht: Kluwer.

    Google Scholar 

  • Skott, J. (2009). Contextualising the notion of ‘belief enactment’. Journal of Mathematics Teacher Education, 12, 27–46.

    Article  Google Scholar 

  • Stein, M. K., Remillard, J., & Smith, M. S. (2007). How curriculum influences student learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 319–369). Charlotte: Information Age.

    Google Scholar 

  • Tall, D. (1989). Concept images, generic organizers, computers & curriculum change. For the Learning of Mathematics, 9(3), 37–42.

    Google Scholar 

  • Tall, D. (1991). The psychology of advanced mathematical thinking. In D. Tall (Ed.), Advanced mathematical thinking (pp. 3–21). Dordrecht: Kluwer.

    Chapter  Google Scholar 

  • Tall, D. (1992). The transition to advanced mathematical thinking: functions, limits, infinity, and proof. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 495–511). New York: Macmillan.

    Google Scholar 

  • Tall, D. (2009). Dynamic mathematics and the blending of knowledge structures in the calculus. ZDM – The International Journal on Mathematics Education, 41(4), 481–492.

    Article  Google Scholar 

  • Tall, D. (2012). A sensible approach to the calculus. To appear in F. Pluvinage & A. Cuevas (Eds.), Handbook on calculus and its teaching. http://homepages.warwick.ac.uk/staff/David.Tall/downloads.html. Accessed 16 June 2014.

  • Tall, D., Smith, D., & Piez, C. (2008). Research on technology and the teaching and learning of mathematics: synthesis, cases and perspectives. Vol. 1: Research synthesis. In M. K. Heid & G. W. Blume (Eds.), Technology and calculus (pp. 207–258). Charlotte: Information Age.

    Google Scholar 

  • Tietze, U. (1992). Der Mathematikunterricht in der Sekundarstufe II – Curriculumsentwicklung und didaktische Forschung. Mathematica Didactica, 15(2), 3–37.

    Google Scholar 

  • Tietze, U., Klika, M., & Wolpers, H. (2000). Mathematikunterricht in der Sekundarstufe II (Vol. 1). Braunschweig: Vieweg.

    Book  Google Scholar 

  • Toeplitz, O. (1927). Das Problem der Universitätsvorlesungen über Infinitesimalrechnung und ihrer Abgrenzung gegenüber der Infinitesimalrechnung an höheren Schulen. Annual Report of the DMV, 36, 90–100.

    Google Scholar 

  • Toh, T. L. (2009). On in-service mathematics teachers’ content knowledge of calculus and related concepts. The Mathematics Educator, 12(1), 69–86.

    Google Scholar 

  • Törner, G. (1999). MAVI-8 Proceedings. 8th European Workshop. Research on Mathematical Beliefs. In G. Philippou (Ed.), Narration as a tool for analyzing beliefs on calculus—a case study (pp. 116–122). Nicosia: University of Cyprus.

    Google Scholar 

  • Zbiek, R. M., Heid, M. K., Blume, G., & Dick, T. P. (2007). Research on technology in mathematics education: the perspective of constructs. In F. K. Lester (Ed.), Second handbook of research in mathematics teaching and learning (pp. 1169–1207). Charlotte: Information age.

    Google Scholar 

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Authors and Affiliations

  1. University of Kassel, Kassel, Germany

    Andreas Eichler

  2. University of Education Freiburg, Freiburg, Germany

    Ralf Erens

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  1. Andreas Eichler
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  2. Ralf Erens
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Correspondence to Andreas Eichler.

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Eichler, A., Erens, R. Teachers’ beliefs towards teaching calculus. ZDM Mathematics Education 46, 647–659 (2014). https://doi.org/10.1007/s11858-014-0606-y

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