Report of the Pedagogy and Content Issues Subgroup
of the Undergraduate Faculty Program, PCMI, July 1998

Jane M. Day


The pedagogy subgroup of the Undergraduate Faculty Program held three types of sessions:

  1. Continuing discussions on the content of linear algebra. This included discussion of various papers and reviews of selected textbooks.

  2. Joint meetings with Math Education Researchers regarding effective teaching techniques, how students learn, and methods to assess learning.

  3. Joint meetings with Math Education Researchers, High School Faculty and other interested PCMI participants about the TIMMS study and the new NCTM Standards, and generally to enhance communication between high school and university level faculty regarding course content and expectations at each level.

What follows is a summary of the activities of each of these groups. Section 4 is references and Section 5 contains a list of participants.




  1. Content of Linear Algebra Courses

  2. Effective Teaching Techniques and Assessment

  3. Joint sessions with High School Teachers and Math Educators

  4. References

  5. Questions to assess students' deeper understanding of concepts

  6. Members of the Pedagogy Subgroup



1. Content of Linear Algebra Courses


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 2-3 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 determinant-free 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 one-to-one 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 one-parameter 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.



2. Effective Teaching Techniques and Assessment


We had many discussions with Math Education researchers Tim Kelly, Richard Lehrer, Jeffrey Barrett, Danny Goroff, Mike Battista, Joan Ferrini-Mundy 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 in-depth 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 open-ended 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. Open-ended 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 R2 and R3. 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 Rn. His work indicates it is better to first define each concept in Rn, 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 calculus-based 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.



3. Joint sessions with High School Teachers and Math Educators


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 7-10 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 R3 before college. They really need to understand vectors, lines and planes in R3 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 3-d 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.



4. References


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.

  1. The Linear Algebra Curriculum Study Group recommendations for a first course in linear algebra, D. Carlson, C.R. Johnson, D. Lay and D. Porter.

  2. Teaching linear algebra: must the fog always roll in? D. Carlson, 39-58.

  3. Some thoughts on a first course in linear algebra at the college level, E. Dubinsky,85-106.

  4. The Linear Algebra Curriculum Study Group recommendations: moving beyond concept definition, G. Harel, 107-126.

  5. Down with determinants! S. Axler, 179-200.

  6. A private universe, M.H. Schneps, Project Star, Derek Bok Center, Harvard University.

  7. Thinking together, collaborative learning in science, Derek Bok Center, Harvard University.

  8. Case number one: force and inertia, a case about teaching introductory physics, G. Goroff and J. Wilkinson. Response to case number one: hockey pucks, monkeys and misconceptions. These two papers are in Change (Nov./Dec. 1993), 22-29.

  9. Viewing some concepts and applications in linear algebra, T. Hern and C. Long, in Visualization in Teaching and Learning Mathematics, ed. W. Zimmerman and S. Cunningham, MAA Notes 19, 1991.

  10. Students' proof schemes: results from exploratory studies, G. Harel and L. Sowder, CBMS Issues in Education, vol. 7, 1991.

  11. Two dual assertions: the first on learning and the second on teaching (or vice versa), G. Harel, American Mathematical Monthly 105 (1998), 497-507.

  12. Force concept inventory, D. Hestenes, M. Wells and G. Swackhamer, Physics Teacher 30 (1992), 141-151.

  13. A mechanics baseline test, D. Hestenes and M. Wells, Physics Teacher 30 (1992), 159-

  14. What does the force concept inventory actually measure? D. Huffman and P. Heller, Physics Teacher 33 (1995), 138-143.

  15. Performance on mechanics concept inventory tests – diagnostics and complementary exam problems, Physics Teacher 35 (1997), 150-155.



5. Questions to assess students' deeper understanding of concepts


Eigenvalues and Eigenvectors

These are some questions that are suitable for interviews with students designed to explore their understanding of eigenvalues and eigenvectors.

  1. What motivates the concepts of eigenvalue and eigenvector for you?

  2. What is the definition of eigenvalue and eigenvector?

  3. Given a matrix A, how do you compute the eigenvalues and eigenvectors?

  4. Can you give a use of eigenvalues and eigenvectors other than the one you mentioned in response to question 1? How about another?

  5. How many different eigenvectors are there associated with a given eigenvalue?

  6. Given a matrix A, can you find enough eigenvectors to form a basis?

  7. What is the difference between the algebraic multiplicity and the geometric multiplicity of an eigenvalue?

  8. What is a generalized eigenvector? Why are they important?Suppose a matrix A is symmetric. What is special about its eigenvalues and eigenvectors?

It is intended that each question would be the start of a conversation. The follow-up questions would be natural follow-ups to the student’s responses.

