They aren't really thinking in the abstract, as they have been well trained and coached on how to do what they do. Each elephant has its own style, and pretty much paints the same thing over and over again, though with some variations. Still, they have incredible focus while they are painting. They cannot pick up the brush themselves, so the trainer hands them the brushes. They do make the decision on when to drop the brush, thereby requesting a new color. Perhaps not abstract, but they are not stupid, either. Clearly some thought goes into what they are doing. Most paint what looks like abstract paintings, others trees or flowers, others still paint animals. Very few can paint more than one "type" of subject.
Paya, however, seems to have the right stuff, painting both trees and his own likeness.
Asian Elephant Art Conservation Project