Cooperative learning tasks in mathematics

By cooperative learning tasks I mean giving particular “jobs” to students during group work; here’s a sampling from this website:

Checker: Checks team members for understanding and agreement
Datakeeper: Keeps track of information generated by group
Helper: Gives help in reading, spelling, problem solving, or using materials
Questioner: Asks questions of instructor or other groups
Reporter: Gives oral reports to the total group
Summarizer: Sums up what the group did or the conclusions the group came to
Validator: Paraphrases what is said for clarity
Writer/Recorder: Writes down ideas and records the task

I tried experimenting with them last year (based on the urging of several people) but I’ve been distinctly unhappy. It feels like the jobs segment up the work in a rote sort of way which gives a student permission to “shut down” when they aren’t needed for something in particular. I’ve still had some luck with engineering-like projects which involve building, but this sort of thing fails for me in general. For example, today I’m having my Algebra I students work on these questions in groups:

You have an
a row by b column matrix
and want to multiply by a
x row by y column matrix.
1. When is this multiplication impossible?
2. If the multiplication is possible, what is the size of the new matrix?

3. When multiplying 2×2 matrices, there is an identity operation (just like multiplying by 1 is an identity operation in arithmetic). What is it?
4. What about for nxn matrices?

5. Give an example (with all work) that shows that multiplying 2×2 matrices is in general not a commutative operation.
6. Even though the commutative law doesn’t apply in general there are specific cases where it works. Give an example of a matrix A and B such that AB = BA.

How would working out answers here divide neatly into jobs?

[EXTRA NOTE: I still think matrices shouldn't be taught before vectors. I'm making my best go at the curriculum I need to do, though. Bert Speelpenning's series on matrices (especially this post) is helping me get some motivation out there.]

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15 Responses

  1. I don’t know how you can make sense of matrices and what they do without understanding vectors. What sense does the equation Ax = b make if you don’t understand x and b as vectors?

    As for giving jobs to students, I’ve tried doing it. With students who have no idea how to work in a group, this can be an okay scaffolding place, but as you note, they can end up doing nothing because “I’m done my job already!” which I really hate. Now, I don’t give jobs to students, I just go around to groups which aren’t working well for some reason and give them some support and encouragement, and hopefully help kids find productive jobs within the group for themselves.

  2. I could be way off base and some better informed educator may argue with me, but I feel as though assigning jobs is, just as you have expressed, unhelpful and counterproductive. Do we really want to produce students that do their job and that is all. In solving a problem of the magnitude that comes up in class, is there really enough seperation to assign jobs without hampering each student’s performance and educational gain from the problem. Isn’t the problem solving process, as a whole, important. After all, it’s not like they are building a house that contains seperate and unrelated systems (such as plumbing and electrical), where subdivision of labor makes sense.

    On a side note, it seems as though some very generic roles may be important. Perhaps a leader to help resolve conflict, make sure conversation is not dominated by one individual, and probe individuals that are not contributing; however, even in this scenario, the position has to be seen as an addition to their role as a group member in the entire solution process. Most of the time, I can serve as this at various points so I will stick with good old fashioned free flowing, semi-chaotic group work.

  3. Even better would be to have the students learn this stuff without the baggage of having to handle group dynamics at the same time. Save the group work for projects that actually benefit from having multiple people working on them at once.

  4. Regarding “baggage of having to handle group dynamics at the same time.”

    It seems to me that the benefits of working in a group (greater feedback , more access to support, etc..) outweigh the drawbacks. Obviously learners also need time working through stuff on their own, but working in a group setting, even on work that is largely doable by the individual does have some benefits.

    Of course, for students who rely entirely on the thinking of the other people in the group working alone is probably best, at least until they can contribute more to the group than they suck out of it.

    • The fairly conceptual ideas on this assignment make the discussion as valuable as the answers, so for these I’d certainly rather have the work done in groups.

  5. I like to use roles/jobs when there’s a cognitive task I want to highlight (like checking, questioning, summarizing). I really like roles for students in audiences listening to other students present, but I’ve seen it work sometimes in small groups too.

