For Jing-Mei's mother, Suyuan, America first represents the opportunity to escape her personal traumas and start a new life. Remember that her mother came to America after suffering a number of tragedies, including the death of her parents and losing her twin daughters. So, for her, America is about having a second chance at life.
Secondly, America represents the fulfillment of personal dreams and ambitions. This is shown clearly in the first line:
My mother believed you could be anything you wanted to be in America.
This belief is significant because it drives the plot of the story. At first, the mother believes that her daughter can be a Chinese version of "Shirley Temple," for example, and then a master at memorizing world capital cities. But it is her desire for Jing-Mei to become a piano prodigy which really leads the story and also creates much of the conflict. It also demonstrates that mother and daughter do not share the same ideas about America: for Suyuan, life is about living the American Dream, while her daughter simply wants the chance to be herself.
In Amy Tan’s short story, “Two Kinds” America represents the opportunity to be whatever you want to be. Having lost her mother, father, first husband, and twin daughters in China, the mother sees America as the place where things could get better. She could erase her past with a new future, a future filled with hope. She works tirelessly and attempts to make her daughter into a child prodigy by finding a talent that suits the girl. Many failed attempts put a great strain on the mother-daughter relationship. Finally, when the daughter asserts her will, the mother is stunned that her daughter did not want to be “something” she simply wanted to be herself. The mother questioned how this could be. “Only two kinds of daughters,” she shouted in Chinese, “Those who are obedient, and those who follow their own mind.” Her daughter rebuffed her. The irony is that, in America, the daughter became what she wanted to be. She was never the best, never truly lived up to her mother’s expectations, but she was herself.