Monday, March 9, 2009

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.


Mr. H said...

Allow students to put their name on papers or leave it off. Who cares? You're not marking the paper so it's not like they stand to lose, if they care to leave it off, they can claim it later (In a small group, handwriting gives this away mostly anyway).

I like the idea of distributing papers and then collecting them - seems like there could be a lot of learning going on. What if you added credit for bringing well thought through, though possibly incorrect work to the fourth hour? That way you could still allow students sufficient time to complete the task while others who hadn't started miss out on credit for having it done with time to revise?

Mitch said...

Time for peer assessment/feedback can be incredibly useful. I like your idea of using it as a way to help your students learn to find mistakes, and you can probably take it a bit further and ask them to give feedback on how they could improve the clarity of their solution (at least for problems, as opposed to exercises). I would not at all be surprised if incorporating an activity of this type increases the number of students who come to class having done at least some of the homework, since they know they'll get input on the problems they've been doing. You probably do want to spend some time discussing what constitutes effective feedback and how to give and receive feedback so that everyone's on the same page.

I think as long as students would be getting a chance to revise their solution before it's turned in for grading, asking them to put their names down is not unreasonable. However, as Mr. H says, they could leave them off and go rummaging, too.

I think after one or two sessions, the problem of students showing up with nothing done will nearly vanish. You'll likely also have others who bring more than one problem they're stuck on to help make up for any that don't bring any. I like the idea of having those who were successful with the assignment make up a mistake for extra credit or something like that.

I think the biggest thing to think about and be careful with is how more timid students will react. Take time to explain why you're doing this: for their learning. When students understand that this is not just their instructor doing something crazy (or that appears to require less preparation), but is intended to enhance their learning, they usually buy in. You might also want to think about ways you can minimize any freeloader problems. I don't think it's all that likely, but a student or two might think "Hey, I can go to discussion section, see lots of other people's half-wrong assignments, copy them down along with the feedback on what's wrong, and then go fix them up and use it for my homework." In an ideal world, students would still learn if they did this, but you might have some who try to cheat the system.

For policies, you might try letting the class help you come up with the policies once you explain the point of the activity. You can come in with your ideas on what is/isn't acceptable and retain veto authority but let them try to decide how things would work best.

OK, I've clearly been reading too many education books this term, as I've got a zillion thoughts tonight, it seems. You might check out Mary Ellen Weimer's Learner-Centered Teaching, as she's got a number of ideas on making the classroom learner-centered that seem to align with things you've posted about.

Kate said...

I like it! Great idea.

I think you should just structure it in a way that seems like it will be workable, and then make adjustments later. It definitely won't work perfectly to start, it will take a few semesters...just don't abandon the idea. Keep refining it.

Maybe to avoid freeloaders, you can require them to bring at least one as a "ticket in the door". If they don't have any they can't figure out, require them to write one out with an intentional mistake. Give them a trivial grade for being prepared for class. If they have done nothing so far, they can't participate and get a 0 for being prepared. Tough noogies.

I don't see anything wrong with students writing their names (on the back?), so they can get them back easily.

sumidiot said...

Thanks, all, for the feedback. Very much appreciated!

It sounds like some sort of credit for bringing in papers is generally recommended to avoid people not bringing anything. I hadn't even thought of @Mitch's point that students could copy other's work with this setup. Perhaps having people look at papers in groups would, via peer pressure, ensure that nobody does this.

I like the reminders that I can modify policies after I start, and can ask for the class to help set up policies. I never think of such things.

Also, @Mitch's extension to the setup, where students give feedback, sounds like a great idea. And I'll have to check out that book you mentioned.

Thanks again! Hopefully I'll get a chance to try this out in an upcoming semester, and will let you all know how it went.