Abacus Model Class 3

[Heartbreak Writhe]

Thispost first appeared at Grading For Growth earlier this month. There is a second part as well which I'll repost here later, or you can just click and read it now. I've added some new thoughts to the end of this post just for rtalbert.org readers!

With summer officially underway, I'm going to be writing for the next two weeks about the grading system I had in place for the semester that just ended, in my Linear Algebra and Differential Equations classes. David wrote a couple of posts on his experiences (part 1, part 2) and mine will be along the same lines.

The first installation today is about the "theory" of the class -- all the background information about the class and my philosophy of teaching it that led to the grading process that I used. Next week, I'll focus on the "practice" -- how it was received by students, what worked and did not work in real-life practice, and what I'll do again and do differently next time.

The class that I taught was MTH 302: Linear Algebra and Differential Equations. This is a four-credit course primarily intended for students in engineering that combines topics from these two gigantic areas of mathematics. I taught two sections, each two hours long, back-to-back on Tuesdays and Thursdays. Enrollment hovered around 30 students in each section, and all the students were second- or third-year engineering majors (with the exception of one lone Accounting major who was working on a math minor).

You'll find versions of this course at many universities that offer engineering degrees, and it's clear that many profs who teach it don't quite know what to make of it. Linear Algebra and Differential Equations, as separate subjects, are significant enough to merit two required courses each, but in MTH 302 we put them both into a single class. There are two ways to handle this kind of compression. One is to build the course around a small, reasonable core of content and learning outcomes that necessarily leaves out some interesting math, and then set up structures to help students learn those things deeply. The other way, which is seemingly far more common, is to leave nothing out, and just Cover All The Things at an enhanced speed and diminished depth, trusting that the "good students" will somehow keep up.

The Cover All The Things approach makes the course tend toward procedural rather than conceptual knowledge (because conceptual knowledge is slow cooking), so the course ends up as a hyper-accelerated flythrough of a cookbook, with a lot of the best recipes missing. The experience for students and instructors alike becomes impoverished, uninspiring, and aimless.

Those three adjectives also describe many of my students' past experiences with math courses, including MTH 302's prerequisites of Calculus 1, 2, and 3. Every student in MTH 302 has completed these; a good portion of my students "completed" them as though escaping through a fire. Don't get me wrong: MTH 302 students are high-achieving and highly capable. But many were simply Calculus survivors, with a survivor mentality about Calculus and other forms of math.

And not to get ahead of myself, but when you've had experiences akin to Covering All the Things in a traditionally-graded system, the survivor mentality goes into overdrive. If you have to learn how to compute things correctly by hand, assessed by one-and-done tests with no feedback loops, every day in class is about survival.

I'd taught Linear Algebra before, and Differential Equations before, but never this weird mashup. I certainly wanted to avoid the negative experiences with the class I'd seen elsewhere. My first order of business, then, was to make the class less weird, by answering the question: What is this class about? I don't mean, What topics does it cover or What does the catalog say. Instead I mean the same thing as when we ask someone what a book or a movie is about.

As I looked at the individual "halves" of the course -- linear algebra, and differential equations -- and how those two subjects interact, I decided that MTH 302 is about modeling systems that undergo change, and seeing what we can learn about those systems from the models. This felt right: Both topics grow out of the need to model real-life systems like ecosystems and spring-mass systems, that change and evolve over time and whose behavior we want to predict.

On a more immediately practical level, I had factors to incorporate into the design of the class at this point. First, as I said, I'd never taught this particular class before, so I wasn't too interested in any radically different approach from what I had used in the past elsewhere. Second, and related, this was a service course for the Engineering school, and I didn't (and still don't) have a great sense of their tolerance level for "far out" pedagogical practices. Third, the students in the course, being engineering majors, were maxed out with a demanding set of courses in their discipline, and I was very hesitant to create a wild new course structure that demanded more cognitive load than necessary.

In particular, I decided at the outset that I would not use ungrading in the course. Since that term is so ambiguous, what I mean is that I would not do what David did in his geometry course or which I did in my Winter 2022 abstract algebra course, where student work got no grades but just feedback, and we collaboratively decided on course grades at the end. I don't think it would have been absolutely wrong to do this with MTH 302; but relative to my newness with this course and what I understood about the constraints on students, it didn't seem like the right call. However: Come back next week for more thoughts on this.

What you'd see if you walked into a typical class would be 6-8 groups of 3-4 students doing heads-down work on a problem that guides them toward understanding of some important learning objective of the course. Sometimes students went to the board to work; other times just working quietly, but very often not quietly, with their friends. My role was to be everywhere all at once, going from group to group to answer questions, prod people along with questions, and check to make sure everyone was OK and progressing.

I've always felt that there were three more-or-less independent axes along which success in a course should be determined: mastery of basic skills, mastery of applying those basic skills to new situations, and what you might call "engagement" or "being in the course". An "A" student is one who can demonstrate consistent excellence along all three axes. A "C" student is one who is "just good enough" on the first two axes and makes a reasonable effort on the third. That 3D axis model seemed to fit particularly well in MTH 302, where there were a lot of basic skills that are important to master, as well as a large helping of applications to master as well.

With the context, philosophy, and assessments all laid out, we can now talk about the grading system. All the criteria for grading different forms of work is spelled out in this document called Standards for Student Work in MTH 302. To summarize:

On the last two, the difference between "Retry" and "Incomplete" is mainly terminology. Either mark could be removed through a reattempt or revision. An "Incomplete" indicates that there was something serious missing from the submission that made it impossible to grade the work: A missing problem, a tangle of significant semantic or math errors, code that won't compile, a Google Doc with the permissions incorrectly set, and so on. If a student's work was incomplete, I'd stop grading immediately, assign the mark, and tell them I'll look closely at their work once they submit something that's complete.

I said we shouldn't try to label the form of grading I was doing here, but if you must add a label, this is pretty much specifications grading, such as I've used in most of my courses since 2017. It features 2- or 3-level grade rubrics on each item, with those items being graded holistically according to specifications that are clearly spelled out.

A student's course grade is the highest row for which all the requirements of the row are satisfied. That establishes the "base grade" of A, B, C, D, or F. There are some rules in the syllabus guiding the assignment of plus and minus grades as well that involve a final exam. The only effect the final exam had on the course grade was to potentially add plus or minus modifiers to the base grade.

Next week, I'll continue this story by writing about what happened when this system made contact with students. Would they like it? Would they be confused by it? Would it make them question their existence and see beyond the universe itself? You'll need to tune in next time to find out.

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