Show your work means that when you are tested on your knowledge of a specific pen-and-pencil algorithm, you are expected to show a trace of execution of that algorithm. You're not being tested on solving the problem, but on knowing the algorithm.
Perhaps math classes should focus less on remembering algorithms and more on inventing your own algorithms by solving novel problems. In that context showing work is much more meaningful.
Then you have parents complaining about the difficulty of "story problems" and claiming math class shouldn't be reading class.
I don't mean to be mean, but people who don't know basic algebra are unlikely to be making algorithmic breakthroughs.
The parent comment probably meant that students would learn how to derive the necessary algorithm from the problem description, rather than memorizing a bunch of rote algorithms without being able to disassemble and reassemble them.
as per the curry howard isomorphism (programs as proofs), there is no difference, except that a few symbols are arbitrarilly bound but without loss of generality (as could be proven by induction).