Swimming can be interpreted as a metaphor for the narrator's movement and progress forward in her life or the lack thereof. In high school, she'd competed at the state level, a story that only later seemed to be "full of drama and chlorine." Looking back, it seems like that was an interesting time in her life, though she didn't fully realize it then.
Now, in Belvedere, the narrator's life is pretty stagnant. Her life is so lacking in excitement and movement, so to speak, that she "marveled at her ability to come up" with phrases like "pilot program" and "early intervention" when she wrote home to her parents and described her actual job. People think her name is Maria, which it, evidently, is not, just as Elizabeth, Kelda, and Jack Jack think of her as a swim coach—which she is not. The lack of movement or progress in the narrator's life seems to be echoed in the manner in which she must "teach" these three old folks how to "swim": by laying on the floor with their faces in bowls of water.
On the other hand, despite the lies she tells them about this being how Olympic swimmers train when there is no pool, the narrator actually does manage to teach them how to complete various strokes and how to breathe while they do so. She is "busy every moment" during the lessons. She is "meticulous" and "hands-on," even teaching them how to dive from her desk to her bed. She motivates them, pushes them, corrects them, and learns from them. This "swimming" then may not technically be swimming, but her three pupils feel as though it is. She feels like a real coach, a "jock," and this is what makes her feel proud.