Actually, yes, we do. This Slate piece suggests that Elana Kagan use her hearings to education people about the US Constitution. That is, not just the document written at the end of the 18th century, but the amendments since then as well. Even if one omits the US Supreme Court's role in interpreting the document, the document itself has been revised, including to correct for defects like how African-Americans were originally treated. Equal protection under the law--pretty fundamental stuff.
Anyhow, I doubt that Kagan will follow the advice given, but this process may just remind people that the Constitution is more than the stuff listed before the Bill of Rights. I hope so.