I suspect that Hemingway wrote "Hills Like White Elephants" based on personal experience of a similar situation. In other words, I think he may have had a relationship with a young woman which resulted in her pregnancy and that he persuaded her to get an abortion. If I am correct, then I also suspect that his purpose in writing the story was to try to get rid of the feelings of guilt he had about the unpleasant incident. He wrote in his short story "Fathers and Sons" that he often wrote to rid his mind of things. Here is the exact quote:
"If he wrote it he could get rid of it. He had gotten rid of many things by writing them."
There are probably many writers who write to get rid of "things." The things may be memories, ideas, feelings, or something else. Vladimir Nabokov in an afterword to Lolita wrote: "Now, I happen to be the kind of author who in starting to work on a book has no other purpose than to get rid of that book . . . . ("On a Book Entitled Lolita")
It may be that some authors feel they have to rid their minds of "things" in order to be free to get new, and hopefully better, ideas from their unconscious minds, or from the muse.
I agree with what others mention. I would also consider adding information about the context in which Hemmingway wrote the story and his purpose, compared to the other stories it was published with.
I use this story, along with several others by Hemingway, to study his writing style. Students are able to discern his short, clipped sentence structure and use of spare details to yet convey the experience these two people are having as they discuss this very sensitive issue. I usually have students work on this story as a class activity so that they can't do any research to start. I like to see who can "get it" and why. At some point in the class I make sure we are all on the same page about the abortion topic, which sends them all back into the story for a "better reading."
I make sure that they know what a White Elephant and what the elephant in the room is. You would be surprised what some kids don't know. Then I have them notice how much of the story is dialogue. We talk about what we learn from dialogue and what we don't. We discuss narrator reliability, characterization and setting. Many kids who don't initially pick up the abortion angle will look up information on the internet. It is pretty hard to keep that a secret, so I just throw it out there to start!
I would focus on ambiguity and the power of that which is left unsaid. I recently read this story again with a graduate school class, and we used the story to focus on the qualities of good writing. In the story, Hemingway offers spare details to the reader and trusts that the reader will grapple with the text to find the meaning that it offers. This is definitely a quality of good writing.
I would focus on the syntax of the story and how so little is said about what the two characters are really discussing. When I have done this story in the past, I usually do not give my students any background info...I basically see what they can get from it. Then, when we discuss that it is a story about abortion, I have them go back through and find evidence to support this assertion. We also get into the phrase "white elephants," which refer to something that is costly but that has little value otherwise, and what this means for the story as a whole.
Here in Korea, my students don't see that word abortion in the story and see the story more as a clash of cultures, with the Unnamed "American" versus the "reasonable [ness] of the Europeans waiting for their train. Then it becomes values of one society and what importance they place on those values versus the other. I also never push them into the abortion position.