A mathematics teacher, Judith, has introduced her class of 14-year-olds to a mathematical investigation about, line segments between points on a square-dot grid. Some way into the activity, she asks Allan, one of the pupils: ‘Right. Can you make any predictions before you start?’ Allan considers the question, and answers: ‘The maximum will probably be, er, the least ’ll probably be ’bout fifteen.’ A first reading may suggest that there is nothing unusual about this interchange; we could, indeed, find many more like it. But step back; view the interchange through the lenses of an anthropologist. If, as it seems, Judith’s intention is to request information, why does her question address Allan’s ability to supply it? And why is his answer, ostensibly a mathematical utterance, so notably devoid of precision? What may we infer about Allan’s attitude to his prediction from the manner in which he formulates it? I set out on this research with a clear aim: to access and describe the mathematical frameworks and private constructions locked away in children’s minds. My concern was to uncover what they ‘knew’ and how they structured that knowledge. I saw this as the most likely kind of outcome (in the spirit of what I shall call the ‘linguistic principle’) of the mathematical conversations in which I planned to engage them. In other words, I began with my attention focused on ‘transactional’ functions of language:
That function which language serves in the expression of content we describe as transactional, and that function involved in expressing social relations we will describe as interactional.