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Women should not literally cover their heads in church or while praying. The head as referred to in 1 Corinthians 11:3 refers to a hierarchy of authority and has nothing in relation to the physical head and a woman’s hair.
1 Corinthians 11:3
But I want you to understand that the head of every man is Christ, the head of a wife is her husband, and the head of Christ is God.
In this verse, it is clear Paul was using the word ‘head’ to refer to authority. That is, Christ is the head of every man, implying He has full authority over a man. And in a marriage setting, the husband is not only the head of the family but has authority over his wife, in that a woman should submit to her own husband. (Ephesians 5:22-24)
The teaching of 1 Corinthians 11:3-16 intermarriage with those of Ephesians 5:22-24. The concept is simple and related to submission in marriage.
I disagree with the literal idea that a woman should cover her head in church. 1 Corinthians 11:15, give a clear hint that Paul was not referring to the covering of the physical head. A woman’s hair is given for physical covering and she does not require a veil.
1 Corinthians 11:15
but if a woman has long hair, it is her glory? For her hair is given to her for a covering.
What did Paul mean by women covering their heads?
What Paul meant by a woman covering her head is that just as a man does not have authority over Christ, so does a wife over her husband. In this regard, a woman should not seek to pray or prophesy if her head (referring to a husband) is not checked or in order, as this will bring shame to her and her husband.
For example, a married woman should not stand to preach while her husband is shaggy dressed or in a condition that may cause people to raise blame the woman. That is why a married woman’s duty is fast to a husband and her household, and not to the church.
This is why Paul encourages women who desire to be fully devoted to the lord to remain single, “for a married woman is anxious about worldly affairs and how to please her husband” (1 Corinthians 7:34). Is it not written that “her desire will be for husband”? (Genesis 3:16)
In summary, a woman is not required to cover her physical head in church while praying or prophesying as her natural hair is there for that purpose. The ‘head’ in this case refers to a hierarchy of authority in a marriage setting in regards to a husband and wife.