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:

- Fill in the blanks: Given two integers $a$ and $b$, we say $a$
**divides**$b$ if there exists ___ such that __ = ____. - Give three examples of integer pairs $a$ and $b$ where $a$ divides $b$ and explain.
- Give three examples of integer pairs $a$ and $b$ where $a$ does not divide $b$ and explain.
- According to the definition, does the integer 0 divide the integer 0? Does 0 divide $b$ if $b$ is any
*nonzero*integer? Explain. - 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:

- Replace the phrase "nonzero element" with a blank and ask students to fill it in.
- 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.) - 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.

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 **w rite clear, measurable learning objectives for the course at both macro and micro levels. **

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…

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

]]>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.

As of today, there are 53 days until the start of Fall semester at my university, and every weekday I am building my classes – two sections of Calculus and one of Discrete Structures – just a little more. And as promised, I will be sharing my processes and the results-in-progress as they get built. In the last post on this topic, I shared my thought process for choosing the "staggered hybrid asynchronous" approach to the course. Since then, I've been spending most of my time working on the heart of the Calculus course: the **learning objectives**.

I think those are in a stable-enough state that I can share those now. But first, I wanted to mention that I'm making *all* my notes and materials are publicly available on GitHub. There are two repositories:

**Calculus**: https://github.com/RobertTalbert/calculus**Discrete Structures for Computer Science**: https://github.com/RobertTalbert/discretecs

Fair warning: Right now (July 9) these repos resemble junk drawers because I'm still roughing things out. But in the next few weeks I think I'll have things put together to the point I can invest time in organizing them. But at any rate, all of this stuff is free for you to use, steal, fork, etc. to your heart's desire.

I've written a lot about the importance of learning objectives. Most recently, I wrote about how having clear, measurable learning objectives is the essential first step in a well-designed online course. This is because I am eventually going to design my learning activities so they align with those objectives and do the same for my assessments, and even use the learning objectives to guide my selection of course materials and technological tools.

**These are decisions to make in sequence, not in parallel, and learning objectives are the first step. **Often in the past, I'd start building a course by deciding what kind of graded work students are going to do, what the textbook is going to be, and what material I'm going to cover, as more or less independent choices. But I've come to realize that's a mistake. I first have to decide what I want students to be able to *do* as a result of their experiences in the course, then work backwards to pick the *right *content, the *right* learning activities, the *right* assessments, and the *right* tools.

There are two levels of learning objectives to consider:

**Course level objectives (CLOs):**These are the global, overarching "big picture" items that students should master as a result of taking the course.**Module level objectives (MLOs):**These are finer-grained objectives focused on specific content tasks, connected to specific units or "modules" of the course.

**Both sets of objectives need to be clear and measurable **(as explained in this post). If you don't like "measurable", substitute "observable". What we *don't* want are objectives that aren't clear from the students' standpoint or which cannot be directly observed, like anything using the verbs "know", "understand", or "appreciate".

Writing the CLOs for the Calculus course was harder than I expected for two reasons. First, it's really hard to avoid "know", "understand", and "appreciate" when writing big-picture objectives. Second, we have a standardized set of course objectives that the department wrote some years ago (click here for direct access):

These are weirdly written – they start with definite integrals and then eventually get to derivatives, which is the opposite order in which the concepts are learned – and there's a couple of "know"/"understand" type objectives there. But this is sort of the law of the land in the department, and I need to ensure these objectives are met.

So job #1 for me was to remix these objectives and state them in a way and an order that makes sense, and which is both clear and measurable. Here's what I came up with:

I opted not to include in the CLOs things that were *process*-oriented, like *Use technology to frame and solve mathematical problems* or *Demonstrate the ability to learn from feedback*. Those are definitely things we will stress, and I want students to be able to do them. But I wanted to keep the CLOs brief and focused. And especially, **whenever I write a learning objective, I am also making a commitment to assess that objective at some point**. Otherwise it's disingenuous to put the objective on the list. At one point I had some technology-oriented CLO on the list but then realized that as currently structured, it didn't make sense in the course to make a way to assess it, so I struck it from the list.

