I did not want to present. Someone had selected my solution to a geometry problem to present at a Mathfest 1996 session. I wasn’t sure who this person was, but I knew already that I did not appreciate them. I was 16 years old, my father was ecstatic at the honor, and I wished my father had never found out, because then I could have played hooky and no one would be the wiser.

That summer, I was a student at the Canada/USA Mathcamp, a residential program whose name describes its purpose. Mathfest and Mathcamp both took place on the campus of the University of Washington-Seattle that year, and someone had the brilliant idea that a few high school students would be made to present their solutions to the problem in the above figure – which featured in the entrance application to Mathcamp – at Mathfest 1996.

When I met my two co-presenters, I found out that none of us wanted to present, nor had we any idea how to present even a measly 10 minutes of mathematics, nor had anyone told us anything except that we were supposed to show up at a particular room on the appointed day. (Mathcamp was a young program at the time, formed on a whim and a dream, with all but nonexistent infrastructure at the time.) The moment was 10 days away, and we dreaded each closing minute.

During this time, we attended daily problem solving classes as a part of Mathcamp. Unbeknownst to us, we had a world class problem solving instructor: Loren Larson. All we knew was that he gave wonderful explanations. I proposed to my co-presenters that we ask him for help. We wondered if we would be bothering him, but our increasing fear of embarrassment (and my filial piety) drove us to approach him after class. To our surprise, he welcomed the interaction and met with us that evening, and again the next day.

I can’t remember all of what he told us, but I do remember that he made presenting mathematics seem fun. In retrospect, he must have granted us immense grace and patience. He explained that we should begin by introducing the problem to the audience, and involving the audience in the process of problem solving. In such a short presentation, this could mean walking through wrong solutions such as drawing the diagonals of the square, and then posing a rhetorical question to the audience such as, “Drawing the diagonals doesn’t work, but would it be possible to find 4 different triangles that would work?” Then, after the audience was invested in the problem, *and only then* — should we walk through our own solutions. Larson also suggested ways to bring flair to our presentation – that in key steps, we should find a way to build up to it, and perhaps even put on a show of surprise as we unveil the solution.

His kindness gave me the momentary gift of relief, even excitement, at the presentation to come. He also gave me the gift of finding joy in talking about mathematics in front of an audience – and eventually, in the years to come, of teaching mathematics.

A few years after Mathfest 1996, as a junior in college, I became a teaching assistant for an Ordinary Differential Equations class, taken primarily by sophomore-level engineering students. From talking to my dormmates, I knew that the students craved examples of how solving equations would ever show up in engineering. From lessons from Larson, I looked out for ways to create anticipation and surprise in the mathematics. I always made sure to explain the problems, invite students to do some initial thinking, and look for ways to have fun. While explaining damped systems one way, I pointed to the springs in the door of our classroom. Then, I explained that were the system not damped … slam! When we solved second order constant coefficient ordinary differential equations, I planned one board to illustrate an electronic circuit, another board to illustrate a spring, and a third board for a table to showcase the beautiful parallels between these systems.

Lee Shulman, in his Presidential Address to the 1985 American Educational Research Association, described “pedagogical content knowledge” – a kind of specialized technical knowledge that teaching requires – as including “the most useful forms of representation of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations – in a word, the ways of representing and formulating the subject that make it comprehensible to others”.

Looking back, when I first began teaching college-level mathematics, I interpreted Larson’s advice as looking for ways to develop pedagogical content knowledge. When the idea was a problem, the teacher needed to find a way to represent the problem so as to be comprehensible to the students. The main technique I picked up from Mathcamp 1996 was to pose reasonable entries into solving the problem, and then show why these might fail or show promise.

Recently, scientists Alan Cowen and Dacher Keltner have suggested that there may be at least 27 distinct varieties of emotion. Citing emotions from their framework, I looked for ways to perform craving, amusement, and surprise for my students, in hopes that they would feel these emotions vicariously through my actions and emoting. Ultimately, I hoped they would leave with awe, calm, and aesthetic appreciation for the mathematics.

I continued developing explanations as aesthetic and emotional performance when I became a graduate student. When I taught ordinary differential equations again in graduate school, this time as the sole instructor, I had built up a mental library of examples, analogies, and illustrations. Shulman observed, in that Presidential Address, that “[s]ince there are no single most powerful forms of representation, the teacher must have at hand a veritable armamentarium of alternative forms of representation, some of which derive from research whereas others originate in the wisdom of practice.” I did not have an armamentarium, but I felt that I was beginning to build one.

Then, in my third year in graduate school, some friends and I launched the Davis Math Circle (which, I found out recently, still exists today). In sessions I taught there, something about my formulation of teaching felt wrong. I wanted the high school students there to experience joy and beauty in mathematics. Yet, the concept of teaching-as-explanatory-performance didn’t fit what seemed to work for the students. What seemed to work best was to give them problems to work on and let them discover the patterns, albeit with scaffolding from the instructor and assistants in the room. I began separating “routine teaching” from “Math Circle teaching”, writing off “Math Circle teaching” as not real teaching and also a mysterious phenomenon. At the same time, I thought that my students who experienced my “routine teaching” would probably benefit from “Math Circle teaching”, but the styles seemed so irreconcilably different that I gave this up as a pipe dream.

After graduate school, I took a post-doctoral fellowship at the University of Michigan, where I joined Deborah Ball’s group for a few years and consequently participated in the Elementary Mathematics Laboratory. This is a summer “turnaround program” where rising fifth graders are selected by their teachers to participate, on the basis of struggling with mathematics. The students in the EML are taught for two weeks by Ball. Lesson design is led by Ball and involves a team of research associates, staff, and graduate students. The content of EML has changed over the years. When I participated, lessons focused on fractions and a “train problem”, a combinatorial problem to which the solution is that no solution exists. Each year that this problem has been used, the students persevere for days to find that there is no solution, and are proud to present their solution of no solution.

The EML features public teaching: behind the students’ seats are an audience of 20-30 adult participants of the program witnessing the teaching in real time. We participate in pre-briefs and de-briefs after each teaching session. Participants range from education grad students to school district leaders to school teachers to mathematics faculty.

My first year participating in EML, another participant pointed out the difference between “teaching *mathematics*” and “teaching *students* mathematics” and pointed out that “good teachers know that they teach *students* mathematics”. Everyone else seemed to concur immediately. I was mystified by this seemingly semantic distinction. I didn’t understand what difference everyone else seemed to instinctively understand. After all, if there are students in front of us, are we not teaching *students*? And shouldn’t the subject have primacy, if we care about and love the subject, as I did? So why emphasize *students* over the *mathematics*? If there is no mathematics, there is no teaching.

At the same time, I couldn’t help but notice that Ball seemed to be teaching intended and substantive content in ways that honored the students’ contributions. She was not performing explanations. Yet the students craved learning and they experienced awe.

The next year, The Algebra Project established a site in Ypsilanti, Michigan. The project focused on a block class of freshman algebra. I began to meet every Saturday with the teacher of the class, a literacy specialist, the math district specialist, and a mathematics faculty member from Eastern Michigan University. Sitting for hours in a coffeeshop, we planned the next week’s instruction. I did not have a car, so on Tuesdays, I woke up at 5am and biked to Ypsilanti, where there was a 24-hour Starbucks less than a mile from the high school where the Algebra Project had a class. I worked at the Starbucks until 20 minutes before first period, when I biked to the high school to observe and assist with the instruction.

The Algebra Project curriculum was nothing like anything I had seen before. It was based on the recently passed Bob Moses’ curriculum design. First, students share a tangible experience. The most well-known example of this idea is the “Trip Line”, where students in early Algebra Project sites, in the Boston area, rode the Red Line T –- a subway line where the train goes back and forth along the same set of stops –- and mapped their trip along this subway line to numbers on the real line. The subway line, like the real line, has only two directions to travel: backwards and forwards. Then students use pictures and writing to represent their experience. From here, the teacher helps students connect their own informal language to formal mathematical language, and finally to symbolic notation for algebra.

I was astounded at how the Algebra Project curriculum wove in students’ stories and deep mathematical ideas. The phrase “teaching *students* mathematics” began to make more sense, but I still wasn’t comfortable enough with it to use it. In this era of my life, I began to always say that, in my own classes, I “taught mathematics” –- out of a sense that I wasn’t actually teaching *students* mathematics.

However, I began to experiment with adapting the curricular design principles of the Algebra Project into my own college teaching. For at least some isolated days in my courses, I looked for ways to build in tangible experiences, and to use students’ informal language as a bridge to the formal mathematics. In real analysis, I asked students to draw “shadows” that neighborhoods cast on the *y*-axis, and used this metaphor to develop the ideas behind continuity and epsilon-delta proofs. In abstract algebra, I asked students to build sets using certain multiplication and addition rules, and used their experiences to define ideals and to raise the question of classifying ideals in various rings. I noticed that the content of these days seemed to stick better with the students, despite the fact that I could not usually perform explanations on these days. Moreover, on these days, the students seemed to tap into the emotions I wished for learning mathematics to bring.

I began to wonder if “teaching *students* mathematics” meant that you were teaching in a way that was highly responsive to students. The mathematics was there, but in a way that was calibrated to the particular students in front of you, in that particular moment of time. It’s not that mathematics is being sidelined, but rather that the teacher is finding a way to integrate the students’ thinking with the intended mathematics.

Each year, I found more ways to incorporate tangible experiences, and bridges from informal language to formal language. Although I have never been able to offer this process for all mathematical topics – and to be honest, I’m not sure that I ever will – I have come to believe that the more that I can offer this process, the more my students appreciate how mathematics is done. It’s more than awe and craving and surprise and joy, although these are present. It’s also an appreciation for the very process of mathematics, of exploration, discovery, conjecture, and proof – and certifying explanations with others in the class. My performance now is not performing explanations, although this still happens on occasion. Rather it is about taking on the role of what I imagine a metaphorical being sitting on a shoulder would say, if this being cared about and loved mathematics, and could help another find this love and care, with humor.

In the intervening years, I have sometimes returned to Shulman’s quote, and to my first encounter with presenting mathematics. Looking back, I wonder if Larson never meant for me to take away the lesson of teaching as performing explanations, but rather the lesson of teaching as inviting others to the process of mathematics. When he taught us to present mathematics, he wasn’t teaching *presenting* to us; he was teaching *us* how to present. More than that, he taught us a slice of what it meant to teach *students* mathematics.

When I first sought to develop pedagogical content knowledge, I believed there might be an infinite collection of powerful forms of representation, contained in a form resembling the “The Book” that Paul Erdős sought to read. Now, when I stumble upon a sequence of activities that seems to work year after year, I wonder whether I’ve found a sequence from this collection. At the same time, I also wonder if, whether or not such a collection exists, teachers begin to build their armamentarium through teaching *students* mathematics, by building bridges to the process of mathematics from what students do and say. Along the way, there may be joy. It’s not about students experiencing vicarious joy through my performance. Instead, the joy comes from the students, and I experience happiness and awe at the seeming miracle that students can interact with a combination of tangible experiences and abstract ideas and find joy.

* Acknowledgements.* I am grateful to Rachel Funk for the conversation that inspired this post, and to Mark Saul and Ben Blum-Smith for editorial feedback.

Covid-19 has left teachers seeking topics that are both engaging and lend themselves to online instruction. As a guiding force for the measures that have reshaped our lives, epidemic modeling stands out as natural. For teachers at the secondary level and those involved in teacher education, this leads to the question: How can an understanding of epidemic modeling be made accessible to students at large?

From the vantage point of evolutionary biology, viewing epidemics as a form of natural selection is a good place to start. An ability to reproduce and mutate rapidly would seem to give the virus a distinct advantage. The tools Humankind can bring to bear include (1) human intellect and (2) a capacity for social organization. Both have figured prominently into efforts to manage Covid-19 and make an appearance in the model to be developed below.

Modern epidemic modeling began with the S.I.R. model created by Kermack and McKendrick in 1927, introducing the use of S, I, and R to designate *susceptible, infected*, and *recovered *demographic variables. Here we consider a population P of fixed size (it undergoes neither births nor deaths, and those recovered enjoy total immunity). In set theoretic terms, this can be thought of as

P = S ∪ I ∪ R

where people pass from S to I and from I to R at prescribed rates. The model yields values for all three variables as a function of time.

The fact that S.I.R. was and continues to be described in terms of a system of differential equations poses a barrier to making its insights accessible to students and citizens at large. The calculus-based approach[1] to our version of S.I.R. calls for an application of Euler’s method to this system. However in the digital age, basic algebra combined with spreadsheet apps such as Googlesheets enable us to sidestep such technical challenges.

While a variety of institutions have addressed the issue of difference vs. differential equations in years gone by, computer technology and Covid-19 have changed our world. Not only is it now possible to introduce such models to students at the secondary level, there is a pressing public health reason to do so. Without claiming that the study of epidemic modeling will serve to inoculate society against disinformation, a basic citizen understanding of such tools seems essential for efforts to control the spread of Covid-19.

