I wonder if our dialect differences change that perception. In NZ English (usually closer to British than American), it's "show your working". "Working" as a noun pretty much exclusively means mathematical steps, rather than the much more general "work", and I don't think I've run into the same mistaken student idea you did (I teach undergrads maths).

I totally disagree about the "fictitious" part though. I think without proper working most interesting problems are impossible to solve.