There is growing interest in designing for movement-based interactions with technology, now that various sensing technologies are available enabling a range of movement possibilities from gestural to whole-body interactions. We present a design methodology of Moving and Making Strange, an approach to movement-based interaction design that recognizes the central role of the body and movement in lived cognition. The methodology was developed through a series of empirical projects, each focusing on different conceptions of movement available within motion-sensing interactive, immersive spaces. The methodology offers designers a set of principles, perspectives, methods, and tools for exploring and testing movement-related design concepts. It is innovative for the inclusion of the perspective of the mover, together with the traditional perspectives of the observer and the machine. Making strange is put forward as an important tactic for rethinking how to approach the design of movement-based interaction.