was it cold?

Illinois in March can be cold.

If I recall correctly you can see steam coming off Judd Nelson's breath as he walks off at the end. So yeah I would say it would have been pretty cold.

You can see at the beginning that it is.

Not necessaril.y

Yes.
Also, layering was cool in the 80's. Also, schools used boilers for heat so they kept the heat down pretty well in the winter to save money. I used to wear my winter jacket all day in school and never got hot.