Building Modern Algebra

It's time for a new semester. Many have already started, although my university decided to delay opening until January 19 (after the MLK holiday) for Covid-19 reasons, so I've been fortunate to have a couple of extra weeks to prepare. As I get my classes ready --- Calculus 1 and Modern Algebra 1 --- I'll be reprising the series of posts from July-August 2020 where I opened the hood on my course design process. I think it's important and potentially helpful to make those processes visible. Even if you're a colleague whose classes have already begun or will begin soon, I hope you can glean something from all this.

Today I'm going to focus on Modern Algebra 1, because while much of the design process that I wrote about with Calculus back in the fall will be the same for this class, there are some major differences. The design process that I wrote about for Calculus cannot simply be reapplied to any other course with the course name changed. Many things stay the same, but some are very different and I think it's illuminating to focus on both parts.

So, what's this Modern Algebra class all about, and what makes it so different?

First, understand that Modern Algebra is known in some places as abstract algebra --- it's not a catchy/cringey term for College Algebra or something on that level. It's a proof-based course on number theory, rings, and fields (we take a "rings-first" approach; group theory is in Modern Algebra 2) intended for third- and fourth-year math majors. This is the starting point for what makes it different from Calculus and Discrete Structures:

• The level and demographic of students is different. Modern Algebra is an upper level course; indeed the entire roster at this point consists of juniors and seniors, whereas Calculus is mostly first- and second-year students with very few upper-level students. Also, almost the entire roster are majoring either in Theoretical Mathematics or in Math Education with a secondary education emphasis. Calculus students tend to come from all over with the plurality coming from Engineering. It's a very different kind of student taking this course than Calculus.
• The background of students is different. The prerequisites for Modern Algebra are our intro-to-proofs course --- widely seen as a rite of passage in our department that shakes up students' entire perception of mathematics --- and either linear algebra or discrete structures. So these students have seen some stuff, in more than one sense. They've definitely had serious experience with advanced mathematical concepts. But in another sense, although we strive to make those courses intellectually stimulating and enjoyable, there's definitely a feeling that Modern Algebra consists of survivors. So students have not only a different background than Calculus students but a different mindset.
• The modality will be different. Last semester, all three of my courses were "staggered hybrid", a complicated setup that ended up roughly equivalent to hyflex. The main thing is that there was a face-to-face component available if students wanted it. Not so this semester. I requested to teach my classes this semester completely online and synchronous. So there are no F2F meetings; we meet twice a week on Zoom for 75 minutes at a time. Not having to juggle between online and F2F meetings and groups simplifies a lot (which is one of the reasons I requested it) but changes much of the course design process too, as you'll see.
• The pedagogical emphasis is different. Calculus and Discrete Structures, both being introductory level courses, tend to focus on skills: Compute this derivative, find the number of ways to count this arrangement, etc. Modern Algebra, being a theoretical subject, has skills embedded in it but the main focus of the course is not on those skills. Modern Algebra is far more focused on processes or big ideas: The ability to write clear and correct proofs about theoretical observations, the ability to draw conclusions from information, the ability to connect abstracted ideas to concrete situations; and so on. Teasing out clear and measurable learning objectives from these big ideas without focusing the course on less-important micro-level skills is the first order of business in designing the course, and perhaps the main challenge in doing so.

These points might resonate with you if you are a faculty colleague, even if you're not in mathematics and perhaps especially so. As I've discussed online teaching, flipped learning, and mastery grading with colleagues in other disciplines, I've often heard something like What you're describing works fine in a math class where it's easy to measure the skills, but what about a philosophy or world history class? I think Modern Algebra has a lot in common with many such classes.

As much as Modern Algebra is different from my Fall classes, there's a lot that's going to remain the same overall:

• The design process still begins with clear, measurable learning objectives. Like I said, the focus of the course is not on skills, so this time it's not as simple as listing out the stuff you want students to be able to do, making those your learning objectives, and then building assessments and activities where they do those things. We do have to think about concrete actions that students should be able to do, but this time the big picture and the big ideas have to be more visible and present.
• Then we'll think about assessment. Once the learning objectives are nailed down, the question is, how are students going to demonstrate acceptable evidence that they are meeting those objectives? We'll revisit my earlier idea of forming a minimal basis of work that accurately and authentically assesses what I think students should be able to do. It will look quite different, because of the nature and especially the fully-online modality of the course.
• Then we'll think about learning activities. Once we have an outline for assessment, we can determine the learning activities. I have had to edit myself several times writing this in order to avoid saying "class activities", because the online modality and the flipped learning setup I'm using mean that there are learning activities that take place both in and outside of our meetings. I have to remember to decouple learning activities (and everything else) from physical location.
• Then we'll think about the grading system. I am sticking with a mastery grading system for Modern Algebra. But based on last semester's experiences, I need to radically simplify it without oversimplifying.
• Then we'll think about course materials and tools. This seems like the easy part, since abstract algebra does not necessarily use a lot of specialized tools as would, say, a Calculus or numerical analysis or computer programming class. But it's turning out to be more complicated than I expected.

And in all of these considerations, I'll need to keep in mind that we're still in a pandemic situation that is wreaking havoc on students' lives. And that means that I need to commit to empathy and support for students while still providing them with a challenging academic experience. And it also means that the social context of the class is radically different than what we're used to, despite all the experiences we've had since the Big Pivot in March. Overall it's a challenging project, and I'm looking forward to sharing what I've come up with and getting your feedback on it.