The axis moves in a cone-shaped pattern, called a precession, with the celestial North Pole describing a full circle every 26,000 years or so.
During this precession, the planet's tilt remains at 23.5 degrees.
The Earth's precession alters the time of the annual vernal and autumnal equinoxes, making them arrive earlier each year.
http://www.space.com/scienceastronomy/planetearth/earth_wobble_010131.html#Scene_1
Making a statement about the change in December ignores the fact that the calender isn't adjusted to make it fully useful when talking about climate changes.