Do Christians and Their Unbaptized Children Pray to the Same God?

The New Calvinists are apparently worried about teaching unbelievers to pray:

While it is wise to be discerning with our children as they grow up and to not give them a false sense of security if they’re not actually Christians, I don’t know of any place that the Bible warns parents to beware of teaching children to pray too early. Rather, we are told to teach them and this includes not just facts, but also practices. By encouraging our children to pray, we are teaching them the language, the practice, and the importance of prayer.

That could also be a reason for bringing back prayer into public schools.

But John Piper is on the same page with Tim Challies:

Yes. I think we should teach our children to pray as soon as they can say anything. … I can’t discern when a child is being spiritually wrought upon by the Lord. … I can’t tell precisely when his faith becomes his own and authentic, I don’t want to wait too long before I start treating him as a believer. …

Would that also apply to David Bowie when he prayed the Lord’s Prayer?

Tim Challies argues, finally, that the Bible is for children praying:

… the Bible doesn’t warn parents against teaching such things to their children too soon. On the other hand, in both the Old and New Testaments, parents (and especially fathers) are told to teach their children to obey the word of God (which includes the practice of prayer). Consider these verses:

“These words that I command you today shall be on your heart. You shall teach them diligently to your children, and shall talk of them when you sit in your house, and when you walk by the way, and when you lie down, and when you rise.” (Deuteronomy 6:6–7)

“Come, O children, listen to me; I will teach you the fear of the LORD.” (Psalm 34:11)

“Discipline your son, and he will give you rest; he will give delight to your heart.” (Proverbs 29:17)

“Fathers, do not provoke your children to anger, but bring them up in the discipline and instruction of the Lord.” (Ephesians 6:4)

The Bible also tells parents that their children should have the sign of the covenant. Infant baptism would fix this problem in a hurry. As non-communicant Christians, why shouldn’t baptized children learn to pray?