Students are inquiring, investigating, discussing, and constructing knowledge (just maybe not the exact knowledge we thought). The put forth their ideas in the community of their peers and justify and defend their thinking. In this sense, students are acting like real mathematicians who speak to one another. They ask questions and comment on another's ideas. They defend their ideas to the community, not just to the teacher.
So, let's not try to fix the mathematics, but rather, work with the mathematician. The point is not to get all of the students to agree with your answer, or to fix the mistakes in their products, but to support their development as mathematicians.