Was Kate really a shrew in The Taming of the Shrew?

When Shakespeare introduces us to Kate in The Taming of the Shrew, she seems fairly innocuous. Upon hearing her father's decree that she marry before her sister, the witty Kate questions whether her father will "make a stale of [her] amongst these mates?" Her reply is telling in two regards.  First, it shows her facility with language, a customarily male trait, setting her up outside the womanly norm. We know from her clever punning on the notion of a stalemate that she is not a stereotypically good woman (meaning: passive and controlled). Second, although we learn early in the story that Kate is perceived as devilish, headstrong, and wild, Kate's remarks reveal that she may not be as bad as they suggest. Her outright contempt for her sister's would-be suitors is bold but not entirely unwarranted. In fact, it speaks well of her judgment.