Originally Posted 05-21-2012
More Reflections: Linear Algebra
Another new class for me this past semester was linear algebra. At UNI this is a Sophomore level course, it is titled “Linear Algebra and Applications” but is sometimes taught as a transition to proof course. Basically, this course is written into our curriculum so vaguely that it can be whatever you want it to be. We have had some conversations about this lately, and I hope that we come to a more focused common understanding of what should be happening in linear algebra.
I tried to run a blended environment with lots of features. I think I got a bit of a Frankenstein’s monster problem as a result.
- Do at least a few serious applications of linear algebra (Google’s PageRank algorithm, curve fitting)
- cover some basics of the geometry of lines and planes in the Euclidean plane and in Euclidean 3-space for intuition
- Discuss matrix algebra as an entry point to modern algebra ideas
- get into the nitty gritty of solving systems enough to “see” the proofs of most of the major theorems (rank-nullity, invertible matrix theorem, etc)
- See linear transformations as a way of organizing the work of linear algebra, and use them as a way of expanding the idea of a function
- introdue the mathematical software package Sage to help handle visualizations and tedious computations. I am a pretty novice user of this software, but I am enthusiastic.
And that is just the list of things I come up with off the top of my head. I am sure I was trying to do a couple of other things at different points of the semester.
To make things more interesting, I also chose to use the open source textbook by Woodruff and Grout. Jason Grout is a Sage developer and helped get me involved in a grant project about using Sage. His book has some nice features, but it is still a work in progress, and he was a week or so behind me in class days.
I didn’t exactly set myself up for a roaring success.
What I Learned
Students are terrible with technology. Sure they can use their phones, but only because they care about that. They are absolutely inept at watching things like syntax and grammar for interactions with Sage (or anything similar).
So prepare for the least technologically inclined student you can imagine. Then go back and prepare for one who makes that student look like Guido van Rossum and Linus Torvalds put together.
If you expect the students to use the software, you have to force them to do it. Make the homework assignments only available as Sage worksheets, for example. Even then some will try to avoid it.
Make sure that the technology instruction is in very small bits. Learn one command or process at a time. Do one each day. Again, force them to become literate by immersion, but do it very gently and very slowly.
You have to stay true to your own strengths. I am sure that the Woodruff-Grout book is wonderful for them. It is their baby. It was fine for me, but it always made me uncomfortable. It just wasn’t my perspective. I haven’t used a textbook in a long time, and now it chafes.
What Will I do Next Time?
I am teaching linear algebra again in the fall. Here is what I will do differently:
- Write my own IBL notes
- distribute those notes as Sage worksheets
- Add explicit instruction on proper Sage usage for linear algebra exploration. A little bit at a time. No, I mean really little bits.
More importantly, my own IBL notes won’t be just a set of tasks. I’ve been thinking about what makes my Eucliean Geometry Course so successful, and one feature is this: I use Euclid’s Elements as a text. That is, students have a reference work, and then I structure problems around it. My task sequence is about coming to grips with the ideas in Euclid by doing related things. I bet I can make something like that work for linear algebra, too. It means basically writing my own supplement to a Schaumm’s outline. I get to propose the interesting tasks with real depth. The routine stuff is in the book. My job is to blend them together. This is going to take a lot of work.
I think the switch to a more pure IBL structure is the only way I can be true to myself and have a successful result. The more I talk in class, the worse things get. All that ever happens then is that I prove that I can talk intelligently about linear algebra. That is not really the point of class, is it?
So, what will I write? Well, I have ordered the latest Schaumm’s outline for the official textbook. I think I will also transcribe my own linear algebra notes (from a class offered at Williams College in 2006–2007) into my own book. And as of right now I am on the lookout for interesting linear algebra problems. I think I might spend some time learning some more functional analysis to find interesting ideas for challenges.
Full Steam Ahead!