New Syllabus for Linear Algebra

I last posted about how I was rethinking the content expectations for linear algebra, so that I could write my own text. I took another step towards getting class ready today: I finished my syllabus.

This was inspired by examples that Susan Hill, our CETL director, handed out at this summer’s Course Design Academy. She had several different syllabus pairs in a before/after set-up. I really liked the way that most of the redesigned and reworked documents looked, so I wanted to write my own, better, syllabus document.

Here it is. I guess you can comment, but I am done with this for now. I have other things to do.

My New Linear Algebra Syllabus

The primary changes for me are these:

  • I am not doing SBG/SBAR this term, since I have so much else to manage.
  • I have streamlined my list of outcomes into four “process goals” and five “Big Questions.”
  • I am going to try out some peer grading for the first time.
  • I am definitely going to use an IBL scheme which is “individual work at home; group discussion in class; solo presentation in class,” but I have not made a big deal of this on the syllabus as being an IBL environment.

Anyway, you can do this too. It just takes several years of flailing about and thinking hard, two weeks of digesting the University, Department, and MAA learning goals, and then about 12 hours of writing and editing over two days.

Learning Goals for Linear Algebra: Content and Process

As part of my reworking of linear algebra, I have been reconsidering the course goals. I went back and read some of what I have written before about the mathematics to be discussed and the working habits on which we will focus. There is also this post from early in last semester. I think I only have small tweaks to make on these counts.

Well, I am a long way from having a fully developed set of Student Learning Objectives and a Student Outcomes Assessment set-up. (The SLO/SOA regime is the language used to sound official at UNI.) I have got a reasonable set-up for the department level technology SLO/SOA. At some point, I should get such an explicit set-up for my course. Maybe next summer I will work on that? Uh, sure, next summer.

Changes to Content Coverage

Last semester, the class got just within shouting distance of the three act content plan I wrote in the post linked above. Our discussion of determinants was superficial, and we just didn’t talk about the finite dimensional spectral theorem. Given that we basically wasted the first two weeks, I think those things can be addressed easily.

I would like to add just the smallest thing. Given all of the things listed here, and the focus I give to treating a matrix as a function, it would be great to add a discussion of the singular value decomposition.

Oh! Also, I didn’t talk about the cross product in Euclidean 3-space at all. I am comfortable with this.

Changes to the Meta-mathematical, process goals

I still see these as primary working habits that students must gain proficiency with to succeed:

  • make and explore examples
  • gain experience with abstraction
  • engage in more careful use of language

I will have to address these goals by writing my course so that students are forced to grapple with them, and also by being explicit with my students that this is expected of them.

And I still have these goals to deal with:

  • learn to use technology in an appropriate way
  • learn to read a textbook for understanding

I have a plan for the technology part. I have to work out something reasonable to help students with reading for understanding. Again, I have to be explicit about expectations and give focused instruction. (Somewhere in the back of my mind I have an evil plan to implement some sort of analog of a grad school foreign language reading exam.)

Linear Algebra Technology Implementation

One of the components of my linear algebra course that has felt like a real struggle is finding a meaningful way to integrate use of technology into the course. By meaningful, I intend something that requires the students to engage with modern computational technology. I’ll have more to say about that below, but a big part of the reason for writing this post is to hash out ideas about what I want to do and how I will do it.

Context: Departmental Student Learning Outcomes

Two or three years ago, the UNI Math Department did a bit of a curriculum review. As part of this, we adopted an official Student Learning Objectives Document (you know, assessment and accountability are everywhere these days) and we discussed tweaks to a few courses to make everything fit.

One of the formal Learning Objectives became this:

Technology specification:

Students will demonstrate basic proficiency with mathematical software. Students will be able to make informed choices about when the use of technology is viable and useful.

And the place we chose to address this learning outcome formally is…Linear Algebra. The main reason for this is that linear algebra is part of our core curriculum: it is part of all three of our major programs. Another reason is the timing: linear algebra comes early enough in each of those programs that we can hope to make use of the technology skills built in later courses, but late enough that we are not impacting many of our client departments heavily with this adjustment.

