Pre-Service Mathematics Teachers’ Understanding of Quadrilaterals and the Internal Relationships between Quadrilaterals: The Case of Parallelograms

Tugce Kozakli Ulger, Menekse Seden Tapan Broutin

APA 6th edition
Ulger, T.K., & Broutin, M.S.T. (2017). Pre-Service Mathematics Teachers’ Understanding of Quadrilaterals and the Internal Relationships between Quadrilaterals: The Case of Parallelograms. European Journal of Educational Research, 6(3), 331 - 345. doi:10.12973/eu-jer.6.3.331

Ulger T.K., and Broutin M.S.T. 2017 'Pre-Service Mathematics Teachers’ Understanding of Quadrilaterals and the Internal Relationships between Quadrilaterals: The Case of Parallelograms', European Journal of Educational Research , vol. 6, no. 3, pp. 331 - 345. Available from:

Chicago 16th edition
Ulger, Tugce Kozakli and Broutin, Menekse Seden Tapan . "Pre-Service Mathematics Teachers’ Understanding of Quadrilaterals and the Internal Relationships between Quadrilaterals: The Case of Parallelograms". (2017)European Journal of Educational Research 6, no. 3(2017): 331 - 345. doi:10.12973/eu-jer.6.3.331


This study attempts to reveal pre-service teachers’ conceptions, definitions, and understanding of quadrilaterals and their internal relationships in terms of personal and formal figural concepts via case of the parallelograms. To collect data, an open-ended question was addressed to 27 pre-service mathematics teachers, and clinical interviews were conducted with them. The factors influential on pre-service teachers’ definitions of parallelograms and conceptions regarding internal relationships between quadrilaterals were analyzed. The strongest result involved definitions based on prototype figures and partially seeing internal relationships between quadrilaterals via these definitions. As a different result from what is reported in the literature, it was found that the fact that rectangle remains as a special case of parallelogram in pre-service teachers’ figural concepts leads them not to adopt the hierarchical relationship. The findings suggested that learners were likely to recognize quadrilaterals by a special case of them and prototypical figures, even though they knew the formal definition in general. This led learners to have difficulty in understanding the inclusion relations of quadrilaterals.

Keywords: Internal relationships between quadrilaterals, personal-formal-figural concept, pre-service mathematics teachers.


Akkoc, H. (2008). Pre-service mathematics teachers’ concept images of radian. International Journal of Mathematical Education in Science and Technology, 39(7), 857-878.

Baki, A., Karatas, I. & Guven, B. (2002). Klinik mulakat yontemiyle problem cozme becerilerinin degerlendirilmesi. V. Ulusal Fen Bilimleri ve Matematik Egitimi Kongresi, 15-18 Eylul, Ankara

Brousseau G. (1988). Le contrat didactique: le milieu, Recherches en Didactique des Mathématiques, vol. 9/3, 309-336.

Clements, D. H., & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 420-464). New York: Macmillan

Craine, T. V., & Rubenstein, R. N. (1993). A quadrilateral hierarchy to facilitate learning in geometry. The Mathematics Teacher86(1), 30-36.

Creswell, J. W. (2007). Qualitative inquiry and research design: Choosing among five approaches. Sage.

De Villiers, M. (1994). The role and function of a hierarchical classification of quadrilaterals. For the Learning of Mathematics, 14, 11-18.

De Villiers, M. (1998) To teach definitions in geometry or teach to define? Proceedings of the 22nd Conference of the International Group for the Psychology of Mathematics Education, 2, 248−255.

De Villiers, M. (2004). Using Dynamic Geometry to Expand Mathematics Teachers’ Understanding of Proof. International Journal of Mathematical Education in Science and Technology.35( 5).703–724

Erdogan, E. O. & Dur, Z. (2014). Preservice mathematics teachers’ personal figural concepts and classifications about quadrilaterals, Australian Journal of Teacher Education, 39(6), 106 – 133.

Erez, M. M., & Yerushalmy, M. (2006). “If You Can Turn a Rectangle into a Square, You Can Turn a Square into a Rectangle...” Young Students Experience the Dragging Tool. International Journal of Computers for Mathematical Learning11(3), 271-299.

Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24 (2),139- 162.

Fischbein, E., & Nachlieli, T. (1998). Concepts and figures in geometrical reasoning. International Journal of Science Education, 20(10), 1193–1211

Forsythe, S. K. (2015). Dragging maintaining symmetry: can it generate the concept of inclusivity as well as a family of shapes? Research in Mathematics Education, 17(3), 198–219.

