One of the projects I was taking on with my teaching this semester was a revamped linear algebra course built around peer instruction and the use of Learning Catalytics, a web-based classroom response platform. I probably owe you a quick update now that it’s nearly mid-semester (what?).

Linear algebra is a strange course in some ways. There are a lot of mechanical skills one has to learn, like multiplying matrices and performing the Row Reduction Algorithm. If you come into linear algebra straight out of calculus with a purely instrumental viewpoint on mathematics, you will almost certainly think that these mechanical skills are the point of linear algebra. But you’d be wrong! It’s the *conceptual* content of the subject that really matters. Like I tell my students, you can answer almost any question in linear algebra by forming a matrix and getting it to reduced row echelon form. But you don’t usually *want* to answer questions this way because there are patterns and theorems that get us to the information we want much more efficiently. So instead you want to understand why and how such things work, and how to apply them to a given situation.

Because of this conceptual emphasis, I wanted to use peer instruction as a central facet of the instruction in the class. Pretty quickly in the design process, I realized that this would require some level of flipping the class — there’s just too much terminology and mechanical skill to master, preliminary to the hard parts of linear algebra but still necessary, and not nearly enough time to cram it all into three 50-minute sessions per week, one of which is a computer lab session. So although I didn’t plan on it, the course ended up needing to be done along the lines of an inverted classroom model.

There was no way I was spending 100+ hours making videos for this course, so for the pre-class assignments I have students doing a combination of reading their textbook and then sampling from a collection of text and video resources that I’m curating for them along the way, then doing some guided exercises to instantiate terms and work out basic mechanical processes before class. So they do a combination of reading and viewing before class — a combination I do not mandate but rather leave up to them, provided they can meet a short list of pre-class learning goals — and then come into class, take a short quiz over their preparation, and then we launch into PI time, which is a rolling sequence of minilectures followed by clicker questions I administer using Learning Catalytics. Then after most classes, they get a short homework assignment (usually mutated versions of book problems) that they do and turn in before the next class. So they usually have two things to hand at the beginning of each class: Preparatory problems for the new class and homework problems for the class just completed. Each of those is short — I have two sections with 30 students each, and it takes me about 20 minutes total to grade each preparatory exercises and about 60–90 minutes total to grade daily homework.

Some interesting things I’ve seen in the class so far:

- Students seem to have a much more positive reaction to the inverted approach than I expected they would. This happened also in my transition-to-proof course. As long as I am giving them quality resources and — very importantly — giving them some guidance in how to digest and make sense out of it, they seem pretty happy to learn some basic things on their own. It’s helped, too, to make sure they realize that I am not expecting them to learn
*everything*on their own, just the basic stuff so we can focus class time on the hard stuff. - As I said, I am curating rather than creating the instructional resources for the course. Sometimes I think creating the stuff from scratch would be less time-consuming. It’s really easy to find videos that are
*this close*to what you really want students to see, but which have some kind of fatal flaw. For example, I found a great video showing how to multiply a matrix to a vector — except that they multiplied the matrix on the right of the vector (\(\mathbf{x} A\)) instead of on the left (\(A \mathbf{x}\)), which for an audience of newbies renders the video so different from what they are reading as to be more harm than good. I even look Khan Academy linear algebra videos occasionally, but in most cases they are so long and aimlessly meandering that I couldn’t expect students to sit through all of it. Curation is hard work. - To help with the curation process, I put an optional question on every preparatory exercise set that asks students to share any really useful resources they found in the process of preparing for class. I give extra credit for any good finds. This incentivizes students to go out and search for additional resources beyond the ones I assigned (text or video), and I just keep the best stuff in a text file for the next time I teach the class.
- Students seem to like peer instruction. They definitely like being able to talk things out with their classmates, and it’s been difficult at times to get the class to quiet down after the discussion following the first round of voting on a question in order to get the second round done. Although PI seems to be helping with student engagement, I’m not sure yet how well it’s helping students actually gain conceptual understanding. I haven’t seen too many cases yet where you have a low percentage of students getting right answers in the first round of voting and then much higher percentages in the second round after discussion. Today we had a review session and in two cases, the percentage getting the correct answer to a question went
*down*significantly after the second round. So this is still a work in progress, and I can do better with the questions I write and my deployment of them.

The technology in the class (Mathematica, Learning Catalytics) is probably worth a separate post. I’ll just say one thing — Google’s tablet devices are really nice machines, but if you are trying to order one of more of these on behalf of a department in a tax-exempt institution, good luck. Google makes you pay all the tax up front and then asks you to submit paperwork to prove you are really a tax-exempt institution, and upon approval they refund your tax. It’s been *two months* since we placed the Nexus 7 orders and we still have no refund.

Overall I think the course is going fine, but needs some halftime adjustments to be the class I really wanted to teach. We have our first test on Friday and a midterm course evaluation in a couple of weeks, so I’ll have more to act upon then.