"Assumptions" play a fundamental role in disciplinary mathematical practice, especially concerning the relativity of truth. However, much is still unclear about ways to help students recognize key aspects of this role. In this paper, we propose a set of principles for task design to introduce students to the role of assumptions in mathematical activity, with particular attention to the following two learning goals: recognize that (1) a conclusion depends on the assumption(s) underlying the argument that led to it; and (2) making the underlying assumption(s) explicit is crucial to reaching consensus on the conclusion. In the context of a 3-year design research study, we first used existing literature to construct an initial version of task design principles which we then empirically tested and refined by designing and implementing two tasks in Japanese school classrooms. One of the tasks was in the area of functions at the secondary level and the other in the area of geometry at the elementary level. We analyze two classroom episodes to discuss the promise and evolution of our proposed task design principles. In addition, our analysis sheds light on the role of the teacher's instructional moves and the students' mathematical knowledge during the implementation of the designed tasks.