Building Modern Algebra: Assessments

In the last article in this series, I wrote about the learning objectives for my upcoming Modern Algebra course. This is the first step in building a course, especially an online course, and I mentioned that the process is significantly different than it was for my Calculus course, because unlike Calculus, Modern Algebra is not really "skills based" and it doesn't make sense to identify 20-25 discrete Learning Targets in the course and focus on those. Instead, the course is about big ideas and the micro-level skills are only important insofar as they are used to demonstrate progress toward mastery of the big ideas.

This makes Modern Algebra similar to courses outside of STEM in many ways. I've never taught a course in the social sciences or humanities, but I have seen pushback from faculty in those disciplines, because they look at learning objectives and see "learning targets", that discrete set of 20-25 skills that need to be checked off, and notice — correctly — that this doesn't fit the ethos of their subject at all. So I'm hopeful that my experiences with Modern Algebra might provide some insight for how learning objectives can be used without reducing a course to a laundry list.

So, those big ideas in Modern Algebra: What are they? I went through the course and the textbook chapter-by-chapter and wrote out the micro-level tasks students will be doing, then took a step back and tried to look for the patterns. I came up with four big areas.

Communication. Students should be highly skilled at communicating their understanding of the structures and results we study in the class – formally and informally, written and oral, in English and in mathematics.

Abstraction. Students should embrace the concept of abstraction and not be afraid of it. Students should be able to compare structures and phenomena in different specific situations and then articulate what they all have in common, and express this in full generality. In many ways this is what algebra is about, and therefore it could be considered the most important goal of the course.

Problem solving. Students should be able to engage in computational thinking as applied to an abstract subject: Decomposing problems into simpler and smaller ones; recognizing patterns among these simpler problems and their solutions; abstracting (again) from these patterns to make general claims; and then using mathematical reasoning to provide proofs and other solutions to the general cases. Notice this is way more than just "write good proofs".

"Comprehension". This one is in quotes because it's a term that I coined to describe a skill set that I think is really important for all abstract mathematics subjects, and I've never seen a term for it before. "Comprehension" is what happens when you take a mathematical definition or theorem statement, and then "unpack" it fully. This looks like any of the following:

• Comprehending definitions: Given a definition of a term, (1) state the definition verbatim (or fill in missing parts of it); (2) construct examples of it, (3) construct non-examples, and (4) either draw conclusions using the definition from given data, or use the definition to rephrase given data.

Example: Consider the term "divides" (applied to two integers).  To comprehend this definition, students might be asked:

1. Fill in the blanks: Given two integers $a$ and $b$, we say $a$ divides $b$ if there exists ___ such that __ = ____.
2. Give three examples of integer pairs $a$ and $b$ where $a$ divides $b$ and explain.
3. Give three examples of integer pairs $a$ and $b$ where $a$ does not divide $b$ and explain.
4. According to the definition, does the integer 0 divide the integer 0? Does 0 divide $b$ if $b$ is any nonzero integer? Explain.
5. Suppose that we know that the integer $x$ can be divided by $5$. Rephrase this statement using the definition of "divides".

If students can do all these things correctly, it's evidence they have "comprehended" the definition in a way that mathematicians themselves learn and use definitions. But this is not the only thing we mathematicians try to comprehend:

• Comprehending mathematical results (theorems, etc.): Given a statement of a result, (1) state the result verbatim (or fill in missing parts of it) and (2) draw conclusions rephrase information using the result and some data; and (3) identify when we cannot use the result.

Example: Here is a typical result from the middle portion of the course, about the cancellation property in a general ring:  

Theorem: Let $R$ be a ring and let $z$ be a nonzero element of $R$ that is not a zero divisor. For all $x,y \in R$, if $zx = zy$, then $x = y$.

