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.