When my husband and I first gave our lives to Jesus (1975) we attended a Four Square church in Fresno, California. They taught that there is an "age of acknowledgement or understanding" for children. Basically, a child could be at the age of 12 or 13 before they were at an age to be expected to understand the gospel and salvation. Where does this doctrine originate from and is there any reference to age in the bible?
Usually this is referred to as the age of accountability, and it's considered to be an age when someone is old enough to be accountable for their sinful behavior. The idea is that before the age of accountability a child is simply excused due to their inability to fully understand the consequences of sinful behavior.
The only real reference to an age of accountability in the Bible is in the book of Numbers when God judged the unbelief of the adults who refused to enter the Land of Promise. God told Moses that all the adults twenty years of age and older would die in the wilderness, and instead their children (those under twenty years of age) would not be held responsible for the rebellion.
That's the only biblical story I can think of where an age of accountability is spelled out, but I don't think I would teach that 20 is the biblical age of accountability. Neither would I say 12, or 13 or any other age for that matter. Why? Because the Bible doesn't list one.
I do believe that there's an age when an individual becomes accountable before God for their actions, but I'm more disposed to the idea that God knows the heart of every person and makes a determination based on a person's cognitive maturity.
The belief in an age of accountability infers that people under a certain age (whatever that may be) are exempt from judgment since they are unable to fully comprehend the seriousness of their sinful condition and therefore unable to make reparation through the blood of Christ. All we really know is that Jesus made statements about children suggesting that they were entrusted to the care of angels. Also He made a special effort to acknowledge and bless children during his earthly ministry.(See Matthew 18:10 and 19:13-14)
Since we know that our God is a God of perfect justice, it is reasonable to presume that children, although not innocent from the stain of sin, enjoy a special consideration regarding their personal culpability.