I have a Scottie, now just under a year old. When walking him, I see lots of children out who are plainly terrified of him. Given his size, I find this a little difficult to understand sometimes. However, quite a number of children run away screaming - which is bound to provoke the dog's interest, if he wasn't before. He's a predator - he can't help chasing things - and if a child runs away, they've indicated they're fair game. I'm not saying that he chases them with evil intent - he really only wants to play - but he wouldn't have been triggered to pay them any attention had they not made a commotion in the first place. (He doesn't chase cats, for instance, because he's never met one that ran away from him.) I know it may be difficult - particularly for a smaller child - but standing still, or retreating slowly, is probably going to cause less of a problem in the long run.
I'd be a bit wary about what puppies you introduce to a frightened child, since those first puppy teeth are like needles - and it takes a while for them to learn bite inhibition (i.e. that they can't close their teeth as hard on human skin as they did on their litter mates). When I first had this dog, he opened quite a deep wound across the palm of my hand whilst I was working on bite inhibition. Now, he likes to nibble fingers, but it's very very gentle. And he does it while he's on his back, with all his sensitive bits exposed, so I like to think of as a mutual trust thing.
I do worry myself, though, when I see people out with dogs that they plainly can't control.