Upper-division physics requires students to use abstract mathematical objects to model measurable properties of physical entities. We have developed activities that engage students in using their own bodies or simple home-built apparatus as metaphors for novel (to the students) types of mathematical objects. These tangible metaphors are chosen to be rich, robust, and flexible so that students can explore several properties of the mathematical objects over an extended period of time. The collaborative nature of the activities and inherent silliness of "dancing" out the behavior of currents or spin ½ states certainly increases the fun in the classroom and may also decrease students' fear of learning about these mathematical objects. We include examples from the electromagnetism, quantum mechanics, and thermodynamics content in the Paradigms in Physics program at Oregon State University.