Possible Clues:

• Worn by corner boy in school as sign of one's stupidity (5'1,3)
• Given out disgracefully by 11 down - it goes over the head of the stupid corner boy (5'1,3)
• Conical headgear of the not too intelligent
• School idiot's hat
• You'd be a fool to wear this
• Something that might go to the head of a slow learner?
• Headgear used to humiliate slow learners