by Jeff Suzuki

Unless you’ve been living under a rock for the past decade, you know that one of the buzzwords in education is *active learning*: Be the guide on the side, not the sage on the stage. One of the more common approaches to active learning is the so-called flipped or inverted classroom. In a flipped classroom, students watch lectures at home, then come to class to do problems. This is actually a 21st century implementation of a very traditional approach to pedagogy, namely reading the textbook before coming to class. Many of us embraced this idea, and shifted our approach to teaching.

Then came the era of social distancing and forced conversion to distance learning. It might seem that those who switched to the flipped classroom model had an advantage: Our lectures are *already* online. And that’s true. But the second part of the flipped classroom involves working problems *in class.* This is now impossible, and those of us who had embraced the flipped classroom model have spent the past few months in existential agony. The “sage on the stage” can still give lectures through Zoom, but the “guide on the side” can’t guide.

**The New Normal?**

And yet…it’s now more important than ever to be the guide on the side.

We don’t know how long the current phase of social distancing will last, but even after it ends, we can expect that distance learning will be the new normal: it’s a trend that began long before the pandemic. And this forces us to deal with a new problem: It is impossible to monitor student activity remotely.

We accept this when we assign homework, and expect students will do the work with their books open, their notes in front of them, and a half-dozen math help sites open in different browser tabs. Before the pandemic, we told ourselves it didn’t matter, since they’d have to do the exam without all these study aids. But in the post-COVID world, there is now no difference between the resources available to students on homework assignments and on exams.

Don’t believe the hype about lockdown browsers (which work fine for the students who don’t have smartphones). Live webcam monitoring can be defeated by taping cheat sheets on the wall behind the computer. And if a student turns in a textbook perfect answer, it’s possible they listened to us when we explained how the answers should be written.

Will students cheat on exams? We’ve found copies of our exam questions posted to Chegg (with answers). This shocked me: Who would pay for a Chegg subscription, when there are so many *free *sites that show all steps to solving a problem and, unlike Chegg, leave no evidence behind?

The bad news is that *every* exam question that can be answered by following an algorithm is now obsolete, because such questions can no longer distinguish between the student who understands the material and the student who knows how to use Google.

Here is where the flipped classroom can be our salvation. A key component of the flipped classroom is letting your students figure things out for themselves, and *not* giving them a step-by-step algorithm for solving a problem.

For example, let’s consider a basic problem in calculus: Finding the derivative of a function. In the internet era, *any* function that can be described algebraically can have its derivative found, with steps, by a free online problem solver. So we have to ask questions that can’t be resolved by typing the problem into www.findthederivativewithstepsfree.com (not, so far as I know, a real website, but a thirty-second Google search will give you a plethora of possibilities).

**Transcending the Machine**

The good news is that computers are good at exactly one type of problem: problems that have algorithmic solutions. If you can *describe* the exact sequence of steps needed to solve a problem, then a computer can implement those steps faster, more accurately, and more cheaply than any human being. The real moral of the story about John Henry is *don’t compete with the machine in the machine’s areas of strength. *Instead, find the things the machine is *bad* at. In this case, the easiest way to neutralize these problem solving sites is to make every problem a word problem.

Of course, “students can’t do word problems.” This is a meaningless objection: at the start of calculus, students can’t integrate, but we still ask them integration questions on the final exam! Our job is to teach these students how to do these things. Here’s where the flipped classroom becomes a key part of the solution. *Don’t* spend class time lecturing: students can view lectures on their own time. Instead, class time should be spent working problems, especially those that can’t be solved by following a sequence of steps.

It’s helpful in the discussion that follows to think of problems as falling into one of two categories:

- Routine problems, where the mathematical question and the relevant information are explicitly given: “Find the derivative of
*y*= tan(3*x*).” - Non-routine problems, where this information is not given explicitly. Roughly speaking, every word problem is non-routine, and such problems form the bulk of the questions in “reform-oriented” textbooks.

**Flipping Your Class, Social Distancing Edition**

Here’s one possible structure for such a class (where “class” means any time you’re working with students in realtime). All of the following takes place before class:

- Students watch assigned lectures on the topic.
- Students complete routine homework problems, using some online homework management (OHM) system. If you’re using a commercial text, there is an OHM associated with your text. If you’re using your own, there are free products (MyOpenMath is my go-to) that can be used.
- Students are also assigned a set of non-routine problems to consider. These don’t have to be separate from the OHM: again, almost every word problem should be considered non-routine.

How should you run class itself? Class time is the most valuable resource available to students; using it efficiently and effectively can be challenging. Here’s a few things that may help.

At the start of class (online or in person), take down a list of student questions. One risk is that the more outspoken students tend to dominate the discussion; taking down a list of all questions at the start of class is a way to make sure that every student has a chance of getting their question answered, and to ensure that a sufficient variety of problems are presented.

Establish from the start that the routine problems have lowest priority: these are problems that should be solvable by students who followed the assigned lecture. It is vitally important that you keep to this rule: The biggest challenge to running a flipped classroom is students who don’t watch the lectures beforehand. Depending on how you’ve set things up and the system you’re using, it might even be possible to determine whether a student has watched the assigned lecture (though trying to do this realtime requires a bit of practice); another option, which I use, is to assign simple 1-point problems that students answer after they’ve watched the lecture. Remember: *Class time is the single most valuable resource available to the students; it should not be spent on things that can be done out of class time.*

One way to efficiently use class time is to focus on the setup. For example, let’s consider the following problem, which probably appears in every calculus text ever written:

*A 25-foot long ladder rests against a wall. The base of the ladder begins sliding away from the wall at 2 ft/second, while the top of the ladder maintains contact with the wall. How rapidly is the top of the ladder falling when the base is 10 feet away from the wall?*

The “sage on the stage” would identify the relevant parameters and write down the mathematical problem to be solved. The “guide on the side” would lead students to the mathematical problem. For this, it’s important to ask leading questions and not give outright answers. For example:

- What’s going to answer the question “How rapidly?” (Students should identify that this is an instantaneous rate of change, so it’s a derivative)
- What other things are changing? (Students should recognize that the distance of the base of the ladder from the wall is also changing, but the length of the ladder is not)
- Is there a relationship we can write between the quantities?
- Which derivative do we want? (Students should identify that they want $\frac{dy}{dt}$; it’s also worth making them explain why $\frac{dy}{dx}$ is not relevant).

In the end, we have the mathematical problem, “Find $\frac{dy}{dt}$ when $x^2 + y^2 = 25$ and $\frac{dx}{dt} = 2.$” At this point, it becomes a routine problem—and if you’ve established that minimal class time will be spent on routine problems, you can leave the problem at this point, perhaps with a directive of “Finish the problem after class.”

It’s worth noting that, at this point, the problem can be handed off to an online calculator, which can then solve the problem. You might even go so far as to point students to the online calculator, lest they develop a mistaken belief that you’re unaware of the existence of such things. This epitomizes the idea that humans should do what humans are good at, namely extracting the mathematical problem to be solved; while machines should do what machines are good at, namely applying an algorithm.

As the preceding example suggests, it’s possible to teach a flipped class with very little change in how you’re already teaching. The main difference is establishing the expectation that students watch lectures before class.

Let’s see how we might take a larger step, using a standard topic: finding the extreme values of a function. A traditional approach might be to have students find derivatives, then critical values, then apply some test to decide whether a critical value corresponds to a maximum or minimum.

In a flipped classroom, students wouldn’t be given this algorithm. Instead, they’d create their own approach, typically through some guided exploration of a question. Coming up with good questions is challenging; fortunately, thirty years of reform calculus have provided us with an abundance of material, and many of these questions have been incorporated into every standard calculus text, so you needn’t write your own.

For example, I like to give students the following question:

*An accelerograph records the acceleration of a train (assumed to be moving in a straight line); some of the data values are shown below. Assuming the velocity of the train at t = 0 was 0 m/s, estimate when the train was moving the fastest; defend your conclusion.*

t (seconds) | 0 | 1 | 2 | 3 | 4 |

a(t) (m/s^{2}) |
3 | 2 | 1 | -1 | -2 |

* *

A sequence of leading questions can guide students to creating their own approach:

- What does the acceleration have to do with the velocity? (The student should identify that it’s the rate of change of the velocity)
- So
*a*(0) = 3 and*a*(4) = -2. What does that tell you? (The student should identify that the velocity is increasing at t= 0, and decreasing at t = 4) - So when is the velocity increasing, and when is it decreasing? (The student should identify the velocity’s increasing at t = 0, 1, 2, and decreasing at t = 3, 4. I usually find I have to ask “Is the velocity increasing or decreasing at
*t*= 1?*t*= 2? ) - So where is the velocity going to be the greatest? (The student will
*probably*say*t*= 2, at which point remind them that they just told you velocity is increasing at t = 2)

and so on, leading to an answer like: *The train’s velocity appears to be increasing until at least t = 2, and is decreasing from t = 3, so there’s a maximum velocity between t = 2 and t = 3.*

What’s worth noticing here is that *none* of these questions can be answered by appeal to a formula or an algorithm. Consequently, any attempt to use an online calculator on this type of question will result in, at best, a nonsense answer. The closest thing to an “algorithm” is recognizing that the change from increasing to decreasing is where the local maximum value will occur, but even then, since that change occurs “offscreen”, students must consider how they know that the change has occurred.

**A Return to Normalcy**

Suppose, against all predictions and the entire trend of human history, we go back to how things were at the beginning of 2020: traditional in-person classes, no social distancing, exams where we could control the resources used by students.

*None* of the preceding needs to change. In fact, *all* of the preceding alterations in our pedagogy and our assessment are worth doing regardless of how we will give exams. The hard truth is that sooner or later, our students will leave the classroom. If what they’ve learned from our classes can be done by a free internet application, then their education is worth a free internet application.

We owe it to our students to give them something more.

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Fostering an understanding and appreciation of the deep, beautiful threads that unite seemingly disparate areas of mathematics is among the most valuable outcomes of teaching. Two such areas that are ripe for bridge building—functions and geometric transformations—are the focus of our NSF project, Forging Connections Through the Geometry of Functions. In this post, we describe the pedagogical benefits of introducing students to functions through the lens of geometric transformations.**Geometric Transformations as Functions**

The most common representations of functions are symbolic and numeric in nature. This emphasis on number limits students’ images of the variety of mathematical relationships that can be represented as functions. As such, it contributes to common student misconceptions. Students may conclude that:

- every function turns an input number into an output number;
- every function can be expressed as an algebraic formula;
- a formula is the primary representation of a function, and all other representations derive from it; and
- the ultimate test of a function requires graphing it in rectangular coordinates and applying the vertical line test.

Although students investigate reflections, translations, rotations, dilations, and glide reflections in a geometry course, they typically do not regard them as functions; the functions they encounter in algebra always have numbers as input and output. We can expand students’ horizons and deepen their concept of function by treating geometric transformations as functions that take a Euclidean point as input and produce another point as output. Coxford and Usiskin pioneered this approach a half century ago in their ground-breaking *Geometry: A Transformation Approach*, but very few of today’s geometry students encounter it.

In Figure 1, a student has used three Web Sketchpad tools to construct the independent variable *x*, the mirror *m*, and the reflected dependent variable *r _{m}*(

*Function notation is meaningful.*The use of function notation gives students language to describe the specific elements that constitute the function: independent variable x, function rule*r*(“reflect in_{m }*m*”), and dependent variable*r*(_{m}*x*) (“the reflection in*m*of*x*”).*Functions need not be algebraic formulas with numeric inputs and outputs.*Point*r*(_{m}*x*) depends on*x*: Students can drag*x*in order to make point*r*(_{m}*x*) move, but cannot drag*r*(_{m}*x*) by itself.*Variables really vary.*As students drag independent variable*x*, red and blue traces memorialize the kinesthetic experience of varying the variable. The traces form a pictorial record of the dynamic interaction and help students analyze the covariation.*Relative rate of change can be observed and described.*By dragging*x*, students observe that*x*and*r*(_{m}*x*) always move at the same speed, but not always in the same direction, and they can investigate how to drag*x*so the variables move in the same direction or in opposite directions.

