Encouraging Apt Engagement?

When I encounter art of any sort, 
I find myself with these questions: 

What am I feeling now?
What is the piece saying?
What's the story (and backstory)?

[Oh, and how is it made?
--among many, many others.]

Now, my students usually want to start with "What does it mean?"

(A fine, though often
misleading or reductive
question to get to know
something, someone,
anything at all.
Where to begin,
then?)

Artists (and fellow audience-members), 
what responses 
and what responsive questions 
would you encourage 
me to encourage 
in my students?