A broadly inclusive formulation of S.I.R. might begin with a problem such as the following. A hospital has an admissions office and a discharge office, both of which keep daily records. Given a Monday morning population of 43 patients, use the hospital’s admission and discharge records to calculate the patient population for the remainder of the week.

Extending this problem to an entire month sets the stage for introducing functional notation. Here one defines I(n) as the hospital population on (the morning of) the n^{th} day, a(n) as the number admitted that day, and d(n) the number discharged. This can be combined with basic spreadsheet instruction to implement the recursive scheme

I(n+1) = I(n) + a(n) – d(n)

By way of relating this simple exercise to the S.I.R. model, we consider a cruise ship that leaves port with a population of 100 passengers, 90 of which are susceptible to a virus and 10 of which are recovered and thereby immune (the crew are all immune). One morning two of the susceptible passengers are diagnosed as infected and the ship is isolated at sea.

To create a record of the ensuing epidemic, the captain uses S(n), I(n), and R(n) for the number of passengers who are susceptible, infected, and recovered (and thereby immune) on the morning of the n^{th} day. But instead of requiring a head count each morning, she keeps daily records of the numbers admitted to and discharged from the sickbay. This leads to three “hospital problems” and the equations

S(n+1) = S(n) – a(n) S(1) = 88

(*) I(n+1) = I(n) + a(n) – d(n) I(1) = 2

R(n+1) = R(n) + d(n) R(1) = 10

So how does one relate this intuitive set of formulas to S.I.R.? Well, the point of epidemic modeling is to *anticipate* the course of an epidemic rather than deducing its history. In the context of (*), this calls for estimating the values of a(n) and d(n) on the basis of the morning head count S(n), I(n), R(n). With all five of these numbers at hand we can use (*) to calculate S(n+1), I(n+1), R(n+1), and then to estimate a(n+1) and d(n+1), and then to …

For the captain of our cruise ship, such a tool might be used to take actions aimed at controlling the epidemic to come. For us, the process of anticipating a(n) and d(n) is where important insights are to be gained. Rather than advanced mathematics, it calls for an understanding of human behavior as well as the disease being modeled. Consider the following line of reasoning:

A disease spreads as the result of contacts (meetings) between pairs of infected and susceptible persons. If a population P of (constant) size N averages m meetings/day, then its infected persons will have I(n) × m meetings per day. But only a fraction of these will be with a susceptible person. Of the I(n) × m meetings, I(n) × m × (S(n)/N) will be with a susceptible person. But not all of these meetings will actually transmit the disease. Letting p denote the probability that a meeting between a susceptible and an infected person will actually transmit the disease, we arrive at

a(n) = I(n) × m × (S(n)/N) x p

S.I.R. makes shorter shrift of d(n).

If the disease lasts r days, the number of patients discharged on the n^{th} day can be taken as

d(n) = I(n)/r.

In the resulting model, the constants m, p, r embody human efforts to control an epidemic. Closing schools and businesses serves to reduce “meetings/day”; masks and social distancing serve to reduce p; medical research can reduce the “recovery time” r. These observations breathe life into the implementation of (*) with

(**) a(n) = I(n) x m x (S(n)/N) x p and d(n) = I(n)/r

In the case of our cruise ship, setting m = 6, p = 0.1, r = 12 and implementing a routine spreadsheet simulation for 60 days yields

By way of building on this classical image from epidemic modeling, one can use the inequality a(n) ≤ d(n) and (**) to arrive at S.I.R.’s condition for herd immunity:

S(n)/N ≤ 1/(mpr).

Efforts to temporarily reduce the size of m, p, and/or r serve to ease the requirement for herd immunity. However, a premature easing of these requirements can serve to give the virus free play. In the graph below, our cruise ship captain has imposed rules that reduce m = 6 to m = 1 at n = 5 and returned to m = 6 at n = 30.

In reducing the infection peak from 50 patients to 36, the captain has also deferred the peak’s occurrence from day 16 to day 43. This may allow for the arrival of vaccines that can be used to modify such a scenario.

The question then becomes, is there value in such work? Our answer is “yes.” It allows

K-12 students to begin a journey that uses mathematical modeling to examine issues that affect their daily lives. It puts students in the driver’s seat by allowing them to use data and to draw inferences as they manipulate variables based on different assumptions. And finally, it makes creative use of technology that is free and readily available.

*These ideas constituted the first Zoom session of an online special study mathematics course we offered at UC Davis this past Winter quarter. They will also be at the heart of an in-person First Year Seminar to be offered at UC Davis next Winter under the heading “Covid-19, By the Numbers”. We will be glad to share further thoughts and materials with anyone having similar interests. *

[1] https://www.maa.org/press/periodicals/loci/joma/the-sir-model-for-spread-of-disease-introduction

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This summer marks the thirtieth year since the end of the Soviet Union. It also marks the passing of one of the great figures of Russian mathematical culture, Nicholas Nikolayevich Konstantinov. This note concerns both events, but cannot do justice to either. Rather, I will here give some personal reminiscences that might contribute to the picture, but not find a place in the historical record. I leave to other sources the task of a more comprehensive account. Here’s my story.

The year was 1987. The Cold War was still smoldering, but no longer raging. I received a phone call from an American teaching colleague: “I got an email message for you from one Professor Konstantinov in Moscow.”

Just the fact that this message had arrived was remarkable. The World Wide Web had yet to appear. Email was new and laptops rare. And it was not yet clear that the internet could form a bridge between the two camps in the global political stalemate. Was a graph of the Eastern and Western computer networks even connected? How did a colleague from Moscow contact me? And why?

I had known for a long time about the remarkable flowering of mathematics in the USSR in the postwar years. Reading Russian, I had access to materials in that language: a subscription to *Kvant (* http://www.kvant.info/ ; http://kvant.mccme.ru/), the journal for pre-college students started by Kolmogorov and other scientists. I also subscribed to *Matematika v Shkole* (http://www.schoolpress.ru/products/magazines/index.php?SECTION_ID=42&MAGAZINE_ID=34945), a journal for teacher of mathematics.

Finally, I possessed a hard-earned personal collection of Russian texts and problem books. Hard-earned, because at that time one had to subscribe in advance to Soviet books that would then ‘be published in future, through a newsletter that listed all such books. I would comb through it weekly and order the ones that looked interesting. Sometimes they came, and sometimes they were of great interest. And sometimes I got a book in Hindi, or a treatise on diseases of cattle. Luckily, the books were uniformly inexpensive. I had access only through the newsletter, *Noviye Knigi SSSR *[New books from the USSR], not through recommendations from working mathematicians or teachers. Through these sources, I learned of the work of Konstantinov and his colleagues.

So what did Nikolai Konstantinov want? That was not clear. Mostly, he wanted to make contact. He had heard of my work, and I of his, through various meetings and publications. I replied to his email, but it was a while before the next contact.

In 1989, I was at an international meeting in Waterloo, Ontario, when someone slid into the seat next to mine and addressed me, in Russian. “What is he saying?” the man inquired. I whispered back a short summary, wondering why he had somehow assumed I would understand him. He replied, as if reading the confusion in my mind, “I am Konstantinov.” He had already read my own name badge.

We sat through the presentations—none of them in Russian or German (Konstantinov’s second language). Then we sat around and over dinner talked about our work, about the difficulties in each of our environments, and about possibilities of collaboration.

Our next contact was in 1990, which was to be the last full year of the existence of the USSR. The Iron Curtain had lifted from Eastern Europe, and many thought that Russia would be next. I got a call directly this time, from someone in Brooklyn. By then a large community of Russian emigrés, mostly Jewish, had settled in New York, and were about to make a significant impact on the math education scene. But not quite yet.

The person who called me was one Irina Speranskaya, who worked in Moscow for a government agency in the nascent area of trade with America. She was visiting New York, and brought me a new book from Konstantinov, with an offer. If I came to Moscow with a group of American students, we would have a Russian-style summer camp, with all expenses (once we arrived in Moscow) borne by them. Miraculously, the National Science Foundation was willing to fund the trip, and I found myself leading 20 US students and five teachers to Moscow for an immersion in Russian pre-college mathematics. I have written accounts elsewhere of this trip (Saul 1992), which contributed to the introduction of a number of Russian cultural traditions to the US.

Well, the Soviet Union fell—just two weeks after the conclusion of the NSF-sponsored summer program—and communication with Russian teachers and mathematicians became more and more common. My personal experience was duplicated by many others. More American teachers and mathematicians came in contact with Russian emigrés and started programs—math circles and math camps—inspired by their experience. And today, if you are reading this essay in the math department of any American university, you can probably walk down the hall and check its accuracy with a colleague who grew up in Russia or the USSR.

Part of what I discovered in the waning USSR, and which Russian mathematicians had long understood, was the remarkable nature of the Soviet mathematical community. It was more of a subculture than a community, or even a counter-culture to the official totalitarian ideology. The origins and characteristics of this phenomenon have been described in detail elsewhere (Gerovitch 2013, Karp 2010, Polyakova 2010, Sossinsky 2010). In brief, the Soviet government attempted to control intellectual life in the country. The arts were heavily, and famously, politicized—even music, perhaps the most abstract of artistic communications. This control was direct and could be brutal. Control of the sciences was often subtler. Certain lines of investigation were encouraged, others stifled. Advances in genetics and computer science, perhaps the two most exciting branches of science in the post-war era, were discouraged or even prohibited. The social sciences were likewise politicized. The physical sciences were largely put at the service of the military (Chan 2015). But even in more academic environments, the need for laboratory facilities was a powerful lever of control.

Mathematics, on the other hand, offered a refuge. One needed no equipment and was given little direction. The applications of one’s work were often sufficiently far removed from the work itself to make the connection between the two difficult for outsiders to fathom. So active minds flocked to mathematics, minds which could have found occupation in other areas had totalitarian forces not been at work. Doing mathematics even became an act of rebellion, of silent refusal to honor the needs of the government. And all this could happen without physical or verbal expression, just by acting as mathematicians or students of mathematics.

So, for example, the social and professional lives of a mathematician were often the same. Summer camps for students, after-school math circles and study groups, all became part of a tradition of enjoying mathematics as people were pushed together by the sometimes harsh totalitarian intellectual climate.

Konstantinov was both a product and a bearer of this unique mathematical culture. I offer here just a few glimpses, from personal recollection, of how it felt to be a part of it.

After the 1991 summer camp, I was invited to to the summer seminar of the International Tournament of the Towns (https://www.turgor.ru/en/). This involved traveling 30 hours by train across the vastness of Russia to Chelyabinsk, the first big city on the Siberian side of the southern Ural Mountains.

This trip was memorable in numerous ways. Konstantinov regaled us with tales of people and events he had known, or known of. There was the mathematician who was the son of a pre-revolutionary railroad magnate, and who recalled traveling around Russia in his youth on a private company railroad car. There was the tale of the runaway train, on the very tracks we were traversing, which rolled from the top of the Ural pass miles down to more inhabited areas. A locomotive was sent to chase and capture it. The locomotive collided forcefully with the train from the rear, to couple with and stop it. The collision was enormous, but prevented the train from devastating a more populated area. We were traveling through the Bashkir Republic, and these Turkic people had a heritage of horsemanship. Konstantinov challenged us to spot a rider on horseback. But all we could see was pipelines from now-exhausted oil wells. Each tale was more interesting than the last, and contributed to a picture of the country and of its mathematical community that few people have glimpsed who have not grown up there.

And the mathematics! We talked for two days about math problems. About ways to classify them. About which were suited for competition and which were not. About logical riddles and their relationship to mathematics. About how contest problems sometimes ended up applied. Three samples stand out in my memory:

An Olympiad problem had been set by Alexey Kanel-Belov a few years before, about packing polyhedra so that their cross-sections tessellated a plane. It turns out that for some such tessellations, the polyhedral blocks forming it will hold each other up when the configuration is lifted. A student solved this problem, and brought it home to his father, an engineer. The father then used it to design tilings for ceilings. (See also Kanel-Belov 2008.)

We discussed a problem about a wire frame forming a cube. Consider the edges as segments. If it is to pass through a plane, what is the smallest length slit you must cut in the plane? That is, suppose the wire frame grew very hot, and had to pass through a piece of paper. What is the smallest ‘length’ of paper that must be burned? This was an interesting problem, but how would the contestants express their solution? They would have to describe the motion of the cube as it passes through the plane. Some motions, even in two dimensions, are difficult to describe. But in three dimensions? We decided not to use this problem.

A third problem was about the “Devil’s Staircase,” a now-classic way of using the Cantor set to define a step function which is continuous. It was decided that there is enough here to offer students who have not had an introduction to analysis. The analytic implications of the results could be appreciated as they learned more.

