This page is some tips for people who want to try using groupwork (a.k.a. "active learning" or "cooperative learning") in teaching math courses. It's based on my experiences using groupwork in both less successful and more successful situations; I hope that reading this page can give other people who are curious about using groupwork the courage to try it out, and can help them avoid some of the mistakes that I made. I expect that some of the comments here might be applicable to non-math courses, especially science courses, but I can't say for sure. If you're new to this, you might want to look at my notes on using experimental teaching methods as well.
The basic idea in groupwork is that students spend some of each class working in small groups, rather than, for example, listening to a lecture for the entire class. These groups typically consist of around 3-5 people. Usually, groups are assigned problems that are fairly simple applications of a technique that has just been introduced: e.g. if you're lecturing on separable differential equations, you give them a couple of examples of such equations to solve. But those aren't the only kinds of possible assignments: if you've introduced a new concept, you could ask a group to come up with examples of that concept ("spend five minutes listing examples of abelian groups"); you could assign discussion problems ("what was the key point behind the proof we just saw?"); and more complicated examples are sometimes workable (I've seen groups come up with proofs, even somewhat complicated ones, for example).
The theory behind this is that students grasp a concept more quickly and more effectively by working through examples of that concept than they would by simply listening to examples of that concept being presented; this is why it's sometimes called "active learning". And when working in groups, students often learn more quickly, because there's less time when they're just stuck: they have help immediately at hand. Also, students who have just understood something can often explain it much more quickly to a fellow student who is stuck than a professor can. This is why it's sometimes called "cooperative learning".
A typical class that I teach using groupwork proceeds as follows:
You may be tempted to omit the example and/or the solutions. In my experience, that isn't a very good idea: professional mathematicians may be able to grasp a simple concept upon first hearing without seeing an example of its being applied, but many students aren't very comfortable with that. (Also, they may have found your introduction less clear than you expected.)
As for the solutions, in an ideal world students will feel confident enough in the solutions that their group came up with not to need solutions presented on the board; and in fact there will always be some students who are bored by the presentation of solutions. In upper-level courses, I sometimes skip this step, depending on time constraints, but in lower-level courses, there are enough students for whom presentation of solutions is important that it's rarely a good idea to skip them. (This also has to do with some students' fears of missing something that they are "officially" supposed to know and might be asked about on the exam.) As for having students present solutions instead of the professor, I urge you to try this. It's a useful skill for students to learn; they'll get better at it as the course goes on. But it's important to avoid micromanaging the students: nobody likes it if somebody else is constantly correcting what a presenter is saying. So sit on your hands!
One obvious difference between planning courses this way and planning traditional lecture courses is that it looks like it takes a lot more time to cover the same amount of material. There are some time savings, though. For one, instead of presenting three examples yourself, you can present one easy example yourself and have the class work through a second one in groupwork. For another thing, you don't have to be quite as complete in everything that you say, since it doesn't have to last until the students start working on the homework a few days later: it only has to last until the students start working in groups a few minutes later. And you don't always have to give complete answers to certain sorts of questions, because they'll be answered in the course of doing the groupwork. It is true, however, that to some extent, you might have to resign yourself to covering less material but covering it better: it is not a technique designed for maximum density, though the difference lessens once you get used to planning courses from a groupwork point of view rather than from a lecture point of view.
A less obvious difference is that it's harder to find suitable breakpoints. While you can't stop whenever you want when lecturing, you usually don't have to go too many minutes between potential breakpoints. However, when doing groupwork, besides the obvious constraint that breaking in the middle of groupwork isn't too useful, there is the less obvious constraint that you should never start a class with groupwork. It just doesn't work very well: students will spend the first five or ten minutes mostly being confused. So if you've ended your previous class with an example, start the next class by briefly reviewing the theory and example. It's not necessary (or particularly desirable) to be complete in this review, but you have to say something to remind students what was going on. It is possible to postpone solutions if absolutely necessary, but not if students are presenting solutions, and it should be avoided whenever possible.
It's possible to do more than one of these lecture/example/groupwork/solution cycles in a class: for example, at the beginning of the quarter, the groupwork and examples may be sufficiently simple that you'll be able to fit in two full cycles. The transition from two cycles to one can be hard, so be prepared for that. (It may depend not so much on the difficulty of the material that you're presenting as on the knowledge from previous classes that the groupwork demands.) Also, some days the groupwork will turn out to be a lot harder than you expected; this happens, it's not the end of the world, but you may have to give a modified version of the same lecture the next day. At least be glad that you know that the students didn't understand what you were talking about: if you'd just been lecturing, you might have remained blithely unaware.
As I mentioned above, groups probably work best with 3-5 people in them, and for the first few weeks I try to enforce that constraint. Also, it may take a little while before students get used to working in groups. So sometimes at the beginning of a groupwork session I make an announcement to the effect that I really want students to move their desks so that they're facing each other, not just adjacent to each other, or I say that the room is too quiet and that I want to hear more talking. A bit of gentle nudging along those lines will suffice, and won't be necessary after the first few weeks.
