Paul says 3 positive things about speaking in tongues:
(a) It is a communing with God through prayer and praise.
(b) The content of tongues is
"mysteries" spoken by the (Holy) Spirit.
Despite these positive aspects about tongues, Paul urges them not to speak in tongues in public worship, unless with interpretation (v.5,13,27). The reason is that tongues edifies the speaker but not the church because it is addressed to God; prophecy edifies the church because it speaks to the people. (Prophecy is not a prepared sermon, but the spontaneous word given by the Spirit for edification.)
If tongues are interpreted, what the person has been speaking to God has now been made intelligible, then others may be edified.
From the perspective of the hearer, unknown languages do not have meaning. Speaking in tongues would sound like foreigners, thus creating a barrier between believers.
In corporate worship, besides worshipping God, the focus is the building up of the church, not personal experience in the Spirit.
It is possible that the one who speaks in tongues also has the gift of interpreting tongues.
Paul will do two things: one apparently for his own sake (pray and praise in tongues in private devotion), the other for the sake of others (pray and praise in Greek in the assembly).
Paul's personal devotion was regularly given to praying and praising in tongues. The edifying of oneself in this way is a good thing; just that it is not the focus of corporate worship.
The childishness of the Corinthians consists of thinking improperly that tongues serves as evidence of their superior spirituality while in fact they evidence all kinds of ethical or behavioural abberations. Paul tells them to cease being childish in their thinking but be as innocent as babies in their behaviour.
Paul does not make a precise citation from Isaiah 28:11-12. His point is that speaking in tongues by foreigners did not effect belief in Israel; they still did not obey.
Some believe that the "sign for unbelievers" refers to the event of Ac 2:5-11 where unbelievers saw the work of God through the miracle of tongues. Yet the word "then" links this verse with the quote from Isaiah where the message is that tongues do not effect belief.
In the Bible, sign functions as an expression of God's attitude; signifying either his disapproval or pleasure. In this case, it is his disapproval. Here, uninterpreted tongues function as a negative sign for unbelievers, not to their advantage but to their disadvantage. Because tongues are unintelligible, unbelievers receive no revelation from God. Even worse, by their response of seeing the work of the Spirit as madness, they are destined for divine judgment. Paul tells them to stop the public use of tongues, since it serves to drive unbelievers away.
In contrast, prophecy functions as a sign for believers because unbelievers may be brought to faith and become believers. Its revelatory character functions as the sign of God's approval.
"Unbeliever" may be translated as seeker or inquirer (see NIV marginal note), someone who is halfway between an unbeliever and a believing Christian.
Through prophecy, the unbeliever is convicted by all those present, is called to account before God by all those present. The Holy Spirit exposes his/her sins and results in his/her conversion.