The conflict is man vs. man, and it plays out in the story between a...
In Amy Tan's short story, "Two Kinds," from her collection, The Joy Luck Club, there are moments of crisis surrounding the conflict in the story between Jing-Mei (June) and her mother.
The conflict is man vs. man, and it plays out in the story between a Chinese mother who wants her Chinese-American daughter to achieve some kind of success. Her mother only wants what is best for her daughter, but her daughter can only see her own resentment, sure her mother is trying to make her something she is not.
The first crisis of the conflict in the story is, as you mention, the piano recital. Jing-Mei has not practiced enough, and yet she somehow believes that her "prodigy" side will step in and magically smooth out the areas of the song Jing-Mei does not know. This, of course, does not happen. Jing-Mei and her mother are devastated and embarrassed by her poor performance.
There is a second moment of crisis, and it still involves the piano. After the fiasco at the recital, Jing-Mei is sure her mother will stop pushing her and that lessons will cease. However, this is not the case. At four o'clock, as Jing-Mei sits watching TV, her mother tells her it is time to practice. Jing-Mei, aware of the power she has to say "no," screams at her mother, refusing to comply. They have a terrible fight, and Jing-Mei looks for something to push her mother over the edge: she says...
'Then I wish I wasn't your daughter. I wish you weren't my mother...'
'Too late change this,' said my mother shrilly...And that's when I remembered the babies she had lost in China, the ones we never talked about. 'Then I wish I'd never been born!' I shouted. 'I wish I were dead! Like them.'
It was like I had said the magic words...and her face went blank, her mouth closed, her arms went slack, and she backed out of the room, stunned...
Jing-Mei's words start out the way many angry adolescents react to authority and their inability to exert enough power over their own lives. However, Jing-Mei is looking for a weapon, something to push her mother over the brink, to make her really angry. So she mentions the twin babies her mother had to leave behind in China, something that has haunted her mom all her life. (See "A Pair of Tickets.")
This is the second moment of crisis within their conflict: the conflict of mother and daughter, of her mother's dreams vs. Jing-Mei's own wishes. Jing-Mei delivers an emotional sucker punch that her mother is not expecting. However, instead of fighting back, Jing-Mei's mother has been touched in such a terrible way, that she loses all her fight and simply backs away. This incident hovers between them for many years.