Well, I'm not an expert that you are looking for, but let me put a bee in your bonnet. The adverb fast refers to the speed whereas quick refers to time. The confusion arises as in AmE, there's no qualm using an adverb fast whereas in BrE it's generally avoided. However, if you want your student to save time, use quick and if you want them to just write speedily, use fast!
There's something interesting. If I write an answer quickly it probably means that I knew the answer and I was quick to respond and write it without any doubt. On the other hand, writing answer fast has nothing to do with my knowledge. It's just my ability to write answer fast (may be blah-blah...)! Anyway, it's just my thought.
Surprisingly, COCAE shows equal number of results for both the phrases (write quickly/fast).
If we go by books, both are interchangeable. McMillian describes fast as quick and quick as fast!
fast - able to move quickly.
quick - able to move fast or do something fast