Conceptualising the function concept : an image functions intervention

Abstract

This thesis comprises three articles submitted for publication, sandwiched between an orientation chapter and a final reflection. The thread that runs through the thesis and connects the three articles is the mathematical concept of function. Students studying Mathematics are expected to have a good understanding of the function concept and its many subtleties and nuanced representations. However, the contrary has been shown to be true by ample research over the last 50 years in countries all over the world. The first article gave details on how photographs or images could be considered as representations of functions. This article used APOS (Action-Process-Object-Schema) theory to determine a genetic decomposition (GD) for the function concept. From this GD, activities were designed for an intervention based on defining and working with image functions,that is working with images/photographs as functions. The Image Functions Intervention (IFI) was then analysed from the APOS theoretical perspective and shown to adhere to the mental structures determined in the GD of the function concept. A first use of the IFI was evaluated by means of a questionnaire and qualitative analysis. The conclusion was that the IFI led to a broadened concept image, specifically regarding what can constitute as a function. The second article analysed the effectiveness of the IFI by means of quantitative analyses. In a randomised control design, an experimental and a control group both completed the Function Concept Inventory (FCI) as a pre-test. The experimental group completed the IFI and then both groups completed the FCI again as post-test. The experimental group showed a significant increase in their scores after the intervention. However, as there was no significant difference between the post-test scores of the experimental and the control groups, it could not be concluded unequivocally that the IFI caused the observed improvements. The last article used qualitative analysis with three data instruments to again gauge the possible effects of the IFI. Specifically, the main objective was to investigate to what extent the IFI could assist participants to develop an object conception of functions. Although multiple participants showed improvement on their understanding, only one participant managed to display a transitioning into the object level of understanding. Overall, the IFI showed merit and it was concluded that the IFI should be adapted and expanded based on the results and conclusions from the three articles. Further research should be undertaken to evaluate and explore the use of the IFI regarding the improvement of function concept understanding.