See here for why I think this problem confuses people: http://www.overcomingbias.com/2008/03/mind-probabilit.html
The main point is that given the way Atwood asked the question the correct answer is 50% which wasn't the answer he intended to discuss and indicates he himself didn't fully understand the subject he was writing about. It wouldn't be the first time either.
In order to get the "unintuitive" outcome, there must be some element of selection (much like the Monty Hall problem has). In your formulation of the question, the GG possibility has already been eliminated because the mathematician answers "yes".
Can you spot the problem?
If my friend had told me "their names are Sarah and--" before the line went dead, I would mentally eliminate BB. Sarah could be any of the four girls. The three remaining options are not equally likely to contain Sarah, because the first one has two girls, and either one could be Sarah. So, the probability of Sarah being the girl in BG is 1/4 and of being the girl in GB is 1/4. The probability of the other child being a girl is also 1/2. It's not so much that BG or GB is eliminated in this case, but that the probability of her being in the GG group is better. But wait a second, I am not trying to find Sarah, I am trying to find the pair of kids that is his pair of kids. Is the probability that a group contains Sarah different from the probability that a group of kids is his? Maybe that's my bad assumption...maybe "at least one is a girl" is some precise formulation I don't understand that means the GG group isn't twice as likely to contain that girl. I'll get back to that. (Possibly her name is "at least once"?).
But what if he instead told me something over the phone that would eliminate the possibility of it being two boys only? What if when he had looked out over the groups of children waiting to be picked up before they blindfolded him and panicked and said "it's not the two boys--"? Is knowing that one of the children is a girl different from knowing they both aren't boys? I think it is, because the latter is a statement about the set of events, and the former is a statement about a single event. When we talk about the pairs of children, the set of events, we are not conveying direct information about the individual events. In this second case he is not conveying information about the individual events and the three remaining choices remain equally likely, 1/3 each.
Looking back to "at least one is a boy"- if we interpret that as a statement about the sets of probabilities, I would more precisely restate Eliezer's question as "Does the set of two children contain at least one boy?". This is why the important part of the story is the word "mathematician". The mathematician is talking about eliminating sets of events when he says "Yes". Ordinary people would just talk about the gender of a child.