In the beginning of the story, how did the beliefs and hopes of Jing-Mei's mother contribute to her desire to see her daughter succeed?

Jing-Mei begins the story by telling the reader that her mother completely buys into an ideal interpretation of the American Dream. Her mother thinks that with hard work, anyone can accomplish anything in America. Her mother even tells Jing Mei, "you can be a prodigy, too." This shows her mother's rather unrealistic idealism. With hard work, Jing-Mei could certainly become a good, or even great, pianist, but a person is a prodigy based upon some innate ability. Jing-Mei is not a prodigy, but her mother insists she can work to achieve the same ability that a prodigy would have. This is not impossible. It's just very difficult. As Jing-Mei is trying to assert herself and develop her own individual personality, her mother's insistence on molding her daughter into a prodigy becomes a burden to her (Jing-Mei).
Considering what her mother has gone through, the promises of the American Dream must have been extremely appealing and uplifting:

America was where all my mother's hopes lay. She had come to San Francisco in 1949 after losing everything in China: her mother and father, her home, her first husband, and two daughters, twin baby girls. But she never looked back with regret. Things could get better in so many ways.

Jing-Mei's mother was determined to succeed. The problem is that she channeled those hopes into molding her daughter. Her mother became too controlling and Jing-Mei eventually rebelled.