We have observed that when people engage in algebraic reasoning,Â they often perceptually and spatially transform algebraic notationsÂ directly rather than first converting the notation to an internal, non spatialÂ representation. We describe empirical evidence for spatialÂ transformations, such as spatially compact grouping, transposition,Â spatially overlaid intermediate results, cancelling out, swapping, andÂ splitting. This research has led us to understand domain models inÂ mathematics as the deployment of trained and strategically craftedÂ perceptual-motor processes working on grounded and strategicallyÂ crafted notations. This approach to domain modeling has alsoÂ motivated us to develop and assess an algebra tutoring systemÂ focused on helping students train their perception and action systemsÂ to coordinate with each other and formal mathematics. Overall, ourÂ laboratory and classroom investigations emphasize the interplayÂ between explicit mathematical understandings and implicit perception actionÂ training as having a high potential payoff for making learningÂ more efficient, robust, and broadly applicable.