Elizabeth Daingerfield Zwicky writes to me, following up on my Maurice Sendak "half-sentence" posting (which I'll have more to say about in a while):
… if I knew how to encourage sentence fragments, I would go for that. Opal's sentences go on for*ev*er. And if I type them for her, and she's watching, and I try to put a period in, so there's a shorter sentence even though it starts with "And"? She says "No, that's not right, it's part of the same sentence. Didn't you hear the 'and'?" Fortunately she doesn't usually watch me type, allowing me to punctuate things as I see fit.
Two things here. First, Opal's attention to the conventions of writing, including her awareness of the stupid No Initial Coordinators advice about written English. Opal is 5, in kindergarten (which she started last month), and is writing on her own, but decidedly imperfectly (she is given, for example, to shifting to a new line when she comes to the edge of the paper, even if that's in the middle of a word), so I'm astonished that she even knows about NIC (and can refer, albeit indirectly, to it), much less cares so deeply about it. Where did she pick up this stuff? Certainly not from her family.
The other thing is Elizabeth's reference to "sentence fragments". Sentences with initial coordinators are not sentence fragments on that account (Elizabeth's sentence beginning "And if" is indeed a sentence fragment, but not because of the "And"). The problem is how to refer to sentences with initial coordinators in an unbiased fashion, without labeling them as a kind of error, as "sentence fragments". Yes, I know, sentence fragments are not in general ungrammatical, but the label has picked up an association with incorrectness. Similar problems arise in other contexts, for instance with regard to "dangling modifiers".
The usual labels are the ones that are widely known by the general public, but they are tainted by their association with error, and that makes it hard to talk about phenomena. In my own practice, I bounce back and forth between the usual labels and neutral, but unfamiliar, terminology, depending on the context.