How Do YOUR Kids Dress For Winter Time?

This time of year many of our children start their day with winds blowing and the temperatures at a cool 35 - 40 degrees.  By lunch time and recess the sun is shining and the temperature makes it to 50.  How do you prepare your kids for days like these?  My rule is my daughter must wear a coat.  Another rule of mine is if you wear your coat outside at school for recess you must keep it on.  None of this taking it off.
I am curious what other parents do when the weather is like this.  I want to trust that my daughter is being taken care of when she is not in my care.  This means if it's in the high 40's and sunny I still expect her to wear her jacket.  Do you?
My husband and I went and had lunch with Grace for her 9th birthday this past Monday.  Recess is right before lunch so it is always fun to see Grace and her friends outside playing.  I was not happy with what I saw when I walked outside.  (I did not make a big deal out of it because this was her special day and I didn't want to embarrass her)  The weather was mid 40's, extremely windy and I was chilly with my winter coat on.  Grace had her coat off with just a thin long sleeve shirt on as she and  several of the other kids did.  What I did notice was the recess teacher on duty had HER winter coat on PLUS her hood up.  GRRRRR
Now, my daughter is old enough to make these decisions for herself.  I understand that Grace made the decision to take her coat off.  I also understand that the teacher said or did nothing but yet she had her coat on, button with her hood on.  I was not pleased by this.
That night Grace and I talked about it and I helped her realize the importance of wearing her coat when it is so cold.  Growing up in Iowa I NEVER went through a phase of not wearing a coat.  It was too damn cold!  I just I don't understand this not wanting to wear a coat thing.  When it is warmer and above 50's we do the large sweatshirt and sleeveless vest.  But when that weather reaches the 40's, windy to me a coat is necessary for my little angel.

So, I ask you...do you make your children wear their coats?  Do you let them decide on their own?