The reason for this seminar-on-wheels lay in the traditions of Soviet mathematics. In the USSR, teachers had very limited access to copying machines of any sort. Among other reasons, these could be used to reproduce unauthorized literature and so worked against control of information by the state. So test questions had to be written on the blackboard or even given orally. This led to traditions in testing—and in contests—which emphasized long-answer ‘Olympiad’ style problems, rather than the short answer problems more typical of American competitions. Many competitions included rounds that were conducted completely orally (for example, see Fomin and Kirichenko, 1994) . And the tradition of math battle or math wrangle (https://www.maa.org/sites/default/files/pdf/sections/math_wrangle.pdf) also evolved partly from this circumstance.

And in fact we were responsible for setting the problems of a math wrangle at the seminar in Chelyabinsk, a gathering of local winners of the Tournament of the Towns. Later, at the camp itself, I witnessed the process of giving the students the problems, a process very different from any American contest I have known. The contestants gathered in a room, and the judges wrote the problem statements on a chalk board. They then explained the problems orally, taking questions from the audience to make sure the problem statements were clear. Finally the students were given three days to solve the problems and present them in math wrangle format.

The Tournament of the Towns was a child of the fertile brain of Konstantinov, offered as an alternative to the rapidly rigidifying structures leading to the International Mathematical Olympiad. Competition is by ‘town’ (city). In the tradition of Russian/Soviet competitions, problems all require solutions written out, and are selected so as to include both novice problem solvers and those with sophisticated background.

Konstantinov’s work was central to numerous other initiatives. In 1978, he started the Lomonosov Tournament, a multi-subject competition named after Mikhail Lomonosov, the 18^{th} century polymath considered by many to be the father of Russian academia. This tournament has been held every year since. In 1990 Konstantinov was one of the founders of the Independent University of Moscow, among the leading institutions of higher learning in mathematics in Russia. And well into his later years, Konstantinov continued working in Moscow High School 179, and helped to edit *Kvant* magazine. Matusov (2017) gives an account of his fresh approach to the classroom, as well as another set of personal reminiscences of Russian/Soviet mathematical culture.

On one of my visits to Moscow, I was fortunate enough to catch a talk by the Russian mathematician Evgeniy Dynkin, visiting Moscow from his position Cornell University. The talk was for high school students, and the topic was a classic problem in probability: A sequence of integers is presented to you, one at a time, then each disappears. You must choose the largest you can. After your choice, the integers stop coming. (This is a model for numerous life experiences—even for high school students–from choosing a spouse or date to finding lodging along a highway.) In classic Russian style, Dynkin was able to break the problem down for his audience. I had seen this sort of exposition before, and was not surprised. What struck me, however, was the collegiality between Dynkin and Konstantinov. They spoke together, both before and after the presentation, about the level of the students, about how the presentation had gone, and about various mathematical and educational activities going on in Moscow. They were clearly members of the same community. It is now becoming more common to find such camaraderie in the American mathematical community.

After the fall of the USSR, when Russians had the opportunity to travel abroad, Konstantinov and I worked together in various places around the world. I recall him balancing on a beam which lay precariously across a swimming pool in Canada. He and I shopped for souvenirs in Australia, where he bought tiny koalas for each student in one of his classes. And in Amman, Jordan, we sat down to dinner at a conference we were both attending. The dinner plates were square. Konstantinov challenged me to find a reason for this shape. His reason? To make it easier to calculate their area.

Konstantinov’s humor, his fresh attitude towards learning, his creative structuring of programs… they will all be missed, even as his legacy continues.

REFERENCES

Chan, Chi Ling (2015). Fallen Behind: Science, Technology, and Soviet Statism. Intersect, 8(3) (1-11). http://ojs.stanford.edu/ojs/index.php/intersect/article/view/691.

Fomin, D., and Kirichenko, A. (1994) *Leningrad Mathematical Olympiads 1987-1991*. Westford, MA: MathPro Press.

Gerovitch, S, (2013). Parallel Worlds: Formal Structure and Informal Mechanisms of Postwar Soviet Mathematics, *Historia Scientiarum*, 22(3),181-200. https://www.academia.edu/5366902

Kanel-Belov, Alexey et al. (2010) Interlocking of Convex Polyhedra: towards a Geometric Theory of Fragmented Solids. Moscow Mathematical Journal., 10:2, 337–342, 2010 (https://arxiv.org/abs/0812.5089 ).

Karp, A. (2010). Reforms and Counter-Reforms: Schools between 1917 and the 1950s, in Karp, A., and Vogeli, B. (eds.) *Russian Mathematics Education: History and World Significance*, Singapore: World Scientific Publishing Co. (43-86)

The Lomonosov Tournament, 1996, *Math. Ed.*, 1997, Issue 1, 79–106 (in Russian) http://www.mathnet.ru/links/075412a9f93379a3b310240dede3b677/mo232.pdf

Matusov, Eugene (2017) Nikolai N. Konstantinov’s Authorial Math Pedagogy for People with Wings, *Journal of Russian & East European Psychology,* 54:1, 1-117, DOI: 10.1080/10610405.2017.1352391 http://dx.doi.org/10.1080/10610405.2017.1352391

Polyakova, T. (2010) “Mathematics Education in Russia before the 1917 Revolution”, in Karp, A., and Vogeli, B. (eds.) *Russian Mathematics Education: History and World Significance*, Singapore: World Scientific Publishing Co. (1-42)

Saul, M. (1992). Love Among the Ruins. *Focus, 12*(1), 1,6,7, https://www.maa.org/sites/default/files/pdf/pubs/focus/past_issues/FOCUS_12_1.pdf. Accessed June 2020.

Sossinsky, A. (2010) “Mathematicians and Mathematics Education: A Tradition of Involvement”, in Karp, A., and Vogeli, B. (eds.) *Russian Mathematics Education: History and World Significance*, Singapore: World Scientific Publishing Co. (187-222)

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Associate Professor of Math

Brandeis University

For many years I taught Calculus with a traditional structure, in which the students’ grades were mostly determined by a few high-stakes exams (a final and a couple of midterms). In my classes, I would tell my students:

- How important it was to practice regularly;
- To carefully review their exams and the solutions;
- That it’s ok to get things wrong and learn from their mistakes;
- That the idea that we can improve through practice applies in math just as it would in anything else they want to learn.

But the structure of my class was giving them a very different message. The structure told them:

- You only
*really*need to study three times during the semester: right before the midterms and the final; - Don’t bother reviewing your work since you will rarely, if ever, get tested on those same problems again;
- You can only do well in the class if you get all the problems (including the very hardest) right on the first try.

It gets worse. Based on the compelling work of Claude Steele, we started reading out a statement before our exams that said:

- “This test has not shown any gender or racial differences in performance or mathematical ability.”

This statement was carefully crafted to be technically true — the test had never been given before, so it couldn’t have shown any biases. We adopted this with the best of intentions: research had shown that making a declaration like this could, like a self-fulfilling prophecy, help to reduce stereotype threat. I hope it worked, at least for some students. But in practice I suspect our tests actually sent a message more like:

- Somehow you have to divine our expectations about how to properly write your solutions out, which (for whatever reason) you’re much more likely to do successfully if your gender and racial identities match those traditionally overrepresented in math.

There were other ways my pedagogical structure was undermining my own messages. I told my students:

- We care primarily about your understanding of the math and communication of it.

But our structure was saying:

- If you can write down some vaguely-related words and symbols, you can probably rack up enough points to pass the class.

Here’s another one. We told students:

- There’s no cap on the number of students who can get an A in this class.

This was technically true. We didn’t stick to a strict curve. But, with many years of experience, we wrote exams for which we could predict the distribution of scores with remarkable accuracy and then set the grades around the median.

You get the idea. The structure of the course was undermining pretty much all of my pedagogical ideals.

What did we do about it? We instituted Outcomes-Based Assessment (also known as Mastery Grading, Standards-Based Grading, or Specifications-Based Grading). There are many different versions of this, but the basic idea is that you have a list of the ideas, skills, techniques, etc., that you want your students to learn over the course of the semester (“content outcomes”, or “outcomes” for short, in our terminology). You have some way for them to demonstrate mastery of those skills. The grading is credit/no-credit (either they have demonstrated mastery or not). And the students can try multiple times to demonstrate mastery of each outcome.

Our initial version of this was copied pretty much wholesale from Jeff Ford of Gustavus Adolphus College. We’ve tweaked it a bit around the edges since then, but it still has the same basic format. I’d also like to give an official shout-out to Eric Hanson, currently at the Norwegian University of Science and Technology, who, as a graduate student at Brandeis, was instrumental in converting our first class (Precalculus).

Here are the components.

**Content Outcomes:**

Our outcomes are fairly fine-grained. Here are a few sample outcomes for our Differential Calculus class:

- Determine information about a function from the graph of its derivative (or vice versa).
- Find and classify the extrema of a function.
- Solve an optimization word problem using the methods of Calculus

It was important to us that our students also be able to apply their knowledge on problems they haven’t seen before, so we include a few outcomes like:

- Solve a challenging problem that combines different skills and/or presents material from Chapters 2-3 + Pre-Reqs in a different way.

We also decided that there were some things that were so fundamental that a student shouldn’t be able to pass the class without mastering them, so we split these out into a category of “Fundamental Outcomes”. Some of these are prerequisites, like factoring polynomials or evaluating trig functions, and some are key elements of the class, like finding the equation of a tangent line or calculating derivatives.

**Assessments:**

We test our students every Friday. On each test, we have a problem corresponding to every outcome we’ve covered so far in the course. Students have to earn credit on two different Fridays to master a particular outcome. After this, they no longer have to do those problems.

Our bar for earning credit is B+/A- level work. If it’s on the border, we ask ourselves questions like, “do they really understand the idea and have they communicated that?” and “do I think they need to spend some more time on this?”. These are the questions that really determine the cutoff

*Illustration by Simon Huynh*

Students’ grades are then determined by how many of the outcomes they master. Here’s the grade breakdown from our Differential Calculus class:

Students need to earn the minimum points for all categories across a given row to earn the letter grade in the left column.

In practice, the grades are pretty much entirely determined by the General Outcomes column. (The homework and participation scores are quite lenient, so that students who are doing their homework and coming to class can easily meet the A-level for those requirements.)

The key idea here is that the outcomes the students have mastered are ones they *really* know. They have demonstrated that they can do these at a B+/A- level. The grades are determined by how *many* of the outcomes they have mastered.

Let’s go back to the beginning and reconsider all those pedagogical ideals. What does our Outcomes-Based Assessment (OBA) structure tell students about them?

- Regular practice is important.

*OBA: You have to be ready for our test every week.*

- Carefully review exams and the solutions.

*OBA: If you didn’t get it right this week, it will be on the test again next week, so you’d better review your work and our feedback.* - It’s ok to get things wrong and learn from mistakes.

*OBA: It doesn’t count against you when you get it wrong, but it does count *for* you when you get it right, even if it takes a while to get there.* - The idea that we can improve through practice applies in math just as it would in anything else they want to learn.

*OBA: You are rewarded just as much if you try a few times and then get it right as if you get it right on the first try.* - “This test has not shown any gender or racial differences in performance or mathematical ability.”

*I wish I could say with authority that OBA helps this. I don’t know yet. It seems like it should, since the whole point is that students can get feedback, learn what the expectations are, and then implement them. We’ll be setting up a multi-year study starting this year to learn more about the impacts (short and long term) of OBA on all of our Calculus students, but particularly on our students of underrepresented backgrounds.* - We care primarily about your understanding of the math and communication of it.

*OBA: This is precisely where we draw the line between when a student gets credit on a particular outcome and when they don’t. Students learn that the only way to succeed in our class is to review, revise, and to seek help when they don’t understand something. With OBA, students are much more likely than ever before to come in asking us to help them *understand* something, because they know that’s the only way they’re going to perform well enough to earn credit.* - There’s no cap on the number of students who can get an A in this class.

*OBA: This is just straight-up true. Any student who meets our threshold gets an A. End of story.*

For all my enthusiasm for this new system, I must admit there are some challenges.

The two biggest obstacles are:

- It takes a lot of work to set it up.
- There’s a lot of proctoring and grading time.

It took a fair amount of work to set up, especially for the first class we converted. (We started with Precalculus, which has a much smaller enrollment and fewer sections than our Calculus classes.) There are many decisions to make and details to iron out. You won’t get it perfect on the first try, so just go for it and adjust as you learn. We also spent a lot of time writing and proofreading problems for tests. Over time, we’re building a large problem bank, but the first semester really requires a lot of work on this.

Since there is a large start-up cost, it makes the most sense to convert classes that you teach regularly so you can reuse your work. But once you get the hang of it, it really seems like a better way to teach so it’s hard to go back to the traditional method.

We decided to test our students every week, and to have no restrictions on how many attempts they get at each outcome. This means that we’re proctoring and grading every week. It also means that if we have even one straggler who is struggling to get credit, we’re writing new problems for them every week. I know other people (including Jeff Ford) who do not test so often and/or restrict the number of times a particular outcome shows up on the tests. I think some restriction is better, both for the sanity of the faculty and because a little more pressure on the students means they have to follow up sooner and can’t just choose to put it off. We’re currently considering the pros and cons of different options for next fall.

