Introduction Mathematics occupies about one-eighth of the secondary schools curriculum, and in primary schools perhaps slightly more. It is widely recognised by pupils, parents and teachers to be an important qualification for employment or for further studies, and it provides a unique type of experience which is an essential component of a complete education. 1

For any subject to be given such a large proportion of time in school, there must be generally high expectations about the contributions which it can make. The above quotation is from a British government report published in 1977, following a period when there had been a desperate shortage of suitably qualified mathematics teachers which restricted pupils' opportunities for studying mathematics. Some secondary schools had used other subject teachers for mathematics or had timetabled some pupils for perhaps only four instead of five mathematics lessons a week, so that there was a lower quality and reduced quantity of teaching. In primary schools, where mathematics lessons are given by most teachers, only about half had an appropriate mathematics qualification from their own schooldays; this meant that a restriction occurred mainly in the quality of the teaching. Despite the efforts made by these secondary and primary school teachers to foster their pupils' mathematical development, the results in many schools could not and did not match people's expectations.