Investigation of Pre-Service Teachers’ Pedagogical Content Knowledge Related to Division by Zero

  • Fatih Karakus Afyon Kocatepe University


Although the topic of division by zero has been widely discussed in the literature, this subject is still confusing for students, pre-service teachers and teachers. Because of this, teachers at various grade levels may encounter difficulties in conveying the concept to their students. Therefore, in order to provide students with a strong conceptual understanding of division by zero, it is important to examine the ways that teachers and pre-service teachers structure their instructional explanations about division by zero. The aim of this study was to explore the instructional explanations given by pre-service teachers concerning division by zero, as well as the effects of teacher training programs on these explanations. The study consisted of a cross-sectional design and was carried out with 197 pre-service teachers of elementary mathematics. To determine the pre-service teachers’ instructional explanations given at different grade levels, a written questionnaire was used. Analysis of the results revealed that although most of the pre-service teachers gave correct answers related to division by zero, few of them provided conceptual-based explanations. Rather, those who gave correct explanations mainly responded with rule-based statements.


Anthony, W., & Walshaw, M. A. (2004). Zero. Teaching Children Mathematics, 11(1), 38.
Bağcı, O. (2015). Ortaokul matematik 7 ders kitabı. Ankara: Tutku Yayıncılık.
Baki, M. (2013). Pre-service classroom teachers' mathematical knowledge and instructional explanations associated with division. Egitim ve Bilim-Education and Science, 38(167), 300-311.
Ball, D. L. (1990). Prospective elementary and secondary teachers’ understanding of division. Journal for Research in Mathematics Education, 21(2), 132-144.
Brousseau, G. (2002). Epistemological obstacles, problems, and didactical engineering. Theory of Didactical Situations in Mathematics: Didactique des Mathématiques, 1970–1990, 79-117.
Bütün, M. (2012). İlköğretim matematik öğretmeni adaylarının uygulanan zenginleştirilmiş program sürecinde matematik öğretme bilgilerinin gelişimi. Unpublished doctoral dissertation, Karadeniz Technical University, Institute of Educational Sciences. Trabzon.
Cankoy, O. (2010). Mathematics Teachers' Topic-Specific Pedagogical Content Knowledge in the Context of Teaching a0, 0! and a÷0. Educational Sciences: Theory and Practice, 10(2), 749-769.
Charalambous, C. Y., Hill, H. C., & Ball, D. L. (2011). Prospective teachers’ learning to provide instructional explanations: how does it look and what might it take?. Journal of Mathematics Teacher Education, 14(6), 441-463.
Cornu, B. (1991). Limits. In Tall, D. Advanced mathematical thinking (pp. 153-166). Dordrecht: Springer.
Crespo, S., & Nicol, C. (2006). Challenging preservice teachers' mathematical understanding: The case of division by zero. School science and mathematics, 106(2), 84-97.
Creswell, J. W. (2012). Qualitative inquiry and research design: Choosing among five approaches (3rd ed.). Thousand Oaks: Sage.
Çelik, D., & Akşan, E. (2013). Preservice mathematics teachers’ perceptions of infinity, indeterminate and undefined. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 7(1), 166-190.
Dede, Y., & Karakuş, F. (2014). The effect of teacher training programs on pre-service mathematics teachers’ beliefs towards mathematics. Educational Sciences: Theory & Practice, 14(2), 804-809.
Even, R. (1990). Subject matter knowledge for teaching and the case of functions. Educational Studies in Mathematics, 21(6), 521-544.
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.
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.
Fennema, E., & Franke, M. L. (1992). Teachers' knowledge and its impact. In D. A. Grouws, (Ed), Handbook of research on mathematics teaching and learning: A project of the National Council of Teachers of Mathematics (pp. 147-164). New York: Macmillan.
Field, A. (2009). Discovering statistics using SPSS (third edition). London: SAGE.
Fraenkel, J. R., Wallen, N. E., & Hyun, H. H. (2012). How to design and evaluate research in education (8th ed.). New York: McGram-Hill.
Green, S.B. & Salkind, N.J. (2005). Using SPSS for Windows and Macintosh analyzing and understanding data (4th ed.). Mahwah, NJ: Pearson Prentice Hall.
