I would like questions that demonstrate why a mathematical tool or technique is useful, and which can be used to introduce that idea. Ideally, this would be a compilation of problems organized by the idea they motivate. Does anyone know of such a list? I am thinking of middle school and high school math, but I think such a list would be useful at any level.
This is similar to fundamental examples, I recommend you read that post as it was very interesting and at least partially answers your question.
I suggest you take a look at the Tricki:
If you're interested in middle school/high school math education, I'm sure you already know about Dan Meyer's excellent blog dy/dan. In case you haven't, he shares (classroom-tested) lesson ideas and media under the label "What Can You Do With This?" (the individual posts are also tagged by subject area).