1 Cor. 11:5 Paul says 'But a woman puts her head to shame when she prays or prophesies with her head uncovered' and in 1 Cor. 14:34 'Women must be silent during your meetings. They must not speak ...' In 1 Cor. 11, women are allowed to speak, pray and prophesy. In 1 Cor. 14, they are to be silent. Is Paul contradicting himself here? Some even consider the verses from 1 Cor. 14 to be inserted later and not from Paul for this reason. 

But if we pay attention to the context we get the following picture. 1 Cor. 11 speaks of praying and prophesying, that is, speaking in the sense of praying, praising, praising and speaking from the heart, speaking inspired by the Spirit. 

However, the speaking in 14:34-35 is "wanting to find out something" as shown in verse 35 "And if they want to find out something, they should ask her husbands at home for clarification; for it is ugly for a woman to speak in the congregation. The speaking here involves asking questions during the evaluation of the spirit gifts (vs.26-33). They should not disturb the congregation with questions during that evaluative speaking, but should ask their husbands at home. The context is the "order" in the congregation. For example, only two or three prophets may also speak, and they may not interrupt each other either. The point here is orderly conduct of the service. Compare this to children asking questions during sacred moments in the service. So again, not a contradiction, but a paradox. The same author's perspective may differ from one context to another.