Workshop — Bringing Modern Math into the Classroom

This Thursday I’ll be running my workshop “Bringing Modern Math into the Classroom” for teachers at Math for America.

In this webinar participants will engage with mathematics at the edge of our understanding. We’ll look at examples of math that’s being invented and discovered right now, and see how it connects to what is happening in classrooms.

We’ll play games, explore patterns, and make conjectures in arithmetic, algebra, and geometry. The goal is for participants to leave not only with a better understanding of how school math and research math are connected, but how to better communicate that connection to students.

This workshop is based on the work I’ve done in my Quantized Academy column for Quanta Magazine. I’ve run similar workshops in past years, and I recently gave a talk on this topic at the NCTM 2020 Virtual Conference. But this week’s workshop is all new, and I’m looking forward to bringing some new ideas and new math to play around with!

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Puzzling Together the Curriculum

To accommodate the different logistical consequences of potential in-person, hybrid, and fully-remote instruction, our school adopted a radically new schedule this year: Classes that meet every other day for periods that are 40% longer, but with an overall reduction of total class time.

The decision made sense from a organizational standpoint, but it made a mess of existing course maps and lessons plans. Trying to reorganize and redistribute content has been an ongoing challenge. It’s no simple thing to break up an existing course and reassemble it in different-sized chunks: You can’t just teach 40% more content because a class is 40% longer. Ideas needs to flow in a sensible way, and some in particular need time to set. Judging how to accomplish this was especially difficult at the start of the year, when it wasn’t even clear how much could be accomplished in a fully remote 55-minute class.

With three months behind me and a much better sense of what I’m doing, I’m feeling more comfortable putting the pieces together. Last week I was struggling to plan a 2-hour block in my trigonometry unit. But after some experimenting, I ended up pulling together material I side-stepped in October (special trigonometric limits), the core material I intended to cover (trigonometric integrals), and wrapped it up by laying the groundwork for some future extensions (Fourier series).

I would never have thought of putting these things together in a normal year. Nor would I have thought of this in September as I mapped out the semester. Back then I wasn’t even sure what would come of special trig limits as I side-stepped them, because it was impossible for me to look ahead.

But with nearly a semester under my belt, it made sense, and it worked. Three months on a steep learning curve can be painful, but it does make a difference.

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Forgetting How to Teach

The holiday break brought some much needed time off. I felt refreshed as I returned to work, but my first day back was a bit disorienting.

I opened my agenda and my SMART Notebook, but forgot to open my lesson plan. I forgot to assign my Geogebra classroom activities ahead of time. I forgot to print out my rosters. I forgot to share my screen. I forgot to check if I was muted.

The muscle memory of teaching I had worked so hard to rebuild had faded after 11 days of vacation. I suppose that’s what vacation’s for, but it’s been many years since I’ve been caught off guard like that after a break.

By the end of the week things it felt like we were back to normal again. In school, at least.

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Remote Intervisitations

I’ve attended and facilitated a lot of professional learning in my career as a teacher and instructional coach. Without question, the most consistently effective professional learning has come from collegial intervisitation.

There’s an undeniable relevance to being in another teacher’s classroom. Every teacher teaches differently: Some teachers do big things differently, like teaching trigonometry before polynomials, or never collecting or reviewing homework. Some teachers do small things differently, like always correcting the statement “A triangle has 180 degrees”, or using a random number generator to call on students.

However they manifest in a classroom, experiencing those differences has an immediate impact on an observing teacher. Maybe you’ll see technique or strategy you’ll want to try. Or maybe something you see will make you think more deeply about your own approach. Intervisitation is so effective because every observation of another teacher is really an observation of yourself. You can’t help but reflect on your practice.

I’ve found intervisitation especially useful during remote learning. Simply experiencing remote teaching from the other side has been invaluable: What’s it like to try to do math virtually? How does it feel to sit in a breakout room? And it’s been very helpful seeing how different teachers try to solve the same problems, like how to properly present mathematics, how to get students engaged, and how to gain access to their thinking.

After a round of intervisitations, our geometry course team had conversations that were both enlightening and heartening. We saw the different ways in which we were each finding success, and the similar ways in which we were all struggling. Teaching is an isolating job, but those common struggles unite us. And we have a better chance of overcoming them together than on our own.

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A New Lesson

I reached a benchmark of sorts this past week. I taught a new lesson.

In a sense, everything I’ve been doing this year is new. Remote teaching means new classroom routines, new assessment strategies, new procedures for facilitating collaboration, all sitting on top of a fundamentally new way to gather and conduct class using new technologies.

But despite all the newness, I’ve mostly been adapting existing instructional materials to fit this new reality. Well, more like because of all the newness. With so much effort being expended adapting to everything that’s new, having quality existing materials to rely on has been invaluable so far.

And in reality, adapting those existing materials to remote learning has been its own challenge. It’s not as though I can just run out the same lesson, the same task, the same problem set as before. Classes are longer, the semester is shorter, the medium is fundamentally different. I can’t simply do what I have done. Putting my existing materials into practice itself has seemed like a new job.

Which is why teaching a new lesson is something of a breakthrough for me. Three months in, I’m a bit more comfortable with what I’m doing and how I’m doing it. I’ve been experimenting with platforms likes Geogebra Classroom and Desmos Activity Builder, which have helped me recreate some of what’s been missing in my classroom, but have also given me an opportunity to take new approaches with old ideas.

Last week I introduced parallelograms with a Geogebra Classroom activity that had kids playing with quadrilaterals and conjecturing about the consequences of parallelism. They were able to explore, make observations, and outline proofs together. In Calculus I changed my approach to integration by substitution, emphasizing the process as a change of variables instead of an algebraic procedure. I designed a Desmos activity that allowed students to experiment and transform integrals and hypothesize about the consequences.

The methodology isn’t new for me: I always try to bring ideas to students through exploration and conjecture. But it’s been hard to figure out how to do that effectively in remote learning. Now I’m getting familiar with new tools that create new opportunities for teaching, and I’m starting to feel comfortable taking advantage of those opportunities.

I love reflecting on how I present mathematical concepts and then redesigning lessons around those new ways of thinking. It’s one of my favorite professional challenges as a teacher, and being able to do it again was a much-needed reminder of how fun teaching can be. My only concern now is what’s going to happen to these lessons when things go back to normal?

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