How do the mother and daughter have the same personality in "Two Kinds?"

In the short story "Two Kinds" by Amy Tan, the mother and daughter at first appear to have very different personalities. For example, the mother, who is an immigrant from China and who lost her family there, wants her daughter to work hard and be a kind of prodigy. The daughter resists her at first and says that "something inside of me began to die," because her mother has subjected her to "raised hopes and failed expectations." 

However, although the daughter at first resists her mother's attempts to make her into a prodigy, she is similar to her mother in that she is also looking for what she refers to as "the prodigy side of me." She looks into the mirror and sees that the person who looks back at her is "angry, powerful." Just as her mother has pushed her to become a prodigy, the daughter begins, like the mother, to be forceful and push back. 

After her mother dies, the daughter rediscovers the old piano that her mother had forced her to play. She notices two pieces side by side--one called "Perfectly Contented," and the other called "Pleading Child," and she realizes that "they were two halves of the same song." In other words, she realizes that she and her mother are two halves of the same being because they are both forceful in their own ways.