Increasing Accessibility of Examples in Abstract Algebra

Using Computer-based Projects -- The Final Report

Submitted July 27, 2000

Peter Blanchard Assistant Professor of Mathematics Denison University blanchard@cc.denison.edu 740-587-6471

Judy Holdener Assistant Professor of Mathematics Kenyon College holdenerj@kenyon.edu 740-427-5266

Overview:

Our collaborative efforts over the past year resulted in the development of six computer-based labs. The labs were designed to enhance student exploration and understanding by making examples, data, and computations more accessible to students. The projects were used as a supplement to our first-semester Abstract Algebra courses. Judy Holdener taught the course in the fall at Kenyon, and Peter Blanchard taught the course in the spring at Denison. All six of the projects rely on the software package GAP (Groups, Algorithms, and Programming), a freely distributed program designed to handle large computations within and relating to groups.

What was planned and what was accomplished?

Our goal, as stated in our proposal, was to develop six computer-based projects to make concrete examples in Abstract Algebra more accessible to the students. We had originally planned to use Maple in one of the projects, and we had a tentative list of ideas for the topics to be covered:

Tentative List of Projects (as given in our proposal)
Project 1:	An Introduction to GAP
Project 2:	A Maple lab introducing rotation and reflection groups visually
Project 3:	A GAP lab covering representations of groups
Project 4:	A GAP lab covering the mathematics of Rubik’s Cube
Project 5:	A GAP lab exploring cosets, normal subgroups, and quotient groups.
Project 6:	Undecided

In the end we developed six projects, but only two of these projects (see An Introduction to GAP and Exploring Rubik's Cube with GAP below) focused on topics from the list above. There were several reasons for these changes. The students responded very well to the first lab covering the Rubik’s cube, and we wanted to emphasize this application further in a later project. We also found it easier to come up with appropriate topics for projects in the midst of teaching the relevant material.

Projects Developed
Project 1:	An Introduction to GAP. Focuses on the basics of GAP, in the context of the group of rotations of a cube; it assumes no prior knowledge of GAP
Project 2:	Subgroups Generated by Subsets Discusses subgroups generated by a subset from two viewpoints: the "top-down" approach using intersections, and the "bottom-up" approach using group closure.
Project 3:	Exploring Rubik's Cube with GAP Investigates the transformation group of Rubik's cube.
Project 4:	Conjugation in Permutation Groups Explores the relationship between the cycle structure of a permutation and cycle structure of its conjugate; Revisits permutations of the Rubik's cube.
Project 5:	Exploring Normal Subgroups Investigates normal subgroups and quotient groups using multiplication tables.
Project 6:	The Number of Groups of a Given Order Explores the number of possible group structures for any given order; the class will need hints and encouragement on the last problem!

Were the goals met?

Both of us agreed that the projects were successful in making examples and computations more accessible to students. Peter Blanchard best summarizes it:

While teaching Math 332 this past semester, I learned that our premise, that examples in abstract algebra are not easily accessible to students, was more true than we had realized. In hindsight, it makes sense. A calculus student has no difficulty conjuring up a function. A student of statistics has no trouble imagining a data set. However, examples in abstract algebra are generally not familiar and not easy to visualize. Another aspect of the difficulty is calculation. An example of abstract algebra may be presented simply as a set with rules for “multiplying” its elements. A natural way to become familiar with the example is do some calculations. These calculations are likely to be unfamiliar, difficult, and possibly tedious.

I believe our GAP-based computer projects helped students address each of these difficulties. One of the most striking examples was the lab on quotient groups. The exercise was clearly pivotal in helping several of my students come to grips with the idea that cosets (set of elements of a system) become elements (of a new system). I recall one illuminating conversation with a student that went something like this:

Professor Blanchard: “The cosets are the elements.”

Student: “The cosets are the elements?”

Professor Blanchard: “Yes, the cosets are the elements.”

Student: “Oh, the cosets are the elements!”

It’s always exciting for a teacher to be able to witness the proverbial light bulb come on above a student’s head.

So true! We need to do more of this hands-on type of learning in upper-level courses. By working through self-paced, open-ended projects, and empowered with the computational capabilities of GAP, students are more able to acquire a working knowledge of the course material. This level of understanding is rarely attained with formal presentations of course material.

The projects also brought some unexpected successes. At Kenyon, the GAP project covering the group theory of the Rubik’s Cube led to a student research project about a different puzzle: TripleCross. Junior math major Adam Knapp used the techniques from the project to explore the group of configurations of this now retired puzzle. He presented his results at the Mathematical Association of America’s sectional meeting at Marshall University in Huntington, WV, in April of 2000, and his talk was very well received!

What was learned about teaching and learning with technology?

As the previous section indicates, we both agree that technology provides the medium for students to gain a deeper understanding of mathematics. The projects served to guide the students through a discovery-based learning of the relevant material. The discovery would not be possible without the access to examples and computationally difficult results. The access would not be possible without technology.

We also found that student collaboration is more likely to occur in a computer lab (as opposed to a traditional classroom). Perhaps this is a result of the more informal environment or maybe it is because of the hands-on nature of the activity. Either way, we found that the in-class collaboration seemed to lead to rich discussions about the material, indicating that students are gaining a deeper understanding of mathematics.

What was learned about collaborative processes?

Implementation of the GAP projects was similar at Kenyon and Denison. Students worked together as a class (in the lab) for three lessons. They also worked together on projects outside of class. There were several benefits to the collaboration. In class, the collaboration meant that students were more likely to ask questions and hence clear up confusions. It also meant that the conversation was truly a conversation, with two sides conversing, as opposed to the one-sided situation where the professor is presenting material to the class. The collaboration outside of class was particularly beneficial to those students who had lesser levels of comfort and sophistication in using computers.

What should we have done differently?

The biggest plans for change involve future implementation of the technology. Throughout its history, GAP has been developed primarily for use on a Unix computing platform. For a time there was a good port of GAP to the DOS/windows environment. At both Denison and Kenyon, we used this port of GAP version 3.4 in the windows environment. At Kenyon, we managed to get GAP installed on the network drives. We were unable to manage it at Denison, so we installed GAP on the local hard drives of several machines in one of our department’s computer labs. Because of the small amount of memory available on these machines, there were some problems with GAP quitting unexpectedly. One of the more computer savvy students in the class installed GAP on the Linux computer in his dorm room. When we were in the lab as a group he would telnet to his dorm computer instead of using the local version! Next time we teach abstract algebra it would be better to use a Linux installation of GAP.

How will these projects be disseminated?

We developed a website for dissemination purposes. Information about our work, including the projects themselves, is currently posted there:

http://www2.kenyon.edu/People/holdenerj/GAPprojects/Mellonproject.htm

We plan to advertise our projects on the Project NExT mailing list. Project NExT (New Experiences in Teaching) is a program designed to support mathematicians new to the teaching profession. There are over 300 people in the program, and many of them will be teaching Abstract Algebra for the first or second time. We also plan present our collaborative work at an upcoming national meeting (probably the American Mathematical Society annual meeting in January).