Teaching

Math that Moves the Needle — Quanta Magazine

My latest column for Quanta Magazine explores a century-old geometry problem that anyone who’s ever performed a three-point turn can appreciate.

Imagine you’re rolling down the street in a driverless car when you see a problem ahead. An Amazon delivery driver got their van halfway past a double-parked UPS truck before realizing they couldn’t make it through. Now they’re stuck. And so are you.

There’s a fun math problem here about how much space you need to turn your car around, and mathematicians have been working on an idealized version of it for over 100 years. It started in 1917 when the Japanese mathematician Sōichi Kakeya posed a problem that sounds a little like our traffic jam. Suppose you’ve got an infinitely thin needle of length 1. What’s the area of the smallest region in which you can turn the needle 180 degrees and return it to its original position? This is known as Kakeya’s needle problem, and mathematicians are still studying variations of it. Let’s take a look at the simple geometry that makes Kakeya’s needle problem so interesting and surprising.

You can read all about the surprising resolution of Kakeya’s needle problem in my full column for Quanta Magazine.

By MrHonner, ago

Sharing Instructional Materials

In a recent thread on Mastodon I was talking with Ben Leis and Tim Ricchuiti about instructional materials and I mentioned that I basically create everything I need from scratch rather than use materials that already exist. The conversation got me thinking about the challenge of creating instructional materials that can be shared.

It certainly does require a lot of work to create my own instructional materials, but it’s work I enjoy. And it really doesn’t occur to me to do it any other way. No instructional materials exist that can properly leverage the collective strengths of our classroom, so I just create materials that do.

As a result, these materials work very well for me but wouldn’t necessarily work well for others. For example, a recent lesson I wrote starts with the following question for students: “What does a ≈ b mean?” I know exactly what I want to happen as a result of that question. I have a sense of how student discourse and collaboration will unfold, and I know how the answer (and the work leading up to it) fits in the development of ideas from intuitive notions of “infinitesimally small” which arose the first day of class to the notion of “arbitrarily close” and the epsilon-delta definition of limit. And as a teacher I know how to manage the action and close the gaps when necessary.

But I doubt that same question (and the same supporting lesson materials) would work for another teacher. Of course anyone can teach this idea in this way, but the instructional design is so tailored to my context that I’m not sure how useful my materials would be to someone else.

You can see the entire thread on Mathstodon here.

By MrHonner, ago

Math Patterns That Go On Forever but Never Repeat — Quanta Magazine

I wrote a column for Quanta Magazine on the recently discovered “hat tile”, the first ever aperiodic monotile!

Have you ever admired how the slats of a hardwood floor fit together so cleanly, or how the hexagons underneath your bathroom rug perfectly meet up? These are examples of geometric tilings, arrangements of shapes that fit snugly together while filling up space. Two-dimensional tilings are admired all around the world, both for their beauty — as seen in the artistry of mosaics in cathedrals and mosques around the world — and for their utility, in walls and floors everywhere.

In math, tilings are often appreciated for their regular patterns. But mathematicians also find beauty in irregularity. It’s this kind of beauty that a retired print technician was seeking when he recently discovered the first “aperiodic monotile”— a single tile that fills up the plane in a non-repeating pattern. To get a handle on this big discovery, let’s start by thinking about a simpler problem: how to tile a line.

You can read the full article for free here.

By MrHonner, ago

Please Teach the Vertical Line Test

I was re-writing an introductory lesson on functions this morning and was reminded of something from years ago.

An influential teacher was telling their followers not to teach the vertical line test because it was confusing. I strongly disagreed. The vertical line test is a great way to meaningfully connect several fundamental ideas: the definition of a function, the definition of a graph, domain and range. The influencer was unmoved, but conceded slightly, saying that the topic should be handled with caution. I responded, “Yes, any time a teacher teaches something they don’t fully understand themselves, they should be cautious!”

It’s sad to know that there are teachers out there not teaching the vertical line test because someone told them it’s too confusing.

Originally posted on Mastodon.

By MrHonner, ago
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