Questions on Cooperative Learning

(These items are from a listserv about teaching.)

1. When you have students work in groups on problem sets, do they all work on the same problems, or does each group get a different set?

I have them work on the same problems so that we can investigate a variety of possible solutions that are used to answer the same question. In introductory level math courses students have a tendency to want to see only one method to solve problems. When students are asked to put their solutions on the board their procedures may differ from those of their fellow students. It is important to establish an environment where multiple solutions are acceptable. I emphasize that students do not have to adopt another method if they are comfortable with their process, but it is also very inappropriate for them to deny someone else's method simply because it is new. I kid with them and suggest they put their fingers in their ears, close their eyes and hum to block all sound, if we are discussing a method which appears to confuse them and they do not want to hear it, rather than saying we shouldn't even discuss the new process.

2. When they put their answers on the board, how is the discussion focused--i.e. if they have worked on the same problems, do you focus on clarifying discrepancies? Do students explain how they got the answers?

I ask students to put their solutions on the board often having several problems presented at the same time. I find it encourages students to go to the board if they are not alone in front of the class. With several students working out problems there is a feeling of camaraderie. Often students will help each other while they are writing out their solutions if they see a mistake in someone elses work or if another student starts having an anxiety attack. Quite common in developmental math classes. I allow time for the students to review the board solutions and ask for questions or comments. If the problem warrants I will ask the author to explain what they did in order to reinforce a particular concept or procedure. This also creates a form of repetition of problem solutions. I will ask for alternative solutions at this point which often requires my calling on someone who I have observed doing the problem differently. Since I use cooperative learning techniques I am easily able to move around the class to observe the students. I need to know in advance what alternatives I am looking for in order to try to encourage students to use other procedures. At this point we are able to observe and discuss as a class the benefits or advantages of each method. Another benefit to the board work occurs when students make mistakes. This usually mortifies students but I stress that this is a very positive thing because it allows me to see what they are doing and it helps their classmates who may tend to make the same mistake. I discuss the value of making a mistake in public and also emphasize that we are among friend who want to help each other. In the past students have been criticized or harassed when they have put work on the board. I assure them that they will never be criticized for making an honest effort. After one such explanation the problem disappears and students willing use the board.

3. If the groups work at a different rate, what do the faster groups do when the slower ones are still working on the problems? How do you make sure that the slower students cover enough of the material?

For the group assignments I make up separate work sheets based upon the text material. For those groups which finish early I "suggest" they review the homework together and check each others answers, especially if they had any questions about a particular problem. I encourage them to work out the solution together before I attempt to answer one partner's question. If they have done the work beforehand as requested then this does not take much time. At this point I encourage them to start on the next section. I kid them that this is their bonus for being so good and working so well together. I keep track of the slower students by the pace at which we go over problems on the board. Remember I have been observing students working together so I am familiar with how much work they were able to accomplish. The slower students to not drive the process but I am able to work with them separately and often other pairs will help their neighbors or slower pairs will seek out help from other people in the class. The process is incredibly synergistic. Lastly, I find that when pairs or groups finish all the material for the class they will tend to socialize. I do not find this a problem as long as they are quiet and do not disturb people around them. It is a nice carrot for them and builds a class spirit. The conversations that I over hear are often pretty serious and include things like where people find time to do the work, how other classes are going, what other teachers are like as well as the usual where are you going Saturday night. Who ever heard of talking in math class as a benefit for doing exceptional work. It is my job to keep them on task until they have completed all the necessary material for that day.

4. Do you prefer to let students choose whom to work with, or do you assign them in any particular ways?

On the first day students usually sit randomly, that is they do not sit with friends or acquaintances, because they come to class at different times and seats are taken. In my advanced classes where students have had me before they will gravitate to familiar faces from the last class. I do not try to make any special arrangements. On the first day I ask them to pair up with someone next to them, preferably someone they do not know so they can meet a new friend. In the next class I ask them to pair up and work with their new buddy. After that I let them change seats to work with whomever they wish. I find there is not much movement but it is important to allow people to change if they find early on that they do not wish to work with the new person. My only requirement is that everyone have a partner before I hand out materials. Sometimes students will group into threes because they become comfortable with each others style and personality. Another school of thought is to group students heterogenously with higher and lower level students matched. I find that this happens to a certain extent by allowing students to move around over time and to encourage them to seek help from other pairs or groups.

5. Your suggestion that students read and do problems *before* class seems quite revolutionary. I've never been able to get many of them to get much out of the book on first reading. Do you discuss tips for how to work with the book? What kinds of issues still need to be discussed after they have read the chapter the first time? Do you end up having to do any lecturing? If not, what do you do instead when students have widespread difficulty with a particular topic or concept?

