Musicaldudepeter said: "In the original production, the curtains parted and Mary Martin travelled down stage while perched on a tree branch. There, still and contemplative, she sang The Sound of Music, a hymn to nature and music. Listen to the original Broadway cast album and you'll hear how much she gets out of the song."
I've heard it, and seen photos from that. It's an interesting take, because it flips the "Problem Like Maria" on its head to some extent... the Maria we see in private there is a gentle, contemplative soul, almost a poet. She's not a flibbertigibet, will-o-the-wisp or clown, she's simply "not like the other nuns" because she doesn't belong there, and they're sensing an ideological conflict more than a behavioral one.
Contrast that with the film's characterization of Maria, where she's running, spinning and dancing, overwhelmed with awe and an almost orgasmic bliss at nature's majesty. This girl IS an uninhibited free spirit who would clash with the strictures of an abbey. She's quirky, she's an early example of the "manic pixie dream girl" archetype, albeit a more positive one. ("I Have Confidence" also lends itself to the endearing-oddball characterization.)