I'll be interested to read any responses to this thread! This is something that I'm just starting to wrap my head around.
However, when I think of whole to part vs part to whole, I think of the way I learned to read. (My dd, too, now that I think of it.) I learned words. Later I learned sounds, though never as well as I probably should have. I do not learn words by sounding them out phonetically. I learned sight words. The word is the whole; the sounds are the parts. It's big picture vs. details.
So in math, I would think that whole would be the mathematical concept and part would be the steps you need to actually solve the problem dealing with the concept.
Does this sound right?