The other time I hear myself barking "Say you're sorry!" is when I'm with a friend and his or her kids, and one of my kids is being obnoxious, and I'm embarrassed. This one I can't really defend: It's a cheap way to signal that I, for one, have some manners; that I know my kids are being trolls and won't let them get away with it, at least not entirely. Forcing an apology is a lot easier than imposing a real punishment. So, it suits for small- to medium-sized infractions that I feel like I should address (or rather shouldn't be seen letting go). Especially, if I'm honest, toward the end of a long day.
Basically, she finds that for most things, it is the same as having adults apologise: it is a communication of someone's acknowledgement of a social wrong. The only difference with kids is that the communication is between parents rather than the wronger and wrongee.
That said, I am not sure if we are going to gather in a 'peace circle' and have our children discuss their feelings when they fight with each other. Not that there is anything wrong with that but it just isn't 'us.'
Instead, these school holidays whenever the inevitable fights between children emerged, I opted for a special 'joint' sending to the corner. This is the place in our house where children go for a time out. The usual rule is that there is no talking in the corner and no communication into and out of it except by the sentencing parent. However, with a 'joint' sending communication in the corner is allowed.
Initially, there was no such talking. Just two children sitting there with increasingly over-acting frowns of frustration, disgust and a vast array of annoyed emotions. Then eventually, the two of them realise their joint predicament and start to wonder what it was that put them there. Apparently, the triviality of it is not lost on them and they are soon giggling and plotting against their true enemy: me. The common cause resolves the conflict and harmony is restored. And the best part, I can do other things all the while. So zero parental effort/involvement equals the same outcome as a ton of parental effort/involvement. For the economist in me, the choice is a no-brainer.