Counterexamples: is one as good as many?

This paper offers a three-level theoretical construct (affective, cognitive and epistemological-cum-pedagogical) that can be used to analyse the role of counterexamples in the teaching and learning of mathematics at university level. It draws on an LTSN-funded study that involved university mathematicians across the UK as educational co-researchers. Based on Focus Group interviews with mathematicians from across the UK (in which they discussed thematically-focused samples of students' mathematical writing and episodes from tutorials) the mathematicians explore the role of counterexamples in terms of the tensions amongst the three dimensions and in terms of how they address these tensions in their teaching. The paper also offers a glimpse into how professional mathematicians use counterexamples in their own research in mathematics. This paper was peer-reviewed and presented at an international conference with a 60% contribution by Iannone.