The important theme of the American Dream is introduced in this excellent short story through the idea of America as a haven where you can start your life over and achieve anything you want to, if you but work hard enough. Note how the very beginning of this story alludes to this concept:
My mother believed you could be anything you wanted to be in America. You could open a restaurant. You could work for the government and get good retirement. You could buy a house with almost no money down. You could become rich. You could become instantly famous.
This idea of America being a place where you can achieve your dreams is one that dominates the entire story, as the mother believes that her daughter could become a prodigy if she but tried hard enough. However, crucially, it is Jing-Mei who does not believe in the American Dream:
For unlike my mother, I did not believe I could be anything I wanted to be. I could only be me.
America therefore represented the land of opportunity to Jing-Mei's mother: a chance of giving her daughter the opportunities that she never had in China. This is an essential part of the conflict between the daughter and mother that drives the story.