    So for the Matrix stuff, a lot of what seemed hard/interesting about the tasks for me is generating & organizing examples to try to draw conclusions. As a problem-solver, I would try to switch between trying stuff and making logical predictions. It’s hard to know what to try, and to feel bold enough to try stuff. Novice problem-solvers might get bogged down in:
    * Not feeling like they can generate examples to try
    * Feeling like there are too many things to try and it’s not worth testing any
    * Not stopping to make predictions or try to logically generate good things to test
    * Not stopping to reflect on the results of a test & use them to generate other candidates

    I wonder about using roles, in this case, in the form of hint cards. If a group gets stuck somewhere, they grab a set of role cards, and the roles are:
    * Brainstormers — whenever the group is stuck, come up with something to try, and try it (1-2 person)
    * Predictors — as the brainstormer is trying something, predict what the results will be (race to predict before the results are in!) and why (half the group) (1-2 persons)
    * Judge — after something has been tried, the brainstormers and predictors both try to convince the judge of the next best guess — the judge gets to choose. The judge will also keep an eye on the brainstormers and predictors to see who is faster and if both groups have accurate, well justified results. (1 person)

  6. I think there may be cultural differences between places that make role-assignment turn out differently in different groups. My students invariably roll their eyes and snigger in an “oh my how very american our teacher is”-way whenever I try role assignments, and then they proceed to completely ignore their new roles during the group work, which usually turns out good anyways. Hrm. But I think group work with roles is a great idea, at least until all students have internalized multiple roles and can play them at will.

    With this task, I’d have a few roles: skeptic (asking questions, challenging validity), recorder (writing down, drawing diagrams), summarizer (“so up to now we’ve seen… and what remains is…”) and leader (making sure group is on task and making progress, etc). Ordinarily I’d have someone with the role to extend and generalize, but you’re providing instructions for that already.

  7. The curriculum I use is called CPM (College Preparatory Mathematics) and students work in study teams with 4 roles:

    Task Manager, Facilitator, Recorder/Reporter, and Resource Manager

    None of the roles has anything to do with the actual solving of the problem, just with how the group functions as a unit. (NB: I teach 6th grade, so I prefer to have students work in pairs by combining the first two and last two roles.) Check out the CPM website for some great ideas about teaching students to work collaboratively.

    http://www.cpm.org/teachers/study.htm

    • In CPM, the roles work out well just as you say, and also because the curriculum is developed to support the process of groups. I would also recommend experimenting with jig-saw or expert groups in the matrix lesson.

  8. I would also recommend experimenting with jig-saw or expert groups in the matrix lesson. For example, say we consider the 3 problem pairs. Start with groups of 3 and give each student one pair of the problems. Then the students from all groups with the same problem pair get together and work on that alone. Finally, they return to their original groups and each has to present their solution to the group.

    • I use jigsaw on occasion, although usually it’s for activities where it is ok for students to skip grappling with a particular problem (say, a set of rate problems that all include similar ideas). In this case I’m not ok with a student just doing #1 and #2 and having the rest of the problems explained to them by peers. Here, to be honest, I’m more interested in the struggle than the answer.

      You can hybridize this slightly by having a “reporter” job which roams around to other groups at a certain point to get advice. The extra layer of indirection usually is enough to force the students to think about all the problems, but also includes some of the good peer-to-peer interaction that comes from a jigsaw.

  9. I am a pre-service teacher and was wondering how you view group work assignments like the one above? Do you think it prevents some students from particiapting? DO you think all students benefit from it?

    • Given the grumpiness of my original post, I’m not sure if I’m the best cheerleader for group work, but–

      how you view group work assignments like the one above

      All teaching involves cost-benefit. The question for any given activity is: what is most important to you? What are you willing to sacrifice?

      Certainly there are things that only can happen in a group context:

      1.) Students teaching each other (given teaching something is the best way to retain it, this is nontrivial).

      2.) Engineering or science-type projects which more than one person is physically required (having one student roll a marble down a ramp while another uses a timer, for instance).

      3.) Some particular insights on difficult problems.

      #3 is by far the most nebulous. It’s quite definite that students can solve more difficult problems together than they can alone, but there are also often situations where one student bears all the load and the others just tag along.

      As for cost, in addition to the “tag along” effect it can be much slower than an individual activity.

      Do you think it prevents some students from particiapting?

      Not if you plan it carefully. The thornier danger is not having all students working at the highest level that they can.

      DO you think all students benefit from it?

      Yes, but with all the caveats above.

      I suppose your subtext here might be “but what about the students who prefer an introspective approach?” but even they will have situations where there is problem they cannot solve alone but can solve with another student.

  10. I use collaboration and group work in most of my classes. Coming from a school that is PBL based – it is important for the kids to work this way in all of their classes. I have found that it is necessary to teach the kids how to work in groups. I start with pairs – and have them check each other and justify their work. They often have to come to a consensus on the answer they are going to turn in. This grows into working in groups of 3 or 4. Starting right off with larger groups before they understand how to collaborate has been frustrating for me in the past! Starting with partners and working up has been more successful.

  11. I *COMPLETELY* agree with you, as my experience has been the same (some kids check out, shut down, or turn into “passengers”).

    At the same time, I think it’s also a matter of personal “fit.” I know teachers who work this way very effectively. I just can’t make it work well for me. Glad to know I am not the only one!

    – Elizabeth (@cheesemonkeysf)

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