Assertiveness Training suggests that there are essentially three different ways that people can relate to one another. They can be: 1) aggressive, 2) passive or 3) assertive. Most people come to assertiveness training already understanding what aggression and passivity mean, but they don't understand assertiveness at all, at first.
Aggression is about dominance. A person is aggressive when they impose their will onto another person and force them to submit, in effect invading that person's personal space and boundary. Violence may be used in this effort, but it is not a necessary component of aggression. Passivity, on the other hand is about submission. Passivity occurs when a person submits to another person's dominance play, putting their own wishes and desires aside so as to pay attention to fulfilling the wishes and desires of their dominant partner. They may not like being dominated (most people don't), but it seems like the smart thing to do at the time (perhaps to avoid the threat of violence or other coercion). Aggression is about domination and invasion; it is fundamentally disrespectful of relationship partner's personal boundaries. Passivity is about submission and being invaded; it is fundamentally disrespectful of one's own personal boundaries.
In contrast to these two fundamentally disrespectful positions, assertiveness is about finding a middle way between aggression and passivity that best respects the personal boundaries of all relationship partners. Assertive people defend themselves when someone else attempts to dominate them, using any necessary method (including force) to repel the invasion attempt. Though they can be strong people who are capable of aggressive domination attempts, they never act in an aggressive manner, however, because they know that to do so would cause them to disrespect their relationship partner's boundaries. Another way to say this is that assertive people use aggression defensively, and never offensively.
There are many classic examples of assertive behavior in history that you can draw upon for guidance and inspiration. The examples of Gandhi and Dr. Martin Luther King come to mind readily, however. Both were leaders of oppressed, invaded groups who were dominated by an upper class (British colonials in the case of Gandhi, and the American white establishment in the case of Dr. King). Both leaders came to a realization that submission to the ruling powers was no longer working and that something drastic had to happen. Both leaders chose a path of non-violent resistance - this is what makes their behavior assertive rather than aggressive and what separates them from run-of-the-mill freedom fighters everywhere. Their commitment to non-violent resistance is what made them great. Both leaders demonstrated and protested against their oppression by the powers that held them down, but did so in a manner that respected the people wielding those powers to not themselves be violently targeted or oppressed. Both stuck to their posture of assertive protest despite becoming targets for escalating violence against their person, their families and the people they represented. In the end, both succeeded in making important reform occur, even if only imperfectly. They were able to make change occur through assertion, and you can do it too.
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It is very hard for people used to acting passively to understand how to act assertively, however. Many people new to assertiveness training mistake aggressiveness for assertiveness. This is because their baseline position is passivity, and they literally cannot conceive that there is any alternative to just giving in to the demands of others other than to "fight fire with fire", usually in the same violent manner that their dominant partners model for them. Such newly "assertive" people will start yelling and screaming back at people who have historically yelled and screamed at them, not realizing in their newly empowered angry state that by acting in this way, they are going far beyond what is necessary for defending themselves, and may enter into the realm of becoming themselves abusive and dominating. This beginners mistake is probably inevitable, and certainly okay to make as a temporary and transitional stage towards better learning how to become assertive, but no one should linger there unnecessarily long. To do so is to substitute aggression for passivity, and to become a bully yourself.