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This book argues that the teaching of elementary linear algebra can be made more effective by emphasizing applications, expositions, and pedagogy.
This volume grew out of the work of the Linear Algebra Curriculum Study Group and the 1993 Special issue on Linear Algebra of the College Mathematics Journal.
Included are the recommendations of the Linear Algebra Curriculum Study Group, with their core syllabus for the first course, and the thoughts of mathematics faculty who have taught linear algebra using these recommendations. It includes elucidation of these ideas, trenchant criticism of them, and a report on putting them into practice.
A valuable resource for anyone teaching linear algebra. This book argues that the teaching of elementary linear algebra can be made more effective by emphasizing applications, exposition, and pedagogy.
* Relevant applications serve as motivation for all students and as sources of stimulating and challenging problems.
* Effective exposition that finds the right way to communicate concepts is especially important in the teaching of linear algebra, often the first course in which students come to grips with abstraction and complexity and with multiple representations of the same idea.
* Attention to pedagogy that takes into account how students learn technology, and new teaching ideas such as cooperative learning can go a long way toward improving the teaching of linear algebra. This attention helps to increase student understanding of the material.
Contains a core syllabus for the first course in linear algebra, and the thoughts of mathematics faculty members who have taught linear algebra using these recommendations.
The idea of "calculus reform" goes back to the Conference/Workshop to Develop Alternative Curriculum and Teaching Methods for Calculus at the College Level held at Tulane University in 1986. With this conference, and the volumes "Toward a Lean and Lively Calculus" and "Calculus for a New Century", we were all confronted with a challenge. The way we had taught calculus wasn't working; given today's students, faculty, and technology, how should we change it?
It is not surprising that the negative aspects of our students' learning of calculus would be seen to appear in other mathematics courses: in remedial algebra and trigonometry, in differential equations, in the related field of statistics, and in particular in linear algebra.
It is our hope that this volume will be in a fact a "Resource for Teaching Linear Algebra" for instructors in our field, and help them help their students to learn our subject better. In "Resources", we present material on many different aspects of the teaching of linear algebra. The titles of the part of the volume illustrate the breadth of the issues we deal with: The Role of Linear Algebra, Algebra as Seen from Client Disciplines, The Teaching of Linear Algebra, Linear Algebra Expositions, and Applications of Linear Algebra. In each part, the articles will be discussed briefly in an introduction.
We begin our volume with a brief history of linear algebra curriculum reform in the United States in recent years.
Workshops on Computing in the Teaching of Linear Algebra
The first "teaching reform" activities in linear algebra were workshops, short courses, mini-course, etc. on the use of computing in teaching linear algebra. By now, many people have organized such activities; the list includes Homer Bechtell, Jane Day, Benny Evans, Eugene Herman and Charles Jepsen, David Hill and David Zitarelli, Jerry Johnson, Don LaTorre, Steve Leon, Koo Rijpkema, and Kermit Sigmon. Of these, the most ambitious have been the NSF-sponsored ATLAST Workshops of Steve Leon. Many faculty have taken part in these activities, and are now using computing in their linear algebra classrooms. A number of them have made presentations about their activities at national or regional or special-interest meetings.
This is now a variety of elementary linear algebra textbooks containing computing exercises (see Jane Day's article in this volume). There are also books and other materials specifically dealing with computing in linear algebra.
The Linear Algebra Curriculum Study Group
"Linear algebra reform" activities beyond computing workshops go back to 1989, when the participants at an NSF-sponsored Summer Short course on Matrices (organized at Laramie by Duane Porter; the principal speaker was Charles Johnson) decided to write down issues they saw facing linear algebra instructors. Porter organized a Panel Discussion on Linear Algebra in the Undergraduate Curriculum at the 1990 Joint Mathematics Meetings in Louisville. The panelists were Irving Katz and John Poole from the Summer Short Course, David Carlson, and David Lay. Three hundred people attended, most for the entire three hour session! A lively audience discussion followed the panelists' presentations.
After the session, Carlson, Johnson, Lay and Porter organized the Linear Algebra Curriculum Study group. The first LACSG activity was an NSF-funded Workshop at the College of William and Mary held August 7-11, 1990, which involved a broadly-based panel of 20 people from academia and industry. A series of recommendations, including a core syllabus for the first course in linear algebra, was prepared and disseminated. It is reproduced here.
There have been special sessions at all Joint Mathematical Meetings since 1991: at San Francisco (1991 and 1995), Baltimore (1992), San Antonio (193), Cincinnati (1994), Orlando (1996) and San Diego (19997). These have been organized by David Lay, Don LaTorre, Steve Leon, and Duane Porter. There have been many speakers on aspects of the teaching of linear algebra, and a great deal of audience interest.
