The complexity of thermodynamics challenges many students as well as faculty. Understanding what a partial derivative represents may be key to reducing the anxiety associated with this topic. In this session, participants engaged with a sequence of activities designed to elucidate the mathematics of thermodynamics through multiple representations of partial derivatives. Activities include: experiments that provide exemplars of measuring thermodynamic quantities involving partial derivatives, thought experiments where students design ways to measure particular partial derivatives representing thermodynamic quantities, a mechanical analogue that physically represents changes that hold specific quantities fixed, and an algebraic formulation of a partial derivative chain rule. Our discussants, Ayush Gupta and Joseph Wagner, each comment on how their different research perspectives can contribute to and are necessary for a holistic understanding of what happens during this kind of curricular sequence.