On the pedagogical insight of mathematicians: 'Interaction' and 'transition from the concrete to the abstract'

In this paper we draw on a 16-month study funded by the Learning and Teaching Support Network in the UK and entitled Mathematicians as Educational Co-Researchers. The study's aims were two-fold. Primarily we intended to explore mathematicians' reflections on issues identified in the literature as highly topical in the area of teaching and learning of undergraduate mathematics. We also wished to explore the conditions under which mutually effective collaboration between mathematicians and researchers in mathematics education might be achieved. Participants were 20 mathematicians from 6 mathematics departments and the study involved a series of Focus Group Interviews where pre-distributed samples of mathematical problems, typical written student responses, observation protocols, interview transcripts and outlines of relevant bibliography were used to trigger an exploration of pedagogical issues. Here we elaborate the theme 'On the Pedagogical Insights of Mathematicians' as it emerged from the data analysis. We do so in two parts: in the first part we present the participants' reflections on issues of interaction and communication within the context of teaching and learning in higher education. The data suggest that the lecturers believe that mathematical learning is achieved more effectively as an interactive process and recognise that lecturing is not a method generally conducive to interaction. However, they discuss ways in which interaction can be achieved and refer to seminars, tutorials and feedback to students' writing as other opportunities for interaction that must not be missed. In the second part, we focus on the lecturers' pedagogical reflections regarding the abstract nature of university mathematics and, in particular, the ways in which teaching can facilitate the transition from the concrete to the abstract. We conclude with a brief evaluation of the project by the mathematicians themselves.