Our practice at Trinity Church is very similar to our sister church, Christ Church (whose pastor, Douglas Wilson, wrote a book, "Against the Church" which I have been quoting and commenting on - and these thoughts parallel his in that book, pp31-32). Our children are included with us in the worship of God on the Lord's Day. One of my own sons said his first word in the middle of a worship service, "Adda!" He was chiming in with the rest of us as we shouted our "Amen!" at the end of a hymn.
Lifting their hands with us as we sing the Doxology (which one little saint just last Lord's Day was doing just before I led us into that part of the liturgy - as though he was sending in signals from the side like a baseball coach), they learn early on that they are a part of the people of God.
That is why it makes complete sense to bring them to the Table as well. Baptized children should be admitted to the Table because they are little saints. We never want to teach them at that time that "we are in" while "they are out." We want them to join with us in the journey of learning all there is in the Lord's Supper, and we want to learn from them what it is to have childlike faith.