By the late 1800s, what most Americans thought was that assimilation would improve the situation of Native Americans. They thought that the Native Americans should be made as much like white Americans as possible. This would improve things for them.
This attitude can be seen in at least two policies that were important at that time. These were the policies of having Indian children go to boarding schools and the Dawes Act of 1887. In both cases, the idea of the policy was to destroy the traditional ways of the Native Americans and to get them to a point where they were more like white Americans. This sort of process is typically known as assimilation.