This paper reports on research from a network of high school and museum partnerships designed to explore techniques for integrating mathematics and physics learning experiences during the first year of high school. The foundation of the curriculum is a problem-based, museum-based, and hands-on approach to mathematics and physics. In this paper, we ask: How does the materiality of this curriculum allow students to think about mathematics in different ways? We first discuss advantages associated with museum-integrated curriculum, and point to how these are related to materialist and embodied theories of learning. We then focus on video data collected during a wing-design lesson, and discuss patterns in students' participation. We show how (1) integrated curriculum creates opportunities for embodied learning experiences in mathematics, (2) gesture and touch are important facets of learning abstract concepts in mathematics and physics, and (3) museum-situated activities contest philosophical assumptions about the nature of mathematics. (Contains 3 figures.)