This study is embedded in the area of research on the Psychology of Mathematics Education currently known as Advanced Mathematical Thinking and is an inquiry on the novice mathematicians' cognition with regard to their encounter with mathematical abstraction in their first year of university studies. It is a qualitative piece of research and its scope covers a wide range of mathematical topics as well as of learning phenomena related to the cognition of advanced mathematics. The theoretical background of the study, regarding research on PME-AMT and Methodology, is presented in Chapters 1 and 2.

Work on the Pilot Study — presented in Chapter 3 — was initiated in Michaelmas 1992. The Pilot was carried out in parallel with the Educational Research Methodology Training for the Research Students of the Department of Educational Studies and lasted until Trinity Term 1993. During this time tutorials to ten first-year undergraduates in one Oxford College were observed. The analysis of the fielnotes kept during observation revealed the potential richness of tutorials as a source of evidence related to the cognition of advanced mathematics.

In the first two weeks of Michaelmas 1993, negotiations with the Mathematical Institute and several Colleges — tutors and students — led to the formation of the body of participants to the Main Study: four tutors and twenty first-year undergraduates. Observation and recording of the tutorials lasted two terms: Michaelmas 1993 and Hilary 1994. During this time the students were also interviewed twice. The procedures of Data Collection are presented in Chapter 4.

Analysis of the collected material started during Data Collection: generally it aimed at the extraction of learning episodes related to the novices' conceptual and reasoning difficulties in their encounter with mathematical abstraction in a range of mathematical topics and at a cross-topical synthesis on these difficulties. Transcribing of the recordings was completed during Trinity and Michaelmas 1994 and the analysis of the material was completed between HIlary 1995 and the submission of the thesis. The procedures of Data Analysis are presented in Chapter 5.

The part of data analysis presented here — and introduced in the Interlude between Chapter 5 and Chapter 6 — is grounded on a selection of learning episodes that are paradigmatical cases of the themes that emerged during analysis. The presentation — Chapters 6-9 — is topical: so Chapter 6 contains a selection of episodes from the area of Foundational Analysis, Chapter 7 from Calculus, Chapter 8 from Linear Algebra and Chapter 9 from Group Theory. The cross-topical synthesis of the findings is presented in Chapter 10 and in the Epilogue I indicate briefly how the themes that emerged in this study may lead to more focused research.

I note that in the Appendices  — that are provided in a separate volume — I present Extracts from the data, as well as other auxiliary material, on which the analysis in Chapters 6-9 is based.

Due to the limitations of space in the thesis, the material presented here is only a part of the collected material: so, for instance, the material relevant to Topology — the fifth area of mathematics explored in this study in addition to the aforementioned four — has been left out. However, as mentioned above and explained in detail in the Methodology Chapters and the Interlude, the material presented here is representative of the collected material.