The poem begins with the speaker on a visit to a public school. In real life, Yeats was an Irish senator who did in fact have to make such visits to schools. As a result, although the speaker isn't explicitly Yeats himself, there is a close connection between the two.
The speaker walks through a classroom, questioning the teacher, who is a nun. This is a convent school—in Ireland, public schools and religion were closely linked, so that it was common for there to be classes taught by nuns. The speaker sees children being taught "to cipher," or do arithmetic, as well as to sing, study history, and sew. These lines set the scene for the poem. As the poem moves on, this scene will provide a contrast to the drift of the speaker's thoughts.
"Among School Children" is written in a loose iambic pentameter (a meter in which each line as five feet in a da-DUM rhythm), as is characteristic of Yeats's poetry. The meter is loose in that many of the lines deviate from this exact pattern. Yeats does this on purpose, as it gives the poem the feeling of spoken language as well as a distinct texture. Rather than feeling entirely predictable and smooth, there's a thorny, knotty texture to the poem. This texture captures the feel of the speaker's own thoughts, doubts, concerns, aspirations, and observations.
Here, for instance, is the first line:
I walk | through the | long school- | room ques- | tioning;
The uneven rhythm of this line suggests a kind of awkwardness to the speaker. Contrast this highly irregular line with the speaker's description of the students:
To cut | and sew, | be neat | in ev- | erything
Here, the perfect iambic pentameter of the lines captures the prim and proper activities of the students, who are "neat in everything," contrasting it with the awkward thorniness of the speaker's own internal narration.