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.