Bert teaches Bernice how to play checkers
Because that sketch was pretty short, here is Bert's signature song (with accompanying dance): "Doin' the Pigeon". I've linked to it before incidentally, but this version includes Bert narrating a film of pigeons.
How unusual of Bert to admit that pigeons may not be that smart, as in the first sketch. I think he's disappointed that Bernice didn't catch on quicker. On the other hand, perhaps she's smarter than he thinks; she moved one of her checkers while Bert was apparently distracted by talking to Ernie.
Perhaps unsurprisingly, they haven't had the same bird play Bernice all these years (I don't think pigeons live that long), but what is a little bit surprising is that they don't even try to get birds that look similar. Perhaps they think that only crazy people like me pay attention to that sort of thing. They're probably right.
As with the goldfish sketch, the unpredictability of the bird's behavior in the first sketch provides Frank Oz with some great improv opportunities and, as always, he rises beautifully to the occasion. In the second sketch, I get the impression that the narration was pre-scripted, but it's still really funny. "See that one back there? The one with the feathers?"