Mathematics education designers and researchers are beginning to appreciate the pedagogical potential of embodied interaction (EI) instructional activities, yet little theory is available to understand its historical roots, sociocognitive mechanisms, and implementation practice. We draw on empirical data (n = 22) from a Grades 4-6 EI interview-based exploratory design for proportionality that involved "What's-my-rule?" remote-controlling of virtual objects on a computer display. Students were guided to reinvent mathematical concepts by reflectively instrumenting entrained perceptuomotor competence with interpolated symbolical artifacts. Analyzing qualitatively for critical dimensions, we propose and demonstrate three interaction milestones enabling or constraining learners' participation in conceptual EI activities: (a) re-seeing particular design features as peripheral-technological rather than central-mathematical; (b) re-seeing some would-be pragmatic actions as epistemic; and (c) re-seeing symbolic artifacts as affording or enhancing the enactment, explanation, or evaluation of hands-in immersive solution procedures. We delineate emergent principles for an effective EI mathematics design framework. (Contains 5 figures and 4 footnotes.)