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...
Here is a screenshot of one of our zoom meetings with Megan, Chriss, Erin, and Rachel:
Blog 1: Erin Rizzie's and Christine von Renesse's Experience...
Note to the reader: This blog ended up more like a conversation. You will see Chrissi and Erin take turns...
Chrissi: Supporting Erin Rizzie in teaching Linear Algebra with more student inquiry was so easy, I almost felt superfluous. We met several times before classes started. We discussed student buy-in, different methods of structuring classes, and different grading methods. Our team in this coaching experience also included Rachel Schwell, who was interested in learning more about being a coach. All three of us taught Linear Algebra in the spring which made it easy to communicate about the material, although Rachel’s course was proof-based and Erin’s and my courses were more application based (Erin and I used David Austin’s electronic book: Understanding Linear Algebra). We decided to only include one more person, Megan Heenahan, who had already been collaborating with Erin since our NE-IBLM fall conference at ECSU. In addition, I exchanged visits with Debbie Borkovitz who I had wanted to see teach and collaborate with for a long time.
Erin: As part of the coaching structure, I visited Chrissi during the Fall semester in her Calculus I class to get a feel for what IBL teaching can look like in real-time. I had very little experience with IBL up to that point: I had sat in on one day of an IBL class several years prior when I had never heard of IBL, I had used some active learning techniques and activities from Active Calculus by Boelkins et. al. one semester a few years ago, and I had participated as a student in a live classroom that Megan led at the Fall 2019 NE-IBLM Conference. Chrissi, Megan, and I also all met virtually in December and January to brainstorm, answer some of my questions, and help me work on a plan to set up the course structure and syllabus.
For me, these meetings and the observation were immeasurably helpful in a few ways: they gave me ideas, guidance, and confidence. First, I took from these meetings and observations some ideas and jumping-off points to be able to imagine how I might integrate more inquiry into my own classroom. Second, I think without Chrissi and Megan’s previous experience and “warnings,” I wouldn’t have taken as seriously the need for students to buy in to my use of inquiry. Based on my experience in two sections, where one seems to have “bought in” more, my feeling is that the whole semester would have been a disaster if I had not implemented some of my mentors’ ideas for the buy-in phase of this course. Finally, on a somewhat more personal level, I felt apprehensive about my ability to effectively implement inquiry-based teaching before working with Chrissi and Megan in December and January. While any new endeavor is uncomfortable in this way, knowing that I had a “team” rooting for me and working to help me succeed made it easier to take that first leap.
Chrissi: When Rachel and I visited Erin in the first month, she already showed the confidence and routine of an experienced IBLer.
Erin: My class was set up this way: students drew playing cards as they entered the room once a week to be placed into randomized groups for activities, discussion, and quizzes. We began the day with a few students randomly selected to share their solutions to the preview assignment, which was assigned at the end of the previous class meeting as a bridge from previous topics into the new day’s ideas. Once the class had discussed the preview solutions, I typically gave a necessary definition and/or theorem to lead into the first small-group activity. Students would spend some time with that activity while I circulated and facilitated the small-group discussions. Once some groups were finishing up, we would discuss the activity as a whole class (I would have several groups’ chosen “spokespeople” talk about their solutions). I tried to lead these discussions toward the main idea that I was hoping for the students to get out of them, whenever possible. Of course, students often needed to go down another path for a while! In any case, once the class had articulated some of the main ideas, I would share the “official mathematician’s version” of whatever definition, theorem, or idea they had arrived at as a class. We would then lead into another activity with a similar set-up: I might give them another definition, if needed, and they would work on the activity for a while. If there was time, we would discuss their solutions as a large group before wrapping up for the day.
Chrissi mentioned that one thing I did in my class was new to her, though it was somewhat natural for me from my experience teaching in large lecture halls. I created a set of Beamer slides for each week, with my intended order of events built in. We would start class with a blank slide stating anything students should do as they come in (i.e., draw a card, or sit with groups from last time) and what preview assignment they should be prepared to discuss at the start of class. Then, using the stylus on the classroom “smart” computer monitor, I would write or have students write their solutions to the preview assignment directly on the blank slide. The next slide(s) typically had the definition or idea that I wanted them to know going into the next activity, so that I could jot down any notes as we talked about them and leave the annotated slides up for them on the projector as they worked through the activity. Next was a blank slide with the activity title at the top, where I would write or have students write their solutions to the group activity. At the end of a week, I had a record of all (or most) of the notes and solutions that we discussed as a class. I would convert these to PDF and upload them to the LMS for students to look back at if they needed to review.
