A "big question" is supposed to do two things in a math class. First, it’s supposed to help students situate knowledge. Second, it’s supposed to make the content more meaningful to students. How does a question have this effect?
My (totally made up) analysis is that we’re trying to bootstrap our math content onto a question that students can quickly recognize as meaningful, and an approach to answering the question that students can quickly recognize as natural. We’re hoping that they care about the question (giving our, say, Alg2 content value) and that they’ll remember the natural approach to answering the question (so that they can associate our, say, Alg2 content with the approach).
A question such as “What simple functions are there?” is no help to students because (a) they’re not interested in the answer and (b) they don’t have any idea how to go about answering it. As a consequence, the question (a) is unable to make Alg2 more meaningful to students and (b) unable to provide students with a framework for their knowledge.
Even excellent metaphors or analogies won't necessarily make great "big questions." Take the idea that functions are analogous to relationships; just as we could catalog human relationships, we could catalog the numerical ones. In question form, that looks like, “What kinds of relationships can numbers have?”
But what makes for a great metaphor or analogy, in this case, doesn't lead to a great question. It fails at both the tasks that a "big question" is supposed to excel at: (a) I don’t think my kids will think that it’s worth answering and (b) I can’t think of a natural way to go about answering the question.
The next step for me is to pick anything--anything at all!--from the three math courses that I teach and try to get some practice finding big questions. Then I'll try to take on NY's Algebra 2.