Be sure to recognize that either situation can be a strength and an area of weakness. Both of these types of personalities need an equal amount of repentant faith on behalf of the child, and an equal amount of grace and redemption by Jesus.
Forgiving Too Quickly
When Child A (remaining nameless, for now) is sinned against, this child usually is able to forgive and forget (well, forget at least). Being able to go back to life as normal is definitely a benefit, as peace is soon restored.
But just because "peace" is apparent, does not mean that the matter has been resolved. The danger of not dealing with the issue with intentionality is that some underlying bitterness may remain. When you can forgive and forget based on your own ability, that is not trusting in Jesus.
Additionally, Child A is also usually just as flippant when Child A is the offender. A quick "I'm sorry" does not indicate a heart that is repentant, especially if the behavior is habitual. We need to help move our children from the offense, to sorrow, to repentance.
When a child "cares too little" about offenses (whether they are the offender or the one offended), they may be missing an opportunity to experience true forgiveness (as the giver or as the receiver). Just as it takes the grace of God to repent, it also takes the same work of the Spirit to forgive. The goal must not be to just get back to a false peace, but to grow in one's faith and trust in the gospel of Jesus Christ.
Not Forgiving Enough
On the other end of the extreme is Child B. This child will hold on to sins. What I appreciate about Child B is the apparent deep sorrow over what the child has done to hurt someone else. Child B's heart is deeply disturbed by this child's own sin.
What concerns me in this case is that Child B has a hard time trusting in Christ's forgiveness. I want this child to fully rejoice in the fact that Jesus promises to forgive our confessed sins (I John 1:9). And I can more easily relate to this child, as I struggle with guilt as well.
Additionally, Child B has a harder time forgiving others when sinned against. This hardness of heart is just as much of a lack of faith as when the child fails trust in Christ's forgiveness. Instead, we need to forgive others completely, just as Jesus did for us (Ephesians 4:32).
For more reading, check out Biblical Parenting vs Gospel Parenting.
Do you have a child who forgives "too quickly," or finds it hard to forgive (whether himself or others)?
**image courtesy of libni via sxc.hu