Two Kinds involves the dream of becoming greater than one already is. Cinderella, Snow White, Sleeping Beauty - they all rise about the situation they are in to become princesses. Here, the mother wants to transform her daughter into a sort of American princess, a musical prodigy that would be the "awe" of all the neighbors. This theme is further pushed when they see a young Chinese girl on the Ed Sullivan show.
Where this story leaves the motif, however, is when the daughter exerts her rebellion against the piano lessons. She causes an abrupt end to the lessons, and to the dream.