Shifting sands: students' understanding of the roles of variables in 'A' level mathematics

Abstract

The move from GCSE to 'A' level mathematics in schools in England and Wales can be, at least partly, characterised by an increase in abstraction and sophistication associated with the introduction of more variables. I have described as 'second variable' situations those in which letters were used in roles that went beyond the single unknown value or the dependent and independent variable, for example - the equation y = mx + c as a general equation of a straight line - the use of (a, b) to describe a general point. This thesis attempts to describe the experiences of students in meeting these situations. The data which I present are drawn from a variety of sources, but come chiefly from a year spent teaching and observing two 'A' level mathematics classes in two different schools. Other sources are my own mathematical work and that of colleagues, in particular two groups of teachers with whom I met to discuss my research. I also refer to my teaching of other groups of students. My conclusions - distinguish between structural and empirical generalisation - identify the shifts in the roles of literal symbols which take place in the solving of some types of problem - describe how stereotyping affects students' treatment of literal symbols and assists in or interferes with solving of problems - list some components of 'second variable thinking'. My research method is qualitative rather than quantitative. It draws on my teaching experience and makes a virtue of the subjectivity of both the researcher and the reader. I offer a number of mathematical exercises to the reader and intend that he should draw on his experience of these exercises in interpreting the thesis. I expect the validity of the thesis to be judged by its coherence and by its capacity to inform the future practice of myself and of readers.