I also visited Megan’s class in the first few weeks, where she did some interesting things that I hadn’t considered before. First, students spent the first part of class working in groups to correct their activities from last time, in colored pen so that Megan could see where they had struggled on their own when she graded their work later. I loved how engaged students were in correcting their work, where often students in other classrooms tend to want to leave behind their errors and move on the next thing. Something else she did that gave me an idea was in how she chose students to talk during whole-class discussions. She had a rule of only calling on a student who hadn’t spoken yet that day, so that all students eventually had a chance to share their ideas with the class. I don’t know why this had never occurred to me, but it was a great way of breaking out of the habit of letting the same students share their answers all the time. I’m planning to try it in some of my future classes.
Chrissi: We met after Rachel and I visited Erin’s class and discussed different options for enhancing whole class discussions, especially since her room had a strange shape and students had trouble hearing each other. We also thought about how to differentiate if some students are stuck and some are done. When classes moved online, we visited each other again, comparing our online solutions. In both her and my classes, the break-out groups still worked well, although her students were “better” at looking up solutions while mine were more likely to talk about non-math things. The whole class discussions felt more guided for both of us. Erin picked groups to present their answers, while I would “randomly” choose a single student to share their thinking. In contrast, Debbie let all her students share simultaneously in the chat which allowed for everyone to contribute. It was fascinating for me to see the different approaches and how they all accomplished different aspects of what we were used to in IBL in the classroom.
Erin: I also visited Chrissi and Rachel’s synchronous online classes after the move to remote delivery. Rachel’s breakout groups in WebEx went well, too, and I could see how well some of her students had gotten to know each other and liked to work together. Her students were somewhat more adept and/or willing than either mine or Chrissi’s to share work visually on the screen via whiteboard (or perhaps WebEx was easier to use for this purpose than Collaborate and Zoom, or Rachel did a better job of setting them up for success in this respect?). This made for some interesting discussions and offshoots of the main activities.
When I visited Chrissi’s class, I loved seeing students’ faces on video and noticed how willing most of them seemed to be to ask and answer questions. Our one-on-one meeting after class was enlightening for me: when Chrissi said that she had been nervous about having a visitor, it helped remind me that even the “experts” are struggling and looking for ways to improve – which helped to put me back into my own growth mindset regarding teaching with inquiry.
Chrissi: Personally, I struggled the most with letting go of my urge to cover everything I had planned and to go with the (different) flow of the online classes without feeling stressed out. Exchanging visits with Debbie and our joint reflections helped me immensely. Hearing Debbie share how she had “let go” of her expectations and worked on supporting her students allowed me to do the same. Once I had shifted my mindset, I was actually able to get to a lot of what I had originally planned, go figure.
Erin: I suppose I struggled with this “coverage issue,” too. I ended up covering everything I had planned on in my syllabus. In hindsight, this may have been to the detriment of the students’ mathematical development. There were times when I could have slowed down a lot and really tried to get students to dive deeper into some definitions. It certainly showed later in the semester as they struggled. For example, some of my students had a hard time with the concept of an eigenspace (which I defined as the null space of A-λI), which made a lot of sense when some of the same students mentioned that they still had trouble with the definition of null space!
Chrissi: I can’t reflect on the semester without thinking about my students: In their final journals, they reflected on how the Zoom classes with break-out groups and whole class discussions really didn’t feel that different from our in person classes. They struggled the most with motivating themselves to get work done, or added family responsibilities and anxieties. I was surprised to hear that according to their reports, most college classes did go the asynchronous route. While I know that there are equity concerns about synchronous approaches, I found that all my students were able to participate meaningfully. Some had to watch the Zoom meetings on their computer and talk on the phone or type in the chat instead of using computer audio. Some students joined without video but they were there and participated. And a few students had to work during the day and watched the recording meetings later and met me in extra Zoom office hours.
Erin: For me, as I think forward to teaching this course again over the summer, I am thinking hard about the ways that I might be able to incorporate inquiry and foster student working relationships in some of the asynchronous components of the course. For my setup in Spring 2020, students’ out-of-class work was generally individual (though I encouraged them to contact their group members outside of class and many did). I am looking toward summer knowing that the set-up I had in Spring will likely not work as well for a fully online, accelerated 5-week course. I think it would be hard to get through the number of in-class activities needed during a 2.5-hour day to keep up, and students would certainly tire of, or tune out altogether from, the same group work structure for such a long time each day. My hope is that whatever I come up with to try out over the summer will inform my teaching over the coming year or two as we face much uncertainty about the mode of delivery of our courses.
Chrissi: My final exams (oral) showed me that almost all of my students learned the procedures and understood most of the concepts. Details of the interesting applications I had to cut unfortunately, but my computer science majors gave me the feedback that they could see how relevant linear algebra is to their major. Specification grading was really helpful in going online as students could keep working on assignments until they mastered the material.
While I didn’t get the energy from teaching online that I get from teaching in person, I learned to enjoy the Zoom meetings. Having visitors in my online classes was scary at first, as I felt as if I was teaching IBL “blindfolded.” But all my visitors helped in the breakout groups and I realized that teaching in a “silo” is never a good idea, especially when I am struggling to learn something new. Collaboration is what allows me to grow.
