Facilitating a Virtual Math Circle

This year I haven’t written much about how my Math Circle is doing. That doesn’t mean it hasn’t been running the whole time.  My routines were shaken by the pandemic and initially I didn’t have the same strong need to document everything.  For some reason I can’t really explain, it was both more nerve wracking and draining, at least initially, to do a zoom session with 20 kids than even in my first year working in person with 15.  But over time, I’ve grown more accustomed to how things work when  everything is over the internet.  I still miss working in person, especially the feedback and sense of immediacy that web conferences do not generate.  However, I don’t approach each afternoon with the same sense of  nervous energy that I did last Autumn and I do have a better sense of how things will likely flow.

One of the routines I’ve built up is, keeping a common document across all the weeks. I give out the same link each week and I add on all the materials and agenda for the day at the end each time.  That way the students can save the link and are more likely to have it each week without me needing to resend it and they can refer back as well if they want to work on something offline.  During each session its my default shared backdrop and workspace.  So I leave lots of white space where everyone can jointly annotate on it.  Sometimes this happens in a group and other times I setup break rooms and have one member share the same document so each room can work together.

Sample screen shot of my joint “workbook”

Its also useful since I can refer back to what I was doing and reconstruct things off the notes and my memory.  

That brings me to the session above. Sometime around February we started to shift and the kids had more requests for weekly topics to discuss as opposed to me setting the agenda each time.   If I have ideas in mind I like to survey interest ahead of time but its gratifying to see evidence of natural curiosity.  Two weeks prior to this one, a few kids had asked to talk about the imaginary number i.  This is actually a fairly hard thing to do given the range of backgrounds: everyone is in 6th-8th grade and taking anything from Math7 to Geometry.  Even at the tail end,  that’s not a lot of  background to easily approach the topic.  And for those still at the pre-algebra level you can’t even assume any formal encounters with i yet like the quadratic formula.  But they key point for me is that kids aren’t living in a vacuum. They know about a lot of these topics, although usually not any specifics and they are curious about them. 

I wanted to honor their request so I spent a lot of time thinking about what would be accessible and how to handle the different levels of background knowledge.  Part of that preparation was reminding myself to let the kids talk as much as possible and to also continually ask  Socratic style questions.  What has worked best for me is just working my way in order through the participant list when doing this type of activity.   It more or less functions like visual random grouping.  All the kids know I’m going to do it so its normalized and it lets me hear feedback from the shyer ones.  (Incidentally, there’s always this moment of trepidation when you worry that someone won’t answer back but in practice its fairly rare which is reassuring.)  I try to also ask a lot of “neutral” questions i.e. what strategies are you trying to do X, what do you already notice about this diagram etc.  that are designed to encourage discussion. 

Simultaneously, I lean into the chat pane a lot. The kids  independently use it more than talk (there are comments occurring almost every minute) when we’re doing things as a large group.  Interestingly, that reverses in the break out rooms.  So I keep an eye on what people are typing and often stop, narrate the question or point and pivot off of it.   This is one of the few areas of online work that feels genuinely useful. I don’t know how you’d replicate this type of  interaction in a live setting.

Returning to specifics, I decided to concentrate on two areas around i that I thought would be really interesting and that would be manageable with least amount of hand waving.  First we delved into “ghost” parabolas, a topic I learned about via Zoe Griffith.  Full details can be found: https://chalkdustmagazine.com/features/the-phantom-parabola/.  What works nicely is we could have a discussion about parabolas and imaginary numbers first where I could have the kids talk about what they already knew. That’s helpful for getting everyone on a more even starting point.  Notably, it was generally well known that i was the square root of -1.

In the online, format I could also have the various graphs pre-inserted into my “workbook” and then I could just scroll when we wanted to refer to them.    This type of graph analysis actually works fairly well over zoom.  I created a table with annotation tools and after talking about what happens with the imaginary numbers I split everyone up to find particularly values.  Next  I called on them to fill in our communal chart and annotated again right on top of the workbook.  After that  was done, we could all share reactions via me doing a round-robin “What do you notice?” style question.  And then everything was primed to reveal the 3-d graphs which managed to generate some genuine virtual “Wows”.  Timewise, everything seems to go slower in this format.  About a half an hour had elapsed by the end of this section.

I was then ready to pivot to the second half where I wanted to focus on multiplying by i as a rotation in the complex plane.  This again is not to hard to explore because everyone already can use the distributive law to compute values and then we can talk about the graphs produced and look for patterns.  To make things easier, I literally embedded a blank graph on one page to overwrite points on.  My aim was to discover the rotational dynamics so I picked various complex numbers to try out and graph. For each one, I had various kids compute the results and then plotted them on the shared page. I picked value that seemed likely to produce patterns and connected lines with various colors to try to emphasize those connections.    Overall, this also worked as expected.  One tangent we ended up on, that I did not expect was discussing what the square root of i would be?  Due to the lack of time, we speculated as a group and I put this down for a topic to more fully explore in the next week.  Its not something, I  would have aimed to talk about on my own but I followed the students lead.

In sum, I’ve been surprised by both how mathematically rich, these type discussions can be and how the kids genuinely have enjoyed them.  I’ve actually been more ambitious in some ways than I was in previous years.