#### Abstract

The aim of this study was to determine the how pre-service mathematics teachers respond to unexpected questions from students about the second derivative. A qualitative research method was used for this purpose, applying a case study framed as an in-depth analysis of how pre-service mathematics teachers respond to students’ ideas. The participants were 39 pre-service mathematics teachers who were in their final year of their mathematics teacher education program. The pre-service teachers, who participated voluntarily, were asked to respond to open-ended questions relating to a scenario that included a teacher, as well as her 12th-grade students. The written responses that the participants provided constitute the data of this study. The results revealed that most of the participants could not effectively answer an unexpected question from students. Nearly half of the participants stated that they could not answer the question. Others ignored it, while some acknowledged the question and attempted to give an answer. Moreover, a small number of the participants made an effort to explain and demonstrate the concept of concavity by drawing the graphs of the function and relating them to the first derivative.

#### Keywords

#### References

- Artigue, M. (1991). Analysis. In D. O. Tall (Ed.),
*Advanced mathematical thinking*(pp.167–198). Dordrecht: Kluwer Academic Publishers. - Aspinwall, L., & Miller, L. D. (2001). Diagnosing conflict factors in calculus through students' writings: One teacher's reflections.
*The Journal of Mathematical Behavior*,*20*(1), 89-107. - Aspinwall, L., Shaw, K. L., & Presmeg, N. C. (1997). Uncontrollable mental imagery: graphical connections between a function and its derivative.
*Educational Studies in Mathematics, 33*(3), 301-317. - Baker, B., Cooley, L. & Trigueros, M. (2000). A calculus graphing schema. T
*he Journal for Research in Mathematics Education*,*31*(5), 557–578. - Breen, S., Meehan, M., O’Shea, A., & Rowland, T. (2018). An analysis of university mathematics teaching using the Knowledge Quartet. In V. Durand-Guerrier, R. Hochmuth, S. Goodchild, & N.M. Hogstad (Eds.),
*Proceedings of the Second Conference of the International Network for Didactic Research in University Mathematics*(pp. 383–392). Kristiansand, Norway: University of Agder and INDRUM. - Bukova-Güzel, E. (2010). An investigation of pre-service mathematics teachers’ pedagogical content knowledge, using solid objects.
*Scientific Research and Essays, 5*(14), 1872-1880. - Corcoran, D. (2013).
*Róisín teaching equivalence of fractions*. Retrieved from http://www.knowledgequartet.org/326/rci-scenario-2/ on June 25, 2019. - 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), 94–116. - García, M., Llinares, S. & Sánchez-Matamoros, G. (2011). Characterizing thematized derivative schema by the underlying emergent structures.
*International Journal of Science and Mathematics Education, 9*(5)*,*1023–1045. - Güler, M., & Çelik, D. (2018). Uncovering the Relation between CK and PCK: An investigation of preservice elementary mathematics teachers’ algebra teaching knowledge.
*REDIMAT-Journal of Research in Mathematics Education,*7(2), 162-194. - Güler, M., & Çelik, D. (2019). How well prepared are the teachers of tomorrow? An examination of prospective mathematics teachers' pedagogical content knowledge.
*International Journal of Mathematical Education in Science and Technology, 50*(1), 82-99. - Kleve, B. (2013).
*Hans teaching fractions greater than one*. Retrieved from www.knowledgequartet.org/317/rci-scenario-4/ on May 20, 2013. - Kula Ünver, S., & Bukova Güzel, E. (2016). Conceptualizing pre-service mathematics teachers’ responding to students’ ideas while teaching limit concept.
*European Journal of Education Studies*, Special Issue (Basic and Advanced Concepts, Theories and Methods Applicable on Modern Mathematics Education), 33-57. - Kula, S. (2011). Examining mathematics pre-service teachers’ subject matter and pedagogical content knowledge by using Knowledge Quartet: The case of limit. [Unpublished master’s thesis]. Dokuz Eylül University, İzmir, Turkey.
- Kula, S. (2014). Conceptualizing mathematics pre-service teachers' approaches towards contingency in teaching process in the context of Knowledge Quartet. [Unpublished doctoral dissertation] Dokuz Eylül University, İzmir, Turkey.
- Kula, S., & Bukova Güzel, E. (2014). Knowledge Quartet’s unit of Contingency in the light of mathematics and mathematics content knowledge.
