Teaching

Questions and Answers

“Never ask a question you don’t know the answer to.” This is an old saw among trial lawyers that I learned growing up watching courtroom dramas on TV. For some reason it’s stuck with me my whole life, but it’s not advice I follow. As a teacher I ask these kinds of questions all the time.

I often ask math questions in class I don’t know the answer to. Sometimes it’s because a student’s curiosity has sparked my own, and I want to share that excitement with others. Sometimes it’s to model not knowing. It’s important for students to become comfortable operating in the space of not knowing, because that’s where mathematicians live.

And every assessment, every quiz, every test I give asks a question I don’t know the answer to: “Do you understand this?” Or, perhaps more to the point, “Did I teach this successfully?” I usually have a feeling about the answer, but I never really know. That’s why I have to ask.

In my end-of-year surveys I asked students a lot of questions I didn’t know the answer to. Did you feel productive in math this year? Do you feel like you learned as much as you would have in a normal math class? Do you feel prepared for next year? I wasn’t sure what I would hear. But I had to ask. That’s what teachers do.

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By MrHonner, 3 years3 years ago

Teaching

The Gift of Hope

I met my calculus class for the final time this week. We commiserated about their awful experience with the AP exam (“I spent so much time typing parentheses!”), did some math (“Don’t you ever take a day off?”), and spent plenty of time in breakout rooms so students could socialize and say goodbye.

Before I said goodbye myself, I thanked them. This group of advanced students were ready to succeed in September. More ready than I was. As I stumbled my way through those first weeks of turning myself into an online math teacher, they did more than their part. They engaged in class when I didn’t give them much to engage with. They took initiative and tried to make breakout rooms productive. They invested themselves in their work and responded to my feedback.

In those first few weeks, they did their jobs better than I did mine. They were learning, despite everything, and that gave me hope. Hope that maybe I just had to figure a few things out and I would be ok. Hope that maybe my 9th graders were learning, too, even though it was harder to tell. Hope that maybe I just might make it to the end of the year and feel like I accomplished something.

Students give so much to their teachers: their enthusiasm, their open-mindedness, their creativity, their skepticism. These gifts are part of what makes teaching so fulfilling. This year my students gave me hope. And I wouldn’t be here without it.

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By MrHonner, 3 years3 years ago

Challenge Teaching

Who Needs Trig Sub? Part 2

I recently wrote about an ingenious integration performed by two of my students. But that wasn’t the whole story. Here’s the rest.

It begins with this integral.

Usually when challenged to evaluate this integral students will try a few substitutions, fail to find an antiderivative, and realize something new is needed. This sets the stage for trigonometric substitution, which we then use to untangle the algebraic obstacles in the integrand.

But this was an unusual year. Not only did two students surprise me with a purely geometric approach, a third student found another completely different solution I had never seen before. She used integration by parts!

This application of integration by parts leads to the following equation.

Which she simplified to this.

This is the point where I would probably abandon integration by parts, because it doesn’t look the situation has improved. But my student did something incredibly inspired. She rewrote the integral

like this

And this is definitely an improvement. The integral on the right is known: it’s just $arcsin(x)$ . And the other integral is the one we were trying to find in the first place. When this is substituted back into the integration by parts equation you get this

The integral we want to evaluate is now on both sides of the equation, so just collect like terms and solve

Before this year I had never seen any alternate derivation of this formula. This year students produced two completely new ones!

I can understand why students had never attempted this technique before. Integration by parts typically comes after trig substitution in the course, so it wouldn’t usually be an option for them. But this year, because of the way I rearranged the curriculum, integration by parts came first. And I’m thrilled that my student thought to try it. And persevered!

It also made me feel good about my integration by parts lesson. Usually the last problem I present in that lesson is the integral

But this year I almost didn’t. This problem requires repeated integration by parts and quite a bit of perseverance to solve. At the end of a long lesson I considered passing on it, but we pushed through. And it was worth it! Seeing this technique set up my student to find her own ingenious integral. And to teach me something new.

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By MrHonner, 3 years3 years ago

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?

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By MrHonner, 3 years3 years ago

Geometry Teaching

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!

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By MrHonner, 3 years3 years ago

Teaching

Occasional Asynchronous Instruction

Untimed Assessments with Resubmissions

Virtual Office Hours

Publishing Class Notes

Private Chat

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