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 orĀ 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?

A Course Announcement: Differential Geometry

In the spring of 2013, I am fortunate to be assigned to teach a course in differential geometry. More about my plans will show up here, of course. I am thinking about several new things to try.

But the first job is to find enough students to run the class. To run a senior level course, I need at least 10 students. Graduate students count double, so if I can find a few of them it will help. Of course, many of our current crop of grad students took this course from me two years ago when they were undergraduates…sigh.

More to the point, I just made a little advertising flyer. I’ll have it sent to the math major email list and posted around the department. Seniors start registering later this week, so this has to go out now! Here it is:

I made the figures with Sage.

What to do with linear algebra? Some Inquiry Based Learning!

Originally Posted 08-17-2012

(Note: the word “some” is important. I am going to try a hybrid this time through. Read on.)

In an effort to keep the momentum up, and maybe embiggen my cyber teaching lounge, I signed up for the Math Educational “New Blogger Initiation” challenge. I am pretty sure I heard about this from the blog of Sam J Shah, Continuous Everywhere but Differentiable Nowhere. The agreement means that I will be writing some specific posts in the next four weeks to prompts the Initiation Team sends me.

For this go round, I thought I’d take some time to write about the next iteration of my linear algebra course. (If you are new here, I basically use this space to think “out loud” about my teaching.)

I have not done any of the bits of preparation that involve actually making things I need for the first day of class. This is bad. I have done a lot of thinking about what happened last semester and how I might adjust things. This is good.

Also, I have spent a fair amount of time looking through linear algebra texts. There was one common theme: I thought they were all going a bit slow. Then it dawned on me.
* My class was too hard. I have unreasonable expectations of the average undergraduate. *

By the way, that is not really news. Or, it is, but it shouldn’t be. I’ve had that revelation many times in the past five years.

Anyway, let’s get down to brass tacks. I’m going to start laying out what I am up to. If you want more of the history of my thinking, there are older posts just waiting to be read.

Here’s the quick background:

  • I took a single linear algebra class in college, but I never attended. When it came time for the final exam, I crammed the whole text in a weekend. This was insane but basically worked. I am still irrationally upset over getting an A- in that course. I know you don’t care.
  • I really learned the material deeply when I studied Lie groups and Lie algebras in grad school. This means I have a weird selection of highly theoretical linear algebra that feels like regular arithmetic. This gets in the way of understanding my students.
  • I taught linear algebra once, about six years ago, at a fancy pants liberal arts college in New England. The students didn’t like it so much. It took a while to internalize why: the course was too hard.
  • I got my first opportunity to teach linear algebra at UNI this past spring. I am test subject for a project (called UTMOST and funded by the NSF) about integrating open source software and literature into the college curriculum, in particular, the mathematical software system Sage.

Last semseter could have gone a lot better, and I am now in the process of preparing for a redo. This is one of the glories of the academic system isn’t it? Every so often, you just get to reboot entirely. Maybe every year, maybe every semester, you get to let go of the baggage attached to one class and start fresh.

I would like to use an Inquiry Based Learning evironment as much as I can, but recent experience has shown me that I am not yet accomplished enough at this style to pull it off with pre-proof classes. So this semester I am aiming for a hybrid. I am stealing an idea that I heard at this year’s IBL session at MathFest in Madison. I forget the speaker’s name, but I am also sure that I heard some version of this once before. (So I am not neglecting to credit the original author of the idea, only the person who reminded me most recently. That is a terrible excuse.)

Here is the plan outlined:

  • Use the Schaumm’s outline series text on Linear algebra. It is succinct, covers everything I need and more, and fills the need for lots of computational work with examples.
  • I will lecture on Mondays. I will explicitly announce how the rapid lectures are keyed to certain portions of the text.
  • On Wednesdays, we will begin with a short check-up quiz focused on computational techniques and low-level understanding/recall. Then we will launch into an IBL format with students presenting.
  • The rest IBL portion (a day and a half per week) will be focused on a sequenc of problems I design around the material to foster deeper understanding. There will be lots of open ended questions.

I will be stealing freely from two geometrically focused books: an old one by Dan Pedoe and a new one by Shiffrin and Adams. I want students to obtain reasonable mental models of what all those symbols mean as pictures. Basically, the mantra is this:

Linear algebra is communicated and conceptualized best by the abstract language 
of vector spaces and linear tranformations, and it is actually computed most easily 
with coordinates and matrices, but the intuition comes from understanding the 
pictures of hyperplanes meeting in space.

I have a ton to do in the next 48 hours: I have been thinking about it all summer, but now it is time to execute the plan. I need the following (in roughly this order):

  1. A reasonable syllabus reflecting what will be covered each week.
  2. A course web site.
  3. A first week lecture.
  4. A second week lecture.
  5. A first week quiz.
  6. A first week of IBL investigations.
  7. An “introduction to Sage” screencast or two. Say, one for setting up an account, and another for trying out some very basic things.
  8. An actual Sage worksheet containing some tutorial material related to the first weeks mathematics and the statements of the first week IBL tasks.
  9. Some ibuprofen when all that is done.

I am also thinking about trying out the learning management system Canvas by Instructure. My campus uses Blackboard at this point, and I have heard nothing but grumbling from my colleagues. I heard a few raves about Canvas at MathFest, and I am curious… If it sets up reasonably fast, I’ll try it.

Sage EDU Days 4

Originally Posted 06-13-2012

Goals for Sage EDU Days 4:

I am at a Sage-focused educational conference (in conjunction with being a test site for
the NSF sponsored UTMOST project). This is primarily a working conference, people come
with things to do and try to pack in as much progress as possible. I have a few specific
goals for my time here.

