I've been in meetings all day, but Weasley is not the only one who noticed you were writing during lessons.

I really have no idea where you boys get your ideas. Long ago, corporal punishment was considered a more than suitable penalty for young wizards who do not mind their studies, though nowadays magical punishments and simple non-magical detentions are deemed more appropriate. Ask Professor Carrow, if either of you dare, for a history of torture and its application to over-inquisitive little boys.

No, Quirrell was not discharged for beating his pupils. I think Draco may have deduced the real reason already, given his eagerness to see Lupin meet the same fate for his diltutionary sentiments.