I gave my geometry students some ChatGPT-generated “proofs” this week to review. There were several examples, each designed to illustrate a different point. One was a “proof” that the diagonals of a rectangle are congruent, which contained several errors. I was proud that several students immediately identified how dangerous it was: “It sounds like it is correct, until you look more closely at it.”
Originally posted on Mastodon.
The Triangle Angle Sum Theorem is one of my favorite topics in Geometry class. It’s a foundational fact about triangles, and in geometry, every problem is a problem about triangles.
I love the standard proof of the theorem, where a line is constructed through a vertex that is parallel to the opposing side. It highlights the crucial role that parallel lines play in our conception of geometry, and it points to the assumptions we make about them as well. With a little nudge, this standard proof is eminently discoverable, and makes for a great classroom activity.
But I also love showing students some non-standard proofs of the theorem. Here’s a demonstration I built in Geogebra meant to mimic a paper folding activity that shows how the angles of a triangle form a straight line.
You should do it with actual paper, too! Here’s a short video. Stick around for the bonus tearing at the end!
Apart from being fun and surprising, what I like about these demonstrations is how they illuminate something important and essential about result: It’s the straight line, not the number 180, that’s important. Plus, the tearing activity works with more than just triangles! Unfortunately it’s not so adaptable to spherical geometry, but that’s another lesson.
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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.
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Tonight I’ll be running a workshop for teachers titled “Building Bridges Through Computational Thinking.”
In the workshop we’ll explore the mathematical and pedagogical benefits in taking a computational approach to mathematics. Through a variety of computational thinking tasks spanning different branches of math, we’ll see how these tasks offer alternate pathways into mathematical ideas, genuine engagement in applied mathematics and mathematical modeling, and opportunities for rich pedagogical variety.
This work is a natural continuation of the work I’ve been doing at the intersection of mathematics and computer science education for the past several years. As always, I’m grateful to be supported by Math for America and MfA’s teacher community in developing and trying out new ideas for students and teachers.
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I’m excited to be heading back to the North Carolina School of Science and Mathematics (NCSSM) for their annual Teaching Contemporary Mathematics (TCM) conference.
I’ll be presenting Building Bridges through Computing, in which I’ll talk about how mathematical computing projects in Python and Scratch can build bridges between theory and practice, the procedural and the abstract, and the simple and the impossible! My talk will focus on Pre Calculus and Calculus topics, and include projects like solving systems of equations, estimating roots, and elementary numerical methods.
I presented at TCM in 2016 on Mathematical Simulations in Scratch and really enjoyed my visit. The conference is focused on big ideas and brings in lots of inspired teachers, and NCSSM is a unique school with incredible programs and exceptional teachers.
TCM 2019 runs January 24-26. You can find out more information here.
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