I identified my self as a pediatrician, and gave a quick talk on peanut allergy and its potential consequences. Children with such a reaction will, within minutes of coming into contact with what they are allergic to, get symptoms including hives, swelling of the face, mouth tongue, difficulty breathing. They can, if left untreated, collapse into shock. I also bluntly explained that anaphylaxis can result in death, and that I had actually witnessed this first hand. The room went silent for a moment and the teacher thanked me for clearing things up.
Some children can get a reaction just by smelling peanut odor.
Others can get a reaction by playing with some one who ate peanut butter without having washed his hands and face. Obviously, the younger the child, the more likely that accidents will happen. They say that in the real world one cannot expect to have such peanut free zones. True enough, but I believe that is absolutely necessary to protect children from inadvertent exposure to peanuts if possible. And peanut-free schools are the best way of achieving this goal.