By Andrews, Paul; Sayers, Judy
Mathematics Teaching , No. 196
Paul Andrews and Judy Sayers analyse teachers' didactic strategies.
In an earlier paper (MT194) we introduced readers to the mathematics education traditions of Europe (METE) project. This was a comparative study, funded by the European Union, of the teaching of mathematics in five European countries, (Flanders, England, Finland, Hungary and Spain) to students in the upper primary (ages 10-12) and lower secondary (12-14) years. These ages were chosen as they represent a time when many students' experiences of mathematics shift from concrete and inductive to abstract and deductive.
The project team video-recorded a single sequence of lessons in each country on each of four key topics. In so doing the team believed it would overcome criticisms of larger projects, which usually videotaped single lessons, including, for example, the temptation for teachers to present 'party piece' lessons. We felt that such concerns diminish when a sequence of lessons on a specified topic is requested. More importantly, the decision allowed us to see how effective teachers, defined by local criteria of effectiveness, in each project country conceptualised and taught key topics of interest to all teachers and researchers. After an initial schoolbased meeting, during which the project aims and aspirations were outlined, each teacher was filmed for four or five successive lessons, starting with the first on a particular topic. The topics, which were representative of the breadth of school mathematics, concerned the teaching of:
* percentages (a topic of arithmetic applicability) at the upper primary level;
* polygons (a routine geometrical topic) at the upper primary level;
* polygons (an opportunity to examine curriculum continuity) at the lower secondary level;
* linear equations (an early topic of formal algebra) at the lower secondary level.
Videotaping focused on all the utterances made by the teacher. Consequently videographers were instructed not only to follow teachers whenever they were talking but also to capture as much board-work as possible. In general a tripod-mounted camera was placed near the rear of the classroom, teachers wore radio microphones and a strategically placed telescopic microphone captured as much student-talk as possible. In some cases two cameras were used, one focused on the teacher and one on the students.
After filming, each videotape was compressed and transferred to CD ROM for copying and distribution, before being coded by its home team against a schedule developed during the first year of the project. Additionally, the first two lessons in each sequence were transcribed and translated into English, allowing colleagues to code other countries' lessons and determine inter-coder reliability. Sadly, the company producing the Finnish data folded mid-way through the programme and so no lesson from that country has yet been coded. Consequently, this paper draws solely on the data sets from Flanders, England, Hungary and Spain.
The schedule used for coding was developed over a period of a year during which a week of live observations was undertaken in each country. One lesson each day was observed by at least one colleague from each country. The same day, supported by a video-recording and home researcher's translations, the lesson was reviewed and discussed to facilitate the development of a descriptive framework for comparing and contrasting the episodes that comprise a mathematics lesson. This process led, after a year's negotiation, to the identification and exemplification of three categories of observable teacher intentions and behaviours which project colleagues believed allowed for a meaningful comparison of lessons. These intentions and behaviours were not construed as exhaustive, exclusive or hierarchical, but as fit for the purpose of categorising and distinguishing between different lessons or lesson forms. The first of these three categories comprised seven generic mathematical learning objectives while the second four conceptions of madiematics underpinning mathematical tasks. …