This paper examines task design that affords deep changes in mathematical thinking in the context of peer interaction. We describe a study in which 60 low-level high-school students solved a proportional reasoning task, the "blocks" task as individuals and/or in dyadic interaction. We show that we could tailor the design of the task in order to create a cognitive conflict among dyads, notwithstanding the strategies used by the students. We show that students' proportional reasoning strategies did not improve as a result of discussion even when guided by an experimenter dedicated at reaching consensus; however the introduction of a hypothesis testing device and the guidance of the experimenter to accommodate divergent views led peers to impressive conceptual change in their discussion and in an individual post-test. Examination of one case of dyadic interaction shows that beyond the value of given characteristics of individuals or of tasks, the process of argumentation that takes place between the peers explains the subsequent gains of the individuals. The conditions under which conceptual change was attained challenge theoretical views on cognitive development and social interactions.