December 2023

Tifin Calcagni

After nine years of teaching math and science in international schools, I took a year off in order to step away from the classroom, avoid burnout, and gain a broader perspective. So I wasn’t teaching when the pandemic started. But I re-entered the classroom in September 2020, taking a new job teaching middle school math at the American School of Brasilia.

The environment of teaching and learning that I entered that fall was worlds away from what I had left one year before. My students were attending classes fully online, meeting for 45 minutes daily on Zoom. My job requirements were the same as they were in schools I taught during pre-pandemic times: to teach the curriculum in a mindful, holistic way, producing kids who were confident and competent in mathematics. My students had been learning online for the previous four months, since the pandemic forced us into isolation in March of that year. I was confident that they would show me the ropes, as they had in other schools I entered as a new teacher.

I was surprised by the indifference I saw in the students, starting the first day. They attended the online classes, but clearly weren’t paying attention. A few weeks in, many were not even logging into Zoom. Others would sign in, turn their cameras off, and not participate.

I worried about the kids who weren’t present for the class. While the media featured “learning loss” from the COVID 19 pandemic front-and-center in their headlines, I was more concerned that my students weren’t developing resilient learning habits. I was also worried that when they eventually tuned in to math class again, they would feel lost, and would come to hate and fear the subject.

Attending school from a computer screen created many barriers. There was background noise. Many family members online at the same time caused slower internet connections. Often, my students had to share computers with their siblings. With all of these challenges, I didn’t necessarily blame them for skipping a class that felt boring and irrelevant, although it was making my job more difficult.

How was I to teach them if they weren’t showing up?

My first goal was to get students to tune in to class. I couldn’t address the standards if I didn’t even have their attention. Instead of a typical lesson with a learning objective on the board, a skill-based, didactic lesson, and a test for understanding at the end, I gave students what I thought was an engaging problem to work on. I made several attempts at creating interesting situational problems that I thought they would like. This included problems involving my students’ interests relating to their favorite sports teams, music groups, and characters from the books they were reading in their humanities class. While these types of questions increased participation from those students who were already actively engaged in class, they didn’t pull in the ones who weren’t attending, or who signed in, turned off their cameras, and weren’t engaged.

I needed another plan.

For the past six years I had been working in an after school program called The Global Math Circle, founded and developed by Bob and Ellen Kaplan. This program engaged kids in deep mathematical thinking by treating them like mathematicians. We would put a challenging problem in front of them, and allow them to explore it in whatever direction the group decided. I always had success getting kids in my math circles to actively participate in the problems, but I hadn’t considered using these techniques in my classroom. For one thing, the topics were often outside of the scope of the math curriculum I was teaching. On top of that, math circles allow students to choose the direction that the circle takes, which would present a challenge when I need to create learning objectives and assess understanding. But now, facing a crisis in student engagement, I considered adopting the math circle practices in my classroom. What did I have to lose?

Without the confusion that existed during the pandemic, I would not have dared to veer so far from standard teaching practices. The possibility of my experiment going poorly and students ending up learning less than they would have using traditional teaching methods would have stopped me. But teachers had been given a sort of grace period during the pandemic. Expectations for teachers were relaxed. Often, schools were happy just to be operating at all. I used this chance to try teaching methods that I would not have otherwise attempted.

Inspired by the captivating challenges in a math circle, I left behind standard textbook problems and ventured into the realm of intriguing, abstract mathematical questions. At the beginning of the following class, I held up a blank slide with a single dot in the center and asked my students, “What do you see?” The responses varied, with some calling it a “point,” others a “dot,” and some described it as a “black circle.” I then progressed to showing a line segment, followed by a square and eventually a cube. “What comes next?” I asked, and encouraged them to sketch their ideas.

I sent them into Zoom breakout rooms to work on the problem together. As I moved through all the breakout rooms expecting silence and blank screens, I was surprised to see every group talking. They were discussing the problem! We came back together, and they shared their ideas with the whole class.

One group proposed a simple pattern as a possible rule. They suggested that each shape is made by duplicating the previous shape and connecting each corner of the original shape with its corresponding corner of the duplicate. They noticed that the cube was two squares connected together at their corners with additional edges. Another group was discussing how it’s impossible to accurately show a three-dimensional shape on a sheet of paper–some of the edges in the drawing of a cube would not be the same length, and some of the angles would not be 90˚. Their idea was to build a 4-dimensional shape in 3-dimensions with some of the side lengths and angles distorted. From there the class decided that a 4-dimensional cube consisted of two, 3-dimensional cubes next to each other, with corresponding corners connected by straight lines.

I was delighted. So many of my previously quiet students had played such an active role in the discussion. But would it continue? Would other, similar problems keep their interest? Or was this enthusiasm a one-off? After all, less than half of my class was there anyway, and there was no way I could engage those who weren’t showing up.

In the next class I asked them, “How many vertices does a 10-dimensional cube have?”

“How are we supposed to know *that*?” was the response.

I shrugged and said, “You figured out what a 4-dimensional cube looks like in a world where it can’t exist.”

One student observed, “The 3-dimensional cube has eight corners, and the 4-dimensional cube has 16 corners.” I wrote this down, and asked for other ideas. We discussed these ideas in the whole-group Zoom meeting until we found the beginnings of a pattern. I then sent them into breakout rooms to see if they could make sense of the pattern we found. If they finished early, I challenged them to find the number of vertices in an n-dimensional cube.

