Teaching

What Would You Keep?

The BC Calculus exam is next week, so I planned for students to take a practice test this past Wednesday. I spent about an hour preparing all the files and forms and Google assignments so students could take the exam and submit their work for me to review.

The first email came on Sunday night. “I won’t be in class on Wednesday because I’ll be taking the AP US History exam.” As more emails trickled in it became apparent that most of my class was going to be absent for the practice test. I should have paid more attention to the school calendar.

In a normal year I would have been upset wasting so much time preparing something I couldn’t use. But not this year. When Wednesday came I posted the work as planned and simply declared it an asynchronous assignment. Everyone did the practice test just as they would have in class, whenever they had the time. Two days later I had my data and everyone had their feedback.

We’re all looking forward to returning to in-person instruction, but there have definitely been some this-works-better-remotely moments over the past year-plus. As the year winds down, I’ve been thinking about what I’d like to keep from this experience when I return to my classroom. Here are a few things that have been on my mind. Some are wishful thinking, some are within my control

Occasional Asynchronous Instruction

Asynchronous instructions isn’t just for adapting when things go wrong. I’ve heard it over and over again this past year: Students love the ability to do their work on their own schedule. Imagine a weekly asynch day where students get meaningful work but are given the flexibility to do it when it makes the most sense for them. They could work at school in a drop-in environment or at home. A weekly asynch day could also give teachers much needed time and space to plan, grade, and collaborate in course teams.

Untimed Assessments with Resubmissions

When emergency remote learning began last year I switched exclusively to untimed assessments and allowed students to re-submit after feedback. Under those circumstances it seemed like the only reasonable option, but it worked so well I stuck with it as my primary assessment strategy this year. I did give timed quizzes throughout the year, and I do look forward to giving in-class tests again, but I’ll definitely find a way to integrate this into my overall assessment strategy.

Virtual Office Hours

After-school tutoring is great, but this year’s virtual office hours were much more flexible and convenient. It’s been so easy for kids to pop in to ask a quick question, get help on an assessment, or retake a quiz. It’s also made it easy for me to coordinate recommendation writing and even catch up with last year’s students. There’s a reason that parent-teacher conferences in New York City will remain virtual next year even as schools reopen. This model works.

Publishing Class Notes

For the first time in nearly 20 years of classroom teaching I posted my class notes everyday. It hasn’t been feasible before now, because the record didn’t exist, but with everything being digital this year it was a snap. And students loved it: It made reviewing easy and it relieved the anxiety of catching everything on the first pass in class. This came up a lot in end-of-year surveys as something that worked well for them.

Private Chat

I came late to this feature in Zoom, but private chat makes something that is very hard to do very easy: Give every student a chance to answer a question without being influenced or judged by others. I spent a lot of time this year trying to figure out how to use digital tools to recreate what I do well as a teacher, but I’ll be thinking about how I can recreate what this digital tool does in an in-person setting.

So, as we wrap this remote/hybrid year and look forward to an in-person fall, what might you keep?

Who Needs Trig Sub?

As a calculus teacher, this is one of my favorite integrals.

This calculation is a fun challenge for students who are so deep into integration techniques they’ve forgotten that not every integral requires antidifferentiation. It’s hard to apply the Fundamental Theorem of Calculus here because the integrand has no obvious antiderivative. But the integral is easy to compute, because it represents the area of a semicircle of radius 1.

The area, and thus the integral, is $\frac{\pi}{2}$ .

I bring this integral back later in the course to motivate the technique of trigonometric substitution.

If you can’t imagine the antiderivative of the integrand the FTC isn’t much help. That’s where a clever (if complicated) change of variables, together with a few trigonometric identities, comes in. Trig substitution solves the algebraic puzzle of integrating the square root of a difference.

But when I taught this topic this year something unexpected happened.

As usual, I presented students with the indefinite integral, expecting it to be inaccessible with their current tools. This would motivate the need for a new technique, and trig substitution would come to the rescue!

But two students didn’t need rescuing. Instead, they figured out a way to evaluate this integral using tools they already had. They did what I often encourage them to do, but in an ingenious way I never would have anticipated: They turned this indefinite integral into an area problem and used geometric reasoning to evaluate it!

They started by reimagining the indefinite integral as a definite integral.

Now here’s the region whose area is given by this definite integral.

The region can be thought of as a right triangle and a sector of a circle.

Because the circle has radius 1, the sides of the triangles are $x$ and $\sqrt{1-x^2}$ .

Which makes the area of the triangle $\frac{1}{2} x\sqrt{1-x^2}$ .

Now the area of a circular sector is equal to $\frac{\theta}{2\pi} \pi r^2$ , where $\theta$ is the central angle. In our diagram

we have $sin\theta = \frac{x}{1}$ , so $\theta = arcsin(x)$ . Since the radius of the circle is 1, the area of the circular sector is

$\frac{1}{2\pi}\pi arcsin(x) = \frac{1}{2}arcsin(x)$

The area of the entire region is then the sum of the areas of the triangle and the sector. But the area is also the value of the definite integral. So they must be equal!

Differentiating both sides (thanks again to the FTC) shows that we really have found an antiderivative of $\sqrt{1-x^2}$ , as required.

And so

Which of course, is the same result we find using trig substitution.

