The pedagogy subgroup of the Undergraduate Faculty Program held three types of sessions:
The UFP participants who were most interested in discussing content and methods formed this Pedagogy Group. First we discussed the recommendations of the Linear Algebra Curriculum Study Group (LACSG) and each participant described the content and clientele of the linear algebra courses they usually teach. We found that the majors, ages and work experience of the students taking linear algebra varied dramatically from university to university, as did the level of abstraction, computer component, type of applications included, etc. We soon decided it would not be realistic for us to try to update the LACSG recommendations, as no uniform syllabus would be appropriate for our courses. But we did agree that anyone teaching linear algebra should find out what their clientele’s needs are and address those as best they can.
We decided to focus our time at PCMI on discussing how particular topics fit into the curriculum, getting better informed about how students learn, and trying to develop new teaching styles that would improve students’ learning.
We discussed Carlson's paper [2], in which he responds to the LACSG recommendatiion for a more matrix oriented course. He proposes a variety of explorations of properties of matrices, none of which appeared in linear algebra texts before 1990, but which have since become standard topics. Another recommendation of the LACSG is that two days be devoted to determinants. Our participants reported they spend 23 days on them, and that the primary use of determinants in our courses is for defining the characteristic polynomial and using that to calculate eigenvalues. Some people also cover Cramer's Rule, and some show how determinants measure area and volume. We were curious about Sheldon Axler’s paper "Down with Determinants" [5], in which he argues forcefully that they not be taught at all and presents a determinantfree development of eigenvalue theory. Tom Hagedorn discussed Axler's main ideas and we agreed that his development is lovely. However for most of our students, his approach is too abstract, and it is not not a practical way for students to understand how to calculate eigenvalues. Also, none of us agreed with Axler that determinants are ugly, useless, and don’t fit in linear algebra. Our students really like them, and they are useful in a variety of applications. Equally important, linear algebra has always had deep connections with combinatorial as well as continuous mathematics, and determinants are the one topic in a first linear algebra course where the connection with combinatorial ideas is apparent.
Dave Morrison led a discussion about articulation – i.e., how well the linear algebra courses we teach coordinate with the subsequent courses where linear algebra is needed. We identified a number of these: multivariable calculus, ODE, physics, scientific computation, statistics, linear programming, economics, (Lagrange multipliers, Markov processes), computer graphics, math courses for teachers, and other math courses such as game theory, graph theory, and coding theory. We found that only a few of us communicate with the people who teach such courses, so we really don't know for sure what they want their students to learn in linear algebra, or whether they are pleased with what we're teaching. We agreed we need to remedy that, and that it will probably work best to do that on a onetoone basis with faculty in other departments.
Roger Howe and Bill Barker (who were teaching courses for the undergraduates at PCMI on Lie theory and continuous symmetry) met with the UFP faculty once, to discuss how topics from representation theory could be integrated into undergraduate courses. Howe includes oneparameter matrix groups in his advanced calculus course, and it seems this would be a lovely topic for a special project in linear algebra or differential equations. Barker’s continuous symmetry course is based on a textbook which he and Howe are presently writing, and this very fresh presentation of geometry would be an attractive and appropriate upper division course at any college.
We had many discussions with Math Education researchers Tim Kelly, Richard Lehrer, Jeffrey Barrett, Danny Goroff, Mike Battista, Joan FerriniMundy and Guershon Harel. They talked about how people learn, and stages of development which a mind goes through as it matures in its ability to understand abstraction. We discussed the value of indepth instruction versus quantity of topics. They suggested ways technology can improve students’ learning. We did not resolve this issue, but the message which Harel in particular sends is that it is pointless for students to study a math topic and not internalize their understanding of it enough that they can remember and use it later. It seemed to most of us that he succeeds at Purdue in covering a remarkable amount in his linear algebra courses, while insisting that students think through all the theory and mature in their ability to understand, verbalize and write proofs.
