Round and round the garden
Like a teddy bear
One step, Two step
Tickly under there!
Mostly sung to tiny tots and toddlers, this rhyme is almost always perfomed with accompanying actions: a finger is traced around the palm of the hand during the first couplet and walked up the arm with the third line, resulting in a tickle under the arm with the final line.
A relatively modern rhyme, Round and Round the Garden was collected in the UK in the late 1940s by Iona and Peter Opie, and certainly does not pre-date 1902, when the term ‘teddy bear’ was coined. It is unlikely that the term came into common use immediately, making the window for the inception of the rhyme rather small.
An earlier rhyme uses the same accompanying actions and similar lyric:
Round about there
Sat a little hare,
A cat came and chased him
Right about there!
The origins of this rhyme are not immediately apparent. Further research required.