Reasoning about abstract relational structures is hard. Even such a simple task as computing the value of arithmetic expressions (e.g., 2x3+8) requires the ability to combine symbols according to formal syntactic rules to generate ever more complex representations—but how can concrete, body-bound agents such as ourselves instantiate formal syntactic computations? External formal notations can play a central role in this instantiation by providing stable physical environments that are easily interpreted by powerful but domain-limited perceptual and motor processes—that is, by serving as diagrams of abstract structures. Compositional reasoning processes that may appear to result from abstract internal symbol systems can instead arise from a relatively simple embodied agent perceiving, acting in and learning about an environment rich with structured symbolic expressions. I will present several experiments and models demonstrating the influence of these experiences on intentionally formal reasoning. Recognizing the importance of the physical structure of symbolic environments urges the construction of a mathematics pedagogy that puts perceptual-motor interactions with dynamic notations at the heart of syntactic understanding. I will present design work and initial pilot experiments illustrating the form such an approach could take.