I wanted to make almost the same comment. I'm convinced that my interest in logic and foundations makes me better at teaching algebra and calculus, because I'm often thinking anyway about why those "obvious" things work the way that they do. (In particular, why don't algebra textbooks discuss the general logical principle of substitution of equals for equals? I tell beginning students that it's the most important lesson of algebra, but it's not in their book!) It's also important to listen to how the students think about things (both prerequisites, and the errors that they're making now) and adapt my explanations to fit them.