**Constructing a Dynagraph**

Having explored reflection and other geometric transformations in two-dimensional Flatland, students then restrict the domain of these transformations into the Lineland (one-dimensional) environment of a number line (Abbott, 1886). They focus in particular on connecting the geometric behavior of dilation and translation to the observed numeric values of their variables on a number line.

In Figure 2, students use the Number Line, Point, and Dilate tools to create a point restricted to the number line and dilate it about the origin to obtain a point labeled *D*_{0,s}(*x*). While this notation may at first seem daunting, it actually may be less mysterious than the traditional *f*(*x*) language. Rather than write out or speak all the words “the **D**ilation (of ** x**) about center point

Students measure the coordinates of *x* and *D*_{0,s}(*x*) and drag *x* to compare the values. When asked to describe how *D*_{0,s}(*x*) moves when *x* is dragged, a student might respond, “As I drag *x*, *D*_{0,s}(*x*) moves faster. It seems to move twice as fast, and I notice its value is always twice the value of *x*. I wonder if its speed is related to the scale factor *s*.” By experimenting with different scale factors, the student concludes that *s* represents the relative speed of *D*_{0,s}(*x*) with respect to *x*, and that the coordinates produced by this dilation satisfy *D*_{0,s}(*x*) = *x*·*s*. Students can then experiment with a translation restricted to the number line and conclude that this new function, translation by a vector parallel to the number line and of directed length *v*, causes the two variables to move at the same speed, and satisfies the equation *T _{v}*(

Students are now ready for a new task: What happens when you dilate *x* and then translate the dilated image; in other words, how does the composite function *T _{v}*(

Students who construct *T _{v}*(

- As x is dragged,
*T*(_{v}*D*_{0,s}(*x*)) moves in the same direction as*x*, but 3 times as fast as*x*. - As x is dragged,
*T*(_{v}*D*_{0,s}(*x*)) is stuck at 2. - As x is dragged,
*T*(_{v}*D*_{0,s}(*x*)) moves at the same speed as*x*, but in the opposite direction. - As x is dragged,
*T*(_{v}*D*_{0,s}(*x*)) moves in the opposite direction as*x*, but twice as fast. - As x is dragged,
*T*(_{v}*D*_{0,s}(*x*)) moves 2 times as fast as*x*, and is always 4 behind*D*_{0,s}(*x*).

**The Cartesian Connection**

To conclude, students create the Cartesian graph of a linear function using geometric transformations. As Figure 4 illustrates, students start with the same initial tools that they used to create a dynagraph, but this activity’s Transfer tool rotates a variable by 90°, transferring it to a vertical number line perpendicular to the original, horizontal number line. After using this tool to rotate *D*_{0,s}(*x*) to a vertical axis and translating by vector *v*, students use the Perpendicular tool to construct lines that keep track of the horizontal location of *x* and the vertical location of *T _{v}*(

**Conclusion**

By using web-based dynamic mathematics software and tools tailored to carefully structured tasks, students can enact geometric transformations as functions, create them, manipulate them, and experiment with them. In the course of their explorations they can develop a solid understanding of geometric transformations, explore connections between geometry and algebra, and construct and shed light on linear functions by using a dynagraph representation.

By beginning with R^{2}→R^{2} functions (transformations in the Euclidean plane) and connecting them to R→R functions in algebra, these activities can help prepare students for later study of complex (C→C) functions, and functions with three-dimensional domains and ranges (R^{3}→R^{3}). A further benefit is the gentle visual introduction of the concept and notation of function composition.

Pedagogically, the constructive nature of activities such as these has the potential to engage students, to give them opportunities to assess their own work, to encourage mathematical discussions, and to help students bridge the gap between the concrete, physical world and the profound elegance of abstract mathematical insights.

**Acknowledgments**

This post is based in part upon work supported by the National Science Foundation under NCSU IUSE award 1712280 (July 2017 through June 2019) and KCP Technologies DRK-12 award ID 0918733 (September 2009 through August 2013). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

We are deeply grateful to McGraw-Hill Education for making Web Sketchpad available for the activities described in this post.

**References**

Abbott, E. A. (1884). *Flatland: A romance of many dimensions*. San Antonio, TX: Eldritch Press.

Coxford, A. F., & Usiskin, Z. (1971). *Geometry: A transformation approach*. Laidlaw Brothers Publishers.

Goldenberg, P., Lewis, P., & O’Keefe, J. (1992). Dynamic representation and the development of a process understanding of function. In E. Dubinsky & G. Harel (Eds.), *The concept of function: Aspects of epistemology and pedagogy* (pp. 235-260). Washington, DC: Mathematical Association of America.

Recently as a graduate student I taught a week-long boot camp for incoming mathematics graduate students at Oregon State University. It was my first foray into teaching under the active learning model and it was a completely transformative experience for me as an instructor. The change in my own attitudes towards teaching and pedagogy were so abrupt, so all-encompassing, that I felt compelled to immediately record my thoughts on the experience.

The purpose of the free, optional boot camp is to ensure that incoming students have a common base of content knowledge and proof techniques. The camp is run by an advanced graduate student, with every aspect at the discretion of the instructor. In designing the boot camp I chose to focus on concepts from real analysis and linear algebra (the subjects of our PhD qualifying exams). Each day I would choose a major idea and orient readings and problems around it. In the mornings we explored the historical development of the idea using guided primary source readings and exercises from the excellent TRIUMPHS projects[1]. The afternoons ran as problem sessions where I posed more sophisticated problems with modern definitions and theorems.

The model of active learning that I chose meant that I did almost no lecturing at all. I assigned readings and exercises to be completed before each session. This meant that the students never came at a definition or theorem completely cold. The readings, exercises, and later problems were discussed first in small groups, and then as a whole class with a student presenting on the board. I believe that the boot camp had a strong positive impact on the students. However, the impact on me was even greater. I am a total convert… an evangelical, born-again, active learning enthusiast. This is my testimonial.

My Own Transformation

I hope never to lecture again. This experience of teaching with active learning was eye opening for me. So many things jumped out at me as reasons to prefer active learning to lecturing. First of all, I noticed that students were much more engaged when their classmates presented on the board than when I presented on the board. The students seemed to feel freer to ask “dumb questions” to their classmates than to me. If they didn’t understand a line in a proof that the student had written they would just ask them to explain more. However, the few times that I wrote on the board I could feel the room tense up and the students’ eyes glaze over (and I pride myself on being an engaging instructor!). The problem is that they felt like what I had to say was gospel, and so they didn’t engage as much or question as much. They just accepted what I wrote, copied it down, and then waited for the next piece of information.

The next thing I noticed is that students often came up with solutions very different than what I had in mind. When there were multiple solutions to the same problem I had students write them all up. We then looked them over, compared them, and discussed the virtues and drawbacks of each attempt. This would never have happened if I had simply written my own proof on the board. Also, it sometimes happened that there would be a subtle error in a proof that made it to the board. These errors prompted great discussions every time. Usually one of the students would notice the error but maybe not know how to fix it. Often some students would still need convincing that there really was an error in the first place. As a class we would discuss the various nuances, and I pretty much never had to swoop in and resolve the mystery. The students would solve it themselves. At the end of these debates I would spend a minute summarizing the debate and emphasizing the main takeaways. When appropriate I would also place the debate in a broader context or relate it to something else we had done that week. These moments allowed me to share my expertise in a way that would not have come out if I had been lecturing.

This led me to the realization that lecturing is a waste of an instructor’s expertise. What is the benefit of having me write down the standard proof of a famous theorem on the board when this is something that the students could find in any textbook? I could perhaps explain the reasoning behind certain moves. But then, if the students have read the proof beforehand, I could still give that explanation, and without wasting time writing out symbols on the board that the students already have in front of them. Even better, I could ask questions or assign exercises that allow the students to discover the subtleties on their own. That is where my experience and knowledge becomes useful. It is in designing the readings, exercises, and discussion questions that best facilitate understanding. It is in answering questions on the fly to help students get beyond some mental block. It is in facilitating thoughtful discussions that bring out the nuances in subtle reasoning. Writing something on the board that the students all have in front of them already is not only a waste of my own expertise, it is a colossal waste of everyone’s time.

Teaching with active learning was way more fun than I had anticipated. I have always enjoyed lecturing, but teaching with active learning was more fun, more engaging, less stressful, and ultimately less work than lecturing. The fun was being able to interact with the students on a more personal level. It was also a fun challenge to meet the students where they were and figure out how to get them on board. Writing proofs on the board can be stressful. Subtle errors easily creep in, and when you are writing on the board you don’t always see the mistakes that you made (even if you would notice them immediately if someone else was writing on the board). With the model of making students write their answers on the board, the stress of presenting is broken down into small pieces and spread out amongst the whole class. Students also gain experience and confidence presenting their work (something that academics are expected to do frequently). And students often enjoy showing off a little bit when they have proved something on their own and they are sometimes quite eager to share.

By the end of the week of boot camp, I trusted the students enough that I could relax a bit on my own preparation. At the beginning of the week I had my own answers to every problem or exercise that I assigned written out in painstaking detail. This was useful as the students were getting accustomed to the format of the class and needed more precise prodding. However, by the end I was no longer worried that no one would be willing to write on the board or that a subtlety would be overlooked if I didn’t micromanage. As a result, by the end I was assigning problems that I knew were doable and that I had an idea of how to do, but that I hadn’t worked out fully on my own. This made class even more fun for me because I got to think along with the class. It was also good for the students because they got to see more into how a relative expert tackles certain problems and why I can do them faster even though I’m not any smarter than them. I would say things like “I’m not sure how that goes, but when I see something like this it always makes me think of…” So my being slightly less prepared actually allowed students to gain new insights into how to problem solve at the graduate level. (Not that I am advocating being under-prepared for class! I am only saying that after a while it was useful to hand the reins over to the class a little more, and that my relaxation of control had its own benefits.)

Conclusion

In the year since the boot camp I have occasionally found myself lecturing (usually by accident), and each time I have deeply regretted it. I can always tell that I’ve started accidentally lecturing by the looks in my students’ faces. They may be following and focused, but they are falling behind. My unease over lecturing is so complete that I now have trouble giving presentations without inserting a significant amount of active learning. This has also worked successfully for me with presentations related to pedagogy as well as with academic presentations to REU students and public outreach. However, I have yet to try this approach in a research talk. My friend and mentor in all things active learning, David Pengelley, has been a great encouragement (bad influence?) in this respect. He is working on incorporating active learning into his research talks. Once he figures out how to do so successfully, perhaps none of us ever need lecture or be lectured to again.

[1] 1https://digitalcommons.ursinus.edu/triumphs/

]]>“The difficulty… is to manage to think in a completely astonished and disconcerted way about things you thought you had always understood.” ― Pierre Bourdieu,

Language and Symbolic Power, p. 207

Proof is the central epistemological method of pure mathematics, and the practice most unique to it among the disciplines. Reading and writing proofs are essential skills (*the* essential skills?) for many working mathematicians.

That said, students learning these skills, especially for the first time, find them *extremely hard*.^{[1]}

Why? What’s in the way? And what are the processes by which students effectively gain these skills?

These questions have been discussed extensively by researchers and teachers alike,^{[2]} and they have personally fascinated me for most of my twenty years in mathematics education.

In this blog post I’d like to examine one little corner of this jigsaw puzzle.

To frame the inquiry, I posit that there are *imported* and *enculturated* capacities involved in reading and writing proofs. Teachers face corresponding challenges when teaching students about proof.

Capacities that are *imported* into the domain of proof-writing are those that students can access independently of whether they have any mathematics training in school or contact with the mathematical community, let alone specific attention to proof.^{[3]} Capacities that are *enculturated* are those that students do not typically develop without some encounter with the mathematics community, whether through reading, schooling, math circles, or otherwise. Examples of imported capacities are the student’s capacity to reason, and fluency in the language of instruction. Enculturated capacities include, for example, knowledge of specific patterns of reasoning common to mathematics writing but rare outside it, such as the elegant complex of ideas behind the phrase, “without loss of generality, we can suppose….”

For imported capacities involved in proof, the teaching challenge is to create conditions that cause students to actually access those capacities while reading and writing proofs.