It only seems revolutionary to them. It seems quite natural to me. However, it is necessary to explain, defend and maniacly cheerlead this approach. I do so by sending my students a letter before the semester starts which explains my class procedures and asks them to get the text and do the first section prior to the first class. Wow! What a response I get. Not the negative one you would expect but they say they are very pleased that I take the time to communicate with them and that I try to establish high expectations from before the beginning of class. In class I start by handing out worksheets to pairs and then follow the above procedures for board work. During the first few classes students will try to talk me out of doing it this way, beg and plead for lectures and try to dissuade me. I patiently discuss the importance of being prepared and how it benefits them. We also discuss their previous experiences with lectures where they go home and have no clue what was said in class, even though the lecture appeared to be informative, humorous and very well organized. Their talking about their experiences helps to sell them on this approach. Another very important point is that I assure them that they are not learning the material on their own and that we will go over more material and understand it better by working on it in class. They can't do that if I am lecturing. Once the fear of not covering the material is removed they relax and for the most part make a reasonable attempt. This all fits into the cooperative learning mode and a mastery approach I use for evaluating their progress in the course. Most resistance to doing math comes from students fear or reprisal in the form of a bad grade. If you can remove that then they begin to learn for themselves.

One qualification is helpful here. It is possible for students to come to class unprepared. Since the class covers all the material needed to understand the concepts a student can see enough and try enough problems in class to learn the material. The problem that arises is that they do not contribute much to their group. Sometimes the group will carry those students for a while and then make it clear they will have to work or find another group. For those students who I cannot inspire or the process does not motivate I suggest they work independently until they are ready to make the effort to help their group. Some students do work better alone and can make up the material after class or by seeking extra help. My only caveat is that I will not lecture because of a few students who think it is easier for them and I will not tutor them if they do not make any effort themselves. I can adopt this approach because I bend over backward to help those students who do try and everyone knows this.

One aspect of all of this discussion which needs separate highlighting is the idea that force of personality is very important in encouraging students to work together and use cooperative learning approaches. If you are uncertain or not convinced about the benefits of CL then the students will react accordingly. I hand out articles on CL and ask them to write about what is happening in class. I ask them to analyze the 7 principles of good teaching and apply them to math classes. In short I sell CL every class because that is how I feel about it. This helps carry along many doubting Thomas's.

I would appreciate any comments or observations about anything I have said here and especially anecdotes from other people who have dealt with these questions.

Ted Panitz, tpanitz@mecn.mass.edu

Your suggestion that students read and do problems *before* class seems quite revolutionary. I've never been able to get many of them to get much out of the book on first reading. Do you discuss tips for how to work with the book? What kinds of issues still need to be discussed after they have read the chapter the first time? Do you end up having to do any lecturing? If not, what do you do instead when students have widespread difficulty with a particular topic or concept?

In my sophomore level geology courses, I have taken to giving a homework assignment before each class. I used to just ask students to read on a particular topic and I gave a weekly quiz as a motivator, but the results were distressingly poor and the quizzes killed motivation instead of building it. Now, I give out a sheet with a reading assignment and 2-3 essay questions, some of which are directly from the text, some of which involve synthesis of material from several sections, and some of which are completely open-ended and intended to provoke thoughts beyond the reading. I give 5 points per day for attempting all the questions. I tell the students that the answers don't have to be right, but they have to have tried. I do not collect the homework, I merely scan it to see that it has been done. The quality of participation in class exercises has improved dramatically since I started doing this last semester.

One of my colleagues commented that this was more like high school than college and I have realized that so little is demanded of these students in high school that they are completely unprepared to deal with the level of organization and work needed to master the complex material we are covering. By breaking it down into daily pieces, I make the task manageable. It seem that there is a critical cutoff between reading 5-10 pages with questions and reading a whole chapter. The former is no big deal, the latter becomes an insurmountable task for reasons that I cannot fathom. I am hoping to gradually increase the level of expectation and decrease the level of organization in homeworks over the course of my two semester sequence in order to prepare these students for upper division courses where they are expected to work more independently.

I now spend class with the students either discussing problems raised by the homework or working on cooperative exercises reinforcing the concepts from the homework. If I were lecturing, I would have spent time covering what they already could have understood from reading. Now, we focus our time on the student's problems, a very efficient means of learning for them. My coverage is no worse and might actually be slightly better than when I used to lecture.

For assessment, I am asking for a portfolio from them and the homeworks will make up a big part of that. It's a chance for me to review their homeworks in more detail if there's a question about their mastery of a skill and it's a chance for them to make up for any they didn't do the first time through.

David Lee Smith, Ph.D.
Director of Academic Operations, Institute for the Advancement of Mathematics and Science Teaching
-and-
Assistant Professor, Department of Geology and Physics
La Salle University
1900 W. Olney Avenue
Philadelphia, PA 19141 USA (mark to my attention at Box 302)
dsmith@lasalle.edu
phone: (215) 951-1706 fax: (215) 951-1488

 

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