The short answer is that it does both; the novel both challenges and supports traditional gender roles. On the one hand, we have Jo March, who is literally described by her sisters Amy and Meg as "boyish." Even Beth admits that Jo is "boyish" and "play[s] brother" to all her sisters. Jo shortens her name, Josephine, to a name that sounds like a boy's, and she even befriends a boy who shortens his name from Theodore Lawrence to Laurie, typically a girl's nickname. Laurie turns out to be the more emotional and romantic of the two, qualities most frequently associated with young women rather than young men, and Jo refuses him when he proposes marriage, though that is not what young ladies are supposed to do when rich and handsome young men propose to them. Instead, Jo strikes out on her own, moving to New York so that she can pursue a career as a writer (while working as a governess to pay the bills). She has intellectual discussions with male students and professors, holding her own in matters of politics and philosophy. She ends up marrying a professor who has no money but a keen intellect and big heart. Her happy ending shows that a woman does not have to fulfill the stereotypical role in order to find happiness. Likewise, Laurie's happy ending shows that a man doesn't have to fulfill the stereotypical male role either (though Laurie does come to embody it more than Jo does the female role).

On the other hand, characters like Meg, Beth, and Amy do seem to fulfill the traditional female gender role to a much greater degree. Meg and Amy both like beautiful things, and they like to be pampered. Beth is angelic, a little homebody who loves all things domestic and really has no desire to leave the house or do anything besides take care of others. These characters do not offer a challenge to tradition, at least not in any way that seems as substantial as Jo's. Meg's attempts to become the perfect housewife sort of backfire when things don't go as planned, and Amy orders Laurie to reform before she will accept him, but these seem minor in comparison even to Jo selling her long hair—cutting it boy-short—to pay for Marmee's trip. Even Beth eventually dies as a result of the disease she contracted while caring for others, a typically female occupation. These characters, with the exception of Beth, have happy endings too, married to the men they love, and so, in this way, the text seems to support the more typical roles.

Ultimately, then, Alcott seems to show that women can and should be able to defy the traditional role if they want to. They can embody it, if they so choose, and they ought to be free to deviate from it if they prefer. Why shouldn't a young woman have short hair? Have a career as a writer? Engage in political and philosophical discussions? Alcott shows that women are just as capable as men—I haven't even mentioned how Marmee holds the entire household and many other households in town together in her husband's absence—and should share the same rights as men, without appearing to be too radical.