Students might be asked:

1. Replace the phrase "nonzero element" with a blank and ask students to fill it in.
2. Consider the ring $\mathbb{Z}_{10}$ and the element $3 \in \mathbb{Z}_{10}$. If $x,y \in \mathbb{Z}_{10}$ and $3x = 3y$, what can we conclude and why? (The "why" must include recognition that $3$ is not a zero divisor.)
3. Stick with the ring $\mathbb{Z}_{10}$ and suppose  $x,y \in \mathbb{Z}_{10}$ and $5x = 5y$. What can we conclude, and why? (Answer: Nothing, if we are looking only at the theorem, because 5 is a zero divisor in this ring. There are some conclusions you might draw, e.g. $x$ and $y$ have the same even/odd parity, but those don't come from the theorem.)

As with definitions, this is how mathematicians "comprehend" proven mathematical results and it's at least as important of a skill as being able to write your own proofs, in my opinion.

It should be said that the first step in this "comprehension" process – stating definitions and theorem statements verbatim – may well be obsolete now. While it's important to internalize these statements, stating definitions and theorems verbatim is a skill that is nearly impossible to assess accurately in an online setting, because students can just look them up. Whether this is a good or bad thing, is irrelevant. We don't operate in a scarcity model of information anymore, and honestly haven't been in one for 20-30 years now, so setting up a course objective whose assessment relies on not having ready access to basic factual information is pointless. And perhaps this isn't such a bad thing, since we can now stress using information rather than recalling it; and in that light maybe this isn't so different from the way professional mathematicians work, despite how we set up our traditional courses.

So those are the big ideas, and all the micro-level tasks in the course are there to serve as a means of building up eventual mastery of these big ideas. I envision this like four big buckets that students are to fill up throughout the course; the only way to do this is by adding water one drop at a time, but the focus is on the water level, not the individual droplets.  

But this article was supposed to be about assessment, so what am I doing there? The assessments in any course are supposed to provide opportunities for students to demonstrate evidence of mastery of the learning objectives which for me is the "buckets". I am planning the following assessments to do this.

• Weekly Practice. These are weekly simple homework sets that will focus on comprehension as described above, as well as communication; and possibly the simple stages of problem solving and abstraction. I'll be giving students activities to do like the examples above.
• Problem Sets. These are all problems that involve figuring out and writing proofs, so they address communication, problem solving, and to some extent abstraction (and comprehension is sort of a prerequisite and a tool). I'm planning on about 6 of these (every other week) with some problems done in groups and some done individually.
• Workshops. These will be weekly discussion board threads where students collectively and openly work on activities involving comprehension, filling in missing explanations or steps in proofs, analyzing written proofs, and engaging in computational thinking. So sort of a mini-version of the weekly practice, and engaging in workshops will help students work independently on their weekly practices. And as I noted here, one thing I learned from Fall 2020 is that if you want social interaction in your online classes, you'll have to engineer it, and this is an effort in that direction.

Those are the main assessments in the course. There are a few smaller ones to go along with these:

• Daily Prep. This is a flipped learning environment and so this is the "Guided Practice" concept for the course. It will involve reading and video, working through demos and exercises, and basic engagement with the bottom-third-of-Bloom concepts of a lesson prior to our meetings.
• Startup and Review Assignments. The last time I taught this course (2016) I was blindsided by how much students needed to review from earlier courses, so I have some asynchronous review activities built in on conditional statements, mathematical induction, functions, set theory, and matrix/complex number arithmetic along with a "Startup" activity that gets them set up on the course tools in week 1. These do not measure progress toward a learning objective but rather formalize familiarity with prerequisites.

Then we have two one-time assessments that are big:

• Proof Portfolio. Some of the problem set problems will be "starred", and at the end of the semester students will choose from among the starred problems to assemble a portfolio of what they consider to be their best proof work. So it's really just a wrapper around the work they are already doing to give them a chance to really show their mastery of the communication and problem solving aspects of the course.
• Project. Students will choose some sort of large-scale application of the course material and do an independent project individually or in pairs on it. That's all the details I have right now, except the topics could be anything — a real life application of the material like a cryptographic system, an application to K-12 teaching, etc. This is what we will do instead of a final exam.

Again, in each of these assessments (except maybe the startup/review) students are doing micro-level tasks but only so that they can fill up the buckets of the big ideas over time.

In the next article, I'll explain the grading system – how all these will be evaluated and how it all fits together for a course grade.