There are some obvious reasons for Jing-mei's mother, Suyuan, to have unreasonably high expectations for her daughter and some that require delving into the mother's past. Any mother likes to brag about her child, and since Suyuan had a close friend whose daughter was a true child prodigy, she felt the pressure to compete. Auntie Lindo's flaunting of Waverly's fame was certainly part of Suyuan's desire for Jing-mei to achieve. Another reason is what Jing-mei's mother tells her: "Only ask you be your best. For you sake." In other words, Suyuan wants the best life possible for her daughter, a worthy and common ambition among parents. Although this doesn't fully explain Suyuan's motivations, it certainly is a factor.
Beyond these surface reasons, Suyuan has deeper psychological motivations at work. Her own traumatic past in China makes her intent on achieving a better life in America and taking advantage of everything available in America--a place where "you could be anything you wanted to be." That much she would acknowledge openly. But subconsciously Suyuan may have been trying to have Jing-mei live out three lives in one. Suyuan had lost two twin baby girls in China. Those girls, to Suyuan's knowledge, had never been able to live a full life, so in Jing-mei, their lives could, in some way, be recognized. These are motivations one can discern from reading this story alone.
Other stories in The Joy Luck Club collection give more information about Suyuan's backstory. When Suyuan was fleeing a Japanese attack in 1944 in China, she was quite ill and thought she would die. She abandoned her twin baby girls, hoping someone would find them. She didn't die; she went on to marry again, move to America, and give birth to Jing-mei, but she didn't know what became of the abandoned twins. Certainly the guilt of having abandoned those babies might lead her to place higher expectations on Jing-mei. If Jing-mei could become famous, it would help make everything Suyuan did in the past worthwhile.
Although it is easy to think of Jing-mei's mother simply as a typical domineering parent, having somewhat selfish reasons for pushing her daughter so hard, Suyuan becomes a more sympathetic character when readers consider her past.
From the beginning of Two Kinds," the mother believes her daughter is a child prodigy. The mother's deep rooted beliefs are shown when she quotes "you can be anything you want in America".
The mother's view is coloured by her observation of other talented children and it is part of her personal pride that her own children are also as talented as these 'others'.
If her daughter succeeds then she also succeeds, the child is an extension of her own ambitions and therefore, further pressure is placed upon the daughter's shoulders. She expects (unrealistically), her child to be a master at dance and piano as well as a genius in terms of her knowledge. She also expects the child to be beautiful in looks, this further shows the blinkered view of the mother's expectations as she shows complete disregard to the child's genetic makeup.