Some other ongoing challenges include:

- Introducing the system to students
- Conveying students’ grades during the semester, particularly around drop decisions.

For most students, this structure is completely foreign. It’s absolutely necessary to spend some extra time introducing it and selling it to students. We hammer home the idea of “high frequency/low stakes” testing. I made a few videos to introduce the pedagogy to our students:

Some students are really anxious about it and have a hard time understanding at first. Most pick up on it after the first week or two, when they have seen all the components work in class.

It helps a LOT to have used the structure before. Our students who have experienced it typically love it, and are happy to endorse and explain it to students who are new to the system. We introduced Outcomes Based Assessment in our Single Variable Calculus sequence for the first time this fall (yes, along with moving online for Covid AND switching to a team-based learning format AND switching to a new textbook — can you say, “gluttons for punishment”?). Several students who took Differential Calculus this fall reached out to me to ask whether we would be using this system for Integral Calculus in the spring. They made it clear that they were only going to take Integral Calculus if we used Outcomes Based Assessment.

I still struggle with communicating to students what their grade is likely to be when we’re midway through the semester. In many ways, our grading is more transparent than the traditional system. Students know exactly where they stand at all times. They know what they already have learned and what they still need to work on. They can see clearly what they still need to do to get a particular letter grade. I built out individual spreadsheets for them to follow along with their progress. But, on the other hand, no one in the class is even passing until quite late in the semester. (I think this is right — after all, they haven’t even learned most of the material yet.) They have a hard time seeing if they’re on track to succeed, especially if they’re not getting everything right on the first try. And I have a hard time predicting what will happen. Will they get those outcomes they’ve been stuck on? Most will, but I can’t guarantee it.

Overall, our grades are higher. It seems that a lot of students who would probably have gotten B’s in the traditional system are able to get to A’s in this system. The system rewards good study habits and guides students on what they need to do so they can do it. And most students feel really comfortable in the last couple of weeks of classes, when they can see things slowing down and they don’t have much left to finish. But it’s hard in the middle to say which way things will break for students who are getting behind. I haven’t solved this problem yet.

This year probably would have been exhausting even without all this extra work we took on by introducing Outcomes Based Assessment. We’re definitely tired and ready to be done with proctoring and grading for the year. But even with all the challenges of online education, I think this structure has helped my students learn more and better than they did under our old, traditional structure.

Yes.

If you got the idea in this article that I did all or most of this on my own, I apologize. It’s not true. This has been a multi-year team effort. Eric Hanson (mentioned above) was the one who wanted to try this out in the first place, and introduced us to Jeff Ford, who generously shared everything with us to help us get started.

My colleague, Keith Merrill, has been vital in making this work, along with many amazing graduate students (some former): Te Cao, Shujian Chen, Tarakaram Gollamudi, Abhishek Gupta, Simon Huynh, Shizhe Liang, Wei Lu, Ray Maresca, Ian Montague, Rose Morris-Wright, Rebecca Rohrlich, Alex Semendinger, Jill Stifano, and Jiajie Zheng.

Our initial implementation in our Precalculus course was supported by a Teaching Innovation Grant from the Provost’s Office and the Center for Teaching and Learning at Brandeis University.

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Maybe it is obvious, but it is something I’ve come to appreciate only after years of experience: mathematics is logic driven, and teaching and learning mathematics is centered on teaching and learning logic. I find this to be true philosophically, but also practically, in my teaching. And even in my own learning.

Philosophically, this point of view has deep roots. Plato’s Academy. Russell and Whitehead. Frege, Tarski. And that’s all I want to say about this area, which is outside my expertise. I leave it to those who think more deeply about the philosophy of logic to forge connections between my experience and their work. I think it is probably enough here to think about the ‘logic’ as concerning just the simplest propositional calculus: implication, negation, and perhaps quantifiers.

Because what I want to say is that in my teaching, the closer I look at difficulties that students have the more likely it is that the difficulty is with these basic aspects of logic. And (conversely!) if students leave my classroom having understood these logical connectives more robustly, I consider that I have succeeded.

Okay. “Deep learning”. (In education, this phrase as a more general and less technical meaning than its use in computer science.) For me, this has a particular and specific mathematical meaning. It refers to learning based on logic, on the connections among statements. Which, I think of as coextensive with mathematics itself.

If we take this point of view, the whole landscape of mathematics is laid out before us, as from the top of a mountain. Too, this view resolves many disputes about the relative importance of skills vs. concepts, etc.

That is: if we are connecting statements, we are doing mathematics. If we are not connecting statements, we are not doing mathematics. We are doing something else. And the statements do not have to be about number or length or functions. Those are the objects on which logic acts in a mathematics classroom.

Of course, logic acts on other objects in other classrooms. We make arguments and build logical structures in studying chemistry, in reading literature, in learning a new language. But the mathematics classroom is the place where we focus directly on these activities, where logic is most quickly and most accurately developed.

As I have noted, this view of course has philosophical roots stretching back to antiquity. More recently, it is the view of Bertrand Russell: “Pure mathematics is the class of all propositions of the form ‘p implies q’…” (see https://todayinsci.com/R/Russell_Bertrand/RussellBertrand-Mathematics-Quotations.htm) . But it also has very practical applications to pedagogy. If a student is struggling, it is logic he is probably struggling with. If you untie the logical knot, lay out the train of thought—particularly of implications—that leads one to the actions taken to solve a problem, then the student will understand and be able to work the problem.

Of course, by ‘lay out the train of thought’ I do not mean ‘give a lucid explanation’. I mean get the student to construct the chain of implications in his or her head. For some (usually graduate) students this may mean giving a very clear lecture. For other populations, it means ‘guide on the side’. I am not claiming that this meaning of ‘deep learning’ implies a particular pedagogy. But it sets a standard for the success of any pedagogy.

I can be even more specific. Russell’s definition of mathematics points to the center of learning of logic: the notion of implication. If my students, after graduating from high school, really understand what it means for one statement to imply another, have been trained to look for such implications, and can judge whether the implication is valid or not—if they can do all that, I don’t care if they know the formula for sin (x+y) or how to measure an inscribed angle. Or even how to perform long division. As Underwood Dudley has provocatively shown us (https://doi.org/10.1080/07468342.1997.11973890), the claim to practicality of mathematics (in the sense of specific mathematical results) is often exaggerated.

But the importance of mathematics, seen as the study of implications, cannot be exaggerated. It is a characteristic of our species. It is what has led us to dominate our environment. It has also led to some incredibly inhuman events. I leave to more serious philosophers to decide whether the phenomenon of human reason is ‘good’ or ‘bad’—or neither. The point is, it is profoundly human.

In making this statement, I disagree with the view that we must ‘humanize’ or ‘re-humanize’ mathematics. Mathematics is, almost by definition, human. It is its uses, and its teaching, whose humanity we must examine.

To be even more precise, and even technical: the definition of implication rests on the distinction between a statement and its converse. So I can go still further in my wild claims to know if I’ve succeeded. If a student, five years after graduation, can distinguish a statement from its converse, in even the most bewildering of logical environments, then I have succeeded with him or her. Don’t think this is so easy: I have caught important mathematicians, or they have caught themselves, confounding a statement with its converse. And of course I have caught myself.

I am not asserting that if you know about the converse then you know mathematics. I am asserting that if you don’t know about the converse, then you do not know mathematics. Or, less aggressively: if you mistake a statement for its converse you are making an error in mathematics.

So, for example in geometry we often teach about the classification of quadrilaterals: trapezoid, parallelogram, rectangle, etc. Students will often say things like: “If we know a parallelogram has equal diagonals, then we know it is a rectangle… or it could be a square.” Venn diagrams, illustrating set inclusion, can certainly help untangle the confusion. But there is also a deeper lesson to draw from this error, one that transfers to, and taps into, other experiences. This deeper lesson emerges in phrasing the statement in canonical ‘if..then’ form: if a parallelogram is a square, then it is a rectangle. But if it is a rectangle, it may or may not be a square.

I find this an important guiding principle in pedagogy at all levels. Even when we teach young children with hands-on tactile experiences, what we are teaching them is about objects which will, sooner or later, be objects subject to reason. For me, this resolves the endless debates about mechanical skills, about fluency or automaticity. And it resolves it in two ways. First, the object of fluency is to be able to reason more easily—more fluently, if you like. When do you concentrate on fluency (‘drill and kill’)? When lack of fluency gets in the way of reasoning. And when do you reach for the calculator? When doing without it will derail your train of thought.

The second way a focus on logic resolves issues about fluency is more directly pedagogical: fluency is best acquired by making logical connections among statements. For example, if a child knows that 8+8 equals 16, she doesn’t have to memorize that 7+9 also equals 16 or that 8+9 equals 17, or that 80 + 80 = 160 or that…

This is the meaning I take from Liping Ma’s “knowledge packages” (1999). I have written elsewhere about how I think her very useful work can be given more meaning (https://www.ams.org/journals/notices/201405/rnoti-p504.pdf). That article was one step towards the view I express here, which I’ve come to only aftera decades of experience. To some readers it may be perfectly obvious, and to others perfectly ridiculous. I would be interested in hearing both reactions.

REFERENCE

Liping Ma, Knowing and Teaching Elementary Mathematics, Lawrence Erlbaum Assoc. Inc., Mahwah, New Jersey; London, 1999.

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Students who have had me for at least one class are familiar with my alter ego, Lamar. If they were to describe him, they may say that he is the poster child for what not to do in mathematics. They may speak to Lamar’s tendency to arrive at the wrong conclusion by making erroneous assumptions and/or using faulty logic. When a mistake in class is made, it is not unusual for someone to say, “That’s totally a Lamar move to make.”

You may wonder how Lamar came to be. Let me take you back to a chilly spring day in March inside a proof-based linear algebra class. When I first saw this material several years ago, it took me a nontrivial amount of time to understand the difference between a scalar 2 and the identity matrix scaled by 2 when performing matrix algebra. In my current Linear Algebra class, this difference is emphasized through the eyes of Lamar as he tries to prove that a square matrix A is invertible if A^2 – 2A + I = 0. Once Lamar’s name leaves my mouth, students become more alert as they watch for the inevitable misstep that Lamar will make. Indeed, for the example mentioned above, Lamar finds that the inverse of A is given by 2-A. The identification of his misstep leads to a fruitful discussion on Matrix Algebra and I mentally pat myself on the back for what I think is a job well done.

A few weeks before Lamar was concocted, I was trying to emphasize the difference between matrix inversion and division. I purposely erroneously tried to define the inverse of a square matrix A by taking the reciprocal of every entry inside the matrix—a classic Lamar move to make. I was hoping that someone would call me out on this mistake and perhaps chastise me (lightly) for having gone down this forsaken path. As I paused on the board to let my mistake permeate into everyone’s head, a student in the back rolled his eyes and questioned my ability to lead the class. He wondered (out loud) if I had the knowledge necessary to teach such a demanding class and that perhaps they should find another TA. He was quick to point out that I could not make such a definition and was offended that I would even offer such a definition.

In that moment, I felt the need to prove myself as a mathematician and to show them that I did indeed earn the right to help guide them through their mathematical journey. To protect the integrity of genuine mathematical inquiry, I would need to wear two hats—the “Terrance” hat and the “Lamar” hat. When I put on my Terrance hat, I am the mathematician who is an expert in this field and who knows all that there is to know about the topic of interest. The development is effortless in hopes of earning their respect and trust. When I wear my “Lamar” hat, I try to humanize the activity of doing math by emphasizing inquiry and how mistakes can be used to develop sound mathematical ideas. I wanted them to experience the same process that I use when I am engaging in mathematical research. I want them to see how important mindset—as opposed to initial ability—is towards reaching the promised land of mathematical understanding. I wanted them to truly experience what being a mathematician is all about.

For the reasons mentioned above when I work with students on novel research projects, I try to be Lamar more than Terrance. No matter their mathematical journey, I invite any student to participate. Because of that policy, I attract a diverse range of students – from those who haven’t even taken a math course at Drake yet and are wondering how math can inform someone about hurricanes all the way to students who have taken every advanced math class that we offer. I try and discard the notion that you must satisfy a bunch of prerequisites to unlock the gates of mathematics (*i.e.*, engage in math research) and I try to involve students in research as soon as possible regardless of where they are in their journey.

I also purposefully pick problems that are outside my area of expertise (though to be frank, students pick these problems) because I want to join them in this area of discovery and showcase a vulnerability that I cannot really show in class (i.e. when I’m wearing my “Terrance” hat) – as a young pre-tenure Black mathematician at a majority institution, I do not want to invite students to question my right to be there, as is sometimes the case. I want students to really feel what it is like to be a mathematician, to question everything, to get stuck, to try again, to revise rough drafts repeatedly until an answer is found if possible. I never really thought about the multiple roles I play at Drake – from the bearer of all knowledge in an analysis class to the guy who had no idea how to “mathematize” a hurricane. I think it helps students to see me as a human and to embrace the questioning and mistakes. Furthermore, I am using this same process of integrating research to recruit students of color and other underrepresented groups by creating a space where mistakes can safely occur.

