The problem I have with the 'IGNORE BULLIES AND BULLYING!' theory of teaching children to cope with harassment is that there isn't any other provision taken in most schools--at least not in the ones I dealt with. My last primary school (the one in the US that I went to for my last three years) called it the 'De-Bugging' system and still use the same set of guidelines to this day. I remember them quite well: 1, Ignore it; 2, Walk Away; 3, Tell a grownup. Other schools have similar systems in place. The problem with this is that it puts ALL responsibility on the victim. None of the things children are advised to do directly impact the problem: the bully.
It honestly is a good idea to do what you can to give the impression that the bully hasn't succeeded in their goal, which is to hurt or upset you, by ignoring it or underreacting. But even when a school DOES take the unusual step of trying to deal with the bully him/herself, they don't usually do it very effectively--in all the cutesy informational videos I remember watching or stories I had to read about it, the end usually consisted of the bully and victim becoming good friends once they put their differences aside. That always leaves the impression that you either hate someone or want to be their buddy. It's okay not to be friends with everyone, but it ISN'T okay to bully anyone.
I just wish I had a better solution to suggest. Unfortunately for me, my only avenue was to fight back--in the end it did work somewhat, and people were less willing to torment me once they knew I could and WOULD seriously fuck them over. I guess I'm lucky because my dad was himself bullied in school and he also got to the end of his rope and snapped and had one good, winning fistfight with each of them that ultimately stopped the bullying completely. He didn't like that he had to do it, and he didn't like that I'd done it either, but he understood why I did it and didn't punish me for it.