Show Work vs. Justify Answers

My current testing protocol is that all of my remedial math classes have multiple-choice tests, but all of my college credit-bearing classes have open-response tests (i.e., not multiple-choice). This is a minor change this year, as previously I felt completely constrained by the various department-level final exams in our system, which are multiple-choice for most everything up through calculus (so as to make it easy for the department staff to score them). 

Anyway, for the in-class tests that I personally give, I recently grappled a bit with exactly what direction I should give in this regard. Of course many instructors use the phrase "Show your work", so much so that students frequently anticipate that as the direction. But does that address a real issue? Some people's work process is just undeniably crappy: scattered, jumbled, incoherent. While that might indeed be their work process, does it really do them or anyone else any good?

What I've recently settled on is this direction: "Justify your answers with well-written math." This gets more to the heart of the matter, that one is using mathematical language to explain why something is to another person. There's a specific syntax and grammar to this (just like French or Russian or anything): any arbitrary "this is the way I do things" doesn't cut it, because we need a shared language to be understood. And it prepares students to read a math book on their own. And help other students in need, and be helped by them. And it allows the instructor to give useful feedback, by identifying a specific logical gap. And probably some other stuff that I'm overlooking right now.

So at the level of College Algebra and above, I've started to grade half-credit on this basis as of this semester (for full credit, students need both the correct answer and properly-written math statements showing small-scale steps). Later in Trigonometry they can deal with more formal identity-proofs, etc., but I think this frames the expectations for students properly at an early point.

Do you agree that this is a much better directive than "show your work"? Can you think of a better phrasing for the requirement?

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