My Previous Attempts

I have taught linear algebra three times at UNI, once in each of the last three years. Each time I had the revision to our Student Learning Objectives in mind, and I tried to do something to address it. This fit nicely with another project I got involved in: UTMOST is a project funded by the NSF and run through the American Institute of Mathematics focused on adoption issues for open-source textbooks and software in the mathematics undergraduate curriculum. This came about right as I was starting to teach linear algebra and I got involved as a “test site.” Fitting in with the project’s aims, I have been learning how to use Sage: and I have tried to incorporate it into my classroom.

At first, this meant giving some large Sage-based homework assignments. These were not quite projects, but they were stand-alone assignments. This was a bad idea. The size and complexity of the assignments meant that students did not really learn how to deal with using Sage because they procrastinated, got frustrated on deadline, and gave up. I learned the hard way that most of my students have very little sense of how to deal with a computer. Even simple tasks like navigating to a web site and making an account were cause for grief and apprehension. It didn’t help that very few of them attended the introductory workshops I held on how to use the software.

Then incorporating software meant giving out weekly homework assignments as Sage worksheets, with embedded instructions. I worked harder at breaking things down into manageable bits to be learned each week. To get the homework and do it, students would have to open Sage and work with a worksheet right away. I made sure to assign problems that were challenging, but workable if you explored using the computer. As I learned the day before the first midterm, most of the students got as far as logging in, and then printed the worksheets and attempted to work out all of the tasks with a pencil and paper.

Last term, I again required students to get their assignments through use of Sage. This time, we used the new cloud service, and I made dedicated tutorial worksheets to go with each reading. I started assigning tasks that explicitly required using the software. (Use Sage to...) This worked better. I gave a take-home midterm that required using the computer, and a few did quite well. But I still found many students avoiding the computer like the plague. I had one admit to me eight weeks into the term that she never bothered to figure out how to log in, and a friend in class sent her a pdf copy of the assignment for each class meeting.

Clearly, we are failing to meet the spirit of the learning goal above.

Going Forward

So it is time for a new plan. I had two disastrous failures, and one mixed experience. But this coming fall I will have two sections of linear algebra, and the curriculum changes that we have proposed officially take effect. It is time for a new, better-informed plan.

Sharpening the Student Learning Objectives

I like the Student Learning Objective statement above. (I helped write it.) But I have come to realize it is inadequate. I don’t have the power to rewrite it unilaterally. But as most of my department seems to be of the opinion that I should just figure this out and do it, I have taken it upon myself to add depth and structure for future use.

First, I added some specific, measurable goals.

Student Learning Goals associated with the Technology Specification

Goal 1: Students can name multiple examples of computer algebra systems for doing work in mathematics.

Goal 2: Students can use one system at the level of beginner, by starting the system, opening a worksheet or development environment, performing basic computations, and making plots.

Goal 3: Students can find information about the capabilities of their chosen system to determine if the system has a particular feature or functionality built-in.
Students can access documentation on how to use unfamiliar features or functionality, and then use that information to make use of that feature.

Goal 4: Students can describe circumstances where use of a computer is a reasonable or appropriate choice to further work in mathematical investigation, and identify features of the circumstance which call for the computer-based work.

I hope these will suit my colleagues. I have asked a few of them for comment, but not heard back much. I choose to believe that this is because it is officially summer.

The Plan for Assessment

The goals don’t mean much if I don’t assess them. So, I plan the following set-up. At the start of the term, I will give the students detailed information about what is expected of them and resources to learn about how to meet those expectations (a simple page on the course web-site with links, a collection of short video tutorials, and other things). Of course, I will also keep using the software in class myself, and I will still give the students the short tutorials that go with the daily assignments.

We will begin the term by using embedded Sage cells in course web pages, but transition to forcing students to log in to the SageMathCloud to get their work.

A few weeks into the term, students will be directed to schedule a short appointment (10-15 minutes) with me, or perhaps the grader, to do a “gateway assessment.” The gateway exam will be an all-or-nothing event. Either the student demonstrates competence on all of the goals, or she does not. I expect that an interview should end as soon as a student fails to demonstrate competence at any stage–there should be no hemming and hawing over these tasks. We will conduct the assessments while sitting at a computer station. I think that the labs in my building are more than sufficient for this. During the interviews, we will ask questions aimed directly at the goals outlined above.