While these questions provide a good starting point, we are reluctant to claim they form anything close to a finished list. The list was compiled in too short a time to assure that.


Systems of Linear Equations

  1. Major concepts.

    1. Algebraic and geometric descriptions of the possible types of solution sets

    2. Algebraic and geometric descriptions of a solution set

    3. The Superposition Principle – i.e., using solutions of Ax = 0, Ax = b1 etc. to construct solutions to other systems.

    4. Rank

    5. Least squares solutions


  2. What we want a student to be able to do:

    1. Describe a linear system in matrix form if it is given as a set of equations, and vice versa.

    2. Explain both algebraically and geometrically why there are only three basic possibilities for the solution set to a linear system (empty, or a single point, or an infinite set of points).

    3. Describe both algebraically and geometrically how the following two sets are alike and how they differ: the solution set of Ax = 0 and the solution set of Ax = b when b ¹ 0.

    4. If A, b1, and b2 are known and solutions of Ax = b1 and Ax = b2 are given, write a solution for Ax = b3 when b3 is in the span of {b1, b2}.

    5. Explain both algebraically and geometrically what a "least squares solution" to a linear system Ax = b means, why such a solution must always exist, and at least one way to calculate one.


  3. Sample questions to determine a student's knowledge. The first 11 would be appropriate for a written test or interview.

    1. Consider this system of equations: x – 4y + 7z = 3 and x – z = -1.

      1. Write the system of equations above as a matrix equation Ax = b (that is, define A and b).

      2. Which one of the following statements is correct?

        1. The system is inconsistent.

        2. The solution set for the system is the single point (-1,-1,0).

        3. The solution set for the system is all vectors of the form t(-1,-1,0), where t can be any real number.

        4. The solution set for the system is all vectors of the form t(1,2,1) + (-1,-1,0), where t can be any real number.

        5. None of the above is correct.

    2. Consider this system of equations: 2x – y + 4z = 2 and x + y – z = 7. What does the solution set look like geometrically?

      1. a single point
      2. a line
      3. a plane
      4. a circle
      5. none of the preceding
      6. not enough information

    3. Suppose A is a 3x3 matrix, y ¹ 0 and Ay = 0. Will there be a solution to Ax = b for every b in R3? Explain.

    4. Two planes in R3 can intersect in various ways. Check the items in the following list that could be correct:

      1. The planes intersect in a single point.
      2. The planes intersect in three points.
      3. The planes intersect in a line.
      4. The planes do not intersect.

    5. Suppose you have three linear equations in three variables. Which of the following could be the solution set for such a system? Check all that are possible:

      1. a single point
      2. the empty set
      3. a line
      4. a parabola
      5. a plane
      6. a circle
      7. all of R3

    6. Suppose A is a 2x2 matrix which has rank one. Which one of the following describes the solution set to Ax = 0?

      1. a line
      2. a single point
      3. a circle
      4. all of R2
      5. none of these
    7. Let A = and b =. Is a solution for the system Ax = b? Explain.

    8. Suppose A is a matrix and v and w are solutions to Ax = 0.

      1. Is 2v also a solution to Ax = 0? Why or why not?

      2. Is v + w also a solution to Ax = 0? Why or why not?

    9. Suppose b¹ 0 and Ax =b is consistent. Describe how the solutions of Ax = b are related to solutions of Ax = 0.

    10. Let A be a 2x3 matrix. Suppose that y0 is a solution to Ax = , y1 is a solution to Ax = , and y2 is a solution to Ax = .

      1. Is 3y0 + y1 + 5y2 a solution to Ax = ?

      2. Write a solution to Ax =.

      3. Write all solutions to Ax =.

      4. Can Ax = b can be solved for every b in R3? Explain.

    11. A certain matrix A has reduced echelon form . Find all solutions to Ax = 0.

    12. A certain matrix A and vector b are such that the augmented matrix [A b] has reduced echelon form . Find all solutions to Ax = b.



      The following questions are more appropriate for a verbal interview, because students learn different methods for solving least squares problems.

    13. Suppose A is a 40x3 matrix and b is a 3x1 column vector. Let W denote the span of the columns of A. Suppose b is not in W, as indicated in the sketch.

      1. Explain what a least squares solution for Ax = b is.

      2. Explain why a least squares solution must exist for Ax = b.

      3. Explain a method for finding a least squares solution for Ax = b.

    14. Suppose b is in W, so Ax = b is consistent, and suppose v is a least squares solution for Ax = b. Is v a true solution of Ax = b? Explain.



