As a classroom teacher of mathematics in English secondary comprehensive schools in the 1970s and early 1980s, I struggled with ways of helping my students to learn mathematics. I enjoyed doing mathematics myself, and I wanted students to have the pleasure that I had in being successful with mathematical problems. I experienced the introduction and implementation of syllabuses involving the ‘new’ mathematics. This gave me a lot of pleasure as I enjoyed working with sets and functions, with Boolean algebra, with matrices, with transformation geometry. Some of my students enjoyed this too, but the vast majority were, I came to realize, as mystified with the ‘new’ maths as with any of the more traditional topics on the syllabus.
Teaching mathematics was difficult, because students found learning mathematics difficult. I started to question what it actually meant to learn mathematics. Explanations or exposition seemed very limited in terms of their effect on students’ learning, so I found myself seeking alternative approaches to teaching mathematical topics, especially for students who were not inclined to like or be successful with mathematics.
I gained considerable personal enjoyment from puzzles, problems and mathematical investigations such as those offered by Martin Gardner (1965, for example) and occasionally used some of these, or modifications of them with students. This was ‘extra’ to my teaching of the mathematics syllabus, and I suspect, on reflection, very much in the mode of the teacher described by Stephen Lerman (1989b, p. 73) who ‘went around the classroom offering advice such as ‘no, not that way, it won’t lead anywhere, try this’. According to Lerman ‘there had been no opportunity for the teacher to discuss or examine…how [an investigation] might differ from “normal” mathematics’. In my earlier teaching it was not so much lack of opportunity to examine this issue as a lack of awareness on my part of how mathematical investigations might be linked to the mathematics on the syllabus.
In 1980, dissatisfaction with the mathematical achievement of many students, and a new post as head of a school mathematics department, made me start to