You're absolutely right in saying that the speaker in William Carlos Williams' short story "Verbal Transcription -- 6 AM" is "talking about so many ordinary life matters." If you consider what exactly she is talking about, however, you may see a pattern. She seems to be talking about the ordinariness of aging and dying. The old cat was killed, for example, and the weak bluejay might be killed if it stayed outside too long.

I agree wtih the comment that the speaker (whom I take to be the wife, with all the domestic focus in the speaker's statements and with the speaker's understanding of a shared, established, routine life with her now deceased husband) is not panicking. She doesn't seem surprised and isn't devastated, and she's very attentive to the doctor (who apparently came over in his pajamas!). Maybe she's still in shock or denial, of course, and is delaying the inevitable confrontation with her loss.

I've never taught the story, but one approach that you may want to consider is having students fill in the gaps. For example, who is the speaker talking to? What has that person just asked? What is the dog doing? And so on. I'd be interested in seeing what other readers come up with. I'm wondering, for example, if at the end of the story the doctor is giving the speaker an injection of something? If not that, why is she talking about giving shots if the husband is already dead? (He is dead, right?)

EDIT: I agree with the husband doesn't have to have died, and the story may make more sense if he's still alive. He's probably the one getting the shots. Exploring the other perspectives -- that of the husband and of the doctor -- may be a really good way to approach the story.