Members of the UMaine Physics Education Research Laboratory are currently conducting a broad investigation of student understanding of thermodynamics and statistical mechanics concepts in upper-level undergraduate thermal physics courses. As part of this work, we are exploring student functional understanding of mathematical concepts required for productive reasoning about the physics. We find evidence that students do not make connections between the physical situations or processes under study and the mathematical concepts or algorithms needed for analysis of the physics. We compare student performance on physics questions requiring interpretation of P-V diagrams and analogous math questions devoid of physical context. We see a distinction between conceptual physics difficulties and difficulties with application of the underlying mathematics. With partial differentiation, we find that although students are able to take partial derivatives easily, many students have difficulty understanding the mathematical and/or physical significance of their differentiation, even after instruction. Evidence from various points in the thermodynamics course indicates a strong context dependence of student responses and difficulties. Results from research are being used to develop instructional materials to address specific difficulties.