Building Modern Algebra: Learning objectives

Last time, I wrote about the Modern Algebra course that I'm teaching this semester and how I'll be writing about how it's being built. This is the first post in that series, and it starts where the course build process starts: with learning objectives.

Back in April 2020, when the Big Pivot was still just a few weeks old and I was thinking about how we might improve our online instruction for the Fall, I wrote that the first step toward excellence in online teaching (or any teaching) is to write clear, measurable learning objectives for the course at both macro and micro levels. Clear, because learners need to know what the target is in order to hit it. Measurable, because we have to know what the target is to know if students have hit it. And we have these objectives in the first place because our courses are not about us; they are about students and what they learn. How do you know if students are learning what they need to learn? You have to start with being clear about what those learning outcomes are.

I won't address the objections that some faculty raise – still, after all this time – to the concept of learning objectives. I've done that before and doing it yet again feels like arguing that the Earth revolves around the Sun.  Instead, I want to write about the learning objectives for the Modern Algebra course, because the process worked out much differently than for Calculus.

The approach with Calculus was simple: Go through the course module-by-module and identify the "micro" level objectives students will encounter. These are things that students should be able to do, but I don't necessarily want to assess every single one of them. I began the course build process by doing this and putting those objectives in a list. Then, from that list of micro-objectives, distill a smaller set of objectives that address the main categories of things students should do. I called those learning targets and I also put those in a list, at the end of the syllabus. The Learning Targets are what I actually assessed, through the use of "Checkpoints" (described in the syllabus; here's a sample one) which used the micro-level objectives not as targets to assess but as raw material for how to assess those targets. I also had some over-arching course-level objectives that described the big ideas of the course.  

I tried this with Modern Algebra, and it didn't work.

It's because Calculus, while it has many conceptual ideas that are important, is a course that can be assessed on the basis of skills. Compute a derivative; look at a graph and state the value of a limit; write out the setup for a Riemann sum. And those tasks that students perform are easily categorized: If I want to assess the ability to "determine the intervals of concavity of a function and find all of its points of inflection" (Learning Target DA.2), then it's simple, I just give them a function and tell them to do exactly that. There is really only one thing students can do to demonstrate their skill: Take the second derivative, set up a sign chart, etc. and if they do this reasonably well, it's evidence of proficiency.

Modern Algebra is different. Modern Algebra has skills embedded in it but is not primarily about those skills. I want students to be able to find all the units and zero divisors of a ring, but not because that skill is relevant or interesting in and of itself, because it isn't. The only reason I want students to be able to carry out that task is in service of some bigger idea. And unlike Calculus where the micro skills map more or less on to just one or at most a small number of big ideas, micro skills in algebra could be used for anything.

Several years ago I taught the second semester of this course, which focuses on group theory. I took the Calculus approach of teasing out every skill that could be important and making sure I assessed them. I ended up with – I am almost ashamed to say it – 67 learning objectives in all. Here they are in all their God-awful glory. At the time I thought I was doing the right thing: If you want students to know something, express it as a learning objective and then assess it. But in retrospect, it's painfully obvious that trying to center the course on skills in this way is nothing but egregious micromanagement, and in the end the students focused laser-like on the micro objectives and missed all the big ideas. And it's not their fault.

So, don't do that.

Here is the approach I am taking this time.

I did go through my course module-by-module (after deciding how the module structure would go, roughly) and wrote down all the micro-level objectives for each module. Here's the list. This process took me about two hours to complete and I think it will save me far more than two hours' time during the semester, since now I have a map of where everything happens in the course and a list of what matters and what doesn't matter content-wise. Advice: If you do nothing else for your courses this semester, do this for each of them.

But, I did not distill these into Learning Targets. The class actually has no learning targets as such, like Calculus does. Instead, I went straight to the course-level objectives. That list is:

After successful completion of the course, students will be able to…

1. Write to communicate the topics of abstract algebra using accepted proof writing conventions, explanations, and correct mathematical notation.
2. Identify fundamental structures of abstract algebra including rings, fields, and integral domains.
3. Comprehend abstract definitions and theorem statements by building examples and non-examples of definitions, and drawing conclusions using definitions and theorems given mathematical information.
4. Demonstrate problem solving skills in the context of abstract algebra topics through consideration of examples, pattern exploration, conjecture, proof construction, and generalization of results.
5. Analyze similarities and differences between algebraic structures including rings, fields, and integral domains.

This is a combination of the official course objectives mandated by my department and my own ideas. Especially, objective 3 — "comprehending" definitions and theorems — is my own creation.

So, I have two layers of course objectives: The topmost layer (above) and the bottom-most layer (the micro-objectives). Therefore the main difference between this and Calculus is that there is no "middle" layer where Calculus' Learning Targets resided.

This makes sense, to me at least, because again Modern Algebra is focused on big ideas and goals and not so much (or at all) on "skills". Insofar as I will assess these objectives, I'll be asking students to do things that provide evidence of proficiency or mastery of the main, course-level objectives. But the focus is not on the things, but rather on the objectives. Students perform tasks in order to make visible their progress toward the course-level objectives; their performance of those tasks works like a progress meter.

Speaking of assessment: Discussion of the grading system comes later, but it's worthwhile to mention it now. This course uses mastery grading but it's much more along the lines of specifications grading than standards-based grading. Sometimes we use all three of those terms as synonyms for each other, but there are actually significant differences.  As I explained above, students will be doing work that shows their progress toward the course objectives, and that work (as I'll detail in another post) will be graded using simple rubrics that use no point values and allow for lots of feedback and revision, and the student's course grade is based on "eventual mastery". But the grading system itself does not have discrete learning targets that are checked off one by one. Instead, students complete "bundles" of tasks, and each bundle maps to a course objective. Doing the work in the course serves to make visible the progress toward mastery of a bundle. But failing to master micro-objective "X" — possibly ever, in the course — does not necessarily imply lack of progress on course objective "Y".

This all seems very theoretical, but in fact I think Modern Algebra has a lot in common with many non-STEM disciplines. Many such courses also focus more on big ideas than on "learning targets", and I can see why some faculty in those disciplines have questions about the idea of Learning Targets. But if you're teaching a literature or philosophy or art history course, your course objectives might not look terribly different than the ones I listed above, and so the interplay between micro-scale and course level objectives might also be similar. I'd love to hear about that in the comments if you're in that situation.

Next time: A little more about assessments.