A year or two ago, I came across the following post on teaching young kids how to apologize:

http://www.cuppacocoa.com/a-better-way-to-say-sorry/

The thing that struck me about it is that in the process of explaining the core of what makes a meaningful apology to children, it does a really nice job of reflecting where, exactly, most public (and, really, private!) apologies tend to fall down.

A good addition to this article.