*by Yvonne Lai (University of Nebraska-Lincoln)*

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.

## 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.

## Teaching *Mathematics* versus Teaching *Students* Mathematics

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.

## Teaching *Students* Mathematics

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.