There is a certain tension that comes from the many hats that I wear as I help students navigate their own mathematical journeys. I find that I am constantly negotiating and renegotiating my identity in the process of learning and teaching through inquiry. On one hand, I do not want to assert so much of my own expertise that it prevents students from developing their own expertise and identities, however; I want to assert enough of my expertise to prove my ability to help students navigate their journeys. I try to walk the line between being an expert in my field in the classroom and the mathematician who sees an interesting problem but does not know what the answer will end up being—assuming of course that there is an answer to begin with! This is in sharp contrast to when I first began my career. My first research project with a student involved a problem that I already knew the answer to. Rather than creating a space for which the student could forge their own mathematical identity, I carefully crafted an experience that I knew would lead to a satisfactory conclusion. Because of the tight control in this experience, I was able to artificially correct their course whenever I felt they strayed too far. That is, I prevented this student from having an authentic experience of what it is like to really be a mathematician—how to think through and reason through a problem of interest.

Nowadays, I purposefully pick problems for which I don’t know what the final solution will be. I don’t exactly know where the problem is going to go or what mathematical tools I will need to acquire in order to reach a satisfactory conclusion. But now, it is tricky, and it does make me nervous sometimes when I do it. I go in and I’m like, “Well, let’s see if we can brainstorm together because I’ll admit I have an idea or two, but I don’t know if they’re actually going to work out.” As part of this experience, students get to see the real me. Here is a mathematician that doesn’t really quite know how these answers are going to work, but he has some ideas and can use his training to help inform his position. I think it’s kind of nice that I can humanize this experience, and they can see like, “Oh, he knows all this stuff, but he still has to figure out how to use the stuff to inform his next steps and his positions,” and things like that. I think that experience is just as important as somebody who is in front of the class, who’s giving them all of this context, who’s an expert in his field, and who’s whipping all this stuff out and weaving all of these different ideas and concepts together that tells this really beautiful story about why things work the way that they do in calculus or linear algebra, or some other math course. Thus, I’ve latched onto that without focusing too much on any possible ramifications. For instance, I wondered if students would think of my classroom persona as a farce. Like he knows all this stuff, but then when it gets to research, he’s clueless. On the other hand, perhaps the response would be something to the tune of “I feel okay now going in and not needing to know the answer because he doesn’t either, and he’s an expert in his field.” So, I have embraced this duality between being this content expert in the classroom, and then being this humble mathematician, who sees this really interesting problem, latches onto it, and then works it out like a mathematician. Thankfully, it’s been a pretty great experience thus far in creating these research spaces for early mathematicians.

Part of this experience is supported by an NSF grant which investigates the ways in which mathematics research projects conducted* early* in students’ college mathematical careers spark students’ engagement and interest in mathematics. The students had the opportunity to pose questions themselves and then develop skills in building innovative quantitative solutions to complex problems. There is a particular focus on students who have been historically excluded from mathematics majors. Participating students were recruited from non-major or pre-major mathematics courses, with a particular emphasis on recruiting students who are people of color.

This past semester, undergraduates pursued two strands of authentic mathematical research, based on their interests:

**How music goes viral on social media:**Here, we attempt to quantify the success of a song by modeling its spread through social media networks such as TikTok, Apple Music and Spotify. By utilizing models in epidemiology (such as those used to model the spread of COVID), we are developing a set of parameters to better understand the conditions a song should satisfy in order to optimize its chances of going viral.**How to create sustainable single use coffee cups.**While single use plastic is a convenient way to enjoy an ice-cold beverage from your favorite coffee shop, its usage comes with a steep environmental price. By combining financial incentives, an optimal redesign of single use plastic cups, and alternative plastic sources, we seek to minimize the total amount of plastic used by companies that rely on these goods to stay in business with the constraint of maintaining some minimum profit.

This material is based upon work supported by the National Science Foundation under Grants DRL1821444 and DRL 2021161. 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.

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We often think about our classes starting on the first day of the semester. But some of our students log on to course management systems and look at the course materials before classes start. I contend that we can start to build relationships with our students well before class begins — with the way we write our syllabi, an initial email to students, and a short first assignment that can be completed before the very first day of class.

A little bit about my own path through college: I went to college right after high school, dropped out in my first semester, went back a year and a half later to finish my first year at a college near home, and then transferred to Colorado College where I (finally!) finished my bachelors degree. To say that my path was bumpy is a gross understatement. I struggled with feeling “behind” my classmates (although, looking back, I doubt anyone realized I was a couple years older than my peers). I had a lot of anxiety about classes, and I often felt like I didn’t belong. (You can hear me talk about this in a pep talk that I recorded for my classes this semester: https://www.youtube.com/watch?v=kenf8E1RuoA)

I’m keenly aware that many of the things I experienced — that I thought I was alone in experiencing — are becoming more common among our students. Student stress and anxiety levels are rising every year. With that in mind, I’ve been working on centering the humanity of my students in my classes. In this blog post, I’ll share a few things I do before the first day of class.

Nancy Wrinkle shared Syllabus Review for Equity-Minded Practice with me (h/t @wrinkle_nancy on twitter). This guide is published by The Center for Urban Education in the School of Education at USC Rossier.

From the guide, here are a few introductory ideas:

What is syllabus review?Syllabus review is an inquiry tool for promoting racial/ethnic equity and equity-minded practice. To achieve this goal, the syllabus review process promotes faculty inquiry into teaching approaches and practices, especially how they affect African American, Latinx, Native American, Pacific Islander, and other racially/ ethnically minoritized students; facilitates a self-assessment of these teaching approaches and practices from a racial/ethnic equity lens; and allows faculty to consider changes that result in more equitable teaching approaches and practice.

What is in the guide?The Syllabus Review Guide is comprised of six parts that provide the conceptual knowledge and practical know-how to conduct equity-minded self-reflection on an essential document in academic life: the syllabus. Throughout the Guide are examples that illustrate the ideas motivating syllabus review, as well as opportunities to practice inquiry and to reflect on how to change your syllabi—and your teaching more generally—so are more equity-minded.

If you’re into Universal Design for Learning (https://udlguidelines.cast.org), you’ll find this resource very helpful for making your syllabi clear, useful, and — my favorite — human. The guide reinforces the idea that you can write stuff in your syllabus specifically to support and encourage students. It doesn’t have to be a boring contract. It can communicate a lot more about the class. For example, it can address:

- The classroom environment (“joyful exploration”)
- Support systems (office hours, campus resources)
- Your values (Federico Ardila’s axioms)

This kind of syllabus review seeks to make the hidden curricula of college visible to students. It’s about transparency as much as it’s about what’s going to be covered and how grades are going to be calculated.

The guide also helps put racial equity front and center. Before using this resource to rework my syllabi, I had not intentionally grappled with “affirm[ing] the belonging of racially/ethnically minoritized students in higher education by representing their experiences in the course materials and by deconstructing the presentation of white students and white experiences as the norm.”

One of the changes I made to my policies specifically addressed this equity-minded competence from the guide:

Views the classroom as a racialized space and actively self-monitors interactions with students of color

(in contrast to “[v]iews the classroom as a utilitarian physical space”). Although I knew I would be teaching remotely when I did my syllabus review, I reflected on the ways a Zoom class space (a ClassZoom, if you will) can be racialized. The previous spring, some of my Black students had confided that keeping cameras on because of professors’ policies felt like it highlighted that there were so few Black students in class (sometimes exactly one). My “aha!” moment was understanding that a “cameras on” policy heightened the stereotype threat that students of color already feel in predominantly white classes. With that in mind, and with the goal of making it clear in my syllabus that active and engaged classroom participation doesn’t require cameras on, I added Remote Classroom expectations to my syllabus, including the following:

Cameras:Whatever you feel comfortable with. I like to see your faces but that’s no reason to require it! If you don’t have your camera on, I request that you make liberal use of Zoom reactions so I don’t feel so alone up here…. I will tell jokes during class (to help you out, I’ll often tell you that they are jokes). This is a great time to use a Zoom reaction.

If the reader is curious to see my entire course policies packet, you can find it here (with the caveat that I still have a lot of homework to do as part of the syllabus review!): https://people.hamilton.edu/cgibbons/files/syllabi/Math325-Policies.pdf

It’s not news that students carry a lot of math anxiety. I like to give them a chance to let me know where they are in their math journey (what classes they’ve taken, what they have liked, what they feel anxious about) by having them complete a Math Autobiography before the first day of class. I also invite them to tell me a little about themselves and about what makes them feel like they belong (or don’t belong) in the classroom. I like to end the survey with a math question we can talk about together on the first day of class.

There are many ways to create a pre-class survey. I’ve chosen to keep the survey class-related (broadly) as a way to let students tell me anything they might want me to know about themselves, their math experiences, what they understand about how they learn best, and their expectations for the class. I’ve gotten the feedback that this survey helps students feel less anxious about the first day of class — especially because they’ve also done a math problem to get ready! Not every student completes the Autobiography before the first day of class, but in my experience over half of students do. (This also helps me with *my* first day of class anxiety — it’s nice to know a little bit about the students before the class meets!)

The specific questions I used this semester for my Modern Algebra class are:

What is your name?

Tell me a bit about yourself.

You can tell me anything you’d like me to know. If you’re having trouble getting started, the following questions might help spark some ideas: Do you have a nickname you’d like me to use? What are your pronouns? Where did you go to high school? How did you decide to attend Hamilton? What have been your favorite subjects in school? Favorite activities? Hobbies?

Tell me a bit about your relationship with math.

Again, you can tell me anything you’d like to know. For example: What math classes have you taken and when? What have your experiences in math classes been like? How do you feel about math? In what ways have you used math outside of school?

Tell me about yourself as a learner.

For example: What makes you feel included (or excluded) in a class? Do you learn best from reading, listening, or doing? Do you prefer to work alone or in groups? What do you do when you get stuck? Do you take notes? Do you procrastinate? Do you read the textbook? Prefer videos?

Tell me about your expectations for this class.

For example: What do you expect from your professor? What do you expect from yourself? What do you expect from your classmates? How does this course fit into your educational and life goals? What makes you feel comfortable in a class setting?

Last question: Tell me how you would solve for x given x^2 – 3x + 2 = 0. (If you’rereallybored, tell me instead how you would solve for x given x^2 – x – 1 = 0.)

This isn’t a quiz; this is where modern algebra picks up from things you might not have thought about for awhile, so it’s a very gentle review for you. Plus, telling me about your thinking helps me know how to start class on day one.

Now that you have a revamped syllabus and a first assignment that lets you get a peek at your students’ identities and experiences before the first day of class, it’s time to give the students a glimpse of *you* before the semester starts.

I like to let the students know the syllabus is available for them to read (along with some incentives for reading it: the grading policy is not strictly points-based, and I want them to ask questions about it on the first day; there are some important dates to put into their calendars; etc). I also like to fill out the Math Autobiography survey as a way of introducing myself to my students and as a way of introducing the assignment to them.

Most of all, I want the first email to convey that I’m excited to begin the semester with them as their collaborator in learning.

Many scholars have written about rehumanizing mathematics (see, for example, Dr. Rochelle Gutierrez’s Why We Need to Rehumanize Mathematics). What I hope to accomplish with my pre-semester relationship-building is twofold. First, I genuinely do want to build relationships with my students; that’s one of the reasons I wanted to teach at a small liberal arts college. Second, I want to convey to my students that they belong in my classroom as their whole, human selves — and that I will also show up as my whole, human self!

]]>*All authors contributed equally to the preparation of the document.*

How do students typically engage with new definitions in undergraduate mathematics classes? Are students provided with a definition, and then instructors help students make sense of it? Do students have opportunities to create their own definitions? Often when instructors choose to involve students in the process of creating a definition, the role of the instructor may be to encourage students to structure or word their definitions in a particular way, with the goal of leading students toward a definition found in a textbook. This can be a daunting task. After all, enacting this kind of lesson means anticipating what students may do or say, deciding when to let students keep talking and when to interject, and responding to unexpected contributions. Designing a lesson that is mathematically substantive but also provides opportunities for students to do a lot of the talking (including students providing feedback to other students) is really hard! Even with the most well-laid plans, surprises can still happen.

