But that's not how "math," as taught to and perceived by middle schoolers, works; as far as they've likely ever seen, a math problem is a question for which the

But that's not how "math," as taught to and perceived by middle schoolers, works; as far as they've likely ever seen, a math problem is a question for which there exists an answer in the form of a sequence of arithmetical operations and a final number or equality. I doubt these students even recognized the concept of a "proof" in this situation- at best, they simply "saw" that it was impossible. But that "seeing" is simply a piece of everyday human reasoning, not the wholly different things that they associate with math classes and happy math teachers, so of course they feel like something is missing. It does sound like the teacher had a good idea- probably it was precisely to get students out of this notion that every math problem has the simple kind of answer they're used to. But it also sounds like the lesson wasn't taught with attention commensurate with its profundity- its easy to forget that these ideas which are so fundamental for us are still alien to most grade-schoolers (or for that matter, many high-school graduates.)