Interlude: An Introduction to Chapters 6-9

Throughout the study the focus was balanced between the topical — that is, related to specific mathematical areas or concepts — and the cross-topical — that is, running through several mathematical areas — aspects of the novices' cognition. The arrangement of the analysis in five topical areas (Foundational Analysis, Calculus, Topology, Linear Algebra, Group Theory) is the overt evidence of the topical focus. Less overtly, this arrangement illustrates metaphorically the novices' journey from formal arithmetic (Foundational Analysis) to the advanced arithmetic of Calculus (enriched with the manipulation of the infinitesimally small or large quantities of the limiting process), then the transition from the numerical to the generalised set-theoretical contexts of Topology and Linear Algebra and, finally, to the total abstraction of Group Theory. This journey was also approximately chronological: the material on the Foundational Analysis comes from the first weeks of observation, whereas the material from Group Theory comes from the last.

The general intention of the study was to carry on with this balanced view between the topical and the cross-topical aspects to the end. As a result, within each area, emerged the, what I call, paradigmatically problematic concepts, that is the concepts towards which most of the students' cognitive concern and difficulty seemed to converge during observation. These were:

and were further explored in the interviews which were structured around reinforcing the evidence provided about them in the tutorials.

However, due to the limitations of space in the thesis this balanced view between the topical and the cross-topical aspects of the novices' cognition had to be abandoned. As a result, the selection of Episodes in Chapters 6-9, while keeping the topical structure of the initial data analysis, is geared towards the presentation of the cross-topical themes that emerged from — and directed — the various stages of analysis, rather than the specific learning difficulties within each topic. I note that some topical aspects of the analysis feature in (Nardi, 1994) and (Nardi, in press). Moreover, again due to the limitations of space, the topical area of Topology was left out of the presentation in the thesis: in the above described continuum (from Foundational Analysis to Group Theory) Topology – along with Linear Algebra – are the two middle areas in terms of the transition from the numerical to the abstract. Having to choose between Topology and Linear Algebra, the extent and strength of the analysed material on the latter and the possible elimination of the former from the first year undergraduate mathematics curriculum in subsequent years, resulted in leaving Topology out.

The Episodes presented in Chapters 6-9 should be read as paradigmatic cases of the cross-topical themes of which a synthesis is presented in Chapter 10 – that is, unless otherwise specified, the Episodes represent trends in the analysed material. Even the description of idiosyncratic cases serves the purpose of accentuating these trends. It is due to the richness and complexity of the naturalistic data collected in this study that the topical and the cross-topical elements are so intertwined that their distinction at times collapses – for instance, within the discourse on the novices' image constructing of new concepts. The topical analytical discourse has been toned down and kept only to the levels in which it serves the development of the cross-topical analytical discourse.

The themes, exemplified in Chapters 6-9 and elaborated upon in Chapter 10, can be concisely described here as features of the novice's encounter with mathematical abstraction. This encounter is seen in this study as an enculturation/cognitive process. The new culture is Advanced Mathematics and it is introduced by an expert mathematician, the tutor. The themes around which the analysis revolves relate to

The above outlined enculturation/mental process into the reasoning mode of Advanced Mathematical Thinking was moreover strongly determined by the teaching environment within which it was observed, the tutorial. Finding that

some of the analysis focused on elaborating the effect of this style on the novices' cognition.

Each Chapter contains 8 Episodes. In Part I of each Chapter a table is provided with a summary of relevant information on the Episodes. In Part II, each Episode is presented in one Section. Each Section consists of

In Part III, I present a concise synthesis of the focal points that were highlighted in the analysis of each Episode. The syntheses in Parts III of Chapters 6-9 constitute the intermediate theorising stage between the presentation of the paradigmatical cases, the Episodes, and the global theorising of Chapter 10. I note that the didactical (relating to teaching) and methodological observations made in the Episodes are synthesised directly in Chapter 10 and not in Parts III of Chapters 6-9.

Throughout the analysis the relevant mathematical material — that is the mathematical questions discussed in the Episodes — has been coded and abbreviated as follows. For example:

means

Problem Sheet Number 2- Course: Continuity and Differentiability - Question 1

means

means

Problem Sheet Number 7 Course: Series and Sequences- Question 1

means

means

In the following table I list the questions from the Michaelmas and Hilary Term problem sheets referred to in these Episodes:

These questions — and their answers — can be found in Appendices For Chapter 6, 7, 8 and 9. The answers to the mathematical problems provided in the Appendices are the solutions — that I have reconstructed either from the fieldnotes or the recordings — discussed in the tutorials.