My bet would be that he heard a child singing the first line, found it very evocative, and elaborated.
I read this poem for the first time in Kenneth Koch and Kate Farrell's Talking to the Sun: An Illustrated Anthology of Poems for Young People (1985) and there it has the dedication "For Miss Kathleen Hanlon," who I always assumed was a real child. Backing out to look at Schwartz's bibliography for the first time, it seems to have been one of the new poems in his Summer Knowledge: New and Selected Poems (1959), although I don't know its date of composition.
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I read this poem for the first time in Kenneth Koch and Kate Farrell's Talking to the Sun: An Illustrated Anthology of Poems for Young People (1985) and there it has the dedication "For Miss Kathleen Hanlon," who I always assumed was a real child. Backing out to look at Schwartz's bibliography for the first time, it seems to have been one of the new poems in his Summer Knowledge: New and Selected Poems (1959), although I don't know its date of composition.