Do They Really Know?

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.

Related Posts

By MrHonner, ago

Pedestrian Algorithms

…17…16…15…

I’m not going to make it. I’m about 75 feet away from where the crosswalk begins, and it’s a boulevard. Plus my back is already damp from commuting with this backpack and shoulder bag. An 18-second trot will only make things worse.

I slow down, but eye the corner. I do have to go north: Should I make the turn? If I do, I run the risk getting caught at the the next light. What’s the right move? Let’s do the math.

If I wait it out here, I’ll be stuck at a newly green light staring at the “DON’T WALK” sign for the maximum possible duration. If I make the turn, there’s no guarantee that I won’t get stuck at the next light, but that’s no worse than what I’m facing now. And there’s a good chance I’ll arrive in the middle of the cycle somewhere and be spared the maximum delay. That would get me to the opposite vertex of my rectangle in less total time than if I stop and wait it out now.

I make the turn. And await the next opportunity to optimize my commute.

By MrHonner, ago

Beyond My Control

Teachers have great power to impact their students, yet so much depends on factors beyond their control. This is one of the many tensions of teaching.

In my first year as a teacher I noticed some students didn’t bring pencils to class. I was dumbfounded. “How are you going to do math without something to write with?” was my naive reaction. Later I realized the more pressing question was “How am I going to teach math if I can’t rely on kids bringing pencils?”

As a public school teacher you become acutely aware of what you rely on. Even the best schools I’ve worked at would run out of paper, or chalk, or chairs. Working hard only to have your plans derailed by something beyond your control really stings.

All of this has shaped my approach to teaching with technology. In many ways I’m a very technology-positive teacher: I was an early adopter of tools like Desmos, Geogebra, and Scratch. But I’ve been reluctant to grow too dependent on technology in my teaching. I’ve had Smartboards for years, but never prepared slides; I’ve had laptop carts, but designed lessons that required internet access sparingly. It’s a very real possibility that I’ll show up to school and the Smartboard or wifi just won’t work. With so much beyond my control, it’s often easier to just avoid the risk.

One of my frustrations in the current remote/hybrid landscape is that I can no longer avoid that risk. Every single moment of my teaching now depends on multiple technologies functioning properly. And teaching well requires not only that they function, but that they and I function together smoothly. Now I find myself depending on a Smartboard and Google Classroom and Zoom and so much more. And I have to learn them all while trying to figure out how to turn a video conference into math class. It’s a bit overwhelming on the best of days. And then my laptop speakers decide to stop working.

There’s a minimalism to teaching and learning math that I’ve always loved. With just a pencil and paper I can become a mathematician. With just one good question I can launch a math class. But now there’s a lot more I have to rely on, and plan for. And it’s all beyond my control.

Related Posts

By MrHonner, ago

Second Time’s the Charm

I’ve resisted comparing pandemic era teaching to my first year in a classroom. I mean, I may be uncomfortable, but I’m not having night terrors.

But after three weeks of teaching digitally I noticed something that hasn’t been true in a very long time: The second time I teach a class goes much, much smoother than the first.

There’s just so much I’m not prepared to prepare for. Did I upload that problem set? Did I change the permissions so everyone can read it? Did I prepare an agenda slide? Is it open in my second screen? Am I sharing my second screen? Did I remember my document camera? Why are breakout rooms not working? How long has Kendra been waiting to get back into the meeting? Oh, wait, am I muted again?

Teaching digitally has stripped me of the procedural expertise I’ve developed, and relied on, the past 20 years of running classrooms. The dozens of automatic decisions and reactions that usually speed class up are now slowing me down.

Luckily my students are patient and understanding. We’ll get up to speed eventually. But this was the first week I let myself dream a little about work getting back to normal. I look forward to being an expert again.

Related Posts

By MrHonner, ago

Weaving Classes into Courses

I’ve heard veteran teachers say that this new era of hybrid and remote learning has them feeling like first-year teachers again. In some ways I feel it too. Seven days into to fully remote instruction and I’m still figuring out how much I can reasonably expect to accomplish in a 55-minute Zoom meeting, how I can most effectively present ideas, how I can best get students interacting with mathematics and each other.

Like in my first year of teaching, I find myself focused on very short-term goals: Getting through today’s class; getting students to engage with a single concept; getting them to demonstrate mastery of one unadorned procedure.

I’m generally energized by the challenges of teaching, but it’s difficult going back. After 20+ years in classrooms I’m used to thinking in terms of threads that weave classes into courses, the small details that bind together a year’s worth of conversations and explorations. It’s hard to get there when you’re unsure about executing the daily details that make class run.

I did it once, and I can do it again. I just hope it doesn’t take me as long this second time.

Related Posts

By MrHonner, ago
Follow

Get every new post delivered to your Inbox

Join other followers: