In Ender's Game, why did the Wiggin family move from the city to rural North Carolina?
The family moves to North Carolina for the good of Valentine and Ender's brother Peter. This is stated plainly in Chapter 9:
Valentine knew why they had moved here. It was for Peter, so that living among trees and small animals, so that nature in as raw a form as Mother and Father could conceive of it, might have a softening influence on their strange and frightening son.
Peter was violent from a young age, even towards his siblings. Before Ender left for battle school, Peter beat up his younger brother. The family hopes the move to a quieter place will curb these violent tendencies, supposedly through the beauty of nature and isolation.
This part of chapter 9 that discusses the move is told through Valentine's perspective. She realizes what relocating means, that her family has given up on seeing Ender again:
And when Mother and Father announced to them that they were leaving the city to move to North Carolina, of all places, Valentine knew that they never expected to see Ender again. They were leaving the only place where he knew to find them. How would Ender find them here, among these trees, under this changeable and heavy sky?
The parents decided to sacrifice their chance of seeing Ender again for the hope that Peter could change. However, Valentine realizes that Peter's brutality has not been "softened" by the move out of the city. She discovers that he has been harming squirrels, and he even threatens to kill her in the woods.
I linked you to the study guide on eNotes for chapter 9 in case you need more context.
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