In the story, Waverley is embarrassed when her mother introduces her to everyone they meet on market day. There are some emotional forces at play here, so let's examine them.
We can gather from the story that Waverley doesn't really enjoy accompanying her mother to market on Saturdays. Since her mother requires her company when Waverley has no chess tournaments to attend, Waverley likely sees this errand as an annoying chore which intrudes on her other desires. Here, we can see that this market day errand is a point of contention between Waverley and her mother.
Furthermore, Waverley also appears to be irritated because she thinks that her mother is using her as a way to show off. As the story tells us, Waverley is an accomplished chess player. By the age of nine, she is already a national chess champion and 'some 429 points away from grand-master status...' However, being used as a prop for bragging rights can prove an embarrassing experience especially if the intentions of the guilty party are blatantly obvious to all.
Waverley's mother is justifiably proud of her daughter, but she bristles when her daughter voices disapproval of her behavior. She fires back that Waverley is probably ashamed to be seen with her own mother and demands to know whether Waverley is embarrassed to be her daughter. Hence, the age old conflict between mothers and daughters play out in this little scenario. Neither Waverley nor her mother can be completely honest with each other, and this generational conflict is further exacerbated by entrenched cultural attitudes.