Some Math Problems Seem Impossible. That Can Be a Good Thing — Quanta Magazine

In my latest column for Quanta Magazine I write about the secret power of impossible math problems.

Construct a convex octagon with four right angles.

It probably says a lot about me as a teacher that I assign problems like this. I watch as students try to arrange the right angles consecutively. When that doesn’t work, some try alternating the right angles. Failing again, they insert them randomly into the polygon. They scribble, erase and argue. The sound of productive struggle is music to a teacher’s ears.

Working on impossible problems has a way of helping us better understand what is possible in math, and the impossible plays an important role in the history of mathematics. In my column I explain using several examples, and include a few extra exercises to play around with. The impossible can be frustrating, but also fun!

The entire article is freely available here.

Forty Phone Calls

Parent-Teacher conferences joined the list of education practices remade by the pandemic this week.

Ours went surprisingly well. Parents signed up in advance, and our administration created a schedule for every teacher detailing whom to call and when. The schedule prioritized students who struggled during the first marking period, which meant conversations that really had to happen were more likely to occur.

Knowing whom I’d be meeting with was very helpful. Usually Parent-Teacher conference night is three hours of pure reaction: A parent walks through the door, I quickly try to identify them, place their child, locate their records, search my mind for what I want to tell them, and then move them along within three minutes because I’ve got 25 other parents waiting in line. This year I was able to prepare talking points and review student work ahead of time. It was much less hectic.

In my conversations I was struck by how grateful parents were. Parents are generally appreciative of the work teachers do, but there was something extra this year. Even though we could speak for only a few minutes, most parents made it a point to explicitly thank me both for what I’m doing in the classroom and for taking the time to call them. In fact, many of these conversations felt like thanking me was itself the point.

I made forty phone calls over the two days. It was exhausting, but worth it: A familiar feeling that made this year’s abnormal Parent-Teacher conferences feel very normal.

Related Posts

I Taught a Good Lesson

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.

Related Posts

Experimental Education

“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.

Related Posts

Media Literacy Week Panel

Tomorrow I’ll be on a panel discussing quantitative literacy as part of Media Literacy Week. The focus will be data and science misinformation, and the panel discussion will run as part of the National Science Teachers Association’s Teacher Tip Tuesdays series.

This is a joint project of the National Science Teachers Association (NSTA), the National Council of Teachers of Mathematics (NCTM), Education Development Center (EDC), and the National Association for Media Literacy Education (NAMLE). Registration is free, and you can find out more here.

UPDATE: You can find the full video of the webinar here.

Follow

Get every new post delivered to your Inbox

Join other followers: