The ending of the story suggests that both of these pieces of music are symbolically related to our lives and the process of growing up, with the titles being very significant in terms of how we develop and teh various stages of relationships that we have with our parents. Let us look at the last paragraph:
And for the first time, or so it seemed, I noticed the piece on the right-hand side. It was called "Perfectly Contented." I tried to play this one as well. It had a lighter melody but the same flowing rhythm and turned out to be quite easy. "Pleading Child" was shorter but slower; "Perfectly Contented" was longer but faster. And after I played them both a few times, I realised they were two halves of the same song.
Both of these pieces of music then relate to specific stages. Note how "Pleading Child" is actually quite short, but because it is slower, it appears to be longer. However, "Perfectly Contented" is longer. Jing-Mei has passed through her "Pleading Child" stage, and now, having gone through the conflict with her mother, is able to reach the stage of being "Perfectly Contented," when she is at peace with herself and with her mother. The story ends with the realisation that both of these stages are necessary and related to each other.
As a child, Jing-Mei resents her mother's seemingly pushy treatment of her. She "pleads" to do things her way, especially when it comes to playing the piano or being able to choose her own "hobbies." Like most children, Jing-Mei wants instant gratification, which is represented by the shorter aspect of "Pleading Child." However when humans seek immediate progress or satisfaction, we are often hindered from achieving our goals; Amy Tan illustrates this truth through Jing-Mei's disastrous piano recital and her tension-filled relationship with her mother.
Later, long after Jing-Mei's mother's death, she comes across the music from her recital. The "Perfectly Contented" piece reflects what she presently views her life to be, and she realizes that while it might take longer to reach perfect contentment, the wait is worth it because then life is lively and full (represented by the song's faster pace).