The Hidden Village (THV) is a motion-capture video game designed for classroom and home use to investigate how physical movements complement geometric thinking and the formulation of transformational proof. THV is currently used by both students and math teachers. Students collaboratively explore conjectures about properties of space and shape through body movements that are directed by interacting with characters in the game. The game collects all group dialog and player movements as players progress through the Hidden Village as they make their way back home. During game play, we observe the close connection between players’ gestures and the quality of their mathematical intuitions, insights, and geometry proofs, thus providing evidence in support of theories of embodied cognition. Players also make use of “extended cognition” as they enlist their team members’ hands and arms when producing collaborative gestures. Students and teachers can create new game content by writing new mathematical conjectures and creating animations for the directed movements that would be used by future players. New embodied conjectures are made available to other players through a cloud-based conjecture library. Analytics on new embodied conjectures help identify the most promising activities for promoting mathematical reasoning and proof. To date, the game has been used by several hundred students and teachers. Movement based mathematical activities can enhance geometric reasoning for a wide range of students, including English Language Learners (ELL). Research informs embodied designs for promoting meaningful STEM education and instruction.