I chose to observe Ben as part of my Phase 3 study because he was an experienced teacher with a declared aim to implement in his classroom ‘an investigative approach to teaching and learning mathematics’. We had run jointly a course for teachers with this title when Ben had been an ESG advisory teacher a few years earlier. 1 He had recently taken up a post as head of mathematics at Compton, a small secondary-modern school in a rural area.
I believed that we had significant common vocabulary. I also knew him to be reflective, taking Locke’s definition—‘the ability of the mind to observe its own operations’. 2 Thus I believed that we could join in critical discussions about an investigative approach related to the classroom practice in which he was engaged.
It was Ben’s second year as head of mathematics at Compton. Before he had arrived, approaches to mathematics teaching had mainly involved direct instruction. 3 Ben had introduced an investigative approach in his own classes, and was in the process of introducing a scheme of work which would encourage other teachers to become involved in investigational work. Year groups in the school were set for mathematics, but there was nevertheless a considerable range of ability in any class.
Ben’s developing scheme of work for mathematics lessons at all levels was not based on any published text, but he used such texts for various purposes at different levels. There was no split between ‘published scheme’ and ‘classwork’ lessons as there had been at both Amberley and Beacham. All lessons were designed by the teacher himself, and in comparison with the other two schools they were all class work lessons. This meant that all students worked from the same starting point, but flexibility within the structure of lessons meant that they could diverge in emphasis once a particular activity had begun. Occasionally, lessons were labelled ‘coursework’ lessons, and in these students developed some area of work into an extended piece of course work for GCSE purposes. I gained little impression of how this pattern extended to other teachers’ lessons within the school.
I observed Ben teaching a Year 10 class, which he had himself chosen for observation because he had already been teaching them for a year and felt they