Workshop — Bringing Modern Math into the Classroom

This Thursday I’ll be running my workshop “Bringing Modern Math into the Classroom” for teachers at Math for America.

In this webinar participants will engage with mathematics at the edge of our understanding. We’ll look at examples of math that’s being invented and discovered right now, and see how it connects to what is happening in classrooms.

We’ll play games, explore patterns, and make conjectures in arithmetic, algebra, and geometry. The goal is for participants to leave not only with a better understanding of how school math and research math are connected, but how to better communicate that connection to students.

This workshop is based on the work I’ve done in my Quantized Academy column for Quanta Magazine. I’ve run similar workshops in past years, and I recently gave a talk on this topic at the NCTM 2020 Virtual Conference. But this week’s workshop is all new, and I’m looking forward to bringing some new ideas and new math to play around with!

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By MrHonner, ago

Puzzling Together the Curriculum

To accommodate the different logistical consequences of potential in-person, hybrid, and fully-remote instruction, our school adopted a radically new schedule this year: Classes that meet every other day for periods that are 40% longer, but with an overall reduction of total class time.

The decision made sense from a organizational standpoint, but it made a mess of existing course maps and lessons plans. Trying to reorganize and redistribute content has been an ongoing challenge. It’s no simple thing to break up an existing course and reassemble it in different-sized chunks: You can’t just teach 40% more content because a class is 40% longer. Ideas needs to flow in a sensible way, and some in particular need time to set. Judging how to accomplish this was especially difficult at the start of the year, when it wasn’t even clear how much could be accomplished in a fully remote 55-minute class.

With three months behind me and a much better sense of what I’m doing, I’m feeling more comfortable putting the pieces together. Last week I was struggling to plan a 2-hour block in my trigonometry unit. But after some experimenting, I ended up pulling together material I side-stepped in October (special trigonometric limits), the core material I intended to cover (trigonometric integrals), and wrapped it up by laying the groundwork for some future extensions (Fourier series).

I would never have thought of putting these things together in a normal year. Nor would I have thought of this in September as I mapped out the semester. Back then I wasn’t even sure what would come of special trig limits as I side-stepped them, because it was impossible for me to look ahead.

But with nearly a semester under my belt, it made sense, and it worked. Three months on a steep learning curve can be painful, but it does make a difference.

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By MrHonner, ago

The Crooked Geometry of Round Trips — Quanta Magazine

My latest column for Quanta Magazine explores what round-the-world trips would look like if we didn’t live on a sphere.

Have you ever wondered what life would be like if Earth weren’t shaped like a sphere? We take for granted the smooth ride through the solar system and the seamless sunsets afforded by the planet’s rotational symmetry. A round Earth also makes it easy to figure out the fastest way to get from point A to point B: Just travel along the circle that goes through those two points and cuts the sphere in half. We use these shortest paths, called geodesics, to plan airplane routes and satellite orbits.

But what if we lived on a cube instead? Our world would wobble more, our horizons would be crooked, and our shortest paths would be harder to find.

Classification of geodesic paths on platonic solids didn’t happen until relatively recently, and the case of the dodecahedron offers quite a surprise! To learn more, read the full article here.

By MrHonner, ago

Forgetting How to Teach

The holiday break brought some much needed time off. I felt refreshed as I returned to work, but my first day back was a bit disorienting.

I opened my agenda and my SMART Notebook, but forgot to open my lesson plan. I forgot to assign my Geogebra classroom activities ahead of time. I forgot to print out my rosters. I forgot to share my screen. I forgot to check if I was muted.

The muscle memory of teaching I had worked so hard to rebuild had faded after 11 days of vacation. I suppose that’s what vacation’s for, but it’s been many years since I’ve been caught off guard like that after a break.

By the end of the week things it felt like we were back to normal again. In school, at least.

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By MrHonner, ago

Proof in a Poem

Inspired by Hannah Hoffman, here’s a poem proof of the irrationality of the square root of 2.

Suppose that root 2 could be taken to be
The integer a over the integer b
Where a and b have no factor in common
So nothing divides both the top and the bottom

Some algebra that’s quite easy to do
Gives a2 is equal to b2 times 2
But if a2 is even, so too must be a
Now each a in a2 has a 2 in our play

So a2 in fact has a factor of 4
But since this equals 2b2 we can say more
This b2 must now have a factor of 2
And just as above we know b has one too

It appears that we are now able to say
That 2 divides b and 2 divides a
But common factors were assumed to be none
So this contradiction shows we are done

Be sure to check out the excellent efforts of Timothy Gowers and Joel David Hamkins as well.

Here’s the original tweet.

By MrHonner, ago
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