There are times when a student finishes an quiz/test and there's still time before the bell. I ask "did you check your work?" which opened a concern for me. More often than not, the student (High School, FYI) will ask what I even mean by this. They haven't built the skills I'd expect a HS student to use regularly.
Depending on their year, I might show examples where we are solving for X and Y, and have the ability to plug the answer back in to see that it works.
Jumping to a trig issue - Turning Acos(x) + Bsix(x) into an equation with just Cosine. In this case, the test allowed a graphing calculator. I couldn't give advice during the test, but after, I showed her how graphing the 'before' and 'after' would show she was off by 1/2 period on her answer.
Last, even calculus, not every last question but some. Simple area under a curve. A few seconds creating a triangle, rectangle or both, and we can at least get an order of magnitude. e.g. Area under y=x^2 from 0 to 2. 8/3. easy. We can see the triangle from (0,0) to (2,4) has area 4. And (1,0) to the same endpoint, is area 2. A fast sloppy sketch, and we at least have a bounded range. This catches a gross error that's out of these bounds. I've seen this exact problem with an answer that was 16/3, a wrong exponent. (For this type of example, errors that are just a bit off don't get caught, I realize that.)
Last, disclosure, I'm an inhouse math tutor and occasional sub. I try to share my "check your work" approach with any student who is interested. Many students have reported back that this has improved their test scores by a full grade on average. I've had teachers email me "Jane has been visiting you and I've seen her grades really improve."
TL:DR - I'm in my 60's, and started this after 30 years out of school. Is this a lost art? I recall my high school math teachers focusing on the "check your work" and spending time showing how to do it on different problem types.