Innovations in Teaching Abstract Algebra

Preface

Over the past decade, the undergraduate abstract algebra classroom has undergone a dramatic transformation. Many faculty who were exposed to new pedagogical techniques during the calculus reform wanted to experiment with those techniques in more advanced classes. A variety of software packages were written or extended for use in abstract algebra. This collection of articles is the outgrowth of several gatherings of mathematicians who were interested in discussing the teaching and learning of abstract algebra. We, the editors of this volume, organized two contributed paper sessions entitled Innovations in Teaching Abstract Algebra at the 1997 and 1999 Joint Mathematics Meetings. One of the editors co-organized the NSF-UFE workshop Exploring Undergraduate Algebra and Geometry with Technology held on the DePauw University campus in June, 1996. We invited participants from these two contributed paper sessions and the workshop, as well as several other mathematicians, to write articles on a variety of new approaches in teaching abstract algebra.

We have chosen the articles that appear in this volume for several different purposes: to disseminate various technological innovations, to detail methods of teaching abstract algebra that engage students, and to share more general reflections on teaching an abstract algebra course. We expect that the reader of this volume will be either a faculty member who is new to the teaching of abstract algebra or a seasoned teacher of algebra who is interested in trying some new approaches. In either case, we hope that the reader will be intrigued and stimulated by these diverse expositions.

This site will be maintained to provide up-to-date sources for all materials, software, and web sites referenced in all of the articles. In fact, to fully appreciate some of the articles the reader may wish to visit this site for the color images that could not be captured in this black and white production.

Engaging Students in Abstract Algebra. We begin this volume with several articles that present individual overviews of course structure. While these are personal examples, we hope that readers can extract useful information for their own classrooms. After a meeting at a NExT session, Laurie Burton, sarah-marie belcastro, and Moira McDermott decided to share their experiences with each other as they each navigated teaching algebra for the first time. The reflections in their article may be particularly appropriate for other first-time algebra instructors. Steve Benson and Brad Findell provide a good discussion of some modified discovery techniques that the reader can implement. Gary Gordon uses geometry to help teach group theory, aided by several dynamic software packages. His article focuses on how the groups of symmetry can illustrate many algebraic concepts. Paul Fjelstad discusses how he has used some concrete experiences (special decks of cards, for example) to motivate students to build various abstract structures. Su Doree shares how she used the theme of the number of solutions of x^2+1 = 0 as a student research project. She includes some guidelines to consider when incorporating such a project.

Using Software to Approach Abstract Algebra. One of the earliest to use software to teach abstract algebra was Ladnor Geissinger, who created Exploring Small Groups (ESG). This DOS-based program is the foundation for Ellen Maycock Parker's volume Laboratory Experiences in Group Theory. Her article illustrates how to integrate a laboratory component into an algebra classroom. Edward Keppelmann and Bayard Webb contribute an article discussing the program Finite Group Behavior (FGB) that they have created. Intended as a successor to ESG, FGB is a Windows-based program that is more flexible than ESG. The programming language ISETL has also made its way into algebra classes, including being the foundation for Learning Abstract Algebra with ISETL by Dubinsky and Leron. Ruth Berger, Karin Pringle, and Robert Smith each contributed articles, with different emphases, based on ISETL. These articles all provide examples of ISETL code and reasons for considering this language.

Although generally considered as a research tool for algebraists, Groups, Algorithms and Programming (GAP) can also be used in the classroom. Juli Rainbolt's article indicates her methods for doing this. In contrast, while some readers may not regard MATLAB as a natural environment for computing in abstract algebra, George Mackiw makes a case for doing so. He illustrates how matrix groups over finite fields, with computations being performed by MATLAB, can provide examples, problems, and opportunities for experimentation. Two other general-purpose computer algebra systems are also used for algebra. Kevin Charlwood indicates how he uses Maple as a vehicle for discovery in his classes. Similarly, Al Hibbard uses Mathematica as the programming environment to execute the AbstractAlgebra packages (which form the foundation for Exploring Abstract Algebra with Mathematica, written by Al and Ken Levasseur). With these packages, one can interactively explore most of the topics that occur in the undergraduate abstract algebra curriculum, providing a visualization of the concepts where possible. Using software to generate examples and to illustrate the abstract material has probably been the most dramatic change in the way we teach abstract algebra.

Learning Algebra Through Applications and Problem Solving. Specific problems often allow an instructor of abstract algebra to explore a concept more creatively. Included here is an article by Michael Bardzell and Kathleen Shannon describing their PascGalois project. They introduce a group-theoretic generalization of Pascal's triangle and explore some of its ramifications. In particular, the coloring provided by their accompanying software (or using the AbstractAlgebra packages) helps students to visualize some interesting patterns. John Wilson takes the Lights Out puzzle and develops a project that analyzes it from an algebraic point of view. This article also incorporates tips for those who want to include similar student projects. Similarly, John Kiltinen explores other puzzles, illustrating how various algebraic concepts (in particular, permutations, conjugates, and commutators) can be seen by studying his puzzles. In Lucy Dechene's article, we see how group-theoretic notions even show up in change ringing (ringing bells in a prescribed fashion). She shows how British bell ringers worked with permutation groups considerably before mathematicians formalized them. An aural rather than a visual approach is yet another way to help students learn.

It has not been our intention to write the definitive volume on how to teach abstract algebra at the beginning of the twenty-first century. Indeed, students can have successful learning experiences in many different types of classrooms. Included here are only a portion of the innovations that are now being developed. We hope, however, that the ideas contained in this volume will stimulate readers to attempt some interesting experiments in their own abstract algebra classrooms. Choose a few ideas and try them out! A classroom that is active and that shows our students how creative and dynamic mathematics can be is an excellent learning environment.

Coherence in a collection of articles such as this one implies a high level of collaboration. The editors would first of all like to thank the authors of the articles for their interest in the project and their patience throughout the editing process. Special thanks go to Chris Christensen (Northern Kentucky University), Joseph Gallian (University of Minnesota Duluth), Gary Gordon (Lafayette College), Steven Hurder (University of Illinois at Chicago), Loren Larsen (St. Olaf College), and Josh Vogler (Central College student) for their careful reading and valuable suggestions. We are grateful for the insightful evaluations done by the Notes Editorial Board and especially wish to thank Sr. Barbara Reynolds for her help as we worked to prepare a final document. We appreciate the continuing support of our institutions, Central College and DePauw University. This project would not have been possible without the hard work of the folks at the Mathematical Association of America, including Don Albers, Elaine Pedreira and Beverly Ruedi. With all this support and encouragement, we have been able to create a volume that will, we think, have an important impact on the teaching of abstract algebra.