Here's an epiphany I just had while my children were discussing the "Twelve Days of Christmas" song.
What object did the true love receive the most of?
My 14 year old math genius son answered it in less than a minute and was able to prove his answer.
My 9 and 8 year old daughters just listened as he explained.
Here's how to use it as a math lesson.
For younger kids, you can start with the correct number of objects for each day. To make it really simple here are some Coloring pages of the objects in the song that you can print out (laminate if you are OCD like that) and use. Or you can use different colored candies, or manipulatives, or you can get really creative and glue the pictures to clothespins or Popsicle sticks and have them put them in order each time you sing the song.
Now, you'll have to remember to print out or cut out the correct number of each item to represent it properly. Once you've sung the song, have your child count each item and find out which object is the correct answer.
When they get a bit older, you can use the song to explain Multiplication. This is doing quick Addition based on groups of the same number of objects. You can also point out that the song has a repeating pattern. For instance there are the same number of objects from the first day as the last. 1x12=12x1
Once they get to higher math, you can have them write algebraic equations to represent the number of objects, or use a cartesian plane to chart the curve. Day one: P=D1, Day two T=D2 or for the purposes of charting Object=Day x Number of ojects (O=D1, 2, 3, etc)
Anyway, that's just my two cents. Hopefully this will make math a bit more fun for your kids, and not ruin the Christmas tradition for them. :)