If you look at the context of 1 Corinthians 11 it's speaking about the levels of authority God ordained.

1Co 11:3 But I would have you know, that the head of every man is Christ; and the head of the woman is the man; and the head of Christ is God.
1Co 11:4 Every man praying or prophesying, having his head covered, dishonoreth his head. (out from under authority)
1Co 11:5 But every woman that prayeth or prophesieth with her head uncovered dishonoreth her head: for that is even all one as if she were shaven. (also out from under authority)
1Co 11:6 For if the woman be not covered, let her also be shorn: but if it be a shame for a woman to be shorn or shaven, let her be covered.

I believe vs. 5 is talking about a woman out from under authority if you look at the context of the scripture. The reason he said for her to shorn or shaven was because that is what harlots did. Harlots had no authority since they were not married.

So you didn't answer my question, do you think women should wear head coverings? I personally don't think so. What are your thoughts?