Going across the grain: mathematical generalisations in a group of low attainers

This paper is a report of a classroom research project whose initial aim was to find out how low attaining students in mathematics would respond to some cognitively challenging prompts usually reserved for higher attaining students. Analysis of classroom incidents revealed the power of a type of mathematical prompt, which can be generalised as "going across the grain of the work". The metaphors of working against and across the grain are also used to describe a particular perception of mathematical structure, the use of unfamiliar methods of interaction with students, and some unorthodox features of the research method. In this study it was found that deep mathematical structure could be encountered by looking across the grain of the work. This finding, although from a very small specific study, is rooted in the characteristics of mathematics. The importance of principles manifested in small classroom studies is discussed briefly.