This may change as the course evolves, and you don't have to do exactly what I do. Just realize that every time you write a learning objective you are also making a commitment to providing a learning activity for students to practice it and a means of assessing it. If you can't follow through with that, drop the objective. Don't write checks that your pedagogy can't cash.

With the course-level objectives in place, we can now think about the module-level objectives. Except first, I have to think about the *modules themselves*. Chunking your course into thematically-focused modules is a best practice in online teaching, because it provides a boundary inside which various learning activities, materials, and assessments can be contained. And that's good for students, because giving learners a pre-built structure that breaks up the course into manageable pieces helps reduce cognitive load and focus attention, both of which are critical for online learners (particularly the most vulnerable ones).

I've always liked breaking my online and hybrid courses into modules that last about one calendar week, because it sets up a nice predictable rhythm where most things happen at the same relative time; then I map the course content into the modules. Our Fall semester starts Monday, August 31 and ends on Friday, December 10. In between we have recesses on September 7, October 26-27, and November 25-26. I opted to take the first two days of classes (August 31-September 1) as a welcome/startup meeting; and I designated the last week of classes (December 7-10) as a catch-up week with no new content. I was able to fit **12 modules of five weekdays each **very neatly into what was left over, with **most modules covering two sections of our textbook **(*Active Calculus* by my colleague Matt Boelkins).

After some experimentation with what should go in each module, I came up with this list:

I like phrasing each module as a question to be answered, so at the end of the module I can ask students to answer it, e.g. *So, how do we find the speed of a moving object? *

A few details about these modules for the math people in the audience:

- Module 3 is a bit dense, but Section 1.8 (L'Hospital's Rule) is going to be done as an independent student project later in the course, not as part of the regular class flow.
- Modules 8 and 12 are just one section, because the sections (Applied Optimization and the Fundamental Theorem of Calculus) are super-dense and historically difficult for students.
- Module 9 has an unusual grouping of sections, but I always felt that implicit differentiation (2.7) is best framed as a prelude to related rates problems (3.5) rather than as an application of the Chain Rule (2.6).

Again, this might change (it's already been overhauled twice since July 1) and your mileage may vary. The important thing here is to make sure you're breaking the course into modules in the first place and that the organization of those modules is consistent and makes sense.

OK, *now* we can think about module-level objectives. In addition to being clear and measurable, **my MLOs have to align with the CLOs**. This basically means that every MLO that I write should flow into one or more of the CLOs, like a tributary creek that flows into a river that eventually empties out into the ocean. An MLO that doesn't fit with the CLOs needs to be reframed or dropped. And the connection between individual MLOs and the CLOs needs to be explicit and clear.

The reason this is important is that students will be asking *Why are we learning this? – *at least we *hope* that they are asking that question – and having a clear connection from any point "on the ground" in the course to the big-picture course objectives will make it easy to answer that question and therefore keep students motivated.

I began the process of writing out MLOs by going section-by-section in the *Active Calculus* book and writing out every task that students should be able to perform after completing the section. Here's the list I came up with – not embedding that this time because it has 68 separate items on it by my count. And that was a problem, because as I said above, whenever we include an objective on an official list we are also making a commitment to assess it at some point. The thought of assessing 68 individual points of skill, and keeping track of student progress toward mastery of those items across two sections of the course, just made me tired.

So I decided to look through the list and group together related tasks into a shorter list of module objectives, while looking ahead at the assessment and grading scheme I wanted to set up. Everyone who knows me or this website knows my commitment to mastery (aka specifications) grading. And as I detailed here, mastery grading entails the use of what I call "learning targets" that represent important specific skills that students will need to master. After thinking about how mastery grading will work in this course, and after some experimentation with the list, **I decided that the learning targets I will eventually assess will be my module-level objectives**.

