Introduction:
In Summer 2020 six to ten NE-IBLM members met once a week to work through activities in the book: "Topology Through Inquiry" by Starbird and Su. We used this experience to reflect on how we learn best in an online setting and implications for our own teaching.
Carly’s Experience:
When I took topology in my first semester of graduate school I struggled with the pace and the content. I often associate this class with my awkward transition from a thriving undergraduate student to a floundering graduate student. When the opportunity to study topology with an inquiry approach arose this summer, I was excited for the chance to revisit the material with a fresh mindset. Time spent digging into the content has certainly helped repair my negative associations with the subject, but the larger benefit came from experiencing IBL from the student perspective. Through this role reversal, I experienced components of IBL that I employ in my own teaching for the first time and was reminded of certain learning strategies that were particularly helpful to me as a student. Here are some of the lessons and observations I took away from the experience:
• I’ve been singing the praises of collaborative learning for many years without having formally experienced it. There are mountains of research that support this and I have anecdotally seen the power of teamwork in my courses but experiencing it has me more convinced than ever that working with others is the secret ingredient to effective learning. Seeing others’ thought process, approach, and sticking points helped me think about the content more deeply and consider different approaches and motivations that hadn’t occurred to me. Notably, I believe that our collaborative community’s success is largely due to the positive and supportive dynamic we intentionally fostered and prioritized.
• Normalizing mistake making was essential and required the help of everyone in our learning community! I was still getting to know most of the people in the group when I shared a proof with a truly rookie mistake about set inclusion. When someone asked about it, I was overcome with imposter syndrome. I had even failed to catch the error while I was presenting my solution! Thankfully, the others jumped in to help. Instead of dismissing my work and moving on to someone else’s “better” proof, they suggested an easy fix and then helped me talk through the edits I was frantically trying to make on the fly. This generous gift of patience and thoughtful feedback was a theme throughout our meetings that I am particularly grateful for. The moments we spent correcting someone’s work were the most memorable to me and were the topics I was more likely to recall and reuse as the content progressed in complexity. Time spent tweaking and unravelling made for slow progress early on but paid off in the long run. Maybe more importantly, time spent building a community of trust made the learning experience more impactful and more fun.
• Being in a student role is more time consuming than I remembered. I needed time before meetings to prepare myself for deep thinking and time after meetings to process everything that had been discussed. And after all of that, I was often exhausted. I found that working on problems directly before our meetings was helpful. This served as a warmup so that everything was fresh in my mind and I could participate fully but this backfired once or twice when I got stuck on something and then didn’t have enough time to complete it before our meeting. I’m not any better at thinking, speaking, and writing at the same time than I was as a grad student. Doing a “brain dump” directly after our meetings helped me solidify the main ideas of our discussions and left me with a good starting point for my next study session. But devoting time to this right away wasn’t always possible and doing it at a later time was less effective. I considered suggesting that my students reserve time before and after class for this purpose. But just as I experienced, this is not always possible. Instead, reserving the first and last few minutes of class time might be more effective and inclusive.
• Meanwhile, when our meetings ended with a big question or without a complete solution, I often found myself pondering the question or problem throughout the following week. It inspired much more depth of thought and kept me interested and excited for our next meeting. I often do this in my classes by accident and assumed that students find it frustrating or unsatisfying. However, I’m now considering ways to intentionally and constructively end class with a cliffhanger.
• Experiencing a missed meeting was surprisingly eye-opening! I watched the meeting recording the following day and found it incredibly helpful to be able to pause, speed up, or rewind parts of the recording. Going at my own pace meant I was able to keep track of questions and comments more effectively than when I was participating synchronously. This gave me the idea to watch parts of previous recordings that I had been present for. I discovered places where I had misunderstood someone’s explanation or entirely missed some important observation. It had not occurred to me that even the students who are present might find value in revisiting the class recording. Additionally, it was fascinating to observe the trajectory of the conversation without me. Had I been present to ask my long list of questions, the discussion would have been very different. This was a good reminder that although I prefer to process new ideas verbally, being able to self-regulate that impulse allows for more voices to be heard which results in more ideas shared.
• It’s still wildly fun to play in a low-stakes mathematical setting, especially with others who are as enthusiastic about playing! Other members of the group came up with incredibly interesting and clever examples and counterexamples which kept me excited to engage further and play more. The time spent with this wonderful community of mathematicians has me feeling ready and reinvigorated to make math equally as joyful for others.