I let the students choose their own groups, and let them vary from day to day. If you want to do otherwise, just keep in mind that not all students come to class every day. Students will naturally tend to work with the same people most of the time, of course. For the first few meetings, it might be a good idea to encourage students to work with different people, so that they get a chance to explore possibilities. Also, about three weeks into the quarter, it might be a good idea to remind students that groups aren't set in stone: if they're unhappy with the way their group is working, maybe they should try sitting in a different part of the classroom and working with another group.
Some students really want to mostly work by themselves. My tendency is to make sure that such students are at least formally working in groups for the first couple of weeks, but to be more lax about this after that.
Some authors propose assigning different students in the group different "roles": it's the job of one student to make sure that everybody's participating, the job of another student to make sure that everybody understand, the job of another student to be prepared to present a solution, or whatever. I'm not a big fan of this, but a minimal version of this has sometimes been useful to me, where each group has one person whose job is to be the "understander", so that person has to make sure that she can really do all the details of the problem, and should drag those details out of other students if necessary. The test of that person's knowledge is that she should be prepared to present a solution on the board (if you're having students present). If you're going to try this, it's important that the same person not be "understander" every day and that that role is chosen before the groups start working, rather than after they've solved the problems.
While the students are working in groups, the instructor should be circulating around the room, prepared to offer help as necessary. You have to draw a fine line here: you don't want to offer too little help, because sometimes groups really do get stuck and need outside assistance, but you also don't want to offer so much help that students ask you for help first rather than asking their fellow group members. (Different groups will have different tendencies in this regard.) My initial tendency was to spend all of my time circulating around the room, since I didn't feel I was doing my job otherwise; I have since learned that it it sometimes appropriate to sit and not do anything. Also, you have to strike a balance in terms of keeping students on track: a bit of social chit-chat is normal and harmless at the beginning of groupwork, but if it crops up later, it may be a sign that that group is stuck. (Or that they've solved the problem(s).)
If you're going to have students present solutions, my big piece of advice is to keep a list of who has presented starting from the very first class, in order to avoid repeats. I don't like to pick somebody to present until I see that their group has actually solved the problem; then I ask one of the students from that group whether she would be willing to present, and give her a few minutes to prepare, with the assistance of the other members of her group. Other instructors may have different styles here. Some flexibility is useful, because not all groups will solve all problems every class.
As far as timing goes, you have some amount of flexibility here. My default is to let groups work until they've all solved all of the problems, but if that wouldn't leave time for solutions, I don't mind stopping some groups before they're done in order to leave time for solutions to be presented. Even the students who haven't solved all of the problems will have benefited from working on them, because they'll be able to learn a lot more from seeing solutions to problems that they've struggled with themselves.
Don't be afraid to try groupwork. Lots of people around the country use it; you can probably find people at your own institution who use it, and it's certainly growing in popularity. It's been tried enough that I can be confident in saying that it won't have any disastrous results. Your students' understanding (and grades) may not shoot up as a result of trying it, especially the first time you try it, but they won't plummet, either. It also has benefits that don't appear in grades: for example, it provides a way for students to meet each other, and to get used to actually talking to other students about math (in a constructive way, as opposed to kvetching about classes or copying each other's homework solutions). In my experience, when it goes well, students really like it: they feel that they understand the material better this way, and they are amazed at how much more painless and pleasant it is than a traditional lecture course. (Even students who don't like it may comment that it makes the time fly by.)
The flip side, though, is that it can take a while to get right. We've all seen lots of examples of lectures, after all, but may not have much experience with using groupwork. There are things you can do to ease your learning process; I've written up some notes on my experiences with using experimental teaching methods that may help in this regard.
Planning a groupwork course isn't any harder than planning a traditional lecture course (once you get used to it). Indeed, sometimes you can get away with saying remarkably little during a meeting of a groupwork course, which means that you have to spend less time actually writing out notes for your lectures; but you do have to spend at least as much time thinking about how each class will go as you would in a lecture course, before you start writing out those notes. So the total preparation time turns out to be about the same, possibly a little less. Also, do try to double-check your groupwork: there will inevitably be days where the groupwork doesn't illustrate the point you thought it did or is more complicated than you realized, but you want to minimize their number.
The comments above are geared towards "service courses". I've actually used it in an upper-level math course with some success; frankly, I found that kind of surprising, and am planning to experiment further. I'm not confident enough with my experiences there to want to make concrete recommendations, however. Also, you can expand the concept to cover homework, for example, if you want; I haven't experimented with that.
There are lots of books on the subject. One relatively practical one is Active Learning, by Johnson, Johnson, and Smith. A more philosophical one that first got me interested in the subject is No Contest, by Alfie Kohn. Both of those books are somewhat idiosyncratic (in different ways), so you may find one or both of them quite annoying. There are lots of other books out there; you may be able to get recommendations from other people at your institution. I don't know of any particularly good books specifically on using groupwork in math courses; if you find any you like, please e-mail me.
I hope this helps those who read it. Please e-mail me with any comments you have, especially suggestions for improving this page.
Last modified: Wed Jun 28 17:35:39 PDT 2006