Foster, C. (2014). Being inclusive. Mathematics in School, 43(3), 12–13.

Fujita, T., & Jones, K. (2006). Primary trainee teachers’ understanding of basic geometrical figures in Scotland. In Proceedings of the 30th Conference of the International Group for PME (pp. 129-136).

Fujita, T. & Jones, K. (2007). Learners’ understanding of the definitions and hierarchical classification of quadrilaterals: towards a theoretical framing. Research in Mathematics Education, 9(1&2), 3-20.

Fujita, T. (2012). Learners’ level of understanding of the inclusion relations of quadrilaterals and prototype phenomenon. The Journal of Mathematical Behavior, 31(1), 60-72.

Govender, R., & De Villiers, M. (2004). A dynamic approach to quadrilateral definitions. Pythagoras, 58, 34–45.

Hancock, B. (2002). An Introduction to Qualitative Research,

Hershkowitz, R. (1989). Visualization in geometry: Two sides of the coin. Focus on Learning Problems in Mathematics, 11(1), 61–76

Hershkowitz, R. (1990). Psychological aspects of learning geometry. In P. Nesher & J. Kilpatrick (Eds.), Mathematics and cognition (pp. 70-95). Cambridge: Cambridge University Press.

Kaur, H. (2015). Two aspects of young children’s thinking about different types of dynamic triangles: prototypicality and inclusion. ZDM Mathematics Education, 47(3), 407–420.

Koseki, K. (Ed.). (1987). The teaching of geometrical proof. Tokyo: Meiji Tosho Publishers (in Japanese).

Levenson, E., Tirosh, D., & Tsamir, P. (2011). Preschool geometry. Theory, research, and practical perspectives. Rotterdam: Sense Publishers.

Linchevski, L., Vinner, S. & Karsenty, R. (1992). To be or not to be minimal? Student teachers' views about definitions in geometry. Proceedings of PME 16 (New Hampshire, USA), Vol 2, pp. 48-55.

Monaghan, F. (2000) What difference does it make? Children views of the difference between some quadrilaterals, Educational Studies in Mathematics, 42(2), 179–196.

Okazaki, M., & Fujita, T. (2007). Prototype phenomena and common cognitive paths in the understanding of the inclusion relations between quadrilaterals in Japan and Scotland. In Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 41-48).

Okazaki, M. (2009). Process and means of reinterpreting tacit properties in understanding the inclusion relations between quadrilaterals. In: Tzekaki, M., Kaldrimidou, M., & Sakonidis, C., (Eds.), Proceedings of the 33rd conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 249–256). Thessaloniki, Greece

Patton, M. Q. (1990). Qualitative evaluation and research methods. SAGE Publications, inc.

ISO 690 Pusey, E. L. (2003). The Van Hiele Model of Reasoning in Geometry: A Literature Review. Mathematics Education Raleigh, Master of Science Thesis, Retrieved 3 February from

Sinclair, N., Bussi, M. G. B., de Villiers, M., Jones, K., Kortenkamp, U., Leung, A., & Owens, K. (2016). Recent research on geometry education: an ICME-13 survey team report. ZDM48(5), 691-719.

Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12( 2), 151-16.

Tall, D. O., Thomas, M. O. J., Davis, G., Gray, E. M., & Simpson, A. P. (2000). What is the object of the encapsulation of a process? Journal of Mathematical Behavior, 18(2), 1 - 19.

Tall, D., Gray, E., Ali, M. B., Crowley, L., DeMarois, P., McGowen, M., …  Yusof, Y. (2001). Symbols and the bifurcation between procedural and conceptual thinking. Canadian Journal of Math, Science & Technology Education, 1(1), 81–104. 

Turnuklu, E., Alayli, F. G. ve Akkas, E. N. (2013). Investigation of prospective primary mathematics teachers’ perceptions and images for quadrilaterals, Educational Sciences: Theory & Practice, 13(2), 1213-1232.

Vinner, S. & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for research in mathematics education, 356-366.

Vinner, S. (1991). The role of definitions in the teaching and learning of mathematics. In D. Tall, (Ed.), Advanced mathematical thinking (pp. 65–81). Dordrecht: Kluwer Academic.

Yildirim, A. & Simsek, H. (2011). Sosyal Bilimlerde Nitel Arastirma Yontemleri (8.baski). Ankara: Seckin Yayincilik.

Zandieh, M. & Rasmussen, C. (2010). Defining as a mathematical activity: a framework for characterizing progress from informal to more formal ways of reasoning, JMB, 29, 55–75.