Because Pearl is the "living A," the embodiment of her parents' sin, she cannot be normal and "pure" like the other children in Boston. Thus, throughout the book she acts as a devilish child. She does not fit in with other children; she is, instead, an outsider like her mother. When the children jeer at her, she does not retreat meekly; she retaliates by throwing rocks at them, demonstrating that even though her mother might try to raise her to be sweet and "godly," her origin and her father's denial negate that possibility.
Additionally, Pearl's abnormality is evident in her precocious and constant reminders to Hester of her sin. Not only must Hester and Dimmesdale remember their "sin" whenever they look at Pearl, the child also uses every possible opportunity to refocus her mother's attention on the literal scarlet letter--whether it be through shaping burrs in an A form or screaming until Hester reattaches her A after meeting Dimmesdale in the forest.
Finally, Pearl is more at home outside, especially in the dark forest, than she is in church. She plays outside while her mother meets with Chillingworth and Dimmesdale but pitches a fit when she is inside the church (meeting house).
All of these seeming "abnormalities" fit Pearl's character because she cannot be "normal" until her father acknowledges his part in her "creation." So, it is only when Dimmesdale stands on the scaffold and confesses in front of others, that Pearl's symbolization as the living "A" ends, and her normal life begins.