A case study of the development of A.C.E. students' concept images of the derivative.
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This research focuses on the development of the concept images of the derivative concept of students enrolled in the in-service programme 'Advanced Certificate in Education' at University of KwaZulu-Natal, Pietermaritzburg campus. In addition, two qualified teachers not enrolled in the programme were included. A theoretical framework which describes the derivative as having three layers - the ratio, limit and function layers - that can be represented by a variety of representations - graphical, rate, physical and symbolic - is used to analyse the development of the students' concept images. This framework was adopted from previous research, but expanded to allow for situations where a student's concept image did not fall into any of the layers or representations. In those cases, the concept image was classified into the non-layer section or the instrumental understanding section. The findings of this research show that of the five ACE students who were interviewed, only one had a profound concept image in all the three layers of the derivative, with multiple representations as. well as connections among representations within the layers. This one student also passed the calculus module with a distinction. The other four students had the ratio layer and graphical representation profound in their concept images, while the other layers and representations were pseudostructural with very few connections. Two of these students passed the calculus module while the other two failed. All the students showed progression in their concept images, which can only be credited to the ACE calculus module. However, it is clear that even upon completion of this module, many practicing teachers have concept images of the derivative which are not encompassing all the layers and more than one or two representations. With the function layer absent, it can be difficult to make sense of maximization and minimization tasks. With the limit layer absent or pseudo-structural , the concept Itself and the essence of calculus escapes the teachers - and therefore also will be out of reach of their learners.