The semester that was, Part 2: A 3x3x3 reflection on Modern Algebra

Last time I wrote about three things I learned, three things that surprised me, and three questions I still had about my Calculus course from Winter 2021 semester. Here's the same, for my Modern Algebra class. Some context for you:

• The subject also goes by "Abstract Algebra". It's not high school-level material.
• This is the first of a two-semester sequence in modern algebra. Unlike many treatments of the subject, at our place we spend the first semester on basic number theory, rings, and fields; then the second semester is on groups.
• Usually the students in the class are split evenly between pre-service teachers and Theoretical Math majors. That held true this semester as well: Of 16 students total, we had 8 Math Education majors, 8 others (combination of Math majors and people from the sciences with a second major in Math).

Three things I learned

1. The need for active learning in a course increases with the level of the course. Active learning is needed in greater quantities in every course we teach because it's the best environment for student learning. That said, it does seem like you can get away with less active learning in a lower-level course, like College Algebra or even Calculus, than you can in a higher level course like Modern Algebra because more complex higher-level mathematics needs more active involvement in order to make sense of it. I'd believed once that upper-level students don't need active learning as much as those in, for example, Calculus because the upper-level students have more "mathematical maturity" and can understand things with less involvement. That's a dumb idea, and all I had to do was look to my own experience as a Ph.D. student to know it. I had plenty of "mathematical maturity" (maybe the only kind of maturity I had at the time) but I made exactly zero progress on my thesis until one of my committee members suggested I hadn't worked out enough concrete examples yet. Whereas, in a Calculus class, active learning is important but many students can learn the subject well without 100% of the class being active — in fact less experienced students probably need a fair amount of direct instruction to make the active involvement fruitful.
2. There seems to be a pattern among successful students in this course. Before taking any proof-based courses at our place, students take an "introduction to proofs" course called MTH 210. There are a few choices for proof-based courses after MTH 210 is completed; the main ones are Discrete Mathematics, Euclidean Geometry, and Modern Algebra. After grades were submitted, I looked through the transcripts of all 16 of my students and checked out their pathway through MTH 210 and into my course. Of the students who had grades of A or B, most of them had one or both of two characteristics: they earned an A or B in MTH 210, or they had successfully completed one of the other proof courses prior to mine. (Or both.) On the other hand, of the students who withdrew or earned D or F in my course, usually neither of those took place — most earned a C in MTH 210 and then came straight into Modern Algebra. There were exceptions and it's not a strong pattern or a huge data set. But it does give a piece of advice that we should pass along to students: If you earned below a B in MTH 210, consider taking another proof-based course before Modern Algebra.
3. Deadlines may be making a comeback in the Fall. I've been using the quota/single deadline system for "big" assignments in my courses recently. This means that the assignments don't have deadlines except for one big one at the end of the term, and students are allowed a quota of up to two submissions per week (two new submissions, two revisions, or one of each). The purpose of that system is to let students work until they feel their work is ready to be evaluated, and to incentivize submitting work early and often rather than punish the opposite. I'm an optimist. But you probably have already guessed what happens: Many students misuse this freedom to wait until week 12 to start submitting their work, they freak out because the end of the semester is upon them, so their work isn't good and they have no chance of revising enough submissions to reach a decent grade in the course. For Modern Algebra, I tried to mitigate this by adding an initial deadline to each problem set — work turned in after a certain date (usually two weeks after it's assigned) isn't accepted, but work that does meet this deadline can be revised as much as needed, and on the student's timetable. This didn't work well either; many students would turn in work that didn't meet the standards, then waited until week 12 to start revising, by which point the trail had gone cold. I'm not entirely sure what the right balance of freedom and structure is here, anymore.

