In The Taming of the Shrew, Katherine is the shrew who needs taming. The question is why she behaves the way she does. My contention is that she sees this kind of outlandish behavior as the only way to capture the attention of the father she loves.
Bianca is, by all accounts, beautiful; she has plenty of suitors and adoring young men seeking to woo her. In order to keep that whirlwind of admiration going, Bianca has obviously learned to manipulate men. Her father is no exception. She is clearly used to getting her own way (note her treatment of the tutors throughout the course of the play), and she's very good at getting it. Once again, her father is no exception.
Kate is wise to her sister's ploys but can not seem to convince her father of Bianca's duplicity. Instead, he has chosen Bianca over Kate. As a consequence, the sisters have practically become mortal enemies and Kate has become a shrew--figuring that her outlandish behavior might be the only chance to capture her father's attention.
It's ironic that Kate's marriage is much more even-keeled than Bianca's marriage, and Baptista is most proud of Kate in the end.