I was again excited as I drifted through the breakout rooms. Every single room was talking about the problem! Some were discussing the patterns of the number of vertices in cubes of different dimensions, but others were discussing multi-dimensions in general. Two of the most vocal students in one of the breakout rooms were students who had been silent in my online math class since the first day. I was feeling really good about the changes I was seeing in my students and their participation levels.

We came back and each group presented what they talked about. Most of the groups figured out the pattern: the number of vertices of cubes in each dimension was found by doubling that of the previous dimension. One group figured out how to find the number of vertices in an n-dimensional cube by using powers of 2, and this excited others in the class.

The invitations to participate in interesting problems were working, and I had re-worked my challenge to myself: find questions that would get my students talking about math, regardless of whether or not they were related to the curriculum. I was using engagement as my thermometer, letting me know whether or not it was a good class.

I kept up this style for the next three months, while the class was conducted through Zoom. I was getting much better attendance, and I had parents write to me saying their child never wanted to miss a math class. I felt like my experiment was working, but I wasn’t yet sure how I was going to transition into the physical classroom when we returned.

We weren’t following the curriculum, and I wasn’t assessing progress on a regular basis. I didn’t feel like I was doing my job as a teacher. While teaching virtually, I justified this because I had much higher attendance than I did when I was using the curriculum, and I figured they were learning more than they would have if they hadn’t been attending class. However, I felt I needed to go back to traditional teaching when we returned to the physical classroom.

It wasn’t long after that return that I realized my old teaching methods weren’t working. “What are you doing? This isn’t how we do math class,” was the feedback I was receiving. My once enthusiastic students rebelled. They were slow to re-engage, weren’t as productive as they had been online, and came to class late. They began to go to the bathroom for long periods of time, and left class as soon as possible. This was such a contrast to the online meetings, where they would be asking questions and conversing long after class was over. The indifference that had appeared in my virtual classroom at the beginning of the school year was rearing its head. I again needed to change something.

The COVID protocols of masks, plastic barriers, noisy classrooms, and rigid assigned seating, created conditions that weren’t any more conducive to learning than being at home. I was able to again justify prioritizing engagement over curriculum to myself. I wanted my students to have a positive math experience, to work together, and to share ideas. For this to happen, I needed to find a way to get them to work in groups.

I assigned groups of three students to a section of the board or a window. Each had their own marker, and I put tape on the floor where they were supposed to stand in an attempt to comply with our COVID protocols. Eventually, they started to re-engage.

One thing hit me: I was having fun. My classroom was engaging, and my students were having a blast doing math together. By this point, I had given up on asking them to master the standards that were part of the curriculum, and I created problems that only vaguely touched on these concepts.

For example, rather than the conventional approach of introducing square roots by explaining that they are the reverse operation of squaring, I presented them with a challenge. I asked them to draw a square on their graph paper with an area of exactly two square units. After some experimentation, they arrived at a square tipped on end, whose vertices aligned with the grid intersections, and whose four sides each extended diagonally through a unit square.

FI then asked them to draw a square that was exactly five square units in area, and they successfully met this challenge as well, by tipping the square slightly on the coordinate grid. With these two squares before them, I posed the question, “What’s the length of each side of these squares?” After a strong collaborative effort, they confidently told me that they couldn’t find an answer. This exploration provided the perfect segue into a discussion about irrational numbers, square roots, and the Pythagorean Theorem.

My assessment strategy aligned with this collaborative and exploratory approach. The assessments were projects that built on the explorations we were doing in class. Students were encouraged to seek assistance from other students or from me, as long as they each submitted original work. I did not stringently track their concept acquisition, but instead emphasized their thinking and reasoning. We had a blast, but there was always the worry in the back of my mind that I wasn’t teaching them the math they needed to progress to their next year’s math class.

And then it was time for the end-of-year standardized test.

I didn’t have high hopes. I had hardly followed the curriculum, and had no idea where any of them fell on any particular concept. I told them not to worry about how they did, because the pandemic affected everyone around the world. All I asked of them is that if they saw a problem they didn’t know how to solve, they would give it a try before clicking on a random multiple choice answer and moving on.

To my surprise, my students did phenomenally. On average, they increased their scores by more than a grade level from the previous year. I was stunned. While I was confident that I was acting in the best interest of my students–they came through the pandemic liking math, were willing to engage in problems that they didn’t understand, and had demonstrated critical reasoning skills–I hadn’t expected that I had taught them the skills necessary to rock a standardized test.

By letting go of my need to control what kids learn, I had actually brought about more learning. The freedom to explore led to deeper, more connected understanding. Putting engagement before the curricular standards, had counterintuitively produced more learning.

I don’t actually know what caused their test scores to rise so dramatically, but I have a hypothesis: these scores went up partially because my students had become used to sitting with problems that they didn’t initially know how to solve. All year I’d been throwing problems at them before I taught them the concepts. I gave them the time and space to struggle together, go down dead-end paths and re-group, start again, and eventually, very slowly, develop confidence in their understanding. Perhaps when they saw problems on the assessment that they didn’t initially understand, they had gained enough confidence to attempt these problems.

I have no idea if this strategy would work for longer than a messed-up pandemic year. Perhaps this pattern of success would not continue. Or perhaps I have stumbled on a way of teaching and learning that challenges widely accepted best practices. I don’t know the answers, or even the right questions to ask. But I do believe that by putting engagement first, by stimulating curiosity in students, and by giving them freedom to explore along their own interest paths, we can reinvent math education to foster excitement for exploration and learning.