I emphasize that reading poetry skillfully means accounting for the words on the page - none of this reading between the lines. They have to actually read the words on the page and so this exercise is really about getting them to slow down, pay attention to the details of the poem, consider how various poetic elements (metaphor, voice, rhythm, pattern, etc.) all come together to create meaning.

NOT WAVING BUT DROWNING (1957) Stevie Smith (1902-1971)

Nobody heard him, the dead man,
But still he lay moaning:
I was much further out than you thought
And not waving but drowning.

Poor chap, he always loved larking
And now he's dead
It must have been too cold for him his heart gave way,
They said.

Oh, no no no, it was too cold always
(Still the dead one lay moaning)
I was much too far out all my life
And not waving but drowning.

NOTE: I use this poem on the first day of introducing poetry. I spend nearly an entire period on just this poem, going over it line by line. What follows is a general outline of the points I cover.

First, I read the poem aloud.
Though we observe that the poem doesn't quite make literal sense, we aim for a description of the action of the poem.
I read each line singularly and ask the students for the word or words from each line that really stand out and grab their attention. I write the words on the board. Once the words are collected (dead, moaning, drowning, cold, heart, etc.), the students get a pretty clear picture of the overall tone of the poem.
We discuss the oddity of a dead man who moans and that we have clear evidence to suggest we should read the poem as a symbol/metaphor since the poem does not make sense on a realistic level - dead men don't moan!
Next, we look at voice and describe the character of the voices present in the three stanzas. So, stanza one seems like news reporting: the voice is not involved in the action, it is distant. Stanza two gives us the voices of the drowned man's friends. We get place (England - chap, larking). We get the sense that these people who knew him don't seem to really care all that much. (I push the image: a drowned man on the beach, the friends standing around over him, "poor chap"). In the third stanza, we get the drowned man himself.
We look at what rhythm adds: the fast pace of "It must have been too cold for him . . ." - how that line is the longest in the poem and yet moves very quickly and has an upbeat bouncy rhythm (which is how I read it to make the point) and that underlines the lack of emotional depth in the response of the friends. We consider how that shallowness is also underlined by the shortest line of the poem following that line, the simple baldness of "They said" (once again our reporter is giving us the facts). Then we look at the weight of "Oh, no no no, it was too cold always" - which is a line that has to be read slow, both because of the heavy use of emphasis (7 out of 10 syllables) and all those long vowels.
We consider repetition, the slight variations of phrasing between the first and third stanzas and what those differences suggest. We consider "too cold always" and "all my life" and how what started out as a literal, though somewhat bizarre image, has changed and deepened in meaning and feeling.
We finally start to talk about what this metaphor might mean. And the students get this pretty quickly: the person who always seems to be joking around but who is in fact quite desperate emotionally. We can discuss just what it is that is cold and what he is drowning in. By now, we have really looked at this poem and they can bring up that the coldness the friends mention is actually reflected in their indifference to his suffering. We consider how much of the poem is about both a lack of feeling, a sense of isolation, and a lack of clear communication: the signs he gives are misunderstood but then there doesn't seem to be much emotional drive on the part of those who knew him to understand.
I might, depending upon the mood of the class, make a few other observations about reading poetry: that a poem often has a fulcrum and we discuss where it is in this poem (generally they agree that it is in the third stanza when we get the impassioned plea of the drowned man).

Students relate well to this poem and that is useful. They don't always want to admit that it is about them, but they have a lot of "friends" that it fits perfectly.

Side note: This poem becomes a useful reference when later in the semester we tackle "The Love Song of J. Alfred Prufrock" - all that business about "Till human voices wake us and we drown."