I have not, yet, decided how much data to keep from these assessments. At a bare minimum, I need to keep a record of which students pass the assessment. But I think I might keep a spreadsheet which records each attempt, the date of those attempts, and how far into the assessment a student gets.

The Assessment Script

The real details hide in the questions I ask to check on my goals. To keep things running smoothly, I have written an “assessment script.” Each question in the script is explicitly tied to one of the four goals. It looks like this:

Technology Specification SOA Script

The following are questions to be asked in determining if a student has met the goals of the Technology Specification.

general questions

[G1.] Can you name some computer algebra systems? How many of those do you know how to use?

[G2.] Choose one of these that you know how to use. Open the program/sign in to the service and then open a new worksheet/start up the computational environment.

[G2.] Use the software to find the first 12 decimal digits of the number 2pi/3 -sqrt(e).

linear algebra specific questions
(replace with something appropriate if used in a different course)

[G2.] Define two 3-vectors a and b and add them.

[G2.] Define a 3×3 matrix A. Use the system to find the determinant and rank of this matrix.

[G2.] Use the computer algebra system to solve the system of linear equations represented by Ax = b.

[G2.] Use the computer algebra system to plot one of the equations from the system Ax = b.

more general questions

[G2.] Save this worksheet/session so that you can access it later.

[G2.] Find a way to share this work with me. You can download and print, email, or use any other way that this system allows you to share your work. How many ways can you share this work?

[G4.] Give an example of a time when you might want to use this computer algebra system instead of just a pen and paper. Explain why this is a time that choice should be made.

[G3.] There is a mathematical construction called <insert new term here>. Show me how you would find out if your chosen computer algebra system has any functionality related to <new term>. Now that you see there is some functionality, show me how you can access the help or documentation of this system to learn how this bit of the software works. Now that you have the documentation, show me how to use this functionality.

For linear algebra, a possible list of ideas for the <new term> includes: minimal polynomial, eigenvector, Cholesky decomposition, polar decomposition, cross product, Jordan form, positive definite. This is just a sampler. The important thing is to choose something new to the student.

Resources I Should Provide

I have started compiling a list of resources I should make available to the students.

Some Discussion on a web page

I will make a page on my course web site that discusses possible computer algebra systems, including Maple, Mathematica, Matlab, graphing calculators, etc.

I will lay out my reasons for choosing Sage, and provide links to resources for using it:

  • the official Sage web site,
  • online documentation,
  • the sage cell server,
  • the cloud service,
  • a few tutorials (from lengthy to short: official one, the SDSU tutorial, my beginner’s tutorial)
  • my youtube channel with short tutorial videos

Video Tutorials to Make

I have been impressed with the short video tutorials that Vincent Knight has made for his students. And recently William Stein made a few that were similar in their tight focus and short length. This seems a good approach: Here is something you want to know how to do, described clearly with an example in two minutes or less.

I want to make some of these, or steal link to some of these, all of which are Sage-specific:

  • How to make an account on SageMathCloud
  • How to use git to pull down all of the course materials
  • How to make a new worksheet and evaluate some cells (basic arithmetic)
  • How to do some basic plotting 2d
  • basic plotting 3d
  • How to make and manipulate vectors and matrices
  • How to share work: printing a pdf, sharing a project with another user, downloading a worksheet
  • How to get help: tab completion, the ? and ?? methods.
  • searching Sage documentation and source code

Well, two thousand words seems like enough. Thanks for those of you who stuck it out so far into this. I welcome all constructive comments and any questions.

Troubles with Tech in Class

At MathFest 2013 in Hartford, I got to participate in the Project NExT activities as a presenter and facilitator. This is a professional development program for new faculty in mathematics run through the Mathematical Association of America. I was a Project NExT fellow way back in 2007. That makes me a “Sun Dot,” because fellows all wear an extra colored dot on their badges at the annual meetings. It was fun to meet so many of the “brown 13 dots.”

