I can see where you're coming from alfiesgirl, but I do think you might be overprotecting her, again which I completely understand. My daughter toddled off to school muffled up to the eyeballs with pants and wellies on with her school skirt, tights and shoes in a bag. It's cold yes, but certainly not unbearably so if you're suitably dressed.
When school closes today -wer'e going to the park to play in it too- daft? Yes possibly but it isnt half fun and bracing ;-)