Teaching

I taught a good lesson this week. Not considering-the-limitations-of-remote/hybrid-learning good. Just really good.

A lesson where all students hit the checkpoints on schedule. Where small group conversations served their intended purpose with precision. Where the variance of whole class conversation was small enough to keep us focused but large enough to keep it interesting.

The lesson was on the Mean Value Theorem. I’ve taught it many times, and it’s usually very successful. But success is a little harder to come by right now, so it was a welcome surprise to see just how successful this lesson was in remote / hybrid learning.

Early in the lesson every group was able to outline the proof of Rolle’s Theorem via its connection to the Extreme Value Theorem. Later on, every group was able to develop a proof of the Mean Value Theorem based on its connection to Rolle’s Theorem. Our final whole-class discussion of the proof of the MVT was driven by several students who have been fairly quiet in class up to this point, which is always a sign of success.

The success of the lesson wasn’t due to any particular innovation. I simply have a better feel for this kind of teaching now: How to structure tasks, which questions to ask, how long to leave students in small groups, how to organize platforms to serve my instructional goals. And I have a better feel for my students, as well. I know who needs a nudge, and who needs to struggle a little more.

It’s been challenging trying to redevelop my feel for teaching this year. Success has come here and there, in brief moments. But this lesson accomplished exactly what I hoped it would. It feels good to finally be able to say that.

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“Now is not the time to be experimenting on students.”

We were discussing whether or not we should give the same final exam we gave last year. I argued that it would offer a rare opportunity to compare this very abnormal situation to a normal one. Others felt it would be unfair to hold students to the same standard without the same preparation. The data would be enlightening, but at what cost? My colleague had a point: It was something of an experiment.

Then again, this is all an experiment. Every new policy we adopt, every new system we put in place, every new technology we try is an experiment in education. We tinker with teaching and grading and assessment and administration and then watch what happens. And we’re not just experimenting on students: teachers, schools, parents, everyone is getting the treatment. The pandemic signed us all up.

This year it seems like everything about my teaching is experimental. At least now I’ve reached the point where I feel like I’m experimenting to innovate. Six weeks ago I felt like I was experimenting out of desperation. Here’s hoping for a speedy end to the trials.

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We had just started basic proof writing in Geometry class. Things were going better than expected: I definitely anticipated proof-writing would be much more difficult to teach remotely, with all its subtle requirements and adjustments. I was pleased that students were getting the hang of it.

I did feel that they needed another class or two to assimilate the basic facts and forms. I had initially budgeted three days, but they needed a few more. I thought about what lies ahead in the semester and weighed my options. The schedule is tight, and every day is accounted for. With fewer overall class meetings, there’s not much wiggle room. But if students don’t achieve the right level of facility now, the schedule in place may not work anyway. I decided to take two more days to continue with basic proofs.

As with most decisions teachers have to make, this required synthesizing a lot of different information: Course sequences and outcomes; interdependencies between content; the relative importance of topics; the time required to master certain skills; and of course, knowledge of the students in front me. It’s the kind of high-level planning decision I make dozens of times a year as a teacher, and one I’m generally comfortable with.

But I’m not comfortable making that decision right now. It’s been difficult to gauge how much can be accomplished in a single class of remote/hybrid learning. And it’s been difficult to predict how long it will take key ideas to take hold for students. With so little experience in this form of teaching, it’s hard to anticipate how the semester will unfold. Trading an extra day here and for a day months from now used to be an easy decision for me to make. It’s not right now.

I think I’ve made the right decision about proofs for my Geometry class. More importantly, I have confidence that we’ll be able to adapt as necessary. I haven’t felt like that often this year. It’s a welcome step back toward normalcy.

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My colleague and I were discussing the Intermediate Value Theorem, an important early result in an AP Calculus course. It’s important for both conceptual and procedural reasons: It’s why continuity is useful, and as the first “Value” theorem to appear, it sets the stage for how to invoke these important results. When applying the IVT students need to make sure that the proper conditions are satisfied: Before you can claim a function takes an intermediate value, you must first be sure the function is continuous and the interval is closed. Attending to these details is important.

This is why we were discussing the IVT in the first place. My colleague was assigning a few problems to make sure students were attending to those details. And when I heard this, I panicked. I taught the IVT last week. Did my students know how to attend to all these details? I wasn’t sure.

I wasn’t sure because, when I taught the IVT last week, I couldn’t walk around class and look over the shoulders of my students to see if they asserted that f(x) was continuous. I couldn’t easily eavesdrop and hear if groupmates were holding each other accountable. Determining whether or not students really know is embedded in my teaching routine, but with my routine disrupted, I’ve been teaching blind. And deaf. I threw the IVT out there and hoped for the best.

The solution was simple enough: Put an IVT problem on my next take-home assessment and make sure they know what they’re doing. But this scared me a little bit. “Do they really know?” is perhaps the most important question a teacher must ask. Overwhelmed by the many challenges of remote / hybrid learning, I haven’t been asking it enough. It’s another routine I have to rebuild.

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