My first responsibility was to run a quick discussion on using technology in college mathematics courses for a small group of fellows. This is something I have actually been thinking about a little bit lately! Regular readers (Hi, Mom!) know that I have participated in a project called UTMOST, and through that I have tried to incorporate Sage into my linear algebra course.

The first step in our conversation was to take a few minutes to write down some questions about teaching with technology to share with the group. I didn’t get the chance to share mine, but I was proud of them. I just found the note card I wrote them on, and I really should recycle it. Fortunately, I have a blog! Regular readers (Hi, Bret!) know that I just write whatever I damn well choose and I don’t care if they read it or not. (Please, keep reading.) So, here is my chance to shout into the aether and be proud of myself.

  1. How do we use technology to liberate class time for “meaningful work” with depth?
  2. How does technology enable or require new questions and activities?

I think it is important to teach the use of computing technology in a discipline-appropriate way. Otherwise, we are presenting a limited view of mathematical work to our students. But introducing the computer (whatever shape it takes) into a classroom has implications for the kind of work we ask our students to do. What are those implications?

Talking to Reflect and Learn: Major Progress

This week I have started to new ongoing conversations that I am really excited about.

A Formal Discussion Group

First, I joined a small “Talking Teaching and Learning” group on campus. This is a multidisciplinary group of people who wish to have a small community for working on issues related to being an educator. One of the ground rules of this group is that the conversations are confidential, so I will just say that I hope to use the group as an accountability mechanism for me. I shared during our first meeting that I will be working on three things in the near future:

  1. developing an assessment method I am happy with using (focus on Math 3600 Euclidean Geometry)
  2. learning to teach Math 2500 Linear Algebra
  3. refining my approach to Math 1100 Math in Decision Making

I will probably talk about some of my thinking here, as I noodle through things, but this will be the last mention of the group. I think this is a fun idea, and I am looking forward to participating.

An Informal Chat over Tea

Today I had a longish discussion with my colleague Scott Peters. Scott teaches political science at UNI, and we sometimes play soccer together. He was curious about what IBL might mean for a social science course! I am so glad I did this today.

Sometimes you just need to start talking and see what comes out of your mouth. Then you can evaluate it and decide if you really mean it. [Hell, that is why this blog exists. Just replace “talking” with “typing.”] The conversation with Scott was nice because he was very thoughtful and because he comes from a very different discipline. This meant we had to talk about and navigate through to the important commonalities that are really about teaching and learning from an inquiry based learning viewpoint without reference to mathematics. I learned some very important lessons from our talk today. They are important enough that I want to write them down, so I can find them again later and feel guilty when I realize I haven’t internalized them well enough. (Hey, look at that. I wrote exactly that thing down two years ago. What was I thinking?)

Lesson One: Introducing Students to asking their own questions should be done in a narrowly focused context

Scott floated the idea of having students pick their own questions to work on–essentially he wanted them to develop their own small program of study. He has tried something like this lately, and was unsatisfied. I reflected upon my EG experiences and shared that I invite the students into the process of asking questions and making conjectures, but I do it in a very deliberate way.

At the beginning of the course, I set all of the questions, and I model making new questions and conjectures during class when the opportunity arises. Sometimes a presentation doesn’t quite hit the mark the student wants, and then an easy way to “salvage” things is to state a theorem encapsulating exactly what the argument does prove and then make a conjecture that covers the gap in the argument. Sometimes an argument looks juicy enough that I just ask the presenter if they can think of any conjectures that come to mind in the context of their work.

Later in the semester, I explicitly ask students to find and prove unstated theorems that are analogous to ones already in our records. For example, after we have proved a bunch of things about rhombi, I introduce the notion of a kite and set them loose. Even later I can work in a more open-ended way. By mid-semester, some of the students have their own observations to share and they ask permission to make conjectures. (Iowa students are so polite and deferential.)

Anyway, the main point is that without realizing it, I have things structured to slowly acculturate the students into doing mathematics including what it means to ask a question and what kind of question we might have a chance of answering. Importantly, the question-asking is also done in very narrow, specific contexts. That allows the students the freedom to practice asking their own questions, but only gives them a big enough sandbox to do so in an appropriate fashion. I get high-quality work out of them because they come to understand what that means first. Apparently, I am so awesome that even I didn’t recognize it.

