Topology Professional Learning Community – Summer 2020

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

Summer Teaching During COVID: Reflections, Lessons Learned, and Moving Forward

 

This reflection is the joint work of five members of the NE-IBLM community, Kyle Evans (Trinity College), Roser Giné (UMASS Lowell), Jean Guillaume (Sacred Heart University), Rachel Schwell (Central Connecticut State University), and Ileana Vasu (Holyoke Community College). The blog is mainly structured as a Q&A with an imaginary interviewer, to allow for easy parsing of course logistics. The latter part of the blog is more open-ended, with anecdotes and final personal reflections from each contributor. Among the topics discussed are course materials, technology, engagement, assessment, and academic honesty. Please feel free to explore and jump around via the links that will be revealed by clicking on "Read More." Also, each contributor’s responses have been color coded to allow you to follow their individual thread if you prefer. We hope you enjoy reading about our experiences! Contact information is at the end if you have follow-up questions for any of us.

Differential Equations Professional Learning Community - Spring 2020

The idea of forming a Differential Equations Working Group emerged during the fall NE-IBLM conference in CT. In spring 2020, four members of NE-IBLM (Max Ahmadov, Ileana Vasu, Mel Henriksen and Mami Wentworth) organized a Differential Equations Working Group, and we were joined by Viktoria Savatorova and a faculty member from Converse College. We met online every other week from January to April to discuss strategies for inquiry-based learning techniques in our introductory differential equations course. 

A part of the discussion was based on the Active Differential Equations curriculum that Mel and I developed over the years. ADE presents a sequence of questions that are designed to engage students in thinking and doing mathematics with short explanations dispersed throughout the curriculum. Although members of the working group were covering different lessons at the time of the meetings, this difference allowed us to talk not just about general questions and suggestions for each other, but ideas and concepts that students struggled with in class.

Max teaching Laplace Transforms with Active Differential Equations at Holyoke Community College in December 2019

After transitioning to online learning in mid-March, the working group provided a nice opportunity for the attendees to share successes and challenges that we faced during a difficult, unprecedented time. Our conversation varied from a feature in Zoom one could employ for effective learning to techniques for assessing student work in an online setting. 

Overall, the working group allowed interested faculty to meet and discuss strategies and struggles with other faculty teaching a similar course. It provided a safe space for faculty to reflect on their teaching style and learn from others. 

If you are interested in joining the Differential Equations Working Group or taking a look at Active Differential Equations, please contact Mami at wentworthm1@wit.edu or Mel at henriksenm@wit.edu. We would love to discuss the content, strategies or anything related to teaching and learning differential equations. 

Linear Algebra with Inquiry, Spring 2020 - Part 4

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 4: Debbie's Experience

Last semester Chrissi and I visited each other's classes when I was about three weeks into remote teaching. I was starting to come out of the acute crisis phase and moving into more of a routine for all three of my classes, albeit one laced with anxiety. In addition to the stressors facing all students during the switch to remote, about two-thirds of my 90 students were international students, predominantly from China, and I had many students who were abroad when remote teaching was announced on our spring break, some who were trying to leave and having flights cancelled, some who were in quarantine with possibly bad Wifi, and some who needed to apply for permission to stay when the dorms officially closed. I was overwhelmed with trying to keep track of where everyone was today and whom I should prioritize reaching out to. As with everyone else teaching at the time, I was tired from all the thought and work involved in trying to learn to teach remotely and decide how to modify the course structures mid-semester. But three weeks in, I had a plan, I had some classroom routines, and I was ready to take advantage of how much easier it was to visit our classes than it would have been in person.

I am used to visiting classes and having visitors -- I was the department chair for many years in a collegial, teaching-focused department at Wheelock College, before Wheelock merged with Boston University two years ago. However, I was nervous about having Chrissi visit, much more so than I'd have been for an in-person visit. I had a bit of imposter syndrome around being a person who supposedly knows a lot about IBL/active learning, because I was doing many more whole class activities than I thought I should be. Students referred to what I thought of as interactive discussions as "lectures" and told me how much they liked them, how they thought I should do more of them, and how helpful the videos of them were. Attendance was way down in my afternoon linear algebra class, which met at 2 p.m. (that's 2 a.m. in China and 3 a.m. in Korea), so many students could only access the live class by watching videos later.

When I visited Chrissi's class, the first thing I noticed was the differences between the populations we were teaching. Her students were mostly white, mostly native English speakers, and mostly tuning in from their bedrooms in Massachusetts. Next I was relieved to see that she was also spending a lot of time in whole class discussions, but she was conducting them differently than I was. She worked through an activity in the book, asking questions and then calling on students to answer them. I was very impressed that students seemed fine with being called on and with saying they didn't know an answer or sharing their confusion; I know that a lot of work goes into creating a classroom culture that is both emotionally safe and intellectually challenging.

