Using example generation to explore undergraduates conceptions of real sequences: a phenomenographic study

This thesis uses an example generation task to explore undergraduate students' understanding of basic sequence properties in Real Analysis. First, a review of the literature looks at three areas of research: the transition to studying mathematics at the tertiary level, examples and the process of example generation, and the learning of Real Analysis. It notes a lack of research on how students interact with simpler de nitions in Analysis, and suggests that an example generation task is an ideal research tool for this purpose. Then, two pilot studies are reported. The rst gave 101 students an example generation task during a lecture. In this task, students were asked to generate examples of sequences that satis ed certain combinations of properties. In the second pilot study a similar task was given to six students in an interview setting with a `think-aloud' protocol. These pilot studies found that many students gave sequences that did not satisfy the requested properties, whilst other students gave examples that were not sequences. The thesis then reports on a main study in which the example generation task was completed by 15 students during an interview, and 147 students during classes. The interview data is analysed phenomenographically, with results presented along four dimensions of variation, where each dimension describes di erent ways of experiencing an aspect of sequence example generation: Using De nitions, Representation of Sequences, Sequence Construction Strategies, and Justi cations. The larger-scale class data is then analysed by Rasch Analysis to objectively rank the questions in order of their di culty, and to show that the interview-based responses re ect those in the wider cohort. By asking students to generate their own examples of sequences, this thesis has furthered what is known about student understanding in two areas. The rst area is how students understand content related to sequences in Analysis. The thesis considers students' understanding of how sequences can be represented, how sequence property de nitions can be combined and how de nitions a ect sequences in di erent ways. The second area is how students interact with example generation tasks, the approaches that are e ective when students are trying to generate examples, and the ways students justify or check their answers