A Good Linear Algebra Activity

Originally Posted 1-21-2012

The big excitement at UNI today was the cancellation of afternoon classes. We had a bit of a snow storm, and it seems that the University had a hard time keeping up with the snow. So, I met with my morning classes (dynamics and linear algebra) but not with my afternoon Math in Decision Making class.

I felt like I wasn’t managing the computer in linear algebra very well, so I ditched it today to focus on ideas. I think I will take the advice of a commenter on Google+ (+Purpose Unites) and only use it sparingly, and instead make it the students responsibility to go to it when they need to do extensive computations. I’ll provide them with support, of course, and I think I have already cleared the hurdle on how to handle basic use with the introductory workshops. I’ll leave the computer work to them. I think this means that some of the homework will be delivered in Sage worksheets–that ought to force them to confront using it. In the future, I might use the computer in class for some visualizations, but otherwise, it is going to be out of the way. To borrow a buzz-word, I am going to a “flipped classroom” as far as technology instruction is concerned.

Today I designed a simple activity that was far too big for a single class meeting, and was also awesome. It will spill into our next meeting, but I already feel much better about how things are going.

The Activity

I told the students that we will play a game. I had them pair up and then handed out a sheet of rules. In essence, each team was to reverse engineer a bunch of systems of three linear equations in three unknowns to satisfy certain conditions on their solution sets. We had already talked about how Gaussian Elimination works (so reduced row echelon form is front and center), and we had discussed how you can read the geometry of a system using the dot product as your guide. This provides two ways to get through the first phase of the experiment.

The second half of the activity is to switch papers with another team and then sort out which of their examples is which. I said I would give small boosts to the participation grade of the two teams that are swiftest to identify systems correctly, and also to those two teams whose systems are hardest to sort out.

All sorts of productive chaos ensued. The activity requires a pretty thorough understanding of systems of linear equations to make your own examples. I had pairs of students coming to good realizations all on their own. For example, the number of pivots in the reduced row echelon form is related to the dimension of the solution set. (We will start to use the word ‘‘rank’’ next class.) Also, if you want your solution set to contain the origin, then the system must be homogeneous. I had many good conversations with students, and I heard many others going on without me. I finally felt like I got the engaged classroom I want.

Now, the activity was challenging, so they aren’t done. It was pretty clear half-way through class that I had asked them to do too much to fit in one meeting. But they worked so well on exactly the right set of ideas that I am not going to complain. I’ve asked them to finish up and come prepared on Monday to do the “switch and solve” first thing.

Oh, and the last five minutes of class was a wash—we all got text messages announcing the cancellation of afternoon business. I had a hard time keeping them focused after that.