Probably the most important thing these researchers conveyed to us is that is essential we listen more to our students, to understand how they are reasoning and where they get stuck. They explained various techniques we could use: "minute" papers, interview techniques, and some cooperative learning methods that can increase the interaction of students with each other and with us. They recommended that we should develop our "interview" skills: practice asking more openended questions that might reveal how a student is thinking, try to listen carefully, respect their ideas, and ask more questions when it appears their understanding is not clear, rather than leaping to tell them the correct answer. This approach can make class time more effective. They also suggested we have formal interviews with a few students occasionally, during office hours, and that we keep notes on these. They pointed out that most of our students have similar backgrounds, so even a small number of such interviews could reveal conceptual difficulties that are common to many students. Jeff Barrett showed a videotape to help us understand how to conduct such interviews. Openended questions would be ones like "explain what linear independence means to you" and "how are you thinking about this problem."
Danny Goroff presented two videotapes [6,7]. The first, "A Private Universe," shows Harvard University graduates who do not understand the fundamental science fact about why the earth has seasons. This led to a discussion regarding whether students know and retain the knowledge we think we are imparting. The second tape, "Thinking Together, Collaborative Learning in Science" illustrated effective teaching techniques with a large lecture class in science using quick questions given to the class, informal short discussions between students and polling of answers.
We had two very provocative discussions with Guershon Harel, who has been teaching linear algebra and doing research on how students learn the subject for more than ten years. One of the LACSG recommendations was that geometric interpretation of concepts should always be included in linear algebra, and most textbooks written since that time include such interpretations. Discussions about the LACSG report continue at professional meetings, and one of the things that many people report is that students seem to understand concepts like span, independence and subspace much better now than they used to. (Presentations of linear algebra used to be heavily algebraic.) So were were somewhat amazed at what Guershon told us. He has become convinced recently that it is a mistake to motivate such concepts by first giving examples in R^{2} and R^{3}. His research indicates that when he does that, many students are never able to let go of these visualizations and handle such concepts effectively in R^{n}. His work indicates it is better to first define each concept in R^{n}, then give examples which include the geometric visualizations in low dimensions as well as some other type examples. The point seems to be, "first impressions are powerful," so we should be careful that students’ first impression of each concept is reasonably complete. We’re not all convinced, because geometric motivation can be so powerful. Indeed, Guershon says that’s the only way he knows how to effectively introduce the method of least squares approximation. But some of us will probably try Guershon’s recommendation with some concepts.
We read and discussed a number of articles suggesting ways to teach linear algebra more effectively [2, 3, 4, 8, 9, 10, 11]. We also read some papers from the physics literature about two "concept inventories" which have been developed by the physics community in the past decade [12, 13, 14, 15]. These have qualitative rather than quantitative questions, and their purpose is to evaluate students' real understanding of force and mechanics after they complete a calculusbased physics course. Apparently these inventories have revealed fundamental misconceptions, among large numbers of students, about Newton's laws  including many students who had performed well in very demanding courses. Considerable analysis has been done of students' answers, and there is much discussion among physics faculty throughout the country about how to change the way physics is taught, to improve students’ fundamental understanding.
Our Pedagogy group had not heard before about this work, and found it intriguing. We talked about trying to develop a similar concept inventory for linear algebra, and we made some preliminary progress toward this goal. We divided into four subgroups to think about these topics: linear systems; inverses; eigenvalues; and subspace, basis and dimension. Each group was to identify the major concepts in its topic and develop some questions that would be suitable for assessing students' deeper understanding of that topic. The reports of these groups are attached. Some of the questions we developed are more appropriate for oral interviews, others for a written inventory. We agreed that developing a concept inventory for linear algebra is a bigger task than we could accomplish. For one thing, our group may not be sufficiently representative, and we as individuals need to find out more about how our students are thinking. So for now, we are simply going to try using questions like these in interviews and on tests.
