### Mathematics education undergraduates’ personal definitions of the notion of angle of contiguity in Kinematics

Zakaria Ndemo^{(1*)}, David Mtetwa

^{(2)}

(1) Department of Science and Mathematics Education, Bindura University of Science Education

(2) Department of Science and Mathematics Education, University of Zimbabwe

(*) Corresponding Author

#### Abstract

The concept of a mathematical definition causes severe difficulties among students during problem solving and proving activities. Students’ difficulties with the use of mathematical definitions often arise from the fact that students are often given those definitions instead of constructing them. With the aim of developing an understanding of the kinds of student teachers evoked concept images of the notion of angle of contiguity, a qualitative case study was conducted at one state university in Zimbabwe. Purposive sampling was used to select 28 mathematics undergraduate student teachers who responded to a test item. Qualitative data analysis was guided by ideas drawn from the theoretical framework of Abstraction in Context and idea of imperative features of a mathematical definition. Student teachers written responses revealed that student teachers personal concept definitions consisted of ambiguous and irrelevant formulations that did not capture the essence of the idea of the angle of contiguity. In some cases their responses were not consistent with the definition of the angle of contiguity. Although there were a few instances of adequate descriptions of the concept, (8 out of 32) these and the inadequate descriptors elicited can contribute significantly towards efforts intended to improve mathematics instruction. Improved mathematics instruction will lead to enhanced conceptualizations of mathematics concepts.

#### Keywords

#### Full Text:

PDF#### References

Baxter, P., & Jack, S. (2008). Qualitative case study methodology: Study design and implementation for novice research. The Research Report, 13(4), 544-559.

Berg, B.L. (2009). Qualitative research methods for the social sciences Topics. Boston: Allyn Brown.

CadawalladerOlsker, T. (2011). What do we mean by mathematical proof. Journal of humanistic mathematics, 1(1), 1-33.

Carlson, M.P., Bloom, I. (2005). The cyclic nature of problem solving: an emergent multidimensional problem-solving-framework. Educational Studies in Mathematics, 58, 58-75.

Corbin, J., & Strauss, A. (2008). Basics of qualitative research. Thousand Oaks: Sage.

Creswell, J.W. (2014). Research design: Qualitative, quantitative and mixed methods approaches. London: Sage.

Davydov, V.V. (1990). Types of generalization in instruction: Logical and psychological problems in structuring school curricula. In J. Patrick (Ed), Soviet Studies in Mathematics Education. Reston: National Council of Teachers for Mathematics (NCTM).

Dreyfus, T., Hoyles, C., Gueudet, G., & Krainer, K. (2014). Solid findings: concept images in students mathematical reasoning. European Mathematical Society Newsletter.

Edwards, B., & Ward, M.B. (2004). Surprises from mathematics education research: students (mis) use of mathematical definitions. American Mathematical Monthly, 411-424.

Gilboa, N., Kidron,I., & Dreyfus, T. (2019). Constructing a mathematical definition: the case of the tangent. International Journal of Mathematics Education in Science and Technology, 50(3), 421-436. https://doi.org/10.1080/0020739X.2018.1516824

Harel, G., & Sowder, L. (1998). Students proof schemes: Results from exploratory studies. In

Schoenfeld, A., Kaput, J., and Dubinsky, E., (Eds.), Research in Collegiate Mathematics Education: Vol. 3. (pp. 234-282). Washington, DC: American Mathematical Society.

Jamillah, Suryadi, D., & Priatna, N. (2019). Students concept images on set: zone of differences between concept image and formal concept definition. International Journal of Advanced Science and Technology, 28(18), 156-166.

Kidron, I. (2011). Constructing knowledge about the notion of limit in the definition of the horizontal asymptote. International Journal of Science and Mathematics Education, 9, 1261-1279. https://doi.org/10.1007/s10763-010-9258-8

Maxwell, J.A., & Mittapalli, K. (2010). Realism as a stance for mixed methods research. In A. Tashakkori and T. Teddlie (Eds), Sage handbook of mixed methods in social behavioural research (pp.145-166). Washington DC: Sage Publications.

Mariotti, M.A., & Fischbein, E. (1997). Defining in classroom activities. Educational Studies in Mathematics, 34, 58-69. https://doi.org/10.1023/A:1002985109323

Miyazaki, M., Fujita, T., & Jones, K. (2017). Students understanding of the structure of a deductive proof. Educational Studies in Mathematics, 94, 223-239. https://doi.org/10.1007/s10649-016-9720-9

Nurwahyu, B. (2014). Concept image definition of students concept understanding. In M. A., Futhani and T. Permadi (Eds), Proceedings of the International Seminar on Mathematics Education and Graph Theory, Islamic University of Malang: Malang.

Pimm, D. (1993). Just a matter of a definition. Educational Studies in Mathematics, 25, 261-277.

Punch, K. F. (2005). Introduction to social science research: Quantitative and qualitative approaches. London: Sage.

Schoenfeld, A.H. (1982). Some thoughts on problem solving research and mathematics education. In F.K. Lester & J. Garofalo (Eds.), Mathematical problem solving: Issues in Research. Franklin Institute Press, Pennsylvania, pp. 27-37.

Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal forResearch in Mathematics Education, 34, 4–36.

Ubuz, B., Dincer, & Bülbül, A. (2013). Argumentation in undergraduate math courses: A study on definition construction. In A.M. Lindmeier, & A. Heinze (Eds.), Proceedings of the 37th Conference of International Group for the Psychology of Mathematics Education: Vol. 4. (pp. 313-320). Kiel University: PME.

VanDormolen, J., & Zaslavsky, O. (2003). The many facets of a definition: the case of a definition: The case of periodicity. Journal of Mathematical Behaviour, 22, 91-106. https://doi.org/10.1016/S0732-3123(03)00006-3

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

Vinner, S., & Hershwitz, R. (1980). Concept images and common cognitive paths in the development of some simple geometric concepts. In R. Karplus (Ed.), Proceedings of the 4th Conference of the International Group for the Psychology of Mathematics Education: (pp. 177-184). Berkeley, CA: PME.

Wilkerson-Jerde, M., & Wilensky, U.J. (2011). How do mathematicians learn math?: resources and acts for constructing and understanding mathematics. Educational Studies in Mathematics, 78, 21-43.

Wilson P. S. (1990). Inconsistent ideas related to definitions and examples. Focus on Learning Problems in Mathematics, 12(3-4), 31-47.

Yin, R. K. (2009). Case study research: design and methods. Thousand Oaks: Sage.

Zaslavsky, O., & Shir, K. (2005). Students conceptions mathematical definition. Journal for Research in Mathematics Education, 36 (4), 317-346. https://doi.org/10.2307/30035043

#### Article Metrics

Abstract view(s): 303 time(s)PDF: 281 time(s)

### Refbacks

- There are currently no refbacks.