When i was about five years old, my dad gave me my first knife.I was taught not to be stupid wih it (carve away from body.....etc) and some basic cutting strokes.
There is a way of teaching a child by listening. If the child hears itself say what you must/must not with a knife then he/she will be more likely to act upon those rules set.
Dad: What do you think will happen if you carve towards yourself?
Child: I will cut myself
(the super-simplified version of a technique called "motivational Interviewing")
If you know and trust your child, then there will be no problem. It is not the childs fault if he/she has an accident, it is just that he/she hasn't been taught properly.
I hope this is clear, i apologise if it is not. Just email me if you have further questions.