Another project of the organizers of the LACSG, also with NSF support, has been to prepare a collection of "Gems of Linear Algebra": especially insightful proofs, longer expositional items, and some problems for students. This is intended to assist instructors in their presentation of fundamental linear algebra ideas in improved ways. This volume is nearing completion at this time.
The Special Issue on Linear Algebra of the College Mathematics Journal
One of the "special" activities in linear algebra curricular discussion was the publication in January 1993 of a Special Issue on Linear Algebra of the College Mathematics Journal, edited at that time by Ann and Bill Watkins. This volume contains a number of articles that appeared first in the Special Issue. It also includes some articles which appeared originally in the American Mathematical Monthly and in other issues of the College Mathematics Journal.
Review (a fair-use review excerpt; the source must be cited):
"It contains much extremely interesting material on specific topics in linear algebra and their teaching (including the use of various packages): so it is, indeed, a resource for the teaching of linear algebra…. This rich and fascinating book will help you respond creatively to it….Every lecturer teaching linear algebra should have a copy." - Philip Mahar, Middlesex University, Queensway, Enfield: Mathematical Gazette
"This book argues that the teaching of elementary linear algebra can be made more effective by emphasizing applications, exposition and pedagogy. Relevant applications serve as motivation for all students and as sources of stimulating and challenging problems. Effective exposition that finds the right way to communicate concepts is especially important in teaching of linear algebra, often the first course in which students come to grips with abstraction and complexity, and with multiple representations of the same idea. Attention to pedagogy that takes into account how students learn, technology, and new teaching ideas such as cooperative learning can go a long way toward improving the teaching of linear algebra. " Zentrallblat fur Mathematik
Part I The Role of Linear Algebra
The Growing Importance of Linear Algebra in Undergraduate Mathematics: Alan Tucker, State University of New York-Stony Brook
Part II Linear Algebra as Seen From Client Disciplines
Matrix Algebra in Economics: Clopper Almon, University of Maryland
The Undergraduate Linear Algebra Curriculum: A View From a Client Discipline, Computer Graphics; Rosemary E. Chang, Silicon Graphics Computing Systems
Linear Algebra for Computer Science Students; Margaret H. Wright, AT&T Bell Laboratories
Linear Algebra Use at Boeing: Implications for Undergraduate Education: David P. Young, Boeing Computer Services
Part III The Teaching of Linear Algebra
Teaching Linear Algebra: Must the Fog Always Roll In?: David Carlson, Dan Diego State University
The Linear Algebra Curriculum Study Group Recommendations for the First course in Linear Algebra: David Carlson, Charles R. Johnson, David C. Lay, A, Duane Porter
A Project on Circles in Space: Carl C. Cowen: Purdue University
Teaching Linear Algebra New Ways: Jane M. Day, San Jose State University
Some Thought on a First Course in Linear Algebra at the College Level: Ed Dubinsky, Purdue University, Purdue University and Education Development Center
The Linear Algebra Curriculum Study Group Recommendations: Moving Beyond Concept Definition; Guershon Harel, Purdue University
Gaussian Elimination in Integer Arithmetic: An Application of the L-U Factorization; Thomas Hern, Bowling Green State University
Iterative Methods in Introductory Linear Algebra; Donald R. LaTorre, Clemson University
Reflections (1988); Robert Mena, California State University Long Beach
Writing About Linear Algebra: Report on an Experiment; Gerald J. Porter
Scenes From Linear Algebra Classes; Shlomo Vitar, the Hebrew University of Jerusalem, Givat Ram
Part IV Linear Algebra Exposition
Down with Determinants!; Sheldon Axler, Michigan State University
Subspaces and Echelon Forms; David C. Lay, University of Maryland
A Geometric Interpretation of the Columns of the (Pseudo)Inverse of A; Melvin J. Maron, University of Louisville, Ghansham M. Manwani, Universidade do Amazonas, Brazil
The Fundamental Theorem of Linear Algebra: Gilbert Strang, Massachusetts Institute of Technology
Part V Applications of Linear Algebra
Some Applications of Elementary Linear Algebra in Combinatorics: Richard A. Brualdi, University of Wisconsin; Jennifer J. Quinn, Occidental College
Arithmetic Matrices and the Amazing Nine-Card Monte; Dean Clark and Dilip K. Datta, University of Rhode Island
A Random Ladder Game: Permutations, Eigenvalues, and Convergence of Markov Chains; Lestor H. Lange, San Jose State University; James W. Miller, MCI Communications Corporation
Linear Algebra and the Affine Planar Transformations; Gerald J. Porter, University of Pennsylvania
Patterns in Linear Algebra; Gilbert Strang, Massachusetts Institute of Technology
Graphs, Matrices, and Subspaces; Gilbert Strang, Massachusetts Institute of Technology