It may just be my fanatical opinion, but I think that clarity is one of the most important attributes of performing mathematics, regardless of level.
For your purposes, let me present an example: Regardless of what course I am teaching, whether it be first year calculus or fourth year topology, I always have students who submit messy, poorly structured, incomprehensible nonsense on their assignments. They typically receive a very poor mark as a result, even if their answer/solution is entirely correct. They of course come and complain that because their answer is correct, they should receive full marks. This is my response:
"A mathematical solution is not just a correct number at the end of a computation: it is a logical sequence of events which is clearly motivated, explained, and justified at every step. If you present non-sense that happens to give you the right number, then while your answer may be correct, your solution is wrong."
Of course, this idea should transcend undergraduate mathematics and manifest in research mathematics as well. A proof is not a proof unless the community can verify your result.