Symbolism, Imagery, Allegory
We're pretty intrigued with how Azkaban is so concerned with the past. So what's the deal with Divination, which is all about predicting the future? Divination plays a big role in the novel – we get more scenes in that class than in nearly any other in this book. Divination seems kind of shifty, when it comes right down to it. Trelawney is basically a hack and none of her students ever manage to predict anything. Trelawney herself only manages one prediction the entire year, and even that isn't really her – she comes off like she's possessed there, like something out of Ghostbusters.
The scenes and images surrounding Divination in the book hint at how the future is pretty much impossible to predict, something Dumbledore actually points out to Harry at the end of the novel (22.3.42). The Divination classroom itself hints at how inconsequential and lacking in substance the "art" is.
At least twenty small, circular tables were crammed inside it, all surrounded by chintz armchairs and fat little poufs. Everything was lit with a dim crimson light [...] (6.1.57)
It's a fluff class, basically, and it's not surprising that the super pragmatic skeptic Hermione storms out of the class in a huff.
So, aside from teaching us that we shouldn't ever call a psychic hotline and expect a useful or accurate answer, what's divination doing in the novel? On a symbolic and thematic level, divination actually works with the book's focus on the past to emphasize the present. That's kind of a wacky idea, so let's break that down. Divination demonstrates that you can't predict the future. And we continually learn that you can't change the past, no matter how much you may want to. Harry can't bring his parents back from the dead with his present actions, and no one can go back in time to prove Sirius's innocence. For all its concern with the past and the future, this book is really about living in the present and accepting that while you can't change the past, you can affect your future, which isn't set in stone.