How can women be forbidden to speak and yet prophesy and speak in tongues?

There is a suggestion that the first verse you quote is Paul quoting the thoughts / writings of the Corinithians. Paul goes on to refute them - hence the apparent contradiction.

For more info on this "rhetorical" interpretation of this passage and a couple of others, you can take a look at the book "Women and worship at Corinth - Paul's Rhetorical Arguments in 1 Corinthians" by Lucy Peppiatt.

http://www.amazon.com/dp/1498201466/

Women DO prophecy (testify of Jesus) and speak in other languages, just not during church.

1Co 14:34 Let your women keep silence IN THE CHURCHES: for it is not permitted unto them to speak;

1.) Prophesying, as defined in Revelations 19:10, is 'testifying.' Rev 19:10 "for the testimony of Jesus is the spirit of prophecy."

2.) Tongues, as defined in Acts 2:4-6, are languages.
Acts 2:4 And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance. Acts 2:5 And there were dwelling at Jerusalem Jews, devout men, out of every nation under heaven. Acts 2:6 Now when this was noised abroad, the multitude came together, and were confounded, because that every man heard them speak in his own language.

How can women be forbidden to speak and yet prophesy and speak in tongues?

The letter to the Corinthians is like a laundry list of things they are doing wrong. Each chapter almost starts a new subject. In context Paul starts writing about spiritual gifts in chapter 12 then he takes a parenthetical excursion into the subject of love. Although the discussion of their confused gatherings starts in chapter 11, the reason for this excursion is to set the foundation for his addressing the subject of their confused gatherings in chapter 14 with an emphasis on the misuse of spiritual gifts. The definition of love given in chapter 13 is essentially selflessness. Much of the confusion he is going to address in chapter 14 comes from selfishness.

In the lengthy description of the problems associated with their confused meetings he also addresses a point of contributory confusion, that of women "speaking".

1 Corinthians 14:34-35 Let your women keep silence in the churches: for it is not permitted unto them to speak; but they are commanded to be under obedience, as also saith the law. And if they will learn any thing, let them ask their husbands at home: for it is a shame for women to speak in the church.

The word used for "speak" here is laleō a common word for talking, speaking, or questioning. Verse 35 sheds a little light on how it is used in this context. It would seem that women were inquiring in such a way that by-passed their husbands and contributed to the total confusion of the meeting.

Many see in this verse a Christian condemnation of women. However, upon closer examination what is being said is that men should not be lazy and neglect their duty to be leaders and teachers in their homes. When husbands neglect their duty, and sort of dump their wives on the church, they harm themselves, their families, and the church as a whole.

1 Corinthians 14:6 Now, brethren, if I come unto you speaking with tongues, what shall I profit you, except I shall speak to you either by revelation, or by knowledge, or by prophesying, or by doctrine?

In verse 6 Paul mentions different sorts of specific speaking that could occur in a meeting. Since Christians did not have a New Testament at this time, the only way they could receive the word of God was by direct revelation called prophesy, prophēteuō = speak under inspiration.

If women speak under inspiration, it would not be the same as general conversation or questioning.

It seems that in Corinthians some women were assuming to much in the equality and liberty that the gospel provided them. They supposed since thy were equal they could 'speak in public gatherings' as though exercising authority over others including men. There may have been other fellowship gatherings where they could exercise gifts of prophesy such as during worship, more private discussions or teaching children, etc. It is difficult with full accuracy to recreate what the primitive church service was actually like, but there does seem to be some part of it when 'non clergy' could speak with gifts that moved them, one at a time, possibly women included.

I assume in this view that prophetic gifts (wise insights into biblical subjects which may or may, or more usually did not, include future predictions) were not seen as 'always operative on demand' as the regular teaching of a person in authority was. Rather the nature of these gifts is that they would 'come and go' like the 'wind', so when a person was 'inspired' to speak, 'allowing' them to do so, was not seen as rebellious at all. They were seen as humble people being moved by God, even 'daughters' could prophecy under this scenereo per Joel 2:28. This respect for the inspiration of the Spirit seems to carry its own meek authority removing the concern over the pretense of 'speaking officially' as those in charge that allowed the more ecstatic impromptu sharing. This 'respect for random inspiration' is clearly seen by the fact that 'one person speaking' will 'stop and remain silent' so that another who receives inspiration can start. This switching from one to the next was regardless of the persons involved and how regularily important they may have otherwise been. (1 Corinthians 14:30).

This ancient practice of allowing people to speak in a service seems more or less lost in most churches today. In fact quite soon in the history of Christianity this practice of the ancient synagogue and more miraculous version added by the outpouring of the Holy Spirit seems to have been abandoned or in some sense possibly not Gods will in some aspects. Instead the preacher just gives a one way sermon with no opportunity for any discussion afterwards (ref this post). There are some charismatic groups that try to re-create it, or even to the more fringe movements support 'sales oriented' woman leading whole ministries, but in my experience this is often even worse and carnally based than the dry charismatic resistant churches which I confess to mostly attend as a lesser of two evils. Pray for me ;)

So one must think this 'speaking' was some kind of 'public', 'authoritative' and 'official' role which Paul did not at all grant women, nor did any other church. This is why he then says about a less private forum such as:

If there is anything they desire to learn, let them ask their husbands at home. For it is shameful for a woman to speak in church. (1 Corinthians 14:35 ESV)

It seems a little harsh to most of us today but we should keep in mind Christianity promoted a status for women and slaves high above what was the norm while not compromising the God ordained structure of the family which most world cultures actually have little difficulty with, recognizing the need for wives to submit and husbands to love and cherish. Actually most wives do not find it that hard to submit to men who genuinely love them. The church was not meant to overthrow these house relationships once various families joined together into community under Christ as their head.

Paul does not forbid women to speak; rather, this is the position held by some Corinthians and it is one of several matters causing division between the believers. After an extended discourse on speaking in tongues [14:33], Paul diverts briefly to another sub-theme [the main theme being division], which has been raised by some Corinthians: the role of women during assembly. A reference in NASB: the New Inductive Study Bible sidebar links 14:34 [The women are to] 'keep silent in the churches' to 1 Corinthians 11:5 'But every woman who has her head uncovered while praying or prophesying disgraces her head'. Significantly, these two issues [sub-themes] are both dealt with through sarcasm- in bold text by me - that shows Paul's true position. Regarding the head covering, Paul ends the discussion with, 1 Corinthians 11:15-16: but if a woman has long hair, it is a glory to her? For her hair is given to her for a covering. But if one is inclined to be contentious, we have no other [such] practice, nor have the churches of God' Then, in Corinthians 14:36, Paul is even more pointed with his sarcasm: 'Was it from you [Corinthians that had complained] that the word of God first went forth? Or has it come to you only? In conclusion, it is clear that Paul does not side with the Corinthian faction that wishes to deny female believers their right to equity with regard to church participation. Jesus taught us to love each other and to be motivated by love; these Corinthians were motivated by ego.