I remember "Old Zip Coon". Times were certainly different then..I have an old 78 c.1916 of Harry C. Brown singing the same tune but with words that I can't repeat here.
Aunt Hessie's White Horse was in Rhythm and Melody I think and was included as an example of syncopation (much was made of this). I think the words may have gone SOMETHING like this:

Can't you see Aunt Hessie's White Horse
Aunt Hessie's White Horse
Aunt Hessie's White Horse
Why don't you ride Aunt Hessie's White Horse
And gee up a trot for me.
Don't you call him slow
Aunt Hessie will make him go
He'll gallop along so fine
He'll make the whole world mine.

Oh..(once more with feeling).

and from Time and Tune

I'm from our village, our village
I'm my father's son (bis)

Over there three houses fine
When I've bought them they'll be mine

I'm from our village, our village
I'm my father's son.

(Time and Tune was for the youngest listeners who might have had trouble reading the word "rhythm")

I also remember Patapan which I was forced to whistle in front of the class.