One way to take on this challenge, and have support as the unexpected arises, is to collaborate with other instructors. The authors of the post are all instructors of geometry courses for prospective high school teachers, who participate in a “GeT: a Pencil” community meeting every other week, and sometimes more often. These community meetings gather university geometry instructors from across the country to collaborate on issues related to the teaching of the geometry course primarily taken by preservice teachers. Among us are mathematics and education faculty, whose academic backgrounds range from mathematical physics to difference equations to hyperbolic geometry to student cognition to teacher education. We saw a pandemic-era opportunity to co-plan and co-teach a common lesson. On Zoom, we can be more than 3000 miles away and learn from each other in the same room. While practices involving the design of lessons (such as the Japanese “lesson study”) have been established for decades in some K-12 settings, it is still rather rare in undergraduate settings, though there are some exceptions.

In this post, we share our experience of developing a lesson that could be taught in any of our courses and how this lesson did not go according to plan. We intended the lesson to focus on creating a new definition. Although the class did not reach a consensus on a definition, the process opened many mathematical questions.

We first show the key example of the concept to be defined. Then we describe why we chose to use this example, how we built a lesson around it, and the unexpected outcomes. Finally, we discuss what we learned (and hope to continue to learn) about collaborative planning and teaching.

Consider the following image:

**Figure 1. Boa Me Na Me Mmoa Wo Adinkra symbol
How would you describe the aesthetic appeal of this figure, mathematically?**

(source: https://csdt.org/culture/adinkra/geometry.html)

Boa Me Na Me Mmoa Wo is an *Adinkra*. Adinkra are symbols created by the Ashanti people of Ghana to represent concepts. Its name in English is “Help me and let me help you”.

Our activity to engage students in constructing definitions focuses on the mathematical properties of this Adinkra symbol that make it visually appealing. Often, as mathematicians, we think of “symmetry” as a way to describe aesthetic elegance. Yet the only standard “symmetry” here is a single reflection. Intuitively, it seems incomplete to describe the “symmetry” of this Adinkra as merely a single reflection. The ethnomathematics educator Ron Eglash suggests that Boa Me Na Me Mmoa Wo exhibits mutuality: “The upper triangle is missing a square, but has an extra circle. The lower triangle is missing a circle, but has an extra square. Each has what the other needs to complete [itself].” Our main task focused on how one might define mutuality. At this point, we encourage the reader to attempt to create a mathematical definition that describes salient aesthetics of the Boa Me Na Me Mmoa Wo symbol.

We had several reasons for co-designing a lesson around mutuality. At the onset, we wanted our students to learn from each other and talk about geometric concepts, definitions, and axiomatic systems in productive ways. We also wanted a lesson that would allow students to compare definitions, and do so in a way that could be connected to secondary geometry from a transformation perspective. We considered several task ideas related to secondary geometry standards for transformations, such as comparing definitions of glide reflections, or identifying symmetries of frieze patterns. We ultimately decided to focus on an activity exploring Adinkra and mutuality because it provided our students (and us) with an opportunity to expand our knowledge about connections to mathematics from non-Eurocentric cultures. Furthermore, because “mutuality” is not a standard symmetry (i.e., described by rotation, reflection, or translation, or a composition thereof), and because it does not (yet!) have a commonly accepted mathematical definition, we saw an opportunity for students to experience genuinely open mathematical inquiry.

We also note that we use Adinkra with the implicit permission of at least some creators of Adinkra. All resources on the site Culturally Situated Design Tools, from which we learned about Adinkra, are disseminated by Ron Eglash with the explicit permission of people he visited to learn about their designs, and knowing that students and teachers may take mathematical directions not necessarily directly aligned with a culture of origin.

We decided to open the lesson with a sorting task, completed individually by each student:

**Figure 2. How might you sort these twelve Adinkra symbols?**

(source for symbols: https://www.adinkrabrand.com/blogs/posts/african-adinkra-symbols-and-meanings)

We designed Google Jamboards with the arrangement shown in Figure 2, featuring the same set of symbols on each, and prompting each student: *Group these symbols by their aesthetic; put each image in exactly one group; then identify names for each group. *

We next put students into teams and asked them to review the individual classifications and discuss: *What did the groupings have in common? How were they different?* These questions had two purposes. First, they could elicit discourse needed to create a definition; for instance, articulating properties, handling disagreements, and coming to consensus. Second, they allowed the instructor to hear students going through this process, to give support as needed, and to adapt later parts of the lesson as needed.

In choosing symbols to include, we included several symbols with rotational and reflectional symmetries, the Boa Me Na Me Mmoa Wo symbol (Figure 1), as well as symbols that we anticipated students might group with it. We hoped to plant a seed for students to see a need to define (and refer to) the standard mathematical symmetries with precision.

Next, we planned for students to formulate a definition of mutuality based on Boa Me Na Me Mmoa Wo, first individually, then in small groups, and then as a whole class. We note that in the end, students formulated and revised individual definitions after reading Eglash’s description quoted above.

We asked the students:

**Figure 3. If you had to create another symbol with that same aesthetic, what would you produce? How would you define an aesthetic category for the Boa Me Na Me Mmoa Wo symbol? **

To prepare for this, we started by brainstorming some potential definitions and experienced for ourselves the uncertainty of what interpretations might arise. For instance, one of us defined mutuality as:

Consider a figure A. If there are two subsets of B, C of A, and two isometries f and g, such that A U f(B) U g(C) has more symmetries than the original figure A, then A has mutuality.

For example, in the Boa Me Na Me Moa Wo symbol, we might take B to be the bottom square, C to be the top circle, f to be an isometry that maps B to the “empty” square, and g to be an isometry that maps C to the “empty” circle. Then A U f(B) U g(C) has a line of symmetry. A second among us defined mutuality as requiring the requirement that the rigid motions used to relate part of the symbol to each other be an involution and yet a third among us defined mutuality as requiring at least one line of reflective symmetry. A fourth among us pointed out that developmentally, students may look at concrete visual features, such as whether a figure has a vertical or horizontal line of symmetry, or whether a figure includes spirals or polygons. We wondered whether students would describe mutuality as the existence of a sequence of actions or operations on the entire figure, or as something that two sub-figures exhibit, and whether they would reason about the black/white contrast as a presence/absence of points or as different colors.

We planned to close the lesson by debriefing the lesson with students, discussing how the process they engaged in might apply to defining activities in secondary geometry classrooms, and providing resources for them to explore Adinkra, including their origins, names and meanings.

Dr. Boyce and Dr. Sears co-taught the first iteration of the lesson in a 150-minute class with six students who were enrolled in a masters degree program in mathematics education. Three of the other co-authors attended as observers. We were fortunate that this class had already developed a warm, welcoming, and supportive environment. This was due in large part to emphasizing “The Five R’s” each class session: rigor, relevance, collaborative responsibility, cultural responsiveness, and authentic relationships. “The Five R’s” represent a point of view that Dr. Sears has emphasized within her classes to support the development of rigorous mathematics and a sense of community in which everyone works collaboratively to co-construct mathematical meaning and develop a conceptual understanding.

Here was our first surprise: None of the students sorted by symmetry! Instead, students had categories such as “ovals”, “stars”, “swirls”, “thick lines”, “4’s”, and “2’s”. Figure 4 shows some of the students’ groups.

**Figure 4. Some groups of symbols created by students**

And here came a benefit of co-teaching: making collaborative in-the-moment decisions. An observer noticed that the students, working in groups of three, were focusing on types of polygons required (i.e., that it should consist of triangles, circles, and squares) in their definitions and we were concerned that the discourse might be headed away from our goal of eliciting transformational reasoning.

Dr. Sears predicted that if Dr. Boyce asked the student who had used the phrases “4’s” and “2’s” to speak up earlier about their reasoning behind their categorization in the whole-class discussion, then this may invoke transformational reasoning for the rest of the students in the class. Dr. Sears observed students in one group come to immediate agreement on “4’s” and “2’s” as a characterization, perhaps because of the practical nature of the description: the number of apparent pieces in the symbol. The students in this group then talked about specific locations of bolder lines and thinner lines in the designs. Noticing the emphasis on concrete visuals, Dr. Sears suggested that Dr. Boyce could ask a question such as, “If I were to reconstruct the grouping, would you know how?” By “reconstruct the grouping”, Dr. Sears meant re-sorting the symbols in a way that would replicate a particular grouping of symbols. Her instinct was that these students would think in terms of geometric transformations to describe relationships between figures. These tactics worked: students then talked about symmetry with respect to x- and y-axes and envisioned lines of reflection on symbols.

We then split students into two groups and asked them to define mutuality. One group wrote the definition: “should consist of an oval with shapes on the end”. The second group defined it as a “shape with one line of symmetry, where the line must be vertical, and where the shape consists of squares, triangles and circles”. The two groups’ definitions focused on what types of shapes could be used to form this particular Adinkra. Although there were opportunities to discuss precision (e.g., what does it mean for shapes to be “on the end”?) and come to a consensus on these definitions, we anticipated the discussion would stray from concept of mutuality as described by Eglash. So, we abandoned our plan for students to compare and revise these definitions. Instead, we decided to set up the task of defining mutuality based on Eglash’s description of the concept, by introducing an unplanned prompt for students to consider individually: “What properties make a definition mathematical?”

Students posted their ideas on a Padlet and then asked each other clarifying questions. The class came to a consensus that one student’s description captured their thoughts: “A definition is mathematical when mathematical vocabulary can be used and to create enough specificity to be proven.” They acknowledged that determining what constitutes “mathematical vocabulary” remained unresolved.

This prompt improved the students’ next attempts, though it also showed us places where we might need to continue building students’ mathematical language and reasoning. When students next read Eglash’s description of mutuality and constructed revised definitions of mutuality, they showed more attention to precision. For instance, one student wrote, “Translating the absence (complementary piece) of both shapes to create symmetry in the whole figure”. A second student wrote, “Two or more shapes are mutual if certain areas of each shape can be translated onto another shape, making all of the shapes congruent.” This student continued, “This applies to shapes that are monochromatic, but what about shapes with multiple colors or something?” A third student wrote, “Moving pieces of the same size, shape, and color around to preserve the symmetry of the original figure.” In these revisions, we can see an attention to geometric transformations and more precise language that was absent in their first attempts.

Finally, a student inquired about the candidate definition in a way that mathematicians would: Wondering about the boundaries of examples and non-examples. This student wondered: “Do areas of the translated pieces have to be equal? Can an exchange between shapes be mutual if one shape gives 100 cm^{2} and receives 1 cm^{2}? Just because an exchange is equitable, does that make it mutual?” The student sketched the figures shown in Figure 5, asking: “Are these two shapes mutual?”

**Figure 5. Do these two figures exhibit mutuality?**

This student’s comments relate to an observation that Eglash made in a recent talk attended by the authors. Eglash mentioned that the point of “mutuality” is that the exchanged objects are unequal, so that exchanges create a system of mutual obligations.

In a whole class discussion following these revisions, Dr. Boyce noted that students all wanted to work with the idea of “complete”, and suggested that some students thought about it in terms of geometric transformations, and others thought of “complete” in terms of congruence.

Two students then publicly debated the meaning of “complete”:

“It’s just an intuition, what it means to be complete.”

“Well, what about ‘whole’? Can ‘whole’ be a mathematical word for ‘complete’?”

“Well a whole doughnut has a hole in it, you know?”

“That’s a different whole, that’s ‘hole’.”

“But is a whole doughnut ‘whole’, is it missing a part, or is it complete?”

“Because it’s a circle without a center piece.”

“But for me, if I were looking for a doughnut, I don’t want to fill that in, I’d be cool with that.”

At this point, there were 5 minutes left to the class, and Dr. Sears pointed out that one might need to begin by defining the “whole” as a way to determine “complete”. Dr. Boyce then closed the class with a brief overview of Adinkras.

As this lesson unfolded, our best-laid plans went awry, with no class consensus on a definition of “mutuality”. Yet the lesson also suggested that areas with no consensus could be openings to further mathematical discussion. For instance, the meaning of “complete” and its dependence on a given “whole”, with the example of a doughnut, could be referenced later as an example of how some mathematical definitions depend on an ambient space. The question of whether mutuality requires “congruence” alludes to the idea that congruence is not the only way to conceive of equivalence. Similarity is also a way to consider equivalence. And ultimately, one might envision a new equivalence relation on shapes based on mutuality.

We met as a group the next morning to debrief after the lesson. What had we learned from the process of planning, facilitating, and reflecting on the outcome of the lesson? We had field notes and screenshots of student work that were collected by observers during the lesson. We were able to record both whole-class and small-group discussions for subsequent viewing, and we reviewed the video-recordings together and documented what we noticed.

Sometimes when teaching a collaboratively designed lesson, there can be pressure to “stick to the lesson plan”. But as our experience shows, instructors may have to make adjustments to the plan, even in the moment. Regardless of co-planning, instructors need to adapt lessons on the fly. When co-teaching, we have the opportunity to figure out modifications with a partner. When co-planning, we have additional opportunities to learn from previous modifications.

When making adjustments, instructors need to keep in mind their intended learning goals, and also whether different intentions might better suit a lesson. In the lesson’s first iteration, the students were able to compare definitions and also increase their precision. In the process of writing definitions with more precision, the students drew on the language of geometric transformations. We believe the improvement in their definitions was based on an in-the-moment modification: asking students to articulate the properties that make a definition mathematical.

