We all want our students to collaborate effectively on problems. Problem is, there's a very fine line between working together to solve a difficult problem, on one hand; and simply copying another student's work, on the other. And, no matter how obvious the difference may be to us, students don't necessarily get it.
It took me a while to learn about this disconnect between my own academic experience, my expectations, and those of my students. At a previous school I became very frustrated and angry with students who seemed like they were copying each others' homework solutions. And, they were copying, without question. In their minds, though, they were merely reporting together the results of their effective collaboration -- that same collaboration that I had encouraged so strongly. So they weren't happy with me for being unhappy with them for following my own instructions. If you follow.
The most important step I took toward resolving this difference of understanding was to re-cast the issue so it wasn't about academic integrity. I couldn't say "don't copy and don't cheat" if the students and I had such good-faith but widely varying ideas of the definitions of "copy" and "cheat." Rather, I had to find a way to give clear guidance to define the line between collaboration and copying, without invoking the emotionally charged language of academic integrity.
What I came up with, and what has served me well for decades now, was the Five Foot Rule. My syllabus states*:
The Five-Foot Rule
We encourage students to help each other. You may even verbally guide a friend step-by-step through his solution to a problem. However, do not under any circumstances just give someone your solution “to look at later”. A friend may, in your presence, look briefly at your work to start himself in the right direction, but no one should ever be using another student’s written solution as a detailed reference.
Thus, when you are actually writing something to be turned in, you must be located at least five feet from any other physics student. Do not do your homework while sitting next to someone; rather, sit well apart from one another in a dorm or conference room; or, have a discussion, then separate yourselves to write up your solution.
* Remember, I teach at a boys' school, so my use of gendered language is deliberate. Please don't flame me.
Students who obey the five-foot rule are hardly likely to be copying, unless they have x-ray vision. The point is for all students to explain the result of collaborative discussion in their own words.
Then, when inevitably you find two identical problem sets, you can avoid making accusations of dishonesty or cheating. You can simply discuss the obvious violation of the five-foot rule, and ask that this rule be adhered to. If the violations continue, and you have to get parents or administrators involved, you are likely to get support.
Accusations of cheating or copying carry harsh implications. Parents instinctively defend their children, logic be damned. Administrators question whether it's worth their political capital to engage in a fight over small-scale homework copying -- especially when you expressly encouraged collaboration!
But you're not accusing anyone of cheating, no, not at all. You're merely asking for cooperation in enforcing and adhering to a straightforward class rule. Just as students are expected to treat laboratory equipment carefully, just as they're expected to show up on time with their homework finished, they're expected to sit five feet from anyone else when they write work to be turned in. That's a reasonable enough procedure to follow that students look like idiots if they protest.
So they don't protest. And they follow the rule. And so they do their own work... often with help, often by rephrasing what a friend told them. But that's fine -- rewriting in one's own words is a significant step toward deep understanding.