Gail Burrill (former president of NCTM) met with us. We were all surprised to hear that some matrix topics have been in most high school texts for 710 years (if our students know these things, they have been keeping it quiet!). She also told us that the new NCTM recommendations for high school mathematics include more linear algebra topics and applications that use matrices. Curriculum material has been written to follow the new guidelines, and this is being used in selected places around the country. This new material will start appearing in major publishers' books within a year or two. We asked about teacher training and Gail admitted that there is no coherent plan to provide that, and this is a problem. We wonder if master teachers could be identified in schools, who could receive special training and then teach their colleagues. Some of the high school teachers at PCMI told us that such training would be quite attractive if it earned some college credit.
Dan Kalman led a discussion between the High School Teachers, Undergraduate Faculty and Math Education Researchers. These teachers were clearly a very dedicated and talented group. They explained what topics they cover in various courses, especially the geometry and matrix topics they teach. We talked about how curricula at the secondary and college levels affect each other, and expressed a fervent wish that students learned some analytic geometry in R^{3} before college. They really need to understand vectors, lines and planes in R^{3} in third semester calculus and in linear algebra. In fact we would much prefer they learn solid geometry in high school rather than calculus or matrix topics. The high school teachers admitted that 3d analytic geometry is not taught any more, but explained that is because of heavy pressure from parents to prepare students for the AP calculus exam. Solid geometry is not on that test so it has been dropped from high school texts. In fact, matrix topics are not essential for that exam either and tend to get skipped when time is limited. It was clear that more communication between high school and college educators could benefit everyone, and the participants were urged to explore avenues to encourage regular exchanges.
The first five references below are from Resources for Teaching Linear Algebra, ed. D. Carlson, C.R. Johnson, D. Lay, D. Porter, A. Watkins and W. Watkins, MAA Notes 42, 1997. This is a collection of 28 papers on many topics related to linear algebra: what clients need; ideas on what and how to teach, including thoughts on incorporating computer use; applications; and special topics.
The following questions are more appropriate for a verbal interview, because students learn different methods for solving least squares problems.
Some problems designed to test students' understanding of these basic concepts:
Consequently, our goal was to develop exercises which tested a student's understanding of these definitions by asking questions which required geometrical intuition and working directly with the definitions.
Daniel Goroff Department of Mathematics Harvard University Science Center 325 Cambridge, MA 021382901 goroff@abel.math.harvard.edu Fernanda Botelho University of Memphis/Institute for Mathematics 514 Vincent Hall 206 Church St. SE Minneapolis, MN 55455 botelho@irna.umn.edu Harold Davenport Department of CS, Mathematics and Statistics Mesa State College Grand Junction, CO 81502 davenpor@mesa5.mesa.colorado.edu Jane Day Dept. of Mathematics and Computer Science San Jose State University San Jose, CA 951920103 day@sjsumcs.sjsu.edu Thomas Hagedorn Department of Mathematics The College of New Jersey 38 Chestnut St. Princeton, NJ 08542 hagedorn@tcnj.edu Paul Hurst Brigham Young University – Hawaii MSC Division Box 1967 Laie, HI 96762 hurstp@buyh.edu

Dan Kalman Department of Mathematics and Statistics American University 4400 Massachusetts Avenue, NW. Washington, DC 200168050 kalman@email.cas.american.edu Andrew Leahy Dept. of Mathematics P.O. Box 110 Knox College Galesburg, IL 61401 aleahy@knox.edu Michael Penkava Math Department University of Wisconsin –Eau Claire Eau Claire, WI 54702 penkawmr@uwec.edu John Polking Department of Mathematics Rice University P.O. Box 1892 Houston, TX 77251 polking@rice.edu Lorenzo Sadun Department of Mathematics C1200 University of Texas Austin, TX 78712 sadun@math.utexas.edu http://www.ma.utexas.edu/users/sadun
