This is a very interesting question for a number of reasons. Firstly it captures the dilemma of the whole nature/nurture debate, which asks the question which is the more powerful influence in our lives - our genes or our upbringing? It is clear that the context of this story is the misunderstandings between a Chinese-born mother who has now emigrated to the US and her American born daughter, who has been brought up in the United States but also within a Chinese family desperate to hold on to at least some of its values.
However, it is clear that if you read this story closely, that Waverly exists in a confusing world where traditional Chinese values are expected but at the same time the influence of the American way is clearly recognised too. For example, consider the gift that the mother gives Waverly before her first chess tournament game:
When my name was called, I leapt up. My mother unwrapped something in her lap. It was her chang, a small tablet of red jade which held the sun's fire. "Is luck," she whispered, and tucked it into my dress pocket.
The role of traditional superstition and charms still clearly is identified as a powerful force, but equally too is mention made to the new "American way". Consider how this operates in the division of the household chores:
That's when my mother decided I no longer had to do the dishes. Winston and Vincent had to do my chores...
"Is new American rules," said my mother. "Meimei play, squeeze all her brains out for win chess. You play, worth squeeze towel."
The "American rules" are thus recognised as a powerful entity in their own right. However, in spite of this, it is clear by the conflict that culminates in the fight between Waverly and her mother that Waverly wants independence and the credit of her success, rather than her mother, in Waverly's opinion, using her success to gain social standing. Thus it appears that the American context in which Waverly was raised has been more powerful than the traditional Chinese values that have formed a part of her upbringing.