I’ve taught this topic for many years and never thought of this approach. I’m grateful to have learned something new from my students, who never fail to impress me with their creativity. And I’m glad I gave them time and space to solve what I thought was an impossible problem! When I teach this next time, I’ll be sure to do it again. And I’ll be sure to share this ingenious integration.

UPDATE: I’ll also be sure to show them this other ingenious solution that a different student came up with!

Teaching

Everything I Didn’t Do

Last year my calculus students worked on creative projects at the end of the term. They submitted videos, podcasts, and art projects related to their favorite topics from the year. Two students wrote and hosted an integration bee. Another baked and decorated a four-layer “Washers Method” cake. We didn’t do that this year.

My geometry students wrote short papers on their favorite topics at the end of last year. They built interactive Geogebra demonstrations to accompany those papers. We didn’t do that, either.

With just a few weeks left in the school year the overwhelming feeling is relief. I’ve made it. I wasn’t sure I would. At the start of the year I often felt like a first-year teacher. I wasn’t sure I could do my job at all, much less at the level I expect. Just making it to this point is an accomplishment.

But the end of the school year is a time for reflection, and it’s hard not to look back and see everything I didn’t do. I didn’t have my students write enough, or take as many photographs, or experiment with computing. These are staples of my classroom, but in a year where just covering the curriculum was a cause for celebration, I simply didn’t have the time or energy or opportunity to get us there.

It’s unusual for me to feel this much regret at the end of the year. But it’s been an unusual year. Thankfully, the end of the school year is also a time to look ahead. And although new challenges certainly lie ahead of us next fall, I’ll be excited to face them.

Teaching

Something I’ve Missed

Monday was my day to be in the building. Before teaching remotely from my empty classroom, I walked upstairs to the gym to say hi to the students who physically came to school. I saw a few kids quietly standing near the ping pong table that had been set up for socially distanced socializing.

One of my calculus students was with them, so I challenged him to a game of ping pong. They didn’t have any ping pong balls, so we searched the empty lockers and he found a fist-sized green squishy ball. He held it up as if to ask “Want to give this a shot?” and I nodded. After about 45 awkward seconds we succeeded in inventing some kind of paddle-ball game. As we played we talked about infinite series, differential equations, and the BC exam. After a few minutes a dozen students were watching. I handed off the paddle and said goodbye, and headed upstairs.

Outside the cafeteria I saw a student standing by himself at another ping pong table. I realized it was a 9th grader I’d been teaching all year but never met in person. I grabbed a paddle and we started playing, chit chatting for the first time after working together for 8 months. A few other students gathered, including one of his classmates. It was 10 o’clock and I had to get back to my empty classroom to teach their class, so I handed off the paddle again. I grabbed an extra ping pong ball and headed back downstairs. Paddle-ball was in full swing, which made me think twice about giving them the ping pong ball I’d snatched for them.

Back near my classroom I noticed a familiar face in the distance. It was a student of mine from last year. He and a few friends had set up an unofficial pod in the lobby and were quietly working at an out of the way table. I took a quick detour to catch up — How’s your year going? How’s Algebra 2? Any plans this summer? — and then hustled back for the start of class.

I love teaching and I love math, but I love this part of my job, too. I love being a small part of the small moments of the daily lives of students. Sharing a smile or a laugh or a serious thought before school. Silently saying hi in the hallway in between classes. Catching up years later, seeing how students have changed and grown, listening to how they remember our shared experience. I figured out the teaching and the math this year, but not this other part. I’ve really missed it.

Teaching

A Bad Lesson

I taught a bad lesson this week. It’s not unusual. I teach hundreds of lessons every year: Some go very well, most are just fine, a few go poorly. A successful school year has its ups and downs, for students and for teachers, and you take the bad days with the good.

This week was the area of a circle. It’s always a challenge teaching a topic that students already know: It can be hard to get students to engage deeply with an essential question like “How do you conceive of the area of curved things when your notion of area is based on straight things?” when they’ve been using $A = \pi r^2$ since 6th grade.

I took a standard approach and had students explore area relationships between a circle of radius 1 and various inscribed and circumscribed polygons. The goal was for students to engage with the essential question while sharpening their area-computing skills and previewing some important arguments they’ll see again in calculus.

Usually this lesson works just fine, with occasional moments of greatness. But this year I had to trim the area unit a bit to accommodate the schedule, so prior to this lesson we just hadn’t done enough with polygon area. As a result, the conversation bogged down in the early stages of the exploration and it was hard to get unstuck. And because the remaining schedule is so tightly packed, there really wasn’t really time to stop and regroup. We just had push through.

It wasn’t a disaster: We did math together, students grappled with and argued about important ideas, connections were made. But the lesson didn’t go as I’d hoped, and that was a downer. On the other hand, it was comforting to realize that this lesson went bad for some very normal reasons. It didn’t fall flat because my wifi went out or because I couldn’t share my screen or because I had no idea how to access student thinking on a computer. The lesson didn’t work for the same reasons most lessons don’t work: I misjudged prior knowledge and tried to execute a plan that was too ambitious and too rigid. It might have been a bad lesson, but it also might have been the most normal lesson I taught all year.

Teaching

What Would You Keep?

Occasional Asynchronous Instruction

Untimed Assessments with Resubmissions

Virtual Office Hours

Publishing Class Notes

Private Chat

Who Needs Trig Sub?

Everything I Didn’t Do

Something I’ve Missed

A Bad Lesson

Follow Mr Honner