Carly Briggs
Bennington College, VT
Debbie's Experience:
Initiating the Summer Topology group was scary for me, but I'm so glad I took the risk. A little over two years ago, Wheelock College, where I'd taught for 25 years, merged with Boston University. Coming to a math department at an R1 school, after so many years at a small teaching-focused school, brought up unhealed thirty-year old pain from when I had left graduate school feeling that I didn't belong in the math community, with a long list of mathematical things I thought I "should" know, a catalogue of skills for hiding what I didn't know, and a belief that I had failed, even though I had received my Ph.D.
The first summer after starting at B.U., I did a lot of reflection about graduate school. I was inspired by this quote from William Yslaz Velez, "If a student finds pleasure and joy in finding a solution, in understanding a new concept, then this emotion is a sign of belonging to the mathematical community. Emotions are not fake and this joy provides the evidence that the student is not an
imposter." I shifted my frame to thinking I had always belonged in math. I worked to understand the complexities of my experience and to forgive myself for missed opportunities. Being around other mathematicians at B.U. showed me that mathematicians weren't arranged in some linear total ordering of knowledge, with me at the bottom, and that I had mathematical things to share with others.
This summer, I thought it could be healing to go back and revisit some math subjects that I felt shame about not knowing more about, and I also wanted to directly address my grief about the many years where I didn't have peers to talk to about math. I chose topology because Mike Starbird and Francis Su's IBL based book had just come out. I've had warm and encouraging conversations with both of them in person, and I respected their work and wanted to delve deeper into it. I actually never took Point-Set Topology. As an undergrad, I had a course on Set Theory and Metric Spaces, and then as a graduate student I took algebraic topology and tried to learn some point-set topology from a book in the library; as a 23 year-old I thought I should be able to learn any math on my own, and I didn't understand that I had been poorly advised on what classes to take.
Last year I was part of an online book group from NE-IBLM, and in the spring I gingerly asked if anyone would be interested in studying some topology this summer. As we discussed the idea, we started thinking about how trying to learn math together might be a great professional development model for getting better at teaching remotely -- something that was not part of my original thought, but that ended up being central to the summer experience. Since then, I've read several pieces about online learning that suggest that teachers preparing to teach online for the first time start by taking an online class as a student, which is good advice, but I think our model was even better.
I approached the group with "everybody belongs" as a guiding principle, and we bonded pretty quickly and started sharing leadership. We joined the group with different needs and goals and with an expansive sense that we could make the group work for everyone. We took turns leading, and we got to play with various technical tools and to see each others' pedagogical moves, with ample time to digress and connect to our teaching situations.
At first I worked ahead, because I was planning to do a whole topology course that summer, but then I accepted that the group had settled on a more leisurely pace, and I was missing out on connection by working on my own. I was disappointed for a while, but then realized that my main goal had been to connect with the group and do math together; it wasn't actually to cover X chapters by August.
In the past few years I have been thinking a lot about how math classes often have future-oriented individual goals and thinking about what it might look like to conceive of their purposes more collectively and more embedded in the present. When we went remote in the spring, it was clear that for some students, our classes were providing structure, connection, and continuity during a scary time; that knowledge helped me calm down and find some purpose in the initial confusion. Moving into the fall, I plan to make the first overarching goal for all my classes, "To provide structure, connection, and joy in the midst of a global pandemic."
In our topology group, we made plenty of time to be present, to enjoy the math, and to connect with each other. I hope to bring some of the same spirit to my fall teaching in this very challenging time. Thanks again to the group for being our best weekly Zoom meeting!
Debra Borkovitz
Boston University, MA
Rebecca's Experience:
The last time I took a graduate-level mathematics course was in 1995, and females were not encouraged to attempt such courses, even then. So, I felt a bit brave signing up for topology, even though I already knew some of the others taking it were kind and supportive. When attempting to complete the exercises, I was often extremely lost, trying to find information and help on the internet and, often, not able to decipher those. But, completely any exercise felt like a triumph, as did following other's proofs, which I could usually do. The biggest take-away for was an understanding of how my students often feel, lacking even an entry point. And how scary it was when I felt too lost to even ask a question. I was in a safe space; I'm typically comfortable asking for help; but I was so concerned that my question would be so basic, it would show me to be the imposter I felt like I was. I finally worked up the nerve to ask, with the extreme encouragement of my classmates, and they were so happy to help. I'm glad to have gotten to practice productive struggle and risk-taking and to understand better how my students might feel in class. I am looking forward to going back through the exercises to see if my tenuous grasp on the material takes a stronger hold. Thank you to the NE-IBLM group for this experience.
Rebecca Mitchell
Pine Manor College, MA