For enculturated capacities, the prima facie teaching challenge is to inculcate them, i.e., to cause the capacities to be developed in the first place. But there is also a prior, less obvious challenge: we have to know they’re there. Since many instructors are already very well-enculturated, our culture is not always fully visible to us. If we can’t see what we’re doing, it’s harder to show students how to do it. (This challenge has the same character as that mentioned by Pierre Bourdieu in the epigraph, although he was writing about sociology.)

When my personal obsession with student difficulty with proofs first developed, I focused on imported capacities. I had many experiences in which students whom I knew to be capable of very cogent reasoning produced illogical work on proof assignments. It seemed to me that the instructional context had somehow severed the connection between the students’ reasoning capacities and what I was asking them to do. I became very curious about why this was happening, i.e., what types of instructional design choices led to this severing, and even more curious about what types of choices could reverse it.

My main conclusion, based primarily on experience in my own and others’ classrooms, and substantially catalyzed by reading Paul Lockhart’s celebrated *Lament* and Patricio Herbst’s thought-provoking article on the contradictory demands of proof teaching, was this: It benefits students, when first learning proof, to have some legitimate uncertainty and suspense regarding what to believe, and to keep the processes of reading and writing proofs as closely tied as possible to the process of deciding what to believe.^{[4]}

I stand by this conclusion, and more broadly, by the view that the core of teaching proof is about empowering students to harness their imported capacities (in the above sense) to the task, rather than learning something wholly new. That said, in the last few years I’ve become equally fascinated by the challenges of enculturation that are part of teaching proof reading and writing. If I’m honest, my zealotry regarding the importance of imported capacities blinded me to importance of the enculturated ones.

What I want to do in the remainder of this blog post is to propose that a particular feature of proof writing is an enculturated capacity. It’s a feature I didn’t even fully notice until fairly recently, because it’s such a second-nature part of mathematical communication. I offer this proposal in the spirit of the quote by sociologist/anthropologist Pierre Bourdieu in the epigraph: to think in a completely astonished and disconcerted way about something we thought we already understood. Naming it as enculturated has the ultimate goal of supporting an inquiry into how students can be explicitly taught how to do it, though this goes beyond my present scope.

I recently encountered an article by Kristen Lew and Juan Pablo Mejía-Ramos, in which they compared undergraduate students’ and mathematicians’ judgements regarding unconventional language used by students in written proofs.^{[5]} One of their findings was that, in their words, “… students did not fully understand the nuances involved in how mathematicians introduce objects in proofs.” (2019, p. 121)

The hypothesis I would like to advance in this post is offered as an explanation for this finding, as well as for a host of student difficulties I’ve witnessed over the years:

*The way we conceptualize the objects in proofs is an enculturated capacity.*

These objects are *weird*. In particular, the sense in which they exist, what they *are*, is weird. They have a different ontology than other kinds of objects, even the objects in other kinds of mathematical work. An aspect of learning how to read and write proofs is getting accustomed to working with objects possessing this alternative ontology.^{[6]} If this is true, then it makes sense that undergraduates don’t quite have their heads wrapped around the way that mathematicians summon these things into being.

The place where this is easiest to see is in proofs by contradiction. When you read a proof by contradiction, you are spending time with objects that you expect will eventually be revealed *never to have existed*, and you expect this revelation to furthermore tell you that *it was impossible that they had ever existed*. That’s bizarro science fiction on its face.

But it’s also true, more subtly perhaps, of objects appearing in pretty much any other type of proof. To illustrate: suppose a proof begins,

Let be a lattice in the real vector space , and let be a nonzero vector of minimal (Euclidean) length in …

Question. *What kind of a thing is ?*

[The camera pans back to reveal this question has been asked by a short babyfaced man wearing a baseball cap, by the name of Lou Costello. His interlocutor is a taller, debonair fellow with a blazer and pocket square, answering to Bud Abbott.]

Abbott: It’s a vector in .

Costello: Which vector?

Abbott: Well, it’s not any *particular* vector. It depends on .

Costello: You just said it was a particular vector and now it’s not a particular vector?

Abbott: No, well, yes, it’s some vector, so in that sense it’s a particular vector, but I can’t tell you which one, so in that sense it’s no particular vector.

Costello: You can’t tell me which one?

Abbott: No.

Costello: Why not?

Abbott: Because it depends on . It’s one of the vectors that’s minimal in length among nonzero vectors in .

Costello: *Which vector?*

Abbott: No *particular* vector.

Costello: But is it some vector?

Abbott: Naturally!

Costello: You said it depends on . What’s ?

Abbott: A lattice in $\mathbb{R}^n$.

Costello: Which lattice?

Abbott: Any lattice.

Costello: Why won’t you say *which* lattice?

Abbott: Because I’m trying to prove something about *all lattices*.

Costello: You mean to say is *every lattice???*

Abbott: No, it’s just one lattice.

Costello: *Which one?!*

For any readers unfamiliar with the allusion here, it is to *“Who’s on First?”*, legendary comedy duo Abbott & Costello’s signature routine.^{[7]} What’s relevant to the present discussion is that the skit is based on Costello asking Abbott a sequence of questions about a situation to which he is an outsider and Abbott is an insider. Costello becomes increasingly frustrated by Abbott’s answers, which make perfect sense from inside the situation, but seem singularly unhelpful from the outside. Abbott for his part maintains patience but is so internal to his situation—enculturated, as it were—that he doesn’t address, or even seem to perceive, the ways he could be misunderstood by an outsider.^{[8]}

My goal with this little literary exercise has been to dramatize the strangeness of the “arbitrary, but fixed” nature of the objects in proofs. Most things we name, outside of proof-writing, don’t have this character. Either they’re singular or plural; one or many; specific or general; not both. Every so often, we speak of a singular that represents a collective (“the average household”, “a typical spring day”), or that is constituted from a collective (“the nation”), but these are still ultimately singular. They are not under the same burden as mathematical proof objects, to be able to stand in for any member of a class. Proof objects aren’t representative members of classes but *universal* members. This makes them fundamentally unspecified, even while we imagine and write about them as concrete things.

There’s an additional strangeness: proof objects, and the classes of which they are the universal members, are themselves often constituted in relation to other proof objects. We get chains, often very long, where each link adds a new layer of remove from true specificity, but we still treat each link in the chain, including the final one, as something concrete. I was trying to hint at this by posing the question “what is it?” about , rather than , in the example above. As consternated as Costello is by , is doubtless more confounding.

I think there are at least two distinct aspects of this that students new to proof do not usually do on their own without some kind of enculturation process. In the first place, the initial move of dealing with everything in a class of objects simultaneously by postulating a “single universal representative” of that class just isn’t automatic. This is a tool mathematical culture has developed. Students need to be trained, or to otherwise catch on, that a good approach to proving a statement of the form “For all real numbers…” might begin, “Let be a real number.”^{[9]}

But secondly, when we work with these objects, their “arbitrary, but fixed” character forces us to hold them in a different way, mentally, than we hold the objects of our daily lives, or even the mathematical objects of concrete calculations. When you read, “Let be a smooth function ,” what do you imagine? A graph? Some symbols? How does your mental apparatus store and track the critical piece of information that can be *any* smooth function on ? Reflecting on my own process, I think what I do in this case is to imagine a vague visual image of a smooth graph, but it is “decorated”—in a semantic, not a visual, way—by information about which features are constitutive and which could easily have been different. The local maxima and minima I happen to be imagining are stored as unimportant features while the smoothness is essential. Likewise, when I wrote, “Let be a lattice in the real vector space ,” what did you imagine? Was there a visual? If so, what did you see? I imagined a triclinic lattice in 3-space. But again, it was somehow semantically “decorated” by information about which features were constitutive vs. contingent. That I was in 3 dimensions was contingent, but the periodicity of the pattern of points I imagined was constitutive. I’m positing that students new to proof do not usually already know how to mentally “decorate” objects in this way.^{[10]}

Here are some specific instances of student struggle that seem to me to be illuminated by the ideas above.

- In the paper of Lew and Mejía-Ramos mentioned above, eight mathematicians and fifteen undergraduates (all having taken at least one proof-oriented mathematics course) were asked to assess student-produced proofs for unconventional linguistic usages. The sample proofs were taken from student work on exams in an introduction to proof class. One of these sample proofs began, “Let .” Seven of the eight mathematicians identified the “Let …” as unconventional without prompting, and the eighth did as well when asked about it. Of the fifteen undergraduate students, on the other hand, only four identified this sentence as unconventional without prompting, while even after being asked directly about it, six of the students maintained that it was not unconventional. I would like to understand better what these six students thought that the sentence “Let ” meant.
- Previously on this blog, I described the struggle of a student to wrap her head around the idea, in the context of – proofs, that you are supposed to write about as though it’s a particular number, when she knew full well that she was trying to prove something for all at once.
- A year and a half ago, I was working with students in a game theory course. They were developing a proof that a Nash equilibrium in a two-player zero-sum game involves maximin moves for both players. It was agreed that the proof would begin by postulating a Nash equilibrium in which some player, say , was playing a move that was not a maximin move. By the definition of a maximin move, this implies that has some other move such that the minimum possible payout for if she plays move is higher than the minimum possible payout if she plays . The students recognized the need to work with this “other move” but had trouble carrying this out. In particular, it was hard for them to keep track of its constitutive attribute, i.e., that its minimum possible payout for is higher than ‘s. They were as drawn to chains of reasoning that circled back to this property of
*as a conclusion*, as they were to chains of reasoning that proceeded forward*from*it. - In the same setting as the previous example, there was a student who, in order to get her mind around what was going on, very sensibly constructed some simple two-player games to look at. I don’t remember the examples, but I remember this: I kept expecting that when she looked at the fully specified games, “what was” would click for her, but it didn’t. Instead, I found
*myself*struggling to be articulate in calling her attention to , precisely because its constitutive attribute was now only one of the many things going on in front of us; nothing was “singling it out.” I found myself working to draw her attention*away*from the details of the examples she’d just constructed in order to focus on the constitutive attribute of . My reflection on this student’s experience was what first pointed me toward the ideas in this blog post:*I mean really, what***is**, anyway, that recedes from view exactly when the situation it’s part of becomes visible in detail? - This semester I taught a course on symmetry for non-math majors. It involved some elementary group theory. An important exercise was to prove that in a group, implies . One student produced an argument that was essentially completely general, but carried out the logic in a specific group, with a specific choice of , and presented it as an example. Here is a direct quote, edited lightly for grammar and typesetting. “For example [take] ; if we will operate on both sides the inverse of we will get . As we have proven that always , we can change the structure of the equation to , [which] shows that x has to be equal to y.” The symbols and refer to one-quarter and three-quarters rotations in the dihedral group . From my point of view as instructor, the student could have transformed this from an illustrative example to an actual proof just by replacing and with and , respectively, throughout. What was the obstruction to the student doing this?

My claim is that *the mathematician’s skill of mentally capturing classes of things by positing “arbitary, but fixed” universal members of those classes, and then proceeding to work with these universal members as though they are actual objects that exist, is an enculturated capacity.*

[1] I trust that any reader of this blog who has ever taught a course, at any level, that serves as its students’ introduction to proof, has some sense of what I am referring to. Additionally, the research literature is dizzyingly vast and there is no hope to do it any justice in this blog post, let alone this footnote. But here are some places for an interested reader to start: S. Senk, How well do students write geometry proofs?, *The Mathematics Teacher* Vol. 78, No. 6 (1985), pp. 448–456 (link); R. C. Moore, Making the transition to formal proof, *Educational Studies in Mathematics*, Vol. 27 (1994), pp. 249–266 (link); W. G. Martin & G. Harel, Proof frames of preservice elementary teachers, *JRME* Vol. 20, No. 1 (1989), pp. 41–51 (link); K. Weber, Student difficulty in constructing proofs: the need for strategic knowledge, *Educational Studies in Mathematics*, Vol. 48 (2001), pp. 101–119 (link); and K. Weber, Students’ difficulties with proof, *MAA Research Sampler #8* (link).

