Explain how Dickinson contradicts popular ideas in the poem "Much Madness."

In this very short poem, the speaker protests that, "to a discerning Eye . . . much Madness is divinest Sense." What she is saying is that many so-called "discerning" people will see things that, to the speaker, are clearly mad and decide that they are "divinest Sense." In the same way, they will look at things that are sensible and see them as "the starkest madness." The speaker says that "the Majority in this, as all, prevail." If you agree with what the majority decides—"assent"—then you are "sane," but if you "demur"—that is, disagree—you are immediately declared dangerous "and handled with a Chain."
In her final line, there may be an element of hyperbole, but we can also interpret this as a protest against the swiftness with which many were committed to mental institutions in Dickinson's time, particularly women, for daring to be different or to express anything other than total conformity with the prevailing opinion.