Punishment is a means to enforce a social standard, not to encourage honest. Natural morality. People forget why an individual bullies to begin with: Neglect, poor self esteem, etc. I do not condone bullying, but it is most certainly not without cause. Simply spraying weed killer on the weeds will not rid them. Fix a problem at its root. Save punishment for misbehaving animals, not humans.
I apologize for the short response; I was rushed on time.
I do believe some kind of disipline is in order. But they may not have a good living situation and need to be patiently taught that that behavior isn't helping anyone, even themselves. In most of my encounters with bullying I have found that they have abusive parents and are in a rough economic state. Just remember that they are people too, not some rabid animal.
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Being of someone who does live and has lived in a very high class area, there are spoiled kids EVERYWHERE. Spoiled kids are one of the major benefactors to the idea of bullying. This idea that they can have anything that they want leads to the perspective that they are of a higher power and the parents feed into that so much. Now don't get me wrong this is not just the rich kids doing all of this but when it comes to my experiences, that is most of what I have seen. This idea of forcing a kid sit down and face a wall? Going to their room where they have a TV, Xbox One. And Playstation? Oh lord whatever shall we do? This love and logic leads to the idea that there is a positive connotation to the abuse of others that may give them what they want. What they really deserve is something that leaves a negative connotation with that same idea. Kids deserve to have that negativity in their life to give them a representation of what the consequence is to do something bad.
It's the action. Even the best of people with the best of intentions need to be made aware when they are bullying someone else. The type of bullying should ultimately define the degree to which the individual is punished, but we need to be careful when assigning the punishment so that the offender knows that it's the action that needs correcting, and that it's not a direct assault on them as a person. Withholding punishment from a bully because "they are a victim too" is really condoning the action, and we don't want to condone bullying.
That question is the same as should a criminal be helped or punished of course they should be punished. A bully only cares for himself because of an inflated ego and therefore helping him would do nothing to solve the problem. If they are the type of bully that they themselves have been bullied and are only taking it out on other people because of there past should also be held accountable. Those types of bullies are the worst and should be held more accountable for their actions because they know what it is like to forced to the ground by other people.