Other forms: insightfully
What English teachers want from students writing papers is insightful analysis –– thinking that sees beyond the obvious and expresses an original thought.
With this word, remember in- and -sight-. When you have the ability to look (sight) inside (in) something––a painting, a discussion, a situation––and find what others aren't seeing, you are being insightful. An insightful person is someone capable of deep, insightful thinking.