The core facts about an invertible nxn matrix A:


Some problems designed to test students' understanding of these basic concepts:

  1. By trial and error, some students working with a particular 3 by 3 matrix A discover several solutions to the equation Ax = [5,8,11]t, including [0,1,0]t and [2,0,0]t . Then one of students suggests finding the inverse of A as a more systematic way to find solutions. Do you agree with this suggestion? Explain why.

  2. A and B are 4 by 4 matrices, that are inverses of each other. Which of the statements below are true? There are may be more than one correct statement. Explain why the ones you select are correct.
    1. If Av=w then Bv=w.
    2. If Av=w then Bw=v.
    3. If Av=w then Aw=v.
    4. If Bv=w then Av=w.
    5. If Bv=w then Aw=v.
  3. Given a 5 by 5 matrix A and a vector b in R5 , which of the following are possibilities for the system Ax=b? (More than one may be possible.)

    1. The system has no solutions.
    2. The system has exactly one solution.
    3. The system has exactly two solutions.
    4. The system has a very large finite number of solutions.
    5. The system has an infinite number of solutions.

  4. Suppose A is a noninvertible square n-matrix and b is a vector in Rn. Is it possible for the equation Ax=b to have a solution? Explain briefly your answer.

  5. Suppose A is an invertible 2 by 2 matrix. Which of the following statements is true? Explain why.

    1. Ae1 is invertible, where e1=[1,0]t.
    2. Ae2= [0,0]t, where e2=[0,1]t.
    3. {Ae1, Ae2} is linearly independent.
    4. Ae1=Ae2.


Linear Combinations, Linear Independence, Span, Basis, Dimension, and Subspace

Our Rationale: We felt that the basic problem that students encounter when seeing these ideas for the first time in a linear algebra course is their unfamiliarity with how to work with definitions in a mathematically precise way. The fact that these unfamiliar definitions are usually encountered one after another in rapid succession---the definition of a span depends on an understanding of linear combinations, the definition of a basis depends on the ideas of linear independence and spanning sets---magnifies this problem.


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.


  1. Given two vectors v1 and v2 in a coordinate plane, sketch the set {a1v1 + a2v2 | 0 < a1,a2 < 2}.

  2. Suppose we have two vectors in R3 that span a plane. Are they independent? Consider a third vector. How would you describe geometrically where to put the third vector so that the resulting set is a linearly independent set? A linearly dependent set?

  3. Given a coordinate plane with vectors v1 and v2 and points A, B, and C drawn in the plane, estimate the constants a1 and a2 such that a1v1 + a2v2 is A, B, or C, respectively.

  4. Suppose a set of vectors spans R3. What can you say about how many vectors are contained in this set? Will every set of elements of R3 with this number of vectors span R3?

  5. Suppose that {v1,...,vn} are linearly dependent. Must v1 be a linear combination of v2,...,vn? Explain.

  6. Suppose v1,v2,v3,v4 are vectors in R2 which span R2. Do v1 and v2 span R2?

  7. Suppose {v1,v2}, {v1,v3}, and {v2,v3} are linearly independent sets. Must the set {v1,v2,v3} be linearly independent?

  8. Two nonzero vectors are independent if and only if the angle between them is neither 0o nor 180o. Can you find conditions on the angles between three nonzero vectors that would tell you whether or not they form an independent set?

  9. Is R2 a subspace of R3? Explain.



6. Members of the Pedagogy Subgroup


Daniel Goroff
Department of Mathematics
Harvard University
Science Center 325
Cambridge, MA 02138-2901


Fernanda Botelho
University of Memphis/Institute for Mathematics
514 Vincent Hall
206 Church St. SE
Minneapolis, MN 55455


Harold Davenport
Department of CS, Mathematics and Statistics
Mesa State College
Grand Junction, CO 81502


Jane Day
Dept. of Mathematics and Computer Science
San Jose State University
San Jose, CA 95192-0103


Thomas Hagedorn
Department of Mathematics
The College of New Jersey
38 Chestnut St.
Princeton, NJ 08542


Paul Hurst
Brigham Young University – Hawaii
MSC Division Box 1967
Laie, HI 96762


Dan Kalman
Department of Mathematics and Statistics
American University
4400 Massachusetts Avenue, NW.
Washington, DC 20016-8050


Andrew Leahy
Dept. of Mathematics
P.O. Box 110
Knox College
Galesburg, IL 61401


Michael Penkava
Math Department
University of Wisconsin –Eau Claire
Eau Claire, WI 54702


John Polking
Department of Mathematics
Rice University
P.O. Box 1892
Houston, TX 77251


Lorenzo Sadun
Department of Mathematics C1200
University of Texas
Austin, TX 78712