The book Structure and Interpretation of Classical Mechanics redefines some of the trickier parts of the standard mathematical notation, and does all of the actual computation in Scheme. They extended the standard Scheme interpreter/compiler to support algebraic manipulation of Scheme programs, which lets them do all of the higher-order computations in Scheme as well (things like transforming between coordinate systems, finding the derivative of a function, computing the Lagrange equations from partial derivatives, etc). Usually the proofs/derivations are shown in the modified standard notation, and then the resulting implementation is shown in Scheme.

I haven't finished the book (turns out I know less calculus than I thought), but the result is pretty effective. You're much less likely to get confused about which things are numbers and which are functions, and which of those functions operate on numbers and which ones operate on other functions, once you see the Scheme implementation of something.