[2] Again, I cannot hope even to graze the surface of this conversation in a footnote. The previous note gives the reader some places to start on the scholarly conversation. A less formal conversation takes place across blogs and twitter. Here is a recent relevant blog post by a teacher, and here are some recent relevant threads on Twitter.

[3] This and the following sentence should be treated as definitions. I am indulging the mathematician’s prerogative to define terms and expect that the audience will interpret them according to those definitions throughout the work. In particular, while I hope I’ve chosen terms whose connotations align with the definitions given, I’m relying on the reader to go with the definitions rather than the connotations in case they diverge. I invite commentary on these word choices.

[4] This is an argument I have made at length in the past on my personal teaching blog (see here, here, here, here, here), and occasionally in a very long comment on someone else’s blog (here). Related arguments are developed in G. Harel, Three principles of learning and teaching mathematics, in J.-L. Dorier (ed.), *On the teaching of linear algebra*, Dordrecth: Kluwer Academic Publishers, 2000, pp. 177–189 (link; see in particular the “principle of necessity”); and in D. L. Ball and H. Bass, Making believe: The collective construction of public mathematical knowledge in the elementary classroom, in D. Phillips (ed.), Yearbook of the National Society for the Study of Education, *Constructivism in Education*, Chicago: Univ. of Chicago Press, 2000, pp. 193–224.

[5] K. Lew & J. P. Mejía-Ramos, Linguistic conventions of mathematical proof writing at the undergraduate level: mathematicians’ and students’ perspectives, *JRME* Vol. 50, No. 2 (2019), pp. 121–155 (link).

[6] Disclaimer: although I am using the word “ontology” here, I am not trying to do metaphysics. The motivation for this line of inquiry is entirely pedagogical: what are the processes involved in students gaining proof skills?

[7] Here’s a video—it’s a classic.

[8] One of the keys to the humor is that the audience is able to see the big picture all at once: the understandable frustration of Costello, the uninitated one, apparently unable to get a straight answer; the endearing patience of Abbott, the insider, trying so valiantly and steadfastly to make himself understood; and, the key idea that Costello is missing and that Abbott can’t seem to see that Costello is missing. I’m hoping to channel that sense of stereovision into the present context, to encourage us to see the objects in a proof simultaneously with insider and outsider eyes.

[9] Annie Selden and John Selden write about the *behavioral knowledge* involved in proof-writing, and use this move as an illustrative example. A. Selden and J. Selden, Teaching proving by coordinating aspects of proofs with students’ abilities, in *Teaching and Learning Proof Across the Grades: A K-16 Perspective*, New York: Routledge, 2009, p. 343.

[10] The ideas in this paragraph are related to Efraim Fischbein’s notion of “figural concepts”—see E. Fischbein, The theory of figural concepts, *Educational Studies in Mathematics* Vol. 24 (1993), pp. 139–162 (link). Fischbein argues that the mental entities studied in geometry “possess simultaneously conceptual and figural characters” (1993, p. 139). Fischbein’s work in turn draws on J. R. Anderson, Arguments concerning representations for mental imagery, *Psychological Review*, Vol. 85 No. 4 (1978), pp. 249–277 (link), which, in a broader (not specifically mathematical) context, discusses “propositional” vs. “pictorial” qualities of mental images. The resonance with the dichotomy I’ve flagged as “semantic” vs. “visual” is clear. I’m suggesting that the particular interplay between these poles that is involved in conceptualizing proof objects is a mental dance that is new to students who are new to proof. (Actually, it is not entirely clear to me that the dichotomy I want to highlight is “semantic” vs. “visual” as much as “general” vs. “specific”; perhaps it’s just that visuals tend to be specific. However, time does not permit to develop this inquiry further here.)

[11] Because this circle of skills involve taking something strange and abstract and turning it into something the mind can deal with as a concrete and specific object, they strike me as related to some notions well-studied in the education research literature: Anna Sfard’s *reification* and Ed Dubinsky’s *APOS theory*—both ways of describing the interplay between process and object in mathematics learning—and the more general concept of *compression* (see, e.g., D. Tall, *How Humans Learn to Think Mathematically*, New York: Cambridge Univ. Press, 2013, chapter 3).

When Yale-NUS College reviewed the curriculum for its Mathematical, Computational, and Statistical (MCS) Sciences major in the autumn of 2018, I spent several weeks reading about mathematics programs at similar institutions. A common learning objective among many of the programs was a variation of “preparing students to become lifelong learners.” I really like this goal because, among many other reasons, it reminds teachers that students are human beings who have lives beyond their studies, and it reminds students that learning is not confined to the early years of one’s life. As I reflect on my life of learning thus far, I cannot help but notice how significantly the way I learn has changed since I was a student. Some of these differences arose naturally with changes in my circumstances over the years, while others could have been addressed while I was still a student.

In this post, I want to share some observations about how my approach to learning has changed since I started working as a professional mathematician, and how I have changed my approach to teaching with the hope of helping my students develop more effective and relevant learning strategies earlier in their mathematical journeys.

*Reading has become my primary mode of knowledge acquisition.*

When I was an undergraduate student, I rarely read mathematics. It wasn’t from a lack of interest in the subject. I remember being enthusiastic about my courses and the joy I felt from solving problems. I simply didn’t read much mathematics. Not *really *anyway. The closest thing I did to reading was scanning through a textbook for a proposition or theorem that could help me link two concepts that would allow me to solve a homework problem. That jigsaw-puzzle approach to reading mathematics lasted well into graduate school. It is dramatically different from my current situation, where the majority of new mathematics I learn, I get from reading. So what changed? Necessity. As a professor, I can go to seminars and conferences to learn more about certain subjects, but not to a degree that is comparable to taking a course. Instead, I spend a lot of time learning on my own and the available format is almost always written text.

When I was a student, the need to read simply wasn’t there. I was fortunate to study at an undergraduate program with many dedicated teachers, who prepared clear, accessible lectures and class activities, so I could successfully complete my coursework without doing the assigned readings. It didn’t become an issue for me until I was a graduate student when I had to look up details of proofs that didn’t fit into lecture notes and read lots of articles for my dissertation research. It was a difficult transition for me.

*Learning new concepts and techniques becomes much easier when I need them to complete an ongoing project.*

When I was a student, I spent a lot of time learning new techniques, diligently practicing them on problem sets… and then forgetting them almost immediately. I don’t think I was particularly unmotivated or lazy – and I completely trusted my professors when they said certain concepts were important – yet I forgot so much of what I learned shortly after learning it. What was going on? On one hand, it is natural to learn new things in stages, picking up a fraction of the content at the first encounter followed by pieces of new information with each subsequent exposure. At each stage of the process, we internalize a portion and forget the rest of what we observe.

On the other hand, I think my struggles were partially related to context. I remember my professors giving clear explanations for why different techniques were developed and how they were used in practice, but there was a disconnect for me because I didn’t have any personal experience developing mathematical techniques, nor did I have an application of my own in mind. Looking back over my career, my most productive learning experiences have come from working on a project where I didn’t have all of the tools I needed and had to learn them on the fly in order to complete the project. In those cases, I didn’t watch a tutorial or listen to a lecture about standard techniques and then practice them on a variety of examples; I started with the problem I was trying to solve, found a technique in the literature that was used to solve similar problems, and figured out how to apply or adapt the technique to my particular situation.

*All of my best ideas have had humble beginnings.*

* *When I was a student, I had a growth mindset about mathematical knowledge but a fixed mindset about mathematical creativity. I believed everyone could have positive, successful, and meaningful experiences with mathematics by learning new techniques but mathematical creativity was an inherent ability that could not be developed. I don’t know why I felt this way, and I can’t recall anyone ever telling me it was the case, yet I remember that impression weighing on me a lot. Whenever I worked on homework sets with other students and someone would figure out how to solve a problem I was stuck on, I always assumed it was because they had some amazing insight that I would not have been capable of finding on my own. I was so preoccupied with trying to figure out whether or not I had what it took to become a successful mathematician that it never occurred to me to ask them how they came up with their idea. Consequently, I spent a lot of time feeling frustrated, not being particularly productive, and waiting for inspiration to strike because that was where I thought creative solutions originated.

Now that I have more experience – and the confidence that comes with it – I can recognize that all my best ideas started with simple observations. And while there is no clear-cut recipe for creativity and innovation, there are concrete things I can do to cultivate situations that make those important kernels of ideas of possible. Instead of dwelling on what I don’t know how to do, I focus on exploring what I can do that might produce a new insight, such as writing out some examples, constructing a conceptual diagram, or drawing a picture.

*All of my proudest accomplishments were made possible through the generous help of people whose experiences and perspectives are different from my own.*

As a student, I found little satisfaction from working in teams, especially with unfamiliar teammates. Team assignments typically went one of two ways for me: either I was confident in my abilities and did the vast majority of the work or I was insecure about my abilities, didn’t want to look stupid, and held back my ideas thinking it was better to appear ignorant than open my mouth and confirm it. In the former scenario, I didn’t mind doing most of the work because I was confident in my ability to succeed, and it often seemed easier to do most of the work myself rather than try to coordinate my teammates’ efforts.

I didn’t see value in exploring different perspectives because there were never any consequences for taking a narrow approach. Like many who have the same privileges as I do (I am a heterosexual white male from North America), I had a limited understanding of how social identities affect group interactions, and I conflated inclusivity with civility. In the latter scenario, I was aware that teamwork required a lot of effort and collaboration. Even though I was willing to put in the work, my insecurities still got the better of me because I didn’t trust my teammates enough to share my ideas openly.

When I look at how the accomplishments I’m most proud of have come about, and how much I have learned in recent years working at an international college in Singapore, I can’t help but wonder how many opportunities to learn and grow I missed out on because I simply wasn’t looking or I didn’t appreciate how much effort goes in to building enough trust to open up a beneficial exchange of ideas.

Here are a few ways I have changed my approach to teaching in response to these observations.

For starters, I no longer rely on lectures or video tutorials for presenting new ideas. Instead, the lion’s share of content delivery comes in the form of reading assignments. To support my students as they adapt to this model, I use the social annotation platform, Perusall, which allows them to highlight passages and ask questions, contribute or link alternative explanations, and propose solutions to “check your understanding” type exercises. They can also upvote annotations of their peers that they find helpful. In addition to developing technical reading skills, Perusall offers the valuable practice contributing to social media debates and online forums like StackExchange in a safe and controlled environment.

To offer my students an authentic learning environment that emulates the typical “on the job” learning that takes place in many technical professions, I have started to build each of my courses around three or four substantial team projects. Instead of asking students to master content and then apply what they have learned to a bigger project, I design the projects in a way that prompts students to learn the relevant material as they go. Each project is assigned on the first day of its respective segment of the course. The students are typically able to understand what the project prompts are asking but are not aware of any obviously relevant tools to get started.

To facilitate effective teamwork, I have adopted the Team-Based Learning (TBL) model, where each lesson has a reading assignment to be completed before class, individual and team readiness assurance tests at the start of each class, and a substantial problem-solving session that enables students to apply and extend their understanding of the tools they will need to successfully complete the project. Students take the readiness assurance tests and work together on the problem-solving sessions within their project teams throughout the duration of the project in order to develop a productive group dynamic.

To encourage and reinforce good habits for mathematical research and creativity, I have started acknowledging and giving credit to teams when they demonstrate important elements of a productive research process, such as generating examples, identifying patterns, asking questions and making conjectures, testing conjectures with new information, drawing connections between relevant topics in the literature to better understand the problem at hand, and re-evaluating an approach based on preliminary findings. Because many of these elements can be difficult to discern in a final written report, I have started asking each team to submit an activity log that documents their progress throughout the project. My rubric for the activity log was heavily influenced by the Creativity-in-Progress Rubric on Proving.

Finally, in addition to research and creativity, I have started to encourage and reinforce good habits for effective and respectful team interactions by asking each team to prepare a mission statement during the first week of the project where they agree on a team name, tentative work schedule, and initial plan of attack. I also ask each team to prepare a set of guidelines for how they will conduct their meetings and a set of criteria for how they will evaluate each other’s contributions to the project.

