Nathaniel Hawthorne does not say much about raising children in The Scarlet Letter; instead, readers must infer some things from the few times he mentions children in town and the few dramatic pictures he paints of them.
In the second chapter, everyone in town has gathered outside the prison door, eagerly anticipating the appearance of the prisoner and her subsequent punishment. This same excitement, says Hawthorne, would have been the case for several other kinds of incidents in this Puritan town:
[I]n that early severity of the Puritan character, ...it might be that a sluggish bond-servant or an undutiful child, whom his parents had given over to the civil authority, was to be corrected at the whipping-post.
It might have also been a heretic who "was to be scourged out of the town," a drunk Indian who was whipped back into the woods, or a witch who would be hanged.
In [any] case, there was very much the same solemnity of demeanour on the part of the spectators; as befitted a people amongst whom religion and law were almost identical, and in whose character both were so thoroughly interfused, that the mildest and severest acts of public discipline were alike made venerable and awful.
From this description, we understand that a for a Puritan child being "undutiful" rises to the same level as witchcraft, drunkenness, and heresy. The practice of publicly whipping a child for being disobedient probably deters overt disobedience in Puritan children.
In chapter six, Hawthorne says,
The discipline of the family, in those days, was of a far more rigid kind than now. The frown, the harsh rebuke, the frequent application of the rod, enjoined by Scriptural authority, were used, not merely in the way of punishment for actual offences, but as a wholesome regimen for the growth and promotion of all childish virtues.
Despite that, Hester is unable to make herself discipline Pearl at all, let alone impose the severe Puritan discipline on her.
The Puritan children's play times consisted of
disporting themselves in such grim fashion as the Puritanic nurture would permit; playing at going to church, perchance; or at scourging Quakers; or taking scalps in a sham-fight with the Indians; or scaring one another with freaks of imitative witchcraft. These are the only games they knew and were allowed to play.
The other children know that Pearl is different, both because of how Pearl acts and from their parents. They mock and taunt Hester with the word they have heard their parents say, and it is horrifying to hear it come from the lips of children: "adulteress."
The truth was, that the little Puritans, being of the most intolerant brood that ever lived, had got a vague idea of something outlandish, unearthly, or at variance with ordinary fashions, in the mother and child; and therefore scorned them in their hearts, and not unfrequently reviled them with their tongues.
Finally, in chapter seven, the children stop their "playing" as Hester and Pearl walk by, and they say:
"Behold, verily, there is the woman of the scarlet letter; and, of a truth, moreover, there is the likeness of the scarlet letter running along by her side! Come, therefore, and let us fling mud at them!"
It is clear that Puritan children are taught early the severity of discipline, sin, and condemnation so rife in the Puritan faith; they also learn how to show their derision and scorn for sin and practice it often on Hester and Pearl.