Semiha Kula Ünver, Esra Bukova-Güzel


The purpose of this study was to conceptualize pre-service mathematics teachers’ responding to students’ ideas, one of the codes of Contingency unit of Knowledge Quartet, while teaching limit concept. The participants were four pre-service secondary mathematics teachers. The data were obtained from the lesson plans, the video records of the participants’ lessons, and the semi-structured interviews. When the data were analysed, the seven sub-codes of the pre-service teachers’ responding to students’ ideas were determined. These sub-codes were named as (a) repeating students’ ideas, (b) approving students’ ideas, (c) explaining and expanding students’ ideas, (d) answering students’ questions, (e) asking how students’ reached their ideas, (f) correcting mistakes in students’ ideas, and (g) ignoring students’ ideas. It is thought that these sub-codes would be helpful to examine pre-service mathematics teachers’ responding to students’ ideas in a detail way.


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contingency; knowledge quartet; limit concept; pre-service mathematics teachers; responding to students’ ideas


Ball, D. L. (2003). What mathematical knowledge is needed for teaching mathematics? Secretary’s Summit on Mathematics. Retrieved November 11, 2010 from

Ball, D. L., & Sleep, L. (2007, January). What is mathematical knowledge for teaching, and what are features of tasks that can be used to develop MKT. In Presentation made at the Center for Proficiency in Teaching Mathematics (CPTM) pre-session of the annual meeting of the Association of Mathematics Teacher Educators (AMTE), Irvine, CA.

Bergthold, T. A. (1999). Patterns of analytical thinking and knowledge use in students' early understanding of the limit concept. Unpublished Doctoral Dissertation, University of Oklahama, Oklahama.

Blaisdell, R. (2012, February). Student understanding in the concept of limit in calculus: how student responses vary depending on question format and type of representation. Proceeding of the 15th Annual Research in Undergraduate Mathematics Education Conference, Portland, Oregon.

Bukova, E. (2006). Öğrencilerin limit kavramını algılamasında ve diğer kavramların ilişkilendirilmesinde karşılaştıkları güçlükleri ortadan kaldıracak yeni bir program geliştirme. Unpublished Doctoral Dissertation, University of Dokuz Eylül, İzmir.

Corbin, J., & Strauss, A. (1990). Grounded theory research: Procedures, canons, and evaluative criteria. Qualitative Sociology, 13(1), 3-21.

Cornu, B. (1991). Limits. In Tall, D. (Ed.), Advanced Mathematical Thinking (pp. 153-166), Boston: Kluwer.

Davies, N., & Walker, K. (2007). Teaching as listening: another aspect of teachers’ content knowledge in the numeracy classroom. Mathematics: essential research, essential practice: Proceedings of the 30th annual conference of Mathematics Research of Australasia, Adelaide: MERGA.

Ding, M. (2007). Knowing mathematics for teaching: a case study of teacher responses to students’ errors and difficulties in teaching equivalent fractions. Doctoral Dissertation, Texas A&M University.

Empson, S. B., & Jacobs, V. R. (2008). Learning to Listen to Children's Mathematics. In D. Tirosh & T. Wood (Eds.), Tools and Processes in mathematics teacher education (pp. 257-281). Rotterdam: Sense Publishers.

Even, R., & Tirosh, D. (1995). Subject-matter knowledge and knowledge about students as sources of teacher presentations of the subject-matter. Educational Studies in Mathematics, 29(1), 1–20.

Glaser, B. G. (1978). Theoretical sensitivity: Advances in methodology of grounded theory. Mill Valley, CA.

Glaser, В., & Strauss, A. (1967). The discovery of grounded theory: Strategies for qualitative research. New York: Aldine.

Graeber, A. O. (1999). Forms of knowing mathematics: What preservice teachers should learn. Educational Studies in Mathematics, 38(1-3), 189-208.

Jordaan, T. (2005). Misconceptions of the limit concept in a mathematics course for engineering students. Unpublished Master Thesis, University of South Africa.