The idea for creating guidelines came from my experience facilitating Intergroup Dialogue (IGD) at Yale-NUS College. IGD is a structured conversation between members of different social identity groups that encourages participants to explore singular and intersecting aspects of their identities while critically examining dynamics of power, privilege, diversity and inequity in society. Because the dialogues can be difficult or contentious, a lot of the groundwork for IGD aims at building trust and creating a space in which people can share their ideas freely without judgment. For instance, at the beginning of each dialogue, the participants prepare a list of guidelines. I adapted those guidelines to fit a team-based learning classroom: The IGD guideline *“We all recognize that participation in this dialogue is voluntary. Everyone who is here wants to be here.”* became *“We all recognize that this course is an elective. Everyone who is here wants to be here.”* Most of the guidelines are common sense statements, but articulating them in a mission statement provides avenues for students to speak their discomfort and overcome obstacles in a responsible and respectful manner.

Here is the first project brief from my Discrete Mathematics course, which is typically taken by second-year prospective MCS majors at Yale-NUS College whose primary interest is computer science. The course meets twice per week for 110 minutes at a time. Each lesson consists of a pre-class reading assignment (8-10 pages of text, approximately 2 hours of interactive reading), in-class readiness assurance tests (20 minutes), and an in-class problem-solving session (90 minutes). The project spans six class meetings, including one lesson each on the Pigeonhole Principle, mathematical induction, and basic enumeration, two lessons on combinatorial proofs and bijections with emphases on the Binomial Theorem and Fibonacci numbers, and one class meeting designated as work time so students have a full week free of reading assignments and problem sets to complete their reports.

The project description presents students with eight seemingly unrelated families of mathematical objects and asks them to find a formula for the number of objects in each family. It then asks them to describe how the families are related based on the formulas they find. Over the course of the project, the students discover that the families are all equinumerous. Indeed, they are all manifestations of the Catalan numbers!

While the project initially appears somewhat daunting, the students typically proceed by generating lots of examples of each family. From there, they tend to observe fairly quickly and conjecture that the number of elements in each family appears to be the same. This is a significant discovery for them since it means that, instead of finding the same formula eight different times, they only need to find the formula for one family and then argue why the different sets are in one-to-one correspondence with one another. That prompts them to review the reading assignments on mathematical induction, recursion, combinatorial proof, and bijections. The diversity of the objects themselves also makes the project well suited for teams made up of students with disparate backgrounds since finding all the connections requires a variety of perspectives.

The overall response to these changes has been positive. A number of students acknowledged the stated goals and embraced the project-based approach straight away. For instance one student wrote:

[The project] was actually a very fun and enjoyable experience, while also providing a good amount of challenge and difficulty. When we first received the project brief, we were genuinely stunned by what we had to do – we didn’t really know where to begin, and everything we tried seemed to be useless. But it was really nice to see us slowly progress, picking at the problem bit by bit, sometimes with no results, sometimes with huge chunks of the problem falling off. I really saw the advantage of having very different minds work on the same problem. I believe my teammates and I, having come from different backgrounds in terms of interests and experiences, approached the problems quite differently, and we were able to really complement each other and bounce off each other’s ideas. All of us contributed in big ways, and together we managed to come out with a closed formula pretty early into the project. Eventually, we managed to link the closed formula to one of the combinatorial objects, and quickly pieced bijections together. Even in the final moments of the project, the group shone through as we all picked on different parts of the project, trying to polish it off as well as we could.

Other students struggled at times, but eventually warmed up to the approach. For instance, a student wrote:

Initially, I felt rather excited about tackling the questions. We made some observations that turned out to be insightful and it felt like the project was going in the right direction. When the team and I got stuck at the later stages of this project, I became frustrated and lost motivation. But my teammates continued to encourage me and kept trying to develop new methods of solving the problems. Through this project, I learned that solving problems is not always a smooth path. It is helpful to acknowledge our frustration and to expect difficulties so that we are less anxious when we are stuck.

The most encouraging feedback I received, however, was from the many (more than 1 in 5 across my sections of Discrete Mathematics) students who explicitly described how empowering the experience was for them in their reflections. For instance, one student wrote:

Throughout this project, I have learnt a lot about how mathematical reasoning happens and [I] have changed a lot of my perceptions about how mathematics is done. Being used to the usual individual problem-solving method in high school, where there is only one right answer and a few preset methods that are best for determining this answer, I have come to love the collaborative approach taken in this project and in the whole Discrete Mathematics course in general. It is truly a vibrant environment for learning, and I am very grateful to have the support and knowledge of my team members. I have always felt my mathematical reasoning skills to be inferior to other upperclassmen or people who reason faster, sharper and more elegantly, but I have come to learn that the final polished product is not all that it appears to be – it is the process that is the most important, and there are lots of things I can contribute within the process while I am working on improving the skills I can use to refine the final solution.

Despite these successes, there is still a lot of room for improvement. A common piece of critical feedback I receive from students is that the reading assignments are very difficult, even with the added support from Perusall. There are a lot of factors at play here such as the choice of text, size of the class, and my (in)ability to effectively respond to the Perusall discussions in real time.

Revising my courses to emphasize reading, research, creativity, and teamwork has been a challenging but rewarding process. I am thankful to the many Yale-NUS students who worked diligently on the projects and offered their thoughtful, constructive feedback, and I am curious to see how my approach extends to other topics besides discrete mathematics and abstract algebra, which may, for one reason or another, be particularly well suited for this type of open-ended, collaborative, team project framework. I hope I have convinced you to try out some of these ideas and I look forward to hearing about the outcomes!

]]>Like many of us, I began teaching online this Spring. Unlike many of us, I began doing so at the start of the semester. I am co-teaching a class at Michigan State, and I live in Nebraska. One of the most useful conversations I had in preparation for this assignment happened in 2013, well before the current coronavirus epidemic. The math department at the University of Nebraska-Lincoln had been considering a synchronous online version of a mathematics course, for rural teachers. I chatted with Ari Nieh, then an instructor for Art of Problem Solving, about what it would take to teach online, especially via chat forum technology. (Ari then became a lecturer in Writing, Rhetoric, and Professional Communication at MIT; and now he is a game designer at Wizards of the Coast.) In the end, that course was run asynchronously (and in many ways consistent with the advice given in a previous post). Nonetheless, much of the advice I received 7 years ago aged well. With Ari’s permission, I share snippets of our conversation in this post, edited for readability, and with commentary from present-day me.

First, here is a summary of the key pieces of advice I took from the conversation.

**Key pieces of advice for teaching online using chat technology**

**Tools for making students feel comfortable in class and that their input is being valued**– careful choice of words to maintain a welcoming tone; consistently responding to questions, whether publicly or privately; and using a fair bit of humor at the beginning of class and during transitions.**Opening a class**– it’s kind of like face-to-face, but word choice is perhaps even more important, because you don’t have tone/body language.**Lectures**– don’t really translate. Make sure to have questions for students. This can take the form of closed questions (e.g., multiple-choice questions or ones where there is one right answer) or open-ended questions.**Whole group discussion**– There are some ways that this is easier, because students can see each others’ thoughts for longer. Also, questions can be answered privately as well as publicly. Giving instructions is perhaps easier than in person, because of persistent text: whatever you say hangs in the air and they keep reading it.**Whole group discussion, continued**– In general, the biggest skill that doesn’t translate is improvisation. You can’t improvise spoken words, so you want to develop skill at improvising written words.**Wrapping up discussions/class**– this is a place where it would be good to have some prepared draft/default text.**Diagrams**– should be prepared ahead of time when possible.

Now for our conversation.

**Yvonne (2013):** So, let’s talk online teaching. The context is this. There’s a class we offer that is 2.5 hours long, that alternates between lecture and working time and discussion, and some time to work on homework problems. We want to translate this online. How should or can this work?

**Ari:** I see. What is the format of the online classroom?

**Y2013:** That is a good question. It hasn’t been determined, but there will be an online chat place for teachers and students to interact.

**Ari:** Our online classroom platform [at Art of Problem Solving] has a setup where everything the students type goes to the teacher first, who has the option of showing it to the whole class or not. In any case, there’s probably some form of moderation?

**Y2013:** Let’s assume that for now.

**Present-day Yvonne (2020):** This kind of moderation is available on Zoom, and potentially on other platforms as well.

**Ari:** Right. Working on shorter-length problems in class works fine online.

Lecturing is actually the hard part. The reason it’s difficult is that it’s much less interactive. If the instructor is just typing stuff which gradually appears on the screen, there’s not much incentive for the student to pay attention instead of deciding, “I’ll just read the whole transcript later.” So lectures must be liberally sprinkled with questions to evaluate comprehension or points for discussion by students.

What’s the topic of the class?

**Y2013:** Geometry from a transformation perspective.

Which brings up another question: How do you handle discussions about diagrams?

**Ari:** We have prepared diagrams for geometry classes.

But if the students want to do something that we haven’t prepared in advance, it doesn’t work too well. You want some sort of interactive blackboard thing for that, I would think.

**Y2013:** Hmm … okay. I’m thinking about a question that we often open the course with: *Given two rectangles in the plane, show that there is always a line that bisects both rectangles simultaneously.*

Should I maybe look for a separate program that students can be logged onto at the same time to draw?

**Ari:** Possibly. Prepared diagrams actually work most of the time. If your chat system supports them sending images, that might be good, too.

But it’s also okay for them to express some idea in words, and then you provide the diagrams which demonstrate it. For instance, suppose some student says, “The lines that bisect a rectangle all have to go through the middle.”

You say, “You mean, like this?” (DIAGRAM)

It’s partly a question of whether they’ll be able to make good diagrams on the fly, which students may or may not be able to.

**Y2020:** In 2013, virtual shared drawing spaces didn’t exist the way they do now, and certainly webcam technology wasn’t as prevalent. All that said, there’s still something nice about being able to ask students to articulate, in words, what they are imagining. There’s an entire literature on how connecting diagrams with the logical constraints behind them may be key to learning geometry (e.g., Duval, 2006; Fischbein, 1993; Jones, 1998; Mesquita, 1998; Presmeg, 2007). In the class that I’m co-teaching currently, which uses Zoom, I swap between making diagrams based on what students are saying and asking students to make drawings and shove them up against the webcam. I find both useful, and the latter especially useful for getting a read on the class as a whole.

**Y2013:** How much have you found that your face-to-face teaching skills translate to online teaching? For instance, we’ve been talking about diagrams and discussions; what about building rapport with students on the first day and throughout, or how to give instructions for what would otherwise have been a handout or slide?

**Ari:** Good question!

So, obviously one can’t build rapport via eye contact, body language, tone of voice, etc. Some tools for making students feel comfortable in class and that their input is being valued include: Careful choice of words to maintain a welcoming tone. Consistently responding to questions, whether publicly or privately. And using a fair bit of humor at the beginning of class and during transitions.

On answering questions. When students ask a question, the teacher gets the question ‘privately’. The teacher can then choose to pass the question to the whole class, or answer it privately. This allows the class to proceed without those questions, which might distract. At the same time, it allows student to get answers to individual questions.

Giving instructions is perhaps easier than in person, because of persistent text: whatever you say hangs in the air and they keep reading it.

In general, the biggest skill that doesn’t translate is improvisation. Instead of improvising spoken words, you need to get good at improvising written words very quickly.

Of course, one can always stick to a script, but it’s nice to have the option of exploring tangents, finding teachable moments within alternate solutions or mistakes, etc.

That said, for lecturing, I strongly recommend having the remarks prepared and using some amount of copy-pasting rather than writing them on the fly.

**Y2013:** Wrapping up discussions and summarizing key points – are those also places where generating some draft or default text ahead of time would be a good idea?

**Ari:** Yes, definitely. Teaching online can be pretty tiring because you have to produce cogent text on the fly. Delegating some of it to your past self is a good idea.

**Y2020:** As one of my friends recently commented, “How is it that talking at students for 20 minutes on Zoom is more tiring than jumping around a classroom for 2 hours?!”

I have also found online teaching to be similarly or more tiring than physical teaching because I’m limited to primarily one mode of interaction: words. Even though there is some facial interaction on Zoom, I haven’t found it to work with my intuition in quite the same way.

