Innovations in Teaching Abstract Algebra

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Overview

This collection of articles is the outgrowth of several gatherings of mathematicians who were interested in discussing the teaching and learning of abstract algebra: two contributed paper sessions entitled Innovations in Teaching Abstract Algebra, at the 1997 and 1999 Joint Mathematics Meetings, and the NSF-UFE workshop Exploring Undergraduate Algebra and Geometry with Technology held on the DePauw University campus in June 1996. Unlike other books and materials that demonstrate one particular software package or pedagogical technique, this volume gives the reader an introduction to a wide range of ideas that have been used to teach abstract algebra over the past decade. The articles that appear here were chosen 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. There is something of interest to stimulate anyone who teaches abstract algebra, whether a seasoned veteran or a relative newcomer. Choose a few ideas and try them out! A classroom that is active and shows students how creative and dynamic mathematics can be is an excellent learning environment.

Since many of the articles rely on colored diagrams, have downloadable materials, or are best read while using some particular software, the editors have created a website that accompanies the volume.

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Editorial Reviews

Andrew B. Perry
"Presenting the fundamentals of abstract algebra to a typical class of undergraduates is easy; making them understand the subject, however, is another matter. While there is no magic formula for teaching abstract algebra, there are many tricks of the trade, and many innovations that have only been explored in the last decade or two. Innovations in teaching Abstract Algebra is a veritable treasure trove of valuable ideas and should prove a useful resource for both novice and experienced mathematics teachers....Innovations in Teaching Abstract Algebra delivers eighteen insightful perspectives in one convenient source. This book is a fantastic resource for any abstract algebra teacher interested in innovate teaching techniques. I strongly recommend it."
MAA Online
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Product Details

  • ISBN-13: 9780883851715
  • Publisher: Mathematical Association of America
  • Publication date: 1/28/2001
  • Series: Maa Notes Ser.
  • Pages: 136
  • Product dimensions: 8.40 (w) x 10.90 (h) x 0.40 (d)

Table of Contents

Preface
I.  Engaging Student in Abstract Algebra
Active Learning in Abstract Algebra: An Arsenal of Techniques; Laurie Burton, sarah-marie belcastro, and Moira McDermott
A Modified Discovery Approach to Teaching and Learning Abstract Algebra; Steve Benson with Brad Findell
On Driving Student to Abstraction; Paul Fjelstad
Using Geometry to Teach Group Theory; Gary Gordon
An Abstract Algebra Research Project: How many solutions does x2+1=0 have?; Suzanne Doree

II. Using Software to Approach Abstract Algebra
Laboratory Experiences in Group Theory: A Discovery Approach; Ellen J. Maycock
Learning Beginning Group Theory with Finite Group Behavior; Edward Keppelman with Bayard Webb
Discovering Abstract Algebra with ISETL; Ruth I. Berger
Teaching Abstract Algebra with ISETL; Karin M. Pringle
Using ISETL and Cooperative Learning to Teach Abstract Algebra: An Instructor's View; Robert S. Smith
Using GAP in an Abstract Algebra Class; Julianne G. Rainbolt
Experiments with Finite Linear Groups Using MATLAB; George Mackiw
Some Uses of Maple in the Teaching of Modern Algebra; Kevin Charlwood
Using Mathematica to Explore Abstract Algebra; Allen C. Hibbard

III. Learning Algebra Through Applications and Problem Solving
The PascGalois Traingle: A Tool for Varnishing Abstract Algebra; Michael J. Budzell and Kathleen M. Shannon
Developing a Student Project in Abstract Algebra: The Lights Out Problem; John Wilson
Leaning Permutation Group Theory via Puzzles; John O. Kiltenen
Ringing the Changes: An Aural Permutation Group; Lucy Dechene
Appendix
Internet Resources for this Volume
About the Authors
Index

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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, an 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 intrigues and stimulated by these diverse expositions.

 We have divided the articles into three broad, but ultimately overlapping areas: Engaging Students in Abstract Algebra, Using Software to Approach Abstract Algebra, and Learning Algebra through Applications and Problem Solving. Below we give an overview of the articles in each section. Additionally, each article begins with a brief abstract. Since some of the articles rely on colored diagrams, have downloadable materials, or are best read while using some particular software, we have created a webs tie that accompanies this volume.

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 x2+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). The 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 to an article discussing the program Finite Group Behavior(FGB) they have created. Intended as 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 indicated 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 topic s 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 Kilinen 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 changing 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.

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