With 2020 in the rearview mirror, we take a moment to pause and reflect on some of what’s happened since the last newsletter. The rising incidence of COVID-19 infections brought with it a massive upheaval for American higher education institutions—forcing students, in many cases, to vacate dorms, and instructors to rapidly transition from hybrid to exclusively online forms of instruction. While this upending of “business as usual” has certainly come with its share of unique and difficult challenges, we have heard from you that it has also provided some surprising and uncommon opportunities. This fall, the ubiquitous use of online and hybrid instruction provided us the possibility and privilege of peering into some of your classrooms. We’d like to share some of what we learned.

We would be remiss if we didn’t acknowledge the challenges that this time period has created for undergraduate students. For one, the disruptions to more routine university life has brought unique logistical challenges for undergraduates. In the moments between problems in one small group, we overheard one student sharing about the struggles she was having with the university’s decision to leave some courses in-person and others remote—putting her in a situation in which her parents insisted she save money and move home only to face several two-hour commutes over snowy roads for her one in-person class. We also observed ways that the sweeping changes to university life has left students feeling uncertain about their futures. In another small group exchange, two students discussed their fears regarding their ability to make progress in their teacher preparation programs with local districts and mentor teachers feeling too overwhelmed to make room for university students’ field placements. Also, we observed ways that these shifts have forced many students to blend their various roles (i.e., in addition to being students, they have familial obligations to their siblings, parents, and sometimes spouses or children) making it more challenging, at times, to focus on their role as students. For example, in the midst of a virtual whole class debate about whether or not Euclid’s third axiom holds in hyperbolic space, we observed one student splitting her attention between the small wiggly child in her lap and taking her turn in the debate—sometimes struggling (understandably) to keep her train of thought.

Yet, in the face of these challenges, we were consistently impressed with the innovative ways that both instructors and students worked collectively to maintain some semblance of normalcy in instruction. We observed instructors and students making use of physical manipulatives to maintain students’ engagement with instructional activities. One instructor told us how, in order to avoid the risk of spreading the virus, she organized and distributed physical manipulatives at the start of the semester to enable students to gain familiarity with the kinds of resources they will eventually need to use in their work as high school teachers. Another instructor supported students over video conferencing to construct their own 450-degree cone out of construction paper and then used these cones in a series of investigations exploring the viability of Euclid’s postulates on the 450-degree cone. 

Beyond the use of physical manipulatives, we also saw familiar technologies being leveraged in creative ways to support teaching and learning in the GeT courses we observed. Technologies as simple as document cameras, video cameras, and even social media were leveraged by instructors and students alike to not only cope with but also thrive in the current situation. On numerous occasions, we even watched as students helped one another and the instructor out by using social media platforms, such as Snapchat and Instagram, to share course files with other students. In some cases, instructors have shared how they have identified new practices during this time that they intend to keep—even after the pandemic is behind us. For example, one instructor shared her plans to continue taking advantage of newly-installed classroom cameras to record students’ mathematical presentations, enabling students to reflect on and improve their communication skills. Another instructor shared plans to maintain virtual office hours for students—noticing a substantial increase in students’ participation.

We also saw instructors and students making use of math-specific technologies to overcome their diverse geographic locations. For example, in a hybrid GeT course, students were managing (in spite of their geographic distance) to collectively communicate and reason about proofs using dynamic geometry software. Crucially, students had learned to harness the functionalities of Geometer’s Sketchpad to remotely walk through the steps of their proofs—simultaneously highlighting statements in their proof and corresponding elements of the given diagram (see Figure 1). 

The students’ used this functionality to participate in a core mathematical practice of listening and critiquing one another’s proofs. Also, this use of Sketchpad enabled a kind of dynamic form of mathematical communication in which a single gesture towards a proof statement is intrinsically linked with the corresponding aspect of the geometric diagram that we think has potential beyond this current season of disruption to normal instruction. In all cases, we saw instructors and students finding ways to use the tools they had available to communicate about new and powerful ideas  for students’ mathematical development.

Overall, what we saw while “sitting in the back of your classrooms” left us curious about the future. We found ourselves wondering what kinds of pedagogical practices students in these courses were having opportunities to “apprentice” into.  We think it’s possible that this cohort of students is more ready than any previous cohort for engaging in online teaching, which may take the field a long way towards the goal of increasing the capacity for teaching high school geometry. We also wondered whether pre-service teachers would translate the kinds of practices they are learning, such as the use of Sketchpad to communicate dynamically about their proofs, into their future professional work. And while we, like you, long for things to return to some semblance of normal, we also found ourselves hopeful regarding the kind of resolve and commitment we saw in both GeT students and instructors. We think it’s possible that undergraduate mathematics instructors and their students will come out of this difficult season stronger and more ready to engage in mathematics education for the 21st century.