I have a vague memory of Shirley Temple skipping up or down the stairs in time to a multiplication table rhyme. "The Littlest Colonel"? Maybe "Captain January"? Sorry, I can't get any closer than that.
Penny, Tom Lehrer's "New Math" actually doesn't have a tune to the math part, just to the chorus. It's more a recitative -- kind of like talking blues, or an early version of rap. If you have the beat, you've got it. By the way, an old college friend of mine is the son of W.V. Quine, a well-respected Harvard professor of mathematical philosophy (I don't know if he's still there, or where Doug is). He told me -- back in 1970 -- that Tom Lehrer was the only person who got an A in his father's class who didn't go right into academics.

Aloha,
Mark