*Turkish Journal of Computer and Mathematics Education*, 5(1), 89-107. - Macdonald, D. (1993). Knowledge, gender and power in physical education teacher education.
*Australian Journal of Education, 37*(3), 259-278. - Miles, M. B., & Huberman, A. M. (1994).
*Qualitative Data Analysis: A Sourcebook of New Methods*(second edition). Beverly Hills: Sage Publications - Orton, A., (1983). Students` understanding of integration.
*Educational Studies in Mathematics,*14(1), 1-18. - Ozgur, Z., Ellis, A. B., Vinsonhaler, R., Dogan, M. F., & Knuth, E. (2019). From examples to proof: Purposes, strategies, and affordances of example use.
*The Journal of Mathematical Behavior, 53*, 284-303. - Patton, M.Q. (2002).
*Qualitative research and evaluation methods*. (3rd ed.). Thousand Oaks, CA: Sage Publications, Inc. - Petrou, M. (2013a). Solving problem using schema-based instruction. Retrieved from www.knowledgequartet.org/329/rci-scenario-1/ on May 20, 2013.
- Petrou, M. (2013b). Christina teaching average. Retrieved from www.knowledgequartet.org/323/rci-scenario-3/ on May 20, 2013.
- Rowland, T. & Zazkis, R. (2013). Contingency in the mathematics classroom: Opportunities taken and opportunities missed.
*Canadian Journal of Science, Mathematics and Technology Education*,*13*(2), 137-153. - Rowland, T. (2013). The Knowledge Quartet: The genesis and application of a framework for analysing mathematics teaching and deepening teachers’ mathematics knowledge.
*Journal of Education*,*1*(3), 15-43. - 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., Huckstep, P., & Thwaites, A. (2003). The Knowledge Quartet.
*Proceedings of the British Society for Research into Learning Mathematics,**23*(3), 97-102. - Rowland, T., Jared, L., & Thwaites, A. (2011). Secondary Mathematics Teachers’ Content Knowledge: The Case of Heidi. In M. Pytlak, T. Rowland and E. Swoboda (Eds.)
*Proceedings of the Seventh Congress of the European Society for Research in Mathematics Education*(pp. 2827-2837). Rzeszow, Poland: University of Rzeszow. - Rowland, T., Thwaites, A., & Jared, L. (2015). Triggers of contingency in mathematics teaching.
*Research in Mathematics Education, 17*(2), 74-91. - Rowland, T., Thwaites, A., & Jared, L. (2011). Triggers of Contingency in Mathematics Teaching. In B. Ubuz (Ed.).
*Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education*(Vol. 4, pp. 73-80). METU: Ankara. - Rowland, T., Turner, F., Thwaites, A., & Huckstep, P. (2009).
*Developing primary mathematics teaching: reflecting on practice with the Knowledge Quartet*. London: Sage. - Sánchez-Matamoros, G., García, M. & Llinares, S. (2006). El desarrollo del esquema de derivada [The development of derivative schema].
*Enseñanza de las Ciencias, 24*(1), 85–98. - Sánchez-Matamoros, G., García, M. & Llinares, S. (2008). La comprensión de la derivada como objeto de investigación en Didáctica de la Matemática [The understanding of derivative as a research topic in Mathematics Education].
*RELIME-Revista Latinoamericana de Investigación en Matemática Educativa, 11*(2), 267–296. - Stump, S. (1999). Secondary mathematics teachers’ knowledge of slope.
*Mathematics Education Research Journal, 11*(2), 124-144. - Tall, D. (1996). Functions and calculus. In A.J. Bishop, K. Clements, J. Kilpatrick and C. Laborde (Eds.),
*International handbook of mathematics education*(pp. 289–325). Dordrecht: Kluwer Academic Publishers. - Thwaites, A., Huckstep, P., & Rowland, T. (2005). The Knowledge Quartet: Sonia’s reflections. In D. Hewitt and A. Noyes (Eds.),
*Proceedings of the Sixth British Congress of Mathematics Education*(pp. 168-175). London: British Society for Research into Learning Mathematics. - Turner, F. (2007). Development in the Mathematics Teaching of Beginning Elementary School Teachers: An Approach Based on Focused Reflections.
*Proceedings of the Second National Conference on Research in Mathematics Education, Mathematics in Ireland*(Vol. 2 pp. 377-386), Dublin: St Patrick's College. - Turner, F. (2009, March). Developing the ability to respond to the unexpected. Informal Paper presented at
*Proceedings of the British Society for Research into Learning Mathematics.*Cambridge: UK.

#### License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Article Info

**Article Type:** Research Article

https://doi.org/10.33902/JPR.2020465074

Journal of Pedagogical Research, 2020 - Volume 4 Issue 3, pp. 359-374

**Published Online: ** 13 Dec 2020

**Views: ** 351 | **Downloads: ** 245

Open Access

How to cite this article