Augmented Reality (AR) technologies allow for holograms to be layered over the real-world, "augmenting" human vision by adding technical information or illustrations onto 3D space. Although AR-based applications are showing positive effects in many systematic reviews and meta-analyses, well-designed, rigorous studies with strong control conditions are lacking. Further, many experimental studies lack process data to illuminate what is happening as students engage with AR. In this pre-registered study, we conducted an experiment where n = 120 high school students were assigned to reason about identical geometry simulations collaboratively either using tablets or AR head-mounted displays (HMDs). We look at their learning and how it was impacted by the dimensionality (2D or 3D) of the shapes they explored, as well as how they engaged with virtual objects using gestures and epistemic actions. AR HMDs were more effective for students getting an initial sense of 3D shapes, but less effective for 2D shapes. For gaining insights into the workings of shapes and formulating justifications of conjectures, we see no evidence AR is more effective, and trends indicating AR may be detrimental to eliciting generalizations. Further, process data showed that students using tablets are more likely to manipulate the geometric shapes in the simulations, while students using the AR HMDs are more likely to use dynamic gestures that simulate these manipulations, which are less constrained by the objects' actual properties. Implications for the future design and use of AR in education are given.