Version from Beth's Notes. This is the version I have heard used in the Silver Burdett music series, although I learned the lyrics "In the air and on the ground."
Beth's Notes found an older version in the 1887 Songs, Games and Rhymes for the Nursery, Kindergarten and Primary School by Eudora Lucas Hailmann.
I found a simliar poem in a publication called The Rosebud, published in 1885. This publication seems to have been a magazine for families to read with their children.
Lastly, and I have NO idea if this was just creative writing or something else, but Lord Byron, under the psuedonym "Horace Hornem" wrote a publication called Waltz; an Apostrophic Hymn (1821) where he refers to a piece of music as a "d______d see-saw up and down sort of tune." I'm not sure if the "see-saw up and down" nursery rhyme was wide spread or if this is just coincidence.
I also found the words "see-saw up and down" in a 1758 publication Imitations of Horace by Thomas Nevile, a book of poetry that imitates the Roman poet Horace.