Sometimes I come across some exam answers which describe a proof sketch or a counterexample very well but are not written formally. Such proofs show that a particular student understands the general picture of the question completely. Furthermore in the case of counterexamples sometimes these descriptive proofs are really interesting and one can find essentially new examples to refute a particular conjecture (or a possible theorem without one of its essential assumptions) using such descriptions.
In the usual process of the course, I encourage my students to understand the general picture of a question/phenomenon and describe it sketchy on their own. I appreciate this approach too much. But I don't know what I should do when I receive such creative descriptive proofs of my best students in an exam, especially when some of the other students write the usual formal proof correctly.
I am worried about "forcing" my students to write their true ideas in exams too formally because I think it could affect their motivation for being creative. On the other hand, formal proofs are necessary parts of formal mathematics and they should adopt themselves with this culture.
Question 1. What should I do with inexact true descriptive proofs in exams? Should I give them a full mark (which I think such proofs deserve it) or considering a full mark could be harmful to encourage them to be more formal?
Question 2. How can I encourage my students to write their ideas as formal as possible but think about them descriptively?