**Ari:** One other thing I should mention about your class plan: I’m not sure that giving students in-class homework time will work. When you do that in person, you can circulate and watch them and there’s lots of direct social pressure to actually work during that time. But if an instructor did that to me online, I feel like I’d probably get up and eat. This is a gut feeling and not based on any particular experience.

**Y2020:** This is perhaps the hardest thing for any instructor seasoned in discussion-based teaching: that in-person social contracts differ from online ones! Social pressure works in different ways, and circulating also feels different. When I teach in person, I often can make in-the-moment changes to my discussion plans based on what I am seeing from students, and how they respond to my questions just to their group. Although I can do this in breakout rooms, it requires a different kind of concentration than visually scanning the room and making connections between different students’ work. As well, in-person, I often send delegates from one group to meet with another group to compare strategies. This doesn’t translate as easily or well, especially as one function of this technique is to give students a physical break by having them walk instead of sitting.

**Y2013:** Thanks so much. I think these are all the questions I have for now.

**Ari:** You’re welcome.

**Y2020:** Reading this over again now, there are some parts that seem charmingly quaint, such as our implicit question about whether there might even exist accessible platforms where people can simultaneously share images. But there are other pieces that ring true: that carefully chosen words are key to helping students feel welcomed. That teaching online can be tiring in its own way, and one way to mitigate that is to have some prepared text. And also that teaching online, though different from physical teaching, can have its perks. A great affordance of online teaching, that you don’t necessarily get so much in physical teaching, is the capacity for students to ask questions privately, and for you and the student to choose between private and public answering. And personally, I’ve found that having students work collaboratively on google docs is far easier than using a document camera to show multiple solutions.

Because I have a toddler at home, I spend a lot of time looking at her shape sorter, and how she will still sometimes put the square peg in the round hole (and it doesn’t fit). On the other hand, the square peg does fit in the octagonal hole, with appropriate rotation (at least in her set). To some degree, I think my difficulties in online teaching are in part due to fitting a square peg of physical teaching into a round role of online teaching. The times that it’s gone best are when I see online teaching not as a lesser version of physical teaching, but as its own kind of teaching with its own special opportunities.

]]>In the midst of the upheaval due to the Coronavirus, students and faculty are transitioning to new virtual classrooms. Many of us haven’t chosen to learn or teach, but here we are, making the best of this new reality.

In this post, I describe some guidelines that may help students manage the transition to online learning as smoothly as possible. Instructors can support students by helping them to learn online, and I encourage instructors reading this to pass it along to your students. I offer these suggestions with a caveat: Some of these ideas may not be feasible for everyone, and that’s ok. We all have unique living, learning, and life situations, and what works for one of us may not work for others. Take what you can, and leave the rest. Keep realistic expectations of yourself, understanding that these circumstances are less than ideal. While the suggestions in this post are directed toward students, I also offer “teaching tips” to help instructors support their students.

__Teaching tip__: Remember the diversity of your students and keep equity issues in mind. Not everyone will have reliable internet access, computer access, time, or a quiet place to study. Students may be caring for children or sick family members, may be sick themselves, may need to work, and may be facing any number of other challenges and stressors.

**Above all else, take care of your physical and mental health. **Self-care isn’t self-indulgent. If your physical or mental health falters, the rest of your work will suffer. Time invested in healthy habits pays off.

*Create healthy habits* for sleep, eating, exercise, and hydration. These basics will help you focus and manage stress, and also have many long-term health benefits. The hardest part of self-care for many of us is getting enough sleep, but this is one of the most important things you can do for yourself.

*Focus on the positive*. While this may sound simplistic in these challenging times, finding ways to laugh, watch funny videos, sing, dance, or do whatever will help you unwind and give you a much-needed break from school and other pressures. Limit the amount of time you spend surfing about how bad things are. If you choose to follow the news, identify a trusted news source, and tune to it judiciously.

*If you struggle with anxiety or depression*, identify resources *in advance *and keep the information handy. This will make it easier to access support if you need it. Do a little research *today *to identify resources for counseling and crisis management available online or by phone. Your college or university has not shut down, only moved to a new normal, and most essential services are still available. You can also identify support groups in the community or online. There are several hotlines and other resources for students in crisis. The counseling center can help you find one to have on hand just in case you need it.

*Create a specific, intentional routine*. Have a structured plan for your days. Consistency in your schedule can help you reduce anxiety and stress.

__Teaching tip__*: **Understand the stress that students are experiencing in this transition. Remain flexible with deadlines, incompletes, extensions, and grades.*

**Make good use of resources***.* What arrangements has your college or university made for access to libraries, advising, and technical support? How can you contact each of your professors? If they have virtual office hours, learn the times and how to reach them. Is there an online forum for asking questions? If you know in advance how to access these resources, it will be easier to seek help when you need it.

__Teaching tips__*: Know which resources are available to students for advising, counseling, library use, crisis intervention, tutoring. Help students connect with these and also with external resources, such as **Virtual Nerd**, **Math is Fun**, **Khan Academy**, or **Math Forum**. Make sure students know when and how they can reach you: Establish virtual office hours and give students other ways to contact you for help. *

**Find a physical workspace that can help you focus and be productive **with as few distractions as possible. Depending on your living situation, this may be particularly challenging, but develop the best plan possible in the circumstances. Think about what kind of environment helps you study, and try to re-create that. Having a regular, designated space helps signal your mind that it’s study time. Aim for adequate lighting, access to your books, calculator, laptop, and other supplies. If you need background noise, or need to block outside noise, consider a white noise app.

*Talk with people around you about what you need***, **including parents, siblings, children, roommates, friends, bosses, and anyone else who has expectations of you or who impacts your daily life. Discuss your educational workload and what you need to meet it. Especially if there are other people in your living space who are studying or working from home, negotiate how you can have the best access possible to the space, technology, time, and quiet you need to meet your responsibilities. You may not be able to achieve the ideal, but planning and talking in advance will get you as close as possible.

**Make and maintain personal connections **to combat isolation. Personal connections can support your physical and mental health and can help you learn. Form a study group, use social media, connect through discussion boards or a Facebook group. Schedule video calls with friends and family. If you find it helpful to study with others, try a virtual or phone-based study session with your group. Keep in touch with instructors and classmates through virtual office hours or study groups so that you can stay up on your coursework.

__Teaching tip__*: Promote community through interactive class activities and authentic communication with students. Recognize and disrupt any racist, xenophobic, or other controversial issues that may arise in your classes. **Making uncomfortable conversations productive* *can help our students and colleagues move forward with greater understanding and inclusion.*

**Find out what is expected of you. **A chart or list of requirements for all of your courses might help you stay organized. Identify changes to course requirements, assignments, and due dates. Record the tools you need to access each course and where they are, including any chats, discussion forums, or other means to ask questions and communicate. Identify if you need to be online at specific times, how you are expected to submit assignments, and the schedule and process for quizzes, exams and other assignments. These details may be different for different courses, so a system for keeping track of what you need to do, and where, how and when, will really help you as you move through the rest of the semester. Ask lots of questions, but also be patient with your instructors! Many of them are new to this online environment as well, and they are figuring all of this out as quickly as they can.

*Get comfortable with the technology in advance*. Test computers, internet connections, software, webcams, headsets, mobile devices, and microphones. Mobile devices may be convenient, but may not always provide all the functionality you need. Know where to get technical support in the event that you need it. *If you don’t have access to something you need, contact your professors as soon as possible*. They may be able to provide alternative assignments or different ways to access the course material.

__Teaching tip__*: **Communicate expectations clearly and often. Use multiple modalities like email, announcements, texts, small peer-support groups, and other means. Offer students multiple ways to participate. Some students have more limited resources than you expect and may need alternative ways to meet course requirements.*

**Use good study habits. **Take notes and study just as you normally would. Close distracting tabs and apps. Humans are not as good at multitasking as you may think! Set yourself up to be able to focus on your work. Limit social media. Give your fullest attention to all course activities (discussions, group work, watching videos).

__Teaching tip__*: Provide students with individualized support and feedback.*

*Manage your time and stay organized. *Setting a schedule can help provide structure, keep you motivated, and prevent you from falling behind. Make sure your schedule is realistic and achievable. Record due dates, exams, and other assignments. Set aside specific times to study, and strive to keep balance by including the things that are most important to you. Allocate sufficient time for self-care. Schedule breaks. Do you need one-on-one time with your family? Build it into your schedule on a regular basis. If unexpected circumstances arise that interfere with your schoolwork, communicate with your professors as soon as you can.

__Teaching tip__*: **Communicate weekly about what is due, when, what tools should be used, and how work should be submitted. Establish weekly routines and rhythms to help you and your students keep on track. *

**Engage actively in online group work.** Group work and collaboration will look a little different online, but is still important to learning and creates a much-needed sense of community. Meet regularly with your team. Check in regularly with a quick text on your group chat. Identify a purpose for each meeting to help your team stay focused and on track. Take notes in a shared document so you can all contribute and keep track of progress. Meet by video when you can, to help you communicate more clearly and stay connected to each other. If someone has been absent from your group meetings, check in on them to make sure they’re ok.

**Remember, as time goes by we will all adjust. **This crisis has disrupted travel plans, ended sports seasons prematurely, confounded important projects, separated friends and family, and overall is a big strain on us all. Remember to take good care of yourself. You will find your way. We all will find our way as we settle in to the new normal. *Until then, take a deep breath, do your best, get some rest, and wash your hands.*

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Many of us are experiencing stress as schools, colleges and universities move instruction out of the classroom. Fortunately, even if distance learning is new to you, it isn’t new, and there is a lot of wisdom to draw on.

This document describes some practical strategies that will hopefully get you started, along with some helpful web-based resources. From there, you can do a deeper dive by accessing the open community on MAA Connect called “Online Teaching and Distance Learning.” MAA members can log in with their member credentials, and anyone who creates a free profile can join this group. This is an extensive platform to exchange ideas with other faculty and to access resources and advice for developing your courses. The STEM faculty blundering through remote teaching in a pandemic FaceBook page is another great place for faculty to share ideas and figure all this out together.

