The vacuum provided by the gap between the demands of the 2 year course and the old 4 year course provided us with a great deal of freedom. Briefly, we decided that the level 1 geometry unit would focus upon some aspects of the geometry of triangles, which would enable the students to develop their knowledge and skills of vector geometry and allow them to revisit many properties from their school experience and to use these in a more demanding context. The level 2 unit would be organised around conics and inversion with an explicit attempt to deal with these aspects with as little appeal as possible to the algebraic aspects of co-ordinate geometry. The fact that there was little previous experience within the mathematics section of teaching these units allowed us considerable freedoms in defining what was to be offered to students and the manner in which we worked. The methodology which I adopted in this study was based upon and supported by that described by John Mason which he has referred to as 'The Discipline of Noticing' and 'Researching from the inside'.
Emphasis is on systematic reflection, as distinct from merely thinking about the event.
The accounts of moments of noticing provide a record of the threads of experience which are most salient for that teacher at that time. Once a corpus of accounts begins to accumulate, it is possible to look back over the accounts to seek common threads. There is a back and forth process of keeping accounts of moments, and looking for threads in those moments. Accounts provide data for detecting threads, and threads detected will tend to influence the subsequent moments which are noticed. Elsewhere, as part of a separate project with Una Hanley and Tansy Hardy, I have written of the difficulty for practitioner research of the production of data.