# Secondary School Students’ Conception of Quadratic Equations with One Unknown

### Abstract

In recent years, quadratic equations have started to become of greater interest to researchers. This study explored what conceptions high school students have about quadratic equations with one unknown using concept definition and concept images as theoretical concept. The data was gathered through semi-structured interviews with 14 eleventh grade high school students. Analysis of data displayed that students could not provide a proper definition of quadratic equations with one unknown, and their definitions were not consistent with the formal (standard) definition of quadratic equations. Moreover, the findings showed that students’ concept image of quadratic equation is quite limited and dominated by ideas factoring. Students’ conception of quadratic equations also showed that participating students lacked three types of prerequisite knowledge, degree of polynomial, variable and equals sign. To enrich students’ concept images both procedurally and conceptually, this study has implications for teachers. The implications of the findings are discussed.

### References

Authors (2015).

Block, J. (2015). Flexible algebraic action on quadratic equations. In K. Krainer, & N. Vondrová (Eds.), Proceedings of the ninth congress of the European Society for Research in Mathematics Education (pp. 391-397). Prague, Czech Republic: Charles University in Prague.

Clark, M. K. (2012). History of mathematics: Illuminating understanding of school mathematics concepts for prospective mathematics teachers. Educational Studies in Mathematics, 81, 67–84.

Corbin, J., & Strauss, A. (2008). Basics of qualitative research: Techniques and procedures for developing grounded theory. Thousand Oaks: Sage Publications

Fox, W., P. (1999). Quadratic connections. PRIMUS, 9(3), 275-278.

Gray, R., & Thomas, M. (2001). Quadratic equation representations and graphic calculators: Procedural and conceptual interactions. In J. Bobis, B. Perry & M. Mitchelmore (Eds.), Numeracy and beyond (Proceedings of the 24th Conference for the Mathematics Education Research Group of Australasia, pp. 257–264). Sydney: MERGA.

Gutiérrez, A., & Jaime, A. (1999). Preservice primary teachers’ understanding of the concept of altitude of a triangle. Journal of Mathematics Teacher Education, 2, 253-275.

Joarder, A., H. (2015) Factorising a quadratic expression with geometric insights. Australian Senior Mathematics Journal, 29(1), 25-31.

López, J., Robles, I., & Martínez-Planell, R. (2016). Students' understanding of quadratic equations. International Journal of Mathematical Education in Science and Technology, 47(4), 552-572.

Ministry of National Education [MoNE] (2013). Ortaöğretim matematik (9, 10, 11 ve 12. sınıflar) dersi öğretim programı. [Secondary mathematics curriculum for grades 9-12]. Ankara, Turkey.

Olteanu, C., & Holmqvist, M. (2012). Differences in success in solving second-degree equations due to the differences in classroom instruction. International Journal of Mathematical Education in Science and Technology, 43(5), 575-587.

Olteanu, C., & Olteanu, L. (2012). Equations, functions, critical aspects and mathematical communication. International Educational Studies, 5(5), 69-78.

Przenioslo, M. (2004). Images of the limit of function formed in the course of mathematical studies at the university. Educational Studies in Mathematics, 55, 103–132.

Radford, L., & Guérette, G. (1996). Second degree equations in the classroom: A Babylonian approach. In V. Katz (Ed.), Using history to teach mathematics: An international perspective (pp. 69–75). Washington: MAA.

Sağlam, R., & Alacacı, C. (2012). A comparative analysis of quadratics unit in Singaporean, Turkish and IMDP mathematics textbooks. Turkish Journal of Computer and Mathematics Education, 3(3), 131–147.

Tall, D., de Lima, R. N., & Healy, L. (2014). Evolving a three-world framework for soling algebraic equations in the light of what a student has met before. The Journal of Mathematical Behavior, 34, 1-13.

Tall, D., & Vinner, S. (1981). Concept images and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169.

Vaiyavutjamai, P., & Clements, M. A. (2006). Effects of classroom instruction on students’ understanding of quadratic equations. Mathematics Education Research Journal, 18(1), 47–77.

Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science and Technology, 14, 293-305.

Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function. Journal for Research in Mathematics Education, 20(5), 356–66.

Vinner, S. & Hershkowitz, R. (1980). Concept images and some common cognitive paths in the development of some simple geometric concepts. In R. Karplus (Ed.), Proceedings of the Fourth International Conference for the Psychology of Mathematics Education, (pp. 177–184). Berkeley: University of California.

Yanik, B., H. (2014). Middle school students’ concept image of geometric translation. Journal of Mathematical Behavior, 36, 33-50.

Zakaria, E., & Maat, M. S. (2010). Analysis of students’ error in learning of quadratic equations. International Education Studies, 3(3), 105–110.

Zaslavky, O. (1997). Conceptual obstacles in the learning of quadratic functions. Focus on Learning Problems in Mathematics, 19(1), 20-44.

Wawro, M., Sweeney, G., F., & Rabin, J. M. (2011). Subspace in linear algebra: investigating students’ concept images and interactions with the formal definition. Educational Studies in Mathematics, 78, 1–19.