The Math of Social Distancing is a Lesson in Geometry — Quanta Magazine

My latest column for Quanta Magazine connects the geometry of social distancing, a topic on everyone’s mind right now, with sphere packing, a problem mathematicians have been working on for hundreds of years.

Determining how to safely reopen buildings and public spaces under social distancing is in part an exercise in geometry: If each person must keep six feet away from everyone else, then figuring out how many people can sit in a classroom or a dining room is a question about packing non-overlapping circles into floor plans.

In two dimensions, the sphere packing problem is really the circle packing problem, and the best way to pack circles in the plane is known. But it was only recently that mathematicians settled the question about how to most efficiently pack spheres in dimensions 8 and 24. And there are still many open questions in other dimensions!

You can read the entire article here.

Remote Learning — Week 14

Everything has changed, so they say. The end of the school year is usually a time of great relief and satisfaction, but there’s something anticlimactic about this transition to summer. Usually it’s accompanied by a comforting sense of resolution. Not this year.

My department spent its final days trying to talk about the fall. What if we see students every other day? What if we only see them every five days? What if we’re entirely remote at the start? Can adjusting our course sequences help? How do we approach assessment? Participation? Collaboration? In a way it’s fitting that we end with as many questions as when we started. Maybe more.

I’ve heard people say remote learning made them feel like they were first-year teachers again. I wouldn’t go quite that far, but rethinking everything from instruction to engagement to assessment definitely brought back memories of my early career, when everything I did seemed like an experiment on the brink of failure.

Navigating Emergency Remote Learning as a teacher has been an unique professional challenge. I’m thankful that I didn’t have to face it alone: I learned alongside many great teachers in New York City and across the country. And I’m lucky that my students were ready to adapt, learn, and succeed, as they always are.

And so ends fourteen weeks of reacting and adapting. I’m glad I made time for these reflections: They’ve helped these past few months, and they’ll help in the fall. But first, I’ll give the comfort of summer a few more chances to sink in.

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Remote Learning — Week 13

As part of their end-of-year portfolios, students completed a survey reflecting on their remote learning experiences. I asked questions about the workload, the resources they found most helpful, and what was missing. Overall I got lots of great feedback, both on things I knew about and things I didn’t.

But the feedback also demonstrates why it can be so difficult to think and talk about education at scale. Take these two responses, sitting right next to each other in my spreadsheet.

“I think that you should consider doing more whole class meetings.”

“I would say to avoid class-wide calls for Remote Learning.”

To me, this exemplifies the core challenge of the practice of teaching: What works for one student may not work for another. What works for one teacher may not work for another. What works today may not work tomorrow.

The fundamental unit of teaching is person-to-person interaction, and those individual interactions are subject to great variation. Of course there are general trends in what the students said in their surveys, and there are general approaches to teaching that work better than others. But to teach all students we need to listen to all students. And we must always be mindful that each experiences instruction in their own way.

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Remote Learning — Week 12

Remote Learning demands a different approach to assessment, so developing a final assessment plan for my courses has been an interesting challenge.

For my 9th grade Geometry class, the New York State Regents exam would have been the final exam for the course. The state cancelled these exams in April and actually absolved students of ever having to take them. This meant I needed to design some kind of final assessment, but it also gave me the freedom to approach it however I wished.

I settled on a portfolio of small items that covered a variety of skills and gave students options in terms of demonstrating what they learned in the course. For their Final Portfolios, each Geometry student had to complete the following:

  • A 2-page mathematical paper on the geometry topic of their choice
  • A dynamic Geogebra demonstration
  • A 1-page written reflection on the year
  • A comprehensive End-of-Year survey
  • A short Exit Interview conduct via video conference

In preparation for the paper, I had students learn to use the equation editor in Google Docs and practice typesetting math and laying out diagrams. The papers ranged in style from recapping theorems and examples from class to remarkably comprehensive overviews of topics like the inscribed angle theorem, rigid motions, and Varignon’s Theorem (my favorite theorem!). Thanks to the earlier practice, for the most part these final papers turned out beautifully.

My Calculus students still had a traditional summative assessment, as they all took the AP exam in May. So for their Final Portfolio I gave them the freedom to design and produce an end-of-year project that was meaningful to them. This isn’t so different from what I often do in advanced courses, but the students were especially creative this year: I’m still working my way through videos, podcasts, computer programs, art projects, and interactive demonstrations. And two students even wrote and hosted a virtual Integration Bee as their final project, looking to recapture a fun in-person experience they were missing out on this year.

This approach has definitely left me with more work to review, evaluate, and respond to at the end of the year. But the balance of work — mathematical, creative, personal — together with the opportunity to see each student to debrief, creates a satisfying sense of closure and accomplishment at the end of what has been a very challenging school year.

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