funny, the first date i ever had in my life started off exactly like you described it. except she didn't make an effort to get near me, it was arranged seating.
my strategy, find a legit excuse to talk to her. then once you do, just start acting like u guys are friends, and before you know it, you will be! its also interesting how the quiet girls aren't really quiet once you get to know them, but they just are in public.