  • Write materials for my linear algebra class in a way that leverages Sage’s capabilities.
    In particular, I will be working on the first unit of my class on the structure of
    Euclidean space as an inner product space. I hope to be producing these as Sage worksheets.

  • Transcribe some of my lecture notes from an old version of linear algebra (taught six
    years ago) into a Sage-enhanced product. I am still thinking about the form of this. Maybe
    an html book with Sage Cells implanted? Maybe a hyperlinked pdf with links leading to a
    Sage Cell server?

Anyway, back to work.

More Research, and the Next LinAlg Activity

Originally Posted 01-24-2012

I had a pleasant and still day in the office. I had no formal teaching duties, but I did have two research oriented meetings, and a nice informal chat with a former student.

Meeting Number One: Undergraduate Project

I am working on a combinatorics/number theory project with an undergraduate student. Our meeting today had a bit of a “wow” moment, too. We have been studying some families of polynomials that “arise naturally” from the problem. The student has found a ton of identities involving these polynomials, all experimentally. We have not succeeded in finding a proof. So, as an alternative to feeling stuck, I thought we might change gears a bit and instead of looking at the coefficients (which is what brought us to this point), we could look at the roots. We finally got some Sage/Python code up and running to compute and plot these roots in a meaningful way, and, ‘pop’, they appeared to lie on a logistic S-shaped curve.
I can’t explain it, but it probably means something deeper is going on.

Meeting Number Two: Professional Research

My colleague Bill Wood and I had an exploratory meeting this afternoon. We attended a workshop on Discrete and Computational Geometry at the Joint Mathematics Meetings earlier this month, and we wanted to find a problem to work on together. After two hours of just talking, I think we had three or four problems that we agree are interesting. Most of them seem a bit out of our reach, but one just might be a good way to get going. So, it looks like I’ll be trying out something new this term. I am excited.

Linear Algebra: The Indoctrination Educating Continues

Feeling good about the last two days of linear algebra, I have designed some more for them to handle tomorrow. The activity is meant as a way to introduce a lot of important language quickly, including the concepts of span, linearly (in)dependent sets, dimension, basis, and determinant. It seems impossible that this will work so well, but I have to try. The focus is again on solving simple sounding problems presented in several equivalent ways, and noting what it all really means.
Also, I have designed a Sage based homework assignment. If either of these goes at all well, I’ll share them later.

The Summer Research Program for Students

I spent some time organizing our summer research program for undergraduate students today, too. This was mostly simple grunt work to prepare to advertise the opportunity. The program has run for three consecutive summers, so much of the organization is on auto-pilot now. Right now, we only have money to support two students–which is the other project I am supposed to be thinking about. Oh, look at the time…

Making Ends Meet

Originally Posted 01-18-2012

Figuring Out Linear Algebra

One of my challenges this semester is implementing a linear algebra class with many new components. The considerations are these:

  • I last taught a linear algebra class five years ago at and a different institution. Therefore, the audience is new to me.
  • I am incorporating use of the open-source mathematical computing system Sage.
  • I am using WeBWorK for “routine” homework assignments.
  • I am using an open-source text which is still “in development.”

This is probably too much. But I am doing it anyway. Another thing which wouldn’t count for anyone else, but counts for me, is that I am trying to blend in an inquiry-based learning approach. This isn’t new to me, though for this material it is.

So far, I find that I haven’t carved out enough time to make the WeBWorK assignments work. This is a major goal of the next few days.

I ran a couple of Sage introductory workshops, one last night, that seemed to go pretty well. The first night went well, with lots of great questions, but was poorly attended. The second night I had a full room, but the crowd was not interested in conversation. One outcome is that I have at least made materials for running these in the future.

The biggest trouble is really that I find I haven’t hit the right stride in class, yet. I can’t seem to get an interactive environment and still include the computer. I am finding it difficult to make activities centered around the material, given that I am roughly following a text. By the way, the text takes the interesting approach that the students should learn all of the computational skills right away, so all of it is in chapter one. Applications come next, and then the theory starts in chapter three. I have already discussed matrix and vector algebra (minimally); the equivalence between solving matrix equations, solving systems of linear equations, looking for a realization of a vector as a linear combination of other vectors and the geometry of intersecting hyperplanes; Gaussian elimination. Next time we will cover rank and determinant, and maybe eigenvectors. Then we’ll have a quiz.

I’ll get there. I suspect that this will be a “muddling through” experience, and then next summer I will try to do something more serious. Maybe I should look at the AIBL grant cycle.

Other Courses

I have two other classes running this term, and they seem to be going well.

Dynamical Systems

My dynamics class is going gangbusters. Almost. I have several students who have had classes from me before, and lots of people who are willing to give presentations and ask questions. I have two or three that I worry about because they have been very quiet. I’ll have to do a personal check on each of them.

Math in Decision Making

(My liberal arts class.) I think that this is going well. I have succeeded in making them confused about things, and then unconfused about some of it. Check!
I was a little aghast that they had no reaction to the weirdness of infinite sets. I mean, they just managed to prove that there is a bijection (we call it a ‘matching’ in class) between the natural numbers and the even natural numbers. I jumped up and down about how weird that is… They couldn’t really muster any emotion. Perhaps this is a bad indicator. At the very least, it means that they haven’t thought deeply about this issue (surprise!), at worst, it means they are just uncritically accepting whatever happens in a math class. I need to work on both of those items.

Other stuff from today

I mentioned in a department meeting that exactly zero people had volunteered to lead a summer research experience in math after my last call for proposals. This was a bit depressing. But I got responses from three colleagues right after the meeting. So, that is looking up.

Tonight, I am looking forward to a soccer game and a night out with friends (for free pie!) to help soothe the cares away.