DG11, ICME-10 Paul Andrews _____________________________________________________________________________________
metaphor in accordance with our desire to illuminate. If the classroom is considered a fabric then the light we need must not be too sharply focused or the beam will highlight only individual threads and show nothing of the fabric's pattern. If the beam is too wide then all it does is alert the observer to the existence of a fabric indistinguishable from other, similar, fabrics. Our intention was to find a beam that offered sufficient illumination to distinguish between the details, patterns and colours, of one fabric and another. This did not mean, for example, that existing frameworks were ignored. Our intention was to acknowledge that the mathematics classroom is a complex set of interactions between teacher, students and mathematics (Kilpatrick et al, 2001). We were also conscious that the subject knowledge (Ma, 1999) and pedagogic content knowledge (Shulman, 1986) of the individual teacher are significant distinguishing factors between those who teach effectively and those who do not, and that observers bring to their work culturally embedded and frequently unconscious assumptions about the nature of effective teaching (Schmidt et al, 1996). It was decided at the outset that the project's instruments could only be developed through the use of systematically conducted live observations involving colleagues from each country. The processes employed reflected those of grounded theory (Glaser and Strauss, 1967). Put simply, lessons were observed and categories of activity that emerged were then tested against further observations in the same and other countries. Each observed lesson highlighted the tension between acknowledging new forms of activity while continuing to provide acceptable accounts of those observed earlier. In respect of this study, lessons were observed on five separate occasions in five countries over the course of eight months. Each phase of observation lasted a week, with one or two lessons observed each day. Each lesson, which was videotaped for later use, was observed by a team of at least five observers, one from each participating country. Afterwards observers returned to the host institution to discuss what had been seen and to relate the lessons’ activities to the developing descriptive framework. This was accompanied by the video recording made the same morning. The first playing of the videotape was explicitly focused on clarifying issues of fact. This allowed colleagues to refine and amend any notes made during the observations and typically lasted upwards of two hours. Once colleagues were satisfied that they understood the dialogue the process of analysis began. The tape was replayed and colleagues invited to code each episode –a discussion on the development of a definition of which follows below - against the current schedule, the tape was played episode by episode according to our emerging sense as to the nature of an episode. Each episode was discussed, particularly discrepancies between individuals’ coding, in order to compare that which was observed with the categories currently thought adequate. This process led to new categories being identified, old ones being amended or discarded and a greater sense of intersubjectivity in respect of the meaning of the descriptive framework. Clarke (2001b) stratifies classroom discourse into six levels - the lesson, the activity, the episode, the negotiative event, the turn and the utterance. For his work with the learners' perspective project, the explicit focus was on the negotiative event, which he describes as comprising those utterances forming the totality of an interaction. Our view was that this equated too closely to the tightly focused illumination described above. The episode, the
6