- In the current situation you don’t need to become an expert in online course delivery. Your course won’t be perfect, and it won’t be the same as it was in the classroom, and that is ok. Give yourself permission to just do the best you can do. Many of your students don’t have much experience learning online, and they are adjusting too. Set realistic expectations for yourself, your course, and your students.
- Keep flexibility and empathy in the forefront. Some students may not have ideal learning or internet environments—they have family responsibilities, lack privacy or quiet space, have unreliable internet access, are in other time zones, need to be online at a library or other public space, or have any number of distractions or obstacles.
- To the extent that your institution allows it, be particularly flexible with deadlines, independent study, and extended incompletes. Focus on what your students need in order to learn, rather than on structure or deadlines. We are in an unusual situation and this flexibility will make it easier on all of you.
- Online learning is one specific means of distance learning, but it is not the only one. Long ago distance learning took place by snail mail. I’m not necessarily advocating that approach, but cite this as a reminder to think outside the box.
- There are many tools available: Zoom, Skype, Dropbox, Blackboard, Canvas, Slack, VoiceThread, email, online chats, video chats, MS teams, Google docs, and many others. Investigate if there are particular platforms or tools that your institution already uses. Even if you are not familiar with them, your students might be and your institution is more likely to offer support for those platforms. Coordinating with your colleagues to use similar tools will allow you to support one another. Keep it simple.
- Does your institution have an office that supports online learning? They’ll be overloaded now, but make sure to check their website to see what they have to offer. Remember, even if you haven’t taught online before, others have, and they will hopefully share their expertise.
- Unless your institution requires it, it is ok to build the course week by week and adjust as you go. You don’t need to have it all figured out in advance.
- Prioritize the learning goals for your course. What are the most important things for your students to come away with? Focus on those, and build up the rest if you can.
- As with any teaching, focus on what you want students to
*learn*rather than what you need to*teach*. - Be transparent. Communicate very clearly with students about expectations, and about why you are making decisions and setting priorities. You can also be open about how you’re all in this together. This will reduce anxiety for everyone.
- Every week, provide a list of deliverables: read this, start this, submit that.
*What*should they be working on,*where*, and*with whom*?*Where*and*how*should their work be submitted? - I always remind students at the start of an online course that even if they are in the course platform frequently, if they don’t speak up and participate, no one knows they’re there. The same applies to instructors—communicate often.
- Are other faculty teaching sections of the same course? It might help to work together.
- Be aware of FERPA and accessibility requirements. Your campus may have resources to guide you with this. If needed, there are services online that will caption or transcribe videos.
- If you use video, use small chunks, no more than about 5 minutes at a time. This allows students to stay focused. Also think about the easy parts of production quality—adjust the lighting to be clear, don’t move around too much, and avoid other distractions that are within your control (until your cat runs across your keyboard or your toddler comes into the room). It might help to use the microphone on your earbuds rather than your computer.
- Test all technology. Do microphones, electronic whiteboards, video cameras work? Can the online platform handle the number of students who will login? But also remember that technical snafus happen. Communicate with your students with humor and you can all figure it out as you go along.
- Consider a combination of synchronous (everyone there at the same time) and asynchronous instruction. Synchronous instruction allows for more direct interaction, but it can be challenging for everyone to make timing work in the new normal. There are many ways to engage students asynchronously instead. Asynchronous instruction is easier on everyone. If you decide to ask everyone to be there at the same time, use the same time slot that your class was originally scheduled for.
- To keep students engaged, have frequent, small assignments with clearly-communicated due dates, and create learning activities that require students to interact at specified intervals. If you are requiring them to engage in collaboration or discussion, communicate clearly about how often you expect them to be online. I usually tell my students that they should be online 2-3 hours per week, on a minimum of 3 separate days.
- Establish netiquette rules up front. Be clear about expectations for respectful and professional communication. “Tone” can easily be misunderstood online.
- Encourage your students to compose their work in Word, LaTeX, by hand, or whatever works in your context,
*before*posting it online. Composing responses directly in a chat box leads to less effective communication. - Students can solve problems on paper, scan or take a picture of the solution, and upload it somewhere. Let them know that you can’t grade it if the image is unclear. Avoid using email for submitting assignments—it gets messy quickly—but provide a clear alternative.
- Ask open-ended questions. Discussion happens when there is struggle or debate, which doesn’t happen easily with yes-no questions. Ask students to interact about
*how*or*why*, not*what*or*whether*. - Some universities offer online tutoring, writing, or other forms of support. Check what is available to your students. You can also refer students to websites like Virtual Nerd, Math is Fun, Khan Academy, or Math Forum. If you hold virtual office hours or offer extra help, try to work with several students at a time so they can support one another and you can use your time effectively.
- There are applets available online for students to create and manipulate graphs. One of my favorites is Desmos, but there are many others.
- Some textbook publishers have online test banks. Google forms has a feature for creating tests and quizzes in an easy-to-use form, and for multiple-choice and fill-in responses, the quizzes can self-grade. In addition, some companies and organizations are providing access to resources during this current crisis. Even if these are not your ideal choice for assessment, you may be able to make them work for you in order to complete this semester. As my advisor would tell me when I was writing my dissertation, “‘Better’ is the enemy of ‘good enough’ .”
- Let your students know that many internet providers are offering free internet service for a fixed period. That does not mean it will be easy for everyone, but it should help many students. Some students will still need to be online in a library or other public space.
- Many group and interactive activities can be adapted to an online setting. For group work, develop ways that all students are held accountable to their group. Assigning group grades is one option (this also reduces your grading load). Some instructors require students to make individual submissions of assignments, and then assign everyone in the group the lowest grade; this is great motivation for them to make sure their group-mates understand what they’re doing.
- No matter what you do to defeat cheating, someone will find a way to work around it. One option is to require students to find a proctor of their own (often a librarian or local teacher) who will attest to their independent work, but this may be challenging for those who are self-isolating or maintaining the recommended social distancing. You could also assume that all work is open book and open notes and design assignments accordingly.
- Talk openly with your students about what they need to know in order to be ready for next semester’s courses. If they don’t do the work honestly, they are really cheating themselves. One contributor to a discussion about online learning hosted by MAA’s Rachael Levy suggested, “I think it’s worth asking what kind of deception we’re trying to prevent. Do we want to keep our students from deceiving us on assessments, or themselves? Because I think the second one is the real danger, and might be better to address directly.”
- Here are a few activities that have worked well with my online students:
- Students write or type the solution to a problem (or problems) using words and sentences and a minimum of mathematical notation. This is challenging! But it also requires a different level of understanding of the mathematics. Communicating about mathematics deepens understanding.
- Students solve a problem and share it online with a partner, a group, or the rest of the class. They provide feedback on a specific number of others’ solutions (for example, if they are in groups of 6, I might require them to comment on 2 other solutions) within a few days. Then they submit a written solution to the problem, using someone else’s strategy. This requires them to communicate with their classmates enough to understand someone else’s reasoning.
- Give them personalized problems to solve. For example, in one assignment about compound interest with regular savings contributions, each student submitted calculations for their personal retirement savings plan. They identified how many years they had until retirement, how much money they wanted to have access to each month, and then calculated their required monthly contributions. This way, everyone could help one another because they were working on the same task, but each had to do independent calculations.

These resources provide solid guidance to help you get started online:

Stanford University Teach Anywhere

The Chronicle of Higher Education: Move Online Now

Get Up and Running with Temporary Remote Teaching: A Plan for Instructors who Lecture

Move Your Course to Remote Delivery

Seven Practices for the Online Classroom

A mathematician describing his experience teaching online

Here you can find ideas for specific online learning activities

You might want to share this one with your students: Tips for students to participate in online group work and projects

YouTube playlist: Ideas for new online faculty

And everyone’s big concern, online assessment:

Self-grading quizzes in Google Forms

And make use of the two excellent resources cited at the top of this blog, MAA Connect and the STEM faculty blundering through remote teaching in a pandemic Facebook page.

This is by no means a comprehensive list of what to do or how to do it. There are as many ways to teach online as there are online instructors (maybe more). Feel free to share your ideas and questions in the comments so we can all help each other manage this transition and provide our students with quality learning opportunities.

Some of these ideas are drawn from resources developed by Open SUNY Center for Online Teaching Excellence of the State University of New York, and the Institute for Teaching, Learning, and Academic Leadership at the University at Albany.

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Ray Levy, Mathematical Association of America

*This is cross-posted in MathValues and Abbe Herzig has written a companion post. Additional resources and future meetings are also available here: https://tinyurl.com/OnlineTalkshop.*

In times of crisis we need community. With schools, colleges and universities mandating online teaching and learning in response to COVID-19, often with only a week of preparation time, people are scrambling for resources and information. Dr. Ray Levy, aås Deputy Executive Director of the MAA, asked an online group whether they would like a Zoom space to discuss online learning. With only 12 hours of notice, Dr. Jeneva Clark helped co-facilitate, and 36 people gathered. The next day, with only a few hours of notice Dr. Abbe Herzig and Dr. Yvonne Lai joined as co-facilitators, and 86 people gathered. Below is some of what we learned in the second meeting.

One point worth emphasizing is that online courses can look very different! Courses may have only text, audio, or video. They may be “synchronous” or “asynchronous”. These terms, originally from communications, to refer to how data is transferred. In online courses, this usually means whether everyone is online at the same time (synchronous) or online at different times (asynchronous). Often a course will contain multiple communication approaches.

Because many of us have not taught in these ways before, we discussed logistics and accessibility. Participants noted that FERPA (Family Educational Rights and Privacy Act) impacts sharing.

In this post, we discuss:

- Logistics of using Zoom for synchronous teaching
- Accessibility
- Writing on the board, online
- Ways to help students stay engaged
- Asynchronous teaching
- How to hold office hours
- Sharing recordings and FERPA
- Open questions

We are writing this because we believe that other mathematics instructors can benefit from the ideas shared by the participants. However, we are not speaking as experts and these are crowd sourced ideas rather than MAA or AMS recommendations! We are only speaking as peers who have had some experience, and who want to share key things that were helpful to us when we began teaching online.

First, many people talked about running courses on Zoom. This is partly because the MAA uses Zoom and we were using it for the meeting. We don’t know in the coming days how Zoom’s infrastructure will hold up with increased demand. However, if you are able to get Zoom or another platform to work for you, here are some things to consider:

- There are different grades of platforms, based on how much you pay. For instance, Zoom’s free version only allows for meetings of a maximum of 40 minutes. Their least expensive paid version (Pro) allows for 24 hour meetings, the option to record meetings. Webinars have some nice features for large groups, but it seems like only Meetings have the whiteboard and breakout room features.
- Make sure that you are in a well-lit room. If you are backlit, the camera software may result in a view of you where your face is completely in the shadows. This means that students who need to read lips won’t be able to. This also means that students will lose nonverbal communication cues.
- As well, practice keeping your head still. If you move around, the camera software can change what is light and dark, which can be distracting or result in backlighting.
- Practice logistics like muting microphones, sharing screens, recording, making sure you know how to see all participants, getting used to different “views” available, or grouping people.
- On Zoom, you can share the window for only one application or share your whole screen. This means that participants may not be able to see your mouse cursor, and it also means that they may or may not see information from any other windows. Watch out for privacy issues with email or pop up notifications.
- It can be helpful to work with a computer attached to an external monitor to give you more room to organize screens.

Once you are comfortable with a conferencing platform, here are some in-class tips:

- When starting class, you will need to allot extra time in the beginning as people “enter” the virtual room. Have some things for students to do during this time. For instance, Ray suggested that you can vary from journaling (“How’s the course going?”) to more personal (“What music should I listen to?” “What’s your favorite TV show?”) . It might be even more important in the next few weeks, as we isolate ourselves, to ask personal questions. Georgia Stuart suggested “quiz yourself” prompts, which could lead to students coming up with questions to ask you and help you see where students are. To show these prompts to students right away, you might put the prompt on a PDF or other kind of file, and screen share that file. (This is similar to the function of “warm up” problems that students are expected to do as they enter the room.)
- On Zoom, the chat box only shows you chats that you were present for! If a student logs in after 10 minutes, they will not see any chats that were sent in the 10 minutes before they arrived. So if there are important links or information to share with the class as a whole, you cannot rely on chat to do this; you need to announce to the class or put an announcement in your course management software.
- You might prepare some documents ahead of time to share, such as PowerPoint or Beamer slides, a google doc, or images. You can use these documents to have prompts for the class, or equations or diagrams. Test run any slides before using them, so you know what they will look like to you and your students. You might schedule a practice session with some friends where you take turn being the host or participant.

Across the group, we had experience with:

- Zoom white board
- Wacom tablet
- iPad with apps Notability, ExplainEverything, or Doceri
- Surface Pro tablet using PowerPoint annotation or Microsoft Whiteboard
- Webcam pointed at paper

Among these, none stood out as a crowd favorite; different instructors had different preferences based on what was easier personally or more familiar.

For students who need to read lips, you will want to make sure that you are in a room with good lighting, and to keep your head relatively still. As discussed above, this is a function of how Zoom and camera software operate together. Appropriate lighting serves everyone well since it can reduce eye strain.

For students who need captioning, options include using transcription services such as otter.ai (free), temi.com ($0.25/minute), or captioning services on YouTube. With all these solutions, you will need to edit the transcript manually, especially for correct technical language use; and you will also want to make sure to share video so as to be FERPA compliant. Because this was new territory to many of us, we were not as a group sure of what is FERPA compliant or not. In general, the more private, the better; and you may want to check with your institution. (If you do so, please share what you find with us!) Video captioning can also be helpful, even in a mathematics course, for students who are language-learners.

We did not talk about students with visual or other accessibility needs, but of course these are important considerations.

Learn about carrying out ADA requirements through your institution. Many institutions have support staff to help building and conducting online courses. The ADA’s website also has guidance, for instance on website accessibility.

