"Thirty-nine years this December," said Professor McGonagall brusquely, snapping her bag shut.
Why do you suppose McGonagall began teaching in December?
Since it seems* that she was taking over for Dumbledore, who was moving to the Headmaster position, I have a few theories. Either something happened to Dippet (the Headmaster prior to Dumbledore) mid-year, or Dumbledore just took too long in finding a replacement for himself, and had to cover classes and Headmaster duties until she was hired part-way through the year. The latter seems unlikely.
Thoughts? Wild theories? McGonagall is a Death Eater?
* - See the timeline of the 1950s** at the Lexicon.
** - I love this time period in the Potterverse, don't you? (Just me, eh? What's your favorite decade, then?)