From Teaching Math to Teaching Students Math

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.

The problem that I did not want to present

The visual of a solution that I did not want to present

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.

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Access To Epidemic Modeling

Kurt Kreith and  Alvin Mendle, University of California, Davis

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.

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In Memoriam N. N. Konstantinov

by Mark Saul

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?

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Outcomes-Based Assessment — Structural Change in Calculus

by Rebecca Torrey

Associate Professor of Math

Brandeis University

Traditional Grading Sends the Wrong Message

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.

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Mathematics as Logic

by Mark Saul

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.

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A Tale of Two Hats (Terrance and Lamar): supporting students in authentic mathematical inquiry

Terrance Pendleton, Drake University

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.

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Building Relationships Before the Semester Begins

By Courtney R. Gibbons (Hamilton College)

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.
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Best-Laid Co-Plans for a Lesson on Creating a Mathematical Definition

By:
Steven Boyce, Portland State University
Michael Ion, University of Michigan
Yvonne Lai, University of Nebraska-Lincoln
Kevin McLeod, University of Wisconsin-Milwaukee
Laura Pyzdrowski, West Virginia University
Ruthmae Sears, University of South Florida
Julia St. Goar, Merrimack College

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.
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Interactive Teaching ​IS​ Possible with Virtual Learning Technologies

By Enes Akbuga, Drake University (Twitter: @enesakbuga; Email: enes.akbuga@drake.edu) and Zachariah Hurdle, Utah Valley University (Email: zhurdle@uvu.edu)

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.

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Mastery Grading for Future Elementary School Teachers

By Emily McMillon and George Nasr (University of Nebraska-Lincoln)

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.

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