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.

One of the most complex issues in teaching mathematics is how to handle examples. On the one hand, examples are important in mathematics because their construction is usually how we make sense of abstract ideas. On the other hand, students can get the wrong idea about examples. They can think that just by seeing enough examples done by a teacher, they will gain understanding of the subject; or that course assessments will be about completing examples just like the teacher did, and so they focus on replicating the teachers' examples instead of learning from them.

Last Fall semester I really felt this struggle as I taught my Modern Algebra 1 class. It's an upper-level course focused on number theory, ring theory, and fields. So there's a lot of abstraction and the only way to truly grasp the subject is to work with a *lot* of examples. Where I fell short in this class, and what I learned from the experience, is something obvious: *When I'm the one doing all the examples, the students aren't learning the math as well as they could.* Or as a colleague of mine put it, *the one doing the math is the one learning the math.* Whenever students ask to "see" more examples, I work them out, and students watch. But watching someone do a thing is not the same as learning the thing.

Why don't we have students generating their own examples more often? And how might such a strategy of student-generated examples look in practice? After my Fall teaching experience, I set out to see what research has been done on these questions, and I found this paper that I wanted to break open here today:

Anne Watson & John Mason (2002) Student‐generated examples in the learning of mathematics, Canadian Journal of Math, Science & Technology Education, 2:2, 237-249, DOI: 10.1080/14926150209556516

Link to paper: https://bit.ly/2SZ5CtX

This paper is a little different than other research articles I've written about here, in that it's a *qualitative* study. Qualitative research focuses not so much (or at all, in this case) on numerical measures of variables and their statistical differences, as it does on making careful observations of phenomena and then systematically analyzing what's observed. You'll often see qualitative research aimed at exploring questions that are difficult or impossible to operationalize, through anthropological-style observations, interviews, surveys, with the agenda of simply asking questions and making sense of the answers.

Some people think this makes qualitative research less rigorous than quantitative research. That's not the case. In my own research experience, doing qualitative research *well* is a lot harder than doing quantitative research; and doing quantitative research poorly is just as easy as doing qualitative research poorly. They're just complementary practices (and you'll often see them mixed together) and sometimes one is simply a better tool for the job.

Back to this study: The authors worked with kids (the study mentions 11- and 12-year olds in a few of the observations) on in-class exercises where students were asked to generate examples of five different kinds:

: Examples that involve executing and reversing processes, and "doing and undoing".**Experiencing structure**: Examples that elicit different kinds of examples of the same concept from different learners, or multiple representations of the same idea, or different questions that give the same answer.**Experiencing and extending the range of variation**: Examples that result in seeing a pattern in the examples that are produced.**Experiencing generality**: Examples asking learners to illustrate new concepts and invent notation or terminology to explain a phenomenon and then compare to standard mathematical notation and conventions.**Experiencing the constraints and meanings of conventions**: Examples that satisfy some conditions but not others, or those that exemplify "what is and what is not" or what cannot be done within specified constraints.**Extending example-spaces and exploring boundaries**

Each kind of example accesses different cognitive aspects of the example-making process. It's appropriate to give examples of each kind of example here.

The *experiencing structure* example given was about solving linear equations. They were asked to start with a value of $x$ stated as an equation (like $x = 5$) and then build up a linear equation by repeatedly doing the same operation to both sides. So start with $x=5$, then add 4 to both sides to get $x + 4 = 9$, then subtract $10x$ from both sides to get $x + 4 - 10x = 9 - 10x$, and so on. Then at some point they stopped and presented their equations and the rest of the class asked to solve them. The question came up --- if you were given this final equation and asked to solve for $x$, how would you do it? Well, for the group that created the example, it was easy --- just reverse all the steps that were used to build up the equation in the first place. For the rest of the class, the process was about figuring out what those steps were. Both groups experienced a structural process that generalizes to solving other linear equations.

The *experiencing and extending the range of variation* task had students give examples of multiplying multi-digit numbers together and making visible their thought processes for how this might work. I've seen this example myself in my own kids' school work. When asked to compute $89 \times 4$, some will compute $80 \times 4 + 9 \times 4$. Others will compute $90 \times 4$ and subtract another $1 \times 4$. By letting kids make the rules and then making their work visible, the entire group is exposed to a multiplicity of examples, some of which might "click" with a student who wouldn't have thought of it otherwise.

In *experiencing generality* tasks the researchers used a method they called "particular-peculiar-general". The specific task they gave students was:

- Write down a particular number that leaves a remainder of 1 when divided by 7.
- Write down a number that leaves a remainder of 1 when divided by 7 which is peculiar in some way.
- Write down a general form of a number that leaves a remainder of 1 when divided by 7.

Students mostly contributed small numbers for the first task like 8 or 15, then weirder ones like 700001 and 1 for the second task. The authors said that discussion ensued about how to handle negative numbers, and that "the third request followed easily from these contributions". (The latter I have to admit I'm skeptical about, because forming the right generalization for these numbers isn't easy.) This is an instance of using examples in a different direction --- not constructing particular instances from general concepts but using particular instances to *arrive at* the general concept, which is something that would be right at home in my Modern Algebra class.

In *experiencing the constraints and meanings of conventions* tasks, the teacher gave students a general idea --- in this case, to represent a function whose output is equal to the input plus 3 --- and have students generate their own representation of the idea. Students came up with some wildly different ways to think about these functions; the paper shows one result involving a sequence of nested boxes that eventually results in the graph of the function $y = x + 3$. After looking at student examples, the idea is to debrief the activity by comparing representations, discussing their pros and cones, and then comparing student representations to mathematically standard representations. The idea is that

If students have had to develop notations for themselves, and compared their usefulness, they are more likely to understand and accept the strengths and idiosyncracies of conventional notations.

And who among us math teachers has not had to deal with students struggling to understand the "idiosyncracies of conventional notations"? I'm looking at you, inverse functions and logarithms.

Finally, the *extending example-spaces and exploring boundaries* tasks are what I've had my students do in the past: Build a sequence of examples that satisfy increasingly strict constrains. In the paper, students were asked to draw a quadrilateral, then a quadrilateral with a pair of equal sides, then a quadrilateral with a pair of equal sides and a pair of parallel sides, then a quadrilateral with all these features and a pair of equal opposite angles. The idea with this kind of example is to explore the space of possible examples of a concept and discern what's possible and what's not possible.

So, what did the researchers find when they gave these kids all these example-generation tasks? Again, while no quantitative data were collected, the researchers uniformly observed that

Students were actively, noisily, and verbally struggling with attempts to reorganize what they knew to fit the kind of example the teacher was seeking. Students were led away from limited perceptions of concepts and towards wider ranges of objects. They restructured their ways of seeing and experienced the creation of mathematical objects and notations.

Some caveats are in order here: This is great, but it is a long way from a systematic analysis of student observations, and unfortunately it's pretty much the only general conclusion that the researchers draw. They also tend to align their observations with their own experiences as mathematicians: *In our own mathematical training, example-generation helped us, and look! It made these kids better too* --- which sounds to me like confirmation bias. I'd like to see this kind of study done again with tighter controls on the observations and analyses, and in fact this has been done --- actually Watson and Mason went on to write an entire book about this subject.

For me, the main importance of this article is that it validates the idea that while instructors may need to give examples to students at times --- and there is some reason to believe that instructor-led examples can be helpful in reducing cognitive load for students --- there is also great value in placing the main work of example generation into the hands of the students. It also sparks ideas for how we might do this on a regular basis in our teaching. Watch this space for some future posts on specific activities for different classes; and leave your own ideas in the comments.

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