Kula, S. (2011). Examining pre-service mathematics teachers’ subject matter and pedagogical content knowledge by using knowledge quartet: the case of limit. Unpublished Master Thesis, University of Dokuz Eylül, İzmir, Türkiye.

Leavit, (2008). German mathematics teachers’ subject content and pedagogical content knowledge. Doctoral Dissertation, University of Nevada, Las Vegas.

Lloyd, G. M., & Wilson, M. (1998). Supporting innovation: The impact of a teacher’s conceptions of functions on his implementation of a reform curriculum. Journal for Research in Mathematics Education, 29(3), 248-274.

Marks, R. (1990). Pedagogical content knowledge: from a mathematical case to a modified conception. Journal of Teacher Education, 41(3), 3-11.

Mhlolo, M. K., & Schäfer, M. (2012). Towards empowering learners in a democratic mathematics classroom: To what extent are teachers’ listening orientations conducive to and respectful of learners’ thinking? Pythagoras, 33(2), 1- 9.

Petrou, M. (2009). Adapting the knowledge quartet in the Cypriot mathematics classroom. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the 6th Congress of the European Society for Research in Mathematics Education (pp. 2020-2029). Lyon: France.

Rowland, T. (2008). The purpose, design and use of examples in the teaching of elementary mathematics. Educational Studies in Mathematics, 69(2), 149-163.

Rowland, T. (2010) Back to the data: Jason, and Elliot’s quarters. In M.M.F. Pinto & T.F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education, 4(pp. 97-104). Belo Horizonte: Brazil.

Rowland, T. Thwaites, A., & Jared, L. (2011). Triggers of contingency in mathematics teaching. In Ubuz, B. (Ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, 4(pp. 73-80). Ankara: Turkey.

Rowland, T., Huckstep, P., & Thwaites, A. (2005). Elementary teachers’ mathematics subject knowledge: the knowledge quartet and the case of Naomi. Journal of Mathematics Teacher Education, 8(3), 255-281.

Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009). Developing primary mathematics teaching: reflecting on practice with the knowledge quartet. London: Sage.

Schoenfeld, A. H. (2006). Problem solving from cradle to grave. In Annales de Didactique et de Sciences Cognitives (Vol. 11, pp. 41-73).

Suurtamm, C., & Vézina, N. (2010). Transforming pedagogical practice in mathematics: Moving from telling to listening. International Journal of Mathematics Teaching and Learning, 31(1), 27-28.

Thwaites, A., Huckstep, P., & Rowland, T. (2005). The knowledge quartet: Sonia’s reflections. In D. Hewitt & A. Noyes (Eds), Proceedings of the Sixth British Congress of Mathematics Education (pp. 168-175). London: England.

Tirosh, D., Even, R., & Robinson, N. (1998). Simplifying algebraic expressions: teacher awareness and teaching approaches. Educational Studies in Mathematics, 35(1), 51–64.

Turner, F. (2009). Developing the ability to respond to the unexpected. In M. Joubert (Ed.), Proceedings of the British Society for Research into Learning Mathematics 29(1), 91-96.

Van Der Valk, T.A.E., & Broekman, H.H.G.B. (1999). The lesson preparation method: A way of investigating pre-service teachers' pedagogical content knowledge. European Journal of Teacher Education, 22(1), 11-22.

Wicks, R., & Janes, R. (2006). Uncovering children's thinking about patterns: teacher-researchers improving classroom practices. In S. Z. Smith, D. S. Mewborn, & M. E. Smith (Eds.), Teachers engaged in research: inquiry into mathematics classrooms, grades pre-k-2 (pp. 211–236). Greenwich, CT: Information Age Publishing.

Yusof, Y. M., & Zakaria, E. (2010). Investigating secondary mathematics teachers’ pedagogical content knowledge: A case study. Journal of Education and Sociology, 1(1), 32-39.



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