This is a great question.
Waverly's mother in the story is always present, even when she is not. In the beginning of the story, Waverly states that her mother taught her the art of invisible strength. She defined this as:
It was a strategy for winning arguments, respect from others, and eventually, though neither of us knew it at the time, chess games.
From this perspective, we can say that Waverly's mother was her first teacher, who taught her a very valuable lesson about life and how to succeed. She made her a more accomplished person in chess and in life in general.
As the story progresses, Waverly wanted more freedom, especially freedom from her mother. So, naturally conflict ensued. When this happened, Waverly began to see her mother as a potential opponent. From this perspective, we can say that as Waverly grew older, she began to see her mother as an opponent. This, too, had an effect on Waverly. Would she exceed her "master"?
Her black men advanced across the plane, slowly marching to each successive level as a single unit. My white pieces screamed as they scurried and fell off the board one by one. As her men drew closer to my edge, I felt myself growing light. I rose up into the air and flew out the window. Higher and higher, above the alley, over the tops of tiled roofs, where I was gathered up by the wind and pushed up toward the night sky until everything below me disappeared and I was alone.