Scott pointed out that one of the things going on is that I am only allowing students the opportunity to ask their own questions after they have begun to get a sense of what a proper mathematical process of finding answers is. He felt that this was lacking in his approach. I can’t say for certain if that is true for him, but [expletive deleted] that is definitely true for me! This is a big part of why my Differential Geometry course failed this term. I gave them all the freedom in the world, which is waaaaay too much. What an eye-opener.

Lesson Two: Replicating the Success of Euclidean Geometry might require more faithful replication of the format

Scott asked me about my textbook choices. This gave me a chance to talk about my rationale for using Euclid’s Elements as a text. For all of you, the abridged version is this:

  1. I want some examples of correct proofs for students to see
  2. I want students to experience reading mathematical literature for understanding
  3. I want to be able to assume something or this class will never get anywhere
  4. I want to infuse the course with a sense of connection to history and wider mathematical culture
  5. I want the students to learn to critique everything, even Euclid (some of his arguments are wonky)
  6. I don’t want to pretend the students don’t know anything, even if they really don’t. (They have all had a geometry course in high school. But mostly they don’t have anything but vague memories.) The Elements acts as a convenient bandaid. The facts we need that they are likely to recall from previous schooling are in there.

I tend to think of the course as a mini research community: I am the grand mathematical guru, and my students are new graduate students who wish to be mentored into the professional mathematics community—but all about a millenium ago, when aspects of this planar geometry stuff is still cutting edge. The Elements is our full suite of reference literature, and then I set a research program for the group around it and extending it.

Scott latched onto the idea of using historical sources as a way to structure the development of his material. I have always liked this idea, but I haven’t done strictly that. And this is another thing that hit me! This is missing from my other courses. I don’t have the “extant research literature” for students to grapple with and use as a foundation. But maybe I need to make that. So, for Differential Geometry or Linear Algebra I could make some synthetic replacement for Euclid’s Elements by looking in the historical record. At this point, I don’t expect to find such a convenient piece of ancient scholarship for other courses that plugs into just the right spot, so I’ll have to create something.

This sounds like a lot of work, but it might be just the thing.

Though right now I have my doubts about using such an approach for Math in Decision Making. I don’t know why. I hope it is not a prejudice on my part. More reflection required.

An Outline for Undergraduate Linear Algebra

Over on the Google+, Dan Drake inverted my challenge to mathematics education faculty. He asks, how can mathematicians share what we have learned?

This is an excellent question, which is pretty hard to answer. Since this is my blog, I’ll follow Pólya’s advice, and attack a simpler problem that I still can’t solve. What is the point of each course in the undergraduate mathematics curriculum?

Linear Algebra

A few weeks ago, I wrote about the real point of linear algebra. As usual, that flash of insight was half-baked. I still stand by it, but I’ve been thinking about the course more as it goes on. Here is a fuller description of what I hope to accomplish in a one-semester linear algebra course at the University of Northern Iowa.

Meta-mathematical goals

Every mathematics course should work on core fundamental skills of mathematical process. In linear algebra, I think the two foremost issues are:

  • making examples for yourself to gain understanding, and
  • gaining comfort with increased levels of abstraction.

Let’s add another item to this list that I think is important:

  • learn the use of modern computational tools to do intensive computations and produce visualizations. (Of course, I am using Sage.)

Content Goals

I would like for students to understand the following.

  • solving systems of equations
  • vector algebra
  • algebra of matrices
  • matrices as transformations
  • the geometry of euclidean space as encapsulated by the dot product
  • determinants
  • Gram-Schmidt & orthonormal bases
  • eigenvectors and eigenvalues
  • “some associated geometry”

I’m not sure anyone would argue with these. Of course, this list leaves a lot out. And I hid a lot inside “some associated geometry.” For starters, the ideas behind a set of vectors being a basis or not are really geometry.

A Skeleton Plan: A Course in Three Acts

Here is how I now view the structure of the course. I am just now finishing up Act Two, so the last bits are just a plan.

