In "Two Kinds," "Pleading Child" and "Perfectly Contented" have a double meaning in the story because they represent the changing feelings of Jing-Mei towards her mother. When she is a young child, for example, Jing-Mei can be characterized as a "Pleading Child" because she fights against her mother's plans to turn her into a child prodigy. She does not want to sing or play the piano; she just wants her mother to accept her as she is.
As Jing-Mei gets older, however, she becomes "Perfectly Contented" because she realizes that her mother never intended to upset her or make her feel inadequate. All she truly wanted was for Jing-Mei to be successful. As an adult, she realizes this when she looks at her piano and feels "proud," as though it were a trophy that she had "won back." In other words, she has finally come to understand her mother's motivation.
Through the last sentence of the story, the author demonstrates the appropriateness of the title, "Two Kinds." As an adult, she realizes and accepts that this conflict between herself and her mother is a normal and natural part of the mother/daughter relationship. As such, "Pleading Child" and "Perfectly Contented" represent the two sides of this often rocky relationship.
That the author had to play them both a "few times" is also significant because it shows that it takes age and experience to really understand that these early conflicts between herself and her mother were just a normal part of growing up.