As a first generation Chinese immigrant, what challenges does the mother face?

In “Rules of the Game,” the relationship between the mother and her daughter is fraught with tension (as in much of Amy Tan’s writing). This tension reveals the struggles that the mother faces as a first-generation immigrant.
Although much of the narrative revolves around the daughter’s discovery of competitive chess, the tension in the story derives from the daughter’s relationship with her mother. This is a common theme in Tan’s writing, as the daughter often becomes the avatar of the mother’s American Dream. This conflict comes to a head in “Rules of the Game” when the daughter fights with her mother at the market:

Why do you have to use me to show off? If you want to show off, then why don't you learn to play chess?

The mother sees chess as both a road to success and as a representation of the struggle of adapting to life in the United States:

"This American rules," she concluded at last. "Every time people come out from foreign country, must know rules. You not know, judge say, Too bad, go back. They not telling you why so you can use their way go forward. They say, Don't know why, you find out yourself. But they knowing all the time. Better you take it, find out why yourself." She tossed her head back with a satisfied smile.

Although the mother is not able to master chess—literally or metaphorically—her daughter is. In the closing moments of the story, the metaphor of chess transforms, and the daughter imagines it as a contest between herself and her mother. Her mother wants so badly for success through her daughter, but this desire damages their relationship.