[M]athematics, like music, needs to be expressed in physical actions and human interactions before its symbols can evoke the silent patterns of mathematical ideas (like musical notes), simultaneous relationships (like harmonies) and expositions or proofs (like melodies). (Skemp, 1983)

Embodied-interaction designs for mathematics learning are computational phenomenalizations of quantitative concepts instantiated as immersive perceptuomotor coupling (Cress, Fischer, Moeller, Sauter, & Nuerk, 2010; Nemirovsky, Tierney, & Wright, 1998). In a recent version of these designs, activities progress from: (a) manipulating objects so as to achieve some qualitative goal endstate of the technological system; to (b) enhancing, explaining, and evaluating these strategies using standard semiotic resources of the discipline (Abrahamson, Trninic, Gutiérrez, Huth, & Lee, 2011; Howison, Trninic, Reinholz, & Abrahamson, 2011). The design rationale, informed by grounded-cognition theory (Hommel, Müsseler, Aschersleben, & Prinz, 2001), is for students to struggle productively with presymbolic embodiments of core learning issues.

These learning environments—in which mathematical concepts are discovered through empirical inquiry, and wherein students’ reasoning is quite readily inferred from their manifest physical actions—offer researchers new entries into historically durable theoretical questions pertaining to cognition and education, such as how mathematical competence emerges through guided participation in problem-solving activities, and what roles instructors and artifacts play in paving students’ cognitive progression from perceptuomotor interaction to reflective abstraction.

In this paper, we develop two complementary constructs—doing-for-seeing (instructor demonstration) and seeing-for-doing (student engaged perception toward imitation)—so as to exemplify the unique affordances of embodied interaction activity as contexts for mathematics-education research. To contextualize these constructs and argue for their methodological utility, we draw on empirical data from our design-based research study of the emergence of proportional schemas via embodied interaction with the Mathematical Imagery Trainer (see Figure 1; *n*=22; Grades 4-6; individual- and paired-student semi-structured clinical interviews).

The conjecture we pursue is that what we call “grounding” in mathematics education research—that mental activity that reform-oriented theoreticians and practitioners collectively desire for students to experience with respect to mathematical concepts—is what happens when students struggle to enact a goal performance. The corollary is that when performances cannot be enacted, concepts are not grounded. This perspective, which we develop by coordinating neo-Piagetian (Vergnaud, 2009), neo-Vygotskian (Saxe & Esmonde, 2005), anthropological (Goodwin, 1994), cognitive-science (e.g., Kirsh, 2009), and semiotic-cultural perspectives (Radford, 2003), enables us to refine our investigation into the microgenesis of mathematical schemas.