Gürbüz, R., Erdem, E., & Gülburnu, M. (2013). Sınıf öğretmenlerinin matematik yeterliklerini etkileyen faktörlerin incelenmesi. Ahi Evran Üniversitesi Kırşehir Eğitim Fakültesi Dergisi, 14(2), 255-272.
Henry, B. (1969). Zero, the troublemaker. The Arithmetic Teacher, 16(5), 365-367.
Işıksal, M. (2006). A study on pre-service elementary mathematics teachers’ subject matter knowledge and pedagogical content knowledge regarding the multiplication and division of fractions. Unpublished doctoral dissertation. Middle East Technical University, Ankara, Turkey.
Kinach, B. M. (2002). A cognitive strategy for developing pedagogical content knowledge in the secondary mathematics methods course: toward a model of effective practice. Teaching and Teacher Education, 18(1), 51-71.
Leinhardt, G. (1990). Towards understanding instructional explanations. Retreived March, 10, 2017 from ERIC database.
Leinhardt, G. (2001). Instructional explanations: A commonplace for teaching and location for contrast. Handbook of research on teaching, 4, 333-357.
Leinhardt, G. (2010). Introduction: explaning instructional explanation. M.K. Stein and L. Kucan (Ed.) Instructional explanations in the disciplines. NY: Springer.
Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. Mahwah, NJ: Erlbaum.
McMillan, J. H., & Schumacher, S. (2006). Research in education. Evidence-based inquiry (6th ed). Boston: Pearson Education.
Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). Thousand Oaks: Sage.
Özmantar, M. F. (2008). Sonsuzluk kavramı: Tarihsel gelişimi, öğrenci zorlukları ve çözüm önerileri. In MF Özmantar, E. Bingölbali ve H. Akkoç. Matematiksel kavram yanılgıları ve çözüm önerileri, (pp.151-180). Ankara: Pegem Akademi.
Quinn, R. J., Lamberg, T. D., & Perrin, J. R. (2008). Teacher perceptions of division by zero. The Clearing House: A Journal of Educational Strategies, Issues and Ideas, 81(3), 101-104.
Rea, L. M., & Parker, R. A. (2014). Designing and conducting survey research: A comprehensive guide. John Wiley & Sons.
Reys, R. E., & Grouws, D. A. (1975). Division involving zero: Some revealing thoughts from interviewing children. School science and mathematics, 75(7), 593-605.
Richardson, V. (1996). The role of attitudes and beliefs in learning to teach. In J. Sikula (Ed.), Handbook of research on teacher education (2nd ed.), (pp. 102-119). New York: Macmillan.
Russell, G., & Chernoff, E. J. (2011). Seeking more than nothing: Two elementary teachers conceptions of zero. The Mathematics Enthusiast, 8(1), 77-112.
Sanders, S. E., & Morris, H. J. (2000). Exposing student teachers’content knowledge: Empowerment or debilitation? Educational Studies, 26(4), 397-408.
Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational researcher, 15(2), 4-14.
Skemp, R. R. (1976). Relational understanding and instrumental understanding. Mathematics teaching, 77(1), 20-26.
Thanhieser, E. (2009). Preservice elementary school teachers’ conceptions of multidigit whole numbers. Journal for Research in mathematics Education, 40, 252-281.
Thompson, A. G. (1992). Teachers' beliefs and conceptions: A synthesis of the research. Macmillan.
Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children's conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 31(1), 5-25.
Toluk Uçar, Z. (2011). Öğretmen adaylarının pedagojik içerik bilgisi: öğretimsel açıklamalar. Turkish Journal of Computer and Mathematics Education, 2(2), 87-102.
Tsamir, P., & Tirosh, D. (2002). Intuitive beliefs, formal definitions and undefined operations: Cases of division by zero. In G. C. Leder, E. Pehkonen, & G. Törner, G. (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 331-344). Dordrecht: Springer.
Tsamir, P., & Sheffer, R. (2000). Concrete and formal arguments: The case of division by zero. Mathematics Education Research Journal, 12(2), 92-106.
Tsamir, P., Sheffer, R., & Tirosh, D. (2000). Intuitions and undefined operations: The cases of division by zero. Focus on Learning Problems in Mathematics, 22(1), 1-16.
Watson, J. M. (1991). Models to show the impossibility of division by zero. School Science and Mathematics, 91(8), 373-376.
Wheeler, M. M., & Feghali, I. (1983). Much ado about nothing: Preservice elementary school teachers' concept of zero. Journal for Research in Mathematics Education, 14(3), 147-155.