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.

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:

1. 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.
2. 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.