In Housekeeping by Marilynne Robinson, what does the concept of housekeeping mean to Sylvie? To the girls' grandma? To Lucille? Why is the idea of housekeeping such an important one in this book?

Marilynne Robinson's book Housekeeping examines the central characters' different approaches toward housekeeping.
The grandmother Sylvia Foster, with whom Ruth and Lucille live, endures traumatic experiences by establishing routines and repeating patterns of established behavior. She does so in attempt to protect the kids from pain and suffering. Housekeeping for her is a series of rituals that serve as a defense mechanism.
In contrast, Sylvie opens all the doors and windows to let the air in, refusing to keep them closed and blurring the boundaries that separate the outdoors and the inside. In doing so, she is embracing transience as a way of life.
Conversely, Lucille, who is Sylvie's niece, is a conformist who values stability and tradition and leaves home to pursue this type of life.
The significance of housekeeping in the book has to do with conventional, domestic female roles in the family and in society. This book suggests that we should not become too attached to these roles as a permanent way of life.
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Though Robinson's choice of title is both direct and concise, the word "housekeeping" is packed with overt yet subtle contextual significance both in regards to femininity at large and as understood by the book's central female characters. As orphans with a childhood history of upheaval and dislocation, Ruth and her sister Lucille develop disparate reactions to the notion of housekeeping. Lucille disavows Sylvie's nebulous, volatile method of running a home, instead electing to idealize the traditional, stable household dynamics she never experienced. Ruth, however, identifies with the vague, dissociative atmosphere that Sylvie creates, submitting deeper and deeper into a dreamlike relationship to the household that cannot reconcile itself with reality. Sylvie's own refusal to acknowledge the inheritance of womanhood by living alone and never having children of her own directly informs her version of housekeeping. She allows both herself and the children a hazy affordance of autonomy until they are so distanced from the conventions of housekeeping that, in the end, Ruth and Sylvie must burn the house down to fully extricate themselves from it.

The title of the novel is ironic from the outset. What does it mean to "keep house"? Obviously after the girls' mother's suicide we learn that metaphorically and literally she could not "keep house." Lucille wants to be part of a conventional world, a safe world, whereas Ruthie explores a world outside of conventions and predictable experiences when she is visited by the mysterious children in the woods. They are spirit children whose voices resonate for her. One of the great conflicts between Lucille and Ruthie is shown when Lucille shakes all of the pressed flowers out of Ruthie's book. It's a heartbreaking moment for Ruthie, as is Lucille's rejection of Ruthie when, with her school friends, she ignores her sister. Most telling is the ending, when they burn down the house and "put an end to housekeeping" and seem destined to wander forever.

In Robinson's novel, housekeeping means different things to different people. To the unconventional, offbeat Sylvie, a free-spirited drifter who arrives to care for her nieces, conventional middle-class notions of housekeeping don't compute. Sylvie marches to her own beat, caring for the girls in her own, odd way. She buys them pretty, sparkly shoes that fall apart quickly, rather than sturdy, sensible ones. She hoards newspapers and doesn't repair things that begin to fall apart. She doesn't feed the girls a healthy, balanced diet. She does share her senses of joy and sorrow with them. She gives them freedom and, rather than sacrificing her life to them, maintains her autonomy, sometimes wandering away for a long time. Lucille, the younger sister of the narrator, Ruth, comes increasingly to reject and hate Sylvie's erratic housekeeping. She wants a completely conventional life and eventually moves in with a "normal" family. Her notion of housekeeping is akin to what you might see on a 1950s sitcom: orderly and repressed. The girls' religious grandmother, who raises them in the early part of the novel, is also conventional in her housekeeping. Robinson questions conventional norms of housekeeping. While the townspeople eventually try to remove the girls from Sylvie, seeing her as unfit, Robinson shows us a quirky but compelling Sylvie, a woman with parenting deficits but also positive qualities. It is a different form of housekeeping. Lucille's very conventional yearnings, while understandable, seem dull and stifling within the context of the novel. Robinson encourages us to interrogate what a home is and what it means to keep one's house. Sometimes, she suggests, the best way to keep house is to burn the house down.