There are two passages which are key to the discussion:
"Let your women keep silent in the churches, for they are not permitted to speak; but they are to be submissive, as the law also says. And if they want to learn something, let them ask their own husbands at home; for it is shameful for women to speak in church. Or did the word of God come originally from you? Or was it you only that it reached? If anyone thinks himself to be a prophet or spiritual, let him acknowledge that the things which I write to you are the commandments of the Lord" (I Corinthians 14:34-37).
"Let a woman learn in silence with all submission. And I do not permit a woman to teach or to have authority over a man, but to be in silence. For Adam was formed first, then Eve. And Adam was not deceived, but the woman being deceived, fell into transgression. Nevertheless she will be saved in childbearing if they continue in faith, love, and holiness, with self-control" (I Timothy 2:11-15).
The two passages cover two different situations. I Corinthians 14 is dealing with activities in church services. I Timothy 2 is dealing with general Christian living. The statements in I Timothy 2 apply to the church services, but they also deal with day-to-day life. But the statements in I Corinthians 14 only apply to the worship service. The distinction is important because the statements in the two passages are not the same.
What is Meant by Silent
It is unfortunate that the word "silence" is used in both passages because there are distinct Greek words being used. In I Corinthians 14:34, "silent" comes from the Greek word sigatosan, which means to be silent, keep still, say nothing, stop speaking, or hold one's peace. Paul defines the word in the passage by stating that women are not permitted to speak and that it is shameful for women to speak in church. The same word is used in I Corinthians 14:28 to tell those who would speak in tongues to be silent if there is no interpreter available. It also appears in I Corinthians 14:30 in the instruction to prophets that they are to remain silent while another prophet is speaking. It is not a complete prohibition of all sound because such would interfere with the command that we are to sing in worship (Ephesians 5:19; Colossians 3:16). Like the commands to be silent to the speakers of tongues and to the prophets, it is an indication certain actions at certain times are not to be done.
In I Timothy 2:11-12 the Greek word is hesuchia behind the English word "silence", which means quietness, rest, or peace. It appears in Acts 22:2 when the Jewish audience became quiet to hear Paul's speech. It also is used in II Thessalonians 3:11-12 when Paul told gossips to stop their gossiping and lead a quiet and productive life. It doesn't refer to absolute silence, but a non-intrusive approach to life. Again, Paul gives a definition by stating that a woman is not permitted to teach a man or to exercise authority over a man.
It appears that women in Corinth were questioning the teachings being given during the church service. Paul states that it is God's command that they be silent in church and if they have questions to wait until they get home to ask their husbands. I have seen commentators twist this command to claim that only married women are not permitted to interrupt services to ask questions, but that unmarried woman can interrupt. If you read Paul's statement again, you see that Paul is stating that women cannot ask questions during church because woman are not permitted to speak in church. He gives an alternative place for married women to ask their questions, but there is no implication that unmarried woman are exempt from the rule of not speaking in church. Thus this command prohibits a woman from conducting a prayer, reading Scripture, or giving a lesson in church. However, it does not mean a woman cannot pray or speak God's word outside of church. In I Corinthians 11:1-16 there is a rule that woman are to have their heads covered while praying and prophesying. Such prayers and prophecies could not be done during the worship service, thus implying that were other times when they could take on those roles but they had to maintain proper decorum by showing submissiveness -- in this case by wearing a head covering.
The passage in I Timothy 2 also rules out a woman giving a lesson to an audience which includes men or other situations where she would be leading men. It would not prohibit a woman from participating in a learning situation, so long as she did not dominate or take the lead. Thus a woman could answer when called upon, or read a passage, but she would be going beyond the limitation if she began instructing others even if she is sitting among the students. I believe that some are overly sensitive about the issue and forbid a woman from even expressing an opinion. However, a woman must be careful not to appear dominating. To respond to a statement by a man with "That can't be right" is placing the woman's opinion over the man's. But to ask "How does that fit with the passage over in _____?" could accomplish a similar end without placing the woman in a dominate position. Ultimately, correction of a man who is teaching wrong will have to be done in a private setting, as demonstrated by Aquila and Priscilla (Acts 18:24-26).
Is a Woman Forbidden to Speak in Mixed Company?
Again, the example of Priscilla, who helped her husband teach Apollos privately, is an example of a woman speaking in a mixed company (Acts 18:24-26). Sapphira, in front of a gathering, answered Peter's questions (Acts 5:8). Rhoda interrupted a gathering of Christians, who had assembled to pray at Mary's house, to announce that Peter was the gate (Acts 12:12-15). Therefore, we conclude that speaking in mixed company is not forbidden. I Timothy 2 limits the type of speaking done in mixed company. I Corinthians 14 forbids addressing the assembly in a particular situation, when in church.
Nor is a woman forbidden from teaching children. "Nevertheless she will be saved in childbearing if they continue in faith, love, and holiness, with self-control" (I Timothy 2:15). A woman's salvation is enhanced when she has children, if her children remain steadfast in godly living. The implication is that she is involved in their rearing and their teaching.
When Are We in Church?
I believe that most women in the church understand these issues, but the area of controversy has been what constitutes "in church." Everyone agrees that when the church is assembled for worship that I Corinthians 14 applies, but what about gatherings for the purpose of studying God's word?
In the context of I Corinthians 14, the emphasis is on a gathering of the whole church (I Corinthians 14:23, 26). This gathering is properly called the church (ekklesia in the Greek, meaning an assembly). In this gathering, acts of worship are being done, including that of teaching (I Corinthians 14:26). Other passages also show the church as a whole acting together in worship (I Corinthians 10:16-17; 11:17-34). Rhoda could speak during the gathering at Mary's house because, while many Christians were gathered, it was not an assembly of the whole church.
The gatherings are times set aside to gather as a church. "When you come together as a church" (I Corinthians 11:18) indicates that Christians can gather without automatically forming a church. In order for a gathering to be "in church" Christians must gather for that purpose. We see the church gathered for other reasons. In Acts 15:4 the church gathered to hear Paul and Barnabas' report. They also determined to send representatives with a message to other churches while gathered (Acts 15:22). These were times when the church acted as a unit, but it wasn't gathered for the purpose of worship. The group acts as a church, but the group is not in church. Thus we read that the whole church (men and women together) being pleased to select deacons (Acts 6:5) and to send a letter with men to deliver it (Acts 15:22). The implication is the Christians, men and women combined, gave their assent in some fashion. This was possible because the local church wasn't gathered at the time for the purpose of being in church.
Is a Time Set Aside by the Congregation for Bible Study "in Church?"
If a church, as a part of their worship assembly, decided to hold an instructional session where the men could ask questions and make comments on the lesson, then that time would be one where the women would have to remain silent. Interactive teaching can be just as much a part of worship as preaching (Acts 2:42; I Corinthians 14:26). It is an opportunity "that all may learn and all may be encouraged" (I Corinthians 14:31).
However, it is also proper to have Bible studies where the entire church is not present. Once again, Aquila and Priscilla are examples of this in Acts 18:24-26. Since this is not an assembly of the entire congregation for the purpose of being in church, the silence of I Corinthians 14 doesn't apply, though rules for quietness in I Timothy 2 still remains. If we could all divide into classes scattered across the city, why could we not divide into classes, but meet in a common location? These classes would not be a part of the worship of the church because the congregation, as a whole, is not gathered together. Even though all the members of the congregation are studying in different rooms, and in those rooms acts of worship (praying, singing, and instruction) are taking place, it remains that the church hasn't called a gathering for the purpose of the entire church to worship. Coming to one location for multiple Bible classes would not constitute being in church.