Roger Ebert describes a scene in the documentary Bully:
"There's only one bully actually identified in the film. A clueless teacher has pulled together two kids who got in a fight and insists they shake hands. One kid is as friendly as an insurance salesman and sticks out his paw with a friendly smile.
The other kid refuses. The teacher sends the friendly kid away and lectures the other one. Of course it's the “nice kid” who is the bully. He probably gets away with more stuff than a con man." (Roger Ebert's review of Bully)
I have been a cluesless teacher myself.
I don't think I'm the only clueless adult, because when I searched at Google Images just now I saw a pretty heavy bias towards depicting bullies as overt, buffoonish stereotypes of Butch from the Our Gang movies. There were few image depictions of the opportunistic clique leaders, the abusive class clowns, that get insufficent scrutiny from teachers.
I'm going to the library now to see if I can find Barbara Coloroso's The Bully, The Bullied, and The Bystander , a book that even has Jiu-Jitsu instructions listed in the table of contents, but also has a large section assessing the people who stand by during bullying, and the people who don't recognize bullying when they see it.