Seems like the best way to think about the date is month-day in that order. Month narrows it down some, and then the day narrows further. If you just say 'it's the ninth today', you have twelve ninths it could be which in some sense, doesn't really narrow things--the possibilities are spread all over the calendar rather than in one group (month)
Then we say the year for the stupid Euros who don't know what year it is