1、allow sb to do sth,allow后有人（宾语）时,用不定式,意思是允许某人做某事,如The teacher allowed us to discuss in class yesterday.
2、allow doing sth,allow后没有人（宾语）时,用动词ing形式,意思是允许做某事,如The teacher allowed discussing in class yesterday.没说允许谁做,所以用动词ing形式.
3、一般不用allow to do sth这种形式