Chrissi came to my morning class the next day, and I had the best Zoom class I'd had up until that point. Partly class went well because I spent extra time planning it due to my nervousness, and partly it was luck (my afternoon class on the same topic didn't go quite as well). In reflecting now, I think visiting Chrissi's class also improved my own class, because seeing her call on students reminded me of how much my students -- including many English Language Learners -- used to talk in class before we went remote. Since I relied so much on non-verbal cues to decide who to gently encourage or who to call on, in moving to remote learning, I'd backed off some on encouraging students to talk, but I pushed more that day.

In remote learning I often used the "3-2-1 Go" Technique, which I learned from Maria Andersen. The day Chrissi visited, I gave students some time in breakout rooms to discuss some true/false questions; then when we came back together I asked students to type whether they thought a question was true or false, but not to hit return until I said, "3-2-1 Go," and then we discussed the question.  A few of my questions that day ended up having much dissent about whether they were true or false; one such question was, "If u and v are both eigenvectors of matrix A, then u + v is also an eigenvector of A." About half the class said true and half false, and I excitedly said that such a distribution meant I'd written a good question. In our debrief the next day, Chrissi pointed out that my giving myself the responsibility to write a good question was a good frame for the students, since it relieved them of feeling the responsibility was on them to always get the right answer. I found Chrissi's way of articulating what I was doing very helpful, and hope to be able to use it more consciously in the future.

We had a lively discussion in the class that Chrissi visited, where many students showed their videos and I was able to circle back to incorrect answers and have students think about what in their thinking might have been true. In my second linear algebra class that day, only two students showed their videos and fewer students volunteered to talk. I did start calling on some people, perhaps emboldened by seeing Chrissi's class, to ask things like, "Can you tell me why you answered false?" Some of these prompts restarted the discussion, and some of them yielded silence.

To write this reflection, I went back and watched the videos of both my classes that day. Now it's clear to me that even if the students labelled the class “a lecture,” I was using active learning strategies, that students were thinking about interesting questions that expanded their conceptual understanding, and they were discussing these questions with each other and with me. I think some of my initial nervousness and fears that I wasn't really doing IBL right were because Chrissi had never visited my classes live, so a part of me wanted to fixate on "this isn't really how I teach!" and then think of some labels for how I teach that I might not be living up to. As Chrissi said in the debrief, it's vulnerable to say that we're new to this, not necessarily very good at it, but we're going to let someone else join us so we can both get better together. I'm pleased that we both took that risk, and that in doing so, we were able to do a little better by our students.

Linear Algebra with Inquiry, Spring 2020 - Part 3

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).

Linear Algebra with Inquiry, Spring 2020 - Part 2

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 2: Megan's Experience

At the fall NE-IBLM meeting I was excited to learn that Erin would also be teaching Linear Algebra and wanted to implement IBL techniques for the first time during the spring 2020 semester. It had been quite a while since I taught linear (before I knew anything about IBL), and I wanted to completely revamp the course. Erin and I started exchanging emails about materials and plans. When she got the faculty coaching fellowship she and Chrissi kindly included me in meetings. It was helpful to be part of these meetings early on. During the meetings I was reminded of IBL techniques and exposed to new ideas. Erin was much more familiar with the course material which helped when choosing a book and in trying to figure out where students would struggle. It is always helpful to hear how others are organizing their course and this was no exception.

After our meetings during the winter break, I felt ready to lead a more active inquiry-based linear algebra course. In person, for the most part, the course went well. It helped that I had had about half of the students before, so they were used to a more active classroom. The students found the activities from Understanding Linear Algebra engaging and challenging. Activities often took longer than I expected, but groups were having meaningful conversations. One thing I need to improve upon is how I ask questions of the students, how I get information out of them, and my whole class discussions. In observing Chrissi’s Math for Liberal Arts course I tried to pay close attention to how she asked questions of the students and her prompts for students. I often feel like my questions of students give too much away and I’ve been told my face gives a lot away when students answer questions.

For me, the whole structure of my course came crashing down when we moved online. Between technology, my emotional state, and my students’ emotional state everything just seemed overwhelming. Focusing on my synchronous class meetings while hearing my daughter in the other room was very hard. Students were struggling with family responsibilities, technology, additional work hours in grocery stores and nursing homes, motivation, and the general state of the world. I used Microsoft Teams, which was great for organizing everything, but not great for transitions from whole class discussions to groups. It felt like everything took 5 times longer and we were getting nowhere. My “mini lectures” started being not so mini and eventually I gave up on the group work (it just didn’t seem productive). Students still completed preview activities, but in class it became a combination of lecture, students working individually, and whole class discussion. Within this small IBL group, I started to feel like an imposter. Of course, when I admitted to the group that I had given up on most of the IBL components of the course the group was extremely supportive. Admitting this to them and hearing their reaction allowed me to make peace with what the course became. It was a good reminder that I am usually my own worst critic and sometimes it is OK, even necessary, to let some things go.

Linear Algebra with Inquiry, Spring 2020 - Part 1

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.