Three things that surprised me

1. How often I wanted students to write code. It's no secret that I believe in using technology as a tool for thinking mathematically, but I was surprised at how often I wished specifically that I'd built programming into the Modern Algebra course. If that sounds weird, remember there are professional-grade tools like SageMath available that excel at working with abstract structures like rings and fields. We already use SageMath in our newly-redesigned linear algebra sequence so there was precedent for including it here, but I chose not to, in order to keep cognitive load manageable. Maybe this was the right call, but we certainly could have used a tool, other than pencil and paper, for exploring concrete examples of the objects we were working with. Coding with something like SageMath also forces you to be precise with your notation, which is something many students in proof-based courses struggle with, and we were no exception.
2. Where gaps come from in student mathematical background. Modern algebra by design pulls together concepts from across the full range of undergraduate and high school math. This is where much of its beauty comes from, and why it can be very tricky to teach — there will be gaps in student knowledge that you don't see coming and didn't prepare for. When I taught the course in 2018, I'd created a big discovery activity about the ring properties of 2x2 matrix multiplication. But about 5 minutes into the activity, I realized that roughly one-third of my students had a completely wrong notion of how matrices were multiplied. (They thought you just multiplied entrywise, i.e. the Hadamard product.) So the activity was a bust. Linear Algebra is a prerequisite for the course, so I assumed that students would remember how to multiply matrices. Hence the saying about "assuming" things. This time, I set up a series of review assignments (set theory, functions, matrices, complex numbers, etc.) to refresh student prerequisite knowledge at the appropriate times and give some early warning about potential big gaps, but I still got ambushed by gaps I didn't detect. The biggest one with this group was an ongoing struggle with the concept of closure of a set under an operation... which is kind of a big one for abstract algebra.
3. The role of semantics in math and the fact we never talk about it. When teaching math, we spend enormous time and energy on the mathematical correctness of statements and computations, but it seems like we never address their semantic correctness. Consider the famous example  Colorless green ideas sleep furiously. The grammar is fine, but it makes no sense: ideas don't have color and they do not sleep; a thing cannot be green and colorless at the same time; and nothing "sleeps furiously". Semantic errors are superabundant in abstract mathematics class and particularly in modern algebra, it seems, and this instance of the course was no exception. For example we can't say that $3 \subseteq \mathbb{Z}$, or that "the integers are closed" (without referencing the operation), or that "the real numbers are associative". These semantic errors were piling up so quickly in the course that I had to repurpose an entire 75-minute class around week 4 to teach my students about semantics and semantic errors. As far as I could tell, this was the first time this concept had ever been discussed explicitly with those students. That is surprising, and alarming.

Three questions I still have

1. Is the traditional sequence and focus of abstract algebra appropriate for most students? Although I'm (sort of) an algebraist by training and love thinking about abstract structures, I am not at all sure that the usual pathway of teaching this course is the right idea for the students I have. I use this book written by three of my colleagues, and while it's a good book and written from the standpoint of active involvement, it's still pretty traditional: Look at the axioms for integer arithmetic, then introduce rings, then introduce subrings, then introduce isomorphisms, etc. It's about abstract structures, which isn't wrong, but I'm not sure it's right either. I'm teaching this course again next year, and I'm contemplating taking it in a completely different direction (while still covering what I am supposed to cover) — like making it about applied cryptography or computational number theory or something, where we start with a practical problem and then invent the abstract structures we need to solve the problem.
2. How do we build strong writing skills? Writing was an issue this semester as well. Every time I give a major writing assignment (such as the semester projects in this course) I have a renewed sense of awe for what my colleagues who teach writing do with students. And yet, building strong writing skills isn't something we can subcontract out to the Writing Department — it has to be built brick by brick throughout the college experience and not just become Somebody Else's Problem. As a writer myself, I tried to bring what I know about good writing practice to the table with my students, but it's not enough. What needs to happen as a recurring, common thread in each college class to impart those skills?
3. What do students really need from a transition-to-proof class? Back to the MTH 210 course I mentioned earlier — it might be time to take a look at it and think if there are topics and concepts in that class that don't really need to be there, excise them, and reinvest the time and energy to really master the essential leftovers on a deep level. And I wonder if we are spending enough time and energy on mathematical habits of mind — like semantic correctness, developing intuition, good habits like constructing examples of definitions when they are introduced, etc. — because we are too busy "exposing" students to concepts they don't really need. What are those essentials? What do students truly need to get out of a course like this?

That's all for now. I am taking a writing break through the end of May for R&R, travel (hooray for being fully vaccinated!), and working on some other projects. See you in a couple of weeks.