I read some time back that if the fathers have not taught their children properly that the sins of the children were on their heads and not the children. For the children are not aware of the sin, but are only doing what was taught them. So if the child is not aware to repent how can the child be held accountable?
In order to answer your questions correctly let’s begin with two scriptures that correlate with your question. In the Book of Mormon we are taught from the words of Jacob,
” Wherefore, ye shall remember your children, how that ye have grieved their hearts because of the example that ye have set before them; and also, remember that ye may, because of your filthiness, bring your children unto destruction, and their sins be heaped upon your heads at the last day” (Jacob 3:10)
Modern revelation via the Doctrine and Covenants specifically uses the phrase you are referring to when the Lord said,
“And again, inasmuch as parents have children in Zion, or in any of her stakes which are organized, that teach them not to understand the doctrine of repentance, faith in Christ the Son of the living God, and of the baptisms and the gift of the Holy Ghost by the laying on of the hands, when eight years old, the sin be upon the head of the parents.” (Doctrine and Covenants 68: 25)
These two scriptures cause me to ponder the words spoken by Dr. Paul Popenoe as quoted by President A. Theodore Tuttle, “Dr. Paul Popenoe said, “Our youth are not products of their own lives, but of what their parents give them. If we can get parents to set a good example, we will take away the greatest stumbling block between generations. (“The Things That Matter Most,” President A. Theodore Tuttle, December 1971 Ensign)
The Prophet Joseph Smith also declared,
“If children are to be brought up in the way they should go, to be good citizens here and happy hereafter, they must be taught. It is idle to suppose that children will grow up good, while surrounded with wickedness, without cultivation. It is folly to suppose that they can become learned without education.” (Discourses of the Prophet Joseph Smith, p. 273)
Thus we come to your question, “So if children are not aware of sin or repentance how can they be held accountable?” The second Article of Faith teaches us that “we believe that man will be punished for their own sins and not for Adam’s transgression.” All of the sons and daughters of God will be held accountable for their own sins, their merits, whether they be good/evil.
The gospel of Jesus Christ sheds some light pertaining to our accountability before God when Moroni declared, “For behold, the Spirit of Christ is given to every man, that he may know good from evil” (Moroni 7:16). In other words, despite the lack of responsibility from parents, children are still blessed with the light of Christ. Children still experience opposition, the ability to choose between truth and error. Children are still taught truth throughout their lives and they will be held accountable before the Lord for the truth they have received and for the truth they have rejected, and vice-versa, for wickedness they have received and for wickedness they have rejected.
The gospel of Jesus Christ also gives insight on how merciful our Father in heaven is toward children who have not been taught when Nephi chastened the Nephites by declaring,
“For behold, [the Lamanites] are more righteous than you, for they have not sinned against that great knowledge which ye have received; therefore the Lord will be merciful unto them; yea, he will lengthen out their days and increase their seed” (Helaman 7:24).
As sons and daughters of God, we will be held accountable (despite the teachings of our parents) according to the decisions we have made, and according to the light and knowledge we have received.