Over the last dozen or so years I've read a few works about family worship and training children in the faith as well as works geared toward guiding families in worship or Bible study. I have also read works about whole-family worship in churches and why we ought not segregate or remove younger children, or other age groups for that matter, from the worship service of the church (by this I don't mean that it is never appropriate to have a space to take young children to look after them during the service - like a crying room or nursery). I am convinced of the importance of both family worship in the home and the inclusion of children of all ages in the Lord's Day worship service of the church. Now, just because I am convinced of these things does not mean we implement them nearly as well or consistently as we ought to, either at home or at church. However, as Chesterton said, "anything worth doing is worth doing badly." In other words, if it is an important thing, don't wait until you are perfect at it before you begin doing it. And when you fall off, get back on right away.
When speaking of culture, society, the work force, consumer markets, the future of a nation, political view points, and many other things, I have often heard people say that "children are the (fill in the blank) of the future." Children are the leaders of the future. Children are the citizens of the future. Children are the hope of the future. As true as this may be to a degree in other areas, I have seen this same train of thought in the church as well. I have often heard Christians and Christian leaders say that children are the church of the future. This is true in a certain sense: Lord willing, when we adults have fallen asleep and are awaiting the resurrection, our children will still be part of the worshipping community of God's people on earth, the body of Christ, the temple of the Spirit, the Church militant. We want our children to grow up and hold fast to the faith and some of them to become leaders in the church. But there is another sense in which "children are the church of the future" is profoundly untrue. In fact, in the way this sentiment is often expressed, it is very dangerously wrong.
When his disciples were preventing little children from approaching Jesus, even parents brining their infants, the disciples believing as they did that Jesus was already busy enough with important adult ministry, Jesus rebuked them: “Let the children come to me, and do not hinder them, for to such belongs the kingdom of God. Truly, I say to you, whoever does not receive the kingdom of God like a child shall not enter it.” (Luke 18:16-17) We often read this as meaning that only those who receive Jesus and his gospel of the kingdom with childlike faith can truly be his disciples and that is certainly part of what Jesus is saying. But that is not all he is saying here. He also says that the kingdom of heaven, the kingdom of God, belongs to little children. And if that is the case, there is no good spiritual or biblical reason to pull children out of the worship service, the service in which God's covenant community enters into his presence through the blood of his Son and in the indwelling presence and unity of the Spirit and speaks to God and hears from God and worships God and is blessed and built up by God in the communion of the saints. They are already citizens of the kingdom; don't forbid them a place with the citizens who only happen to be more chronologically advanced than they.
If Sunday Lord's Day worship was primarily a teaching time, at least in many churches much of the teaching would not be aimed at young children and so they would miss a fair bit of it (though not nearly as much as we sometimes think). But Lord's Day worship is not primarily teaching time, even thought teaching plays an important part.
If Lord's Day worship was primarily an entertainment time, admittedly the type of entertainment would not be the most effective for engaging young children. But Lord's Day worship is not entertainment. It is interaction. God's people go not only to receive, and certainly not to be passive observers. God's people go to be in back-and-forth conversation with their God, to speak to him and be spoken to by him.
Lord's Day worship is a time of conversation between God and his people, of interaction between the King and the citizens of his kingdom, between a loving Father and his children, between a husband and his bride. The Lord's Day worship service is where God's people are formed through teaching, yes, but also and much more through the pattern of covenantal interaction with our Lord of the covenant. The Sunday morning liturgy is a family ritual, not unlike a dinner time routine, wherein God's children, no matter their age, all come into his house and gather together to converse with, be blessed by, share a meal with, and to ascribe glory to their Heavenly Father.
Alexander Schmemann, an Orthodox theologian and liturgist, has some good thoughts on children and church here.