Students understanding of the fundamental theorem of calculus : an exploration of definitions, theorems and visual imagery

The aim of this research was to investigate students' understanding of the Fundamental Theorem of Calculus (FTC). The FTC was chosen as the basis of this research because it is one of the most important topics taught in calculus, establishing the link between the concepts of differentiation and integration. Data was collected from first year undergraduate students at the Federal University of Rio de Janeiro, in Brazil. The sample comprised students from three areas: mathematics, computer sciences and engineering. A pilot study was applied in 1994 and the main study in 1995 and 1996. Questionnaire, interviews based on responses to the questionnaire and computer-task based interviews were used. The data were analysed using both quantitative and qualitative methods. The results show that some of the students' obstacles to understanding the FTC are related to difficulties with the concepts of function, continuity, derivative and integral. The definitions of these concepts were not clear in their minds and they frequently made use of images that contained only partial aspects of the definitions or were based on some particular examples. This hampered the students when they met new examples that did not fit with their pre-formed images. It was also found that definitions and theorems were so fragmented in the students' minds that there was no way they could appreciate a proof, exemplified by the proof of the FTC. Their conceptions of proof reflected the fact that they were not used to thinking of proving as fundamental to generalising a proposition, and in examining a proof, it was difficult for them to see the central ideas behind it. These results are closely associated with students' habits: they tend not to pay attention to theoretical aspects, memorising algorithmic procedures without reflecting on their applicability