Introduction
Many faculty were interested in teaching linear algebra with inquiry in Spring 2020 so we had a mixture of visits and collaborations happening:
Chrissi von Renesse (Westfield State, MA) and Rachel Schwell (CCSU, CT) are both members of the NE-IBLM leadership team, and co-creators (along with the other members of the team) of the Faculty Fellowship & Coaching Program. Erin Rizzie (UConn, CT) was an applicant to this program specifically for linear algebra for spring 2020, and as Chrissi and Rachel were both teaching linear algebra that semester, we created a team of three: Chrissi as lead mentor, Erin as faculty fellow, and Rachel as apprentice mentor. Megan Heenehan (ECSU, CT) had been informally working with Erin and Chrissi and so she became a fourth member of the team in an unofficial capacity. Debbie Borkovitz (Boston University, MA) was also teaching linear algebra using the same book as Chrissi, Erin, and Megan, and so Debbie and Chrissi exchanged visits during the online phase of the spring semester. Our linear algebra classes differed in sizes: Erin had the largest with two classes of about 35 students, Chrissi had 20, Rachel had 15, Megan had 30 and Debbie had two classes of about 30 students each. We reflected together on our experiences in a series of blogs...
Blog 3: Rachel's Experience:
I was assigned to teach Linear Algebra at CCSU in the spring semester of 2020. We have two versions of this course, a more computational one for engineering and science majors that does not have any proof-based prerequisite, and a more abstract/theoretical one for math majors that has Discrete Math (serving as a transitions course) as a prerequisite. This course was the latter of the two. I had about 15 students after the attrition of the first few weeks, most of whom attended regularly when we were on ground.
I used the OER “Linear Algebra” by Hefferon along with guided IBL activities I created. I also had a presentation system, where students would select a problem from a list I provided to present to the class. This list also included thoroughly explaining proofs of theorems that were already given in the book, but with many details needing to be filled in. This was my third time teaching this course using this system so I felt confident in the set-up.
However, I also knew from experience to expect varying levels of preparedness for abstract and/or creative argumentation. In particular, for most of the students this is their first time actually applying the proof techniques they learned in Discrete Math, and perhaps the first time they have been asked to write a proof without a template. This feels like jumping in the deep end for many of them. Because of this, students often need “training wheels” in the beginning in the form of hints on the homework. As they get more practice throughout the course via IBL, the hints slowly fade away. In particular, the presentations seem to help a great deal with this because not only do they build each individual presenter’s confidence, the audience is able to believe they too are capable of taking off the training wheels by witnessing it in their peers. That being said, I still suffer from high attrition rates in the beginning of this course in particular, and I am definitely still working on that.
Before the transition to online, Chrissi and I were able to visit Erin’s class once. I was shocked to learn this was her first major attempt at IBL. She had so much structure in place, including systems for organizing both groups and participation. I found this very courageous on her part, to jump in head first to such big changes. She was also clearly committed to this structure – she did not fall back on speaking for the students even when responses were slow or delayed. If there was a way in which I could sense any inexperience on Erin’s part, it would be that I could tell she wasn’t totally confident her students would perform/contribute as she hoped. In other words, she didn’t necessarily expect them to rise to the situation but was still in the stage of hoping they would. This distinction can go a long way with student buy-in. It can also improve dramatically after the first attempt at IBL; once an instructor has had one successful experience, they can expect to have another.
Another observation I made during this in-person visit to Erin’s class led me to reflect on my own teaching. Erin’s class was much larger than mine, maybe double the size, and many students were able to go the whole period without contributing out loud. It reminded me that as observers of a class, we notice the students who are not participating much more than we do when we are standing in front of the class. This has even been my experience visiting classrooms on my own campus as well, across multiple disciplines. In particular, I am certain this happens in my own classroom more than I realize or would like to admit, and it often takes more effort than I feel I have the energy for to truly ensure full participation. I personally found this easier online because I simply went down the list of participants in order and called on them. For some reason I found that more natural than I find calling students in order based on their physical positions in a classroom.
Chrissi and I had planned one more in-person visit to Erin’s class, and Erin had planned to visit mine. While the transition to remote learning forced us to alter those plans slightly, Erin and I were indeed able to visit each other’s classes virtually. One thing I noticed about both of our classes was that in some cases, participation in group work was harder to determine because we could not see or hear students if they were working things out quietly on their own papers to later share and discuss, but that in other cases it was actually easier because each breakout group was an isolated little bubble in which it was very clear when someone was speaking or not.
Another observation I made during my virtual visit to Erin’s class was regarding the activities themselves. The textbook Erin was using did not contain all the topics she was required to cover, so at this point she was using activities she created herself. Overall, I was impressed, and would definitely want to use them myself if I were covering those topics (which I didn’t this particular semester). But maybe more importantly, I really gained some valuable insight in terms of activity-creation, being able to participate with the students as someone who did not already know the narrative of the mathematical topic the worksheet was aiming to tell. I noticed that even though several of the groups successfully worked through the questions on the activity, at the end they were still unsure of what they had actually done. The activities were extremely successful in getting students from point A to point B algebraically or computationally, but didn’t necessarily communicate the big picture. (In fairness, Erin did discuss that once she brought everyone back together.) But it reminded me of myself in my grad student days or even as a colloquium attendee now: I can usually follow each individual step, but if I don’t understand why we did them, I will feel as though I haven’t understood. This prompted me to wonder if I myself overlook this important piece in creating my IBL activities, not just in linear algebra, but in all my courses. It reminded me that the big picture is an essential part of the IBL process, not to be overlooked or minimized.
Overall I really enjoyed and benefited from the experience. While my role was initially that of an experienced IBL user who was meant to be learning to mentor another faculty member, once we transitioned to remote learning I felt it became more of a peer situation as I was equally inexperienced in an online setting. In fact, when Erin first asked to visit my class online, I wanted to wait a couple of weeks so that I could get back to feeling like the experienced role model I was supposed to be. But then, Chrissi communicated to us how unsure she herself was feeling, and how much more self-conscious she felt having a visitor than she would have in an on-ground visit, and I realized that the point of the visit shouldn’t be for me to present perfection, but rather a snapshot of how I am handling the situation. Once I accepted that, I no longer felt nervous for a visitor because, to be honest, my ego was no longer wrapped up in how well I “performed”. I was able to acknowledge that there would almost definitely be imperfections, and that in fact, those imperfections would be much better discussion points. I also realized that a true pedagogical role model is always learning, and that if anything I would demonstrate more professionalism by being totally open to that in this situation.
Finally, I must acknowledge that I felt I had an excellent group of students in linear algebra this semester. I am proud of how well the course ultimately went, but I do not feel I can take full credit for that because those students showed up prepared and eager to discuss with their friends/classmates. While all of my classes this semester did have at least somewhat successful group work, the ultimate overall learning was not as strong as it was in linear algebra - abstract algebra in particular for me was a disappointment. My goal moving forward is to dissect what really made this experience work so I can pass it on to all future classes (online or not).
