Mathematical fluency: the nature of practice and the role of subordination

There has been considerable debate recently in the UK about claims that students are arriving on mathematics degree courses without that same fluency in calculus and algebraic skills as students had many years ago. A joint report from the London Mathematical Society, the Institute of Mathematics and its Applications, and the Royal Mathematical Society (LMS et al, 1995) says that there is a serious lack of essential technical fluency - the ability to undertake numerical and algebraic calculation with fluency and accuracy (p2) as one of three problems they highlight. This follows a series of articles in the national press (for example, Barnard and Saunders, 1994; Ernest, 1995), and a similar debate where there were claims that the introduction of GCSE, replacing 'O' level and CSE, meant students weren't so fluent in some algebraic skills that were taken as pre-requisite for an 'A' level pure mathematics course. Tahta (1985) has commented, with reference to notation, We do not pay enough attention to the actual techniques involved in helping people gain facility in the handling of mathematical symbols (p49). The joint report from LMS et al (1995) calls for ... an urgent and serious examination of what levels of ‘traditional’ numerical and algebraic fluency are needed as a foundation for students’ subsequent mathematical progress, and how such levels of fluency can be reliably attained (p14, their emphasis). I consider traditional ways in which attempts have been made to help students become fluent, and offer a model for ways in which fluency can be achieved with a more economic use of students’ time and effort than through the traditional model of exercises based on repetition. Examples of impressive learning from everyday life can offer insight into possible ways forward inside a mathematics classroom and I begin with an example of impressive learning that we have all achieved (unless someone has had an accident, illness or a disability which has prevented them. In which case an equally impressive alternative can be substituted).