The main idea of this very complex poem is the ultimate unity of body and soul. In Western philosophy, it has been common to separate the two. The ancient Greek philosopher Plato provides the most famous example in this regard. He argued that the soul was somehow more real than the body; it was eternal, whereas the body was prone to aging, illness, and decay. Yeats disagrees with this, and takes Plato to task for regarding nature as nothing more than the frothy foam on the ocean of life:
Plato thought nature but a spume that plays/Upon a ghostly paradigm of things
For Yeats, life is much more complex than Plato and other philosophers would have us believe. Life is full of opposites, and yet a basic, primordial unity still remains. The whole of reality is composed of many parts, but the whole is very much greater than the sum of those parts. A chestnut tree, for example, doesn't simply consist in leaf, blossom, or trunk, but a unity of all three. And that unity is the spirit of the tree, the thing that gives the tree its life.
As with much of Yeats's poetry, "Among School Children" deals with the issue of aging. But unlike Plato and many others before him, Yeats makes no effort to separate the body from the soul. It doesn't matter whether we're eminent Greek philosophers, little babies bouncing on our mothers' knees, or famous Irish poets, old age comes to us all eventually, and it is pointless mourning the inevitable loss of youth and beauty. The soul should not be tortured over the state of the body, and vice versa. They exist together in a state of harmony, just as the bodily movements of the dancer are at one with the dancer herself:
O body swayed to music, O brightening glance,
How can we know the dancer from the dance?