Act One: Ways To See A System of Equations

The first part of the course focuses on understanding the various ways to interpret a system of $m$ linear equations in $n$ unknowns.

  1. The ‘row picture’: view the system as defining m hyperplanes in \mathbb{R}^n
  2. The ‘column picture’: view the system as defining an equation expressing some vector as a linear combination of n vectors in \mathbb{R}^m.
  3. The ‘transformational picture’: The system defines a “matrix-vector equation” Ax = b.

As part of this, we study the geometry involved, make lots of pictures and try to get a sense of how these things hang together. We will introduce the idea of vector and matrix algebra, including the difficulties of matrix algebra, the dot product, and the cross product in \mathbb{R}^3. There are basic properties to investigate, and some problems to formulate, including the basic problem of invertibility for square matrices.

Act Two: Gaussian Elimination and its Uses

This part of the course is all about how Gaussian Elimination works, and the myriad ways we can leverage it to learn about all of the problems we’ve set ourselves. I think you can hit things like this:

  1. Solving systems of equations
  2. linear dependence and independence, the idea of a basis
  3. the column space, kernel, row space and left kernel appear
  4. the LU decomposition of a matrix
  5. matrices as products of elementary matrices
  6. the rank-nullity theorem
  7. the determinant (I use a geometric definition)
  8. the invertible matrix theorem
  9. the general idea of a vector space and of subspaces
  10. the nature of a matrix as a function
Act Three: Inner Product spaces

This final part of the course is devoted to extra structure we can glean using the tools we have built so far. The major ideas I think one should hit are these:

  1. Gram-Schmidt and orthonormal bases
  2. orthogonal matrices
  3. the QR decomposition? (maybe not, but it is right there)
  4. eigenvectors and eigenvalues
  5. the finite dimensional spectral theorem
The Appendix: Cool Stuff

If there happens to be time…

  1. The PageRank algorithm
  2. Curve fitting
  3. Image Compression


At this point, I think I have a decent set of materials for Act One, a realized plan of attack for Act Two (with materials that need some work), vague plans for Act Three, and some stuff for the appendix already built.

I believe that I will take some versions of my materials and turn them into Sage-enhanced monographs. This will take a significant amount of work, but I can reuse that stuff to support future instances of the course. I hope those future instances will be more inquiry-based.

A Small Personal Breakthrough: the real goal of linear algebra

Originally Posted 09-30-2012

What is each course about?

Think of a course you are teaching right now. What is it really about?

I have been grading linear algebra homework tonight, and I finally had a substantive thought about what the class really should be.

Maybe I am funny, but I have a lot of trouble figuring out the point of the courses I teach, and until I do, everything is just a hash. Without a guiding principle, I have no decent way of making choices and structuring course work.

And I can’t seem to think up a reasonable answer to the question “What is this course about?” until sometime in the middle of the second or third iteration (at the earliest). Apparently I am slow. But it only really dawned on me that I needed this information when I figured it out for Euclidean Geometry a few years ago.

For example, my Euclidean Geometry class is really about the nature and role of mathematical definitions, and how that gives us a foundation for the axiomatic method. It took a few semesters to realize that this is what the students didn’t know and what I was really helping them learn.

I think that linear algebra is about the process of making your own examples and abstracting out the patterns. Seriously. Undergraduate linear algebra is all about this “simple” process of solving systems of linear equations using Gaussian Elimination. The rest of the semester is about context, motivation, interpretation and abstraction of that basic process. And it seems the the mathematical skill that my students lack is the ability to make their own examples for intellectual profit.

Well, it is not that they don’t know how to make an example. It is that they don’t seem to understand that this is a foundational skill. What do you do when you have a new term to comprehend? Or a theorem to understand? Make Examples.

Linear algebra seems like a good place to teach this. The whole course can be understood by having enough examples at your fingertips and learning to use them as prototypes. All of the crazy abstraction is just organizational language for how to apply the prototype examples.

Tell me, am I crazy? What do you all think? If this works, I have a way to organize and refine my course. The next time through, everything will go much smoother because I’ll know how to orient myself. I can make choices with the main goal in mind.