Students who have been sent home may not have the same kind of time or space or bandwidth that they previously had. For instance, they may have additional household responsibilities, or they may not have reliable internet access. Here is a questionnaire for students, adapted from one developed by Christina Weaver:

- Where do you expect to be from now through [end of the semester]? (City, State, and time zone). If you expect to be in more than one city/state, please list that too.
- On a scale of 1 (really slow/unreliable) to 5 (really fast/reliable), how would you rate the internet connection that you expect to have while away?
- Do you expect to be available during all of our usual class times (keep in mind time zone)?
- Are there any online meeting software / video sharing apps (other than Canvas/Google) that you use and recommend? If so, tell me about them!
- What else do you want me to know? (You can tell me logistics here, or additional responsibilities, or anything you might be feeling right now.) I will not share this information with anyone else [at our college] without your permission. [Note from Ray: please be aware if you are a mandatory reporter that you may be obligated to report certain things.]

To help students stay engaged, Georgia recommended:

- Have regular due dates.
- Check in with individual students over the semester.
- Have a platform for students to talk to each other and answer each others? questions.
- Create learning activities that require students to interact at specified intervals. Be explicit with due dates.

These serve to help students’ executive functioning, the skills needed to plan, prioritize, focus, remember instructions, and handle multiple tasks. Georgia found this resource very helpful for thinking about teaching strategies: Universal Design for Learning’s resource on Executive Functioning in Online Environments.

For student chat platforms, Georgia has used Microsoft Teams and the paid version of Slack, and allows for picture sharing and video conversations. Ray recommended Piazza. Teams were provided through Georgia’s university, and Slack (paid) were deemed FERPA compliant by her university, but other universities may decide differently. You may need to consult with your university before using a new online tool that may expose your roster information.

You want to be careful in public spaces such as chats, to be kind and constructive. Ray and Georgia both shared stories of comments they thought were gentle, but chilled chat discussion to the point that students did not use it anymore. Both strongly urged us to stay away from language such as, “You forgot to ____” or “You didn’t _____”.But even comments such as “Did you remember to ___?” have chilled chat rooms.

A method that Georgia has found helpful is to chat with a student privately about it, and then ask that student they would be okay for her to respond publicly, or to delete the comment. When students are responding to each other’s questions (which can be a huge time-saver), setting expectations about kindness can be key.

In general, don’t ask yes/no questions nor instruct; ask open ended questions. Why did you do it like that? Can you explain to me why that works?

Ray recommended reading through chats at least before quizzes or exams, to see whether there are comments or concepts that you want to address. Although you want to read through the chats regularly, there isn’t a need to check it every second. Perhaps let students know that you will aim to check at particular times. You can think of this as part of your ‘office hours.’ Some people may prefer to disable this function because of the extra time consideration.

All the following advice comes from Georgia who has taught a completely asynchronous course, where students learned to program in R and learned concepts of statistics.

She recommends developing videos, no longer than 12-15 minutes long. This is really important, so that students can focus on only one major idea per video.

In her experience, students in asynchronous courses can get behind easily if assignments are too big. To help with this, she uses the strategies listed above for course engagement. Small chunks of work with very explicit deadlines can especially help.

Online courses can blur any work-life boundary, especially for instructors that aren’t used to teaching online. With Microsoft Teams, students may send questions at all hours of the day. They may be working on their homework at 2am. You want to be careful to set boundaries, say giving yourself 24 hours to respond, especially because it’s a chat platform where usually we expect responses right away.

You can hold office hours on Zoom using a personal meeting room. Dianna Torres and others said that they have times that students know that they will be on. Dianna’s students share their computer screens with her to ask questions about homework. Some have students hold their work up to the camera.

Yvonne has held office hours by appointment on Skype, trying to schedule at least two students at a time so they can have each other to talk. Something she has found useful is to have a movable webcam. She uses this to point at paper that she writes on, and also to give “privacy”. For instance, if she wanted to give a student time to work out an idea, she has found it helpful to ask: “Do you want some time to think about this by yourself?” Usually, students say that they do. She then asks, “Would it be helpful for me to face away?” Students usually say that it would be helpful. She then says, “I will leave the audio on to help me follow up, but otherwise I will give you this time.” She then turns the webcam to point to the keyboard and listens for a lull in conversation and also for points that she wants to probe.

Briefly, and to over simplify, FERPA means that you cannot share students’ identities, assessments, or grades with anyone but the student. When teaching online, you may be recording sessions that include student talk. Sharing these sessions may violate FERPA.

According to Zoom’s website, “Zoom enables FERPA/HIPAA compliance and provides end-to-end 256-bit encryption.” To share Zoom videos, use course management software (e.g., Canvas, Blackboard). Check with your institution about FERPA requirements, ADA, and resources available to you and to students. Some universities have online tutoring available, for example.

- Assessment at scale: How do we minimize cheating on exams, especially for large courses?
- Math resource centers: How do we convert tutoring centers to online platforms?
- Group work: How do we do group work and keep students engaged?
- FERPA: What are all the implications? Legal ramifications of incidental disclosure of information are still unfolding. Instructors should probably contact their University legal department for their institution’s guidelines.

A few words on assessment. This was perhaps one of the most sobering moments of discussion. Several people asked, “Are we setting up students for failure if we allow cheating?”

We realized people are ready to have thoughtful conversations about why we test, who is harmed by issues with academic integrity, and where things should be on the scale of “let it go” to “strict” in this unusual time.

In some large lecture courses, some instructors have seen students ace online homework while failing proctored exams. While there are many reasons for this, including test-induced anxiety and impostor syndrome, there is unfortunately also the reason that students have a friend next to them who answers online homework questions for them.

On the other hand, Yvonne has had the experience of homework grades predicting both midterm and final exam grades, even when homework allowed collaboration, the midterms were proctored, and the final exam was take-home.

We hope you find at least some of this helpful. We also hope the act of gathering, sharing ideas and concerns, and struggling together will be constructive. May you find the community and support you need as we work through these next months.

]]>“I am sad this class is going to be over,” said one student. “What am I going to do with myself?” asked another during the last week of an Intermediate Algebra class that I taught last summer at the Lincoln Correctional Center (LCC) with Meggan Hass, then a University of Nebraska graduate student.

Meggan and I were sad, too. It’s not often that we hear these types of comments from students, but as I have learned, the unexpected can happen when one teaches in prison.

Here is my story.

It started with a visit by Bryan Stevenson, author of *Just Mercy*, to Nebraska Wesleyan University, where I work. Stevenson is a lawyer who defends incarcerated individuals, many of whom are on death row. In his talk, he urged us to:

“Get proximate. Get uncomfortable. Change the narrative. Have hope.”

I was sold. In the coming months, I put in a request to adjust my Spring 2018 sabbatical.

When Stevenson called for the audience to “get proximate”, he was encouraging us to be closer to those who are suffering or excluded. In response, I arranged to teach classes at the Nebraska State Penitentiary and at LCC during my sabbatical.

The classes were non-credit bearing, so I had liberties in the content. I wanted to pick a topic that wouldn’t require a strong math background but that would be interesting. The 8-week course I developed was “Combinatorics and Probability.”

Not knowing anything about the prison system, I asked basic questions of corrections staff: Will paper and pencils be provided? (yes) Can I move around freely in the class? (yes) Are calculators available? (yes) Can they work in groups? (yes) What can I bring into the facility? (notes, book, pencils, calculator) Can I bring dice and playing cards? (no) How can we determine who is accepted into the class? (basic math proficiency, no recent misconduct)

I was scheduled to meet my students for two hours each week, but the prison staff would regularly let the classes run long. For my students, the longer, the better. The chance to think about something different and be away from the drudgery of the units was welcomed. My students never missed a chance to tell me how much they loved to learn and how education was what they needed more than anything.

Although I had carefully planned the course before it began, I made big adjustments once it started. My students’ backgrounds were quite disparate, creating some unease for me. But I adjusted the material by giving them more time to “count” things using brute force, with the idea that it would facilitate the process of making conjectures. I also asked students with stronger backgrounds to help those with weaker ones. That worked beautifully.

Towards the end of the 8 weeks, one student asked me what surprised me most about teaching them. He wondered if I was scared. No, I wasn’t scared. However, I had a preconceived notion that asking students to participate would make them feel vulnerable; that made me uncomfortable as I walked into class on the first day. The big surprise for me was how willing they were to ask and answer questions. In fact, my main time management issue was trying to address all of their questions. Most questions were about the specific content, but some extended into vocabulary and notation: “Did the question mark first show up in mathematics or in other writing?” (I always researched those questions I could not answer and reported back to them. Incidentally, the history of the question mark is fascinating, and I encourage you to research it.) To cover the main concepts adequately, I occasionally omitted a few examples.

The centerpiece of Bryan Stevenson’s directive to “change the narrative” is to talk honestly about the historical roots of racism and poverty. For me, changing the narrative was more personal. It was about refocusing where I put my energies; I wanted to continue teaching in prisons.

The following academic year, I taught a course in Algebra 1 at LCC. At the prompting of the assistant warden there, I subsequently co-taught, with Meggan Hass, an Intermediate Algebra course in Summer 2019, through a program at the University of Nebraska-Lincoln (UNL), supported by private funding.

Most of the students who had taken Algebra 1 with me applied to take the Intermediate Algebra course. All applicants needed to have a GED or high school diploma. Beyond that, UNL gave me full discretion in choosing participants, so I created a simple application form and held interviews.

Among other things, I asked about the hardest thing that they had ever done. The responses gave a sense of how varied their backgrounds and perspectives were: “Working at Long John Silver’s,” “Your Algebra 1 class,” “Fitting pipes for an oil rig.” But each talked about these hard things with the kind of conviction that conveyed their seriousness about this educational opportunity.

As Meggan and I began preparing the course, a question we had to answer was how much “discovering math” we wanted to put into our course. For example, we needed to decide if we wanted our students to “discover” the rules of exponents working with others, or if it would be better to offer more guidance. The challenge I had had when letting the students in Algebra 1 spend time discovering math was that some students got distracted or overwhelmed. So, we made worksheets for each section and worked through these as a class, asking students many questions along the way. As an example, to show that 3^{4}3^{5}=3^{(4+5)}, we asked our students to write all of the factors of 3 on the left side of the equation to see why it would be the same number of factors of 3 on the right.

I have a fascination with words and so for each class period, we had a “word of the day.” We delighted in vocabulary such as *vociferously*, *chagrined*, *vexillology*, *verisimilitude*, *polyglot*, *polymath*, *apoplectic*, *belie*, *ebullient*, *capacious*, and *pacific*. Riveting discussions ensued with some of the students reaching for a dictionary to uncover the word origins.

On the last day of Combinatorics and Probability, we held a certificate ceremony. I prepared certificates with one of my favorite quotes:

“We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills”

– President John Fitzgerald Kennedy

I chose this quote because it connected to their eagerness to learn and their willingness to challenge themselves.

The students asked if they could speak. They talked about how important education was to them. One shared that few people are willing to go into the prisons to volunteer and that it was significant when people did. They asked me to tell people about what I was doing so that it would encourage more to do the same. It was emotional.

These thoughts were echoed at the end of the Intermediate Algebra class, too. In this class, there were 26 homework assignments, 3 quizzes, 3 exams, and one final exam. Of these, only one homework assignment was not turned in by one student. In fact, this student had completed it, but accidentally left it in his unit. None of the homework turned in was incomplete and it was done with exceptional care. Attendance was almost perfect; we had one excused absence and no unexcused absences during the entire course.

On the last day of class, the students presented Meggan and me with homemade pop-up cards. They each wrote a personal note of gratitude. We were equally grateful to have been their teachers. In their course evaluations, they talked about the class teaching them much more than math – one said he finally could believe in himself. They shared that the class made them feel like human beings.

It gave them hope.

There are many reminders that you are in a prison when you teach there. But, just like in any class I teach on my campus, each student comes to class with a different history, each student learns differently and each challenges me as a teacher. In my prison classes, I was stretched as a teacher in consequential ways; I had to adjust quickly to different backgrounds, find different ways to explain material and rally around my students to help them build their confidence. I came away seeing the immense value there is in offering incarcerated individuals the opportunity to learn.

* * *

*Note from the author*: If you think you might be interested in such an opportunity, and you would like an experience you might fit into a summer, I am happy to share my course materials from Combinatorics and Probability (class worksheets and homework sets). Email me at kpfabe@nebrwesleyan.edu. They are imperfect but can serve as a starting place.