INTRODUCTION: I am currently teaching two sections of Topics in Contemporary Mathematics, a course that covers a lot of graph theory. I am constantly being challenged to present the content in creative ways because students in this course typically do not want to sit through hours of lecture. I find providing them with hands-on activities works very well. Graph theory, it turns out, is a wonderful topic for students to discover on their own in group activities.

The collaborative learning structure I use is a variation of the Jigsaw model that I learned in the On Course I Workshop, and it is appropriate for any content that can be subdivided into three to five units of information. So, while I use it to teach graphs, an English instructor could use this same structure to teach three to five grammar concepts or a psychology instructor could use it to teach three to five theories of personality development.


  • To help students become independent learners, able to investigate new academic content on their own
  • To teach students the value of interdependence when learning new content
  • To have students learn to effectively present course content to their peers


  • Any content divided into three to five separate units (such as reading assignments) or separate packets of information
  • A set of study questions for each unit/packet of information


1. Divide the class into numbered groups. (I have them count off.) Determine how many groups to create by the number of units/packets of information. Have the ‘ones’ congregate in a specified area of the classroom; the ‘twos,’ ‘threes,’ etc., should also go to their designated areas.

2. Distribute the reading assignments or information packets (with study questions) to all students. (I give a complete Hamilton Circuit Algorithm packet to every student so they will each have a copy for their notes. See below for source.) You could have students read the information for homework and continue with the collaborative activity in the following class period.

3. Assign each group a different concept to master. For example, I assign Group #1 the Brute Force Algorithm, Group #2 the Nearest Neighbor Algorithm, Group #3 the Repetitive Nearest Neighbor Algorithm, and Group #4 the Cheapest Link Algorithm.

4. Have students work in their group to learn the content of the unit/packet and answer all of their study questions. Emphasize the importance of all group members gaining a clear understanding of their group’s material because they will each be presenting their information to students in other groups. This realization typically motivates greater participation. I allow an entire 50-minute class period for this group-study process to occur. During this time, I walk from group to group, answering questions and making sure that students are learning their material properly.

5. At the beginning of the next class period, have the groups meet again for a short period of time to clear up any last minute questions they might have (perhaps needing your input).

6. Form the second set of groups by having students count off again, this time within their own group. When the second set of groups form, you want each new group to have one person presenting each unit/packet of information. To determine how they should count off the second time, determine the number of students in the largest group. If it is 5, have students count off in 5’s, if it is 6, have students count off in 6’s, etc. As you did previously, have the 1’s congregate in an assigned area, the 2’s, 3’s, etc. so all groups have a place to meet. All of the groups should have at least one person presenting each separate unit/packet (or in my case, algorithm). [Editor’s note: To minimize confusion about group designations, you may want to have this second count-off be alphabetical, creating groups A, B, C, etc.]

7. When there is more than one student presenting the same unit/packet, tell them to share the presentation so everyone is participating. Walk around the room and listen to the presentations. This way, if a student leaves out an important idea, you can pick it up during the wrap up. If a group is missing one or more people to present a unit/packet, you as the instructor can fill in that role, or you can have the students merge with other groups.

8. Leave about 15 minutes at the end of class for wrap up. During the wrap up, briefly review the content and highlight any ideas that may be especially important. Also, review any ideas you feel the students did not explain thoroughly. You can gauge this need while walking around the room during the presentations to the groups. This gives the students a chance to bring up any points they feel were unclear to them. This also helps students see how they could have presented their information differently or more clearly.


The purpose of this activity is to help students become independent learners, while at the same time understanding the value of interdependence in learning new course content. I found this activity to be very successful in achieving the purpose. Although students were learning new content without direct instruction from me, they did have the help of their group members. In both the learning groups and the teaching groups, my students seemed to become very engaged in the activity and all groups appeared to have good participation. Students were asking each other to clarify ideas, and the students who picked up the concepts faster were able to explain the ideas to the students who were not catching on as fast. Students had to ask each other questions and explore the textbook for answers if they wanted to fully understand the material. This experience, I believe, drove home the value of both independence and interdependence in learning.

I believe the biggest challenge occurred in the teaching groups when students had to present the material to each other. Some students found the material to be very easy and were not as detailed in their presentation. They discovered they had to be more thorough when their presentations sparked many questions from members of their groups. This was a good lesson for these students because it really tested whether they understood how to break down the algorithm into a step-by-step process.

I found the wrap up to be an important part of the activity, because it satisfied the needs of the students who didn’t trust the knowledge of their peers. Some students shared with me their concern that their peers would not teach them the correct algorithm. They found, however, during wrap up that their peers did indeed teach it correctly. They simply needed to hear me confirm that it was right. Many students also found the extra benefit of meeting other class members since they were placed in random groups.

At the end of the activity, I had the students write a journal entry on what they thought about the experience. I was surprised and pleased to read so many positive comments. Most observed that the activity was a nice change of pace. One student wrote, “It forced me to participate when I would normally sit back and just listen.” Another commented, “I feel I learned the material better since I had to figure it out on my own and then explain it to my classmates.” Some students commented that their peers explained things more simply than the instructor and it was easier for them to understand. Overall, students observed that the experience of having to teach the material to their classmates forced them to take the learning of the concept more seriously. Since this is the first time I have taught this course, I cannot say whether students learned the information better than they would have by lecture. I can say, however, from tests and quizzes, that the students performed as well on this material as they did on other material that I presented with lecture. Because of this, I believe the students learned as much content as they would have from lecture, and I noticed a definite difference in the overall attitude of the class. There was clearly more energy and I observed a greater interest in what was going on in the class.


I have always been afraid to have students work in this fashion. I worried that students would miss out on the content somehow if I were not spoon-feeding it to them. I discovered that I was wrong. As noted above, tests and quizzes confirmed that the students understood the material presented collaboratively just as well as material presented by lecture. Students are capable of using their resources, such as the text and their classmates, to discover new content. I believe I may actually be more effective if my role in some instances is as a guide to their discovery versus the boring lecturer putting everyone to sleep. This format really makes students take responsibility for their learning. If they do not do enough investigation, they are not going to know the content. This activity made for an enjoyable class and a good experience over all. I believe I have more confidence now that this type of learner-centered activity can be very valuable in the classroom. I can already see that this activity could be repeated with many different portions of my course. The content in my course seems to lend itself to this format.


This activity is a variation of the jigsaw process that I learned at the On Course I Workshop.


I used a Hamilton Circuit Algorithms packet downloaded from the Tannenbaum website that goes along with the text, Excursions in Modern Mathematics, 4th edition, by Peter Tannenbaum and Robert Arnold.

HANDOUT: Study Questions for the Hamilton Circuit Algorithms packet.

1. How many different routes did you have?
2. How many different routes are there if you don’t count those which are just the same thing in the opposite direction twice?
3. How many would there be if there were six cities?
4. How many would there be if there were six cities without counting opposites twice?
5. How many would there be if there were 8 cities?
6. How many would there be if there were 10 cities?
7. How many would there be if there were 15 cities? Use your calculator to get the exact number.
8. How many would there be if there were n cities?

–Beth Rinehart, Faculty, Mathematics, Goucher College, MD

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