One intention of this lesson was for students to come to consensus on a class definition. This did not happen, and perhaps it could not have in the time that we had. However, something arguably more important happened. Namely, the students began to see areas where a draft definition could be improved. The way that the students took up the mathematical ambiguity of “complete” has hallmarks of genuine mathematical inquiry. They drew on their mathematical knowledge to articulate properties of a new concept, they identified areas where equivalence could be interpreted in different ways, and they identified where their definition needed more precision. As we move forward to subsequent iterations, we will continue to reflect on how students take up precision, what might hamper precision, and what will support precision. Just as importantly, we will attend to where mathematical inquiry is happening and how to help students see the doors that they open.

It is a rare opportunity for multiple instructors to plan a lesson together, see the result together, and learn about teaching and learning together. Because of the need for online teaching, we were able to collaborate across six different states. We benefited from each others’ different experiences and expertise when planning the lesson. For instance, the Adinkra context was one that Dr. Boyce had previously used. The focus on definition came from Dr. McLeod, with support from others. The sorting task resembled a task that Dr. Lai had written in an entirely different context. The kinds of questions we planned to ask students, and the emphasis on collaboration through the “five R’s”, came from Dr. Sears’s experiences. Together we were able to envision a lesson idea that was more powerful than what we could have designed individually. In the coming months, as we see the lesson unfold in more of our classes, we will learn more about teaching and learning than we could have individually. Most importantly, perhaps especially in these times, we also found a teaching community in each other through this experience.

Acknowledgments. The reported work is supported by NSF DUE-1725837 and NSF DUE-1937512. All opinions are those of the writers and do not necessarily represent the views of the Foundation. We are grateful to Mark Saul and Carolyn Abbott for helpful comments.

]]>Many academics and teachers have been struggling with facilitating classes virtually. The 2020 global pandemic has brought many challenges and disruptions to teaching, but opportunities to explore and learn as well. This blog post discusses what we have learned so far, with the hope that these reflections are useful to other higher education instructors.

We teach in two very different university settings. Enes teaches at Drake University, which is a small liberal arts college in Iowa. Zach teaches at Utah Valley University, a public school that is the largest in the state (and open-access, as well). Since the spring of 2020, we have been collaborating on opportunities to use and explore some technological tools. Via frequent discussions over the past year on the new teaching and learning space, we shared some of the successes and frustrations throughout the experience. Specifically in this post, we share some of the highlights of facilitating synchronous class sessions using video conferencing tools. From what we have learned so far, most students enjoy real-time, synchronous, virtual interactions and perhaps prefer that over non-synchronous interactions. Like most instructors, we found Zoom (or likewise) to be a useful tool in facilitating online mathematics courses that is an experience shared across other institutions. The main motivation for this post is to share some of our experiences teaching mathematics online as well as talk about our thoughts on the possibilities of interactive teaching pedagogies.

First and foremost, providing breakout sessions during synchronous class sessions is a must. Our students have been enjoying breakout sessions. Students provided insights that these sessions are great opportunities for them to interact with their peers and learn from them. We design these sessions to resemble the small-group work of our face-to-face class formats. Students generally work on a specific class handout or activity and collaborate with each other during that time. But how to best lead this process? We often discussed the possibility of using online tools such as Google documents or Desmos so that we could facilitate class discussions better and get instant feedback from students.

Enes experimented with this strategy in his precalculus courses at Drake University. He created a Google document (see examples in Figure 1) that has a class activity on it (similar to class handouts including instructions), and this is linked to a Google Form where students access the activity. This is given at the beginning of each online class session (but also provided on a learning management system for those who miss the class).

After a quick check-in with students, and any other business that needs discussing, students are put into the breakout sessions of 3 or 4 students to work on the day’s activity. Students are encouraged to collaborate while working on the given activity with their group members, but are asked to submit individual answers to Google forms (see Figure 2).

We found using Google forms to collect student work to be the simplest when it comes to facilitating class discussions, easily collecting work, and getting instant feedback from the students. Whenever Enes pulled the class back from breakout sessions, he utilized the responses to these documents to shape the whole-class discussions (see Figure 3; the responses are instantly listed on a spreadsheet automatically created by Google). This was useful for bringing up issues about particular mathematical concepts. For instance, while looking at Column H in Figure 3, one can see that some students do not use ordered pairs for representing x and y intercepts. This issue can then be brought to the attention of the whole class.

Overall, we found this strategy to be helpful when facilitating not only small-group but also whole-class discussions in online environments. We are aware that there are some other tools that can be utilized for this purpose (e.g., Desmos Classroom Activities) but we found Google to be easier to navigate and manage data (student submissions).

Next, we would like to provide some insights into specific pedagogies that we have implemented during our online class sessions to improve student engagement and peer interaction. One such pedagogy is assigning roles to students during breakout sessions. This semester, Enes has assigned his students into groups of 4 or 5, and each group has a group captain and a timer. He has asked group captains to be in charge of groups when the instructor (or TA) was absent from the group, and to make sure that students remained on track. This responsibility is more of a motivational and informal role assigned to the participants, and doesn’t really have a formal assessment mechanism. The group captains ask questions to their group members and keep track of their progress through the group work. They also make sure that everyone’s voice is heard during these times. Timers make sure that there is enough time left on each task, and that the group has allocated enough time to each part of the assignment.

Another pedagogy we think is important is considering how to assign students into groups. Enes has been experimenting with Zoom’s pre-assigned group feature. With this, it is possible to automatically assign students to pre-arranged groups during each Zoom session. This reduces the time spent on assigning people into rooms during live sessions. Another advantage of this is that the instructor can decide to keep the same students in the same groups for a certain period of time or throughout the semester. This way, the same assigned roles can take effect each time students are put into breakout sessions. Enes has observed that when students get to know each other and work together, they prefer to continue to be in the same groups for the rest of the semester. One can simply upload a list of names to Zoom in advance and this way, once breakout sessions are activated, students are always assigned to the indicated rooms in the list (to learn more about this: https://it.umn.edu/services-technologies/how-tos/zoom-create-manage-pre-assigned-breakout).

We believe that breakout sessions provide more equitable and inclusive learning opportunities for all students. Math educators have long been advocating for including small-group portions for instructions to provide learning opportunities that do not typically occur in traditional classrooms (see Yackel, Cobb, & Wood, 1991 for just one example). We think that breakout sessions during online environments are a way to provide that opportunity.

While we previously were discussing some tools and pedagogies for breakout sessions including small-group work, we now want to shift our attention to large-scale classroom teaching practices, and we have experimented with a few pedagogies.

First of all, we like to ask students to turn on their cameras and click the gallery view. In order to encourage camera use (but not mandate it, as most universities prohibit) we have made remarks such as: “It feels weird to talk to a blank screen,” “I can’t tell if you are talking to me or sleeping,” or “Is there anybody out there?” We noticed that students appreciated honest and natural comments like these, through their own responses. In terms of encouraging students to put the gallery mode on, Enes would even ask random students to comment and describe other students’ camera views to make sure they can easily see others. So far, we think we had great success in students turning their cameras on in his class, as around 80% of students participated (although this has declined towards the end of semester). However, we recently found out about a report of the environmental impact of people turning their cameras on during virtual meetings and the results were somewhat concerning (https://phys.org/news/2021-01-camera-virtual-environmental.html). This made us question whether to always require students to turn on their cameras or just ask on certain occasions (e.g., when speaking). This issue is definitely up for consideration.

Enes has been facilitating online class sessions in such a way as to reduce direct instruction (or lecturing) and increase student engagement and interaction. He is almost completely avoiding lecturing or directly talking. He provides pre-recorded videos and slides to provide detailed information before class sessions. He then mostly spends time on discussing key information and the bigger picture of concepts while letting participants discuss and contemplate important matters during live classes. This provides students the time and opportunity to think and reason about mathematical concepts.

One interesting practice Enes saw the mathematician Dave Kung (http://www.davekung.com; Twitter: @dtkung) use in a talk provided by Project NExT (https://www.maa.org/programs-and-communities/professional-development/project-next) was asking the participants to move their heads on the camera in certain directions to get visual feedback. We have tried that technique in our classes when posing questions or asking them if they are ready for a certain class activity. Most of the students found that super fun, although camera mirroring caused a great deal of confusion among the class!

Another interesting pedagogical technique that Enes recently found out about was during Maria Anderson’s (https://busynessgirl.com; Twitter: @busynessgirl) talk at the Project NExT meeting in 2020. She would pose a question and ask the participants to come up with an answer to be typed in Zoom’s chat box. However, she wouldn’t ask people to submit answers right away. The facilitator asked students to not answer for a couple of minutes in order to provide opportunities for the participants to compose an answer. Then, she would count down from three for everyone to submit their answers at exactly the same time. The participants were excited about this and the chatbox would become a constant stream of answers! (We have even heard of this being unofficially called the Waterfall Method, at times.) Enes experimented with this in his precalculus course recently and noticed that students would get super engaged about trying to come up with some answers. He has observed that even the students who wouldn’t previously have attempted to answer questions are now willing to at least try and come up with something to contribute, which increases the student participation overall.

In conclusion, we have found that online teaching is less than ideal and is something we are still learning along the way. However, we believe that there are pedagogies that exist in facilitating active learning environments. The key is that technology does provide opportunities for getting connected with students and having them interact with one another.

Yackel, E., Cobb, P., & Wood, T. (1991). Small-group interactions as a source of learning opportunities in second-grade mathematics. Journal for research in mathematics education, 22(5), 390-408.

]]>We—Emily McMillon and George Nasr—are graduate students at the University of Nebraska-Lincoln. We implemented mastery based testing for two sections of a course on geometry for pre-service elementary teachers during the Spring 2020 semester, and found that our students

- looked over mistakes on assessments to improve their understanding,
- felt less stress and testing anxiety,
- experienced increased confidence in mathematics and greater growth mindset,
- viewed exams as an opportunity to show knowledge, and
- reflected on the purpose of assessment in student learning.

In this post, we will discuss what led us to try mastery based testing for this student population, how we implemented mastery based testing in our courses, and some student survey responses.

We first heard about Mastery Grading late in 2019 when Austin Mohr gave a talk at UNL on the topic. At the time, we were both teaching mathematics courses for pre-service elementary teachers. Hearing about Mastery Grading, we both, independently, thought this type of grading would be excellent for pre-service teachers. Hence, with permission from our department, we decided to implement mastery grading in two sections of the same course in the Spring 2020 semester.

Before describing what exactly Mastery Grading is, we would like to discuss some general learning goals we find valuable in a course for future elementary school teachers. Our first goal is to guarantee that our students fully understand most of the course concepts upon leaving the class. We feel that it is particularly crucial that students in an education program fully understand concepts, given that they are responsible for being able to articulate similar concepts to their future students.

A second goal is to encourage students to revisit and reflect on their previous work and mistakes. It is particularly imperative that future teachers understand that mathematical ability can be improved upon, as studies have shown that elementary teachers pass on their views of mathematics to their students.

Our third goal is to broaden the scope of students’ understanding of the purpose of assessments beyond a numeric score. As future teachers, it is important that they are at the very least aware of different styles of assessment, and, ideally, critically assess different styles of assessment to determine which is ideal for their own future students.

Overall, we believe it is important that elementary education mathematics classes are designed in a way that encourages future teachers to continue working on concepts until they have demonstrated understanding. We want our students leaving these classes feeling confident that they have truly mastered the concepts that they may one day teach for themselves. We also want assessments to be seen as low-stakes opportunities for students to show us the progress they have made, while also incentivizing them to look back at their mistakes and try to understand what it is they have yet to learn. We believe this can be accomplished with Mastery Grading.

Mastery Grading is a grading scheme by which students are expected to show complete understanding of course objectives. This is done by offering multiple opportunities throughout the semester to reattempt course objectives for all or nothing credit. There is no penalty for students taking longer to master a course objective. The goal is for students to eventually show that they understand the material, not for students to necessarily demonstrate complete understanding of material the first time it is assessed.

There are many variations on mastery based grading; our implementation as described below is but one example. Many additional resources are available online. We found the following blog very helpful and so pass it along to the interested reader: https://mbtmath.wordpress.com/.

We believe that Mastery Grading helps achieve the three goals we mentioned in the preceding section on our motivation. Mastery is designed to encourage students to revisit concepts to receive full credit for learning them. In a point-based class, students can earn partial credit for partially learning something and then may never have to revisit that concept again. In this way, students will ideally leave this course with a robust understanding of the course content.

Another feature of mastery is that it only rewards students points for a problem once they have shown full understanding of the underlying concept. This incentivizes learning from mistakes and has the potential to help students cultivate a growth mindset toward mathematics. We also feel that mastery provides students with another perspective on how to run a class and assign grades.

