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Responding to an increasingly urgent need for collaboration between mathematicians and mathematics educators, the study reported in this paper engages mathematicians as educational co-researchers in a series of themed Focus Group interviews where a pre-distributed sample of mathematical problems, typical written student responses, observation protocols, interview transcripts and outlines of relevant bibliography is used as a trigger for reflection upon and exploration of pedagogical issues. In the extract exemplified here, a part of the sample that includes a question involving the concept of det(M), the determinant of an nxn matrix, triggers a conversation that reveals various images of the concept held by members of the group as well as their beliefs about personal and shared concept images. (Contains 1 figure.) [For complete proceedings, see ED500858.]

This paper reports from a research project at Oxford in the UK that focused (a) on university mathematics teachers' conceptualizations of first-year undergraduate teaching related to observation of their teaching; and (b) on issues relating the conceptualizations to mathematics as a discipline. This research builds on a qualitative study of learning difficulties of first year undergraduates in their encounter with the abstractions of advanced mathematics within a tutorial-based pedagogy. Six tutors' responses to and interpretations of such difficulties were studied in semi-structured interviews conducted during an 8-week university term and following minimally-participant observation of their tutorials. This paper describes a 4-stage spectrum of pedagogical development (SPD) that emerged from the analysis of the tutors' conceptualizations of the students' difficulties, descriptive accounts of the strategies they employ in order to facilitate their students' overcoming of these difficulties, and self-evaluative reflective accounts regarding their teaching practices. This study then exemplify the third and fourth stages of SPD thought a discussion of characteristic examples from the interview data. In these stages the tutors' strategies begin to resemble less a traditional induction process and more a process of facilitating the students' construction of mathematical meaning. This discussion employs tools from sociocultural, enactivist and constructivist theories on the teaching and learning of mathematics. In particular, the data used here exemplify certain tutor strategies to achieve consideration of students' needs. (Author)