I think this is an important issue, and wrestling with it has informed a lot of what I do as a community college lecturer in mathematics. Helping students make the transition from the BS they get in many K-12 programs, and getting to share real math with them for the first time, is a great challenge and responsibility.
What gets me a surprising amount of traction is to address this explicitly, as the first thing on my syllabus. Top goal: "Read and write math properly with variables." I verbally quiz them on this on day two. I touch back on it almost every day. I explicate how I'll be grading for this on tests, and show grading examples from old tests. Why? The professional writing (a) provides a shared language, (b) allows them to read any math book, (c) serves as an explanation to other students and colleagues, and (d) makes it easy to find and fix errors and disputes.
Example interaction from yesterday (day 9 of the fall semester): One student has garbled the writing of a polynomial multiplication; I point this out, and she does the "But I got the right answer" bit. I ask, "But what's the number one goal for the course?". She says, "I don't know" (which even she can tell is not a good response), and almost all the rest of the class calls out, "Reading and writing math properly".
So this makes the expectation very clear, and by talking about it as item #1, I get the majority of the class on my side, and the community standard (peer pressure) works greatly in my favor. There aren't many silver bullets in teaching, but I'm delighted at how effectively this one works for me.
(P.S.: While the above is math-specific, I think the basic idea of setting a reading/writing/justifying "top goal" can work in many classes. E.g., in my C++ programming course I start with a quote from Bjarne Stroustrup, "Design and programming are human activities; forget that and all is lost", and then likewise emphasize making one's code readable to other programmers via a common style. As Ken Bain writes in What the Best College Teachers Do, "Finally, the best educators often teach students how to read the materials..." [Ch. 4]).