This article is the second paper in a series of papers on studies focusing on teaching mathematical reasoning-and-proving in elementary mathematics classroom. Participants are in-service teachers enrolled in a continuing university education program in mathematics. Results from the first paper suggested the method of imaginary dialogues to have the potential to support in-service teachers in engaging their students in mathematical reasoning-and-proving, and Balacheff’s taxonomy of proofs to support in-service teachers in identifying students’ argumentation. This study is on the following years’ in-service teachers in the program. It examines their perceptions of the usefulness of two constituent parts of this approach, and what insights students’ written dialogues might provide. The study draws on G. J. Stylianides’ analytic framework for reasoning-and-proving. Main data were obtained from a questionnaire taken by 32 in-service teachers and follow-up interviews with four of them. The study reveals engaging students to reason, argue, and prove, while making students’ argumentation visible for teachers was perceived the most useful with imaginary dialogues. The teachers’ increasing awareness of levels of argumentation, was perceived to be the most useful with getting exposed to Balacheff’s distinctions. Keywords: Balacheff’s four levels of proofs, mathematical reasoning-and-proving, written imaginary dialoguesNivå
