Children and the Lord's Supper

On Holy Sunday this year, the children at my church were allowed to participate in Communion. This doesn't usually happen, and I was extremely happy. Much as I love my church, I really disagree with many of our childrens ministry practices. It is not that children are specifically not allowed to take part in communion, but that the structure of our Sunday morning service means they are not able to.
Mentioning this in an after church chat revealed that not everyone agrees with me (shock horror), which, of course has resulted in me musing, and finally deciding to blog about it. :)

What I believe: Children are a part of Jesus' church, and therefore shouldn't be excluded from any aspect of communal worship. They should be allowed to have full participation in the sacraments. My understanding is that there are two sacraments, Holy Communion and Baptism.
(Side note: I do not believe in infant baptism, although I think it is OK if others want to do it, I won't be baptising my kids as babies. What I am referring to here is children making the decision to be baptised. For example I was baptised when I was about 7 or 8, because I wanted to make that commitment to God.)

Why I believe this: I guess I have a few different perspectives on this.

On a personal level, I took communion as a child, and it was a hugely significant part of my faith journey. My earliest memory of church is being given communion, and mum or dad (I assume) saying 'this is the special juice/bread that reminds us how much Jesus loves us'. The reason I am a Christian today is because faith was such an integral part of my life as a child. I have always known that God loves me, because I was told that. The transition between believing because I trusted my parents and believing because I had experienced this love myself was seamless. There were no classes I had to go through, or steps I had to take before I could be fully a part of God's family. I was born there, and stayed there my whole life thus far. My family's attitude towards communion was very important in this journey.

When I read the bible, I cannot find anywhere where children are forbidden from taking part in communion.
I do find Jesus saying "Let the little children come to me, and do not hinder them, for the kingdom of heaven belongs to such as these." (Matthew 19:14). I know that this is not referring to communion, but it makes me think that if the Kingdom of Heaven belongs to them, then surely we shouldn't be holding them back from participating in any part of it?
1 Corinthians 11:17-34 deals with abuse of the Lord's Supper. Many people use this to back up their claims that children should not be involved in Communion. Basically they say that children are too young too understand exactly what communion signifies, and are therefore unworthy of taking it. I would argue that none of us can fully understand salvation, and are continuously growing in our understanding of it.

*spends 20 minutes googling for articles against children taking part in communion so she has something to argue against. fails*

OK, so can people who disagree please post why, coz I really don't understand why people believe kids shouldn't be involved in Communion.

I do want to have a better understanding of this. Maybe I am wrong. I have been wrong once or twice before...