Park City Mathematics Institute
Learning about Geometric Groups
Project Abstract

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Reflections from the Learning about Geometric Groups SSTP Working Group, aka "Groups Group"
Bond Caldaro, Tina Cardone, Brandon Hatfield, Brian Hopkins, Kayty Himmelstein, Erik Johnson, Marvin Jones Jr., Joe Ochiltree, and Gabe Rosenberg

The Groups Group attended the 1pm course taught by Jen Taback for the Undergraduate Summer School. At 2pm each day, we debriefed, reviewed and expanded on class notes, and worked together on the frequent problem sets.

The course built on the typical undergraduate treatment of finite groups by developing Cayley graphs for several infinite groups. New properties, applicable to infinite groups, were introduced in conjunction with important examples, so that definitions such as "almost convex" were motivated. The primary examples were the families of free groups, Baumslag-Solitar groups (which serve as discrete models of hyperbolic space), and lamplighter groups.

Each participant wrote a brief reflection on the experience, excerpts below. Every participant explained that they enjoyed this mathematically challenging and rewarding experience. There were also insightful comments about aspects of the lecture and our follow-up sessions on the participants' teaching.

"Taking this class has been useful in helping me to reflect on my teaching practice. As a student in a class with content that was extremely challenging for me, I found myself adopting many of the behaviors I have observed among my students: feeling 'stupid' and incapable of learning the material, being embarrassed to ask what seemed like "basic" questions, and attempting to avoid being called on. In the fall, I hope to share this experience with my students and discuss with ways that my behaviors might have interfered with my learning. Being explicit with my students about this might, I hope, allow them to be more reflective about when and why they are adopting these behaviors. I would also like to share with my students some of the positive academic behaviors I have re-learned from being in this class, including studying with others, forcing myself to do "extra" practice to ensure I understood the material independently, and exposing myself to the material in multiple ways (reading from the book, attending class, discussing with others)."

"One thing I have learned in my teaching career is that it is very difficult for English Language Learners to take lecture notes: their language processing capabilities are simply not advanced enough to be able to write and listen to academic content simultaneously. For myself, I felt that I often had to choose between writing and trying to understand --- a tricky balance. I made the decision to spend more time making sense of what was being said and less time dutifully recording the contents of the chalkboard into my notebook. What helped amazingly was the discussion sessions that we held after the lectures. In these sessions, I felt no restrictions in asking even the most basic questions, and we made it a point to make sense of the lectures together. This was something that I probably never did during my undergrad education taking math classes. Looking back, I wish somebody would have structured similar experiences to help me learn to learn more from lectures. This was certainly a valuable experience that connected to and expanded on my math background. As a student, I have a renewed realization that there is so much more out there for me to learn. And as a teacher, I come away more firmly resolved to give students the structure they need to be successful, and to always put meaning-making at the center of my classroom."

In addition to the work of group facilitator Brian Hopkins, the expertise of participant Gabe Rosenberg helped the group very much.

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This material is based upon work supported by the National Science Foundation under DMS-0940733 and DMS-1441467. 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.