Geometry Matters is a required course for most elementary education majors at UNL. The course covers geometry and measurement and follows chapters 10-14 of Sybilla Beckmann’s textbook Mathematics for Elementary Teachers. This course is part of a three-course sequence that covers chapters 1-14 of the aforementioned textbook. The first course in the sequence is Math Matters, which must be taken prior to Geometry Matters, covers chapters 1-7 in the textbook. The other course in the sequence, Math Modeling, covers chapters 8-10 and can be taken at any point.

The course is taught by faculty, lecturers, and advanced graduate teaching assistants, depending upon instructor availability in any given semester. The course grade is usually determined by some combination of assessment scores, homework scores, and written project scores (so-called “Habits of Mind” problems). Students tend to do well in the course — in the last six years, pass rates have ranged from 79% to 100%, with most semesters having over 90% of students pass the course. Hence, grades and pass rates were not a reason we decided to implement Mastery Grading.

We divided the course content into 18 Learning Outcomes. Student grades were based 60% on mastery of these outcomes, with homework and project problems making up the remaining 40%. Grading for individual Learning Outcomes was for all or nothing credit, and homework and project problems were graded with a traditional points-based system. Our original plan was to test outcomes 1–7 on the first assessment, 1–13 on the second assessment, and 1–18 on the third assessment. The final would not cover new material but would be a final opportunity to master previously not mastered outcomes. In addition, we planned to offer occasional opportunities to take one to two outcomes as “quizzes” in class as the opportunities arose.

As these courses were taught during the Spring 2020 semester, we were forced to move the courses online in March of 2020. We chose to make some modifications to the course assessment structure to better work in the online, asynchronous format required by our university.

Before the move to online, we had given the first assessment as well as two mastery quizzes. The second assessment had to be taken online. We decided to eliminate the third assessment and instead replace it with weekly mastery quizzes that would each test a single new concept and offer an opportunity for students to reattempt up to two learning outcomes they had not yet mastered. Recall that quizzes were made up of exam-level problems—the only difference between these and exams was the quantity of problems. The final exam remained as previously scheduled, albeit online.

The following is a description of one of our 18 Learning Outcomes assessing areas of polygons other than rectangles, which spans sections 12.3 and 12.4 of our textbook.

- Be able to determine the area of triangles and parallelograms in various ways, including by making reference to the moving and additivity principles of area.
- Be able to use the area formulas for triangles and parallelograms to determine areas and solve problems.

That is, to earn points for this learning outcome, students would have to show mastery of both parts A and B.

Here is a sample two-part problem assessing this learning outcome, along with work from a student that did not master the concept on their first try. Students knew they may use the formulas for the areas of standard shapes such as rectangles, triangles, and parallelograms. Students also knew they were expected to express reasoning for their conclusions, and to substantiate their reasoning with ideas such as principles of area.

On part (a), the student was very close and would have earned most points for this part, but we would have liked the student to say that you can form a rectangle out of two triangles of equal area, and hence, half of the area of the rectangle is the area of either triangle. One can infer from the dashed lines the student drew on the triangle provided that they are thinking about this as two triangles forming a rectangle, but being explicit in their explanation was critical for us to ensure their understanding.

However, the work on part (b) is what really led us to feel it was critical to have the student spend more time reviewing this outcome. The point of this part was for the student to recognize the shaded region could be decomposed into a triangle and parallelogram, and that adding the area of these shapes would yield the area of the original region. The student’s attempt still showed some understanding of how one can decompose and move regions in an effort to figure out the area, which showed a desire to use principles of area. However, if one carefully checks, it is not possible to fit both triangles the student creates in the specified regions, and this is critical for the student’s method to work. (Though coincidently, they do get the right answer for the area.)

Recalling that each learning outcome was worth 15 points. We would say the student would have earned around 8 points on this problem had this been graded with points. However, due to the mastery grading system, this student had a second chance to demonstrate their understanding. Below, we show a second version of a problem for this learning outcome and the student’s response.

We recognize that part (a) has some unconventional notational choices, but we feel it is clear the student showed comprehension of the underlying concept being assessed. On part (b), we see clear improvement from the first attempt in the student’s ability to find the area of a large shape by decomposing it into smaller, familiar regions. This student earned certification of mastery on this attempt.

We would like to give a brief overview of how our students did. By the end of the semester, 37 of our 42 students had mastered at least 17 of our 18 learning outcomes, and no student mastered fewer than 14 learning outcomes. Many concepts were mastered by students on their first attempt, and the majority of students needed at most two attempts to master a concept.

At the end of the semester, we surveyed our students on their experiences in the course. There was no concrete incentive to complete the survey, but 41 out of our 42 students completed the form.

This survey consisted of two parts — a series of Likert questions, and a series of open-response questions.

We asked students to respond to the following three statements on a scale of 1 to 5, with 1 meaning “strongly disagree” and 5 meaning “strongly agree”.

- I feel like mastery grading allowed me to demonstrate my understanding of the course content.
- Mastery based grading influenced me to look at exams and try to understand my mistakes.
- This course has made me more confident in my ability to learn math.

Below are the results.

We noticed two students who wrote overwhelmingly positive things in the next part of the survey but responded with “strongly disagree” to these questions, so we infer that these responses to the Likert survey were the opposite of their intended responses.

We also wanted to give students a chance to share, in their own words, how their experiences with mastery grading impacted their experiences. We asked our students a few questions regarding their experiences with mastery grading. We also asked them to compare these assessments with points-based assessments they’d had in Math 300 (a prerequisite course also about mathematics for future elementary teachers) and to reflect on how these experiences would impact their future teaching.

Working through the responses, we found several themes that were shared among many students, which we now discuss, categorized into expected and unexpected results.

Content Understanding: As instructors, we noticed that the work being turned in by our students was of exceptionally high quality as compared to previous semesters of mathematics courses we had taught for future teachers. A few students remarked on their personal feelings that they had taken more away from the course than they might have under a points-based system.

- “Even though we didn’t have a final, I think I would have been able to pass a final easily because I actually remember the learning outcomes. This is probably due to doing the homework and actually caring to learn what I did wrong and how I can fix it. In the past, I just did the word for an ‘A’ and didn’t really bother to learn it.”
- “I felt less pressure to cram studying and to be perfect. I felt like I studied to actually understand the material.”
- “[Mastery grading] made me care more about my learning rather than stressing over a test score. I was more willing to put in the work and less motivated to use shortcuts.”

**Learning from Mistakes:** We found that mastery grading encouraged our students to look back at their mistakes on their exams. Of our survey respondents, 16 mentioned learning from mistakes as a positive takeaway of the course in their open survey responses. Many commented that they would have never looked back at mistakes they made on exams in other classes. The following two quotes are representative of the types of responses in this category. Some students mentioned specific learning outcomes they learned best, while others gave more general responses indicating that looking back at their mistakes benefited their learning.

- “I felt I learned how to do [Learning Outcome 5] the best during this course. I learned this because I failed the first time and I had to go back and figure out what I was doing wrong.”
- “I had multiple chances to show that I could master concepts and could prove that I can learn from mistakes and better myself in math.”

**Math Confidence and Growth Mindset:** As one may expect from our third Likert question, several students indicated feeling more confident in mathematics. Students mentioned how they were able to learn content they did not think they would have been able to learn at the start of the semester. We also found some encouraging comments about students’ development of their growth mindset. In total, 8 respondents explicitly mentioned math confidence or an increased growth mindset in their responses. A representative comment is:

- “It isn’t the end of the world if you don’t pass on the first try, you just have to keep trying.”

One of the interesting results was that some students even commented on growth mindset oriented toward their future students, as in the comment that follows.

- “I will always tell my students to keep trying and they will get it eventually, sometimes it just takes more time and effort!”

**Stress and Anxiety:** 15 students indicated in their responses that exams felt a lot less stressful since they could redo their mistakes. Several of our students admitted to struggling with testing anxiety and said that this grading scheme gave them some relief to that. It should be noted that several students commented that at the beginning of the semester, the “all or nothing” nature of these exams seemed daunting. However, all these students said that things improved once they became more familiar with the grading scheme and started passing outcomes.

**Exams as Opportunities:** Mastery grading also affected how at least 6 respondents felt about exams. In particular, they felt that exams were an opportunity to show their knowledge and understanding as opposed to a hurdle to be overcome. The following quotes represent these responses.

- “I knew that my instructor was looking for key factors that indicated I knew the material [on assessments].”
- “[Mastery] was based on creating a genuine understanding of the content. I feel that traditional math assessments can sometimes be more discouraging with trick questions, and this is more transparent with concrete goals and objectives.”

In our experience, students sometimes view mathematics exams as being antagonistic, unfair, or that our goal as instructors is to trick them or make them get a lower grade. To us, these responses show that students saw this grading scheme as being friendlier and more conducive to allowing them to demonstrate their understanding.

**Student Learning:** Our future teachers also demonstrated an immense capacity to think about their future students. It appears mastery grading encouraged some to think more carefully about what their concrete goals are as instructors, such as with the following student.

- “[Mastery] has helped me realize that as a teacher, I want to always ask myself, `What do I really want them to know? How do I want them to show it?’ … There are outside factors that may have messed them up in the moment, but the mastery of that skill is what I should be looking for.”

They also showed an ability to preemptively empathize with their potential students. In particular, many students who admitted to struggling with testing and/or math anxiety commented on wanting to try mastery based grading with their own students as a way to alleviate their students’ testing anxiety.

- “Sometimes students learn at a different pace, [and it’s] not fair to give one shot at something [where] if they don’t do well they can’t redeem themselves.

Other students perhaps did not struggle with testing anxiety, but still saw the importance of giving students multiple opportunities to demonstrate their knowledge.

- “I would want my students who struggle with math exams [to] benefit as much as they can like I did!”

**Challenges:** It is important for us to acknowledge that not all feedback we received was positive. There were two common themes among those that found issues with Mastery Grading. First, students did not enjoy having to redo outcomes when they thought they misunderstood only a small portion of the outcome or made only a small error. Second, students still wished they could get some partial credit for the ideas for which they did demonstrate a good understanding. A small number of students commented that having multiple chances led them to care less about any individual assessment, and so they studied less. We also noted a trend that students who had taken more “traditional” math courses, i.e., calculus sequences courses, seemed more frustrated by having to retake outcomes when they made fairly small errors.

We believe we accomplished two of our three main goals. Students seemed to be successful in understanding to course content. In addition, students appeared encouraged to learn the content and felt motivated to understand their mistakes. We even saw that students felt more confident with mathematics and demonstrated a growth mindset. However, we are less confident that we broadened the scope of students’ understanding of the purpose of assessments beyond a numeric score, although based on some student comments, it appears that our students at least started thinking about this.

There were a few less expected results that we were delighted to see. Students generally reported feeling less anxious about exams since they knew they would have multiple opportunities to show what they know. Students felt assessments gave them the chance to accurately show their knowledge. They also reflected on their experiences with mastery and how it might inform their future teaching.

We feel that future teachers were the most amenable to this style of grading as they themselves tend to value the opportunity to grow and learn.

More to this point, we have already seen evidence of how mastery has affected their future experiences. We highlight one piece of evidence here. During Fall 2020, the semester after we implemented mastery grading, some of our students took another math for future teachers class with our colleague, Kelsey Quigley. At one point during the semester, Kelsey offered an opportunity for her students to earn points back on their first exam. As she discussed logistical considerations with them, a student suggested that the way they should earn points back would be to redo the problems they individually did not do well on, as opposed to the problems the class did not do well on overall. They said, “It’s like you’re mastering the concepts you missed versus going back and doing the ones that you understood.” Kelsey had the impression that this student gained this perspective through their experiences with mastery based testing in our course.

The challenges mentioned in the previous section are important to address. While many of the challenges presented by the students are inherent to mastery grading, we feel that there are a few things that instructors can do to address the issues.

- Have regular conversations with your students. Mastery is likely to be new, so having these conversations can help them understand how to feel. Be transparent. Tell them why you’re doing this.
- Positive feedback may compensate instead of partial credit. While it is not the same as getting points, you at least send the message that you recognize the good work they did.
- You can discuss how not getting partial credit doesn’t mean you didn’t do anything good, and how it doesn’t hurt to continue to practice skills you already understand.

What we described is not the only way to approach mastery. Some implementations have multiple “Levels” of mastery, so in that sense students earn partial credit.

In conclusion, we found mastery grading to be a rewarding experience both for us as instructors as well as for our students. This testing style felt like a perfect fit for pre-service teachers, and we would encourage any instructors of pre-service elementary teachers to consider giving mastery based grading a try in their courses.

**Acknowledgements**

We wish to thank Allan Donsig and Michelle Homp for backing our desire to teach this class using mastery-based testing, Wendy Smith for her help in designing our study and methods of data collection, and Yvonne Lai for her helpful feedback and guidance in writing this article. Finally, we would like to thank Austin Mohr for introducing us to this testing method and inspiring us to try it ourselves.

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