Collaboration on math homework assignments?

Question

There is considerable evidence that pair programming, when executed properly, both increases the accuracy of the code produced and enhances the learning of both participants. I wonder if anyone has explored similar pairing for math homework assignments, say, in a college Discrete Math course? The two domains are rather different, and it is quite possible the advantages disappear in a course where proof assignments are not uncommon.

[Updated to specify graded assignments.]

I use group work in the classroom, but have not permitted it for graded homework assignments. If anyone has experience, positive or negative, with permitting collaboration on graded assignments, or can point me to relevant literature, I would appreciate it.

Answer 1

As I see it, the key issue with allowing collaboration on homework assignments is how to suitably assign grades, and in particular how to be seen to be 'fair' in the process of doing so. The approach I've been introduced to is to use peer assessment to modify the marks from the mark assigned to the group piece of work to an individual mark for each student. The idea is that the students report on how well each person contributed to the group, by whatever criteria of 'well' you set up.

WebPA is a system designed to implement this process. The documentation contains some discussion of the relevant research.

Answer 2

What level of mathematics are you looking at? When you say discrete math is this an undergrad course for general students, math majors or a grad course?

My experience has been that for undergrad courses for general students it was enough to make sure they DID their homework whether or not they collaborated seemed to make little difference as an indicator of test performance.

For grad courses the problems were usually hard enough that we would collaborate and just each write up our own solution. That again would usually be enough to distinguish between those that actually came up with the solutions vs. those that just listened to them.

I think the trickiest part is math majors. There the problems might be easy enough to be fairly easy to copy while still being hard enough that you want to the students to do more then just write them down.

EDIT In response to clarified/amended OP.

When I TA'd/taught I would grade in a very minimalist way (choose 5 problems and grade them 0/1 add that up and add up to 2 points for neatness and up to 3 points for completeness). In this case I felt that whether or not they collaborated was not particularly important with regards to the homework grade (which would usually comprise at most 20% of their course grade). This is usually appropriate if each problem is fairly easy (derivatives, integrals, combinations, permutations etc.).

If the problems are hard enough to necessitate a decent write up (a page+ in mathematical notation) you can probably get away with letting the students collaborate and insist they write the problems up separately. This works better then you would expect since usually they come up with the solution together, but actually do write them up separately (copy and paste would be obvious and since they've got their notes it's easy enough to write up on their own right? wrong) and that surprisingly gives you more then enough to grade since if they didn't understand the solution properly they will make mistakes in the write up.

I still think there is a middle ground here where the problems aren't quite hard enough for write ups to make a difference yet they are hard enough that just checking they copied it down correctly isn't enough. I admit I don't know what to do then.

Answer 3

In order to take full advantage of group homework, I think an instructor needs to think carefully about the purpose of such an assignment and how it fits in with the general goals of the course. The folks who teach calculus at the University of Michigan, for example, have thought a lot about this.

For instance, when I teach Calculus I assign students individual homework using WebWork (these are the more straightforward computational problems and the like) and assign team problems in the style of University of Michigan Calculus Problems (these are messy real-world data-driven problems). The team problems, in addition to being messy, also have more structure: groups consist of four people with one of four rotating roles, the write-ups have to be written in carefully and rigorously and clear expectations are laid out about how much time these should take. So the main goals of these team problems are: (1) they are different kinds of problems than just ones that deliver content, (2) they require students to write careful explanations, (3) they encourage students to think about team work in the abstract and (4) they give students the opportunity for peer instruction.

Answer 4

I would consider how intricate your homework is. For "drill", more of the advantage comes from solo trials. For "projects", collaboration may have an advantage, or at least less of a disadvantage.* I would also consider if there are ways to get some social interaction without full joint responsibility (half a loaf, quarter of a loaf). "Trade papers and grade." "Do you agree with Johnny's answer? Anything to add/change?"

I also like the suggestion of separate parts of the homework, because it allows some grading independence. Perhaps you can do something like a part A and part B (separate) that merge into a joint part C. Not really sure how to implement, just brainstorming.

*I would add that professors tend to overvalue "project type" homeworks and undervalue "drill" because the former is closer to their research role...and just kind of more interesting to them, than duller problems. But that's a different battle to fight. (There's a reason why Gene Hackman makes the kids do drill rather than just scrimmage all practice long, in Hoosiers. And gets static for it from the townspeople.)