Finding Mistakes

Of all the questions I get in the office hours for my calculus classes, the most frequent are probably from students who have worked through a problem and gotten the wrong answer, but can't find their mistake. I sit down with these students and go through each line of their work, ideally getting them to explain each of their steps to me. Sometimes, students are able to spot their own errors when we do this. Frequently, though, they can't.

While it's generally not terribly difficult for me to find errors, it's a skill I believe I've developed after several years in my math classes. It's a skill I'd like for my students to develop. Recently, I struck on an idea for how to run class that might help students find mistakes.

Our calculus classes are accompanied by an additional class period, called the fourth hour or discussion section. Mostly what happens during this time is that students ask questions from the homework, and the TA works them. Or, at least, gives some hints for how to work them. Sometimes the TA for the discussion section is just the instructor, sometimes it is another graduate student. While students certainly appreciate the chance to ask these questions and get answers to their homework, this setup has always frustrated me.

Part of the problem with this setup is the partition of the class into students who have started problems but gotten stuck or made a mistake, and students who have not started problem. The students who have not started are waiting for as many answers in the discussion as possible before doing whatever few remaining problems there are on their own. My hope is that these students do poorly on the exam, if I'm honest. The other students, the ones who ask the questions, because they have started their work, are also not gaining much from most of the time spent answering their question. This is because their mistake shows up, or they got stuck, mid-way through the problem, so all the time used in class getting to that point of the solution isn't much help. However, starting mid-way through the problem won't work, since most of the rest of the class will be lost.

I think a better plan would be to have students bring in their work, and spend most, if not all, of discussion sections working on finding mistakes. I'm trying to think about implementation details for this, and thought I'd see about getting some feedback here. I envision students writing each problem that they worked on but didn't get correct on a separate sheet of paper, and bringing those to discussion sections. Then, during class, the papers would all be gathered up and redistributed to all the students. Depending on how many papers there are, students might break into groups to tackle a paper, or perhaps collections of papers (or they could work alone). With all of those eyes, bugs are, famously, shallow. Groups would make notes on the paper about errors, or tips on how to proceed, and then papers would be returned to their owner. If, after this time, nobody can find a mistake on a paper, or everybody is stuck at some point in the same problem, the TA can talk through the problem with the class.

So, a few questions.

1. Do I have students write their names on the papers, so that it's easy to get them returned? Or does this violate some sort of anonymity that should be preserved? Would writing some fixed number on the paper be better? Or perhaps initials? Maybe have students write something on their papers that they can identify, but other students wont, and then when passing papers back, just hand back all the papers at once, to be passed around so students can grab their own?
2. What about students who haven't started the assignment? Or completed it successfully? Should I set things up so the assignment is due very shortly after the discussion section, encouraging students to have looked at it before-hand? Or will this lead to more students in office hours, avoiding postponing getting their problems fixed, and thus defeating the purpose of the group mistake-finding exercise? Should I have students who have already finished pick a problem and write up a fake solution, artificially introducing an error, to give somebody (whoever ends up with their paper) a chance to try to find the mistake?
3. What could go wrong with this setup? What policies should be put in place? What do I need to be careful of, or think more about?

Thanks for any feedback.