The arguments have gone on for centuries: what does Paul mean when he writes in the New Testament that Christ is the head of the church? What does he mean when he says a husband is the head of his wife?
The typical battle lines on this have to do with translation: the word kephale in Greek is in question. Complementarians, or more accurately, hierarchists, say it means “authority.” Egalitarians argue it means “source,” like the head of a river.
Carrie Miles argues for a third, simpler, and clearer way. Read her latest paper, For the husband is the head of the wife as Christ is the head of the church: What way is that?
