ERG SES F 04, Parallel Session F 04
Content knowledge is considered as one of the important aspects of professional knowledge for teachers. Shulman (1986) describes content knowledge as “the amount and organization of knowledge per se in the mind of the teacher” (p.9). The fact that understanding the structure of the subject matter is more important than the knowledge of the facts or concepts of a domain is also emphasized (Shulman, 1987). It is widely accepted that teachers who have insufficient content knowledge tend to focus more on procedures rather than the development of conceptual understanding (Hill, Rowan & Ball, 2005). Therefore, whilst students become successful in operational understanding, they do not develop a sufficient relational understanding in mathematical concepts. Basing on this premise, exploring pre-service teachers’ understanding of mathematical concepts becomes a more important issue. In the present study, pre-service mathematics teachers’ conceptions of continuity of a function are aimed to be explored.
The concept of continuity is a fundamental concept for calculus. Research studies highlight that students have considerable difficulties in understanding the concept (Bezuidenhout, 2001; Cornu, 1991; Vinner, 1982). For instance students have difficulties in combining properties of continuity and functions (Wilson, 1994). They generally have a tendency to consider the connectedness of a graph as the continuity of a function (Cornu, 1991) and they think the existence of a limit at a point verify the continuity at the same point (Williams, 1991). It was also reported that students mainly focus on the value of a function at a given point instead of considering it as a function property (Bezuidenhout, 2001; Vinner, 1992). Hence students’ interpretation of continuity of a function as revealed by previous research is affected by; (i) representation type used, (ii) meaning of the word “continuity” in everyday language, (iii) the idea of “connectedness of a graph” held by students such as having breaks, jumps, or gaps, (iv) informal language used in the instruction of continuity (such as a graph having no ‘gaps’, being ‘all in one piece’). While research studies well documented students’ difficulties, more studies about the pre-service teachers’ conceptions of continuity are needed.
Bezuidenhout J (2001). Limits and continuity: Some conceptions of first year students. International Journal of Mathematics Education in Science and Technology 32(4): 487-500. Cornu B (1991). Limits. In D. Tall (Ed.), Advanced Mathematical Thinking (pp. 153- 166). Dordrecht: Kluwer Academic Publishers. Hill H, Rowan B, & Ball, D.L. (2005). Effects of teachers' mathematical knowledge for teaching on student achievement. American Educational Research Journal, 42(2): 371-406. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1-22. Vinner S (1992). The function concept as a prototype for problems in mathematics learning. In E. Dubinsky and G. Harel (Eds.), The Concept of Function: Aspects of Epistemology and Pedagogy. Washington, DC: Mathematical Association of America. pp. 195-213. Wilson MR (1994). One preservice secondary teacher's understanding of function: The impact of a course integrating mathematical content and pedagogy. Journal of Research in Mathematics Education 25(4): 346-370.
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