Teacher beliefs and the didactic contract on visualisation.

This paper explores secondary teachers’ views on the role of visualisation in the justification of a claim in the mathematics classroom and how these views could influence instruction. We engaged 91 teachers with tasks that invited them to: reflect on/solve a mathematical problem; examine flawed (fictional) student solutions; and, describe, in writing, feedback to students. Eleven teachers were also interviewed. Here we draw on the interviews and the responses to one Task (which involved recognising a line as a tangent to a curve at an inflection point) in order to explore the influence on the teachers’ feedback to students of: persistent images of the tangent line; beliefs about the sufficiency of a visual argument; and, beliefs about the role of visual arguments in student learning. We focus particularly on the influence on the didactical contract regarding mathematical reasoning that teachers with a variation of beliefs about the role of visualisation are likely to offer their students. We conclude with a concise description of a didactical contract which maintains a role for proof in the mathematics classroom that is not disjoint from the creative parts of visually-based classroom activity and that reflects an essential intellectual need. We also conclude with crediting the combination ‘task engagement-followed by-interview’ for the identification of subtle issues regarding the teachers’ pedagogical and epistemological beliefs and for the raising of their awareness of these issues.