The way I thought about it was like this: The CLOs are your big-picture items. The MLOs are the finer-grained content tasks directly connected to the CLOs, and those will be assessed through graded work. The CLO's on the other hand are assessed not directly but indirectly through mastery of the MLOs. And the super-fine grained objectives from my list of 68 – I started calling them "micro-objectives" – are also not directly assessed but are incremental steps along the way to mastering the MLOs.

With this in mind, I was able to come up with a list of **24 Learning Targets/module-level objectives **for the course:

The list originally was closer to 30, but keeping in mind that I will be assessing any learning objective I officially publish, and also keeping in mind that I hate grading, I took pains to remove or consolidate several of my original targets to get the list as short as possible.

Some of these are designated as **Core **targets. I'll explain more about that when I post about the grading system, but basically these 10 targets are what I consider essential knowledge for Calculus, and a student must demonstrate mastery of these in order to be eligible for a grade of C or higher.

Note that the MLOs/learning targets are (I think) clearly stated and measurable. I like to phrase them in the form of "I can..." to get students thinking about what they *can* do rather than what they *can't* do.

To make the connection between the MLOs and CLOs clear, I introduced a simple naming system that encodes the relationship by giving each CLO a one- or two-letter identifier; for example the CLO *Calculate, use, and explain the concept of limits *was designated **L** (for limits), and the CLO *Use derivatives to solve authentic real-life application problems *was designated **DA **(Derivative applications). Then, the MLOs are given a designation that includes the CLO they are connected to along with a number. Here's the above list, remixed and renamed in this way:

So now, there's a clear way to see how each learning target connects to the big picture. To see how each "micro-objective" connects to each learning target, I'll be doing that in students' individual pre-class assignments – I'll post examples when I get to that point.

There was one final link that I wanted to see: Where each learning target appeared in the course. I knew how each one connected to a CLO, but how did they connect to the sections in the textbook? I went through each learning target and each section of the book and made this nifty course map.

This revealed a few things I didn't already see. First, one learning target (DC.2) extends across two different modules; I'll need to think later how the assessment will work there. Second, Module 12 at the very end of the course is stacked with learning targets, so I'll need to take care and provide extra support when we get there – and also think about alternative assessments, since there's only two weeks left in the course when we start that module.

I put a lot of effort into the learning objectives for a course because if you get those right, and give yourself a strong structure at the beginning, it makes a lot of things 10x easier later. For example when dealing with the difficult question of *What should I have students do during the face-to-face meetings if we're socially distanced?* the answer is: *What learning objectives will they be working toward? *The answer acts as a filter that focuses pedagogical choices down to only those that are truly relevant. On the other hand, if I don't give these due attention, I'll likely end up duplicating effort later or waste a lot of time and energy on the wrong questions.

I also often geek out and go overboard with this, so if you feel overwhelmed by the above, here's the basic gist:

- Write out clear, measurable/observable
**course level objectives**first – the big-picture items that successful students will be able to do as a result of their experiences in the course. Keep in mind that**introducing a learning objective commits you to providing practice and assessment on that objective**, so don't go crazy here. Keep it brief and focused. - Break the course down into smaller
**modules**that have a coherent narrative or topical focus. (This can be done first if desired.) - For each module, determine a short list of
**module level objectives**that are also clear and measurable that represent fine-grained atomic-level tasks – but not*too*atomic. Remember: Brief and focused. **Make the alignment between the module level objectives and the course level objectives clear**. Use a nomenclature system like I did, or a concept map, or a bulleted list, etc.

Also, be prepared to revise your learning objectives as you build the course. You are allowed to change your mind and probably will do so. But not forever, because we need a stable, final list of learning objectives to move on to the next phase of building – which is about** learning activities.** Details on how that's shaping up for calculus, next time.

**MY PERSONAL CHALLENGE TO YOU: Take one of your courses for Fall semester and focus for the next 3-4 days on creating the course- and module-level objectives for it. Put those in a public place and share the link in the comments. **